Question

Suppose that a consumer consumes two goods X and Y and derives utility according the following utility function where U = 25X2/5Y 3/5 where α = 2/5 and β = 3/5 a. If Px is the price of good X and Py is the price of good Y and the consumer’s income is M. Derive the demand functions for the two goods X and Y b. If Px is shs 15 and Py is shs 10 and the consumer has shs.800 to spend on the two goods what are the optimal quantities of X and Y that maximize the consumer’s utility? c. Using the information in b above show that the values of α and β represent the proportion of the consumer’s income spent on good X and good Y respectively

Answer 1

$U= 25X^\frac{2}{5}Y^\frac{3}{5}$

$\alpha= \frac{2}{5}$

$\beta= \frac{3}{5}$

a) Demand functions

Budget = $P_xX+ P_yY= m$

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

$Mu_x= 10X^\frac{-3}{5}Y\frac{3}{5}$

$Mu_y= 15X^\frac{2}{5}Y\frac{-2}{5}$

$\frac{10X^\frac{-3}{5}Y\frac{3}{5}}{15X^\frac{2}{5}Y\frac{-2}{5}}= \frac{P_x}{P_y}$

$\frac{10Y}{15X}= \frac{P_x}{P_y}$

$15P_xX=10P_yY$

$X= \frac{0.67P_yY}{P_x}$

$Y= \frac{1.5P_xX}{P_y}$

Plug into the budget Equation

$P_xX+P_yY=m$

$P_x(\frac{0.67P_yY}{P_x})+P_yY=m$

$0.67P_yY+P_yY=m$

$\frac{2}{3}P_y+P_yY=m$

$5P_y= 3m$

$Y^*= \frac{3m}{5P_y}$

$P_xX+P_y(\frac{1.5P_xX}{P_y})=m$

$2.5P_xX=m$

$X^*= \frac{m}{2.5P_x}$

$= \frac{m} {\frac{5}{2}P_x}$

$X^*= \frac{2m}{5P_x}$

b) Optimal Quantities

$Y^*= \frac{3m}{5P_y}= \frac{3\times800}{5\times10}$

$=\frac{2400}{50}= 48$

$X^*= \frac{m}{2.5P_x}= \frac{800}{2.5\times 5}= 64$

Amount spend on good X=$64\times5=320$

Amount spend on good Y=$48\times 10= 480$

Proportion on X= $\frac{320}{800}=0.4=\alpha$

Proportion on Y= $\frac{480}{800}=0.6=\beta$