Question

A consumer with income M, derives utility from consuming good X and Y.
He has a Cobb-Douglas utility function, U(X,Y) = X^1/2 Y^1/2
The prices of good X and Y are Px and Py respectively.
a) Derive his demand functions for X and Y.
b) Derive the price elasticity of demand for X.
c) Is X a normal good?

Need help in part b

Answer 1

$$Solution$$

$U(X,Y)=X^{0.5}Y^{0.5}\\$

Solve the consumers optimization problem by maximizing subject to the budget constraint.

$P_x\cdot X+P_y\cdot\ Y\leq\ m\\$

Then, lets use the langrange Theorem to rewrite the constrained optimization problem into a non constrained form,

$\ Max\ L(X, Y, \lambda)={X^{0.5}}{Y^{0.5}}+ \lambda(m-P_xX-P_yY$

The first order condition(necessary) will result in

$0.5XY^{0.5}=\lambda\ P_x......(i)\\ 0.5YX^{0.5}=\lambda P_y.......(ii)$

Combining 1 and 2 will result in

$0.5P_yY=0.5P_xX\\ x=\frac{0.5m}{0.5+0.5P_x}\implies is\ the\ demand\ function$

Price elasticity

$=\frac{0.5}{1p}/\frac{0.5}{1p}=1$

c)X is a normal good since it has a positive income elasticity of demand.