Question

Only 8.2 and 8.3 needs to be answered. I solved 8.1

Question 8: 8.1. Consider a transformation of the utility function in Question 7 using u In(u). In other words, the new utility function u' = In(u) = În(xay) = a × ln(x) + b × ln(y). What is MRSy of this new utility function? Is it the same as or different from MRSy 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). 11-112 b). 11, 1/112 c). 11, 1987 X 11-507 d). 11, = eu 8.3. Explain why taking a monotonic transformation of a utility function does not change the marginal rate of substitution (MRS)

Answer 1

8.2)

Here, we will check for all three conditions on individual basis.

For First one, U'=U^2

Not differentiating both side with respect to x and y we will arrive at below solution -

dU'/dx=2U*dU/dx

dU'/dy=2U*dU/dy

Hence we can say that

MRS of U'=MRS of U

Now for second case, U'=1/U^2

U' = U^-2

Not differentiating both side with respect to x and y we will arrive at below solution -

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

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

Hence we can say that,

MRS of U'= MRS of U

For Third one, U'=1987U-507

Therefore, we can say that MRS of U'=MRS of U as differentiation will result the same yield.

For Fourth one, U'=exp(u)

Not differentiating both side with respect to x and y we will arrive at below solution -

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

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

MRS of U'= MRS of U

8.3)

if we will consider a monotonic transformation of utility, then the properties of indifference curve of original utility will be equal to new utility. And due to same, it will not change the marginal rate of substitution.