Question

Suppose that the consumer’s utility function is given by 1/ 2 1/ 4 U  10X Y . If price of good X and Y are 4 and 2

respectively and income constraint is Birr 100.

a. The demand equation for X and Y.

b. The utility maximizing levels of X and Y.

c. The maximum utility.

d. Compute both X ,Y Y ,X MRS and MRS . Is there any difference between the two?

Answer 1

$U(10X^\frac {1}{2}Y^\frac{1}{4})$

$P_x=4$

$P_y=2$

Income= 100

a)

$\frac{Mu_x}{Mu_y}=\frac{P_x}{P_y}$

$Mu_x=5x^\frac{-1}{2}Y^\frac{1}{4}$

$Mu_y=2.5x^\frac{1}{2}Y^\frac{-3}{4}$

$\frac{5x^\frac{-1}{2}Y^\frac{1}{4}}{2.5x^\frac{1}{2}Y^\frac{-3}{4}}=\frac{P_y}{P_x}$

$\frac{5Y}{2.5X}=\frac{P_x}{P_y}$

$Y=\frac{2.5XP_x}{5P_y}$

=$\frac{0.5XP_x}{P_y}$ ...........(i)

$X=\frac{5P_yY}{2.5P_x}$

$X=\frac{2P_y}{P_x}$ .........(i

$100=P_x(\frac{2P_yY}{P_x})+P_yY$

$100=2P_yY+P_yY$

$100=P_yY(2+1)$

$P_yY=\frac{100}{3}$

$Y^*=\frac{100}{3P_y}...... Demand function for Y$

$100=P_xX+P_y(\frac{0.5P_xX}{P_y})$

$P_xX=\frac{100}{1.5}$

$X^*=\frac{100}{1.5P_x}$.......Demand function of X

b) Maximizing Levels of X and Y

$\frac{5x^\frac{-1}{2}Y^\frac{1}{4}}{2.5x^\frac{1}{2}Y^\frac{-3}{4}}=\frac{4}{2}$

$\frac{5Y}{2.5X}=\frac{4}{2}$

Y=X

$P_x\times X= P_y\times Y= m$

$4Y+2Y=100$

$Y=\frac{100}{6}=\frac{50}3$

X= $\frac{50}3$

c) Maximum Utility= $\frac {50}{3}+\frac{50}{3}=\frac{100}{3}=33.33$

d) Marginal rate of substitution of the X and Y

$MRS_{x_y}= \frac{\delta U}{(\delta_x\delta_y)}$

$MRS_{x_y}= \frac{0.125}{5\times 2.5}$ =0.01

$MRS_{y_x}= \frac{0.125}{2.5\times 5}$ =0.01

There is no difference in the MRS