Chapter 31: Exchange

Transcription

1 Econ 401 Price Theory Chapter 31: Exchange Instructor: Hiroki Watanabe Summer / 53 1 Introduction General Equilibrium Positive & Normative Pure Exchange Economy 2 Edgeworth Box 3 Adding Preferences 4 Normative: Pareto Optimal Allocation 5 Positive: Equilibrium UMP Finding an Equilibrium Allocation 6 First Fundamental Theorem of Welfare Economics 7 Market Mechanism Excess Demand & Excess Supply Prices Change to Clear Both Market 8 Second Fundamental Theorem of Welfare Economics 9 Summary 2 / 53

2 General Equilibrium Demand for cheesecake ϕ C (p C, p T, m) depends on p C, p T and m. Supply of cheesecake p C = MC(y). Each agent takes market prices as given and determines his/her demands and supplies. The price adjusted to clear the market, and at the equilibrium price, no agent would desire to change his/her actions. Partial equilibrium: the price of the good being studied are assumed to remain fixed. General equilibrium: All prices are variable. 3 / 53 Positive & Normative 1 Positive analysis: describing what is. E.g., equilibrium. 2 Normative analysis: describing what should be. E.g., efficiency (Pareto optimality). 4 / 53

3 Pure Exchange Economy Pure exchange economy: There is no firm. Two commodities with two consumers. Instead of income, the agents are endowed with the two commodities. 5 / 53 1 Introduction General Equilibrium Positive & Normative Pure Exchange Economy 2 Edgeworth Box 3 Adding Preferences 4 Normative: Pareto Optimal Allocation 5 Positive: Equilibrium UMP Finding an Equilibrium Allocation 6 First Fundamental Theorem of Welfare Economics 7 Market Mechanism Excess Demand & Excess Supply Prices Change to Clear Both Market 8 Second Fundamental Theorem of Welfare Economics 9 Summary 6 / 53

4 Two consumers: Dharma and Greg. Dharma is endowed with ω D (ω D C, ωd T ). Greg is endowed with ω G (ω G C, ωg T ). Dharma s consumption bundle is x D (x D C, xd T ). Greg s consumption bundle is x G (x G C, xg T ). 7 / 53 A pair of bundles x D, x G is called an allocation. How many commodities are there in the economy as a whole? Feasible Allocation An allocation (x D, x G ) (x D C, xd T, xg C, xg ) is said to be T feasible if x D C + xg C = ωd C + ωg C x D T + xg T = ωd T + ωg T. They trade commodities to end up having the final allocation (x D, x D, x G, x G ). C T C T How do we represent the allocation (x D C, xd T, xg C, xg T )? 8 / 53

5 Suppose ω D (ω D C, ωd ) = (6, 4). T ω G (ω G C, ωg ) = (2, 2). T 9 / 53 D T 6 6 G T 4 2 O D 6 8 D C O G 2 8 G C 10 / 53

6 6 D T G C 8 2 O G 4 2 O D 6 8 D C G T 6 11 / / 53

7 Feasible allocation (x D, x G ) = (x D C, xd T, xg C, xg T ) is represented as: 13 / / 53

8 1 Introduction General Equilibrium Positive & Normative Pure Exchange Economy 2 Edgeworth Box 3 Adding Preferences 4 Normative: Pareto Optimal Allocation 5 Positive: Equilibrium UMP Finding an Equilibrium Allocation 6 First Fundamental Theorem of Welfare Economics 7 Market Mechanism Excess Demand & Excess Supply Prices Change to Clear Both Market 8 Second Fundamental Theorem of Welfare Economics 9 Summary 15 / 53 Q: What kind of trade pattern are we going to observe? Q: What is it that Dharma and Greg try to achieve? A: UMP. 16 / 53

9 Figure: 17 / / 53

10 1 Introduction General Equilibrium Positive & Normative Pure Exchange Economy 2 Edgeworth Box 3 Adding Preferences 4 Normative: Pareto Optimal Allocation 5 Positive: Equilibrium UMP Finding an Equilibrium Allocation 6 First Fundamental Theorem of Welfare Economics 7 Market Mechanism Excess Demand & Excess Supply Prices Change to Clear Both Market 8 Second Fundamental Theorem of Welfare Economics 9 Summary 19 / 53 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? 20 / 53

11 Figure: 21 / 53 Since each consumer can refuse to trade, the only possible outcomes from exchange are Pareto-improving allocations. But which particular Pareto-improving allocation will be the outcome of trade? 22 / 53

12 Figure: 23 / / 53

13 Figure: 25 / 53 Pareto Optimal Allocation A feasible allocation (x D, x G ) (x D, x D, x G, x G ) is C T C T 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 Dharma cannot get strictly better-off without hurting Greg (and vice versa). Note there is no price involved. 26 / 53

14 Example: Efficiency Does Not Imply Equity Leave the spice cake aside for a while. Suppose there is a whole cheesecake (ω D C + ωg C = 1). What are the feasible allocations? Which allocation (x D C, xg ) is Pareto optimal? C 1 (x D C, xg ) = (.5,.5). C 2 (x D C, xg ) = (.6,.6). C 3 (x D C, xg ) = (.4,.6). C 4 (x D C, xg ) = (.00001,.99999). C 5 (x D C, xg ) = (1, 0). Doesn t sound fair. C 6 (x D C, xg ) = (.45,.45), throw away.1. C 7 (x D C, xg ) = (1.2,.2). C 27 / 53 Under standard assumptions: PO Tangency. Tangency PO. What is the slope of indifference curve again? 28 / 53

15 At a PO allocation, Dharma is willing to give up MRS D (x D C, x D ) cups of tea to obtain one more T cheesecake. At a PO allocation, Greg is willing to give up one slice of cheesecake for MRS G (x G ) cups of tea. C, x G T If indifference curves are tangent to each other: MRS D (x D C, x D T ) = MRSG (x G C, x G T ). A set of feasible allocations where indifference curves are tangent is called contract curve. 29 / / 53

16 Exercise: Finding Suppose preferences are: Initial endowments are u D (x D C, xd T ) = xd C xd T u G (x G C, xg T ) = xg C xg T. ω D (ω D C, ωd ) = (6, 2) T ω G (ω G C, ωg ) = (2, 2). T Marginal rate of substitution is Derive the contract curve. MRS D (x D C, xd T ) = xd T x D C MRS G (x G C, xg T ) = xg T x G C. 31 / 53 Steps: 1 Write (x G C, xg T ) in terms of xd C and xd T allocation feasible. 2 Equate MRS s (i.e., MRS D = MRS G.) to maintain the 32 / 53

17 4 : x T D =.5xC D 3 Tea (x T D ) D Cheesecake (x C ) 33 / 53 1 Introduction General Equilibrium Positive & Normative Pure Exchange Economy 2 Edgeworth Box 3 Adding Preferences 4 Normative: Pareto Optimal Allocation 5 Positive: Equilibrium UMP Finding an Equilibrium Allocation 6 First Fundamental Theorem of Welfare Economics 7 Market Mechanism Excess Demand & Excess Supply Prices Change to Clear Both Market 8 Second Fundamental Theorem of Welfare Economics 9 Summary 34 / 53

18 UMP Ok, PO is what it should be. Q: If we leave Dharma and Greg alone, are they going to mimic the PO allocation at the end of the day? Consider trade in perfectly competitive markets. Each consumer is a price-taker trying to maximize her own utility given p C, p T and her own endowment. What is Dharma s budget? He can cash in all his endowment and then use that cash as his income: m D = p C ω D C + p Tω D T. Similarly m G = p C ω G T + p Tω G T. What are their UMP then? 35 / 53 UMP Dharma s UMP max u D (x D x D =(x D C,xD T ) C, xd T ) s.t. p Cω D C + p Tω D T = p Cx D C + p Tx D T. Greg s UMP max u G (x G x G =(x G C,xG T ) C, xg T ) s.t. p Cω G C + p Tω G T = p Cx G C + p Tx G T. Btw the budget constraint goes through the initial endowment ω D (ω D C, ωd ). (Why?) T Also, recall that the slope of budget constraint represents the relative price ( p C ) p T 36 / 53

19 UMP Figure: 37 / 53 Finding an Equilibrium Allocation Givenp C and p T, Dharma s net demands for commodities 1 and 2 are Similarly for Greg x D C ωd C, x D T ωd T. x G C ωg C, x G T ωg T. An equilibrium occurs when prices p C and p T cause both the markets for commodities 1 and 2 to clear; i.e. x D C + x G C = ωd C + ωg C x D T + x G T = ωd T + ωg T. 38 / 53

20 Finding an Equilibrium Allocation Exercise: Finding the Equilibrium Allocation Suppose preferences are: Initial endowments are u D (x D C, xd T ) = xd C xd T u G (x G C, xg T ) = xg C xg T. ω D (ω D C, ωd ) = (6, 2) T ω G (ω G C, ωg ) = (2, 2). T Marginal rate of substitution is MRS D (x D C, xd T ) = xd T x D C MRS G (x G C, xg T ) = xg T x G C Find the equilibrium allocation (x ED, x EG ) (x ED C, xed T, xeg C, xeg T ) when p (p C, p T ) = (1, 2).. 39 / 53 Finding an Equilibrium Allocation 1 Compute income m D and m G from their endowment. 2 Write UMP: m D = p C ω D C + p Tω D T = 10 m G = p C ω G C + p Tω G T = 6. max x D C,xD T ud (x D C, xd T ) s.t. p Cx D C + p Tx D T = 10 max x G C,xG T ug (x G C, xg T ) s.t. p Cx G C + p Tx G T = 6. 3 Sove UMP to obtain the equilibrium allocation (x ED C, xed T, xeg C, xeg T ) = (5, 5 2, 3, 3 2 ). 40 / 53

21 On the contract curve: At equilibrium MRS D (x D ) = MRS G (x G ). MRS D (x ED ) = MRS G (x EG ) = p C p T. PO allocations are on the contract curve under standard assumptions. 41 / 53 First Fundamental Theorem 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 Theorem translates to "don t mess with the perfect competition." or equivalently "laissez-faire". Recall taxation creates the DWL. 42 / 53

22 When does the 1st Theorem fail to hold? Monoply & oligopoly (violates perfect competition). 43 / 53 1 Introduction General Equilibrium Positive & Normative Pure Exchange Economy 2 Edgeworth Box 3 Adding Preferences 4 Normative: Pareto Optimal Allocation 5 Positive: Equilibrium UMP Finding an Equilibrium Allocation 6 First Fundamental Theorem of Welfare Economics 7 Market Mechanism Excess Demand & Excess Supply Prices Change to Clear Both Market 8 Second Fundamental Theorem of Welfare Economics 9 Summary 44 / 53

23 Excess Demand & Excess Supply Figure: 45 / 53 Excess Demand & Excess Supply Note x D C + x G T < ωd C + ωg C x D T + x G T > ωd T + ωg T So at the given prices p C and p T there is 1 an excess supply of cheesecake. 2 an excess demand for tea. Neither market clears so the prices p C and p T do not cause a general equilibrium. 46 / 53

24 Prices Change to Clear Both Market Since there is an excess demand for tea, p T will rise. Since there is an excess supply of cheesecake, p C will fall. The slope of the budget constraint is p C /p T so the budget constraint will pivot about the endowment point and become less steep. 47 / 53 Prices Change to Clear Both Market 48 / 53

25 1 Introduction General Equilibrium Positive & Normative Pure Exchange Economy 2 Edgeworth Box 3 Adding Preferences 4 Normative: Pareto Optimal Allocation 5 Positive: Equilibrium UMP Finding an Equilibrium Allocation 6 First Fundamental Theorem of Welfare Economics 7 Market Mechanism Excess Demand & Excess Supply Prices Change to Clear Both Market 8 Second Fundamental Theorem of Welfare Economics 9 Summary 49 / 53 Second Fundamental Theorem 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. 50 / 53

26 Proof. Pick any PO allocation (x D a PO allocation Set the price at C, x D T, x G C, x G MRS D (x D C, x D T ) = MRSG (x G C, x G T ). p C p T = MRS D (x D C, x D T ) T ). Since x is and set the initial endowment at ω D = x D, ω G = x G. Market equilibrium should lead to (x ED, x EG ) = (x D, x G ). 51 / 53 1 Introduction General Equilibrium Positive & Normative Pure Exchange Economy 2 Edgeworth Box 3 Adding Preferences 4 Normative: Pareto Optimal Allocation 5 Positive: Equilibrium UMP Finding an Equilibrium Allocation 6 First Fundamental Theorem of Welfare Economics 7 Market Mechanism Excess Demand & Excess Supply Prices Change to Clear Both Market 8 Second Fundamental Theorem of Welfare Economics 9 Summary 52 / 53

27 Pareto optimal allocation and MRS. Market mechanism automatically achieves Pareto optimal allocation. 1st Theorem: Equilibrium PO. 2nd Theorem: PO Equilibrium. 53 / 53