Question

1. Chuck has the following quasi-linear utility function: a) Derive Chuck's demand curve for x as a function of P,and P b) Derive Chuck's demand for for y c) Is y a normal good?

Answer 1

according to the question:

a)

$U(x,y) = \sqrt x+ y\\ \\ subject \, to \, \\ p_x x + p_y Y = M\\ where \, p_x \, and \, p_y \, are\, the\, prices\, of\, x\, and\, y\, respectively.\\ \textit{L} = \sqrt x+ y - \lambda (p_x x + p_y Y -M) \\$

$\\ \frac{\partial L}{\partial x} = \frac{1}{2 \sqrt x} - \lambda p_x = 0 \, (1) \\ \\ \frac{\partial L}{\partial y} = 1 - \lambda p_y = 0 \, (2) \\$

$\frac{\partial L}{\partial \lambda} = p_x x + p_y y =M \, (3) \\$

Using equations (1) and (2), we get

$\frac{1}{2 \sqrt x} = \lambda p_x \\ \frac{1}{p_y} = \lambda \\ \frac{1}{2 \sqrt x} = \frac{1}{p_y} p_x \\ x = (\frac{p_y}{2 p_x})^2$

Demand function of x :

$x = (\frac{p_y}{2 p_x})^2$

It depends on the prices of y and x. Since its a quasilinear function in x so demand of x does not depend on income.

b) Demand function of y:

Using the budget constraint equation and demand function of x :

$p_x x + p_y Y = M \\ and \, \\ x = (\frac{p_y}{2 p_x})^2 \\ p_x (\frac{p_y}{2 p_x})^2 + p_y y = M\\ \frac{(p_y)^2}{4p_x} + p_y y =M\\ y = \frac{M}{p_y} - \frac{p_y}{4p_x}$

Thus, demand for y depends on prices as well as on income.

c) A normal good is a good whose demand increases as the consumer's income increases.

To check whether y is a normal good or not, we have to evaluate the income elasticity of y.

$y = \frac{M}{p_y} - \frac{p_y}{4p_x} \\ \frac{dy}{dM} = \frac{1}{p_y}$

Income elasticity = % change in demand / % change in M

$E_m = \frac{dy}{dM} \frac{M}{y}$

$E_m = \frac{dy}{dM} \frac{M}{y} \\ E_m = \frac{1}{p_y} \frac{M}{y}$

Em is positive since M, y and p_y are positive numbers. So, y is the normal good.