Question

A consumer has a utility function given by lnU =5linx subscrip1 +3lnx subscript 2 if the budget constraint is given by 10x subscript 1 +14x subscript 2 =124 find the optimal quantity of the two commodities

Answer 1

Need to find-

optimal quantity of the two commodities

Given in the question-

$lnU =5lnx_1 +3lnx_2$

$lnU = lnx_1^5 + lnx_2^3$

$lnU = ln(x_1)^5\times(x_2)^3$

$U = (x_1)^5\times(x_2)^3.................(1)$

Budget equation-

$10x_1 + 14x_2 = 124 ...............(2)$

Equilibrium condition -

$\frac{MU_x}{MU_y} =\frac{ P_1}{P_2}$

$MU_x= 5\times(x_1)^4\times(x_2)^3$

$MU_y = 3(x_1)^5\times(x_2)^2$

$\frac{MU_x}{MU_y }= \frac{(5\times(x_1)^4\times(x_2)^3)}{(3(x_1)^5\times(x2)^2)}$

$\frac{MU_x}{MU_y} =\frac{ 5x_2}{3x_1}$

$\frac{P_1}{p_2}= \frac{10}{14} = \frac{5}{7}$

$\frac{5x_2}{3x_1} =\frac{ 5}{7}$

$7x_2 = 3x_1$

Using this relation and putting the in equation "2"

So,

$x_1 = 7$

$x_2 = 3$

Answer -

Optimal bundle will be (7,3)