1. the new utility function u′ = ln(u) = ln(x^a. y^b)=a*ln(x)+b*ln(y).
what is mrs_x, y of this new utility function?
2. will the mrs be still the same for each of the following transformation? explain without directly solving for mrs.

a) u′=u2
b) u′=1/u2
c) u′=1987*u−507
d) u′=eu

Explanation:

To begin, we will have to check for all 3 conditions on individual basis

Case one;

U'=U²

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

Case Two;

U'=1/U²

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

Case Three;

U'=1987U-507

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

Case Four;

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

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 have effect on the marginal rate of substitution.