Question

What is the MRS for the following utility function U(x1, x2) = ln(x1) + ln(x1 + x2) (show all the work). Does it satisfy the law of diminishing MRS?

Answer 1

The Given function can be an indifference curve with a constant utility considered for various combinations of x₁ and x₂ . The variables are consumption levels of 2 goods. It implies that x₁ >0 and x₂ >0

MRS can be found using the MU of x₁ and x₂as follows

MRSx₁x₂ = (2x₁ + x₂)/x₁

Diminishing MRS occurs in the case of a convex utility function (when second order derivatives with respect to x₁ and x₂ are positive).

BUT, Second order derivatives with respect to x₁ and x₂ are positive (concave function)

Also, change in MRS with respect to change in x₁ and x₂ (First order derivatives) would be negative for diminishing MRS.

But, change in MRS with respect to change in x₁ and x₂ (First order derivatives) is positive.

Therefore, given utility function does not satisfy the law of diminishing MRS.

It exhibits Increasing MRS.