Question

Suppose a consumers preferences can be represented by the utility function U (x,y)=min(2x,y).also suppose the consumer has 300 dollar to spend and the price of good x is px =3 dollar and the price of good y is py =1 dollar.if the consumer maximize their utility subject to their budget constraint,how much of good x and how much of good y will the consumer purchase?

Answer 1

Maximize U (x,y)=min(2x,y)

Subject to 3x + y ≤ 300

x≥0, y≥0

we then form a lagrangian function to help us solve the problem

L= 2xy + λ(3x+y-300)

$\frac{∂L}{∂x}$ = 2y +3λ =0...............equation 1

$\frac{∂L}{∂y}$ = 2x + λ =0.................equation 2

$\frac{∂L}{∂λ}$ = 3x+y-300 =0.............equation 3

making λ the subject in equations 1 and 2 and then equating them

λ = $\frac{-2y}{3}$

λ= -2x

$\frac{-2y}{3}$ =-2x

x = $\frac{4y}{3}$

substituting x = $\frac{4y}{3}$ in equation 3

3x+y =300

$\frac{3(4y)}{3}$ +y =300

5y = 300

y = 60

substituting y = 60 in x = $\frac{4y}{3}$

x = $\frac{4\times60}{3}$

x= 80

Therefore the consumer will purchase 80 units of X and 60 units of y