Answer to Question #297748 in Macroeconomics for Kim

Question

a) Given a utility function

u = x_{1} ^ 0.5 * x_{2} ^ 0.25

If P_{1} = 2 and P_{2} = 4 and Y=100, compute optimal X_{1} and X_{2} that will maximize utility

Answer 1

$U= X_1^{0.5}X_2^{0.25}$

For utility maximization,

$\frac{Mu_1}{Mu_2}= \frac{P_1}{P_2}$

$Mu_1= 0.5X_1^{-0.5}X_2^{0.25}$

$Mu_2= 0.25X_1^{0.5}X_2^{-0.75}$

$\frac{0.5 X_ 2}{0.25X_1}= \frac{2}{4}$

$0.5X_1= 2X_2$

$X_1= 4X_2$

$X_2= 0.5X_1$

Budget line

$2X_1+4X_2= 100$

$2(4X_2)+ 4X_2= 100$

$X_2= 8.33$

$2X_1+ 4(0.5X_1)= 100$

$X_1= \frac{100}{4}= 25$