Question

2.) 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 Y 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.
3.) 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.

Answer 1

The utility function is:

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

The prices of goods x and y are $P_x = \$5\text{ and } P_y = \$10$ respectively. The consumer will maximize her utility at the point where:

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

Therefore:

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

Thus:

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

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

$400 = 5x + 10y$

Substituting equation (i) into the budget constraint:

$400 = 2y + 10y\\[0.3cm] 400 = 12y\\[0.3cm] \color{red}{y^* = \dfrac{100}{3}}$

But $x = 2y$ . Therefore:

$x = 2\left(\dfrac{100}{3}\right)\\[0.3cm] \color{red}{x^* = \dfrac{200}{3}}$