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Equal Treatment Property

2011. Q8, 2002, Q7, 1996, Q3 1. The equal treatment property is informally stated as a situation in which identicals are treated identically. In the context of pure exchange economy, give a precise and formal statement of this intuition. Answer: Pure exchange economy is summarized by ({(Xi ,I i )}i=1

, {Yi }J , ) where j=1

each consumer i = 1, . . . , I (I > 0) is characterized by consumption set Xi RL (L commodities) and rational (complete and transitive) preference relation i dened on Xi , each rm is j = 1, . . . , J is characterized by a nonempty and closed technology or production set Yj = RL j J (economys only technological possibility is that of free disposal) and the endowments (initial resources of commodities) are given by = (1 , . . . , L ) L R Now that we understand the pure exchange economy we can dene equal treatment property for pure exchange economy. Let H = {1, . . . , H} stand for a set of types of consumers with h H having preferences h and endowments h . Suppose there are N consumers of each type. Then the equal treatment property implies that all consumers of the same type get the same consumption bundle: xhm = xhn , 1 m, 0 < n N, 1 h H, that is the mth consumer of type h gets the same consumption bundle as the nth consumer of type h. 2. Dene the notion of competitive equilibrium for a pure exchange economy. Does this solution concept exhibit the equal treatment property? Justify your answer in either case. Do the properties of preferences have any relevance for your answer? Answer: Given a pure exchange economy ({(Xi , i )}I , {Yi }J , ), an allocation (x, y) i=1 j=1 and a price vector p = (p1 , . . . , pL ) constitute a competitive equilibrium if (a) For every j, yj maximizes prot in Yj : p yj p yj yj Yj

(b) For every i, x is maximal (maximizes consumers well-being) for i budget set : {xi Xi : p xi p i } (c) Markets must clear:i

i

in the

x = + i

j

yj

Yes, under certain conditions about preferences, CE implies e.t. Given that preferences are continuous, strictly convex and strongly monotone, any competitive equilibrium belongs to the core. Under the same conditions, any allocation in the core exhibits the equal treatment property. Notice that these are not i statements, so they hold under certain conditions but not the other way around. So properties of preferences determine whether the equilibrium exhibits equal treatment property or not. Intuitive Proof: Suppose not. CE and not e.t. Then two consumers of the same type are getting dierent bundles. Then they will block. But we know CE implies core. Contradiction. 3. Dene the notion of a Pareto Optimal (P.O) allocation, and indicate whether it exhibits the equal treatment property? If so provide a proof. If not, develop an argument as to why we should have any interest in this concept. Answer: An allocation (x, y) is feasible ifi

xi = +

j

yj .i

Denote feasible set by F = (x, y) RL(I+J) :

xi = +

j

yj

A feasible allocation (x, y) is P.O if there does not exist (x , y ) in the feasible set s.t xi xi i and xi xi for some i. An allocation is P.O if there is no waste. Note that P.O concept does not concern itself with distributional issues. For example, an allocation in a pure exchange economy that gives all of societys endowments to one consumer who has strongly monotone preferences is necessarily P.O. So does not necessarily exhibit equal treatment property, but it is nonetheless minimal notion of economic eciency. However, under certain conditions P.O does imply equal treatment. Given that preferences are locally non-satiated, convex and continuous, Xi is convex, 0 Xi for all i, every production set is convex, and a P.O allocation (x, y), then there exists 0 = 0 s.t. (x, y, p) is a competitive equilibrium. Given that preferences are continuous, strictly convex and strongly monotone, any competitive equilibrium belongs to the core. Under the same conditions, any allocation in the

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core exhibits the equal treatment property. Therefore, only in a very restrictive case, P.O would imply equal treatment. 4. Dene the notion of a core for a pure exchange economy. Does this solution concept exhibit the equal treatment property in the context of a pure exchange economy which is replicated in the sense that there are equal number of agents of a nite number of types? Justify your answer in either case. You may assume that the preferences of each type of agent are strictly convex. Answer: Suppose preferences are continuous, strictly convex and strongly monotone. A coalition S I improves upon or blocks the feasible allocation x = (x1 , . . . , xI ) RLI if for all i S we can nd xi 0 s.t. xi xi for all i SiS

xi Y +

iS

i

A feasible allocation x = (x1 , . . . , xI ) RLI has the core property if there is no coalition of consumers (S I) that can improve upon this allocation. The core is the set of allocations that have the core property. Let H = {1, . . . , H} stand for a set of types of consumers with h H having preferences h and endowments h . Suppose there are N consumers of each type. Denoting by hn the nth individual of type h, suppose that the allocation x = LHN belongs to the (x11 , . . . , x1n , . . . , x1N , . . . , xH1 , . . . , xHn , . . . , xHN ) R+ core of the economy composed of N consumers of each type, for a total number of IN = N H (N-replica economy). Then x has the equal treatment property, that is, all consumers of the same type get the same consumption bundle: xhm = xhn , 1 m, 0 < n N, 1 h H, that is the mth consumer of type h gets the same consumption bundle as the nth consumer of type h. So Yes! Core implies e.t. in N-replica economy under given conditions. 5. Let N = (n1 , . . . , nH ) be the set of dierent number of dierent types of consumers. Then the total number of consumers is N H (dot product), x = L(HN ) (x11 , . . . , x1n1 , . . . , xH1 , . . . , xHnH ) R+ . ??? Counterexamples for each ?

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