Microeconomics IV. First Semster, Course

Transcription

1 Microeconomics IV Part II. General Professor: Marc Teignier Baqué Universitat de Barcelona, Facultat de Ciències Econòmiques and Empresarials, Departament de Teoria Econòmica First Semster, Course

2 PART II. General So far, partial equilibrium analysis: Each market studied independently; only own price changes taken into account; ignoring eects of changes in other markets. In general, however, demand and supply of dierent markets interact with each other: Goods may be complements or substitutes; Consumers' income may depend on prices of many markets. In this part, we study the equilibrium of all markets simultaneously. Simplications: perfect competition, one or two goods. First, pure exchange economies. Second, introduction of production decisions.

3 Part II Contents Net Demand Slutsky 1. Consumers with initial endowments Net demand 1.3 The Slutsky equation revisited 2. General equilibrium in a pure exchange economy 3. General equilibrium in a economy 4. General equilibrium in an economy with production

4 TOPIC 5. General equilibrium in an exchange economy Net Demand Slutsky So far, consumers' income taken as exogenous and independent of prices. In reality, consumers' income coming from exchange by sellers and buyers. How are incomes generated? How does the value of income depend upon commodity prices? How can we put all this together to explain better how price changes aect demands?

5 Net Demand Slutsky 5.1 Consumers with initial endowments In this chapter, consumers get income from endowments. This makes the budget set denition change slightly. Denition The list of resource units with which a consumer starts is her endowment, denoted by ω = (ω 1,ω 2 ). Denitions Given p 1 and p 2, the budget constraint for a consumer with an endowment ω = (ω 1,ω 2 ) is p 1 x 1 + p 2 x 2 = p 1 ω 1 + p 2 ω 2. and the budget set is formally dened as {(x 1, x 2 ) x 1 0, x 2 0 and p 1 x 1 + p 2 x 2 p 1 ω 1 + p 2 ω 2 }.

6 Net Demand Slutsky and Budget Sets Graphically, the endowment point is always on the budget constraint. Hence, price changes pivot the constraint around the endowment point. x 2 p 1x 1 p 2x 2 p 1 1 p 2 2 Budget Set before price change Budget Set after the price change p1 ' x1 p ' 2x2 p1 ' 1 p ' 2 2 x 1

7 Net Demand Slutsky Net demand Denition The dierence between nal consumption and initial endowment of a given good i, x i of good i. The sum of the values of net demands is zero: ω i, is called net demand p 1 x 1 + p 2 x 2 = p 1 ω 1 + p 2 ω 2 p 1 (x 1 ω 1 ) + p 2 (x 2 ω 2 ) = 0. x 2 At prices (p 1,p 2 ) the consumer sells units of good 2 to acquire more of good 1. The net demand of good 1 is, therefore, positive and the net demand dof good 2 is negative. x 2 * x 2 At prices (p 1,p 2 ) the consumer sells units of good 1 to acquire more units of good 2. The net demand of good 1 is, therefore, negative,and the net demand of good 2 is positive. ii x 2 * x 1 * 1 ' 1 ' ' 1 ' 2 2 x 1 px px p p p1( x1 1) p2( x2 2) 0 x 1 * x 1

8 Net Demand Slutsky Price oer curve Denition Price-oer curve contains all the utility-maximizing gross demands for which the endowment is exchanged. x 2 Sell good 1, buy good 2 Buy good 1, sell good 2 x 1

9 Net Demand Slutsky Slutsky equation revisited In an endowment economy, the overall change in demand caused by a price change is the sum of a pure substitution eect, an (ordinary) income eect, and an endowment income eect. Pure Substitution Eect: eect of relative prices change. Income Eect: eect of original bundle cost change. Endowment Income Eect: change in demand due only to the change in endowment value. Price change from (p 1,p 2 ) to (p 1, p 2 ): x 2 Pure substitution effect Ordinary income effect Endowment income effect x 2 2 x 2

10 Net Demand Slutsky Slutsky equation revisited (2) Overall change in demand of normal good (demand increases with income) caused by own price change: When income is exogenous, both the substitution and (ordinary) income eects increase demand after an own-price fall; hence, a normal good's ordinary demand curve slopes down (thus, Law of Downward-Sloping Demand always applies to normal goods when income is exogenous). When income is given by initial endowments, endowment-income eect decreases demand if consumer supplies that good (negative net demand); thus, if the endowment income eect osets the substitution and the (ordinary) income eects, the demand function could be upward-sloping!

11 Part II Contents Edgeworth Core Walras 1. Consumers with initial endowments 2. General equilibrium in a pure exchange economy 2.1 Edgeworth box 2.2 The Core 2.3 Competitive equilibrium 2.4 theorems 2.5 Walras law 3. General equilibrium in a economy 4. General equilibrium in an economy with production

12 5.2 General equilibrium in a pure exchange economy Edgeworth Core Walras Two consumers, A and B, two goods, 1 and 2. of goods 1 and 2: ( ω A = ω1 A,ω 2 A ), ω B = ( ) ω1 B,ω 2 B. Given a price vector (p1,p 2 ), consumers choose their favorite aordable allocation (as in topic 1): p 1 x A 1 + p 2x A 2 p 1ω A 1 + p 2ω A 2 p 1 x B 1 + p 2x B 2 p 1ω B 1 + p 2ω B 2 Prices must be such that allocations chosen are feasible: x A 1 + x B 1 ω A 1 + ω B 1, x A 2 + x B 2 ω A 2 + ω B 2

13 Edgeworth Core Walras Edgeworth Box Edgeworth box is diagram showing all possible allocations of the available quantities of goods 1 and 2 between the two consumers. The dimensions of the box are the quantities available of the goods. The allocations depicted are the feasible allocations. Height = A B 2 2 A B Width = 1 1

14 Edgeworth Core Walras Endowment allocation The endowment allocation is the before-trade allocation: B ω 1 A O B ω 2 B ω 2 A ω 2 + Endowment B A ω2 ω 2 allocation O A ω A ω 1 A B 1 + ω1

15 Edgeworth Core Walras Feasible reallocations Which reallocation will consumers choose? Feasible. Pareto-improving over the endowment allocation. An allocation is feasible if and only if x A 1 + x B 1 ω A 1 + ω B 1 x A 2 + x B 2 ω A 2 + ω B 2 All points in the box, including the boundary, represent feasible allocations of the combined endowments: A ω 2 + B ω 2 x A 2 O A x B 1 O B x B 2 x1 A A B ω1 + ω1

16 Edgeworth Core Walras Pareto-improving allocations An allocation is Pareto-improving over the endowment allocation if it improves the welfare of a consumer without reducing the welfare of another. Preferences of consumers A and B: x A 2 Preferences consumer A B x 1 Preferences of consumer B 1 B O B B 2 2 A O A A 1 x 1 A B x 2

17 Edgeworth Core Walras Pareto-improving allocations An allocation is Pareto-improving over the endowment allocation if it improves the welfare of a consumer without reducing the welfare of another. 1 B O B A 2 O A Set of Pareto improving allocations 1 A B 2 Since each consumer can refuse to trade, the only possible outcomes from exchange are Pareto-improving allocations.

18 Edgeworth Core Walras Contract curve An allocation is Pareto-optimal if the only way one consumer's welfare can be increased is to decrease the welfare of the other consumer. The set of all Pareto-optimal allocations is called contract curve. Pareto optimal allocations are marked by. Convex indifference curves are tangent at. 1 B O B A 2 O A The contract tcurve 1 A B 2

19 Edgeworth Core Walras The Core The core is the set of all Pareto-optimal allocations that are welfare-improving for both consumers relative to their own endowments. 1 B O B A 2 O A A The Core: Pareto optimal 1 trades not blocked by A or B. B 2 Rational trade should achieve a core allocation.

20 Edgeworth Core Walras Trade in competitive markets Specic core alloation achieved depends upon the manner in which trade is conducted. In perfectly competitive markets, each consumer is a price-taker trying to maximize her own utility given (p 1, p 2 ) and her own endowment: A x A 2 Consumer A optimization px 1 1 A px 2 A 2 p1 1 A p2 2 A A x * 2 2 A O A x * A A x1 A Similarly for consumer B.

21 Edgeworth Core Walras Trade in competitive markets At equilibrium prices p 1 and p 2, both consumers maximize their own utility and both markets clear: x A 1 + x B 1 = ω A 1 + ω B 1 x A 2 + x B 2 = ω A 2 + ω B 2 Budget constraint for consumer B B x * 1 B 1 O B A x * 2 A 2 O A A x * 1 A 1 allocation Budget constraint for consumer A B x * 2 B 2

22 Edgeworth Core Walras First fundamental theorem of welfare economics Theorem Given that consumers' preferences are well-behaved, trading in perfectly competitive markets implements a Pareto-optimal allocation of the economy's endowment. Note: Indierence curves are tangent, which implies that the equilibrium allocation is Pareto optimal.

23 Edgeworth Core Walras Second fundamental theorem of welfare economics Theorem Given that consumers' preferences are well-behaved, for any Pareto-optimal allocation, there are prices and an allocation of the total endowment that makes the Pareto-optimal allocation implementable by trading in competitive markets. In other words, any Pareto-optimal allocation can be achieved by trading in competitive markets provided that endowments are rst appropriately rearranged. L assignació Pareto òptima no es pot implementar a través del mercat competitiu desde el punt de dotació inicial però si desde el punt de dotació inicial. O B O A

24 Edgeworth Core Walras Walras' Law Theorem If consumer's preferences are well-behaved, so that for any positive prices (p 1, p 2 ) consumers spend all their budget, the summed market value of excess demands is zero. This is Walras' Law. p 1 x A 1 + p 2 x A 2 = p 1 ω A 1 + p 2 ω A 2 p 1 (x A 1 + x B 1 ω A 1 ω B 1 p 1 x B 1 + p 2 x B 2 = p 1 ω B 1 + p 2 ω B 2 ) ( ) + p 2 x2 A + x2 B ω2 A ω2 B = 0

25 Edgeworth Core Walras Implications of Walras' Law One implication of Walras' Law for a two-commodity exchange economy is that if one market is in equilibrium then the other market must also be in equilibrium. p 1 (x A 1 + x B 1 ω A 1 ω B 1 )+p 2 ( x A 2 + x B 2 ω A 2 ω B 2 ) = 0 If x A 1 + x B 1 = ω A 1 + ω B 1, then x A 2 + x B 2 = ω A 2 + ω B 2. Another implication of Walras' Law for a two-commodity exchange economy is that an excess supply in one market implies an excess demand in the other market. p 1 (x A 1 + x B 1 ω A 1 ω B 1 )+p 2 ( x A 2 + x B 2 ω A 2 ω B 2 ) = 0 If x A 1 + x B 1 < ω A 1 + ω B 1, then x A 2 + x B 2 > ω A 2 + ω B 2.

26 Part II Contents Setup 1. Consumers with initial endowments 2. General equilibrium in a pure exchange economy 3. General equilibrium in a economy 3.1 economy 3.2 Competitive equilibrium 3.3 theorems 4. General equilibrium in an economy with production

27 Setup 6.1 General equilibrium in a Robinson Cruose economy Add input and output markets, rms' technologies. 's Economy: One agent:. Endowment: a xed quantity of time. Decision: use time for labor (production of coconuts) or leisure. Technology: coconuts are obtained from labor according to the production function C = f (L). Coconuts function 0 Feasible production plans 24 Labor (hours)

28 Setup 's preferences Indierence curves in the leisure-coconut diagram: coconut is a good, leisure is a good: Coconuts More preferred 0 24 Leisure (hours) Indierence curves in the labor-coconuts diagram: coconut is a good, labor is a bad. Coconuts More preferred 0 24 Labor (hours)

29 Setup 's choice Robinson chooses time allocation and, as a result, his consumption of coconuts: Coconuts MRS = MP L C* function Labor Output t Leisure 0 L* Labor (hours) Leisure (hours)

30 Competitive equilibrium in the Robinson economy Setup Robinson esquizofrenia: We rst consider Robinson as a prot-maximizing rm, who takes prices as given and decides how much hours to hire and how much to produce. Then, we consider Robinson as a utility-maximizing consumer who gets the rm prots and decides his hours of work and his consumption of coconuts. Let p be the coconuts price and w the wage rate. Use coconuts as the numeraire good; i.e. price of a coconut = 1.

31 Setup Robinson as a rm Optimization problem of the rm: given w, choose labor demand and coconut supply to maximize prots: maxπ = C wl = f (L) wl MP (L ) = w L Labor demanded: L, output supplied: C = f (L ). Graphically, rm demands L such that production function tangent to isoprot line: Coconuts C* * w = MP L Isoprofit line: * C * wl * function 0 L* 24 Labor (hours)

32 Setup Robinson as consumer Optimization problem of the consumer: choose labor supply and coconut demand to maximize utility subject to the budget constraint: max U (C,L) s.t. C = π + wl U (C,L)/ L C,L U (C,L)/ C = w Labor supplies: L, coconuts demanded: C. Graphically, consumer chooses C and L such that the indierence curve is tangent to the budget constraint: Coconuts C* * MRS = w Budget constraint: C * wl. 0 L* 24 Labor (hours)

33 Setup Market equilibrium In equilibrium, wage rate must be such that quantity labor demanded = quantity labor supplied (quantity output supplied = quantity output demanded) Coconuts MRS = w = MP L C* * 0 L* 24 Labor (hours)

34 Setup First Fundamental Theorem of Economics Theorem If consumers' preferences are convex and there are no externalities in consumption or production, a competitive market equilibrium is Pareto ecient. Pareto eciency: MRS = MP: Competitive equilibrium achieves Pareto eciency: w is the common slope of the isprot line and the budget constraint. Coconuts MRS = MP 0 24 Labor (hours)

35 Second Fundamental Theorem of Economics Setup Theorem If consumers' preferences are convex, rms' technologies are convex, and there are no externalities in consumption or production any Pareto ecient economic state can be achieved as a competitive market equilibrium.

36 Setup Non-convex technologies The First still holds if rms have non-convex technologies since it does not rely upon rms' technologies being convex. The Second Theorem does not hold if rms have non-convex technologies. Coconuts MRS = MP L If competitive i equilibrium i exists, the common slope is the relative wage rate w that t implements the Pareto efficient plan by decentralized pricing. Coconuts MRS = MP L. This Pareto optimal allocation cannot be implemented by a competitive equilibrium Labor (hours) 0 24 Labor (hours)

37 Part II Contents PPF Eciency 1. Consumers with initial endowments 2. General equilibrium in a pure exchange economy 3. General equilibrium in a economy 4. General equilibrium in an economy with production 4.1 The Possibilities and the comparative advantage 4.2 Eciency 4.3 Competitive equilibrium

38 PPF Eciency 6.2 General equilibrium in an economy with production In this section, introduction more inputs/goods into the GE analysis. Denition. The Possibilities set represents the production combination that are feasible given the technology. The upper bound is called Possibilities Frontier (PPF). Cocos Frontera de possiblitats de producció (FPP) Conjunt de possiblitats de producció Peixos

39 PPF Eciency The marginal rate of transformation Denition. The slope of the Possibilities set is the Marginal Rate of Transformation. Thus, it gives the extra amount we can produce of a good if we give up a unit of the other good. Cocos Peixos An example: C = 2L c, P = 3L p, L c + L p = 10 P 3 + C 2 = 10, RMT = 2 3

40 The combined production possibilities set Let's introduce a new worker in the economy: Friday (V ). Suppose the abilities of Friday are the following: PPF Eciency C = 5L c, P = 5 2 L p, L c + L p = 10 2P + C = 10, RMT = C C 50 Viernes P 25 P

41 PPF Eciency The comparative advantage Robinson has a comparative advantage in the production of sh, Friday on the coconuts production: Robinson has to give up 2/3 of coconuts for each extra sh produced. Friday has to give up 2 coconuts for each extra sh produced. The combined production possibilities set is the amount is the total feasible production by both workers given the existing technology As we add more workers, the PPF keeps smoothing out: C C P P

42 Pareto eciency The PPF shows the technically eciency allocations. Whare the Pareto ecient allocations? Given a point (P, C ) on the PPF, the Pareto-ecient allocation (in terms of consumption) is the one where the indierence curves are tangent: PPF Eciency Cocos C P' V O V C RC C V O RC P' RC P' Peix

43 Pareto eciency (2) What are the allocations that are Pareto ecient also in terms of production? If the MRS are not equal to the MRT, then it is possible to increase the welfare of one consumer without changing the other consumer welfare by changin the production point. PPF Eciency Cocos C O V P' V O V C V O RC P' ' Peixos

44 PPF Eciency Pareto eciency (3) Thus, the Pareto ecient allocations are where the MRS of the two consumers are equal to the MRT: Cocos C P V MRS RC = MRS V = MRT MRS = MRPT O V C RC C V O RC PRC P Peixos

45 PPF Eciency Competitive equilibrium The competitive equilibrium in this economy consists on a situation where: Consumers, Robinson and Friday, maximize their utility. Firms maximize their benets. All markets clear (labor, sh, coconuts). Optimization problems: Consumers: max C i,l i U (C i, P i ) s.a. p c C i + p f F i = π i + w i L i, i = RC, F Producers: max L RC,L V π = p c C + p f F w RC L RC w F L F

46 Producer optimization The prot maximization occurs when MRT = p f p c PPF Eciency Cocos Maximització beneficis p Pendent = p p c Pi Peixos

47 PPF Eciency Eciency of the competitive equilibrium When consumers optimize, they choose the basket such that MRS i = p f p c, i = RC, F Hence, the decentralized allocation of resources in a competitive equilibrium is ecient! Cocos C P V p RMS p O V p p RMT C RC C V O RC PRC P Peixos

48 THE END