Question

Imagine that two consumers, A and B, have identical utility functions equal to U=x^(1/4)y^(3/4). Person A has j units of x and k units of y. Person B has m units of x and n units of y. The variables j,k,m,n will be chosen from the set of any positive whole numbers.

Part a) Show that this allocation of x and y is inefficient.

Part b) Propose a trade that would be a Pareto improvement.

Answer 1

Both A and B have identical utility functions given by:
$U = x^{1/4}y^{3/4}$
Person A gets j units of x and k units of y.
Person B gets m units of x and n units of y.

Let us find the marginal rate of substitution for both A and B.
$MRS(x,y) = - \frac{\partial u/\partial x}{\partial u/\partial y}$

Thus,
$MRS(x,y) = - \frac{y}{3x}$

This marginal rate of substitution holds for both Person A and Person B.
As long as j $\neq$ m and k $\neq$ n, the allocation will never be efficient since the values of MRS will differ and one person will always prefer a good over the other person.

A trade becomes Pareto optimal and leads to an improvement if goods are traded in a manner that their MRS values become equal and there is no scope for further improvement.