Question

1. Bernice’s preferences over consumption bundles (X, Y) are summarized by the utility function U (X, Y) = X(Y+ 1)².

a.Derive an algebraic expression for the marginal utility MUx (X, Y) of good X.

b.Derive an algebraic expression for the marginal utility MUy (X, Y) of good Y.

c.   Use your answers from parts (a) and (b) to derive an algebraic expression for Bernice’s marginal rate of substitution (MRS) of good Y for good X. If Bernice is currently consuming 3 units of good X and 5 units of good Y, what is the value of her MRS?

Answer 1

U(X, Y) = X(Y + 1)²

a) Marginal utility of X is calculated by differentiating the utility function with respect to X by keeping Y constant.

So, marginal utility of good X = X(0) + (Y + 1)²(1) = (Y + 1)² is the answer.

b) Marginal utility of good Y is calculated by differentiating the utility function with respect to Y keeping X constant.

So, marginal utility of good Y = (Y + 1)²(0) + 2X(Y + 1) = 2X(Y + 1) is the answer.

c) Marginal rate of substitution of good Y for good X is the rate at which the consumer is willing to sacrifice units of good X in order to obtain one more unit for good Y.

Marginal rate of substitution = marginal utility of good Y / marginal utility of good X

= [ 2X(Y + 1) ] / [(Y + 1)²]

= 2X / (Y + 1) is the answer.

If X = 3 and Y = 5, then, MRS = 2X / (Y + 1)

= (2 x 3) / (5 + 1)

= 6/6 = 1 is the answer.