Question

Consider a consumer with preferences represented by the utility function:

u(x; y) = x^1/4y^1/2

Suppose the consumer has income M = 10 and the prices are px = 1 and py = 2.

(a) Are goods x and y both desirable?

(b) Are there implications for the utility maximization problem for the consumer from your finding in a? If so, explain in detail.

Answer 1

The utility function is U(x,y) = x^(1/4)y^(1/2)

The budget equation is Px*x + Py*y = M

Px = 1 and Py =2 and M=10

Thus, the budget equation is 20 = x + 2y

The marginal utility of x = MUx = (1/4)*x^((1/4)-1)*y^(1/2) = (1/4)*x^(-3/4)*y^(1/2)

The marginal utility of y = MUy = (1/2)*x^(1/4)*y^((1/2)-1) = (1/2)*x^(1/4)*y^(-1/2)

(a)

Yes, the goods x and y are desirable. This is because both x and y are included in the utility function which is not linear, indicating that both consumption of x and y matters for utility maximization

(b)

The implication is that both x and y are to be consumed in order to maximize the utility.

That is both x and y contribute to utility maximization.

At utility maximization equilibrium, both x and y have values greater than zero.

At equilibrium,

MRSx,y = Px/Py

MUx/MUy = Px/Py

(1/4)*x^(-3/4)*y^(1/2)/ (1/2)*x^(1/4)*y^(-1/2) = 1/2

x/2y = 1/2

x=y

x + 2y = 10

x + 2x = 10

3x = 10

x = 10/3 and y= 10/3

Thus, both x and y contribute to utility maximization.