Lecture 2A: General Equilibrium

Transcription

1 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm Econ Urban Economics Lecture A: General Equilibrium Instructor: Hiroki Watanabe Spring Hiroki Watanabe 1 / 79 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm 1 Introduction General Equilibrium Positive & Normative Analysis Pure Exchange Economy Edgeworth Box 3 Adding Preferences Represent Preferences on the Edgeworth Box Edgeworth Box Dos and Don ts Normative Analysis: Pareto Optimal Allocation ontract urve Positive Analysis: Equilibrium Utility Maximization Problem Finding an Equilibrium Allocation First Fundamental heorem of Welfare Economics 7 Market Mechanism Excess Demand & Excess Supply Price hange to lear Both Markets Second Fundamental heorem of Welfare Economics 9 Summary 11 Hiroki Watanabe / 79 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm General Equilibrium Partial equilibirum analysis: 1 Liz consumes cheesecake and tea L = ( L, L ). Her demand function is φ L (p, p, m L ). 3 She will respond to a rise in p by reducing her quantity demanded L (law of demand). his is what we have learned in 1. Is this the end of the story? 11 Hiroki Watanabe 3 / 79

2 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm General Equilibrium No. 3 She will respond to a rise in p by reducing her quantity demanded L (law of demand). She will shift her consumption from cheesecake to tea. hat will raise p, which, in turn, will raise L = φl (p, p, m L ) if they are substitutes. Partial equilibrium analysis doesn t deal with this ricochet effect. 11 Hiroki Watanabe / 79 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm General Equilibrium But the effect does exist. General equilibrium models analyze several markets all at once, addressing interactive relationship among different markets. Not only p but p is also determined within the model. (Note that we can still see how L responds to p in the partial equilibrium analysis as long as p is exogenous). 11 Hiroki Watanabe / 79 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm General Equilibrium What does general equilibrium analysis have to do with Econ? Partial equilibrium analysis cannot describe inter-locational relationships in the economy. Locational choice usually sinks in in the form of commuting cost and/or transportation cost. hat necessitates a simultaneous analysis of at least two markets (one for land and other for composite goods). More on this in the following lectures. Wait and see. 11 Hiroki Watanabe / 79

3 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm Positive & Normative Analysis Definition 1.1 (Positive and Normative Analysis) 1 Positive analysis describes the condition the economy is in. opics include equilibrium and price. Normative analysis: describes the condition the economy should be in. opics include efficiency. 11 Hiroki Watanabe 7 / 79 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm Pure Exchange Economy Pure exchange economy: here is no firm. wo commodities with two consumers. Instead of income, the agents are endowed with the two commodities. 11 Hiroki Watanabe / 79 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm 1 Introduction General Equilibrium Positive & Normative Analysis Pure Exchange Economy Edgeworth Box 3 Adding Preferences Represent Preferences on the Edgeworth Box Edgeworth Box Dos and Don ts Normative Analysis: Pareto Optimal Allocation ontract urve Positive Analysis: Equilibrium Utility Maximization Problem Finding an Equilibrium Allocation First Fundamental heorem of Welfare Economics 7 Market Mechanism Excess Demand & Excess Supply Price hange to lear Both Markets Second Fundamental heorem of Welfare Economics 9 Summary 11 Hiroki Watanabe 9 / 79

4 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm wo consumers: Liz and Kenneth. Liz is endowed with ω L (ω L, ωl ). Kenneth is endowed with ω K (ω K, ωk ). Liz s consumption bundle is L ( L, L ). Kenneth s consumption bundle is K ( K, K ). 11 Hiroki Watanabe / 79 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm A pair of bundles L, K is called an allocation. How many commodities are there in the economy as a whole? Definition.1 (Feasible Allocation) An allocation ( L, K ) ( L, L, K, K ) is said to be feasible if L + K = ωl + ωk L + K = ωl + ωk. 11 Hiroki Watanabe 11 / 79 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm hey trade commodities to end up with the final allocation ( L, L, K, K ). Liz receives (or gives if negative) L ωl cups of tea from Kenneth. ωk cheesecakes and cups of tea from Liz. cheesecakes and L ωl Kenneth receives/gives K K ω K 11 Hiroki Watanabe 1 / 79

5 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm If, in particular, the allocation is feasible, these net trade volumes above sum up to zero: ( L ωl ) + ( K ωk ) ( L ωl ) + ( K ω K ) = or simply put, ω ω =. 11 Hiroki Watanabe 13 / 79 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm Suppose ω L (ω L, ωl ) = (, ). ω K (ω K, ωk ) = (, ). 11 Hiroki Watanabe 1 / 79 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm Are the following bundles feasible? L L = 3 K =, y =, z =, = 1, =. 3 K 11 Hiroki Watanabe 1 / 79

6 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm Fine, but it ll take forever to find all the feasible allocations. Graphical representation? Rotate Kenneth s bundle through 1 degrees and put it alongside with Liz s. 11 Hiroki Watanabe 1 / 79 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm heesecake x K (slices) x y z v w ea x K (cups) ea x L (cups) heesecake x L (slices) 11 Hiroki Watanabe 17 / 79 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm What do feasible allocations have in common on the graph? Kenneth s bundle (and his graph) is redundant if an allocation = ( L, L, K, K ) is feasible. Whatever left is Kenneth s: K = K = K ωl + ωk L ω L + ωk. L 11 Hiroki Watanabe 1 / 79

7 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm L K ωl ωk O L L O K K 11 Hiroki Watanabe 19 / 79 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm L K O K ω L ω K O L L K 11 Hiroki Watanabe / 79 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm L K O K ω K ωl O L L 11 Hiroki Watanabe 1 / 79 K

8 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm he box is called the Edgeworth box. Any allocation in the Edgeworth box is feasible. Any allocation outside of the Edgeworth box is not. 11 Hiroki Watanabe / 79 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm 1 Introduction General Equilibrium Positive & Normative Analysis Pure Exchange Economy Edgeworth Box 3 Adding Preferences Represent Preferences on the Edgeworth Box Edgeworth Box Dos and Don ts Normative Analysis: Pareto Optimal Allocation ontract urve Positive Analysis: Equilibrium Utility Maximization Problem Finding an Equilibrium Allocation First Fundamental heorem of Welfare Economics 7 Market Mechanism Excess Demand & Excess Supply Price hange to lear Both Markets Second Fundamental heorem of Welfare Economics 9 Summary 11 Hiroki Watanabe 3 / 79 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm Represent Preferences on the Edgeworth Box Q: What kind of trade patterns are we going to observe? Q: What is it that Liz and Kenneth try to achieve? A: Maximize their utility levels. 11 Hiroki Watanabe / 79

9 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm Represent Preferences on the Edgeworth Box heesecake x K (slices) ea x K (cups) ea x L (cups) heesecake x L (slices) 11 Hiroki Watanabe / 79 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm Represent Preferences on the Edgeworth Box Liz prefers bundles to the northeast. How about Kenneth? 11 Hiroki Watanabe / 79 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm Represent Preferences on the Edgeworth Box Liz: u(x L, xl )=cl Kenneth: u(x K, xk )=ck heesecake x K (slices) ea x L (cups) ea x K (cups) heesecake x L (slices) 11 Hiroki Watanabe 7 / 79

10 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm Edgeworth Box Dos and Don ts Do yourselves (and my A and me) a favor when drawing Edgeworth boxes: 1 When drawing Kenneth s axes and indifference curves, flip your paper upside down. Flip it back, and you ll get labels like and. Kenneth s Indifference urve But I don t mind your having Kenneth s labels upside down at all so long as they are in the right place. For that reason, make it your habit to always mark the origins O L and. O K K 11 Hiroki Watanabe / 79 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm Edgeworth Box Dos and Don ts K L O L ω K ω L O K K 11 Hiroki Watanabe 9 / 79 L Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm Edgeworth Box Dos and Don ts Do extend the axes and indifference curves beyond the rectangle. he agent s preferences are defined over R + not just over the box. 11 Hiroki Watanabe 3 / 79

11 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm Edgeworth Box Dos and Don ts Liz: u(x L, xl )=cl Kenneth: u(x K, xk )=ck heesecake x K (slices) ea x L (cups) ea x K (cups) heesecake x L (slices) 11 Hiroki Watanabe 31 / 79 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm Edgeworth Box Dos and Don ts 3 If you have a colored pen, reserve it for Kenneth. You don t have to buy tons of colored pens though. We can only handle two agents in the box. In the following (and the next lecture + problem sets / exams), I ll do some dont s and won t do some dos. My apology. 11 Hiroki Watanabe 3 / 79 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm 1 Introduction General Equilibrium Positive & Normative Analysis Pure Exchange Economy Edgeworth Box 3 Adding Preferences Represent Preferences on the Edgeworth Box Edgeworth Box Dos and Don ts Normative Analysis: Pareto Optimal Allocation ontract urve Positive Analysis: Equilibrium Utility Maximization Problem Finding an Equilibrium Allocation First Fundamental heorem of Welfare Economics 7 Market Mechanism Excess Demand & Excess Supply Price hange to lear Both Markets Second Fundamental heorem of Welfare Economics 9 Summary 11 Hiroki Watanabe 33 / 79

12 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm An allocation of the endowment that improves the welfare of a consumer without reducing the welfare of another is a Pareto-improving allocation. Where do the Pareto-improving allocations lie? 11 Hiroki Watanabe 3 / 79 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm heesecake x K (slices) ea x L (cups) 1 1 Liz: u(x L, xl )=cl Kenneth: u(x K, xk )=ck Endowment ω ea x K (cups) 1 1 heesecake x L (slices) 11 Hiroki Watanabe 3 / 79 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm Since each consumer can refuse to trade, the only possible outcomes from exchange are Pareto-improving allocations. Which particular Pareto-improving allocation will be the outcome of trade? 11 Hiroki Watanabe 3 / 79

13 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm heesecake x K (slices) 1 1 ea x L (cups) ea x K (cups) 1 1 heesecake x L (slices) 11 Hiroki Watanabe 37 / 79 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm ea x L (cups) heesecake x K (slices) heesecake x L (slices) 11 Hiroki Watanabe 3 / 79 ea x K (cups) Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm ea x L (cups) heesecake x K (slices) heesecake x L (slices) 11 Hiroki Watanabe 39 / 79 ea x K (cups)

14 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm Definition.1 (Pareto Optimal Allocation) A feasible allocation ( L, K ) ( L, L, K, K ) is said to be a Pareto optimal allocation or efficient if there is no feasible Pareto-improving allocations. An allocation where convex indifference curves are "only just back-to-back" is Pareto-optimal. An allocation where Liz cannot get strictly better-off without hurting Kenneth (and vice versa). Note there is no price involved. 11 Hiroki Watanabe / 79 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm Discussion. (Efficiency and Equity) Leave the tea aside for a while. Suppose there is a whole cheesecake (ω L + ωk = 1) in the economy. What are the feasible allocations? Which allocation ( L, K ) is Pareto optimal? 1 ( L, K ) = (.,.). ( L, K ) = (.,.). 3 ( L, K ) = (.,.). ( L, K ) = (.1,.99999). ( L, K ) = (1, ). ( L, K ) = (.,.), throw away.1. 7 ( L, K ) = (1.,.). 11 Hiroki Watanabe 1 / 79 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm ontract urve Under standard assumptions: PO angency. angency PO. For exceptions see Example. and Example. What is the slope of indifference curve again? 11 Hiroki Watanabe / 79

15 9 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm ontract urve At a PO allocation, Liz is willing to give up MRS L ( L, L ) cups of tea to obtain one more cheesecake. At a PO allocation, Kenneth is willing to give up one slice of cheesecake for MRS K ( K, K ) cups of tea. If indifference curves are tangent to each other: MRS L ( L, L ) = MRSK ( K, K ). A set of feasible allocations where indifference curves are tangent is called contract curve. 11 Hiroki Watanabe 3 / 79 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm ontract urve ea x L (cups) 1 1 Liz: u(x L, xl )=cl Kenneth: u(x K, xk )=ck ontract urve heesecake x K (slices) heesecake x L (slices) 11 Hiroki Watanabe / 79 ea x K (cups) Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm ontract urve Example.3 (Finding ontract urve: obb-douglas) Suppose preferences are: L ( L, L ) = L L K ( K, K ) = K K. Initial endowments are ω L ωl =, ω ω K ωk =. L ω K Marginal rate of substitution is MRS L ( L, L ) = L L MRS K ( K, K ) = K. K Derive and sketch the contract curve. 11 Hiroki Watanabe / 79

16 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm ontract urve Steps: 1 Find the total endowment. Write ( K, K ) in terms of L and L to maintain the allocation feasible. 3 Equate MRS s (i.e., MRS L ( ) = MRS K ( )). 11 Hiroki Watanabe / 79 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm ontract urve ea x L (cups) heesecake x K (slices) Liz: u(x L 3. )=cl Kenneth: u(x K, xk )=ck 3 ontract urve heesecake x L (slices) 11 Hiroki Watanabe 7 / ea x K (cups) Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm ontract urve Example. (Finding ontract urve: Perfect Substitute) Suppose preferences are: L ( L, L ) = L + L K ( K, K ) = K + K. Initial endowments are ω L ωl =, ω K ω L ωk = ω K Derive and sketch the contract curve.. 11 Hiroki Watanabe / 79

17 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm ontract urve Steps: 1 Find the total endowment. Find MRS L ( L, L ). 3 Find MRS K ( K, K ). Observe that MRS s won t equate. heck the four edges for corner solutions. 11 Hiroki Watanabe 9 / 79 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm ontract urve ea x L (cups) heesecake x K (slices) Liz: u(x L, xl )=cl. Kenneth: u(x K, xk )=ck PO Allocations ea x K (cups). 3. heesecake x L (slices) 11 Hiroki Watanabe / 79 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm ontract urve Liz appreciates cheesecake more than Kenneth does. Kenneth appreciates tea more than Liz does. Any allocation with L > will be Pareto-improved upon by some other allocations. Any allocation with K > will be Pareto-improved upon by some other allocations. Pareto optimal allocations are on the south and east edges. 11 Hiroki Watanabe 1 / 79

18 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm ontract urve Example. (Finding ontract urve: Quasi Linear) Suppose preferences are: L ( L, L ) = L + L K ( K, K ) = K + K. Initial endowments are ω L ωl =, ω ω K ωk =. L ω K Marginal rate of substitution is Derive the contract curve. MRS L ( L, L ) = 1 L MRS K ( K, K ) = 1. K 11 Hiroki Watanabe / 79 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm ontract urve Steps: 1 Follow the same step as Example.3 heck the four edges for corner solutions. 11 Hiroki Watanabe 3 / 79 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm ontract urve ea x L (cups) heesecake x K (slices) Liz: u(x L, xl )=cl Kenneth: u(x K, xk )=ck ontract urve PO Allocations ea x K (cups). heesecake x L (slices) 11 Hiroki Watanabe / 79

19 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm ontract urve On the south edge ( L = ): For L ω / MRS L ( L, L ) MRSK ( K, K ). Kenneth appreciates tea more than Liz. here is no Pareto-improving allocations. For L > ω / MRS L ( L, L ) < MRSK ( K, K ). Liz appreciates tea more than Kenneth. here are Pareto-improving allocations. On the north edge ( L = ω ): vice versa. 11 Hiroki Watanabe / 79 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm 1 Introduction General Equilibrium Positive & Normative Analysis Pure Exchange Economy Edgeworth Box 3 Adding Preferences Represent Preferences on the Edgeworth Box Edgeworth Box Dos and Don ts Normative Analysis: Pareto Optimal Allocation ontract urve Positive Analysis: Equilibrium Utility Maximization Problem Finding an Equilibrium Allocation First Fundamental heorem of Welfare Economics 7 Market Mechanism Excess Demand & Excess Supply Price hange to lear Both Markets Second Fundamental heorem of Welfare Economics 9 Summary 11 Hiroki Watanabe / 79 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm Utility Maximization Problem We know what the ideal allocations are. Q: If we leave Liz and Kenneth alone, will they reach a PO allocation on their own? onsider trade in perfectly competitive markets. Each consumer is a price-taker trying to maximize her own utility given p, p and her own endowment. What is Liz s budget? She can cash in all her endowment and then use that cash as her income: m L := p ω L + p ω L. Similarly m K := p ω K + p ω K. What is their utility maximization problem then? 11 Hiroki Watanabe 7 / 79

20 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm Utility Maximization Problem Liz s UMP Kenneth s UMP max L =( L, L ) L ( L, L ) subject to p ω L + p ω L = p L + p L. max K =( K, K ) K ( K, K ) subject to p ω K + p ω K = p K + p K. Btw the budget constraint goes through the initial endowment ω L (ω L, ωl ). (Why?) Also, recall that the slope of budget constraint represents the relative price ( p ) p 11 Hiroki Watanabe / 79 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm Utility Maximization Problem ea x L (cups) 1 Budget onstraint u L (x L, xl )=ul (ω L, ωl ) u L (x L, xl )=ul (x LE, xle ) Endowment x E 1 heesecake x L (slices) 11 Hiroki Watanabe 9 / 79 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm Finding an Equilibrium Allocation Given p and p, Liz s net demands for cheesecake and tea are L ωl, L ωl. Similarly for Kenneth K ωk, K ωk. An equilibrium occurs when prices p and p make both the markets cleared, i.e. LE + KE = ωl + ωk LE + KE = ωl + ωk. 11 Hiroki Watanabe / 79

21 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm Finding an Equilibrium Allocation Question.1 (Finding the Equilibrium Allocation) Suppose preferences are: Initial endowments are L ( L, L ) = L L K ( K, K ) = K K. ω L (ω L, ωl ) = (, ) ω K (ω K, ωk ) = (, ). Marginal rate of substitution is MRS L ( L, L ) = L L MRS K ( K, K ) = K. K Find the equilibrium allocation ( LE, KE ) ( LE, LE, KE, KE ) when p (p, p ) = (1, ). 11 Hiroki Watanabe 1 / 79 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm Finding an Equilibrium Allocation 1 ompute income m L and m K from their endowment. Write UMP: m L := p ω L + p ω L = m K := p ω K + p ω K =. max L, L L ( L, L ) s.t. p L + p L = max K, K K ( K, K ) s.t. p K + p K =. 3 Solve UMP to obtain the equilibrium allocation ( LE, LE, KE, KE ) =,, 3, Hiroki Watanabe / 79 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm On the contract curve: At the equilibrium MRS L ( L ) = MRS K ( K ). MRS L ( LE ) = MRS K ( KE ) = p p. PO allocations are on the contract curve under standard assumptions. 11 Hiroki Watanabe 3 / 79

22 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm heorem.1 (First Fundamental heorem of Welfare Economics) Given that consumers preferences are well-behaved, trading in perfectly competitive markets implements a Pareto-optimal allocation of the economy s endowment. Recall there were no deadweight loss in a perfectly competitive market. 1st heorem translates to "don t mess with the perfect competition." or equivalently "laissez-faire". Recall taxation creates the DWL. 11 Hiroki Watanabe / 79 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm When can the 1st heorem fail to hold? It may not work this way if we don t have a perfectly competitive market: 1 Product homogeneity (Product differentiation) Small agents (IO) 3 Free mobility (Urban economics) Perfect knowledge (Information economics) Exclusion (Public finance) Rivalry (Public finance) 7 Free entry (IO) 11 Hiroki Watanabe / 79 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm 1 Introduction General Equilibrium Positive & Normative Analysis Pure Exchange Economy Edgeworth Box 3 Adding Preferences Represent Preferences on the Edgeworth Box Edgeworth Box Dos and Don ts Normative Analysis: Pareto Optimal Allocation ontract urve Positive Analysis: Equilibrium Utility Maximization Problem Finding an Equilibrium Allocation First Fundamental heorem of Welfare Economics 7 Market Mechanism Excess Demand & Excess Supply Price hange to lear Both Markets Second Fundamental heorem of Welfare Economics 9 Summary 11 Hiroki Watanabe / 79

23 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm Excess Demand & Excess Supply heesecake x K (slices) 1 1 ea x L (cups) ω φ K (p, p ) φ L (p, p ) ea x K (cups) 1 1 heesecake x L (slices) 11 Hiroki Watanabe 7 / 79 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm Excess Demand & Excess Supply Suppose that the ongoing price is p = p = (p, p ). Under this price, utility-maximizing bundle is L = φ L (p, p ) φl (p ) φ L (p ) L and K = φ K (p K, p ) φk (p ) φ K (p ) 11 Hiroki Watanabe / 79 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm Excess Demand & Excess Supply Note φ L (p ) + φ K (p ) > ω L + ωk φ L (p ) + φ K (p ) < ω L + ωk It looks like their demand φ L ( ) and φ K ( ) cannot be fulfilled in this economy... Is this a market failure? 11 Hiroki Watanabe 9 / 79

24 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm Excess Demand & Excess Supply No. Market failure is a thing but it refers to something else. he situation above is just a disequilibrium. urrently, at the given prices p and p there is 1 an excess demand for cheesecake. an excess supply of tea. Neither market clears so the price p does not realize the equilibrium. 11 Hiroki Watanabe 7 / 79 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm Price hange to lear Both Markets onsider the individual markets: Since there is an excess demand for cheesecake, p will rise. Since there is an excess supply of tea, p will fall. he slope of the budget constraint is p /p so the budget constraint will pivot about the endowment point and become steeper. he price changes till it clears both markets, at which point, there is no reason to update the price anymore. 11 Hiroki Watanabe 71 / 79 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm Price hange to lear Both Markets he equilibrium price realizes MRS L ( LE ) = MRS K ( KE ) = pe. p E Market mechanism leads to the equilibrium, which happens to be efficient, by adjusting the price accordingly ( heorem.1 ). 11 Hiroki Watanabe 7 / 79

25 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm Price hange to lear Both Markets heesecake x K (slices) 1 1 ea x L (cups) ω φ K (p, p ) φ K (p E, pe )=φl (p E, pe ) φ L (p, p ) ea x K (cups) 1 1 heesecake x L (slices) 11 Hiroki Watanabe 73 / 79 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm 1 Introduction General Equilibrium Positive & Normative Analysis Pure Exchange Economy Edgeworth Box 3 Adding Preferences Represent Preferences on the Edgeworth Box Edgeworth Box Dos and Don ts Normative Analysis: Pareto Optimal Allocation ontract urve Positive Analysis: Equilibrium Utility Maximization Problem Finding an Equilibrium Allocation First Fundamental heorem of Welfare Economics 7 Market Mechanism Excess Demand & Excess Supply Price hange to lear Both Markets Second Fundamental heorem of Welfare Economics 9 Summary 11 Hiroki Watanabe 7 / 79 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm heorem.1 (Second Fundamental heorem of Welfare Economics) Given that consumers preferences are well-behaved, for any Pareto-optimal allocation there are prices that makes the Pareto-optimal allocation implementable by trading in competitive markets provided that endowments are first appropriately rearranged amongst the consumers. 11 Hiroki Watanabe 7 / 79

26 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm Proof. Pick any PO allocation ( L PO allocation Set the price at MRS L ( L p p, L, L, K ) = MRSK ( K, K ). = MRS L ( L, L ), K ). Since is a and set the initial endowment at ω L = L, ω K = K. Market equilibrium should lead to ( LE, KE ) = ( L, K ). 11 Hiroki Watanabe 7 / 79 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm 1 Introduction General Equilibrium Positive & Normative Analysis Pure Exchange Economy Edgeworth Box 3 Adding Preferences Represent Preferences on the Edgeworth Box Edgeworth Box Dos and Don ts Normative Analysis: Pareto Optimal Allocation ontract urve Positive Analysis: Equilibrium Utility Maximization Problem Finding an Equilibrium Allocation First Fundamental heorem of Welfare Economics 7 Market Mechanism Excess Demand & Excess Supply Price hange to lear Both Markets Second Fundamental heorem of Welfare Economics 9 Summary 11 Hiroki Watanabe 77 / 79 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm Pareto optimal allocation and MRS. Market mechanism automatically achieves Pareto optimal allocation. 1st heorem: Equilibrium PO. nd heorem: PO Equilibrium. 11 Hiroki Watanabe 7 / 79

27 Intro Edgeworth Preferences Pareto Optimality Equilibrium 1st hm Market Mech nd hm Map du Jour Is St. Louis a freak of nature...? Source 11 Hiroki Watanabe 79 / 79