Question

Given the following information: utility function is U(x,y)=〖4x〗^0.5 y^0.5 , price of good X is N$5, the price of good Y is N$10 and the consumer income N$400.
What is the level of quantity demanded of good X when the price of good X and Y is N$4 and N$10 respectively? Let us assume good X is on the x-axis.

Answer 1

We have a utility function:

$$E(x,y) = 4x^{0.5}y^{0.5}$$

A consumer with this utility function will maximize her utility by producing at the point where:

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

From the utility function:

$$MU_x = \dfrac{\delta U(x,y)}{\delta x} = 2x^{-0.5}y^{0.5}\\[0.3cm] MU_y = \dfrac{\delta U(x,y)}{\delta y} = 2x^{0.5}y^{-0.5}$$

The price of good x is $5 and the price of good y is $10. Therefore:

$$\dfrac{2x^{0.5}y^{-0.5}}{2x^{-0.5}y^{0.5}} = \dfrac{5}{10}\\[0.3cm] \dfrac{y}{x} = \dfrac{1}{2}\\[0.3cm] x = 2y............(i)$$

The consumer's income is $400. Thus, the budget line is:

$$400 = 5x + 10y$$

Substituting equation (i) into the budget constraint, we get:

$$400 = 5(2y) + 10y\\[0.3cm] 400 = 20y\\[0.3cm] \color{red}{y^* = 20}$$

Since $x = 2y$ , then:

$$x^* = 2(20)\\[0.3cm] \color{red}{x^* = 40}$$