1 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)}Ii=1 , {Yi}Jj=1, ω̄) where

• each consumer i = 1, . . . , I (I > 0) is characterized by consumption set Xi ⊂RL (L commodities) and rational (complete and transitive) preference relation

�i defined on Xi,

• each firm is j = 1, . . . , J is characterized by a nonempty and closed technology

or production set Yj = −RL ∀j ∈ J (economy’s only technological possibility

is that of free disposal) and

• the endowments (initial resources of commodities) are given by ω̄ = (ω̄1, . . . , ω̄L) ∈RL

Now that we understand the pure exchange economy we can define 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 ω̄h. 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. Define 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)}Ii=1 , {Yi}Jj=1, ω̄), an allocation (x∗, y∗)and a price vector p = (p1, . . . , pL) constitute a competitive equilibrium if

(a) For every j, y∗j maximizes profit in Yj: p · yj ≤ p · y∗j ∀yj ∈ Yj

(b) For every i, x∗i is maximal (maximizes consumer’s well-being) for �i in the

budget set : {xi ∈ Xi : p · xi ≤ p · ω̄i}

(c) Markets must clear:∑

i x∗i = ω̄ +

∑j y∗j

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 al-

location in the core exhibits the equal treatment property. Notice that these are

not iff 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 different bundles. Then they will block. But we know CE implies

core. Contradiction.

3. Define 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 if∑

i xi = ω̄ +∑

j yj.

Denote feasible set by F ={

(x, y) ∈ RL(I+J) :∑

i xi = ω̄ +∑

j yj

}A feasible allocation (x, y) is P.O if there does not exist (x′, y′) in the feasible set

s.t x′i � xi ∀i and x′i � 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 society’s 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 efficiency. However, under certain conditions P.O does imply equal

treatment.

Given that preferences are locally non-satiated, convex and continuous, Xi is con-

vex, 0 ∈ Xi for all i, every production set is convex, and a P.O allocation (x∗, y∗),then there exists 0 6= 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. Define 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 finite

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 find xi ≥ 0 s.t.

• xi � xi∗ for all i ∈ S

•∑

i∈S xi ∈ Y +{∑

i∈S ωi

}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 ω̄h. Suppose there are N consumers of each type.

Denoting by hn the nth individual of type h, suppose that the allocation x∗ =

(x11∗, . . . , x1n∗, . . . , x1N∗, . . . , xH1∗, . . . , xHn∗, . . . , xHN∗) ∈ RLHN+ belongs to the

core of the economy composed of N consumers of each type, for a total number of

IN = NH (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 different number of different types of con-

sumers. Then the total number of consumers is N · H (dot product), x∗ =

(x11∗, . . . , x1n1∗, . . . , xH1∗, . . . , xHnH∗) ∈ R

L(H·N)+ . ???

Counterexamples for each ?

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