Question

8.1. Consider a transformation of the utility function in Question 7 using In(u). In other words the new utility function u' = In(u) = In(xay!) = x In(x) + b × ln(y). What is MRSr.y of this new utility function? Is it the same as or different from MRS,y you found in Q7.3? Explain. 8.2.Will the MRS be still the same for each of the following transformation? Explain without directly solving for MRS. a), u, = u2 b). 1/ = 1/1.2 c). 1/-1987x 11-507 d), u' = c" 8.3. Explain why taking a monotonic transformation of a utility function does not change the marginal rate of substitution (MRS).

Answer 1

Ans 8.1

U=x^a+y^b

MRS(x,y)=(dU/dx)/(dU/dy)

dU/dx=ax^(a-1)

dU/dy=by^(b-1)

MRS(x,y)=(a/b)*(x^(a-1)/y^(b-1))

When U'(x,y)=alnx+blny

dU'/dx=a/x and dU'/dy=b/y

MRS=(a/b)(y/x)

Both MRS expressions are not same.

Ans 2)

Let's check case by case

U'=U^2

dU'/dx=2U*dU/dx and dU'/dy=2U*dU/dy

MRS of U'=MRS of U

U'=U^-2

dU'/dx=(-2U^(-3))dU/dx and dU'/dy=(-2U^(-3))dU/dy

MRS of U'= MRS of U

U'=1987U-507

Therefore MRS of U'=MRS of U

U'=exp(U)

dU'/dx=exp(U) dU/dx

dU'/dy=exp(U) dU/dy

MRS of U'= MRS of U

Ans 8.3

All given functional form of Utility these are homogeneous and monotonic functions and these transformations are homothetic hence when we have homothetic functions as transformed one properties of original functional form remaims the same.

Hence properties of indifference curve of original utility function will be similar to the properties of indifference curve of new utility curve