Question

(a) Calculate the marginal utility of x1 and x2 for the following utility function u (x1; x2) = x 1 x 2

(b) What must be true of and for the consumer to have a positive marginal utility for each good?

(c) Does the utility function above exhibit a diminishing marginal rate of substitution? Assume that and satisfy the conditions from Part b. (Hint: A utility function exhibits a diminishing marginal rate of substitution if the derivative of the marginal rate of substitution is positive).

Answer 1

(a) Marginal utility is mathematically defined as the rate of change of Utility for one extra unit of quantity change.

U = x1x2

MU1 = dU/dx1 = x2

MU2 = dU/dx2 = x1

(b) For the consumer to have positive marginal utility it must be true that the good he consumes is a good that gives positive utility only and consuming more of this good is not harmfull in any manner. This means that good should be 'GOOD' .For e.g cigarate is not a good commodity. It causes disutility to both user and to the surrounding. A good commodity could be many things like home made food, fruits, education etc.

(c) A good which has diminishing mrs means that IC is convex shaped and the consumer is willing to part with less of one good to get an additional unit of other good. Mathematically MRSx1,x2 = MUx1/MUx2 = x2/x1 (from a)

dMRS/dx = 0 ( because both marginal utilities are quantity of goods x1 and x2. So the MRS is a constant and that is why the derivative is 0). This implies constant marginal substitution rate of 0.

This utility function exhibits weak convexity preferences and can be said to give diminishing returns under this preferences only.