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6-2: the 2 2 × Exchange Economy ©John Riley 4 May 2007 6-2: THE 2× 2 EXCHANGE ECONOMY In this section we switch the focus from production efficiency to the allocation of goods among consumers. We begin by focusing on a simple exchange economy in which there are two consumers, Alex and Bev. Each consumer has an endowment of two commodities. Commodities are private. That is, each consumer cares only about his own consumption. Consumer h h 2 h h h, h= AB , has an endowment ω , a consumption set X = R+ and a utility function U( x ) that is strictly monotonic. Pareto Efficiency With more than one consumer, the social ranking of allocations requires weighing the utility of one individual against that of another. Suppose that the set of possible utility pairs (the “utility possibility set”) associated with all possible allocations of the two commodities is as depicted below. B U2 45 line A W U+ U = k 1 2 U 1 6.2-1: Utility Possibility Set Setting aside the question of measuring utility, one philosophical approach to social choice places each individual behind a “veil of ignorance.” Not knowing which consumer you are going 1 to be, it is natural to assign a probability of 2 to each possibility. Then if individuals are neutral Section 6.2 page 1 ©John Riley 4 May 2007 towards risk while behind the veil of ignorance, they will prefer allocations with a higher expected utility 1+ 1 2U1 2 U 2 . This is equivalent to maximizing the sum of utilities, a proposal first put forth by Jeremy Bentham. In Fig. 6.2-1, the Benthamite criterion picks the point B. A more recent philosophical argument developed by John Rawls argues that, behind the veil of ignorance consumers will be highly averse to risk and thus place most weight on the worst possible outcome. Rawls takes the extreme position of infinite risk aversion so that all weight is placed on the worst possible outcome. The social criterion then become the Max Min criterion. Maximizing the minimum utility is achieved by moving out along the 45 line to the utility possibility frontier; the point W in the figure. Economists tend to be agnostic when it comes to theorizing about social choice rankings. Instead they focus on minimizing unnecessary waste. The utility allocation A in the figure is wasteful or “inefficient” because there are alternative allocations of goods which would make both individuals better off. Both the Benthamite allocation and the Rawlsian allocations are said to be Pareto Efficient (or simply “efficient”) since the only way to raise the utility of one individual is by reducing the utility of the other. Generalizing to more than two individuals we have the following definition. Pareto Efficiency A feasible allocation of commodities is Pareto efficient if there is no other feasible allocation that is strictly preferred by at least one consumer and is weakly preferred by all consumers. × ω= ω ω For the special 2 2 case, Alex and Bev must share the aggregate endowment (1 , 2 ) . Any feasible consumption bundle for Alex, xA = x must lie in the “Edgeworth Box” created by drawing perpendiculars from the aggregate endowment to the axes. Bev’s consumption bundle is then xB =ω − x . Thus Bev’s consumption bundle is measured from the North-West corner of the Edgeworth Box. We can therefore draw the indifference curves for both consumers through the allocation x. At the endowment point ω A , Alex’s marginal rate of substitution MRS A(ω A ) Section 6.2 page 2 ©John Riley 4 May 2007 exceeds MRSBB()ω= MRS B ( ω − ω A ) . Thus Alex is willing to give up more of commodity 2 than Bev for an additional unit of commodity 1. It follows that there is a set of allocations (shaded in the figure) that are strictly preferred by both consumers. x 2 ω B 1 OB ω ω (1 , 2 ) ω A ω B 2 2 A ω BB BB U() x= U ()ω AA AA U() x= U ()ω x1 A ω A O 1 Fig. 6.2-2: Edgeworth Box Diagram Note that the axes for the two consumers and the indifference curves extend beyond the boundaries of the Edgeworth Box. This is important. The consumers’ preferences are defined over their consumption sets (the positive consumption bundles) without reference to the availability of commodities. Suppose we consider all alternative allocations {xA , x B } such that UxBB()≥ Ux BB () and choose the allocation {xˆA , x ˆ B } that maximizes Alex’s utility. Mathematically, we solve the following optimization problem. MaxUx{()|()AA Ux BB≥ Ux BB (), x AB + x = ω } . xA, x B In the diagram, we seek Alex’s most preferred allocation that is in the shaded region of points at least weakly preferred by Bev. Consider any other allocation {xA , x B } which is weakly preferred Section 6.2 page 3 ©John Riley 4 May 2007 by Bev. That is UxBB()≥ Ux BB () . Since xˆ A is optimal for Alex, it cannot be the case that UxAA()> Ux AA ()ˆ . Then the allocation is Pareto Efficient. x2 OB A ˆ A x x UxAA()= Ux AA ()ˆ UxBB()= Ux BB () x O A 1 Fig. 6.2-3: Pareto Efficient Allocations By changing the initial allocation we can then map out the entire set of Pareto Efficient allocations. Example: Identical CES Preferences If preferences are CES with elasticity of substitution σ , both consumers have a marginal rate of h x σ h h = 2 1/ substitution, MRS() x k (h ) . From Fig. 6.3-2, if a PE allocation is in the interior of the x1 Edgeworth Box the indifference curves of the two consumers must have the same slope. Then xA x B 2= 2 A B . x1 x 1 Section 6.2 page 4 ©John Riley 4 May 2007 Appealing to the Ratio Rule 1 and then setting demand equal to supply, xA x B x A+ x B ω 2= 2 = 22 = 2 . A B A+ B ω x11 x xa x 11 Thus, in a Pareto efficient allocation each consumer is allocated a fraction of the aggregate endowment. It follows that for each consumer the marginal rate of substitution is h ω MRSh () x* = k (2 ) 1/ σ . (6.2-1) ω 1 Walrasian Equilibrium for an Exchange Economy In a Walrasian equilibrium all consumers are price-takers. Consumer h, h= AB , , with endowment ω h has preferences represented by the utility function Uh( x h ) where xh is his private consumption. His wealth is therefore Wh= p ⋅ ω h and so he chooses a consumption bundle xh( p , W h ) that solves Max{()| Uh x p⋅≤ x W h =⋅ p ω h } x There is no production. Let p ≥ 0 be a price vector of this exchange economy. Define ω= ∑ ω h to be the vector of total endowments in the economy and xp()=∑ xpph (, ⋅ ω h ) to h h be total (or “market”) demand. Then the vector of excess demands, ep()= xp () − ω . Definition: Market Clearing Prices ≥ Let ej ( p ) be the excess demand for commodity j at the price vector p 0 . The market for ≤ = commodity j clears if ej ( p ) 0 and pj e j ( p ) 0 . = That is, either excess demand for commodity j is zero or it is negative and the price p j 0 . a b a+ b 1 If 1= 1 = k then 1 1 = k + b2 b 2 a2 b 2 Section 6.2 page 5 ©John Riley 4 May 2007 Walras Law We now show that the market value of excess demands must be zero. First note that peppx⋅()()(( =⋅−=⋅ω p∑ xhh − ω )) = ∑ ( pxp ⋅−⋅ hh ω ) . h h By hypothesis utility is strictly increasing. Therefore each consumer spends all his wealth, that is p⋅ xh = p ⋅ ω h . Then the right hand expression is zero. Hence + = pe11() p pe 22 ()0 p . Thus, if one market clears, so must the other. This is known as Walras Law. 2 Definition: Walrasian Equilibrium The price vector p ≥ 0 is a Walrasian equilibrium price vector if all markets clear. For our two commodity model, it follows from Walras Law that we need only consider market = clearing in one market. Take any price vector p( p1 , p 2 ) . Given an endowment allocation {ωA , ω B ) , each consumer chooses his utility maximizing consumption bundle. As depicted, in Figure 6.2-4, Alex wants to trade from the endowment point N to C A , while Bev wishes to trade from N to C B . Thus, there is excess supply of commodity 1. The Walrasian auctioneer therefore lowers the price of commodity 1 (relative to commodity 2) and so flattens the budget line until supply equals demand. The Walrasian equilibrium E is depicted in Fig. 6.2-5. 2 With H consumers and n commodities an identical argument establishes that p⋅ e( p ) = 0 . Thus if n-1 markets clear, the n-th market must also clear. Section 6.2 page 6 ©John Riley 4 May 2007 x2 B O ω ω (1 , 2 ) C A C B N ωA ω A (1 , 2 ) A O x1 Fig. 6.2-4: Excess supply of commodity 1 x2 B O ω ω (1 , 2 ) E N B= B B U() x U () x O A x1 Fig. 6.2-5: Walrasian equilibrium Section 6.2 page 7 ©John Riley 4 May 2007 Equilibrium and Efficiency From the figure it is clear that any redistribution of commodities that raises the utility of Alex ( the North-East shaded region) must reduce Bev’s utility.
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