Question

Consider a market with two goods, x and z that has the following utility function U = x^0.2z^0.8

a) What is the marginal rate of substitution?

b) As a function of the price of good x (px), the price of good z (pz) and the income level (Y ), derive the demand functions for goods x and z.

Answer 1

a) Marginal rate of substitution is the rate at which the consumer is willing to give up one good in order to obtain one more unit of the other good.

In this case, marginal rate of substitution = marginal utility of good x / marginal utility of good z

Marginal utility of the two goods can be calculated by differentiating the utility function with respect to that good keeping the other good constant.

Marginal utility of good x = 0.2x^-0.8z^0.8

Marginal utility of good z = 0.8x^0.2z^-0.2

So, marginal rate of substitution = 0.2x^-0.8z^0.8 / 0.8x^0.2z^-0.2

= 0.2x^-1/0.8z^-1 = 2z/8x = z/4x is the answer.

b) The budget line equation of the above question can be written as:

p_xx + p_zz = Y

The demand functions of the two goods can be calculated by using the optimal condition that is,

Marginal rate of substitution = p_x / p_z

So, z/4x = p_x / p_z

z = 4x ( p_x / p_z)

Putting the value of z in the budget line equation above,

p_xx + p_z [4x ( p_x / p_z)] = Y

p_xx + 4p_xx = Y

5p_xx = Y

x = Y / 5p_x is the answer.

Now, putting the value of x in the equation of z derived above,

z = 4x ( p_x / p_z)

z =  4 X (Y / 5p_x) X ( p_x / p_z)

z = 4Y / 5p_z is the answer.