Question

given the following information U(x y)=x^0.5,y^0.5,the initial price of good X is $10 and good Y is $5:consumers income $1000.calculate the optimal bundle at initial bundle with good X on horizontal axis

Answer 1

Optimal bundle will occur where:

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

The utility function is:

$$U(x, y)=x^{0.5}y^{0.5}$$

Therefore:

$$MU_x = 0.5x^{-0.5}y^{0.5}\\[0.3cm] MU_y = 0.5x^{0.5}y^{-0.5}$$

In our question, $P_x = \$10, \quad P_y = \$5$ . Therefore:

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

The consumer's income is $1,000. Therefore, the budget constraint is:

$$1000 = 10x + 5y$$

Substituting equation 1 into the budget constraint:

$$1000 = 10x + 5(2x)\\[0.3cm] 1000 = 20x\\[0.3cm] \color{red}{x^* = 50}$$

Since $y = 2x$ , then:

$$y^* = 2(50)\\[0.3cm] \color{red}{y^* = 100}$$