Question

The exercise is asking to describe a competitive equilibrium using the proposition

Exercise 10 Consider the special case in uhich everyone has the same indif- ference curve map and the same endoument-that is, (wnl, wna) = (IK/N,w5/N) for all n. Describe a CE One important result that we can state (and prove) is a version of Adam Smith's invisible-hand proposition Proposition 1 If(n, n) forn 1,2 for n= 1,2, , N is Pareto efficient. N is a CE allocation, then(n, ên) Here is an outline of a proof by contradiction. Because (n, en2) for n = 1,2, , N is a CE allocation, there is a price p = (P1,P2) such that nen2) for 1,2, V and is a CE. If ne2 for 1,2, N is not Pareto efficient, then there exists another allocation, call it (n,n2) for n-1,2, N that is Pareto superior to (ni,) for n1,N and is feasible It follows from Pareto superiority that there is some person, say person i, who strictly prefers (ai,a2) to (a1, a). Therefore

Answer 1

A competitive allocation is an equilibrium condition where the interaction of the profit maximizing producers and utility maximizing consumers in the competitive markets with freely determined prices arrive at an equilibrium. The quantity supplied equals the quantity demanded in this equilibrium.

Pareto efficient condition is that condition from which it is impossible to reallocate the allocation so as to make one individual or preference better off without making atleast other individual or preference worse off.

However, the competitive equilibrium given here, (c^n1,c^n2)is not the Pareto efficient outcome, hence there will definitely exist another equilibrium which will be the Pareto efficient outcome. There exists another allocation (c~n1,c~n2) for n=1,2...N which is Pareto superior to (c^n1,c^n2) for n=1,2,...N and which is also feasible.

An outcome will be Pareto superior if one possible outcome is preferred to another possible outcome just in case even if one member in the group prefers the second outcome i.e one member of the group gets an utility from the second outcome which is greater than the first outcome. And for other members of the group the utility which they get from the second outcome is not less than the utility which they get from the first outcome.

Therefore, from Pareto efficiency if there is some person, person i who strictly prefers (c~n1, c~n2) to (c^n1,c ^n2), therefore, where prices are (p^1,p^2), so,

p^1c~n1 + p^2c~n2 > p^1wi1 + p^2wi2.

This equation states the equilibrium (c~n1,c~n2) could afford the allocated endowments.

Therefore, the Pareto efficient outcome would be (c~n1,c~n2).