Question

Suppose a consumer x and y has the following utility function U=x^0.4y^0.8 if Price of good x and y 10 and 20 and income constraint is 500$. Find the quantities of x and y maximize utility

Answer 1

The utility function is:

$$U=x^{0.4}y^{0.8}$$

Utility will be maximized when:

$$\dfrac{MU_x}{MU_y} = \dfrac{P_x}{P_y}$$

From the utility function:

$$MU_x =0.4 x^{-0.6}y^{0.8}\\[0.3cm] MU_y = 0.8x^{0.4}y^{-0.2}$$

The prices are:

$$P_x = 10\\[0.3cm] P_y = 20$$

Therefore:

$$\dfrac{0.4x^{-0.6}y^{0.8}}{0.8x^{0.4}y^{-0.2}} = \dfrac{10}{20}\\[0.3cm] y = x ..............\text{Eqn 1}$$

Suppose the consumer buys x units at Px = 10 and y units at Py. If his income is $500, the budget line is:

$$10x + 20y = 500$$

Substituting equation 1 into the budget line:

$$10x + 20x = 500\\[0.3cm] 30x = 500\\[0.3cm] \color{red}{x^* = \dfrac{50}{3}}$$

Since $x = y$ , then:

$$\color{red}{y^* = \dfrac{50}{3}}$$