Question

Consider the following utility functions:

(i) u(x,y) = x²y

(ii) u(x,y) = max{x,y}

(iii) u(x,y) = √x + y

(a) For each example, with prices p_x = 2 and p_y = 4 find the expenditure minimising bundle to achieve utility level of 10.

(b) Verify, in each case, that if you use the expenditure minimizing amount as income and face the same prices, then the expenditure minimizing bundle will maximize your utility.

Answer 1

a) (i) Since the utility function is quasi-concave, we use the equilibrium condition, i.e. MRS equals price ratio. So, the Marshallian demand functions are as follows:

Using these demand function in the utility function, we get the indirect utility function as follows:

To achieve a utility of 10, we insert v=10 in the above equation to get the expenditure minimizing value:

(ii) Since this utility function is not quasi-concave, we cannot use the usual condition, i.e. MRS equals price ratio. As the marginal utility of both the goods is equal, the consumer will consume the good which is cheaper (good x in this case). So, the Marshallian demand functions are:

The corresponding indirect utility function is:

To achieve a utility of 10, the expenditure minimizing value is

(iii) MRS equals price ratio implies

Using this in budget equation, we get

The correponding indirect utility function is

The expenditure minimizing value is

b) It is easy to check that expenditure minimizing bundle is utility maximizing bundle.