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 1a? If so, explain in detail.

(c) Derive the utility maximizing bundle.

Answer 1

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

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

1*x + 2*y = 10

x + 2y = 10

The absolute slope of the budget line assuming x is measured along X-axis and y is measured along Y-axis = Px/Py = 1/2

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

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

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

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

The marginal rate of substitution = MRSx,y = MUx/MUy = (1/4)*x^(-3/4)*y^(1/2)/(1/2)*x^(1/4)*y^(-1/2)

=y/2x

(a)

Yes, both good x and y are desirable. It is only when the goods are desirable they form the basis to derive utility from their respective consumption.

Since x and y are included in the utility function that has non-linear form, this is indicative of the fact that both the goods are desirable.

(b)

The implication is that utility maximization would have both goods contributing to the utility maximization.

(c)

At equilibrium,

Px/Py = MRSx,y

(1/2) = y/2x

y=x

Substituting it in the budget equation, the result is:

x + 2y = 10

x + 2x = 10

3x = 10

x = 10/3 and y = 10/3

Thus, utility maximization bundle is x=10/3 and y=10/3