Make a beautiful website − TEORI EKONOMI MIKRO DOSEN: DR. ARDITO BHINADI, SE., M.SI JURUSAN ILMU EKONOMI, FAKULTAS EKONOMI, UPN “VETERAN” YOGYAKARTA 2013 RANCANGAN PEMBELAJARAN SEMESTER ( RPS) Program Studi /Jurusan Matakuliah / Kode SKS / Semester Mata Kuliah Prasyarat Dosen : : : : : EKONOMI PEMBANGUNAN/ILMU EKONOMI TEORI EKONOMI MIKRO / 3 (tiga x 50 menit)/ II (dua) Ekonomi Mikro Pengantar Dr. H. Ardito Bhinadi, M.Si I.Deskripsi Mata Kuliah: Matakuliah ini membahas sejumlah teori ekonomi mikro dari teori konsumen, teori produsen, berbagai bentuk pasar dan eksternalitas. II.Kompetensi Umum : Pada akhir perkuliahan mahasiswa diharapkan mampu memahami dan menjelaskan model-model ekonomi, pilihan dan permintaan, produksi dan penawaran, pasar kompetitif, kekuatan pasar, penetapan harga di pasar input, dan kegagalan pasar. III. Analisis Instruksional Terlampir IV. Strategi Pembelajaran : Pembelajaran menggunakan metoda ceramah dan diskusi dengan harapan muncul sensitifitas mahasiswa terhadap masalah mikro ekonomi. Materi perkuliahan didasarkan pada beberapa buku dan studi kasus yang harus difahami oleh mahasiswa. Dosen menyampaikan materi dalam bentuk dalam power point. Media yang digunakan adalah papan tulis, LCD, dan Laptop. V. Rencana Pembelajaran Mingguan Pertemuan Ke 1 (Satu) 2 (Dua) 3 (Tiga) 4 (Empat) Kompetensi Pokok/Sub-pokok Bahasan Model-Model Ekonomi Metoda Pembelajaran Ceramah dan diskusi Media Pembelajaran Papan tulis, LCD, Laptop, Metoda Evaluasi Pertanyaan kuis/umpan balik Referensi Mahasiswa mampu memahami preferensi dan utilitas konsumen. Preferensi dan Utilitas Mahasiswa Presentasi, Ceramah dan diskusi Papan tulis, LCD, Laptop, Pertanyaan kuis/umpan balik Ch 3 Mahasiswa mampu efek substitusi dan pendapatan. Efek Substitusi dan Pendapatan Mahasiswa Presentasi, Ceramah dan diskusi Papan tulis, LCD, Laptop, Pertanyaan kuis/umpan balik Ch 5 Mahasiswa mampu memahami hubungan permintaan antar barang. Hubungan Permintaan Antar Barang Mahasiswa Presentasi, Ceramah dan diskusi Papan tulis, LCD, Laptop, Pertanyaan kuis/umpan balik Ch 6 Mahasiswa mampu memahami berbagai model ekonomi. Ch1 1 5 (Lima) 6 (Enam) 7 (Tujuh) Pertemuan Ke 8 (Delapan) 9 (Sembilan) 10 (Sepuluh) 11 (Sebelas) 12 (Dua Belas) 13 (Tiga Belas) Mahasiswa mampu memahami fungsi-fungsi produksi Fungsi-Fungsi Produksi Mahasiswa Presentasi, Ceramah dan diskusi Papan tulis, LCD, Laptop, Pertanyaan kuis/umpan balik Ch 9 Mahasiswa mampu memahami fungsi-fungsi biaya. Fungsi-Fungsi Biaya. Mahasiswa Presentasi, Ceramah dan diskusi Papan tulis, LCD, Laptop, Pertanyaan kuis/umpan balik Ch 10 Mahasiwa mampu menghitung maksimisasi laba. Maksimisasi Laba Mahasiswa Presentasi, Ceramah dan Diskusi Papan tulis, LCD, Laptop Pertanyaan umpan balik Ch 11 Media Pembelajaran Papan tulis, LCD, Laptop Metoda Evaluasi Pertanyaan umpan balik Referensi Kompetensi Mahasiwa mampu memahami model persaingan keseimbangan parsial. Mahasiwa mampu memahami keseimbangan umum dan kesejahteraan. Mahasiwa mampu memahami monopoli. Ujian Tengah Semester Pokok/Sub-pokok Metoda Bahasan Pembelajaran Model Persaingan Diskusi dan Keseimbangan Kuis Parsial Ch 12 Keseimbangan Umum dan Kesejahteraan Diskusi dan Kuis Papan tulis, LCD, Laptop Pertanyaan umpan balik Ch 13 Monopoli Diskusi dan Kuis Papan tulis, LCD, Laptop Pertanyaan umpan balik Ch 14 Mahasiwa mampu memahami persaingan tidak sempurna. Persaingan Tidak Sempurna Diskusi dan Kuis Papan tulis, LCD, Laptop Pertanyaan umpan balik Ch 15 Mahasiwa mampu memahami pasar tenaga kerja Pasar Tenaga Kerja Diskusi dan Kuis Papan tulis, LCD, Laptop Pertanyaan umpan balik Ch 16 Mahasiwa mampu memahami informasi asimetris. Asimetris Informasi Diskusi dan Kuis Papan tulis, LCD, Laptop Pertanyaan umpan balik Ch 18 2 14 (Empat Belas) Mahasiwa mampu memahami eksternalitas dan barang publik. Eksternalitas dan Barang Publik Diskusi dan Kuis Papan tulis, LCD, Laptop Pertanyaan umpan balik Ch 19 Ujian Akhir Semester 1. Sumber Referensi Nicholson, Walter and Christopher Snyder, 2008. Microeconomic Theory, Basic Principles and Extensions, Tenth Edition, Thomson South-Western, United Stated of America. 2. Komponen Penilaian 1.Ujian Tengah Semester 2.Ujian Akhir Semester 3.Partisipasi Kelas 4.Tugas-Tugas = = = = 30% 30% 20% 20% 3 Microeconomic Theory Basic Principles and Extensions, 9e Chapter 1 ECONOMIC MODELS By WALTER NICHOLSON Slides prepared by Linda Ghent Eastern Illinois University Copyright ©2005 by South-Western, a division of Thomson Learning. All rights reserved. 1 Theoretical Models 2 Verification of Economic Models • Economists use models to describe economic activities • There are two general methods used to verify economic models: – direct approach • While most economic models are abstractions from reality, they provide aid in understanding economic behavior 3 • establishes the validity of the model’s assumptions – indirect approach • shows that the model correctly predicts realworld events 4 1 Verification of Economic Models • We can use the profit-maximization model to examine these approaches – is the basic assumption valid? do firms really seek to maximize profits? – can the model predict the behavior of real-world firms? Features of Economic Models • Ceteris Paribus assumption • Optimization assumption • Distinction between positive and normative analysis 5 Ceteris Paribus Assumption • Ceteris Paribus means “other things the same” • Economic models attempt to explain simple relationships 6 Optimization Assumptions • Many economic models begin with the assumption that economic actors are rationally pursuing some goal – focus on the effects of only a few forces at a time – other variables are assumed to be unchanged during the period of study 7 – consumers seek to maximize their utility – firms seek to maximize profits (or minimize costs) – government regulators seek to maximize public welfare 8 2 Optimization Assumptions Positive-Normative Distinction • Optimization assumptions generate precise, solvable models • Positive economic theories seek to explain the economic phenomena that is observed • Optimization models appear to be perform fairly well in explaining reality • Normative economic theories focus on what “should” be done 9 The Economic Theory of Value 10 The Economic Theory of Value • The Founding of Modern Economics • Early Economic Thought – the publication of Adam Smith’s The Wealth of Nations is considered the beginning of modern economics – distinguishing between “value” and “price” continued (illustrated by the diamond-water paradox) – “value” was considered to be synonymous with “importance” – since prices were determined by humans, it was possible for the price of an item to differ from its value – prices > value were judged to be “unjust” • the value of an item meant its “value in use” • the price of an item meant its “value in exchange” 11 12 3 The Economic Theory of Value • Labor Theory of Exchange Value The Economic Theory of Value • The Marginalist Revolution – the exchange values of goods are determined by what it costs to produce them • these costs of production were primarily affected by labor costs • therefore, the exchange values of goods were determined by the quantities of labor used to produce them – the exchange value of an item is not determined by the total usefulness of the item, but rather the usefulness of the last unit consumed • because water is plentiful, consuming an additional unit has a relatively low value to individuals – producing diamonds requires more labor than producing water 13 The Economic Theory of Value • Marshallian Supply-Demand Synthesis 14 Supply-Demand Equilibrium Price – Alfred Marshall showed that supply and demand simultaneously operate to determine price – prices reflect both the marginal evaluation that consumers place on goods and the marginal costs of producing the goods • water has a low marginal value and a low marginal cost of production Low price • diamonds have a high marginal value and a high marginal cost of production High price 15 Equilibrium QD = Qs S The supply curve has a positive slope because marginal cost rises as quantity increases P* D Q* The demand curve has a negative slope because the marginal value falls as quantity increases Quantity per period 16 4 Supply-Demand Equilibrium Supply-Demand Equilibrium • A more general model is qD = 1000 - 100p qS = -125 + 125p qD = a + bp Equilibrium qD = qS qS = c + dp 1000 - 100p = -125 + 125p Equilibrium qD = qS 225p = 1125 a + bp = c + dp p* = 5 q* = 500 p* 17 Supply-Demand Equilibrium A shift in demand will lead to a new equilibrium: ac d b 18 Supply-Demand Equilibrium Price An increase in demand... S Q’D = 1450 - 100P Q’D = 1450 - 100P = QS = -125 + 125P 225P = 1575 P* = 7 Q* = 750 …leads to a rise in the equilibrium price and quantity. 7 5 D’ D 500 750 19 Quantity per period 20 5 The Economic Theory of Value • General Equilibrium Models The Economic Theory of Value • The production possibilities frontier can be used as a basic building block for general equilibrium models • A production possibilities frontier shows the combinations of two outputs that can be produced with an economy’s resources – the Marshallian model is a partial equilibrium model • focuses only on one market at a time – to answer more general questions, we need a model of the entire economy • need to include the interrelationships between markets and economic agents 21 A Production Possibility Frontier Quantity of food (weekly) Opportunity cost of clothing = 1/2 pound of food 10 9.5 Opportunity cost of clothing = 2 pounds of food 4 2 3 4 12 13 22 A Production Possibility Frontier • The production possibility frontier reminds us that resources are scarce • Scarcity means that we must make choices – each choice has opportunity costs – the opportunity costs depend on how much of each good is produced Quantity of clothing (weekly) 23 24 6 A Production Possibility Frontier A Production Possibility Frontier • Suppose that the production possibility frontier can be represented by dy 1 4 x 2x (225 2x 2 )1/ 2 ( 4 x ) dx 2 2y y 2x 2 y 2 225 • when x=5, y=13.2, the slope= -2(5)/13.2= -0.76 • To find the slope, we can solve for Y • when x=10, y=5, the slope= -2(10)/5= -4 y 225 2x 2 • the slope rises as y rises • If we differentiate dy 1 4 x 2x (225 2x 2 )1/ 2 ( 4 x ) dx 2 2y y 25 The Economic Theory of Value • Welfare Economics – tools used in general equilibrium analysis have been used for normative analysis concerning the desirability of various economic outcomes • economists Francis Edgeworth and Vilfredo Pareto helped to provide a precise definition of economic efficiency and demonstrated the conditions under which markets can attain that goal 27 26 Modern Tools • Clarification of the basic behavioral assumptions about individual and firm behavior • Creation of new tools to study markets • Incorporation of uncertainty and imperfect information into economic models • Increasing use of computers to analyze data 28 7 Important Points to Note: Important Points to Note: • Economics is the study of how scarce resources are allocated among alternative uses • The most commonly used economic model is the supply-demand model – shows how prices serve to balance production costs and the willingness of buyers to pay for these costs – economists use simple models to understand the process 29 Important Points to Note: 30 Important Points to Note: • The supply-demand model is only a partial-equilibrium model • Testing the validity of a model is a difficult task – a general equilibrium model is needed to look at many markets together – are the model’s assumptions reasonable? – does the model explain real-world events? 31 32 8 Axioms of Rational Choice • Completeness Chapter 3 – if A and B are any two situations, an individual can always specify exactly one of these possibilities: PREFERENCES AND UTILITY • A is preferred to B • B is preferred to A • A and B are equally attractive Copyright ©2005 by South-Western, a division of Thomson Learning. All rights reserved. 1 Axioms of Rational Choice 2 Axioms of Rational Choice • Transitivity • Continuity – if A is preferred to B, and B is preferred to C, then A is preferred to C – assumes that the individual’s choices are internally consistent – if A is preferred to B, then situations suitably “close to” A must also be preferred to B – used to analyze individuals’ responses to relatively small changes in income and prices 3 4 1 Utility Utility • Given these assumptions, it is possible to show that people are able to rank in order all possible situations from least desirable to most • Economists call this ranking utility – if A is preferred to B, then the utility assigned to A exceeds the utility assigned to B U(A) > U(B) • Utility rankings are ordinal in nature – they record the relative desirability of commodity bundles • Because utility measures are not unique, it makes no sense to consider how much more utility is gained from A than from B • It is also impossible to compare utilities between people 5 6 Utility Utility • Utility is affected by the consumption of physical commodities, psychological attitudes, peer group pressures, personal experiences, and the general cultural environment • Economists generally devote attention to quantifiable options while holding constant the other things that affect utility • Assume that an individual must choose among consumption goods x1, x2,…, xn • The individual’s rankings can be shown by a utility function of the form: utility = U(x1, x2,…, xn; other things) – this function is unique up to an orderpreserving transformation – ceteris paribus assumption 7 8 2 Economic Goods Indifference Curves • In the utility function, the x’s are assumed to be “goods” • An indifference curve shows a set of consumption bundles among which the individual is indifferent – more is preferred to less Quantity of y Quantity of y Combinations (x1, y1) and (x2, y2) provide the same level of utility Preferred to x*, y* ? y* y1 ? Worse than x*, y* y2 Quantity of x Quantity of x 9 x* U1 x1 x2 10 Marginal Rate of Substitution Marginal Rate of Substitution • The negative of the slope of the indifference curve at any point is called the marginal rate of substitution (MRS) • MRS changes as x and y change – reflects the individual’s willingness to trade y for x Quantity of y Quantity of y MRS dy dx U U1 y1 At (x1, y1), the indifference curve is steeper. The person would be willing to give up more y to gain additional units of x At (x2, y2), the indifference curve is flatter. The person would be willing to give up less y to gain additional units of x y1 y2 y2 U1 Quantity of x x1 x2 11 U1 Quantity of x x1 x2 12 3 Indifference Curve Map Transitivity • Each point must have an indifference curve through it • Can any two of an individual’s indifference curves intersect? Quantity of y The individual is indifferent between A and C. The individual is indifferent between B and C. Transitivity suggests that the individual should be indifferent between A and B Quantity of y Increasing utility U3 U2 C U1 < U2 < U3 B U2 A But B is preferred to A because B contains more x and y than A U1 U1 Quantity of x Quantity of x 13 Convexity 14 Convexity • A set of points is convex if any two points can be joined by a straight line that is contained completely within the set Quantity of y • If the indifference curve is convex, then the combination (x1 + x2)/2, (y1 + y2)/2 will be preferred to either (x1,y1) or (x2,y2) Quantity of y This implies that “well-balanced” bundles are preferred to bundles that are heavily weighted toward one commodity The assumption of a diminishing MRS is equivalent to the assumption that all combinations of x and y which are preferred to x* and y* form a convex set y1 (y1 + y2)/2 y* y2 U1 Quantity of x x* 15 U1 Quantity of x x1 (x1 + x2)/2 x2 16 4 Utility and the MRS Utility and the MRS • Suppose an individual’s preferences for hamburgers (y) and soft drinks (x) can be represented by MRS = -dy/dx = 100/x2 • Note that as x rises, MRS falls – when x = 5, MRS = 4 – when x = 20, MRS = 0.25 utility 10 x y • Solving for y, we get y = 100/x • Solving for MRS = -dy/dx: MRS = -dy/dx = 100/x2 17 Marginal Utility 18 Deriving the MRS • Suppose that an individual has a utility function of the form • Therefore, we get: utility = U(x,y) dy MRS dx • The total differential of U is U U dU dx dy x y Uconstant U x U y • MRS is the ratio of the marginal utility of x to the marginal utility of y • Along any indifference curve, utility is constant (dU = 0) 19 20 5 Diminishing Marginal Utility and the MRS • Intuitively, it seems that the assumption of decreasing marginal utility is related to the concept of a diminishing MRS – diminishing MRS requires that the utility function be quasi-concave Convexity of Indifference Curves • Suppose that the utility function is utility x y • We can simplify the algebra by taking the logarithm of this function • this is independent of how utility is measured – diminishing marginal utility depends on how utility is measured U*(x,y) = ln[U(x,y)] = 0.5 ln x + 0.5 ln y • Thus, these two concepts are different 21 22 Convexity of Indifference Curves Convexity of Indifference Curves • If the utility function is • Thus, U(x,y) = x + xy + y • There is no advantage to transforming this utility function, so U * 0.5 y MRS x x U * 0.5 x y y U 1 y MRS x U 1 x y 23 24 6 Convexity of Indifference Curves Convexity of Indifference Curves • Suppose that the utility function is • Thus, utility x 2 y 2 U * 2x x MRS x U * 2y y y • For this example, it is easier to use the transformation U*(x,y) = [U(x,y)]2 = x2 + y2 25 Examples of Utility Functions • Cobb-Douglas Utility utility = U(x,y) = 26 Examples of Utility Functions • Perfect Substitutes utility = U(x,y) = x + y xy where and are positive constants Quantity of y The indifference curves will be linear. The MRS will be constant along the indifference curve. – The relative sizes of and indicate the relative importance of the goods U3 U1 27 U2 Quantity of x 28 7 Examples of Utility Functions • CES Utility (Constant elasticity of substitution) • Perfect Complements utility = U(x,y) = min (x, y) utility = U(x,y) = x/ + y/ Quantity of y The indifference curves will be L-shaped. Only by choosing more of the two goods together can utility be increased. when 0 and utility = U(x,y) = ln x + ln y when = 0 – Perfect substitutes = 1 – Cobb-Douglas = 0 – Perfect complements = U3 U2 U1 Quantity of x Examples of Utility Functions 29 Examples of Utility Functions • CES Utility (Constant elasticity of substitution) 30 Homothetic Preferences • If the MRS depends only on the ratio of the amounts of the two goods, not on the quantities of the goods, the utility function is homothetic – The elasticity of substitution () is equal to 1/(1 - ) – Perfect substitutes MRS is the same at every point – Perfect complements MRS = if y/x > /, undefined if y/x = /, and MRS = 0 if y/x < / • Perfect substitutes = • Fixed proportions = 0 31 32 8 Homothetic Preferences Nonhomothetic Preferences • For the general Cobb-Douglas function, the MRS can be found as • Some utility functions do not exhibit homothetic preferences utility = U(x,y) = x + ln y U x 1y y MRS x U x y 1 x y U 1 MRS x y U 1 y y 33 The Many-Good Case The Many-Good Case • Suppose utility is a function of n goods given by • We can find the MRS between any two goods by setting dU = 0 utility = U(x1, x2,…, xn) dU 0 • The total differential of U is dU 34 U U dxi dx j xi x j • Rearranging, we get U U U dx1 dx2 ... dxn x1 x2 xn U x i MRS( x i for x j ) U dx i x j dx j 35 36 9 Multigood Indifference Surfaces Multigood Indifference Surfaces • We will define an indifference surface as being the set of points in n dimensions that satisfy the equation • If the utility function is quasi-concave, the set of points for which U k will be convex – all of the points on a line joining any two points on the U = k indifference surface will also have U k U(x1,x2,…xn) = k where k is any preassigned constant 37 38 Important Points to Note: Important Points to Note: • If individuals obey certain behavioral postulates, they will be able to rank all commodity bundles • The negative of the slope of the indifference curve measures the marginal rate of substitution (MRS) – the ranking can be represented by a utility function – in making choices, individuals will act as if they were maximizing this function – the rate at which an individual would trade an amount of one good (y) for one more unit of another good (x) • MRS decreases as x is substituted for y • Utility functions for two goods can be illustrated by an indifference curve map – individuals prefer some balance in their consumption choices 39 40 10 Important Points to Note: Important Points to Note: • A few simple functional forms can capture important differences in individuals’ preferences for two (or more) goods – Cobb-Douglas function – linear function (perfect substitutes) – fixed proportions function (perfect complements) – CES function • It is a simple matter to generalize from two-good examples to many goods – studying peoples’ choices among many goods can yield many insights – the mathematics of many goods is not especially intuitive, so we will rely on twogood cases to build intuition • includes the other three as special cases 41 42 11 Demand Functions • The optimal levels of x1,x2,…,xn can be expressed as functions of all prices and income • These can be expressed as n demand functions of the form: Chapter 5 INCOME AND SUBSTITUTION EFFECTS x1* = d1(p1,p2,…,pn,I) x2* = d2(p1,p2,…,pn,I) Copyright ©2005 by South-Western, a division of Thomson Learning. All rights reserved. 1 • • • xn* = dn(p1,p2,…,pn,I) 2 Homogeneity Demand Functions • If we were to double all prices and income, the optimal quantities demanded will not change • If there are only two goods (x and y), we can simplify the notation x* = x(px,py,I) – the budget constraint is unchanged y* = y(px,py,I) xi* = di(p1,p2,…,pn,I) = di(tp1,tp2,…,tpn,tI) • Prices and income are exogenous • Individual demand functions are homogeneous of degree zero in all prices and income – the individual has no control over these parameters 3 4 1 Homogeneity Homogeneity • With a Cobb-Douglas utility function utility = U(x,y) = • With a CES utility function utility = U(x,y) = x0.5 + y0.5 the demand functions are 1 I 1 I x* y* 1 px / py px 1 py / px py x0.3y0.7 the demand functions are x* 0 .3 I px y* 0.7 I py • Note that a doubling of both prices and income would leave x* and y* unaffected • Note that a doubling of both prices and income would leave x* and y* unaffected 5 6 Increase in Income Changes in Income • If both x and y increase as income rises, x and y are normal goods • An increase in income will cause the budget constraint out in a parallel fashion • Since px/py does not change, the MRS will stay constant as the worker moves to higher levels of satisfaction Quantity of y As income rises, the individual chooses to consume more x and y B C A U1 U3 U2 Quantity of x 7 8 2 Increase in Income Normal and Inferior Goods • If x decreases as income rises, x is an inferior good • A good xi for which xi/I 0 over some range of income is a normal good in that range As income rises, the individual chooses to consume less x and more y Quantity of y Note that the indifference curves do not have to be “oddly” shaped. The assumption of a diminishing MRS is obeyed. C B U3 U2 A • A good xi for which xi/I < 0 over some range of income is an inferior good in that range U1 Quantity of x 9 10 Changes in a Good’s Price Changes in a Good’s Price • A change in the price of a good alters the slope of the budget constraint – it also changes the MRS at the consumer’s utility-maximizing choices • When the price changes, two effects come into play • Even if the individual remained on the same indifference curve when the price changes, his optimal choice will change because the MRS must equal the new price ratio – the substitution effect • The price change alters the individual’s “real” income and therefore he must move to a new indifference curve – substitution effect – income effect – the income effect 11 12 3 Changes in a Good’s Price Changes in a Good’s Price Quantity of y Suppose the consumer is maximizing utility at point A. Quantity of y To isolate the substitution effect, we hold “real” income constant but allow the relative price of good x to change If the price of good x falls, the consumer will maximize utility at point B. The substitution effect is the movement from point A to point C B A The individual substitutes good x for good y because it is now relatively cheaper C A U2 U1 U1 Quantity of x Quantity of x Substitution effect Total increase in x 13 Changes in a Good’s Price Quantity of y A Changes in a Good’s Price Quantity of y The income effect occurs because the individual’s “real” income changes when the price of good x changes B 14 The income effect is the movement from point C to point B C U2 U1 If x is a normal good, the individual will buy more because “real” income increased An increase in the price of good x means that the budget constraint gets steeper The substitution effect is the movement from point A to point C C A B U1 The income effect is the movement from point C to point B U2 Quantity of x Quantity of x Substitution effect Income effect Income effect 15 16 4 Price Changes for Inferior Goods Price Changes for Normal Goods • If a good is normal, substitution and income effects reinforce one another – when price falls, both effects lead to a rise in quantity demanded – when price rises, both effects lead to a drop in quantity demanded 17 • If a good is inferior, substitution and income effects move in opposite directions • The combined effect is indeterminate – when price rises, the substitution effect leads to a drop in quantity demanded, but the income effect is opposite – when price falls, the substitution effect leads to a rise in quantity demanded, but the income effect is opposite 18 Giffen’s Paradox A Summary • If the income effect of a price change is strong enough, there could be a positive relationship between price and quantity demanded • Utility maximization implies that (for normal goods) a fall in price leads to an increase in quantity demanded – an increase in price leads to a drop in real income – since the good is inferior, a drop in income causes quantity demanded to rise 19 – the substitution effect causes more to be purchased as the individual moves along an indifference curve – the income effect causes more to be purchased because the resulting rise in purchasing power allows the individual to move to a higher indifference curve 20 5 A Summary A Summary • Utility maximization implies that (for normal goods) a rise in price leads to a decline in quantity demanded • Utility maximization implies that (for inferior goods) no definite prediction can be made for changes in price – the substitution effect causes less to be purchased as the individual moves along an indifference curve – the income effect causes less to be purchased because the resulting drop in purchasing power moves the individual to a lower indifference curve 21 The Individual’s Demand Curve • An individual’s demand for x depends on preferences, all prices, and income: – the substitution effect and income effect move in opposite directions – if the income effect outweighs the substitution effect, we have a case of Giffen’s paradox 22 The Individual’s Demand Curve Quantity of y As the price of x falls... px …quantity of x demanded rises. x* = x(px,py,I) px’ • It may be convenient to graph the individual’s demand for x assuming that income and the price of y (py) are held constant px’’ px’’’ U1 x1 23 I = px’ + py x2 x3 I = px’’ + py U2 U3 Quantity of x I = px’’’ + py x x’ x’’ x’’’ Quantity of x 24 6 The Individual’s Demand Curve • An individual demand curve shows the relationship between the price of a good and the quantity of that good purchased by an individual assuming that all other determinants of demand are held constant Shifts in the Demand Curve • Three factors are held constant when a demand curve is derived – income – prices of other goods (py) – the individual’s preferences • If any of these factors change, the demand curve will shift to a new position 25 Shifts in the Demand Curve • A movement along a given demand curve is caused by a change in the price of the good – a change in quantity demanded • A shift in the demand curve is caused by changes in income, prices of other goods, or preferences – a change in demand 27 26 Demand Functions and Curves • We discovered earlier that x* 0 .3 I px y* 0.7 I py • If the individual’s income is $100, these functions become x* 30 px y* 70 py 28 7 Demand Functions and Curves Compensated Demand Curves • The actual level of utility varies along the demand curve • As the price of x falls, the individual moves to higher indifference curves • Any change in income will shift these demand curves – it is assumed that nominal income is held constant as the demand curve is derived – this means that “real” income rises as the price of x falls 29 30 Compensated Demand Curves Compensated Demand Curves • An alternative approach holds real income (or utility) constant while examining reactions to changes in px • A compensated (Hicksian) demand curve shows the relationship between the price of a good and the quantity purchased assuming that other prices and utility are held constant • The compensated demand curve is a twodimensional representation of the compensated demand function – the effects of the price change are “compensated” so as to constrain the individual to remain on the same indifference curve – reactions to price changes include only substitution effects 31 x* = xc(px,py,U) 32 8 Compensated Demand Curves Holding utility constant, as price falls... Quantity of y slope px px ' py slope …quantity demanded rises. px ' ' py Compensated & Uncompensated Demand px At px’’, the curves intersect because the individual’s income is just sufficient to attain utility level U2 px’ px’’ px’’ slope px ' ' ' py x px’’’ xc xc U2 x’ x’’ x’’’ x’ Quantity of x x’’ x’’’ x’’ Quantity of x Quantity of x 33 Compensated & Uncompensated Demand Compensated & Uncompensated Demand At prices above px2, income compensation is positive because the individual needs some help to remain on U2 px 34 px At prices below px2, income compensation is negative to prevent an increase in utility from a lower price px’ px’’ px’’ px’’’ x x xc x’ x* xc x*** Quantity of x 35 x’’’ Quantity of x 36 9 Compensated Demand Functions Compensated & Uncompensated Demand • For a normal good, the compensated demand curve is less responsive to price changes than is the uncompensated demand curve – the uncompensated demand curve reflects both income and substitution effects • Suppose that utility is given by utility = U(x,y) = x0.5y0.5 • The Marshallian demand functions are x = I/2px y = I/2py • The indirect utility function is – the compensated demand curve reflects only substitution effects utility V ( I, px , py ) I 0.5 x 2p py0.5 37 Compensated Demand Functions Compensated Demand Functions • To obtain the compensated demand functions, we can solve the indirect utility function for I and then substitute into the Marshallian demand functions x Vpy0.5 px0.5 y 38 x Vpy0.5 px0.5 y Vpx0.5 py0.5 • Demand now depends on utility (V) rather than income • Increases in px reduce the amount of x demanded Vpx0.5 py0.5 – only a substitution effect 39 40 10 A Mathematical Examination of a Change in Price A Mathematical Examination of a Change in Price • Our goal is to examine how purchases of good x change when px changes • Instead, we will use an indirect approach • Remember the expenditure function x/px minimum expenditure = E(px,py,U) • Differentiation of the first-order conditions from utility maximization can be performed to solve for this derivative • However, this approach is cumbersome and provides little economic insight 41 A Mathematical Examination of a Change in Price xc (px,py,U) = x [px,py,E(px,py,U)] – quantity demanded is equal for both demand functions when income is exactly what is needed to attain the required utility level 42 A Mathematical Examination of a Change in Price x x c x E
px px E px xc (px,py,U) = x[px,py,E(px,py,U)] • We can differentiate the compensated demand function and get • The first term is the slope of the compensated demand curve x x x E
px px E px c x x c x E
px px E px • Then, by definition – the mathematical representation of the substitution effect 43 44 11 A Mathematical Examination of a Change in Price The Slutsky Equation • The substitution effect can be written as x x c x E
px px E px substituti on effect • The second term measures the way in which changes in px affect the demand for x through changes in purchasing power x c x px px U constant • The income effect can be written as income effect – the mathematical representation of the income effect x E x E E px I px 45 46 The Slutsky Equation The Slutsky Equation • The utility-maximization hypothesis shows that the substitution and income effects arising from a price change can be represented by • Note that E/px = x – a $1 increase in px raises necessary expenditures by x dollars – $1 extra must be paid for each unit of x purchased x substituti on effect income effect px x x px px 47 x U constant x I 48 12 The Slutsky Equation x x px px x U constant The Slutsky Equation x I x x px px • The first term is the substitution effect x U constant x I • The second term is the income effect – always negative as long as MRS is diminishing – the slope of the compensated demand curve must be negative – if x is a normal good, then x/I > 0 • the entire income effect is negative – if x is an inferior good, then x/I < 0 • the entire income effect is positive 49 A Slutsky Decomposition 50 A Slutsky Decomposition • We can demonstrate the decomposition of a price effect using the Cobb-Douglas example studied earlier • The Marshallian demand function for good x was 0 .5 I x ( p x , py , I ) px 51 • The Hicksian (compensated) demand function for good x was x c ( px , py ,V ) Vpy0.5 px0.5 • The overall effect of a price change on the demand for x is x 0 .5 I px px2 52 13 A Slutsky Decomposition A Slutsky Decomposition • This total effect is the sum of the two effects that Slutsky identified • The substitution effect is found by differentiating the compensated demand function • We can substitute in for the indirect utility function (V) substituti on effect substituti on effect 0.5(0.5 Ipx0.5 py0.5 )py0.5 p 1.5 x 0.25I px2 0. 5 x c 0.5Vpy px p1x.5 53 A Slutsky Decomposition • Calculation of the income effect is easier income effect x 0.5I 0.5 x 0.25I I px2 px px • Interestingly, the substitution and income effects are exactly the same size 55 54 Marshallian Demand Elasticities • Most of the commonly used demand elasticities are derived from the Marshallian demand function x(px,py,I) • Price elasticity of demand (ex,px) ex ,px x / x x px px / px px x 56 14 Marshallian Demand Elasticities Price Elasticity of Demand • The own price elasticity of demand is always negative • Income elasticity of demand (ex,I) e x ,I x / x x I I / I I x – the only exception is Giffen’s paradox • The size of the elasticity is important • Cross-price elasticity of demand (ex,py) ex ,py – if ex,px < -1, demand is elastic – if ex,px > -1, demand is inelastic – if ex,px = -1, demand is unit elastic x / x x py py / py py x 57 Price Elasticity and Total Spending 58 Price Elasticity and Total Spending • Total spending on x is equal to ( p x x ) x px x x[ex,px 1] px px total spending =pxx • Using elasticity, we can determine how total spending changes when the price of x changes ( p x x ) x px x x[ex,px 1] px px 59 • The sign of this derivative depends on whether ex,px is greater or less than -1 – if ex,px > -1, demand is inelastic and price and total spending move in the same direction – if ex,px < -1, demand is elastic and price and total spending move in opposite directions 60 15 Compensated Price Elasticities • It is also useful to define elasticities based on the compensated demand function Compensated Price Elasticities • If the compensated demand function is xc = xc(px,py,U) we can calculate – compensated own price elasticity of demand (exc,px) – compensated cross-price elasticity of demand (exc,py) 61 Compensated Price Elasticities • The compensated own price elasticity of demand (exc,px) is exc,px x c / x c x c px px / px px x c • The compensated cross-price elasticity of demand (exc,py) is e c x ,py x c / x c x c py
py / py py x c 62 Compensated Price Elasticities • The relationship between Marshallian and compensated price elasticities can be shown using the Slutsky equation px x p x c px x ex,px xc x x px x px x I • If sx = pxx/I, then ex,px exc,px sx ex,I 63 64 16 Compensated Price Elasticities Homogeneity • The Slutsky equation shows that the compensated and uncompensated price elasticities will be similar if • Demand functions are homogeneous of degree zero in all prices and income • Euler’s theorem for homogenous functions shows that – the share of income devoted to x is small – the income elasticity of x is small 0 px x x x py I px py I 65 Homogeneity 66 Engel Aggregation • Dividing by x, we get • Engel’s law suggests that the income elasticity of demand for food items is less than one 0 ex,px ex,py ex,I • Any proportional change in all prices and income will leave the quantity of x demanded unchanged – this implies that the income elasticity of demand for all nonfood items must be greater than one 67 68 17 Engel Aggregation Cournot Aggregation • We can see this by differentiating the budget constraint with respect to income (treating prices as constant) 1 px 1 px • The size of the cross-price effect of a change in the price of x on the quantity of y consumed is restricted because of the budget constraint • We can demonstrate this by differentiating the budget constraint with respect to px x y py I I x xI y yI py s x e x , I s y ey , I I xI I yI 69 Demand Elasticities Cournot Aggregation • The Cobb-Douglas utility function is I x y 0 px x py px px px 0 px 70 U(x,y) = xy (+=1) • The demand functions for x and y are x px x p y px y x x py px I x I px I y x 0 s x ex,px s x sy ey ,px I px y I py s x ex,px sy ey ,px s x 71 72 18 Demand Elasticities Demand Elasticities • Calculating the elasticities, we get ex ,px – homogeneity x px I p 2 x 1 px x p x I px e x , py e x ,I • We can also show ex,px ex,py ex,I 1 0 1 0 – Engel aggregation p x py 0 y 0 py x x x I I 1 I x px I px s x e x , I s y ey , I 1 1 1 – Cournot aggregation s x ex,px sy ey ,px ( 1) 0 s x 73 74 Demand Elasticities Demand Elasticities • We can also use the Slutsky equation to derive the compensated price elasticity • The CES utility function (with = 2, = 5) is U(x,y) = x0.5 + y0.5 exc,px ex,px sx ex,I 1 (1) 1 • The compensated price elasticity depends on how important other goods (y) are in the utility function 75 • The demand functions for x and y are x I px (1 px py1 ) y I py (1 px1py ) 76 19 Demand Elasticities Demand Elasticities • We will use the “share elasticity” to derive the own price elasticity es x ,px • Thus, the share elasticity is given by esx ,px s x px 1 ex,px px s x • Therefore, if we let px = py • In this case, sx py1 px py1 s x px px
px s x (1 px py1 )2 (1 px py1 )1 1 px py1 px x 1 I 1 px py1 ex,px es x ,px 1 1 1 1.5 1 1 77 Demand Elasticities 78 Demand Elasticities • Thus, the share elasticity is given by • The CES utility function (with = 0.5, = -1) is es x ,px U(x,y) = -x -1 - y -1 • The share of good x is sx px x 1 0.5 0.5 I 1 py px 0.5 py0.5 px1.5 s x px px
0.5 0.5 2 0.5 0.5 1 px s x (1 py px ) (1 py px ) 0.5 py0.5 px0.5 1 py0.5 px0.5 • Again, if we let px = py 79 ex,px esx ,px 1 0.5 1 0.75 2 80 20 Consumer Surplus Consumer Welfare • One way to evaluate the welfare cost of a price increase (from px0 to px1) would be to compare the expenditures required to achieve U0 under these two situations • An important problem in welfare economics is to devise a monetary measure of the gains and losses that individuals experience when prices change expenditure at px0 = E0 = E(px0,py,U0) expenditure at px1 = E1 = E(px1,py,U0) 81 82 Consumer Welfare Consumer Welfare Suppose the consumer is maximizing utility at point A. Quantity of y • In order to compensate for the price rise, this person would require a compensating variation (CV) of If the price of good x rises, the consumer will maximize utility at point B. A CV = E(px1,py,U0) - E(px0,py,U0) B U1 The consumer’s utility falls from U1 to U2 U2 Quantity of x 83 84 21 Consumer Welfare Consumer Welfare The consumer could be compensated so that he can afford to remain on U1 Quantity of y CV is the amount that the C • The derivative of the expenditure function with respect to px is the compensated demand function individual would need to be compensated A E ( px , py ,U0 ) B px U1 x c ( px , py ,U0 ) U2 Quantity of x 85 Consumer Welfare 86 px Consumer Welfare • The amount of CV required can be found by integrating across a sequence of small increments to price from px0 to px1 p1x p1x px0 px0 welfare loss px1 CV dE x c ( px , py ,U0 )dpx – this integral is the area to the left of the compensated demand curve between px0 and px1 When the price rises from px0 to px1, the consumer suffers a loss in welfare px0 xc(px…U0) x1 87 x0 Quantity of x 88 22 The Consumer Surplus Concept Consumer Welfare • Because a price change generally involves both income and substitution effects, it is unclear which compensated demand curve should be used • Do we use the compensated demand curve for the original target utility (U0) or the new level of utility after the price change (U1)? • Another way to look at this issue is to ask how much the person would be willing to pay for the right to consume all of this good that he wanted at the market price of px0 89 The Consumer Surplus Concept 90 Consumer Welfare px • The area below the compensated demand curve and above the market price is called consumer surplus When the price rises from px0 to px1, the actual market reaction will be to move from A to C The consumer’s utility falls from U0 to U1 px1 – the extra benefit the person receives by being able to make market transactions at the prevailing market price px C A 0 x(px…) xc(...U0) xc(...U1) x1 91 x0 Quantity of x 92 23 Consumer Welfare px px1 Consumer Welfare Is the consumer’s loss in welfare best described by area px1BApx0 [using xc(...U0)] or by area px1CDpx0 [using xc(...U1)]? C B A px0 D xc(...U 0) Is U0 or U1 the appropriate utility target? px px1 We can use the Marshallian demand curve as a compromise C B A px0 D x(px…) xc(...U 0) xc(...U1) x1 x0 The area px1CApx0 falls between the sizes of the welfare losses defined by xc(...U0) and xc(...U1) xc(...U1) x1 Quantity of x x0 Quantity of x 93 94 Welfare Loss from a Price Increase Consumer Surplus • We will define consumer surplus as the area below the Marshallian demand curve and above price • Suppose that the compensated demand function for x is given by – shows what an individual would pay for the right to make voluntary transactions at this price – changes in consumer surplus measure the welfare effects of price changes 95 x c ( px , py ,V ) Vpy0.5 px0.5 • The welfare cost of a price increase from px = 1 to px = 4 is given by 4 CV Vpy0.5 px0.5 2Vpy0.5 px0.5 1 px 4 p X 1 96 24 Welfare Loss from a Price Increase Welfare Loss from Price Increase • Suppose that we use the Marshallian demand function instead • If we assume that V = 2 and py = 2, CV = 222(4)0.5 – 222(1)0.5 = 8 • If we assume that the utility level (V) falls to 1 after the price increase (and used this level to calculate welfare loss), CV = 122(4)0.5 – 122(1)0.5 = 4 x( px , py , I ) 0.5Ipx-1 • The welfare loss from a price increase from px = 1 to px = 4 is given by 4 Loss 0.5 Ipx-1dpx 0.5 I ln px 97 Welfare Loss from a Price Increase 1 px 4 px 1 98 Revealed Preference and the Substitution Effect • If income (I) is equal to 8, • The theory of revealed preference was proposed by Paul Samuelson in the late 1940s • The theory defines a principle of rationality based on observed behavior and then uses it to approximate an individual’s utility function loss = 4 ln(4) - 4 ln(1) = 4 ln(4) = 4(1.39) = 5.55 – this computed loss from the Marshallian demand function is a compromise between the two amounts computed using the compensated demand functions 99 100 25 Revealed Preference and the Substitution Effect Revealed Preference and the Substitution Effect • Consider two bundles of goods: A and B • If the individual can afford to purchase either bundle but chooses A, we say that A had been revealed preferred to B • Under any other price-income arrangement, B can never be revealed preferred to A 101 Suppose that, when the budget constraint is given by I1, A is chosen Quantity of y A must still be preferred to B when income is I3 (because both A and B are available) A If B is chosen, the budget constraint must be similar to that given by I2 where A is not available B I3 I1 I2 Quantity of x 102 Negativity of the Substitution Effect Negativity of the Substitution Effect • Suppose that an individual is indifferent between two bundles: C and D • Since the individual is indifferent between C and D – When C is chosen, D must cost at least as much as C pxCxC + pyCyC ≤ pxCxD + pyCyD • Let pxC,pyC be the prices at which bundle C is chosen • Let pxD,pyD be the prices at which bundle D is chosen – When D is chosen, C must cost at least as much as D pxDxD + pyDyD ≤ pxDxC + pyDyC 103 104 26 Negativity of the Substitution Effect Negativity of the Substitution Effect • Suppose that only the price of x changes (pyC = pyD) • Rearranging, we get pxC(xC - xD) + pyC(yC -yD) ≤ 0 (pxC – pxD)(xC - xD) ≤ 0 pxD(xD - xC) + pyD(yD -yC) ≤ 0 • Adding these together, we get (pxC – pxD)(xC - xD) + (pyC – pyD)(yC - yD) ≤ 0 • This implies that price and quantity move in opposite direction when utility is held constant – the substitution effect is negative 105 106 Mathematical Generalization Mathematical Generalization • If, at prices pi0 bundle xi0 is chosen instead of bundle xi1 (and bundle xi1 is affordable), then • Consequently, at prices that prevail when bundle 1 is chosen (pi1), then n p x i 1 0 i n p x n 0 i pi0 xi1 i 1 1 0 i i n pi1xi1 i 1 i 1 • Bundle 0 has been “revealed preferred” to bundle 1 107 • Bundle 0 must be more expensive than bundle 1 108 27 Strong Axiom of Revealed Preference Important Points to Note: • If commodity bundle 0 is revealed preferred to bundle 1, and if bundle 1 is revealed preferred to bundle 2, and if bundle 2 is revealed preferred to bundle 3,…,and if bundle K-1 is revealed preferred to bundle K, then bundle K cannot be revealed preferred to bundle 0 • Proportional changes in all prices and income do not shift the individual’s budget constraint and therefore do not alter the quantities of goods chosen – demand functions are homogeneous of degree zero in all prices and income 109 Important Points to Note: 110 Important Points to Note: • When purchasing power changes (income changes but prices remain the same), budget constraints shift • A fall in the price of a good causes substitution and income effects – for a normal good, both effects cause more of the good to be purchased – for inferior goods, substitution and income effects work in opposite directions – for normal goods, an increase in income means that more is purchased – for inferior goods, an increase in income means that less is purchased • no unambiguous prediction is possible 111 112 28 Important Points to Note: Important Points to Note: • A rise in the price of a good also causes income and substitution effects • The Marshallian demand curve summarizes the total quantity of a good demanded at each possible price – for normal goods, less will be demanded – for inferior goods, the net result is ambiguous – changes in price prompt movements along the curve – changes in income, prices of other goods, or preferences may cause the demand curve to shift 113 Important Points to Note: 114 Important Points to Note: • Compensated demand curves illustrate movements along a given indifference curve for alternative prices – they are constructed by holding utility constant and exhibit only the substitution effects from a price change – their slope is unambiguously negative (or zero) 115 • Demand elasticities are often used in empirical work to summarize how individuals react to changes in prices and income – the most important is the price elasticity of demand • measures the proportionate change in quantity in response to a 1 percent change in price 116 29 Important Points to Note: Important Points to Note: • There are many relationships among demand elasticities • Welfare effects of price changes can be measured by changing areas below either compensated or ordinary demand curves – own-price elasticities determine how a price change affects total spending on a good – substitution and income effects can be summarized by the Slutsky equation – various aggregation results hold among elasticities – such changes affect the size of the consumer surplus that individuals receive by being able to make market transactions 117 118 Important Points to Note: • The negativity of the substitution effect is one of the most basic findings of demand theory – this result can be shown using revealed preference theory and does not necessarily require assuming the existence of a utility function 119 30 The Two-Good Case • The types of relationships that can occur when there are only two goods are limited • But this case can be illustrated with twodimensional graphs Chapter 6 DEMAND RELATIONSHIPS AMONG GOODS Copyright ©2005 by South-Western, a division of Thomson Learning. All rights reserved. 1 2 Gross Complements Quantity of y Gross Substitutes When the price of y falls, the substitution effect may be so small that the consumer purchases more x and more y In this case, we call x and y gross complements y1 y0 U1 U0 Quantity of y When the price of y falls, the substitution effect may be so large that the consumer purchases less x and more y In this case, we call x and y gross substitutes y1 y0 x/py < 0 U1 x/py > 0 U0 x0 x1 x1 Quantity of x 3 x0 Quantity of x 4 1 Substitutes and Complements A Mathematical Treatment • The change in x caused by changes in py can be shown by a Slutsky-type equation x x py py y U constant substitution effect (+) • For the case of many goods, we can generalize the Slutsky analysis x I xi xi p j p j U constant xi I for any i or j income effect (-) if x is normal combined effect (ambiguous) xj 5 Substitutes and Complements • Two goods are substitutes if one good may replace the other in use – this implies that the change in the price of any good induces income and substitution effects that may change the quantity of every good demanded 6 Gross Substitutes and Complements • The concepts of gross substitutes and complements include both substitution and income effects – examples: tea & coffee, butter & margarine • Two goods are complements if they are used together – two goods are gross substitutes if xi /pj > 0 – examples: coffee & cream, fish & chips – two goods are gross complements if xi /pj < 0 7 8 2 Asymmetry of the Gross Definitions Asymmetry of the Gross Definitions • One undesirable characteristic of the gross definitions of substitutes and complements is that they are not symmetric • It is possible for x1 to be a substitute for x2 and at the same time for x2 to be a complement of x1 • Suppose that the utility function for two goods is given by U(x,y) = ln x + y • Setting up the Lagrangian L = ln x + y + (I – pxx – pyy) 9 Asymmetry of the Gross Definitions 10 Asymmetry of the Gross Definitions gives us the following first-order conditions: L/x = 1/x - px = 0 • Inserting this into the budget constraint, we can find the Marshallian demand for y pyy = I – py L/y = 1 - py = 0 – an increase in py causes a decline in spending on y L/ = I - pxx - pyy = 0 • Manipulating the first two equations, we get pxx = py 11 • since px and I are unchanged, spending on x must rise ( x and y are gross substitutes) • but spending on y is independent of px ( x and y are independent of one another) 12 3 Net Substitutes and Complements Net Substitutes and Complements • The concepts of net substitutes and complements focuses solely on substitution effects – two goods are net substitutes if xi p j 0 xi p j U constant – two goods are net complements if xi p j U constant x j pi U constant 13 Even though x and y are gross complements, they are net substitutes y1 y0 U1 Since MRS is diminishing, the own-price substitution effect must be negative so the cross-price substitution effect must be positive U0 x0 x1 0 U constant Gross Complements Quantity of y • This definition looks only at the shape of the indifference curve • This definition is unambiguous because the definitions are perfectly symmetric 14 Substitutability with Many Goods • Once the utility-maximizing model is extended to may goods, a wide variety of demand patterns become possible • According to Hicks’ second law of demand, “most” goods must be substitutes Quantity of x 15 16 4 Substitutability with Many Goods Substitutability with Many Goods • To prove this, we can start with the compensated demand function • In elasticity terms, we get eic1 eic2 ... einc 0 xc(p1,…pn,V) • Since the negativity of the substitution effect implies that eiic 0, it must be the case that • Applying Euler’s theorem yields p1 xic x c x c p2 i ... pn i 0 p1 p2 pn e c ij 0 j i 17 18 Composite Commodity Theorem Composite Commodities • Suppose that consumers choose among n goods • The demand for x1 will depend on the prices of the other n-1 commodities • If all of these prices move together, it may make sense to lump them into a single composite commodity (y) • In the most general case, an individual who consumes n goods will have demand functions that reflect n(n+1)/2 different substitution effects • It is often convenient to group goods into larger aggregates – examples: food, clothing, “all other goods” 19 20 5 Composite Commodity Theorem • Let p20…pn0 represent the initial prices of these other commodities – assume that they all vary together (so that the relative prices of x2…xn do not change) • Define the composite commodity y to be total expenditures on x2…xn at the initial prices Composite Commodity Theorem • The individual’s budget constraint is I = p1x1 + p20x2 +…+ pn0xn = p1x1 + y • If we assume that all of the prices p20…pn0 change by the same factor (t > 0) then the budget constraint becomes I = p1x1 + tp20x2 +…+ tpn0xn = p1x1 + ty – changes in p1 or t induce substitution effects y = p20x2 + p30x3 +…+ pn0xn 21 Composite Commodity Theorem p20…pn0 • As long as move together, we can confine our examination of demand to choices between buying x1 and “everything else” 22 Composite Commodity • A composite commodity is a group of goods for which all prices move together • These goods can be treated as a single commodity – the individual behaves as if he is choosing between other goods and spending on this entire composite group • The theorem makes no prediction about how choices of x2…xn behave – only focuses on total spending on x2…xn 23 24 6 Example: Composite Commodity Example: Composite Commodity • Suppose that an individual receives utility from three goods: utility U ( x, y , z ) 1 1 1 x y z • The Lagrangian technique can be used to derive demand functions – food (x) – housing services (y), measured in hundreds of square feet – household operations (z), measured by electricity use x I p x p x p y p x pz z • Assume a CES utility function y I py py px py pz I pz pz px pz py 25 Example: Composite Commodity 26 Example: Composite Commodity • If we assume that the prices of housing services (py) and electricity (pz) move together, we can use their initial prices to define the “composite commodity” housing (h) • If initially I = 100, px = 1, py = 4, and pz = 1, then • x* = 25, y* = 12.5, z* = 25 – $25 is spent on food and $75 is spent on housing-related needs h = 4y + 1z • The initial quantity of housing is the total spent on housing (75) 27 28 7 Example: Composite Commodity Example: Composite Commodity • Now x can be shown as a function of I, px, and ph x • If py rises to 16 and pz rises to 4 (with px remaining at 1), ph would also rise to 4 • The demand for x would fall to I py 3 px ph • If I = 100, px = 1, py = 4, and ph = 1, then x* = 25 and spending on housing (h*) = 75 29 Example: Composite Commodity 100 100 7 1 3 4 • Housing purchases would be given by Ph h* 100 100 600 7 7 30 Household Production Model • Since ph = 4, h* = 150/7 • If I = 100, px = 1, py = 16, and pz = 4, the individual demand functions show that x* = 100/7, y* = 100/28, z* = 100/14 • Assume that individuals do not receive utility directly from the goods they purchase in the market • Utility is received when the individual produces goods by combining market goods with time inputs – raw beef and uncooked potatoes yield no utility until they are cooked together to produce stew • This means that the amount of h that is consumed can also be computed as h* = 4y* + 1z* = 150/7 x* 31 32 8 Household Production Model Household Production Model • Assume that there are three goods that a person might want to purchase in the market: x, y, and z • The individual’s goal is to choose x,y, and z so as to maximize utility utility = U(a1,a2) – these goods provide no direct utility – these goods can be combined by the individual to produce either of two homeproduced goods: a1 or a2 subject to the production functions a1 = f1(x,y,z) a2 = f2(x,y,z) and a financial budget constraint • the technology of this household production can be represented by a production function pxx + pyy + pzz = I 33 34 Household Production Model The Linear Attributes Model • Two important insights from this general model can be drawn • In this model, it is the attributes of goods that provide utility to individuals • Each good has a fixed set of attributes • The model assumes that the production equations for a1 and a2 have the form – because the production functions are measurable, households can be treated as “multi-product” firms – because consuming more a1 requires more use of x, y, and z, this activity has an opportunity cost in terms of the amount of a2 that can be produced 35 a1 = ax1x + ay1y + az1z a2 = ax2x + ay2y + az2z 36 9 The Linear Attributes Model The ray 0x shows the combinations of a1 and a2 available from successively larger amounts of good x a2 x y The ray 0y shows the combinations of a1 and a2 available from successively larger amounts of good y z The Linear Attributes Model • If the individual spends all of his or her income on good x x* = I/px • That will yield a1* = ax1x* = (ax1I)/px The ray 0z shows the combinations of a1 and a2 available from successively larger amounts of good z a2* = ax2x* = (ax2I)/px a1 0 37 The Linear Attributes Model x* is the combination of a1 and a2 that would be obtained if all income was spent on x a2 38 The Linear Attributes Model All possible combinations from mixing the three market goods are represented by the shaded triangular area x*y*z* a2 x x y* is the combination of a1 and a2 that y x* would be obtained if all income was spent on y y x* y* y* z z* is the combination of a1 and a2 that would be z obtained if all income was spent on z Z* 0 z* a1 0 39 a1 40 10 The Linear Attributes Model A utility-maximizing individual would never consume positive quantities of all three goods a2 x Individuals with a preference toward a1 will have indifference curves similar to U0 and will consume only y and z y U1 z U0 0 Individuals with a preference toward a0 will have indifference curves similar to U1 and will consume only x and y a1 41 Important Points to Note: The Linear Attributes Model • The model predicts that corner solutions (where individuals consume zero amounts of some commodities) will be relatively common – especially in cases where individuals attach value to fewer attributes than there are market goods to choose from • Consumption patterns may change abruptly if income, prices, or preferences change 42 Important Points to Note: • When there are only two goods, the income and substitution effects from the change in the price of one good (py) on the demand for another good (x) usually work in opposite directions – the sign of x/py is ambiguous • the substitution effect is positive • the income effect is negative • In cases of more than two goods, demand relationships can be specified in two ways – two goods are gross substitutes if xi /pj > 0 and gross complements if xi /pj < 0 – because these price effects include income effects, they may not be symmetric • it is possible that xi /pj xj /pi 43 44 11 Important Points to Note: Important Points to Note: • Focusing only on the substitution effects from price changes does provide a symmetric definition • If a group of goods has prices that always move in unison, expenditures on these goods can be treated as a “composite commodity” whose “price” is given by the size of the proportional change in the composite goods’ prices – two goods are net substitutes if xi c/pj > 0 and net complements if xi c/pj < 0 – because xic /pj = xjc /pi, there is no ambiguity – Hicks’ second law of demand shows that net substitutes are more prevalent 45 46 Important Points to Note: • An alternative way to develop the theory of choice among market goods is to focus on the ways in which market goods are used in household production to yield utility-providing attributes – this may provide additional insights into relationships among goods 47 12 Production Function • The firm’s production function for a particular good (q) shows the maximum amount of the good that can be produced using alternative combinations of capital (k) and labor (l) Chapter 9 PRODUCTION FUNCTIONS q = f(k,l) 1 Copyright ©2005 by South-Western, a division of Thomson Learning. All rights reserved. Marginal Physical Product • To study variation in a single input, we define marginal physical product as the additional output that can be produced by employing one more unit of that input while holding other inputs constant marginal physical product of capital MPk marginal physical product of labor MPl q fk k q fl l 3 2 Diminishing Marginal Productivity • The marginal physical product of an input depends on how much of that input is used • In general, we assume diminishing marginal productivity MPk 2f 2 fkk f11 0 k k MPl 2f 2 fll f22 0 l l 4 1 Diminishing Marginal Productivity Average Physical Product • Because of diminishing marginal productivity, 19th century economist Thomas Malthus worried about the effect of population growth on labor productivity • But changes in the marginal productivity of labor over time also depend on changes in other inputs such as capital • Labor productivity is often measured by average productivity APl output q f (k, l ) labor input l l • Note that APl also depends on the amount of capital employed – we need to consider flk which is often > 0 5 A Two-Input Production Function A Two-Input Production Function • Suppose the production function for flyswatters can be represented by q = f(k,l) = 600k 2l2 • The marginal productivity function is MPl = q/l = 120,000l - 3000l2 k 3l3 which diminishes as l increases • This implies that q has a maximum value: • To construct MPl and APl, we must assume a value for k 120,000l - 3000l2 = 0 40l = l2 l = 40 – let k = 10 • The production function becomes q = 60,000l2 - 1000l3 6 7 • Labor input beyond l = 40 reduces output8 2 A Two-Input Production Function A Two-Input Production Function • To find average productivity, we hold k=10 and solve • In fact, when l = 30, both APl and MPl are equal to 900,000 APl = q/l = 60,000l - 1000l2 • Thus, when APl is at its maximum, APl and MPl are equal • APl reaches its maximum where APl/l = 60,000 - 2000l = 0 l = 30 9 10 Isoquant Map Isoquant Maps • To illustrate the possible substitution of one input for another, we use an isoquant map • An isoquant shows those combinations of k and l that can produce a given level of output (q0) • Each isoquant represents a different level of output – output rises as we move northeast k per period q = 30 f(k,l) = q0 q = 20 11 l per period 12 3 Marginal Rate of Technical Substitution (RTS) • The slope of an isoquant shows the rate at which l can be substituted for k k per period - slope = marginal rate of technical substitution (RTS) RTS > 0 and is diminishing for kA increasing inputs of labor A q = 20 l per period lA • The marginal rate of technical substitution (RTS) shows the rate at which labor can be substituted for capital while holding output constant along an isoquant RTS (l for k ) B kB Marginal Rate of Technical Substitution (RTS) RTS and Marginal Productivities • Take the total differential of the production function: f f dq dl dk MPl dl MPk dk l k MPl dl MPk dk q q0 14 RTS and Marginal Productivities • Because MPl and MPk will both be nonnegative, RTS will be positive (or zero) • However, it is generally not possible to derive a diminishing RTS from the assumption of diminishing marginal productivity alone • Along an isoquant dq = 0, so dk dl q q0 13 lB RTS (l for k ) dk dl MPl MPk 15 16 4 RTS and Marginal Productivities RTS and Marginal Productivities • Using the fact that dk/dl = -fl/fk along an isoquant and Young’s theorem (fkl = flk) • To show that isoquants are convex, we would like to show that d(RTS)/dl < 0 • Since RTS = fl/fk dRTS (fk2fll 2fk fl fkl fl 2fkk ) dl (fk )3 dRTS d (fl / fk ) dl dl dRTS [fk (fll flk dk / dl ) fl (fkl fkk dk / dl )] dl (fk )2 17 RTS and Marginal Productivities • Intuitively, it seems reasonable that fkl = flk should be positive – if workers have more capital, they will be more productive • But some production functions have fkl < 0 over some input ranges – when we assume diminishing RTS we are assuming that MPl and MPk diminish quickly enough to compensate for any possible negative cross-productivity effects 19 • Because we have assumed fk > 0, the denominator is positive • Because fll and fkk are both assumed to be negative, the ratio will be negative if fkl is positive 18 A Diminishing RTS • Suppose the production function is q = f(k,l) = 600k 2l 2 - k 3l 3 • For this production function MPl = fl = 1200k 2l - 3k 3l 2 MPk = fk = 1200kl 2 - 3k 2l 3 – these marginal productivities will be positive for values of k and l for which kl < 400 20 5 A Diminishing RTS A Diminishing RTS • Because 2 fll = 1200k • Cross differentiation of either of the marginal productivity functions yields 6k 3l fkk = 1200l 2 - 6kl 3 fkl = flk = 2400kl - 9k 2l 2 this production function exhibits diminishing marginal productivities for sufficiently large values of k and l which is positive only for kl < 266 – fll and fkk < 0 if kl > 200 21 A Diminishing RTS 22 Returns to Scale • Thus, for this production function, RTS is diminishing throughout the range of k and l where marginal productivities are positive – for higher values of k and l, the diminishing marginal productivities are sufficient to overcome the influence of a negative value for fkl to ensure convexity of the isoquants 23 • How does output respond to increases in all inputs together? – suppose that all inputs are doubled, would output double? • Returns to scale have been of interest to economists since the days of Adam Smith 24 6 Returns to Scale Returns to Scale • Smith identified two forces that come into operation as inputs are doubled – greater division of labor and specialization of function – loss in efficiency because management may become more difficult given the larger scale of the firm • If the production function is given by q = f(k,l) and all inputs are multiplied by the same positive constant (t >1), then Effect on Output Returns to Scale f(tk,tl) = tf(k,l) Constant f(tk,tl) < tf(k,l) Decreasing f(tk,tl) > tf(k,l) Increasing 25 26 Constant Returns to Scale Returns to Scale • It is possible for a production function to exhibit constant returns to scale for some levels of input usage and increasing or decreasing returns for other levels – economists refer to the degree of returns to scale with the implicit notion that only a fairly narrow range of variation in input usage and the related level of output is being considered 27 • Constant returns-to-scale production functions are homogeneous of degree one in inputs f(tk,tl) = t1f(k,l) = tq • This implies that the marginal productivity functions are homogeneous of degree zero – if a function is homogeneous of degree k, its derivatives are homogeneous of degree 28 k-1 7 Constant Returns to Scale Constant Returns to Scale • The marginal productivity of any input depends on the ratio of capital and labor (not on the absolute levels of these inputs) • The RTS between k and l depends only on the ratio of k to l, not the scale of operation • The production function will be homothetic • Geometrically, all of the isoquants are radial expansions of one another 29 30 Returns to Scale Constant Returns to Scale • Along a ray from the origin (constant k/l), the RTS will be the same on all isoquants • Returns to scale can be generalized to a production function with n inputs q = f(x1,x2,…,xn) k per period The isoquants are equally spaced as output expands • If all inputs are multiplied by a positive constant t, we have f(tx1,tx2,…,txn) = tkf(x1,x2,…,xn)=tkq – If k = 1, we have constant returns to scale – If k < 1, we have decreasing returns to scale – If k > 1, we have increasing returns to scale q=3 q=2 q=1 l per period 31 32 8 Elasticity of Substitution Elasticity of Substitution • The elasticity of substitution () measures the proportionate change in k/l relative to the proportionate change in the RTS along an isoquant %(k / l ) d (k / l ) RTS ln( k / l )
%RTS dRTS k / l ln RTS is the ratio of these k per period proportional changes measures the RTSA A • The value of will always be positive because k/l and RTS move in the same direction 33 Elasticity of Substitution RTSB (k/l)A (k/l)B B curvature of the isoquant q = q0 l per period 34 Elasticity of Substitution • If is high, the RTS will not change much relative to k/l • Generalizing the elasticity of substitution to the many-input case raises several complications – the isoquant will be relatively flat – if we define the elasticity of substitution between two inputs to be the proportionate change in the ratio of the two inputs to the proportionate change in RTS, we need to hold output and the levels of other inputs constant • If is low, the RTS will change by a substantial amount as k/l changes – the isoquant will be sharply curved • It is possible for to change along an isoquant or as the scale of production changes • Both RTS and k/l will change as we move from point A to point B 35 36 9 The Linear Production Function • Suppose that the production function is The Linear Production Function Capital and labor are perfect substitutes q = f(k,l) = ak + bl • This production function exhibits constant returns to scale k per period RTS is constant as k/l changes f(tk,tl) = atk + btl = t(ak + bl) = tf(k,l) = slope = -b/a • All isoquants are straight lines – RTS is constant –= q2 q1 q3 l per period 37 Fixed Proportions 38 Fixed Proportions No substitution between labor and capital is possible • Suppose that the production function is q = min (ak,bl) a,b > 0 k/l is fixed at b/a k per period • Capital and labor must always be used in a fixed ratio – the firm will always operate along a ray where k/l is constant =0 q3 q3/a • Because k/l is constant, = 0 q2 q1 39 l per period q3/b 40 10 Cobb-Douglas Production Function Cobb-Douglas Production Function • Suppose that the production function is • The Cobb-Douglas production function is linear in logarithms q = f(k,l) = Akalb A,a,b > 0 • This production function can exhibit any returns to scale ln q = ln A + a ln k + b ln l – a is the elasticity of output with respect to k – b is the elasticity of output with respect to l f(tk,tl) = A(tk)a(tl)b = Ata+b kalb = ta+bf(k,l) – if a + b = 1 constant returns to scale – if a + b > 1 increasing returns to scale – if a + b < 1 decreasing returns to scale 41 42 A Generalized Leontief Production Function CES Production Function • Suppose that the production function is q = f(k,l) = [k + l] / 1, 0, > 0 • Suppose that the production function is – > 1 increasing returns to scale q = f(k,l) = k + l + 2(kl)0.5 – < 1 decreasing returns to scale • Marginal productivities are • For this production function = 1/(1-) – = 1 linear production function – = - fixed proportions production function – = 0 Cobb-Douglas production function 43 fk = 1 + (k/l)-0.5 fl = 1 + (k/l)0.5 • Thus, RTS fl 1 (k / l )0.5 fk 1 (k / l )0.5 44 11 Technical Progress Technical Progress • Methods of production change over time • Following the development of superior production techniques, the same level of output can be produced with fewer inputs • Suppose that the production function is – the isoquant shifts in q = A(t)f(k,l) where A(t) represents all influences that go into determining q other than k and l – changes in A over time represent technical progress • A is shown as a function of time (t) • dA/dt > 0 45 Technical Progress 46 Technical Progress • Differentiating the production function with respect to time we get • Dividing by q gives us dq / dt dA / dt f / k dk f / l dl
q A f (k, l ) dt f (k, l ) dt dq dA df (k, l ) f (k, l ) A dt dt dt dq / dt dA / dt f k dk / dt f l dl / dt
q A k f (k, l ) k l f (k, l ) l dq dA q q f dk f dl dt dt A f (k, l) k dt l dt 47 48 12 Technical Progress Technical Progress • For any variable x, [(dx/dt)/x] is the proportional growth rate in x • Since – denote this by Gx • Then, we can write the equation in terms of growth rates Gq GA f k f l Gk Gl k f (k, l ) l f (k, l ) f k q k eq,k k f (k, l ) k q f l q l eq,l l f (k, l ) l q Gq GA eq,kGk eq,lGl 49 Technical Progress in the Cobb-Douglas Function Technical Progress in the Cobb-Douglas Function • Suppose that the production function is q = A(t)f(k,l) = 50 A(t)k l 1• If we assume that technical progress occurs at a constant exponential () then A(t) = Ae-t q = Ae-tk l 151 • Taking logarithms and differentiating with respect to t gives the growth equation ln q ln q q q / t
Gq t q t q 52 13 Technical Progress in the Cobb-Douglas Function Important Points to Note: (ln A t ln k (1 ) ln l ) t ln k ln l (1 ) Gk (1 )Gl t t Gq • If all but one of the inputs are held constant, a relationship between the single variable input and output can be derived – the marginal physical productivity is the change in output resulting from a one-unit increase in the use of the input • assumed to decline as use of the input increases 53 Important Points to Note: 54 Important Points to Note: • The entire production function can be illustrated by an isoquant map • Isoquants are usually assumed to be convex – the slope of an isoquant is the marginal rate of technical substitution (RTS) – they obey the assumption of a diminishing RTS • it shows how one input can be substituted for another while holding output constant • it is the ratio of the marginal physical productivities of the two inputs • this assumption cannot be derived exclusively from the assumption of diminishing marginal productivity • one must be concerned with the effect of changes in one input on the marginal productivity of other inputs 55 56 14 Important Points to Note: Important Points to Note: • The elasticity of substitution () provides a measure of how easy it is to substitute one input for another in production • The returns to scale exhibited by a production function record how output responds to proportionate increases in all inputs – if output increases proportionately with input use, there are constant returns to scale 57 – a high implies nearly straight isoquants – a low implies that isoquants are nearly L-shaped 58 Important Points to Note: • Technical progress shifts the entire production function and isoquant map – technical improvements may arise from the use of more productive inputs or better methods of economic organization 59 15 Definitions of Costs • It is important to differentiate between accounting cost and economic cost Chapter 10 – the accountant’s view of cost stresses outof-pocket expenses, historical costs, depreciation, and other bookkeeping entries – economists focus more on opportunity cost COST FUNCTIONS Copyright ©2005 by South-western, a division of Thomson learning. All rights reserved. 1 Definitions of Costs 2 Definitions of Costs • Capital Costs • Labor Costs – accountants use the historical price of the capital and apply some depreciation rule to determine current costs – economists refer to the capital’s original price as a “sunk cost” and instead regard the implicit cost of the capital to be what someone else would be willing to pay for its use – to accountants, expenditures on labor are current expenses and hence costs of production – to economists, labor is an explicit cost • labor services are contracted at some hourly wage (w) and it is assumed that this is also what the labor could earn in alternative employment • we will use v to denote the rental rate for capital 3 4 1 Definitions of Costs Economic Cost • Costs of Entrepreneurial Services – accountants believe that the owner of a firm is entitled to all profits • revenues or losses left over after paying all input costs – economists consider the opportunity costs of time and funds that owners devote to the operation of their firms • The economic cost of any input is the payment required to keep that input in its present employment – the remuneration the input would receive in its best alternative employment • part of accounting profits would be considered as entrepreneurial costs by economists 5 Two Simplifying Assumptions • There are only two inputs 6 Economic Profits • Total costs for the firm are given by – homogeneous labor (l), measured in laborhours – homogeneous capital (k), measured in machine-hours total costs = C = wl + vk • Total revenue for the firm is given by total revenue = pq = pf(k,l) • entrepreneurial costs are included in capital costs • Inputs are hired in perfectly competitive markets – firms are price takers in input markets 7 • Economic profits () are equal to = total revenue - total cost = pq - wl - vk = pf(k,l) - wl - vk 8 2 Cost-Minimizing Input Choices Economic Profits • Economic profits are a function of the amount of capital and labor employed – we could examine how a firm would choose k and l to maximize profit • “derived demand” theory of labor and capital inputs • To minimize the cost of producing a given level of output, a firm should choose a point on the isoquant at which the RTS is equal to the ratio w/v – it should equate the rate at which k can be traded for l in the productive process to the rate at which they can be traded in the marketplace – for now, we will assume that the firm has already chosen its output level (q0) and wants to minimize its costs 9 Cost-Minimizing Input Choices • Mathematically, we seek to minimize total costs given q = f(k,l) = q0 • Setting up the Lagrangian: 10 Cost-Minimizing Input Choices • Dividing the first two conditions we get w f / l RTS (l for k ) v f / k L = wl + vk + [q0 - f(k,l)] • The cost-minimizing firm should equate the RTS for the two inputs to the ratio of their prices • First order conditions are L/l = w - (f/l) = 0 L/k = v - (f/k) = 0 L/ = q0 - f(k,l) = 0 11 12 3 Cost-Minimizing Input Choices • Cross-multiplying, we get Cost-Minimizing Input Choices • Note that this equation’s inverse is also of interest fk fl v w • For costs to be minimized, the marginal productivity per dollar spent should be the same for all inputs w v fl fk • The Lagrangian multiplier shows how much in extra costs would be incurred by increasing the output constraint slightly 13 Cost-Minimizing Input Choices 14 Cost-Minimizing Input Choices Given output q0, we wish to find the least costly point on the isoquant The minimum cost of producing q0 is C2 k per period k per period C1 C3 Costs are represented by parallel lines with a slope of w/v This occurs at the tangency between the isoquant and the total cost curve C1 C3 C2 C2 C1 < C2 < C3 k* q0 q0 l per period 15 l* The optimal choice is l*, k* l per period 16 4 Contingent Demand for Inputs • In Chapter 4, we considered an individual’s expenditure-minimization problem Contingent Demand for Inputs • In the present case, cost minimization leads to a demand for capital and labor that is contingent on the level of output being produced • The demand for an input is a derived demand – we used this technique to develop the compensated demand for a good • Can we develop a firm’s demand for an input in the same way? – it is based on the level of the firm’s output 17 The Firm’s Expansion Path 18 The Firm’s Expansion Path • The firm can determine the costminimizing combinations of k and l for every level of output • If input costs remain constant for all amounts of k and l the firm may demand, we can trace the locus of costminimizing choices The expansion path is the locus of costminimizing tangencies k per period The curve shows how inputs increase as output increases E q1 – called the firm’s expansion path q0 q00 19 l per period 20 5 The Firm’s Expansion Path Cost Minimization • The expansion path does not have to be a straight line • Suppose that the production function is Cobb-Douglas: – the use of some inputs may increase faster than others as output expands • depends on the shape of the isoquants • The expansion path does not have to be upward sloping – if the use of an input falls as output expands, that input is an inferior input q = kl • The Lagrangian expression for cost minimization of producing q0 is L = vk + wl + (q0 - k l ) 21 22 Cost Minimization Cost Minimization • The first-order conditions for a minimum are • Dividing the first equation by the second gives us L/k = v - k -1l = 0 w k l 1 k RTS v k 1l l L/l = w - k l -1 = 0 L/ = q0 - k l = 0 • This production function is homothetic – the RTS depends only on the ratio of the two inputs – the expansion path is a straight line 23 24 6 Cost Minimization Cost Minimization • Suppose that the production function is CES: • The first-order conditions for a minimum are q = (k + l )/ L/k = v - (/)(k + l)(-)/()k-1 = 0 • The Lagrangian expression for cost minimization of producing q0 is L/l = w - (/)(k + l)(-)/()l-1 = 0 L/ = q0 - (k + l )/ = 0 L = vk + wl + [q0 - (k + l )/] 25 26 Cost Minimization Total Cost Function • Dividing the first equation by the second gives us • The total cost function shows that for any set of input costs and for any output level, the minimum cost incurred by the firm is w 1 v k 1 1 k l 1/ k l C = C(v,w,q) • This production function is also homothetic • As output (q) increases, total costs increase 27 28 7 Marginal Cost Function Average Cost Function • The average cost function (AC) is found by computing total costs per unit of output average cost AC(v ,w, q ) • The marginal cost function (MC) is found by computing the change in total costs for a change in output produced C(v ,w, q ) q marginal cost MC(v ,w, q ) C(v ,w, q ) q 29 30 Graphical Analysis of Total Costs Graphical Analysis of Total Costs • Suppose that k1 units of capital and l1 units of labor input are required to produce one unit of output Total costs With constant returns to scale, total costs are proportional to output AC = MC C C(q=1) = vk1 + wl1 Both AC and MC will be constant • To produce m units of output (assuming constant returns to scale) C(q=m) = vmk1 + wml1 = m(vk1 + wl1) C(q=m) = m C(q=1) Output 31 32 8 Graphical Analysis of Total Costs Graphical Analysis of Total Costs • Suppose instead that total costs start out as concave and then becomes convex as output increases Total costs C Total costs rise dramatically as output increases after diminishing returns set in – one possible explanation for this is that there is a third factor of production that is fixed as capital and labor usage expands – total costs begin rising rapidly after diminishing returns set in Output 33 Graphical Analysis of Total Costs Average and marginal costs Shifts in Cost Curves MC is the slope of the C curve MC AC min AC 34 If AC > MC, AC must be falling If AC < MC, AC must be rising • The cost curves are drawn under the assumption that input prices and the level of technology are held constant – any change in these factors will cause the cost curves to shift Output 35 36 9 Some Illustrative Cost Functions Some Illustrative Cost Functions • Suppose we have a fixed proportions technology such that • Suppose we have a Cobb-Douglas technology such that q = f(k,l) = k l q = f(k,l) = min(ak,bl) • Production will occur at the vertex of the L-shaped isoquants (q = ak = bl) • Cost minimization requires that w k v l C(w,v,q) = vk + wl = v(q/a) + w(q/b) v w C(w ,v , q ) a a b k 37 Some Illustrative Cost Functions l q • Now we can derive total costs as C(v,w, q ) vk wl q1/ Bv / w / / w / v / where B ( ) / / • A similar method will yield k q 1 / 38 Some Illustrative Cost Functions • If we substitute into the production function and solve for l, we will get 1/ w l v / which is a constant that involves only the parameters and w / v / 39 40 10 Some Illustrative Cost Functions Properties of Cost Functions • Homogeneity • Suppose we have a CES technology such that q = f(k,l) = (k – cost functions are all homogeneous of degree one in the input prices + l )/ • cost minimization requires that the ratio of input prices be set equal to RTS, a doubling of all input prices will not change the levels of inputs purchased • pure, uniform inflation will not change a firm’s input decisions but will shift the cost curves up • To derive the total cost, we would use the same method and eventually get C(v,w, q ) vk wl q1/ (v / 1 w / 1 )( 1) / C(v,w, q ) q1/ (v 1 w 1 )1/ 1 41 42 Properties of Cost Functions Properties of Cost Functions • Nondecreasing in q, v, and w • Concave in input prices – cost functions are derived from a costminimization process • any decline in costs from an increase in one of the function’s arguments would lead to a contradiction 43 – costs will be lower when a firm faces input prices that fluctuate around a given level than when they remain constant at that level • the firm can adapt its input mix to take advantage of such fluctuations 44 11 Concavity of Cost Function Properties of Cost Functions At w1, the firm’s costs are C(v,w1,q1) • Some of these properties carry over to average and marginal costs Costs If the firm continues to buy the same input mix as w changes, its cost function would be Cpseudo Cpseudo – homogeneity – effects of v, w, and q are ambiguous C(v,w,q1) Since the firm’s input mix will likely change, actual costs will be less than Cpseudo such as C(v,w,q1) C(v,w 1,q1) w w1 45 Input Substitution Input Substitution • Putting this in proportional terms as • A change in the price of an input will cause the firm to alter its input mix • We wish to see how k/l changes in response to a change in w/v, while holding q constant k l 46 s (k / l ) w / v ln( k / l ) (w / v ) k / l ln( w / v ) gives an alternative definition of the elasticity of substitution – in the two-input case, s must be nonnegative – large values of s indicate that firms change their input mix significantly if input prices change w v 47 48 12 Partial Elasticity of Substitution • The partial elasticity of substitution between two inputs (xi and xj) with prices wi and wj is given by sij Size of Shifts in Costs Curves • The increase in costs will be largely influenced by the relative significance of the input in the production process • If firms can easily substitute another input for the one that has risen in price, there may be little increase in costs ( x i / x j ) w j / w i ln( xi / x j ) (w j / w i ) xi / x j ln( w j / w i ) • Sij is a more flexible concept than because it allows the firm to alter the usage of inputs other than xi and xj when input prices change 49 Technical Progress 50 Technical Progress • Improvements in technology also lower cost curves • Suppose that total costs (with constant returns to scale) are • Because the same inputs that produced one unit of output in period zero will produce A(t) units in period t Ct(v,w,A(t)) = A(t)Ct(v,w,1)= C0(v,w,1) C0 = C0(q,v,w) = qC0(v,w,1) • Total costs are given by Ct(v,w,q) = qCt(v,w,1) = qC0(v,w,1)/A(t) = C0(v,w,q)/A(t) 51 52 13 Shifting the Cobb-Douglas Cost Function Shifting the Cobb-Douglas Cost Function • The Cobb-Douglas cost function is • If v = 3 and w = 12, the relationship is C(v,w, q ) vk wl q1/ Bv / w / C(3,12, q ) 2q 36 12q where B ( ) / / • If we assume = = 0.5, the total cost curve is greatly simplified: – C = 480 to produce q =40 – AC = C/q = 12 – MC = C/q = 12 C(v,w, q ) vk wl 2qv 0.5w 0.5 53 Shifting the Cobb-Douglas Cost Function 54 Contingent Demand for Inputs • If v = 3 and w = 27, the relationship is • Contingent demand functions for all of the firms inputs can be derived from the cost function C(3,27, q ) 2q 81 18q – C = 720 to produce q =40 – AC = C/q = 18 – MC = C/q = 18 – Shephard’s lemma • the contingent demand function for any input is given by the partial derivative of the total-cost function with respect to that input’s price 55 56 14 Contingent Demand for Inputs • Suppose we have a fixed proportions technology • The cost function is Contingent Demand for Inputs • For this cost function, contingent demand functions are quite simple: v w C(w ,v , q ) a a b k c (v ,w , q ) C(v ,w , q ) q v a l c (v ,w , q ) C(v ,w , q ) q w b 57 Contingent Demand for Inputs 58 Contingent Demand for Inputs • For this cost function, the derivation is messier: • Suppose we have a Cobb-Douglas technology • The cost function is k c (v ,w , q ) C(v,w, q ) vk wl q1/ Bv / w / 59 C q 1/ Bv / w / v w q 1/ B v / 60 15 Contingent Demand for Inputs l c (v ,w , q ) C q 1/ Bv / w / w w q 1/ B v Short-Run, Long-Run Distinction • In the short run, economic actors have only limited flexibility in their actions • Assume that the capital input is held constant at k1 and the firm is free to vary only its labor input • The production function becomes / • The contingent demands for inputs depend on both inputs’ prices q = f(k1,l) 61 62 Short-Run Total Costs Short-Run Total Costs • Short-run total cost for the firm is • Short-run costs are not minimal costs for producing the various output levels SC = vk1 + wl – the firm does not have the flexibility of input choice – to vary its output in the short run, the firm must use nonoptimal input combinations – the RTS will not be equal to the ratio of input prices • There are two types of short-run costs: – short-run fixed costs are costs associated with fixed inputs (vk1) – short-run variable costs are costs associated with variable inputs (wl) 63 64 16 Short-Run Marginal and Average Costs Short-Run Total Costs k per period Because capital is fixed at k1, the firm cannot equate RTS with the ratio of input prices • The short-run average total cost (SAC) function is SAC = total costs/total output = SC/q • The short-run marginal cost (SMC) function is k1 q2 SMC = change in SC/change in output = SC/q q1 q0 l1 l2 l per period l3 65 66 Relationship between ShortRun and Long-Run Costs Relationship between ShortRun and Long-Run Costs SC (k2) Total costs SC (k1) Costs C SMC (k0) q0 q1 q2 Output 67 SAC (k0) MC AC The long-run C curve can be derived by varying the level of k SC (k0) SMC (k1) q0 q1 SAC (k1) The geometric relationship between shortrun and long-run AC and MC can also be shown Output 68 17 Relationship between ShortRun and Long-Run Costs • At the minimum point of the AC curve: – the MC curve crosses the AC curve • MC = AC at this point – the SAC curve is tangent to the AC curve • SAC (for this level of k) is minimized at the same level of output as AC • SMC intersects SAC also at this point Important Points to Note: • A firm that wishes to minimize the economic costs of producing a particular level of output should choose that input combination for which the rate of technical substitution (RTS) is equal to the ratio of the inputs’ rental prices AC = MC = SAC = SMC 69 Important Points to Note: 70 Important Points to Note: • The firm’s average cost (AC = C/q) and marginal cost (MC = C/q) can be derived directly from the total-cost function • Repeated application of this minimization procedure yields the firm’s expansion path – the expansion path shows how input usage expands with the level of output – if the total cost curve has a general cubic shape, the AC and MC curves will be ushaped • it also shows the relationship between output level and total cost • this relationship is summarized by the total cost function, C(v,w,q) 71 72 18 Important Points to Note: Important Points to Note: • All cost curves are drawn on the assumption that the input prices are held constant • Input demand functions can be derived from the firm’s total-cost function through partial differentiation – when an input price changes, cost curves shift to new positions • the size of the shifts will be determined by the overall importance of the input and the substitution abilities of the firm – technical progress will also shift cost curves 73 – these input demands will depend on the quantity of output the firm chooses to produce • are called “contingent” demand functions 74 Important Points to Note: • In the short run, the firm may not be able to vary some inputs – it can then alter its level of production only by changing the employment of its variable inputs – it may have to use nonoptimal, highercost input combinations than it would choose if it were possible to vary all inputs 75 19 The Nature of Firms • A firm is an association of individuals who have organized themselves for the purpose of turning inputs into outputs • Different individuals will provide different types of inputs Chapter 11 PROFIT MAXIMIZATION – the nature of the contractual relationship between the providers of inputs to a firm may be quite complicated Copyright ©2005 by South-Western, a division of Thomson Learning. All rights reserved. 1 2 Modeling Firms’ Behavior Contractual Relationships • Some contracts between providers of inputs may be explicit • Most economists treat the firm as a single decision-making unit – may specify hours, work details, or compensation – the decisions are made by a single dictatorial manager who rationally pursues some goal • Other arrangements will be more implicit in nature • usually profit-maximization – decision-making authority or sharing of tasks 3 4 1 Profit Maximization Profit Maximization • A profit-maximizing firm chooses both its inputs and its outputs with the sole goal of achieving maximum economic profits • If firms are strictly profit maximizers, they will make decisions in a “marginal” way – examine the marginal profit obtainable from producing one more unit of hiring one additional laborer – seeks to maximize the difference between total revenue and total economic costs 5 Output Choice 6 Output Choice • Total revenue for a firm is given by • The necessary condition for choosing the level of q that maximizes profits can be found by setting the derivative of the function with respect to q equal to zero R(q) = p(q)q • In the production of q, certain economic costs are incurred [C(q)] • Economic profits () are the difference between total revenue and total costs d dR dC ' (q ) 0 dq dq dq (q) = R(q) – C(q) = p(q)q –C(q) dR dC dq dq 7 8 2 Output Choice Second-Order Conditions • To maximize economic profits, the firm should choose the output for which marginal revenue is equal to marginal cost • MR = MC is only a necessary condition for profit maximization • For sufficiency, it is also required that d 2 d' (q ) 0 2 dq q q * dq q q * dR dC MR MC dq dq • “marginal” profit must be decreasing at the optimal level of q 9 Profit Maximization Marginal Revenue Profits are maximized when the slope of the revenue function is equal to the slope of the cost function revenues & costs 10 C R The second-order condition prevents us from mistaking q0 as a maximum • If a firm can sell all it wishes without having any effect on market price, marginal revenue will be equal to price • If a firm faces a downward-sloping demand curve, more output can only be sold if the firm reduces the good’s price marginal revenue MR(q ) q0 q* dR d [ p(q ) q ] dp pq dq dq dq output 11 12 3 Marginal Revenue Marginal Revenue • If a firm faces a downward-sloping demand curve, marginal revenue will be a function of output • If price falls as a firm increases output, marginal revenue will be less than price • Suppose that the demand curve for a sub sandwich is q = 100 – 10p • Solving for price, we get p = -q/10 + 10 • This means that total revenue is R = pq = -q2/10 + 10q • Marginal revenue will be given by 13 MR = dR/dq = -q/5 + 10 14 Marginal Revenue and Elasticity Profit Maximization • To determine the profit-maximizing output, we must know the firm’s costs • If subs can be produced at a constant average and marginal cost of $4, then • The concept of marginal revenue is directly related to the elasticity of the demand curve facing the firm • The price elasticity of demand is equal to the percentage change in quantity that results from a one percent change in price MR = MC -q/5 + 10 = 4 q = 30 eq,p 15 dq / q dq p dp / p dp q 16 4 Marginal Revenue and Elasticity Marginal Revenue and Elasticity • This means that MR p q dp q dp 1 p1 p1 dq p dq eq,p – if the demand curve slopes downward, eq,p < 0 and MR < p – if the demand is elastic, eq,p < -1 and marginal revenue will be positive eq,p < -1 MR > 0 eq,p = -1 MR = 0 eq,p > -1 MR < 0 • if the demand is infinitely elastic, eq,p = - and marginal revenue will equal price 17 The Inverse Elasticity Rule The Inverse Elasticity Rule • Because MR = MC when the firm maximizes profit, we can see that 1 MC p1 e q ,p 18 p MC 1 p eq,p p MC 1 p eq,p • The gap between price and marginal cost will fall as the demand curve facing the firm becomes more elastic 19 • If eq,p > -1, MC < 0 • This means that firms will choose to operate only at points on the demand curve where demand is elastic 20 5 Average Revenue Curve Marginal Revenue Curve • If we assume that the firm must sell all its output at one price, we can think of the demand curve facing the firm as its average revenue curve • The marginal revenue curve shows the extra revenue provided by the last unit sold • In the case of a downward-sloping demand curve, the marginal revenue curve will lie below the demand curve – shows the revenue per unit yielded by alternative output choices 21 Marginal Revenue Curve price 22 Marginal Revenue Curve As output increases from 0 to q1, total revenue increases so MR > 0 As output increases beyond q1, total revenue decreases so MR < 0 • When the demand curve shifts, its associated marginal revenue curve shifts as well – a marginal revenue curve cannot be calculated without referring to a specific demand curve p1 D (average revenue) output q1 MR 23 24 6 The Constant Elasticity Case The Constant Elasticity Case • We showed (in Chapter 5) that a demand function of the form • This means that R = pq = kq(1+b)/b q = apb and has a constant price elasticity of demand equal to b • Solving this equation for p, we get p = (1/a)1/bq1/b = kq1/b MR = dr/dq = [(1+b)/b]kq1/b = [(1+b)/b]p • This implies that MR is proportional to price where k = (1/a)1/b 25 Short-Run Supply by a Price-Taking Firm price Short-Run Supply by a Price-Taking Firm price SMC SAC p* = MR q* 26 SMC SAC p* = MR SAVC SAVC Maximum profit occurs where p = SMC Since p > SAC, profit > 0 output q* 27 output 28 7 Short-Run Supply by a Price-Taking Firm Short-Run Supply by a Price-Taking Firm price price SMC SMC p** SAC p* = MR SAC p* = MR SAVC SAVC If the price rises to p**, the firm will produce q** and > 0 q* q** If the price falls to p***, the firm will produce q*** output Profit maximization requires that p = SMC and that SMC is upward-sloping p*** q*** q* output <0 29 Short-Run Supply by a Price-Taking Firm 30 Short-Run Supply by a Price-Taking Firm • The positively-sloped portion of the short-run marginal cost curve is the short-run supply curve for a price-taking firm – it shows how much the firm will produce at every possible market price – firms will only operate in the short run as long as total revenue covers variable cost • Thus, the price-taking firm’s short-run supply curve is the positively-sloped portion of the firm’s short-run marginal cost curve above the point of minimum average variable cost – for prices below this level, the firm’s profitmaximizing decision is to shut down and produce no output • the firm will produce no output if p < SAVC 31 32 8 Short-Run Supply by a Price-Taking Firm price Short-Run Supply • Suppose that the firm’s short-run total cost curve is SMC SC(v,w,q,k) = vk1 + wq1/k1-/ SAC SAVC The firm’s short-run supply curve is the SMC curve that is above SAVC where k1 is the level of capital held constant in the short run • Short-run marginal cost is SMC(v ,w, q, k1 ) output SC w (1 ) / / q k1 q 33 Short-Run Supply Short-Run Supply • The price-taking firm will maximize profit where p = SMC SMC • SAVC < SMC for all values of < 1 • Therefore, quantity supplied will be – there is no price low enough that the firm will want to shut down /(1 ) /(1 ) 1 k p • To find the firm’s shut-down price, we need to solve for SAVC SVC = wq1/k1-/ SAVC = SVC/q = wq(1-)/k1-/ w (1 ) / / q k1 p w q 34 /(1 ) 35 36 9 Profit Functions Profit Functions • A firm’s economic profit can be expressed as a function of inputs • A firm’s profit function shows its maximal profits as a function of the prices that the firm faces = pq - C(q) = pf(k,l) - vk - wl • Only the variables k and l are under the firm’s control ( p,v,w ) Max (k, l ) Max[ pf (k, l ) vk wl ] k ,l k ,l – the firm chooses levels of these inputs in order to maximize profits • treats p, v, and w as fixed parameters in its decisions 37 Properties of the Profit Function 38 Properties of the Profit Function • Homogeneity • Nondecreasing in output price – the profit function is homogeneous of degree one in all prices • with pure inflation, a firm will not change its production plans and its level of profits will keep up with that inflation 39 – a firm could always respond to a rise in the price of its output by not changing its input or output plans • profits must rise 40 10 Properties of the Profit Function Properties of the Profit Function • Nonincreasing in input prices • Convex in output prices – if the firm responded to an increase in an input price by not changing the level of that input, its costs would rise • profits would fall – the profits obtainable by averaging those from two different output prices will be at least as large as those obtainable from the average of the two prices ( p1,v ,w ) ( p2 ,v ,w ) p p2 1 ,v ,w 2 2 41 Producer Surplus in the Short Run Envelope Results • We can apply the envelope theorem to see how profits respond to changes in output and input prices • Because the profit function is nondecreasing in output prices, we know that if p2 > p1 ( p,v ,w ) q( p,v ,w ) p (p2,…) (p1,…) • The welfare gain to the firm of this price increase can be measured by ( p,v ,w ) k ( p,v ,w ) v ( p,v ,w ) l ( p,v ,w ) w 42 welfare gain = (p2,…) - (p1,…) 43 44 11 Producer Surplus in the Short Run Producer Surplus in the Short Run SMC price If the market price is p1, the firm will produce q1 p2 p1 If the market price rises to p2, the firm will produce q2 q1 output q2 SMC price The firm’s profits rise by the shaded area p2 p1 q1 q2 output 45 Producer Surplus in the Short Run Producer Surplus in the Short Run • Mathematically, we can use the envelope theorem results p2 p2 p1 p1 46 • We can measure how much the firm values the right to produce at the prevailing price relative to a situation where it would produce no output welfare gain q( p)dp ( / p)dp ( p2 ,...) ( p1,...) 47 48 12 Producer Surplus in the Short Run Producer Surplus in the Short Run SMC price Suppose that the firm’s shutdown price is p0 p1 • The extra profits available from facing a price of p1 are defined to be producer surplus p1 producer surplus ( p1,...) ( p0 ,...) q( p)dp p 0 p0 output q1 49 Producer Surplus in the Short Run Producer Surplus in the Short Run SMC price Producer surplus at a market price of p1 is the shaded area p1 50 p • Producer surplus is the extra return that producers make by making transactions at the market price over and above what they would earn if nothing was produced – the area below the market price and above the supply curve 0 q1 output 51 52 13 Producer Surplus in the Short Run Profit Maximization and Input Demand • Because the firm produces no output at the shutdown price, (p0,…) = -vk1 – profits at the shutdown price are equal to the firm’s fixed costs • This implies that producer surplus = (p1,…) - (p0,…) = (p1,…) – (-vk1) = (p1,…) + vk1 – producer surplus is equal to current profits plus short-run fixed costs 53 Profit Maximization and Input Demand • A firm’s output is determined by the amount of inputs it chooses to employ – the relationship between inputs and outputs is summarized by the production function • A firm’s economic profit can also be expressed as a function of inputs (k,l) = pq –C(q) = pf(k,l) – (vk + wl) 54 Profit Maximization and Input Demand • The first-order conditions for a maximum are /k = p[f/k] – v = 0 /l = p[f/l] – w = 0 • These first-order conditions for profit maximization also imply cost minimization – they imply that RTS = w/v • A profit-maximizing firm should hire any input up to the point at which its marginal contribution to revenues is equal to the marginal cost of hiring the input 55 56 14 Profit Maximization and Input Demand Input Demand Functions • In principle, the first-order conditions can be solved to yield input demand functions • To ensure a true maximum, secondorder conditions require that Capital Demand = k(p,v,w) Labor Demand = l(p,v,w) kk = fkk < 0 ll = fll < 0 • These demand functions are unconditional kk ll - kl2 = fkkfll – fkl2 > 0 – capital and labor must exhibit sufficiently diminishing marginal productivities so that marginal costs rise as output expands – they implicitly allow the firm to adjust its output to changing prices 57 Single-Input Case Single-Input Case • This reduces to • We expect l/w 0 – diminishing marginal productivity of labor 1 p fll • The first order condition for profit maximization was l w • Solving further, we get l 1 w p fll /l = p[f/l] – w = 0 • Taking the total differential, we get dw p 58 • Since fll 0, l/w 0 fl l dw l w 59 60 15 Two-Input Case Two-Input Case • For the case of two (or more inputs), the story is more complex – if there is a decrease in w, there will not only be a change in l but also a change in k as a new cost-minimizing combination of inputs is chosen • When w falls, two effects occur – substitution effect • if output is held constant, there will be a tendency for the firm to want to substitute l for k in the production process – output effect • a change in w will shift the firm’s expansion path • the firm’s cost curves will shift and a different output level will be chosen • when k changes, the entire fl function changes • But, even in this case, l/w 0 61 62 Substitution Effect k per period Output Effect If output is held constant at q0 and w falls, the firm will substitute l for k in the production process Because of diminishing RTS along an isoquant, the substitution effect will always be negative A decline in w will lower the firm’s MC Price MC MC’ Consequently, the firm will choose a new level of output that is higher P q0 l per period 63 q0 q1 Output 64 16 Output Effect Cross-Price Effects Output will rise to q1 k per period • No definite statement can be made about how capital usage responds to a wage change Thus, the output effect also implies a negative relationship between l and w – a fall in the wage will lead the firm to substitute away from capital – the output effect will cause more capital to be demanded as the firm expands production q1 q0 l per period 65 Substitution and Output Effects 66 Substitution and Output Effects • We have two concepts of demand for any input • Differentiation with respect to w yields – the conditional demand for labor, – the unconditional demand for labor, l(p,v,w) lc(v,w,q) • At the profit-maximizing level of output l( p,v ,w ) l c (v ,w, q ) l c (v ,w, q ) q
w w q w substitution effect lc(v,w,q) = l(p,v,w) output effect total effect 67 68 17 Important Points to Note: Important Points to Note: • In order to maximize profits, the firm should choose to produce that output level for which the marginal revenue is equal to the marginal cost • If a firm is a price taker, its output decisions do not affect the price of its output – marginal revenue is equal to price • If the firm faces a downward-sloping demand for its output, marginal revenue will be less than price 69 70 Important Points to Note: Important Points to Note: • Marginal revenue and the price elasticity of demand are related by the formula • The supply curve for a price-taking, profit-maximizing firm is given by the positively sloped portion of its marginal cost curve above the point of minimum average variable cost (AVC) 1 MR p1 e q ,p – if price falls below minimum AVC, the firm’s profit-maximizing choice is to shut down and produce nothing 71 72 18 Important Points to Note: Important Points to Note: • The firm’s reactions to the various prices it faces can be judged through use of its profit function • The firm’s profit function yields particularly useful envelope results – differentiation with respect to market price yields the supply function – differentiation with respect to any input price yields the (inverse of) the demand function for that input – shows maximum profits for the firm given the price of its output, the prices of its inputs, and the production technology 73 74 Important Points to Note: Important Points to Note: • Short-run changes in market price result in changes in the firm’s short-run profitability • Profit maximization provides a theory of the firm’s derived demand for inputs – these can be measured graphically by changes in the size of producer surplus – the profit function can also be used to calculate changes in producer surplus 75 – the firm will hire any input up to the point at which the value of its marginal product is just equal to its per-unit market price – increases in the price of an input will induce substitution and output effects that cause the firm to reduce hiring of that input 76 19 Market Demand • Assume that there are only two goods (x and y) Chapter 12 – An individual’s demand for x is THE PARTIAL EQUILIBRIUM COMPETITIVE MODEL Quantity of x demanded = x(px,py,I) – If we use i to reflect each individual in the market, then the market demand curve is n Market demand for X xi ( px , py , Ii ) Copyright ©2005 by South-Western, a division of Thomson Learning. All rights reserved. i 1 1 2 Market Demand Market Demand • To construct the market demand curve, PX is allowed to vary while Py and the income of each individual are held constant • If each individual’s demand for x is downward sloping, the market demand curve will also be downward sloping To derive the market demand curve, we sum the quantities demanded at every price px px Individual 1’s demand curve Individual 2’s demand curve px Market demand curve px* x1 x1 * X x2 x x2 * x X* x x1* + x2* = X* 3 4 1 Shifts in the Market Demand Curve Shifts in Market Demand • Suppose that individual 1’s demand for oranges is given by x1 = 10 – 2px + 0.1I1 + 0.5py • The market demand summarizes the ceteris paribus relationship between X and px – changes in px result in movements along the curve (change in quantity demanded) – changes in other determinants of the demand for X cause the demand curve to shift to a new position (change in demand) and individual 2’s demand is x2 = 17 – px + 0.05I2 + 0.5py • The market demand curve is X = x1 + x2 = 27 – 3px + 0.1I1 + 0.05I2 + py 5 Shifts in Market Demand 6 Shifts in Market Demand • If py rises to 6, the market demand curve shifts outward to X = 27 – 3px + 4 + 1 + 6 = 38 – 3px • To graph the demand curve, we must assume values for py, I1, and I2 • If py = 4, I1 = 40, and I2 = 20, the market demand curve becomes – note that X and Y are substitutes • If I1 fell to 30 while I2 rose to 30, the market demand would shift inward to X = 27 – 3px + 4 + 1 + 4 = 36 – 3px X = 27 – 3px + 3 + 1.5 + 4 = 35.5 – 3px – note that X is a normal good for both buyers 7 8 2 Generalizations Generalizations • Suppose that there are n goods (xi, i = 1,n) with prices pi, i = 1,n. • Assume that there are m individuals in the economy • The j th’s demand for the i th good will depend on all prices and on Ij • The market demand function for xi is the sum of each individual’s demand for that good m X i xij ( p1,..., pn , I j ) j 1 • The market demand function depends on the prices of all goods and the incomes and preferences of all buyers xij = xij(p1,…,pn, Ij) 9 Elasticity of Market Demand • The price elasticity of market demand is measured by eQ,P 10 Elasticity of Market Demand • The cross-price elasticity of market demand is measured by QD (P, P ' , I ) P P QD eQ,P QD (P, P ' , I ) P ' P ' QD • The income elasticity of market demand is measured by • Market demand is characterized by whether demand is elastic (eQ,P <-1) or inelastic (0> eQ,P > -1) eQ,I 11 QD (P, P ' , I ) I I QD 12 3 Timing of the Supply Response • In the analysis of competitive pricing, the time period under consideration is important – very short run Pricing in the Very Short Run • In the very short run (or the market period), there is no supply response to changing market conditions – price acts only as a device to ration demand • no supply response (quantity supplied is fixed) • price will adjust to clear the market – short run • existing firms can alter their quantity supplied, but no new firms can enter the industry – the supply curve is a vertical line – long run • new firms may enter an industry 13 Pricing in the Very Short Run Price S When quantity is fixed in the very short run, price will rise from P1 to P2 when the demand rises from D to D’ P2 P1 14 Short-Run Price Determination • The number of firms in an industry is fixed • These firms are able to adjust the quantity they are producing – they can do this by altering the levels of the variable inputs they employ D’ D Q* Quantity 15 16 4 Perfect Competition Short-Run Market Supply • A perfectly competitive industry is one that obeys the following assumptions: • The quantity of output supplied to the entire market in the short run is the sum of the quantities supplied by each firm – there are a large number of firms, each producing the same homogeneous product – each firm attempts to maximize profits – each firm is a price taker – the amount supplied by each firm depends on price • The short-run market supply curve will be upward-sloping because each firm’s short-run supply curve has a positive slope • its actions have no effect on the market price – information is perfect – transactions are costless 17 18 Short-Run Market Supply Curve To derive the market supply curve, we sum the quantities supplied at every price P Firm A’s supply curve P sB sA P Firm B’s supply curve Market supply curve S Short-Run Market Supply Function • The short-run market supply function shows total quantity supplied by each firm to a market n Qs (P,v ,w ) qi (P,v ,w ) P1 i 1 q1A quantity q1B quantity Q1 Quantity q1A + q1B = Q1 19 • Firms are assumed to face the same market price and the same prices for inputs 20 5 Short-Run Supply Elasticity • The short-run supply elasticity describes the responsiveness of quantity supplied to changes in market price eS,P A Short-Run Supply Function • Suppose that there are 100 identical firms each with the following short-run supply curve qi (P,v,w) = 10P/3 % change in Q supplied QS P % change in P P QS • Because price and quantity supplied are positively related, eS,P > 0 (i = 1,2,…,100) • This means that the market supply function is given by 100 100 10P 1000P 3 i 1 3 Qs qi i 1 21 A Short-Run Supply Function 22 Equilibrium Price Determination • In this case, computation of the elasticity of supply shows that it is unit elastic • An equilibrium price is one at which quantity demanded is equal to quantity supplied QS (P,v ,w ) P 1000 P
1 P QS 3 1000P / 3 – neither suppliers nor demanders have an incentive to alter their economic decisions eS,P • An equilibrium price (P*) solves the equation: QD (P *, P ' , I ) QS (P *,v,w ) 23 24 6 Equilibrium Price Determination Equilibrium Price Determination • The equilibrium price depends on many exogenous factors The interaction between market demand and market supply determines the equilibrium price Price S – changes in any of these factors will likely result in a new equilibrium price P1 D Quantity Q1 25 26 Market Reaction to a Shift in Demand Market Reaction to a Shift in Demand If many buyers experience an increase in their demands, the market demand curve will shift to the right Price S If the market price rises, firms will increase their level of output Price SMC SAC P2 P1 D’ Equilibrium price and equilibrium quantity will both rise P2 P1 This is the short-run supply response to an increase in market price D Q1 Q2 Quantity q1 27 q2 Quantity 28 7 Shifts in Supply and Demand Curves Shifts in Supply and Demand Curves • Demand curves shift because • When either a supply curve or a demand curve shift, equilibrium price and quantity will change • The relative magnitudes of these changes depends on the shapes of the supply and demand curves – incomes change – prices of substitutes or complements change – preferences change • Supply curves shift because – input prices change – technology changes – number of producers change 29 30 Shifts in Supply Small increase in price, large drop in quantity Price Shifts in Demand Large increase in price, small drop in quantity Price S’ Large increase in price, small rise in quantity Price S’ S Small increase in price, large rise in quantity Price S S S P’ P’ P P’ P’ P P P D’ D D Q’ Q Elastic Demand Quantity Q’ Q Inelastic Demand D’ D Quantity 31 Q Q’ Elastic Supply D Quantity Q Q’ Inelastic Supply Quantity 32 8 Changing Short-Run Equilibria • Suppose that the market demand for luxury beach towels is Changing Short-Run Equilibria • Suppose instead that the demand for luxury towels rises to QD = 10,000 – 500P QD = 12,500 – 500P • Solving for the new equilibrium, we find and the short-run market supply is QS = 1,000P/3 P* = $15 Q* = 5,000 • Setting these equal, we find • Equilibrium price and quantity both rise P* = $12 Q* = 4,000 33 Changing Short-Run Equilibria • Suppose that the wage of towel cutters rises so that the short-run market supply becomes 34 Mathematical Model of Supply and Demand • Suppose that the demand function is represented by QD = D(P,) QS = 800P/3 • Solving for the new equilibrium, we find – is a parameter that shifts the demand curve • D/ = D can have any sign P* = $13.04 Q* = 3,480 – D/P = DP < 0 • Equilibrium price rises and quantity falls 35 36 9 Mathematical Model of Supply and Demand Mathematical Model of Supply and Demand • The supply relationship can be shown as QS = S(P,) – is a parameter that shifts the supply curve • S/ = S can have any sign • To analyze the comparative statics of this model, we need to use the total differentials of the supply and demand functions: dQD = DPdP + Dd dQS = SPdP + Sd – S/P = SP > 0 • Maintenance of equilibrium requires that • Equilibrium requires that QD = QS dQD = dQS 37 38 Mathematical Model of Supply and Demand Mathematical Model of Supply and Demand • Suppose that the demand parameter () changed while remains constant • The equilibrium condition requires that • We can convert our analysis to elasticities eP , DPdP + Dd = SPdP D P SP DP • Because SP - DP > 0, P/ will have the same sign as D 39 eP , D P P SP DP P D Q P (SP DP ) Q eQ, eS,P eQ,P 40 10 Long-Run Analysis Long-Run Analysis • In the long run, a firm may adapt all of its inputs to fit market conditions – profit-maximization for a price-taking firm implies that price is equal to long-run MC • Firms can also enter and exit an industry in the long run – perfect competition assumes that there are no special costs of entering or exiting an industry • New firms will be lured into any market for which economic profits are greater than zero – entry of firms will cause the short-run industry supply curve to shift outward – market price and profits will fall – the process will continue until economic profits are zero 41 42 Long-Run Competitive Equilibrium Long-Run Analysis • Existing firms will leave any industry for which economic profits are negative – exit of firms will cause the short-run industry supply curve to shift inward – market price will rise and losses will fall – the process will continue until economic profits are zero 43 • A perfectly competitive industry is in long-run equilibrium if there are no incentives for profit-maximizing firms to enter or to leave the industry – this will occur when the number of firms is such that P = MC = AC and each firm operates at minimum AC 44 11 Long-Run Competitive Equilibrium Long-Run Equilibrium: Constant-Cost Case • We will assume that all firms in an industry have identical cost curves • Assume that the entry of new firms in an industry has no effect on the cost of inputs – no firm controls any special resources or technology – no matter how many firms enter or leave an industry, a firm’s cost curves will remain unchanged • The equilibrium long-run position requires that each firm earn zero economic profit • This is referred to as a constant-cost industry 45 46 Long-Run Equilibrium: Constant-Cost Case Long-Run Equilibrium: Constant-Cost Case Suppose that market demand rises to D’ This is a long-run equilibrium for this industry Price SMC P = MC = AC Price MC Price SMC Price MC Market price rises to P2 S S AC AC P2 P1 P1 D’ D q1 A Typical Firm Quantity Q1 Total Market D 47 Quantity q1 A Typical Firm Quantity Q1 Q2 48 Quantity Total Market 12 Long-Run Equilibrium: Constant-Cost Case Long-Run Equilibrium: Constant-Cost Case In the short run, each firm increases output to q2 Price SMC Price MC In the long run, new firms will enter the industry Economic profit > 0 Economic profit will return to 0 Price SMC Price MC S S AC S’ AC P2 P1 P1 D’ D’ D q1 q2 Quantity A Typical Firm D 49 Quantity Q1 Q2 Total Market The long-run supply curve will be a horizontal line (infinitely elastic) at p1 SMC Q1 Q3 50 Quantity Total Market • Suppose that the total cost curve for a typical firm in the bicycle industry is Price MC Quantity Infinitely Elastic Long-Run Supply Long-Run Equilibrium: Constant-Cost Case Price q1 A Typical Firm S S’ TC = q3 – 20q2 + 100q + 8,000 AC • Demand for bicycles is given by P1 LS D’ QD = 2,500 – 3P D q1 A Typical Firm Quantity Q1 Q3 51 Quantity 52 Total Market 13 Infinitely Elastic Long-Run Supply Shape of the Long-Run Supply Curve • To find the long-run equilibrium for this market, we must find the low point on the typical firm’s average cost curve • The zero-profit condition is the factor that determines the shape of the long-run cost curve – where AC = MC AC = q2 – 20q + 100 + 8,000/q MC = 3q2 – 40q + 100 – this occurs where q = 20 – if average costs are constant as firms enter, long-run supply will be horizontal – if average costs rise as firms enter, long-run supply will have an upward slope – if average costs fall as firms enter, long-run supply will be negatively sloped • If q = 20, AC = MC = $500 – this will be the long-run equilibrium price 53 54 Long-Run Equilibrium: Increasing-Cost Industry Long-Run Equilibrium: Increasing-Cost Industry • The entry of new firms may cause the average costs of all firms to rise Suppose that we are in long-run equilibrium in this industry Price SMC – prices of scarce inputs may rise – new firms may impose “external” costs on existing firms – new firms may increase the demand for tax-financed services P = MC = AC Price MC S AC P1 D 55 q1 Quantity A Typical Firm (before entry) Q1 56 Quantity Total Market 14 Long-Run Equilibrium: Increasing-Cost Industry Long-Run Equilibrium: Increasing-Cost Industry Suppose that market demand rises to D’ Positive profits attract new firms and supply shifts out Market price rises to P2 and firms increase output to q2 Price SMC MC Entry of firms causes costs for each firm to rise SMC’ Price Price MC’ Price S S S’ AC’ AC P2 P3 P1 P1 D’ D’ D q1 q2 Quantity A Typical Firm (before entry) D 57 Quantity Q1 Q2 Total Market Price MC’ S LS p3 D’ D A Typical Firm (after entry) Quantity Q3 58 Quantity • permits the development of more efficient transportation and communications networks p1 Q1 Q3 – new firms may attract a larger pool of trained labor – entry of new firms may provide a “critical mass” of industrialization S’ AC’ q3 Q1 Total Market • The entry of new firms may cause the average costs of all firms to fall The long-run supply curve will be upward-sloping SMC’ Quantity Long-Run Equilibrium: Decreasing-Cost Industry Long-Run Equilibrium: Increasing-Cost Industry Price q3 A Typical Firm (after entry) 59 Quantity 60 Total Market 15 Long-Run Equilibrium: Decreasing-Cost Case Long-Run Equilibrium: Decreasing-Cost Industry Suppose that market demand rises to D’ Suppose that we are in long-run equilibrium in this industry Price SMC P = MC = AC Price MC Market price rises to P2 and firms increase output to q2 Price SMC Price MC S S AC AC P2 P1 P1 D q1 Quantity A Typical Firm (before entry) D’ D q1 q2 Quantity A Typical Firm (before entry) 61 Quantity Q1 Total Market Long-Run Equilibrium: Decreasing-Cost Industry 62 Quantity Q1 Q2 Total Market Long-Run Equilibrium: Decreasing-Cost Industry Positive profits attract new firms and supply shifts out The long-run industry supply curve will be downward-sloping Entry of firms causes costs for each firm to fall Price SMC’ Price Price SMC’ S MC’ Price S MC’ S’ S’ AC’ AC’ P1 P1 P3 D’ D q1 q3 Quantity A Typical Firm (before entry) Q1 Total Market Q3 63 Quantity P3 D q1 q3 Quantity A Typical Firm (before entry) Q1 D’ LS 64 Q3 Quantity Total Market 16 Classification of Long-Run Supply Curves Classification of Long-Run Supply Curves • Constant Cost • Decreasing Cost – entry does not affect input costs – the long-run supply curve is horizontal at the long-run equilibrium price – entry reduces input costs – the long-run supply curve is negatively sloped • Increasing Cost – entry increases inputs costs – the long-run supply curve is positively sloped 65 Long-Run Elasticity of Supply • The long-run elasticity of supply (eLS,P) records the proportionate change in longrun industry output to a proportionate change in price eLS ,P % change in Q QLS P % change in P P QLS • eLS,P can be positive or negative – the sign depends on whether the industry exhibits increasing or decreasing costs 67 66 Comparative Statics Analysis of Long-Run Equilibrium • Comparative statics analysis of long-run equilibria can be conducted using estimates of long-run elasticities of supply and demand • Remember that, in the long run, the number of firms in the industry will vary from one long-run equilibrium to another 68 17 Comparative Statics Analysis of Long-Run Equilibrium Comparative Statics Analysis of Long-Run Equilibrium • Assume that we are examining a constant-cost industry • Suppose that the initial long-run equilibrium industry output is Q0 and the typical firm’s output is q* (where AC is minimized) • The equilibrium number of firms in the industry (n0) is Q0/q* • A shift in demand that changes the equilibrium industry output to Q1 will change the equilibrium number of firms to 69 n1 = Q1/q* • The change in the number of firms is Q1 Q0 q* – completely determined by the extent of the demand shift and the optimal output level for 70 the typical firm n1 n0 Comparative Statics Analysis of Long-Run Equilibrium Comparative Statics Analysis of Long-Run Equilibrium • The effect of a change in input prices is more complicated • The optimal level of output for each firm may also be affected • Therefore, the change in the number of firms becomes – we need to know how much minimum average cost is affected – we need to know how an increase in longrun equilibrium price will affect quantity demanded Q Q n1 n0 *1 *0 q1 q0 71 72 18 Rising Input Costs and Industry Structure Rising Input Costs and Industry Structure • Suppose that the total cost curve for a typical firm in the bicycle industry is • At q = 22, MC = AC = $672 so the longrun equilibrium price will be $672 • If demand can be represented by TC = q3 – 20q2 + 100q + 8,000 QD = 2,500 – 3P and then rises to then QD = 484 • This means that the industry will have 22 firms (484 22) TC = q3 – 20q2 + 100q + 11,616 • The optimal scale of each firm rises from 20 to 22 (where MC = AC) 73 Producer Surplus in the Long Run 74 Producer Surplus in the Long Run • In the long-run, all profits are zero and there are no fixed costs • Short-run producer surplus represents the return to a firm’s owners in excess of what would be earned if output was zero – owners are indifferent about whether they are in a particular market – the sum of short-run profits and fixed costs 75 • they could earn identical returns on their investments elsewhere • Suppliers of inputs may not be indifferent about the level of production in an industry 76 19 Producer Surplus in the Long Run Producer Surplus in the Long Run • In the constant-cost case, input prices are assumed to be independent of the level of production • Long-run producer surplus represents the additional returns to the inputs in an industry in excess of what these inputs would earn if industry output was zero – inputs can earn the same amount in alternative occupations • In the increasing-cost case, entry will bid up some input prices – suppliers of these inputs will be made better 77 off Ricardian Rent – the area above the long-run supply curve and below the market price • this would equal zero in the case of constant costs 78 Ricardian Rent • Long-run producer surplus can be most easily illustrated with a situation first described by economist David Ricardo – assume that there are many parcels of land on which a particular crop may be grown • the land ranges from very fertile land (low costs of production) to very poor, dry land (high costs of production) 79 • At low prices only the best land is used • Higher prices lead to an increase in output through the use of higher-cost land – the long-run supply curve is upward-sloping because of the increased costs of using less fertile land 80 20 Ricardian Rent Ricardian Rent The owners of low-cost firms will earn positive profits Price MC Price The owners of the marginal firm will earn zero profit Price Price MC AC AC S S P* P* D Quantity q* Low-Cost Firm Q* D 81 Quantity q* Total Market Q* Quantity Marginal Firm 82 Quantity Total Market Ricardian Rent Ricardian Rent • Firms with higher costs (than the marginal firm) will stay out of the market Each point on the supply curve represents minimum average cost for some firm Price For each firm, P – AC represents profit per unit of output – would incur losses at a price of P* • Profits earned by intramarginal firms can persist in the long run Total long-run profits can be computed by summing over all units of output S – they reflect a return to a unique resource P* • The sum of these long-run profits constitutes long-run producer surplus D Q* 83 Total Market Quantity 84 21 Ricardian Rent Ricardian Rent • The long-run profits for the low-cost firms will often be reflected in the prices of the unique resources owned by those firms – the more fertile the land is, the higher its price • Thus, profits are said to be capitalized inputs’ prices • It is the scarcity of low-cost inputs that creates the possibility of Ricardian rent • In industries with upward-sloping longrun supply curves, increases in output not only raise firms’ costs but also generate factor rents for inputs – reflect the present value of all future profits 85 Important Points to Note: 86 Important Points to Note: • In the short run, equilibrium prices are established by the intersection of what demanders are willing to pay (as reflected by the demand curve) and what firms are willing to produce (as reflected by the short-run supply curve) – these prices are treated as fixed in both demanders’ and suppliers’ decision-making processes 87 • A shift in either demand or supply will cause the equilibrium price to change – the extent of such a change will depend on the slopes of the various curves • Firms may earn positive profits in the short run – because fixed costs must always be paid, firms will choose a positive output as long as revenues exceed variable costs 88 22 Important Points to Note: Important Points to Note: • In the long run, the number of firms is variable in response to profit opportunities – the assumption of free entry and exit implies that firms in a competitive industry will earn zero economic profits in the long run (P = AC) – because firms also seek maximum profits, the equality P = AC = MC implies that firms will operate at the low points of their long-run average cost curves • The shape of the long-run supply curve depends on how entry and exit affect firms’ input costs – in the constant-cost case, input prices do not change and the long-run supply curve is horizontal – if entry raises input costs, the long-run supply curve will have a positive slope 89 Important Points to Note: 90 Important Points to Note: • Changes in long-run market equilibrium will also change the number of firms – precise predictions about the extent of these changes is made difficult by the possibility that the minimum average cost level of output may be affected by changes in input costs or by technical progress 91 • If changes in the long-run equilibrium in a market change the prices of inputs to that market, the welfare of the suppliers of these inputs will be affected – such changes can be measured by changes in the value of long-run producer surplus 92 23 Perfectly Competitive Price System • We will assume that all markets are perfectly competitive Chapter 13 – there is some large number of homogeneous goods in the economy GENERAL EQUILIBRIUM AND WELFARE Copyright ©2005 by South-Western, a division of Thomson Learning. All rights reserved. • both consumption goods and factors of production 1 – each good has an equilibrium price – there are no transaction or transportation costs – individuals and firms have perfect information 2 Assumptions of Perfect Competition Law of One Price • A homogeneous good trades at the same price no matter who buys it or who sells it • There are a large number of people buying any one good – each person takes all prices as given and seeks to maximize utility given his budget constraint – if one good traded at two different prices, demanders would rush to buy the good where it was cheaper and firms would try to sell their output where the price was higher • There are a large number of firms producing each good – each firm takes all prices as given and attempts to maximize profits • these actions would tend to equalize the price of the good 3 4 1 Edgeworth Box Diagram General Equilibrium • Assume that there are only two goods, x and y • All individuals are assumed to have identical preferences – represented by an indifference map • The production possibility curve can be used to show how outputs and inputs are related 5 Edgeworth Box Diagram Labor for y Capital Oy in y production Capital for y Total Capital Ox Labor in x production Total Labor 6 • Many of the allocations in the Edgeworth box are technically inefficient – it is possible to produce more x and more y by shifting capital and labor around • We will assume that competitive markets will not exhibit inefficient input choices • We want to find the efficient allocations Capital for x A Capital in x production – any point in the box represents a fully employed allocation of the available resources to x and y Edgeworth Box Diagram Labor in y production Labor for x • Construction of the production possibility curve for x and y starts with the assumption that the amounts of k and l are fixed • An Edgeworth box shows every possible way the existing k and l might be used to produce x and y – they illustrate the actual production outcomes 7 8 2 Edgeworth Box Diagram Edgeworth Box Diagram Point A is inefficient because, by moving along y1, we can increase x from x1 to x2 while holding y constant Oy • We will use isoquant maps for the two goods – the isoquant map for good x uses Ox as the origin – the isoquant map for good y uses Oy as the origin Total Capital y1 y2 • The efficient allocations will occur where the isoquants are tangent to one another A Ox 9 Edgeworth Box Diagram x2 x1 10 Total Labor Edgeworth Box Diagram We could also increase y from y1 to y2 while holding x constant by moving along x1 Oy At each efficient point, the RTS (of k for l) is equal in both x and y production Oy y1 p4 y2 A y2 Total Capital Total Capital y1 x2 p3 x4 y3 p2 y4 x1 x3 p1 x2 x1 Ox Total Labor 11 Ox Total Labor 12 3 Production Possibility Frontier • The locus of efficient points shows the maximum output of y that can be produced for any level of x Production Possibility Frontier Each efficient point of production becomes a point on the production possibility frontier Quantity of y Ox y4 y3 – we can use this information to construct a production possibility frontier p1 p2 • shows the alternative outputs of x and y that can be produced with the fixed capital and labor inputs that are employed efficiently p4 y1 x1 13 The negative of the slope of the production possibility frontier is the rate of product transformation (RPT) p3 y2 x2 x3 x4 Oy Quantity of x 14 Rate of Product Transformation Rate of Product Transformation • The rate of product transformation (RPT) between two outputs is the negative of the slope of the production possibility frontier • The rate of product transformation shows how x can be technically traded for y while continuing to keep the available productive inputs efficiently employed RPT (of x for y ) slope of production possibility frontier RPT (of x for y ) dy (along OxOy ) dx 15 16 4 Shape of the Production Possibility Frontier Shape of the Production Possibility Frontier • The production possibility frontier shown earlier exhibited an increasing RPT • Suppose that the costs of any output combination are C(x,y) – along the production possibility frontier, C(x,y) is constant – this concave shape will characterize most production situations • We can write the total differential of the cost function as • RPT is equal to the ratio of MCx to MCy dC C C dx dy 0 x y 17 Shape of the Production Possibility Frontier Shape of the Production Possibility Frontier • Rewriting, we get RPT 18 • As production of x rises and production of y falls, the ratio of MCx to MCy rises dy C / x MCx (along OxOy ) dx C / y MCy – this occurs if both goods are produced under diminishing returns • increasing the production of x raises MCx, while reducing the production of y lowers MCy • The RPT is a measure of the relative marginal costs of the two goods – this could also occur if some inputs were more suited for x production than for y production 19 20 5 Shape of the Production Possibility Frontier Opportunity Cost • But we have assumed that inputs are homogeneous • We need an explanation that allows homogeneous inputs and constant returns to scale • The production possibility frontier will be concave if goods x and y use inputs in different proportions • The production possibility frontier demonstrates that there are many possible efficient combinations of two goods • Producing more of one good necessitates lowering the production of the other good – this is what economists mean by opportunity cost 21 22 Concavity of the Production Possibility Frontier Opportunity Cost • The opportunity cost of one more unit of x is the reduction in y that this entails • Thus, the opportunity cost is best measured as the RPT (of x for y) at the prevailing point on the production possibility frontier • Suppose that the production of x and y depends only on labor and the production functions are x f (lx ) lx0.5 y f (ly ) ly0.5 • If labor supply is fixed at 100, then lx + ly = 100 – this opportunity cost rises as more x is produced • The production possibility frontier is 23 x2 + y2 = 100 for x,y 0 24 6 Concavity of the Production Possibility Frontier • The RPT can be calculated by taking the total differential: dy ( 2x ) x dx 2y y 2xdx 2ydy 0 or RPT • The slope of the production possibility frontier increases as x output increases Determination of Equilibrium Prices • We can use the production possibility frontier along with a set of indifference curves to show how equilibrium prices are determined – the indifference curves represent individuals’ preferences for the two goods – the frontier is concave 25 26 Determination of Equilibrium Prices Determination of Equilibrium Prices If the prices of x and y are px and py, society’s budget constraint is C Quantity of y C There is excess demand for x and excess supply of y Quantity of y C Output will be x1, y1 y1 The price of x will rise and the price of y will fall y1 Individuals will demand x1’, y1’ y1 ’ excess supply y1 ’ U3 U2 U1 x1 x1 ’ U3 C U2 px slope py Quantity of x U1 27 x1 ’ x 1 C slope px py Quantity of x 28 excess demand 7 Determination of Equilibrium Prices Comparative Statics Analysis The equilibrium prices will be px* and py* Quantity of y C* C The equilibrium output will be x1* and y1* y1 y1 * y1 ’ U3 U2 U1 C* x x1 * x1 ’ 1 slope px* py* – we would move in a clockwise direction along the production possibility frontier C slope px py Quantity of x • The equilibrium price ratio will tend to persist until either preferences or production technologies change • If preferences were to shift toward good x, px /py would rise and more x and less y would be produced 29 Comparative Statics Analysis • Technical progress in the production of good x will shift the production possibility curve outward 30 Technical Progress in the Production of x Technical progress in the production of x will shift the production possibility curve out Quantity of y – this will lower the relative price of x – more x will be consumed The relative price of x will fall More x will be consumed • if x is a normal good U3 – the effect on y is ambiguous U2 U1 31 x1 * x2 * Quantity of x 32 8 General Equilibrium Pricing General Equilibrium Pricing • Suppose that the production possibility frontier can be represented by • Profit-maximizing firms will equate RPT and the ratio of px /py x 2 + y 2 = 100 RPT • Suppose also that the community’s preferences can be represented by x px y py • Utility maximization requires that U(x,y) = x0.5y0.5 MRS y px x py 33 General Equilibrium Pricing • Equilibrium requires that firms and individuals face the same price ratio RPT 34 The Corn Laws Debate • High tariffs on grain imports were imposed by the British government after the Napoleonic wars • Economists debated the effects of these “corn laws” between 1829 and 1845 x px y MRS y py x or x* = y* – what effect would the elimination of these tariffs have on factor prices? 35 36 9 The Corn Laws Debate Quantity of manufactured goods (y) The Corn Laws Debate If the corn laws completely prevented trade, output would be x0 and y0 The equilibrium prices will be px* and py* y0 Quantity of manufactured goods (y) Removal of the corn laws will change the prices to px’ and py’ Output will be x1’ and y1’ y1 ’ Individuals will demand x1 and y1 y0 y1 U2 U2 U1 slope U1 slope px* py* Quantity of Grain (x) x0 x1 ’ x0 x1 px ' py ' Quantity of Grain (x) 37 The Corn Laws Debate Quantity of manufactured goods (y) exports of goods The Corn Laws Debate Grain imports will be x1 – x1’ These imports will be financed by the export of manufactured goods equal to y1’ – y1 y1 ’ y0 y1 U2 U1 slope x1 ’ x0 imports of grain x1 38 px ' py ' • We can use an Edgeworth box diagram to see the effects of tariff reduction on the use of labor and capital • If the corn laws were repealed, there would be an increase in the production of manufactured goods and a decline in the production of grain Quantity of Grain (x) 39 40 10 The Corn Laws Debate The Corn Laws Debate A repeal of the corn laws would result in a movement from p3 to p1 where more y and less x is produced Oy y1 p4 Total Capital y2 p3 – the relative price of capital will fall – the relative price of labor will rise x4 y3 p2 y4 x3 p1 x1 Ox x2 Total Labor • If we assume that grain production is relatively capital intensive, the movement from p3 to p1 causes the ratio of k to l to rise in both industries 41 • The repeal of the corn laws will be harmful to capital owners and helpful to laborers 42 Political Support for Trade Policies Existence of General Equilibrium Prices • Trade policies may affect the relative incomes of various factors of production • In the United States, exports tend to be intensive in their use of skilled labor whereas imports tend to be intensive in their use of unskilled labor • Beginning with 19th century investigations by Leon Walras, economists have examined whether there exists a set of prices that equilibrates all markets simultaneously – free trade policies will result in rising relative wages for skilled workers and in falling relative wages for unskilled workers 43 – if this set of prices exists, how can it be found? 44 11 Existence of General Equilibrium Prices Existence of General Equilibrium Prices • Suppose that there are n goods in fixed supply in this economy • We will write this demand function as dependent on the whole set of prices (P) – let Si (i =1,…,n) be the total supply of good i available – let pi (i =1,…n) be the price of good i • The total demand for good i depends on all prices Di (p1,…,pn) for i =1,…,n Di (P) • Walras’ problem: Does there exist an equilibrium set of prices such that Di (P*) = Si for all values of i ? 45 46 Excess Demand Functions Excess Demand Functions • The excess demand function for any good i at any set of prices (P) is defined to be • Demand functions are homogeneous of degree zero EDi (P) = Di (P) – Si • This means that the equilibrium condition can be rewritten as – this implies that we can only establish equilibrium relative prices in a Walrasiantype model • Walras also assumed that demand functions are continuous EDi (P*) = Di (P*) – Si = 0 – small changes in price lead to small changes in quantity demanded 47 48 12 Walras’ Law Walras’ Law • A final observation that Walras made was that the n excess demand equations are not independent of one another • Walras’ law shows that the total value of excess demand is zero at any set of prices • Walras’ law holds for any set of prices (not just equilibrium prices) • There can be neither excess demand for all goods together nor excess supply n P ED (P ) 0 i 1 i i 49 50 Walras’ Proof of the Existence of Equilibrium Prices Walras’ Proof of the Existence of Equilibrium Prices • The market equilibrium conditions provide (n-1) independent equations in (n-1) unknown relative prices • Start with an arbitrary set of prices • Holding the other n-1 prices constant, find the equilibrium price for good 1 (p1’) • Holding p1’ and the other n-2 prices constant, solve for the equilibrium price of good 2 (p2’) – can we solve the system for an equilibrium condition? • the equations are not necessarily linear • all prices must be nonnegative • To attack these difficulties, Walras set up a complicated proof 51 – in changing p2 from its initial position to p2’, the price calculated for good 1 does not 52 need to remain an equilibrium price 13 Walras’ Proof of the Existence of Equilibrium Prices Walras’ Proof of the Existence of Equilibrium Prices • Using the provisional prices p1’ and p2’, solve for p3’ • The importance of Walras’ proof is its ability to demonstrate the simultaneous nature of the problem of finding equilibrium prices • Because it is cumbersome, it is not generally used today • More recent work uses some relatively simple tools from advanced mathematics – proceed in this way until an entire set of provisional relative prices has been found • In the 2nd iteration of Walras’ proof, p2’,…,pn’ are held constant while a new equilibrium price is calculated for good 1 – proceed in this way until an entire new set of prices is found 53 Brouwer’s Fixed-Point Theorem 54 Brouwer’s Fixed-Point Theorem f (X) • Any continuous mapping [F(X)] of a closed, bounded, convex set into itself has at least one fixed point (X*) such that F(X*) = X* Suppose that f(X) is a continuous function defined on the interval [0,1] and that f(X) takes on the values also on the interval [0,1] 1 Any continuous function must cross the 45 line f (X*) This point of crossing is a “fixed point” because f maps this point (X*) into itself 45 0 55 X* 1 x 56 14 Brouwer’s Fixed-Point Theorem Brouwer’s Fixed-Point Theorem • A mapping is a rule that associates the points in one set with points in another set • A mapping is continuous if points that are “close” to each other are mapped into other points that are “close” to each other • The Brouwer fixed-point theorem considers mappings defined on certain kinds of sets – let X be a point for which a mapping (F) is defined • the mapping associates X with some point Y = F(X) – if a mapping is defined over a subset of ndimensional space (S), and if every point in S is associated (by the rule F) with some other point in S, the mapping is said to map S into itself 57 – closed (they contain their boundaries) – bounded (none of their dimensions is infinitely large) – convex (they have no “holes” in them) 58 Proof of the Existence of Equilibrium Prices Proof of the Existence of Equilibrium Prices • Because only relative prices matter, it is convenient to assume that prices have been defined so that the sum of all prices is equal to 1 • Thus, for any arbitrary set of prices (p1,…,pn), we can use normalized prices of the form • These new prices will retain their original relative values and will sum to 1 pi ' pi • These new prices will sum to 1 n p ' 1 i 1 n p i i 1 pi ' pi pj ' pj 59 i 60 15 Proof of the Existence of Equilibrium Prices Free Goods • We will assume that the feasible set of prices (S) is composed of all nonnegative numbers that sum to 1 – S is the set to which we will apply Brouwer’s theorem – S is closed, bounded, and convex – we will need to define a continuous mapping of S into itself • Equilibrium does not really require that excess demand be zero for every market • Goods may exist for which the markets are in equilibrium where supply exceeds demand (negative excess demand) – it is necessary for the prices of these goods to be equal to zero – “free goods” 61 62 Mapping the Set of Prices Into Itself Free Goods • The equilibrium conditions are • In order to achieve equilibrium, prices of goods in excess demand should be raised, whereas those in excess supply should have their prices lowered EDi (P*) = 0 for pi* > 0 EDi (P*) 0 for pi* = 0 • Note that this set of equilibrium prices continues to obey Walras’ law 63 64 16 Mapping the Set of Prices Into Itself Mapping the Set of Prices Into Itself • Two problems exist with this mapping • First, nothing ensures that the prices will be nonnegative • We define the mapping F(P) for any normalized set of prices (P), such that the ith component of F(P) is given by – the mapping must be redefined to be F i(P) = pi + EDi (P) F i(P) = Max [pi + EDi (P),0] • The mapping performs the necessary task of appropriately raising or lowering prices – the new prices defined by the mapping must be positive or zero 65 66 Application of Brouwer’s Theorem Mapping the Set of Prices Into Itself • Second, the recalculated prices are not necessarily normalized • Thus, F satisfies the conditions of the Brouwer fixed-point theorem – they will not sum to 1 – it will be simple to normalize such that – it is a continuous mapping of the set S into itself • There exists a point (P*) that is mapped into itself • For this point, n F (P ) 1 i i 1 – we will assume that this normalization has been done 67 pi* = Max [pi* + EDi (P*),0] for all i 68 17 Application of Brouwer’s Theorem A General Equilibrium with Three Goods • This says that P* is an equilibrium set of prices • The economy of Oz is composed only of three precious metals: (1) silver, (2) gold, and (3) platinum – for pi* > 0, pi* = pi* + EDi (P*) EDi (P*) = 0 – For pi* = 0, – there are 10 (thousand) ounces of each metal available • The demands for gold and platinum are pi* + EDi (P*) 0 EDi (P*) 0 D2 2 69 A General Equilibrium with Three Goods D3 p p2 2 3 18 p1 p1 70 A General Equilibrium with Three Goods • This system of simultaneous equations can be solved as • Equilibrium in the gold and platinum markets requires that demand equal supply in both markets simultaneously 2 p2 p3 11 p1 p1 p2/p1 = 2 p3/p1 = 3 • In equilibrium: p2 p3 11 10 p1 p1 – gold will have a price twice that of silver – platinum will have a price three times that of silver – the price of platinum will be 1.5 times that of gold p p2 2 3 18 10 p1 p1 71 72 18 A General Equilibrium with Three Goods Smith’s Invisible Hand Hypothesis • Because Walras’ law must hold, we know • Adam Smith believed that the competitive market system provided a powerful “invisible hand” that ensured resources would find their way to where they were most valued • Reliance on the economic self-interest of individuals and firms would result in a desirable social outcome p1ED1 = – p2ED2 – p3ED3 • Substituting the excess demand functions for gold and silver and substituting, we get p1ED1 2 pp p2 p22 p2 p3 p2 2 3 2 3 8 p3 p1 p1 p1 p1 ED1 2 p2 p p p22 2 32 2 8 3 2 p1 p1 p1 p1 73 74 Smith’s Invisible Hand Hypothesis Pareto Efficiency • Smith’s insights gave rise to modern welfare economics • The “First Theorem of Welfare Economics” suggests that there is an exact correspondence between the efficient allocation of resources and the competitive pricing of these resources • An allocation of resources is Pareto efficient if it is not possible (through further reallocations) to make one person better off without making someone else worse off • The Pareto definition identifies allocations as being “inefficient” if unambiguous improvements are possible 75 76 19 Efficiency in Production • An allocation of resources is efficient in production (or “technically efficient”) if no further reallocation would permit more of one good to be produced without necessarily reducing the output of some other good • Technical efficiency is a precondition for Pareto efficiency but does not guarantee Pareto efficiency 77 Efficient Choice of Inputs for a Single Firm • A single firm with fixed inputs of labor and capital will have allocated these resources efficiently if they are fully employed and if the RTS between capital and labor is the same for every output the firm produces 78 Efficient Choice of Inputs for a Single Firm Efficient Choice of Inputs for a Single Firm • Assume that the firm produces two goods (x and y) and that the available levels of capital and labor are k’ and l’ • The production function for x is given by • Technical efficiency requires that x output be as large as possible for any value of y (y’) • Setting up the Lagrangian and solving for the first-order conditions: x = f (kx, lx) L = f (kx, lx) + [y’ – g (k’ - kx, l’ - lx)] L/kx = fk + gk = 0 • If we assume full employment, the production function for y is y = g (ky, ly) = g (k’ - kx, l’ - lx) L/lx = fl + gl = 0 79 L/ = y’ – g (k’ - kx, l’ - lx) = 0 80 20 Efficient Choice of Inputs for a Single Firm Efficient Allocation of Resources among Firms • From the first two conditions, we can see that • Resources should be allocated to those firms where they can be most efficiently used fk g k fl gl – the marginal physical product of any resource in the production of a particular good should be the same across all firms that produce the good • This implies that RTSx (k for l) = RTSy (k for l) 81 Efficient Allocation of Resources among Firms 82 Efficient Allocation of Resources among Firms • The allocational problem is to maximize • Suppose that there are two firms producing x and their production functions are x = f1(k1, l1) + f2(k2, l2) subject to the constraints k1 + k2 = k’ l1 + l2 = l’ f1(k1, l1) f2(k2, l2) • Assume that the total supplies of capital and labor are k’ and l’ 83 • Substituting, the maximization problem becomes x = f1(k1, l1) + f2(k’ - k1, l’ - l1) 84 21 Efficient Allocation of Resources among Firms Efficient Allocation of Resources among Firms • First-order conditions for a maximum are • These first-order conditions can be rewritten as x f f f f 1 2 1 2 0 k1 k1 k1 k1 k 2 f1 f 2 k1 k 2 x f1 f2 f1 f2
0 l1 l1 l1 l1 l2 85 Efficient Choice of Output by Firms f1 f2 l1 l2 • The marginal physical product of each input should be equal across the two firms 86 Efficient Choice of Output by Firms • Suppose that there are two outputs (x and y) each produced by two firms • The production possibility frontiers for these two firms are • The Lagrangian for this problem is L = x1 + x2 + [y* - f1(x1) - f2(x2)] and yields the first-order condition: f1/x1 = f2/x2 yi = fi (xi ) for i=1,2 • The rate of product transformation (RPT) should be the same for all firms producing these goods • The overall optimization problem is to produce the maximum amount of x for any given level of y (y*) 87 88 22 Efficient Choice of Output by Firms Efficient Choice of Output by Firms Firm A is relatively efficient at producing cars, while Firm B is relatively efficient at producing trucks If each firm was to specialize in its efficient product, total output could be increased Cars Cars RPT RPT 2 1 100 Cars 1 1 100 Trucks 50 Firm A Cars RPT 100 Trucks 50 Firm B 89 Theory of Comparative Advantage RPT 2 1 1 1 100 Trucks 50 Firm A Trucks 50 Firm B 90 Efficiency in Product Mix • The theory of comparative advantage was first proposed by Ricardo • Technical efficiency is not a sufficient condition for Pareto efficiency – countries should specialize in producing those goods of which they are relatively more efficient producers – demand must also be brought into the picture • In order to ensure Pareto efficiency, we must be able to tie individual’s preferences and production possibilities together • these countries should then trade with the rest of the world to obtain needed commodities – if countries do specialize this way, total world production will be greater 91 92 23 Efficiency in Product Mix Efficiency in Product Mix • The condition necessary to ensure that the right goods are produced is Output of y Suppose that we have a one-person (Robinson Crusoe) economy and PP represents the combinations of x and y that can be produced P MRS = RPT – the psychological rate of trade-off between the two goods in people’s preferences must be equal to the rate at which they can be traded off in production Any point on PP represents a point of technical efficiency P Output of x 93 Efficiency in Product Mix Output of y Efficiency in Product Mix Only one point on PP will maximize Crusoe’s utility At the point of tangency, Crusoe’s MRS will be equal to the technical RPT P U3 U2 • Assume that there are only two goods (x and y) and one individual in society (Robinson Crusoe) • Crusoe’s utility function is U = U(x,y) • The production possibility frontier is U1 P 94 T(x,y) = 0 Output of x 95 96 24 Efficiency in Product Mix Efficiency in Product Mix • Crusoe’s problem is to maximize his utility subject to the production constraint • First-order conditions for an interior maximum are L U T 0 x x x • Setting up the Lagrangian yields L U T 0 y y y L = U(x,y) + [T(x,y)] 97 98 Competitive Prices and Efficiency Efficiency in Product Mix • Combining the first two, we get U / x T / x U / y T / y or MRS ( x for y ) L T ( x, y ) 0 dy (along T ) RPT ( x for y ) dx 99 • Attaining a Pareto efficient allocation of resources requires that the rate of trade-off between any two goods be the same for all economic agents • In a perfectly competitive economy, the ratio of the prices of the two goods provides the common rate of trade-off to which all agents will adjust 100 25 Competitive Prices and Efficiency Efficiency in Production • Because all agents face the same prices, all trade-off rates will be equalized and an efficient allocation will be achieved • This is the “First Theorem of Welfare Economics” • In minimizing costs, a firm will equate the RTS between any two inputs (k and l) to the ratio of their competitive prices (w/v) – this is true for all outputs the firm produces – RTS will be equal across all outputs 101 102 Efficiency in Production Efficiency in Production • A profit-maximizing firm will hire additional units of an input (l) up to the point at which its marginal contribution to revenues is equal to the marginal cost of hiring the input (w) • If this is true for every firm, then with a competitive labor market pxfl = w 103 pxfl1 = w = pxfl2 fl1 = fl2 • Every firm that produces x has identical marginal productivities of every input in the production of x 104 26 Efficiency in Production Efficiency in Production • Recall that the RPT (of x for y) is equal to MCx /MCy • In perfect competition, each profitmaximizing firm will produce the output level for which marginal cost is equal to price • Since px = MCx and py = MCy for every firm, RTS = MCx /MCy = px /py • Thus, the profit-maximizing decisions of many firms can achieve technical efficiency in production without any central direction • Competitive market prices act as signals to unify the multitude of decisions that firms make into one coherent, efficient pattern 105 106 Efficiency in Product Mix Efficiency in Product Mix • The price ratios quoted to consumers are the same ratios the market presents to firms • This implies that the MRS shared by all individuals will be equal to the RPT shared by all the firms • An efficient mix of goods will therefore be produced 107 Output of y x* and y* represent the efficient output mix slope px* py* P Only with a price ratio of px*/py* will supply and demand be in equilibrium y* U0 x* P Output of x 108 27 Departing from the Competitive Assumptions Laissez-Faire Policies • The correspondence between competitive equilibrium and Pareto efficiency provides some support for the laissez-faire position taken by many economists – government intervention may only result in a loss of Pareto efficiency • The ability of competitive markets to achieve efficiency may be impaired because of – imperfect competition – externalities – public goods – imperfect information 109 110 Imperfect Competition Externalities • Imperfect competition includes all situations in which economic agents exert some market power in determining market prices • An externality occurs when there are interactions among firms and individuals that are not adequately reflected in market prices • With externalities, market prices no longer reflect all of a good’s costs of production – these agents will take these effects into account in their decisions • Market prices no longer carry the informational content required to achieve Pareto efficiency 111 – there is a divergence between private and social marginal cost 112 28 Public Goods Imperfect Information • Public goods have two properties that make them unsuitable for production in markets – they are nonrival • additional people can consume the benefits of these goods at zero cost • If economic actors are uncertain about prices or if markets cannot reach equilibrium, there is no reason to expect that the efficiency property of competitive pricing will be retained – they are nonexclusive • extra individuals cannot be precluded from consuming the good 113 114 Distribution Distribution • Although the First Theorem of Welfare Economics ensures that competitive markets will achieve efficient allocations, there are no guarantees that these allocations will exhibit desirable distributions of welfare among individuals • Assume that there are only two people in society (Smith and Jones) • The quantities of two goods (x and y) to be distributed among these two people are fixed in supply • We can use an Edgeworth box diagram to show all possible allocations of these goods between Smith and Jones 115 116 29 Distribution Distribution OJ UJ 1 • Any point within the Edgeworth box in which the MRS for Smith is unequal to that for Jones offers an opportunity for Pareto improvements UJ2 US4 UJ3 Total Y US3 UJ4 – both can move to higher levels of utility through trade US2 US1 OS Total X 117 Distribution Contract Curve OJ • In an exchange economy, all efficient allocations lie along a contract curve UJ1 UJ2 – points off the curve are necessarily inefficient US4 UJ3 • individuals can be made better off by moving to the curve US3 UJ4 • Along the contract curve, individuals’ preferences are rivals US2 A OS Any trade in this area is an improvement over A 118 US1 119 – one may be made better off only by making the other worse off 120 30 Exchange with Initial Endowments Contract Curve OJ UJ1 • Suppose that the two individuals possess different quantities of the two goods at the start UJ2 US4 UJ3 – it is possible that the two individuals could both benefit from trade if the initial allocations were inefficient US3 UJ4 US2 A US1 Contract curve OS 121 122 Exchange with Initial Endowments Exchange with Initial Endowments • Neither person would engage in a trade that would leave him worse off • Only a portion of the contract curve shows allocations that may result from voluntary exchange Suppose that A represents the initial endowments UJA A 123 OJ OS USA 124 31 Exchange with Initial Endowments Exchange with Initial Endowments OJ OJ Only allocations between M1 and M2 will be acceptable to both Neither individual would be willing to accept a lower level of utility than A gives UJA UJA M2 M1 A A USA OS 125 The Distributional Dilemma • If the initial endowments are skewed in favor of some economic actors, the Pareto efficient allocations promised by the competitive price system will also tend to favor those actors – voluntary transactions cannot overcome large differences in initial endowments – some sort of transfers will be needed to attain more equal results 127 USA OS 126 The Distributional Dilemma • These thoughts lead to the “Second Theorem of Welfare Economics” – any desired distribution of welfare among individuals in an economy can be achieved in an efficient manner through competitive pricing if initial endowments are adjusted appropriately 128 32 Important Points to Note: Important Points to Note: • Preferences and production technologies provide the building blocks upon which all general equilibrium models are based – one particularly simple version of such a model uses individual preferences for two goods together with a concave production possibility frontier for those two goods • Competitive markets can establish equilibrium prices by making marginal adjustments in prices in response to information about the demand and supply for individual goods – Walras’ law ties markets together so that such a solution is assured (in most cases) 129 Important Points to Note: 130 Important Points to Note: • Competitive prices will result in a Pareto-efficient allocation of resources – this is the First Theorem of Welfare Economics • Factors that will interfere with competitive markets’ abilities to achieve efficiency include – market power – externalities – existence of public goods – imperfect information 131 132 33 Important Points to Note: • Competitive markets need not yield equitable distributions of resources, especially when initial endowments are very skewed – in theory any desired distribution can be attained through competitive markets accompanied by lump-sum transfers • there are many practical problems in implementing such transfers 133 34 Monopoly • A monopoly is a single supplier to a market • This firm may choose to produce at any point on the market demand curve Chapter 14 MODELS OF MONOPOLY Copyright ©2005 by South-Western, a division of Thomson Learning. All rights reserved. 1 Barriers to Entry 2 Technical Barriers to Entry • The reason a monopoly exists is that other firms find it unprofitable or impossible to enter the market • Barriers to entry are the source of all monopoly power • The production of a good may exhibit decreasing marginal and average costs over a wide range of output levels – in this situation, relatively large-scale firms are low-cost producers • firms may find it profitable to drive others out of the industry by cutting prices • this situation is known as natural monopoly • once the monopoly is established, entry of new firms will be difficult – there are two general types of barriers to entry • technical barriers • legal barriers 3 4 1 Technical Barriers to Entry Legal Barriers to Entry • Another technical basis of monopoly is special knowledge of a low-cost productive technique • Many pure monopolies are created as a matter of law – with a patent, the basic technology for a product is assigned to one firm – the government may also award a firm an exclusive franchise to serve a market – it may be difficult to keep this knowledge out of the hands of other firms • Ownership of unique resources may also be a lasting basis for maintaining a monopoly 5 Creation of Barriers to Entry • Some barriers to entry result from actions taken by the firm – research and development of new products or technologies – purchase of unique resources – lobbying efforts to gain monopoly power • The attempt by a monopolist to erect barriers to entry may involve real resource costs 6 Profit Maximization • To maximize profits, a monopolist will choose to produce that output level for which marginal revenue is equal to marginal cost – marginal revenue is less than price because the monopolist faces a downward-sloping demand curve • he must lower its price on all units to be sold if it is to generate the extra demand for this unit 7 8 2 Profit Maximization Profit Maximization • Since MR = MC at the profit-maximizing output and P > MR for a monopolist, the monopolist will set a price greater than marginal cost MC Price The monopolist will maximize profits where MR = MC AC P* The firm will charge a price of P* C Profits can be found in the shaded rectangle MR Q* D Quantity 9 10 The Inverse Elasticity Rule The Inverse Elasticity Rule • The gap between a firm’s price and its marginal cost is inversely related to the price elasticity of demand facing the firm • Two general conclusions about monopoly pricing can be drawn: P MC 1 P eQ,P – a monopoly will choose to operate only in regions where the market demand curve is elastic • eQ,P < -1 – the firm’s “markup” over marginal cost depends inversely on the elasticity of market demand where eQ,P is the elasticity of demand for the entire market 11 12 3 Monopoly Profits Monopoly Profits • Monopoly profits will be positive as long as P > AC • Monopoly profits can continue into the long run because entry is not possible • The size of monopoly profits in the long run will depend on the relationship between average costs and market demand for the product – some economists refer to the profits that a monopoly earns in the long run as monopoly rents • the return to the factor that forms the basis of the monopoly 13 Monopoly Profits No Monopoly Supply Curve Price Price MC MC • With a fixed market demand curve, the supply “curve” for a monopolist will only be one point AC AC P*=AC P* 14 – the price-output combination where MR = MC C MR Q* Positive profits D MR Quantity Q* Zero profit D Quantity 15 • If the demand curve shifts, the marginal revenue curve shifts and a new profitmaximizing output will be chosen 16 4 Monopoly with Linear Demand • Suppose that the market for frisbees has a linear demand curve of the form Monopoly with Linear Demand • To maximize profits, the monopolist chooses the output for which MR = MC • We need to find total revenue Q = 2,000 - 20P or TR = PQ = 100Q - Q2/20 P = 100 - Q/20 • Therefore, marginal revenue is • The total costs of the frisbee producer are given by MR = 100 - Q/10 while marginal cost is C(Q) = 0.05Q2 + 10,000 MC = 0.01Q 17 Monopoly with Linear Demand • Thus, MR = MC where Monopoly with Linear Demand • To see that the inverse elasticity rule holds, we can calculate the elasticity of demand at the monopoly’s profitmaximizing level of output 100 - Q/10 = 0.01Q Q* = 500 18 P* = 75 • At the profit-maximizing output, C(Q) = 0.05(500)2 + 10,000 = 22,500 AC = 22,500/500 = 45 = (P* - AC)Q = (75 - 45)500 = 15,000 eQ,P 19 Q P 75 20 3 P Q 500 20 5 Monopoly with Linear Demand Monopoly and Resource Allocation • The inverse elasticity rule specifies that • To evaluate the allocational effect of a monopoly, we will use a perfectly competitive, constant-cost industry as a basis of comparison P MC 1 1 P eQ,P 3 • Since P* = 75 and MC = 50, this relationship holds – the industry’s long-run supply curve is infinitely elastic with a price equal to both marginal and average cost 21 22 Monopoly and Resource Allocation Price Monopoly and Resource Allocation If this market was competitive, output would be Q* and price would be P* Under a monopoly, output would be Q** and price would rise to P** P** MC=AC P* Price Consumer surplus would fall Producer surplus will rise Consumer surplus falls by more than producer surplus rises. P** MC=AC P* D D MR Q** There is a deadweight loss from monopoly MR Q* Quantity Q** 23 Q* Quantity 24 6 Welfare Losses and Elasticity Welfare Losses and Elasticity • Assume that the constant marginal (and average) costs for a monopolist are given by c and that the compensated demand curve has a constant elasticity: • The competitive price in this market will be Q = Pe where e is the price elasticity of demand (e < -1) Pc = c and the monopoly price is given by c Pm 1 1 e 25 26 Welfare Losses and Elasticity Welfare Losses and Elasticity • The consumer surplus associated with any price (P0) can be computed as • Therefore, under perfect competition P0 P0 CSc CS Q(P )dP P edP c e 1 e 1 and under monopoly e 1 CS e 1 e 1 0 P P e 1P e 1 0 27 c 1 1 e CSm e 1 28 7 Welfare Losses and Elasticity Welfare Losses and Elasticity • Taking the ratio of these two surplus measures yields • Monopoly profits are given by CSm 1 CSc 1 1 e
c m PmQm cQm c Qm 1 1
e e 1 e • If e = -2, this ratio is ½ – consumer surplus under monopoly is half what it is under perfect competition 29 c
c c e m 1 1 1 1 1 1
e e e e 1 1 e 30 Welfare Losses and Elasticity Monopoly and Product Quality • To find the transfer from consumer surplus into monopoly profits we can divide monopoly profits by the competitive consumer surplus • The market power enjoyed by a monopoly may be exercised along dimensions other than the market price of its product
m e 1 1 CSc e 1 1 e e 1 • If e = -2, this ratio is ¼ e 1 e – type, quality, or diversity of goods • Whether a monopoly will produce a higher-quality or lower-quality good than would be produced under competition depends on demand and the firm’s costs e 31 32 8 Monopoly and Product Quality • Suppose that consumers’ willingness to pay for quality (X) is given by the inverse demand function P(Q,X) where P/Q < 0 and P/X > 0 Monopoly and Product Quality • First-order conditions for a maximum are P P (Q, X ) Q CQ 0 Q Q – MR = MC for output decisions • If costs are given by C(Q,X), the monopoly will choose Q and X to maximize P Q CX 0 X X = P(Q,X)Q - C(Q,X) 33 – Marginal revenue from increasing quality by one unit is equal to the marginal cost of 34 making such an increase Monopoly and Product Quality Monopoly and Product Quality • The level of product quality that will be opted for under competitive conditions is the one that maximizes net social welfare • The difference between the quality choice of a competitive industry and the monopolist is: Q* SW P(Q, X )dQ C(Q, X ) 0 • Maximizing with respect to X yields Q* SW PX (Q, X )dQ C X 0 0 X 35 – the monopolist looks at the marginal valuation of one more unit of quality assuming that Q is at its profit-maximizing level – the competitve industry looks at the marginal value of quality averaged across all output levels 36 9 Price Discrimination Monopoly and Product Quality • A monopoly engages in price discrimination if it is able to sell otherwise identical units of output at different prices • Whether a price discrimination strategy is feasible depends on the inability of buyers to practice arbitrage • Even if a monopoly and a perfectly competitive industry chose the same output level, they might opt for diffferent quality levels – each is concerned with a different margin in its decision making – profit-seeking middlemen will destroy any discriminatory pricing scheme if possible • price discrimination becomes possible if resale is costly 38 37 Perfect Price Discrimination Perfect Price Discrimination • If each buyer can be separately identified by the monopolist, it may be possible to charge each buyer the maximum price he would be willing to pay for the good Under perfect price discrimination, the monopolist charges a different price to each buyer Price The first buyer pays P1 for Q1 units P1 The second buyer pays P2 for Q2-Q1 units P2 MC – perfect or first-degree price discrimination • extracts all consumer surplus • no deadweight loss D The monopolist will continue this way until the marginal buyer is no longer willing to pay the good’s marginal cost Quantity 39 Q1 Q2 40 Q2 10 Perfect Price Discrimination • Recall the example of the frisbee manufacturer • If this monopolist wishes to practice perfect price discrimination, he will want to produce the quantity for which the marginal buyer pays a price exactly equal to the marginal cost 41 Perfect Price Discrimination • Therefore, P = 100 - Q/20 = MC = 0.1Q Q* = 666 • Total revenue and total costs will be 666 Q* R P (Q )dQ 100Q 0 Q2 40 0 55,511 c(Q) 0.05Q 2 10,000 32,178 • Profit is much larger (23,333 > 15,000) 42 Market Separation Market Separation • Perfect price discrimination requires the monopolist to know the demand function for each potential buyer • A less stringent requirement would be to assume that the monopoly can separate its buyers into a few identifiable markets • All the monopolist needs to know in this case is the price elasticities of demand for each market – can follow a different pricing policy in each market – third-degree price discrimination – set price according to the inverse elasticity rule • If the marginal cost is the same in all markets, Pi (1 43 1 1 ) Pj (1 ) ei ej 44 11 Market Separation Market Separation If two markets are separate, maximum profits occur by setting different prices in the two markets • This implies that Price 1 ) ej Pi Pj (1 1 ) ei (1 The market with the less elastic demand will be charged the higher price P1 P2 • The profit-maximizing price will be higher in markets where demand is less elastic MC MC D D MR Quantity in Market 1 MR Q1* 0 Q2* Quantity in Market 2 45 Third-Degree Price Discrimination 46 Third-Degree Price Discrimination • Suppose that the demand curves in two separated markets are given by • Optimal choices and prices are Q1 = 9 P1 = 15 Q1 = 24 – P1 Q2 = 6 P2 = 9 Q2 = 24 – 2P2 • Profits for the monopoly are • Suppose that MC = 6 • Profit maximization requires that = (P1 - 6)Q1 + (P2 - 6)Q2 = 81 + 18 = 99 MR1 = 24 – 2Q1 = 6 = MR2 = 12 – Q2 47 48 12 Third-Degree Price Discrimination Third-Degree Price Discrimination • The allocational impact of this policy can be evaluated by calculating the deadweight losses in the two markets – the competitive output would be 18 in market 1 and 12 in market 2 DW 1 = 0.5(P1-MC)(18-Q1) = 0.5(15-6)(18-9) = 40.5 DW 2 = 0.5(P2-MC)(12-Q2) = 0.5(9-6)(12-6) = 9 49 Third-Degree Price Discrimination • If this monopoly was to pursue a singleprice policy, it would use the demand function Q = Q1 + Q2 = 48 – 3P • So marginal revenue would be MR = 16 – 2Q/3 • Profit-maximization occurs where Q = 15 P = 11 50 Two-Part Tariffs • The deadweight loss is smaller with one price than with two: DW = 0.5(P-MC)(30-Q) = 0.5(11-6)(15) = 37.5 • A linear two-part tariff occurs when buyers must pay a fixed fee for the right to consume a good and a uniform price for each unit consumed T(q) = a + pq • The monopolist’s goal is to choose a and p to maximize profits, given the demand for the product 51 52 13 Two-Part Tariffs Two-Part Tariffs • Because the average price paid by any demander is • One feasible approach for profit maximization would be for the firm to set p = MC and then set a equal to the consumer surplus of the least eager buyer p’ = T/q = a/q + p this tariff is only feasible if those who pay low average prices (those for whom q is large) cannot resell the good to those who must pay high average prices (those for whom q is small) – this might not be the most profitable approach – in general, optimal pricing schedules will depend on a variety of contingencies 53 Two-Part Tariffs Two-Part Tariffs • Suppose there are two different buyers with the demand functions • With this marginal price, demander 2 obtains consumer surplus of 36 q1 = 24 - p1 q2 = 24 - 2p2 • If MC = 6, one way for the monopolist to implement a two-part tariff would be to set p1 = p2 = MC = 6 q1 = 18 54 q2 = 12 55 – this would be the maximum entry fee that can be charged without causing this buyer to leave the market • This means that the two-part tariff in this case would be T(q) = 36 + 6q 56 14 Regulation of Monopoly Regulation of Monopoly • Many economists believe that it is important for the prices of regulated monopolies to reflect marginal costs of production accurately • An enforced policy of marginal cost pricing will cause a natural monopoly to operate at a loss • Natural monopolies such as the utility, communications, and transportation industries are highly regulated in many countries – natural monopolies exhibit declining average costs over a wide range of output 57 58 Regulation of Monopoly Regulation of Monopoly Because natural monopolies exhibit decreasing costs, MC falls below AC Price An unregulated monopoly will maximize profit at Q1 and P1 P1 C1 C2 AC P2 MR Q1 If regulators force the monopoly to charge a price of P2, the firm will suffer a loss because P2 < C2 MC Q2 D Price Suppose that the regulatory commission allows the monopoly to charge a price of P1 to some users Other users are offered the lower price of P2 The profits on the sales to highprice customers are enough to cover the losses on the sales to low-price customers P1 C1 C2 AC MC P2 Quantity Q1 59 Q2 D Quantity 60 15 Regulation of Monopoly Regulation of Monopoly • Another approach followed in many regulatory situations is to allow the monopoly to charge a price above marginal cost that is sufficient to earn a “fair” rate of return on investment • Suppose that a regulated utility has a production function of the form q = f (k,l) – if this rate of return is greater than that which would occur in a competitive market, there is an incentive to use relatively more capital than would truly minimize costs • The firm’s actual rate of return on capital is defined as s pf (k, l ) wl k 61 Regulation of Monopoly 62 Regulation of Monopoly • If =0, regulation is ineffective and the monopoly behaves like any profitmaximizing firm • If =1, the Lagrangian reduces to • Suppose that s is constrained by regulation to be equal to s0, then the firm’s problem is to maximize profits = pf (k,l) – wl – vk L = (s0 – v)k subject to this constraint • The Lagrangian for this problem is which (assuming s0>v), will mean that the monopoly will hire infinite amounts of capital – an implausible result L = pf (k,l) – wl – vk + [wl + s0k – pf (k,l)] 63 64 16 Regulation of Monopoly Regulation of Monopoly • Because s0>v and <1, this means that • Therefore, 0<<1 and the first-order conditions for a maximum are: pfk < v L pfl w (w pfl ) 0 l • The firm will hire more capital than it would under unregulated conditions L pfk v (s0 pfk ) 0 k – it will also achieve a lower marginal productivity of capital L wl s0 pf (k, l ) 0 65 66 Dynamic Views of Monopoly Important Points to Note: • Some economists have stressed the beneficial role that monopoly profits can play in the process of economic development • The most profitable level of output for the monopolist is the one for which marginal revenue is equal to marginal cost – these profits provide funds that can be invested in research and development – the possibility of attaining or maintaining a monopoly position provides an incentive to keep one step ahead of potential competitors 67 – at this output level, price will exceed marginal cost – the profitability of the monopolist will depend on the relationship between price and average cost 68 17 Important Points to Note: Important Points to Note: • Relative to perfect competition, monopoly involves a loss of consumer surplus for demanders • Monopolies may opt for different levels of quality than would perfectly competitive firms • Durable good monopolists may be constrained by markets for used goods – some of this is transferred into monopoly profits, whereas some of the loss in consumer surplus represents a deadweight loss of overall economic welfare – it is a sign of Pareto inefficiency 69 70 Important Points to Note: Important Points to Note: • A monopoly may be able to increase its profits further through price discrimination – charging different prices to different categories of buyers • Governments often choose to regulate natural monopolies (firms with diminishing average costs over a broad range of output levels) – the ability of the monopoly to practice price discrimination depends on its ability to prevent arbitrage among buyers – the type of regulatory mechanisms adopted can affect the behavior of the regulated firm 71 72 18 Pricing Under Homogeneous Oligopoly • We will assume that the market is perfectly competitive on the demand side Chapter 15 – there are many buyers, each of whom is a price taker TRADITIONAL MODELS OF IMPERFECT COMPETITION • We will assume that the good obeys the law of one price – this assumption will be relaxed when product differentiation is discussed Copyright ©2005 by South-Western, a division of Thomson Learning. All rights reserved. 1 Pricing Under Homogeneous Oligopoly 2 Pricing Under Homogeneous Oligopoly • We will assume that there is a relatively small number of identical firms (n) • The inverse demand function for the good shows the price that buyers are willing to pay for any particular level of industry output – we will initially start with n fixed, but later allow n to vary through entry and exit in response to firms’ profitability P = f(Q) = f(q1+q2+…+qn) • The output of each firm is qi (i=1,…,n) • Each firm’s goal is to maximize profits – symmetry in costs across firms will usually require that these outputs are equal i = f(Q)qi –Ci(qi) i = f(q1+q2+…qn)qi –Ci 3 4 1 Oligopoly Pricing Models Oligopoly Pricing Models • The Cournot model assumes that firm i treats firm j’s output as fixed in its decisions • The quasi-competitive model assumes price-taking behavior by all firms – P is treated as fixed – qj/qi = 0 • The cartel model assumes that firms can collude perfectly in choosing industry output and P • The conjectural variations model assumes that firm j’s output will respond to variations in firm i’s output – qj/qi 0 5 6 Quasi-Competitive Model Quasi-Competitive Model • Each firm is assumed to be a price taker • The first-order condition for profitmaximization is i /qi = P – (Ci /qi) = 0 P = MCi (qi) (i=1,…,n) • Along with market demand, these n supply equations will ensure that this market ends up at the short-run competitive solution Price If each firm acts as a price taker, P = MCi so QC output is produced and sold at a price of PC MC PC D MR QC 7 Quantity 8 2 Cartel Model Cartel Model • The assumption of price-taking behavior may be inappropriate in oligopolistic industries • In this case, the cartel acts as a multiplant monopoly and chooses qi for each firm so as to maximize total industry profits – each firm can recognize that its output decision will affect price • An alternative assumption would be that firms act as a group and coordinate their decisions so as to achieve monopoly profits = PQ – [C1(q1) + C2(q2) +…+ Cn(qn)] n f (q1 q2 ... qn )[q1 q2 ... qn ] Ci (qi ) i 1 9 10 Cartel Model Cartel Model • The first-order conditions for a maximum are that P P (q1 q2 ... qn ) MC(qi ) 0 qi qi If the firms form a group and act as a monopoly, MR = MCi so QM output is produced and sold at a price of PM Price PM • This implies that MC MR(Q) = MCi(qi) • At the profit-maximizing point, marginal revenue will be equal to each firm’s marginal cost D MR QM 11 Quantity 12 3 Cartel Model Cournot Model • There are three problems with the cartel solution – these monopolistic decisions may be illegal – it requires that the directors of the cartel know the market demand function and each firm’s marginal cost function – the solution may be unstable • each firm has an incentive to expand output because P > MCi • Each firm recognizes that its own decisions about qi affect price – P/qi 0 • However, each firm believes that its decisions do not affect those of any other firm – qj /qi = 0 for all j i 13 Cournot Model 14 Cournot Model • The first-order conditions for a profit maximization are • Each firm’s output will exceed the cartel output i P P qi MCi (qi ) 0 qi qi – the firm-specific marginal revenue is larger than the market-marginal revenue • Each firm’s output will fall short of the competitive output • The firm maximizes profit where MRi = MCi – the firm assumes that changes in qi affect its total revenue only through their direct effect on market price 15 – qi P/qi < 0 16 4 Cournot’s Natural Springs Duopoly Cournot Model • Price will exceed marginal cost, but industry profits will be lower than in the cartel model • The greater the number of firms in the industry, the closer the equilibrium point will be to the competitive result • Assume that there are two owners of natural springs – each firm has no production costs – each firm has to decide how much water to supply to the market • The demand for spring water is given by the linear demand function Q = q1 + q2 = 120 - P 17 Cournot’s Natural Springs Duopoly 18 Cournot’s Natural Springs Duopoly • Because each firm has zero marginal costs, the quasi-competitive solution will result in a market price of zero • The cartel solution to this problem can be found by maximizing industry revenue (and profits) – total demand will be 120 – the division of output between the two firms is indeterminate = PQ = 120Q - Q2 • The first-order condition is /Q = 120 - 2Q = 0 • each firm has zero marginal cost over all output ranges 19 20 5 Cournot’s Natural Springs Duopoly Cournot’s Natural Springs Duopoly • The profit-maximizing output, price, and level of profit are • The two firms’ revenues (and profits) are given by Q = 60 P = 60 1 = Pq1 = (120 - q1 - q2) q1 = 120q1 - q12 - q1q2 = 3,600 • First-order conditions for a maximum are 2 = Pq2 = (120 - q1 - q2) q2 = 120q2 - q22 - q1q2 • The precise division of output and profits is indeterminate 1 120 2q1 q2 0 q1 2 120 2q2 q1 0 q2 21 Cournot’s Natural Springs Duopoly 22 Cournot’s Natural Springs Duopoly • These first-order equations are called reaction functions • We can solve the reaction functions simultaneously to find that – show how each firm reacts to the other’s output level q1 = q2 = 40 P = 120 - (q1 + q2) = 40 1 = 2 = Pq1 = Pq2 = 1,600 • In equilibrium, each firm must produce what the other firm thinks it will 23 • Note that the Cournot equilibrium falls between the quasi-competitive model and the cartel model 24 6 Conjectural Variations Model Conjectural Variations Model • In markets with only a few firms, we can expect there to be strategic interaction among firms • One way to build strategic concerns into our model is to consider the assumptions that might be made by one firm about the other firm’s behavior • For each firm i, we are concerned with the assumed value of qj /qi for ij – because the value will be speculative, models based on various assumptions about its value are termed conjectural variations models • they are concerned with firm i’s conjectures about firm j’s output variations 25 Conjectural Variations Model • The first-order condition for profit maximization becomes Price Leadership Model • Suppose that the market is composed of a single price leader (firm 1) and a fringe of quasi-competitors P i P q j P qi MCi (qi ) 0 qi qi j i q j qi – firms 2,…,n would be price takers – firm 1 would have a more complex reaction function, taking other firms’ actions into account • The firm must consider how its output decisions will affect price in two ways – directly – indirectly through its effect on the output decisions of other firms 26 27 28 7 Price Leadership Model Price Leadership Model We can derive the demand curve facing the industry leader D represents the market demand curve Price Price SC SC SC represents the supply decisions of all of the n-1 firms in For a price of P1 or above, the leader will sell nothing P1 the competitive fringe For a price of P2 or below, the leader has the market to itself P2 D D Quantity Quantity 29 30 Price Leadership Model Between P2 and P1, the demand for the leader (D’) is constructed by subtracting what the fringe will supply from total market demand Price SC P1 PL D’ P2 MC’ MR’ QL Price Leadership Model D The leader would then set MR’ = MC’ and produce QL at a price of PL Price SC P1 D’ P2 MC’ QC 31 The competitive fringe will produce QC and total industry output will be QT (= QC + QL) PL MR’ Quantity Market price will then be PL QL QT D Quantity 32 8 Stackelberg Leadership Model Price Leadership Model • This model does not explain how the price leader is chosen or what happens if a member of the fringe decides to challenge the leader • The model does illustrate one tractable example of the conjectural variations model that may explain pricing behavior in some instances • The assumption of a constant marginal cost makes the price leadership model inappropriate for Cournot’s natural spring problem – the competitive fringe would take the entire market by pricing at marginal cost (= 0) – there would be no room left in the market for the price leader 33 Stackelberg Leadership Model • There is the possibility of a different type of strategic leadership • Assume that firm 1 knows that firm 2 chooses q2 so that 34 Stackelberg Leadership Model • This means that firm 2 reduces its output by ½ unit for each unit increase in q1 • Firm 1’s profit-maximization problem can be rewritten as q2 = (120 – q1)/2 1 = Pq1 = 120q1 – q12 – q1q2 • Firm 1 can now calculate the conjectural variation 1/q1 = 120 – 2q1 – q1(q2/q1) – q2 = 0 q2/q1 = -1/2 35 1/q1 = 120 – (3/2)q1 – q2 = 0 36 9 Stackelberg Leadership Model Product Differentiation • Solving this simultaneously with firm 2’s reaction function, we get • Firms often devote considerable resources to differentiating their products from those of their competitors q1 = 60 q2 = 30 P = 120 – (q1 + q2) = 30 1 = Pq1 = 1,800 2 = Pq2 = 900 • Again, there is no theory on how the leader is chosen – quality and style variations – warranties and guarantees – special service features – product advertising 37 38 Product Differentiation Product Differentiation • The law of one price may not hold, because demanders may now have preferences about which suppliers to purchase the product from – there are now many closely related, but not identical, products to choose from • We must be careful about which products we assume are in the same market 39 • The output of a set of firms constitute a product group if the substitutability in demand among the products (as measured by the cross-price elasticity) is very high relative to the substitutability between those firms’ outputs and other goods generally 40 10 Product Differentiation Product Differentiation • We will assume that there are n firms competing in a particular product group • Because there are n firms competing in the product group, we must allow for different market prices for each (p1,...,pn) • The demand facing the ith firm is – each firm can choose the amount it spends on attempting to differentiate its product from its competitors (zi) • The firm’s costs are now given by pi = g(qi,pj,zi,zj) • Presumably, pi/qi 0, pi/pj 0, pi/zi 0, and pi/zj 0 total costs = Ci (qi,zi) 41 Product Differentiation Product Differentiation • The ith firm’s profits are given by i = piqi –Ci(qi,zi) • In the simple case where zj/qi, zj/zi, pj/qi, and pj/zi are all equal to zero, the first-order conditions for a maximum are i p C pi qi i i 0 qi qi qi i p C qi i i 0 zi zi zi 42 43 • At the profit-maximizing level of output, marginal revenue is equal to marginal cost • Additional differentiation activities should be pursued up to the point at which the additional revenues they generate are equal to their marginal costs 44 11 Product Differentiation Spatial Differentiation • Suppose we are examining the case of ice cream stands located on a beach • The demand curve facing any one firm may shift often – assume that demanders are located uniformly along the beach – it depends on the prices and product differentiation activities of its competitors • The firm must make some assumptions in order to make its decisions • The firm must realize that its own actions may influence its competitors’ actions • one at each unit of beach • each buyer purchases exactly one ice cream cone per period – ice cream cones are costless to produce but carrying them back to one’s place on the beach results in a cost of c per unit traveled 45 Spatial Differentiation Spatial Differentiation • A person located at point E will be indifferent between stands A and B if L Ice cream stands are located at points A and B along a linear beach of length L A E 46 pA + cx = pB + cy where pA and pB are the prices charged by each stand, x is the distance from E to A, and y is the distance from E to B B Suppose that a person is standing at point E 47 48 12 Spatial Differentiation Spatial Differentiation • The coordinate of point E is L a x y x b a+x+y+b=L A E x pB pA Lab x c x 1 p pA L a b B 2 c y 1 p pB L a b A 2 c B 49 Spatial Differentiation 1 p p pA2 (L a b ) p A A B 2 2c B pB (b y ) 1 p p pB2 (L a b)pB A B 2 2c 50 Spatial Differentiation • Each firm will choose its price so as to maximize profits • Profits for the two firms are A p A (a x ) pB pA cy c A 1 p p (L a b ) B A 0 pA 2 2c c B 1 p p (L a b ) A B 0 pB 2 2c c 51 52 13 Spatial Differentiation Spatial Differentiation • These can be solved to yield: L ab pA c L 3 a ab pB c L 3 x y b Because A is somewhat more favorably located than B, pA will exceed pB • These prices depend on the precise locations of the stands and will differ from one another A E B 53 Spatial Differentiation 54 Entry • If we allow the ice cream stands to change their locations at zero cost, each stand has an incentive to move to the center of the beach • In perfect competition, the possibility of entry ensures that firms will earn zero profit in the long run • These conditions continue to operate under oligopoly – any stand that opts for an off-center position is subject to its rival moving between it and the center and taking a larger share of the market – to the extent that entry is possible, long-run profits are constrained – if entry is completely costless, long-run profits will be zero • this encourages a similarity of products 55 56 14 Entry Entry • If firms are price takers: • Whether firms in an oligopolistic industry with free entry will be directed to the point of minimum average cost depends on the nature of the demand facing them – P = MR = MC for profit maximization, P = AC for zero profits, so production takes place at MC = AC • If firms have some control over price: 57 Entry Monopolistic Competition Firms will initially be maximizing profits at q*. Since P > AC, > 0 Price MC AC P* Since > 0, firms will enter and the demand facing the firm will shift left Entry will end when = 0 P’ d mr’ q’ mr q* qm d’ – each firm will face a downward-sloping demand curve – entry may reduce profits to zero, but production at minimum average cost is not ensured 58 Firms will exhibit excess capacity = qm - q’ • The zero-profit equilibrium model just shown was developed by Chamberlin who termed it monopolistic competition – each firm produces a slightly differentiated product and entry is costless • Suppose that there are n firms in a market and that each firm has the total cost schedule ci = 9 + 4qi Quantity 59 60 15 Monopolistic Competition Monopolistic Competition • To find the equilibrium n, we must examine each firm’s profit-maximizing choice of pi • Because • Each firm also faces a demand curve for its product of the form: qi 0.01(n 1)pi 0.01 p j j i 303 n • We will define an equilibrium for this industry to be a situation in which prices must be equal – pi = pj for all i and j i = piqi – ci the first-order condition for a maximum is i 303 0.02(n 1)pi 0.01 p j 0.04(n 1) 0 pi n j i 61 Monopolistic Competition Monopolistic Competition • This means that pi 0.5 p j j i n 1 • The equilibrium n is determined by the zero-profit condition 303 2 0.02(n 1)n pi qi ci 0 • Applying the equilibrium condition that pi = pj yields pi 62 30,300 303 4(303) 4(303) 9 n 2 (n 1) n n 30,300 4 (n 1)n • P approaches MC (4) as n gets larger • Substituting in the expression for pi, we find that 63 n 101 64 16 Monopolistic Competition Monopolistic Competition • The final equilibrium is • If each firm faces a similar demand function, this equilibrium is sustainable pi = pj = 7 qi = 3 – no firm would find it profitable to enter this industry i = 0 • In this equilibrium, each firm has pi = ACi, but pi > MCi = 4 • Because ACi = 4 + 9/qi, each firm has diminishing AC throughout all output ranges 65 • But what if a potential entrant adopted a large-scale production plan? Contestable Markets and Industry Structure Perfectly Contestable Market • Several economists have challenged that this zero-profit equilibrium is sustainable in the long run – the low average cost may give the potential entrant considerable leeway in pricing so as to tempt customers of existing firms to 66 switch allegiances • A market is perfectly contestable if entry and exit are absolutely free – the model ignores the effects of potential entry on market equilibrium by focusing only on actual entrants – need to distinguish between competition in the market and competition for the market 67 – no outside potential competitor can enter by cutting price and still make a profit • if such profit opportunities existed, potential entrants would take advantage of them 68 17 Perfectly Contestable Market This market would be unsustainable in a perfectly contestable market Price MC Because P > MC, a potential entrant can take one zero-profit firm’s market away and encroach a bit on other firms’ markets where, at the margin, profits are attainable AC P* P’ d mr’ q’ mr q* q’ d’ Perfectly Contestable Market • Therefore, to be perfectly contestable, the market must be such that firms earn zero profits and price at marginal costs – firms will produce at minimum average cost – P = AC = MC • Perfect contestability guides market equilibrium to a competitive-type result Quantity 69 70 Perfectly Contestable Market Perfectly Contestable Market • If we let q* represent the output level for which average costs are minimized and Q* represent the total market demand when price equals average cost, then the equilibrium number of firms in the industry is given by In a perfectly contestable market, equilibrium requires that P = MC = AC Price AC1 AC2 AC3 The number of firms is completely determined by market demand (Q*) and by the output level that minimizes AC (q*) AC4 P* n = Q*/q* – this number may be relatively small (unlike the perfectly competitive case) 71 D q* 2q* 3q* Q*=4q* Quantity 72 18 Barriers to Entry Barriers to Entry • If barriers to entry prevent free entry and exit, the results of this model must be modified • The completely flexible type of hit-andrun behavior assumed in the contestable markets theory may be subject to barriers to entry – barriers to entry can be the same as those that lead to monopolies or can be the result of some of the features of oligopolistic markets • product differentiation • strategic pricing decisions – some types of capital investments may not be reversible – demanders may not respond to price differentials quickly 73 74 A Contestable Natural Monopoly A Contestable Natural Monopoly • Suppose that the total cost of producing electric power is given by • If the producer behaves as a monopolist, it will maximize profits by C(Q) = 100Q + 8,000 – since AC declines over all output ranges, MR = 200 - (2Q)/5 = MC = 100 Qm = 250 Pm = 150 this is a natural monopoly • The demand for electricity is given by m = R - C = 37,500 - 33,000 = 4,500 QD = 1,000 - 5P 75 • These profits will be tempting to would-be entrants 76 19 A Contestable Natural Monopoly A Contestable Natural Monopoly • If there are no entry barriers, a potential entrant can offer electricity customers a lower price and still cover costs • If electricity production is fully contestable, the only price viable under threat of potential entry is average cost – this monopoly solution might not represent a viable equilibrium Q = 1,000 - 5P = 1,000 – 5(AC) Q = 1,000 - 5[100 + (8,000/Q)] Q2 - 500Q + 40,000 = 0 (Q - 400)(Q - 100) = 0 77 A Contestable Natural Monopoly • Only Q = 400 is a sustainable entry deterrent • Under contestability, the market equilibrium is Qc = 400 Pc = 120 • Contestability increased consumer welfare from what it was under the monopoly 79 situation 78 Important Points to Note: • Markets with few firms offer potential profits through the formation of a monopoly cartel – such cartels may, however, be unstable and costly to maintain because each member has an incentive to chisel on price 80 20 Important Points to Note: Important Points to Note: • In markets with few firms, output and price decisions are interdependent • The Cournot model provides a tractable approach to oligopoly markets, but neglects important strategic issues – each firm must consider its rivals’ decisions – modeling such interdependence is difficult because of the need to consider conjectural variations 81 Important Points to Note: 82 Important Points to Note: • Product differentiation can be analyzed in a standard profitmaximization framework • Entry conditions are important determinants of the long-run sustainability of various market equilibria – with differentiated products, the law of one price no longer holds and firms may have somewhat more leeway in their pricing decisions – with perfect contestability, equilibria may resemble perfectly competitive ones even though there are relatively few firms in the market 83 84 21 Allocation of Time • Individuals must decide how to allocate the fixed amount of time they have • We will initially assume that there are only two uses of an individual’s time Chapter 16 LABOR MARKETS – engaging in market work at a real wage rate of w – leisure (nonwork) Copyright ©2005 by South-Western, a division of Thomson Learning. All rights reserved. 1 Allocation of Time 2 Allocation of Time • Assume that an individual’s utility depends on consumption (c) and hours of leisure (h) • Combining the two constraints, we get utility = U(c,h) • An individual has a “full income” of 24w c = w(24 – h) c + wh = 24w • In seeking to maximize utility, the individual is bound by two constraints – may spend the full income either by working (for real income and consumption) or by not working (enjoying leisure) l + h = 24 c = wl • The opportunity cost of leisure is w 3 4 1 Utility Maximization Utility Maximization • The individual’s problem is to maximize utility subject to the full income constraint • Setting up the Lagrangian L = U(c,h) + (24w – c – wh) • Dividing the two, we get U / c w MRS (h for c ) U / h • To maximize utility, the individual should choose to work that number of hours for which the MRS (of h for c) is equal to w • The first-order conditions are L/c = U/c - = 0 – to be a true maximum, the MRS (of h for c) must be diminishing L/h = U/h - = 0 5 Income and Substitution Effects 6 Consumption Income and Substitution Effects The substitution effect is the movement from point A to point C • Both a substitution effect and an income effect occur when w changes – when w rises, the price of leisure becomes higher and the individual will choose less leisure – because leisure is a normal good, an increase in w leads to an increase in leisure • The income and substitution effects move 7 in opposite directions B The income effect is the movement from point C to point B C A U2 U1 The individual chooses less leisure as a result of the increase in w Leisure substitution effect > income effect 8 2 Consumption Income and Substitution Effects A Mathematical Analysis of Labor Supply The substitution effect is the movement from point A to point C The income effect is the movement from point C to point B B C A U1 U2 Leisure The individual chooses more leisure as a result of the increase in w substitution effect < income effect • We will start by amending the budget constraint to allow for the possibility of nonlabor income c = wl + n • Maximization of utility subject to this constraint yields identical results – as long as n is unaffected by the laborleisure choice 9 A Mathematical Analysis of Labor Supply 10 Dual Statement of the Problem • The dual problem can be phrased as choosing levels of c and h so that the amount of expenditure (E = c – wl) required to obtain a given utility level (U0) is as small as possible • The only effect of introducing nonlabor income is that the budget constraint shifts out (or in) in a parallel fashion • We can now write the individual’s labor supply function as l(w,n) – solving this minimization problem will yield exactly the same solution as the utility maximization problem – hours worked will depend on both the wage and the amount of nonlabor income – since leisure is a normal good, l/n < 0 11 12 3 Dual Statement of the Problem • A small change in w will change the minimum expenditures required by Dual Statement of the Problem • This means that a labor supply function can be calculated by partially differentiating the expenditure function E/w = -l – this is the extent to which labor earnings are increased by the wage change – because utility is held constant, this function should be interpreted as a “compensated” (constant utility) labor supply function lc(w,U) 13 Slutsky Equation of Labor Supply Slutsky Equation of Labor Supply • The expenditures being minimized in the dual expenditure-minimization problem play the role of nonlabor income in the primary utility-maximization problem lc(w,U) = l[w,E(w,U)] = l(w,N) • Partial differentiation of both sides with respect to w gives us l c l l E
w w E w 14 15 • Substituting for E/w, we get l c l l l l l l w w E w n • Introducing a different notation for lc , and rearranging terms gives us the Slutsky equation for labor supply: l l w w l U U 0 l n 16 4 Cobb-Douglas Labor Supply Cobb-Douglas Labor Supply • Suppose that utility is of the form • The Lagrangian expression for utility maximization is U c h L = ch + (w + n - wh - c) • The budget constraint is • First-order conditions are c = wl + n L/c = c-h - = 0 and the time constraint is l+h=1 – note that we have set maximum work time to 1 hour for convenience L/h = ch- - w = 0 L/ = w + n - wh - c = 0 17 18 Cobb-Douglas Labor Supply Cobb-Douglas Labor Supply • Dividing the first by the second yields • Substitution into the full income constraint yields h h 1 c (1 )c w wh c = (w + n) h = (w + n)/w – the person spends of his income on consumption and = 1- on leisure – the labor supply function is 1 c l (w , n ) 1 h (1 ) 19 n w 20 5 Cobb-Douglas Labor Supply Cobb-Douglas Labor Supply • Note that if n = 0, the person will work (1-) of each hour no matter what the wage is • If n > 0, l/w > 0 – the individual will always choose to spend n on leisure – Since leisure costs w per hour, an increase in w means that less leisure can be bought with n – the substitution and income effects of a change in w offset each other and leave l unaffected 21 Cobb-Douglas Labor Supply 22 CES Labor Supply • Suppose that the utility function is • Note that l/n < 0 – an increase in nonlabor income allows this person to buy more leisure • income transfer programs are likely to reduce labor supply • lump-sum taxes will increase labor supply 23 U (c, h ) c h
• Budget share equations are given by sc c 1 w n (1 w ) sh wh 1 w n (1 w ) – where = /(-1) 24 6 Market Supply Curve for Labor CES Labor Supply To derive the market supply curve for labor, we sum the quantities of labor offered at every wage • Solving for leisure gives h w n w w 1 w Individual A’s supply curve w sA and l(w, n ) 1 h Individual B’s supply curve w Total labor supply curve sB S w* w 1 n w w 1 lA* l lB* l l l* lA* + lB* = l* 25 26 Market Supply Curve for Labor Note that at w0, individual B would choose to remain out of the labor force w Individual A’s supply curve w sA Individual B’s supply curve w Total labor supply curve sB S w0 l l Labor Market Equilibrium • Equilibrium in the labor market is established through the interactions of individuals’ labor supply decisions with firms’ decisions about how much labor to hire l As w rises, l rises for two reasons: increased hours of work and increased labor force participation 27 28 7 Labor Market Equilibrium real wage Mandated Benefits At w*, the quantity of labor demanded is equal to the quantity of labor supplied At any wage above w*, the quantity of labor demanded will be less than the quantity of labor supplied S w* D l* At any wage below w*, the quantity of labor demanded will be greater than the quantity of labor supplied quantity of labor 29 • A number of new laws have mandated that employers provide special benefits to their workers – health insurance – paid time off – minimum severance packages • The effects of these mandates depend on how much the employee values the benefit 30 Mandated Benefits Mandated Benefits • Suppose that, prior to the mandate, the supply and demand for labor are • Suppose that the government mandates that all firms provide a benefit to their workers that costs t per unit of labor hired lS = a + bw lD = c – dw • Setting lS = lD yields an equilibrium wage of w* = (c – a)/(b + d) – unit labor costs become w + t • Suppose also that the benefit has a value of k per unit supplied – the net return from employment rises to 31 w+k 32 8 Mandated Benefits Mandated Benefits • Equilibrium in the labor market then requires that • If workers derive no value from the mandated benefits (k = 0), the mandate is just like a tax on employment a + b(w + k) = c – d(w + t) – similar results will occur as long as k < t • This means that the net wage is w ** • If k = t, the new wage falls precisely by the amount of the cost and the equilibrium level of employment does not change c a bk dt bk dt w * bd bd bd 33 Mandated Benefits 34 Wage Variation • If k > t, the new wage falls by more than the cost of the benefit and the equilibrium level of employment rises • It is impossible to explain the variation in wages across workers with the tools developed so far – we must consider the heterogeneity that exists across workers and the types of jobs they take 35 36 9 Wage Variation Wage Variation • Human Capital • Compensating Differentials – differences in human capital translate into differences in worker productivities – workers with greater productivities would be expected to earn higher wages – while the investment in human capital is similar to that in physical capital, there are two differences • investments are sunk costs • opportunity costs are related to past investments 37 – individuals prefer some jobs to others – desirable job characteristics may make a person willing to take a job that pays less than others – jobs that are unpleasant or dangerous will require higher wages to attract workers – these differences in wages are termed compensating differentials 38 Monopsony in the Labor Market Monopsony in the Labor Market • In many situations, the supply curve for an input (l) is not perfectly elastic • We will examine the polar case of monopsony, where the firm is the single buyer of the input in question • The marginal expense (ME) associated with any input is the increase in total costs of that input that results from hiring one more unit – the firm faces the entire market supply curve – to increase its hiring of labor, the firm must pay a higher wage 39 – if the firm faces an upward-sloping supply curve for that input, the marginal expense will exceed the market price of the input 40 10 Monopsony in the Labor Market Monopsony in the Labor Market • If the total cost of labor is wl, then MEl Note that the quantity of labor demanded by this firm falls short of the level that would be hired in a competitive labor market (l*) Wage MEl wl w w l l l S • In the competitive case, w/l = 0 and MEl = w • If w/l > 0, MEl > w w* The wage paid by the firm will also be lower than the competitive level (w*) w1 D 41 Monopsonistic Hiring l1 Labor l* 42 Monopsonistic Hiring • Suppose that a coal mine’s workers can dig 2 tons per hour and coal sells for $10 per ton – this implies that MRPl = $20 per hour • The firm’s wage bill is wl = l2/50 • The marginal expense associated with hiring miners is MEl = wl/l = l/25 • If the coal mine is the only hirer of miners in the local area, it faces a labor supply curve of the form • Setting MEl = MRPl, we find that the optimal quantity of labor is 500 and the optimal wage is $10 l = 50w 43 44 11 Labor Unions Labor Unions • We will assume that the goals of the union are representative of the goals of its members • In some ways, we can use a monopoly model to examine unions • If association with a union was wholly voluntary, we can assume that every member derives a positive benefit • With compulsory membership, we cannot make the same claim – the union faces a demand curve for labor – as the sole supplier, it can choose at which point it will operate – even if workers would benefit from the union, they may choose to be “free riders” • this point depends on the union’s goals 45 46 Labor Unions Wage Labor Unions The union may wish to maximize the total wage bill (wl). This occurs where MR = 0 S Wage The union may wish to maximize the total economic rent of its employed members This occurs where S w2 l1 workers will be w1 l2 workers will be hired and paid a wage of w1 D MR l1 Labor This choice will create an excess supply of labor 47 MR = S hired and paid a wage of w2 D MR l2 Labor Again, this will cause an excess supply of labor 48 12 Labor Unions Wage Modeling a Union The union may wish to maximize the total employment of its members This occurs where D=S S l = 50w l3 workers will be w3 hired and paid a wage of w3 • Assume that the monopsony has a MRPL curve of the form MRPl = 70 – 0.1l D • The monopsonist will choose to hire 500 workers at a wage of $10 MR l3 • A monopsonistic hirer of coal miners faces a supply curve of Labor 49 Modeling a Union 50 A Union Bargaining Model • If a union can establish control over labor supply, other options become possible • Suppose a firm and a union engage in a two-stage game – first stage: union sets the wage rate its workers will accept – second stage: firm chooses its employment level – competitive solution where l = 583 and w = $11.66 – monopoly solution where l = 318 and w = $38.20 51 52 13 A Union Bargaining Model A Union Bargaining Model • Assuming that l* solves the firm’s problem, the union’s goal is to choose w to maximize utility • This two-stage game can be solved by backward induction • The firm’s second-stage problem is to maximize its profits: U(w,l) = U[w,l*(w)] = R(l) – wl • The first-order condition for a maximum is R’(l) = w and the first-order condition for a maximum is U1 + U2l’ = 0 U1/U2 = l’ 53 54 A Union Bargaining Model Important Points to Note: • This implies that the union should choose w so that its MRS is equal to the slope of the firm’s labor demand function • The result from this game is a Nash equilibrium • A utility-maximizing individual will choose to supply an amount of labor at which the MRS of leisure for consumption is equal to the real wage rate 55 56 14 Important Points to Note: Important Points to Note: • An increase in the real wage rate creates income and substitution effects that operate in different directions in affecting the quantity of labor supplied • A competitive labor market will establish an equilibrium real wage rate at which the quantity of labor supplied by individuals is equal to the quantity demanded by firms – this result can be summarized by a Slutsky-type equation much like the one already derived in consumer theory 57 Important Points to Note: 58 Important Points to Note: • Monopsony power by firms on the demand side of the market will reduce both the quantity of labor hired and the real wage rate • Labor unions can be treated analytically as monopoly suppliers of labor – the nature of labor market equilibrium in the presence of unions will depend importantly on the goals the union chooses to pursue – as in the monopoly case, there will be a welfare loss 59 60 15 Properties of Information • Information is not easy to define Chapter 18 – it is difficult to measure the quantity of information obtainable from different actions – there are too many forms of useful information to permit the standard pricequantity characterization used in supply and demand analysis THE ECONOMICS OF INFORMATION Copyright ©2005 by South-Western, a division of Thomson Learning. All rights reserved. 1 Properties of Information 2 The Value of Information • Studying information also becomes difficult due to some technical properties of information • In many respects, lack of information does represent a problem involving uncertainty for a decision maker – the individual may not know exactly what the consequences of a particular action will be – it is durable and retains value after its use – it can be nonrival and nonexclusive • Better information can reduce uncertainty and lead to better decisions and higher utility • in this manner it can be considered a public good 3 4 1 The Value of Information The Value of Information • Assume an individual forms subjective opinions about the probabilities of two states of the world • Assume that information can be measured by the number of “messages” (m) purchased – “good times” (probability = g) and “bad times” (probability = b) – g and b will be functions of m • Information is valuable because it helps the individual revise his estimates of these probabilities 5 6 The Value of Information The Value of Information • First-order conditions for a constrained maximum are: • The individual’s goal will be to maximize E(U) = gU(W g) + bU(W b) L gU ' (Wg ) pg 0 Wg subject to I = pgW g + pbW b + pmm L bU ' (Wb ) pb 0 Wb • We need to set up the Lagrangian L = gU(W g) + bU(W b) + (I-pgW g-pbW b-pmm) 7 L I pgWg pbWb pm m 0 8 2 The Value of Information The Value of Information • First-order conditions for a constrained maximum are: • The first two equations show that the individual will maximize utility at a point where the subjective ratio of expected marginal utilities is equal to the price ratio (pg /pb) • The last equation shows the utilitymaximizing level of information to buy dWg dWb L gU ' (Wg ) bU ' (Wb ) m dm dm d g dWg d b U (Wg ) U (Wb ) pg dm dm dm dWb pb pm 0 dm 9 10 Asymmetry of Information Information and Insurance • The level of information that a person buys will depend on the price per unit • Information costs may differ significantly across individuals • There are a number of information asymmetries in the market for insurance • Buyers are often in a better position to know the likelihood of uncertain events – some may possess specific skills for acquiring information – some may have experience that is relevant – some may have made different former investments in information services 11 – may also be able to take actions that impact these probabilities 12 3 Moral Hazard Moral Hazard • Moral hazard is the effect of insurance coverage on individuals’ decisions to take activities that may change the likelihood or size of losses • Suppose a risk-averse individual faces the risk of a loss (l) that will lower wealth – parking an insured car in an unsafe area – choosing not to install a sprinkler system in an insured home – the probability of a loss is – this probability can be lowered by the amount the person spends on preventive measures (a) 13 Moral Hazard 14 Moral Hazard • The first-order condition for a maximum is • Wealth in the two states is given by E U (W1 ) (1 )U ' (W1 ) U (W2 ) U ' (W2 ) 0 a a a U ' (W2 ) (1 )U ' (W1 ) [U (W2 ) U (W1 )] a W1 = W 0 - a W2 = W 0 - a - l • The individual chooses a to maximize E(U) = E = (1-)U(W 1) + U(W 2) 15 – the optimal point is where the expected marginal utility cost from spending one additional dollar on prevention is equal to the reduction in the expected value of the utility loss that may be encountered in bad times 16 4 Behavior with Insurance and Perfect Monitoring Behavior with Insurance and Perfect Monitoring • Suppose that the individual may purchase insurance (premium = p) that pays x if a loss occurs • Wealth in each state becomes • The person can maximize expected utility by choosing x such that W 1 = W 2 W1 = W 0 - a - p W2 = W 0 - a - p - l + x • A fair premium would be equal to p = x E (1 )U ' (W1 )1 l U (W1 ) a a a U ' (W2 )1 l U (W2 ) 0 a a 17 Behavior with Insurance and Perfect Monitoring • Since W 1 = W 2, this condition becomes 1 l • The first-order condition is a – at the utility maximizing choice, the marginal cost of an extra unit of prevention should equal the marginal reduction in the expected loss provided by the extra spending – with full insurance and actuarially fair premiums, precautionary purchases still occur 19 at the optimal level 18 Moral Hazard • So far, we have assumed that insurance providers know the probability of a loss and can charge the actuarially fair premium – this is doubtful when individuals can undertake precautionary activities – the insurance provider would have to constantly monitor each person’s activities to determine the correct probability of loss 20 5 Moral Hazard Adverse Selection • In the simplest case, the insurer might set a premium based on the average probability of loss experienced by some group of people • Individuals may have different probabilities of experiencing a loss • If individuals know the probabilities more accurately than insurers, insurance markets may not function properly – no variation in premiums allowed for specific precautionary activities • each individual would have an incentive to reduce his level of precautionary activities – it will be difficult for insurers to set premiums based on accurate measures of expected loss 21 22 Adverse Selection W2 Adverse Selection W2 certainty line Suppose that one person has a probability of loss equal to H, while the other has a probability of loss equal to l certainty line Assume that two individuals have the same initial wealth (W*) and each face a potential loss of l W *- l G W*- l E W* F W1 E W* 23 Both individuals would prefer to move to the certainty line if premiums are actuarially fair W1 24 6 Adverse Selection W2 Adverse Selection The lines show the market opportunities for each person to trade W 1 for W2 by buying fair insurance certainty line F slope G W*- l E slope (1 H ) H (1 l ) l The low-risk person will maximize utility at point F, while the high-risk person will choose G W1 W* • If insurers have imperfect information about which individuals fall into low- and high-risk categories, this solution is unstable – point F provides more wealth in both states – high-risk individuals will want to buy insurance that is intended for low-risk individuals – insurers will lose money on each policy sold 25 26 Adverse Selection W2 Adverse Selection One possible solution would be for the insurer to offer premiums based on the average probability of loss W2 Point M is not an equilibrium because further trading opportunities exist for low-risk individuals certainty line F H G W*- l M E W* certainty line Since EH does not accurately reflect the true probabilities of each buyer, they may not fully insure and may choose a point such as M W1 F H G W*- l M UH N E W* 27 UL An insurance policy such as N would be unattractive to highrisk individuals, but attractive to low-risk individuals and profitable for insurers W1 28 7 Adverse Selection Adverse Selection • If a market has asymmetric information, the equilibria must be separated in some way – high-risk individuals must have an incentive to purchase one type of insurance, while low-risk purchase another Suppose that insurers offer policy G. High-risk individuals will opt for full insurance. W2 certainty line F G W*- l UH E W* Insurers cannot offer any policy that lies above UH because they cannot prevent high-risk individuals from taking advantage of it W1 29 Adverse Selection Adverse Selection The best policy that low-risk individuals can obtain is one such as J W2 certainty line F G W*- l J UH E W* 30 The policies G and J represent a separating equilibrium W1 31 • Low-risk individuals could try to signal insurers their true probabilities of loss – insurers must be able to determine if the signals are believable – insurers may be able to infer accurate probabilities by observing their clients’ market behavior – the separating equilibrium identifies an individual’s risk category 32 8 The Principal-Agent Relationship Adverse Selection • Market signals can be drawn from a number of sources • One important way in which asymmetric information may affect the allocation of resources is when one person hires another person to make decisions – the economic behavior must accurately reflect risk categories – the costs to individuals of taking the signaling action must be related to the probability of loss – patients hiring physicians – investors hiring financial advisors – car owners hiring mechanics – stockholders hiring managers 33 34 The Principal-Agent Relationship The Principal-Agent Relationship • In each of these cases, a person with less information (the principal) is hiring a more informed person (the agent) to make decisions that will directly affect the principal’s own well-being • Assume that we can show a graph of the owner’s (or manager’s) preferences in terms of profits and various benefits (such as fancy offices or use of the corporate jet) • The owner’s budget constraint will have a slope of -1 – each $1 of benefits reduces profit by $1 35 36 9 The Principal-Agent Relationship Profits The Principal-Agent Relationship If the manager is also the owner of the firm, he will maximize his utility at profits of * and benefits of b* * Profits The owner-manager maximizes profit because any other ownermanager will also want b* in benefits b* represents a true cost of doing business * U1 U1 Owner’s constraint b* Owner’s constraint Benefits b* Benefits 37 The Principal-Agent Relationship 38 The Principal-Agent Relationship • Suppose that the manager is not the sole owner of the firm • The new budget constraint continues to include the point b*, * – suppose there are two other owners who play no role in operating the firm – the manager could still make the same decision that a sole owner could) • $1 in benefits only costs the manager $0.33 in profits • For benefits greater than b*, the slope of the budget constraint is only -1/3 – the other $0.67 is effectively paid by the other owners in terms of reduced profits 39 40 10 The Principal-Agent Relationship The Principal-Agent Relationship Given the manager’s budget constraint, he will maximize utility at benefits of b** Profits Agent’s constraint * ** U2 Profits for the firm will be *** U1 *** Owner’s constraint b* b** Benefits • The firm’s owners are harmed by having to rely on an agency relationship with the firm’s manager • The smaller the fraction of the firm that is owned by the manager, the greater the distortions that will be induced by this relationship 41 Using the Corporate Jet 42 Using the Corporate Jet • A firm owns a fleet of corporate jets used mainly for business purposes • Suppose that all would-be applicants have the same utility function – the firm has just fired a CEO for misusing the corporate fleet U(s,j) = 0.1s0.5 + j where s is salary and j is jet use (0 or 1) • All applicants have job offers from other firms promising them a utility level of at most 2.0 • The firm wants to structure a management contract that provides better incentives for cost control 43 44 11 Using the Corporate Jet Using the Corporate Jet • Because jet use is expensive, = 800 (thousand) if j =0 and = 162 if j =1 • If the directors find it difficult to monitor the CEO’s jet usage, this could mean that the firm ends up with < 0 • The owner’s may therefore want to create a contract where the compensation of the new CEO is tied to profit – the directors will be willing to pay the new CEO up to 638 providing that they can guarantee that he will not use the corporate jet for personal use – a salary of more than 400 will just be sufficient to get a potential candidate to accept the job without jet usage 45 The Owner-Manager Relationship • Suppose that the gross profits of the firm depend on some specific action that a hired manager might take (a) net profits = ’ = (a) – s[(a)] • Both gross and net profits are maximized when /a = 0 – the owners’ problem is to design a salary structure that provides an incentive for the manager to choose a that maximizes 47 46 The Owner-Manager Relationship • The owners face two issues – they must know the agent’s utility function which depends on net income (IM) IM = s[(a)] = c(a) = c0 • where c(a) represents the cost to the manager of undertaking a – they must design the compensation system so that the agent is willing to take the job • this requires that IM 0 48 12 The Owner-Manager Relationship The Owner-Manager Relationship • One option would be to pay no compensation unless the manager chooses a* and to pay an amount equal to c(a*) + c0 if a* is chosen • Another possible scheme is s(a) = (a) – f, where f = (a) – c(a*) – c0 • The manager will choose a* and receive an income that just covers costs IM = s(a*) – c(a*) – c0 = (a*) – f – c(a*) – c0 = 0 • This compensation plan makes the agent the “residual claimant” to the firm’s profits – with this compensation package, the manager’s income is maximized by setting s(a)/a = /a = 0 49 50 Asymmetric Information Hidden Action • Models of the principal-agent relationship have introduced asymmetric information into this problem in two ways • The primary reason that the manager’s action may be hidden is that profits depend on random factors that cannot be observed by the firm’s owner • Suppose that profits depend on both the manager’s action and on a random variable (u) – it is assumed that a manager’s action is not directly observed and cannot be perfectly inferred from the firm’s profits • referred to as “hidden action” – the agent-manager’s objective function is not directly observed • referred to as “hidden information” 51 (a) = ’(a) + u where ’ represents expected profits 52 13 Hidden Action Hidden Information • Because owners observe only and not ’, they can only use actual profits in their compensation function • When the principal does not know the incentive structure of the agent, the incentive scheme must be designed using some initial assumptions about the agent’s motivation – a risk averse manager will be concerned that actual profits will turn out badly and may decline the job • The owner might need to design a compensation scheme that allows for profit-sharing – will be adapted as new information becomes available 53 Important Points to Note: 54 Important Points to Note: • Information is valuable because it permits individuals to increase the expected utility of their decisions • Information has a number of special properties that suggest that inefficiencies associated with imperfect and asymmetric information may be quite prevalent – individuals might be willing to pay something to acquire additional information – differing costs of acquisition – some aspects of a public good 55 56 14 Important Points to Note: Important Points to Note: • The presence of asymmetric information may affect a variety of market outcomes, many of which are illustrated in the context of insurance theory • If insurers are unable to monitor the behavior of insured individuals accurately, moral hazard may arise – being insured will affect the willingness to make precautionary expenditures – such behavioral effects can arise in any contractual situation in which monitoring costs are high – insurers may have less information about potential risks than do insurance purchasers 57 58 Important Points to Note: Important Points to Note: • Informational asymmetries can also lead to adverse selection in insurance markets • Asymmetric information may also cause some (principal) economic actors to hire others (agents) to make decisions for them – the resulting equilibria may often be inefficient because low-risk individuals will be worse off than in the full information case – market signaling may be able to reduce these inefficiencies 59 – providing the correct incentives to the agent is a difficult problem 60 15 Externality • An externality occurs whenever the activities of one economic agent affect the activities of another economic agent in ways that are not reflected in market transactions Chapter 19 EXTERNALITIES AND PUBLIC GOODS – chemical manufacturers releasing toxic fumes – noise from airplanes – motorists littering roadways Copyright ©2005 by South-Western, a division of Thomson Learning. All rights reserved. 1 Interfirm Externalities 2 Beneficial Externalities • Consider two firms, one producing good x and the other producing good y • The production of x will have an external effect on the production of y if the output of y depends not only on the level of inputs chosen by the firm but on the level at which x is produced • The relationship between the two firms can be beneficial – two firms, one producing honey and the other producing apples y = f(k,l;x) 3 4 1 Externalities in Utility Public Goods Externalities • Externalities can also occur if the activities of an economic agent directly affect an individual’s utility • Public goods are nonexclusive – externalities can decrease or increase utility • It is also possible for someone’s utility to be dependent on the utility of another – once they are produced, they provide benefits to an entire group – it is impossible to restrict these benefits to the specific groups of individuals who pay for them utility = US(x1,…,xn;UJ) 5 Externalities and Allocative Inefficiency 6 Externalities and Allocative Inefficiency • Externalities lead to inefficient allocations of resources because market prices do not accurately reflect the additional costs imposed on or the benefits provided to third parties • We can show this by using a general equilibrium model with only one individual • Suppose that the individual’s utility function is given by utility = U(xc,yc) where xc and yc are the levels of x and y consumed • The individual has initial stocks of x* and y* 7 – can consume them or use them in production 8 2 Externalities and Allocative Inefficiency Externalities and Allocative Inefficiency • Assume that good x is produced using only good y according to xo = f(yi) • Assume that the output of good y depends on both the amount of x used in the production process and the amount of x produced • For example, y could be produced downriver from x and thus firm y must cope with any pollution that production of x creates • This implies that g1 > 0 and g2 < 0 yo = g(xi,xo) 9 Externalities and Allocative Inefficiency 10 Finding the Efficient Allocation • The economic problem is to maximize utility subject to the four constraints listed earlier • The Lagrangian for this problem is • The quantities of each good in this economy are constrained by the initial stocks available and by the additional production that takes place L = U(xc,yc) + 1[f(yi) - xo] + 2[g(xi,xo) - yo] xc + xi = xo + x* + 3(xc + xi - xo - x*) + 4(yc + yi - yo - y*) yc + yi = xo + y* 11 12 3 Finding the Efficient Allocation • The six first-order conditions are • Taking the ratio of the first two, we find MRS = U1/U2 = 3/4 L/xc = U1 + 3 = 0 • The third and sixth equation also imply that L/yc = U2 + 4 = 0 L/xi = 2g1 + 3 = 0 MRS = 3/4 = 2g1/2 = g1 L/yi = 1fy + 4 = 0 L/xo = -1 + 2g2 - 3 = 0 L/yo = -2 - 4 = 0 Finding the Efficient Allocation 13 • Optimality in y production requires that the individual’s MRS in consumption equals the marginal productivity of x in the production of y 14 Finding the Efficient Allocation Finding the Efficient Allocation • To achieve efficiency in x production, we must also consider the externality this production poses to y • Combining the last three equations gives • This equation requires the individual’s MRS to equal dy/dx obtained through x production MRS = 3/4 = (-1 + 2g2)/4 = -1/4 + 2g2/4 – 1/fy represents the reciprocal of the marginal productivity of y in x production – g2 represents the negative impact that added x production has on y output • allows us to consider the externality from x production MRS = 1/fy - g2 15 16 4 Inefficiency of the Competitive Allocation Inefficiency of the Competitive Allocation • Reliance on competitive pricing will result in an inefficient allocation of resources • A utility-maximizing individual will opt for MRS = Px/Py • But the producer of x would choose y input so that Py = Pxfy Px/Py = 1/fy and the profit-maximizing producer of y would choose x input according to Px = Pyg1 17 Production Externalities • This means that the producer of x would disregard the externality that its production poses for y and will overproduce x 18 Production Externalities • The downstream firm has a similar production function but its output may be affected by chemicals that firm x pours in the river • Suppose that two newsprint producers are located along a river • The upstream firm has a production function of the form x = 2,000lx0.5 y = 2,000ly0.5(x - x0) (for x > x0) y = 2,000ly0.5 (for x x0) where x0 represents the river’s natural capacity for pollutants 19 20 5 Production Externalities Production Externalities • Assuming that newsprint sells for $1 per foot and workers earn $50 per day, firm x will maximize profits by setting this wage equal to the labor’s marginal product 50 p x 1,000lx0.5 lx • lx* = 400 • If = 0 (no externalities), ly* = 400 21 22 Production Externalities Production Externalities • Suppose that these two firms merge and the manager must now decide how to allocate the combined workforce • If one worker is transferred from x to y, output of x becomes • If = -0.1 and x0 = 38,000, firm y will maximize profits by 50 p • When firm x does have a negative externality ( < 0), its profit-maximizing decision will be unaffected (lx* = 400 and x* = 40,000) • But the marginal product of labor will be lower in firm y because of the externality y 1,000ly0.5 ( 40,000 38,000)0.1 ly 50 468ly0.5 x = 2,000(399)0.5 = 39,950 • Because of the externality, ly* = 87 and y output will be 8,723 and output of y becomes y = 2,000(88)0.5(1,950)-0.1 = 8,796 23 24 6 Production Externalities Production Externalities • If firm x was to hire one more worker, its own output would rise to • Total output increased with no change in total labor input • The earlier market-based allocation was inefficient because firm x did not take into account the effect of its hiring decisions on firm y x = 2,000(401)0.5 = 40,050 – the private marginal value product of the 401st worker is equal to the wage 25 • But, increasing the output of x causes the output of y to fall (by about 21 units) • The social marginal value product of the additional worker is only $29 26 Solutions to the Externality Problem Solutions to the Externality Problem Price • The output of the externality-producing activity is too high under a marketdetermined equilibrium • Incentive-based solutions to the externality problem originated with Pigou, who suggested that the most direct solution would be to tax the externality-creating entity MC’ Market equilibrium will occur at p1, x1 S = MC If there are external costs in the production of x, social marginal costs are represented by p1 MC’ D Quantity of x 27 x1 28 7 Solutions to the Externality Problem Price MC’ S = MC p2 tax A Pigouvian Tax on Newsprint A tax equal to these additional marginal costs will reduce output to the socially optimal level (x2) The price paid for the good (p2) now reflects all costs • A suitably chosen tax on firm x can cause it to reduce its hiring to a level at which the externality vanishes • Because the river can handle pollutants with an output of x = 38,000, we might consider a tax that encourages the firm to produce at that level D Quantity of x x2 29 A Pigouvian Tax on Newsprint • Output of x will be 38,000 if lx = 361 • Thus, we can calculate t from the labor demand condition 30 Taxation in the General Equilibrium Model • The optimal Pigouvian tax in our general equilibrium model is to set t = -pyg2 (1 - t)MPl = (1 - t)1,000(361)-0.5 = 50 – the per-unit tax on x should reflect the marginal harm that x does in reducing y output, valued at the price of good y t = 0.05 • Therefore, a 5 percent tax on the price firm x receives would eliminate the externality 31 32 8 Taxation in the General Equilibrium Model Taxation in the General Equilibrium Model • With the optimal tax, firm x now faces a net price of (px - t) and will choose y input according to • The Pigouvian tax scheme requires that regulators have enough information to set the tax properly py = (px - t)fy • The resulting allocation of resources will achieve – in this case, they would need to know firm y’s production function MRS = px/py = (1/fy) + t/py = (1/fy) - g2 33 34 Pollution Rights Pollution Rights • An innovation that would mitigate the informational requirements involved with Pigouvian taxation is the creation of a market for “pollution rights” • Suppose that firm x must purchase from firm y the rights to pollute the river they share • The net revenue that x receives per unit is given by px - r, where r is the payment the firm must make to firm y for each unit of x it produces • Firm y must decide how many rights to sell firm x by choosing x output to maximize its profits – x’s choice to purchase these rights is identical to its output choice y = pyg(xi,xo) + rxo 35 36 9 Pollution Rights The Coase Theorem • The first-order condition for a maximum is y/xo = pyg2 + r = 0 r = -pyg2 • The equilibrium solution is identical to that for the Pigouvian tax – from firm x’s point of view, it makes no difference whether it pays the fee to the government or to firm y • The key feature of the pollution rights equilibrium is that the rights are welldefined and tradable with zero transactions costs • The initial assignment of rights is irrelevant – subsequent trading will always achieve the same, efficient equilibrium 37 The Coase Theorem 38 The Coase Theorem • Suppose that firm x is initially given xT rights to produce (and to pollute) • Profits for firm y are given by y = pyg(xi,xo) - r(xT - xo) – it can choose to use these for its own production or it may sell some to firm y • Profit maximization in this case will lead to precisely the same solution as in the case where firm y was assigned the rights • Profits for firm x are given by x = pxxo + r(xT - xo) = (px - r)xo + rxT x = (px - r)f(yi) + rxT 39 40 10 The Coase Theorem Attributes of Public Goods • The independence of initial rights assignment is usually referred to as the Coase Theorem • A good is exclusive if it is relatively easy to exclude individuals from benefiting from the good once it is produced • A good is nonexclusive if it is impossible, or very costly, to exclude individuals from benefiting from the good – in the absence of impediments to making bargains, all mutually beneficial transactions will be completed – if transactions costs are involved or if information is asymmetric, initial rights assignments will matter 41 Attributes of Public Goods 42 Attributes of Public Goods • A good is nonrival if consumption of additional units of the good involves zero social marginal costs of production • Some examples of these types of goods include: Exclusive Yes Rival No 43 Yes Hot dogs, cars, houses Bridges, swimming pools No Fishing grounds, clean air National defense, mosquito control 44 11 Public Goods and Resource Allocation Public Good • A good is a pure public good if, once produced, no one can be excluded from benefiting from its availability and if the good is nonrival -- the marginal cost of an additional consumer is zero 45 Public Goods and Resource Allocation • Resulting utilities for these individuals are UA[x,(yA* - ysA)] UB[x,(yB* - ysB)] • The level of x enters identically into each person’s utility curve – it is nonexclusive and nonrival • each person’s consumption is unrelated to what he contributes to production • each consumes the total amount produced 47 • We will use a simple general equilibrium model with two individuals (A and B) • There are only two goods – good y is an ordinary private good • each person begins with an allocation (yA* and yB*) – good x is a public good that is produced using y x = f(ysA + ysB) 46 Public Goods and Resource Allocation • The necessary conditions for efficient resource allocation consist of choosing the levels of ysA and ysB that maximize one person’s (A’s) utility for any given level of the other’s (B’s) utility • The Lagrangian expression is L = UA(x, yA* - ysA) + [UB(x, yB* - ysB) - K] 48 12 Public Goods and Resource Allocation Public Goods and Resource Allocation • The first-order conditions for a maximum are L/ysA = U1Af’ - U2A + U1Bf’ = 0 L/ysB = U1Af’ - U2B + U1Bf’ = 0 • We can now derive the optimality condition for the production of x • From the initial first-order condition we know that U1A/U2A + U1B/U2B = 1/f’ • Comparing the two equations, we find MRSA + MRSB = 1/f’ U2B = U2A 49 • The MRS must reflect all consumers because all will get the same benefits 50 Failure of a Competitive Market Failure of a Competitive Market • Production of x and y in competitive markets will fail to achieve this allocation • For public goods, the value of producing one more unit is the sum of each consumer’s valuation of that output – with perfectly competitive prices px and py, each individual will equate his MRS to px/py – the producer will also set 1/f’ equal to px/py to maximize profits – the price ratio px/py will be too low – individual demand curves should be added vertically rather than horizontally • Thus, the usual market demand curve will not reflect the full marginal valuation • it would provide too little incentive to produce x 51 52 13 Inefficiency of a Nash Equilibrium Inefficiency of a Nash Equilibrium • Suppose that individual A is thinking about contributing sA of his initial y endowment to the production of x • The utility maximization problem for A is then • The first-order condition for a maximum is choose sA to maximize UA[f(sA + sB),yA - sA] U1Af’ - U2A = 0 U1A/U2A = MRSA = 1/f’ • Because a similar argument can be applied to B, the efficiency condition will fail to be achieved – each person considers only his own benefit 53 The Roommates’ Dilemma The Roommates’ Dilemma • Suppose two roommates with identical preferences derive utility from the number of paintings hung on their walls (x) and the number of granola bars they eat (y) with a utility function of Ui(x,yi) = x1/3yi2/3 54 (for i=1,2) • Assume each roommate has $300 to spend and that px = $100 and py = $0.20 55 • We know from our earlier analysis of Cobb-Douglas utility functions that if each individual lived alone, he would spend 1/3 of his income on paintings (x = 1) and 2/3 on granola bars (y = 1,000) • When the roommates live together, each must consider what the other will do – if each assumed the other would buy paintings, x = 0 and utility = 0 56 14 The Roommates’ Dilemma The Roommates’ Dilemma • If person 1 believes that person 2 will not buy any paintings, he could choose to purchase one and receive utility of • We can show that this solution is inefficient by calculating each person’s MRS U1(x,y1) = 11/3(1,000)2/3 = 100 MRSi while person 2’s utility will be U i / x y i U i / y i 2 x • At the allocations described, U2(x,y2) = 11/3(1,500)2/3 = 131 • Person 2 has gained from his free-riding position MRS1 = 1,000/2 = 500 MRS2 = 1,500/2 = 750 57 58 The Roommates’ Dilemma The Roommates’ Dilemma • Since MRS1 + MRS2 = 1,250, the roommates would be willing to sacrifice 1,250 granola bars to have one additional painting • To calculate the efficient level of x, we must set the sum of each person’s MRS equal to the price ratio – an additional painting would only cost them 500 granola bars – too few paintings are bought MRS1 MRS2 y1 y 2 y1 y 2 px 100
2x 2x 2x py 0.20 • This means that y1 + y2 = 1,000x 59 60 15 Lindahl Pricing of Public Goods The Roommates’ Dilemma • Substituting into the budget constraint, we get • Swedish economist E. Lindahl suggested that individuals might be willing to be taxed for public goods if they knew that others were being taxed 0.20(y1 + y2) + 100x = 600 x=2 y1 + y2 = 2,000 • The allocation of the cost of the paintings depends on how each roommate plays the strategic financing 61 game Lindahl Pricing of Public Goods 62 Lindahl Pricing of Public Goods • Suppose that individual A would be quoted a specific percentage (A) and asked the level of a public good (x) he would want given the knowledge that this fraction of total cost would have to be paid • The person would choose the level of x which maximizes utility = UA[x,yA*- Af -1(x)] – Lindahl assumed that each individual would be presented by the government with the proportion of a public good’s cost he was expected to pay and then reply with the level of public good he would prefer 63 • The first-order condition is given by U1A - AU2B(1/f’)=0 MRSA = A/f’ • Faced by the same choice, individual B would opt for the level of x which satisfies MRSB = B/f’ 64 16 Lindahl Pricing of Public Goods Shortcomings of the Lindahl Solution • An equilibrium would occur when A+B = 1 • The incentive to be a free rider is very strong – the level of public goods expenditure favored by the two individuals precisely generates enough tax contributions to pay for it MRSA + MRSB = (A + B)/f’ = 1/f’ – this makes it difficult to envision how the information necessary to compute equilibrium Lindahl shares might be computed • individuals have a clear incentive to understate their true preferences 65 66 Important Points to Note: Important Points to Note: • Externalities may cause a misallocation of resources because of a divergence between private and social marginal cost • If transactions costs are small, private bargaining among the parties affected by an externality may bring social and private costs into line – traditional solutions to this divergence includes mergers among the affected parties and adoption of suitable Pigouvian taxes or subsidies – the proof that resources will be efficiently allocated under such circumstances is sometimes called the Coase theorem 67 68 17 Important Points to Note: Important Points to Note: • Public goods provide benefits to individuals on a nonexclusive basis no one can be prevented from consuming such goods • Private markets will tend to underallocate resources to public goods because no single buyer can appropriate all of the benefits that such goods provide – such goods are usually nonrival in that the marginal cost of serving another user is zero 69 70 Important Points to Note: • A Lindahl optimal tax-sharing scheme can result in an efficient allocation of resources to the production of public goods – computation of these tax shares requires substantial information that individuals have incentives to hide 71 18
