Question

3. The following utility function is known as CES (constant elasticity of substi- tution) function where α > 0 β > 0 (a) Is this function homothetic? (b) How does the MRS,y depend on the ratio x/y? Specifically, show that the MRSy is strictly decreasing in the ratio x/y for all values δ < 1, increasing in the ratio x/y for all values δ > 1 and constant for δ 1. (c) Show that if x = y, the MRS of this function depends only on the relatively sizes of α and β

Answer 1

A).

So, here the utility function is given by, “U=[aX^d + bY^d]^1/d, where “a, b > 0”.

Now, let’s assume another function “Z(X, Y)= aX^d + bY^d”, => Z(tX, tY)= a(tX^d) + b(tY^d).

=> Z(tX, tY)= t^d[aX^d + bY^d] = t^d*Z, => “Z” is homogeneous of degree “d”.

So, here “U” can be written as, “U=Z^1/d, => dU/dZ = (1/d)*Z^1/d-1 > 0 for all “d >0”. So, here “U” is a homothetic function.

B).

So, given the utility function, “U(X, Y) = [aX^d + bY^d]^1/d”,

=> Ux = (1/d)*[aX^d + bY^d]^1/d-1*a*X^d-1.

=> Uy = (1/d)*[aX^d + bY^d]^1/d-1*b*Y^d-1, => MRS = Ux/Uy.

=> MRS = {(1/d)*[aX^d + bY^d]^1/d-1*a*X^d-1}/{(1/d)*[aX^d + bY^d]^1/d-1*b*Y^d-1}.

=> MRS = {a*X^d-1}/{b*Y^d-1} = (a/b)*(X/Y)^d-1. So, we can see that “MRS” depends on the ratio “X/Y”.

Now, for “d<1”,=> d-1<0, => X/Y having negative power, => as (X/Y) increases MRS decreases. Similarly if “d > 1”,=> d-1 > 0, => X/Y having positive power, => as (X/Y) increases MRS also increases. Now, for “d=1” the MRS is given by “a/b”, => MRS is constant.

C).

The MRS is given by.

=> MRS = (a/b)*(X/Y)^d-1, => MRS = (a/b) for “X=Y”. So, here MRS depends on the relative value of “a” and “b”.