Utility function

Question

Given a consumer's utility function: U (x,y) = In(x - 3) + In(y)

And their budget constraint: 3x + 2y = 100

How much of good y should this consumer purchase? (Round to two decimal places only if necessary)

Answer 1

Use Lagrange Multipliers where g(x,y)=3x+2y-100 is our constraint function:

L(x,y,λ) = U(x,y) + λg(x,y) = [ln(x-3)+ln(y)] + λ[3x+2y-100] = ln(yx-3y) + 3xλ + 2yλ - 100λ

∂L/∂x = y/(yx-3y) + 3λ = 1/(x-3) + 3λ

∂L/∂y = (x-3)/(yx-3y) - 2λ = 1/y + 2λ

Find critical points

1/(x-3) + 3λ = 0 --> λ = -1/(3x-9)

1/y + 2λ = 0 --> λ = -1/(2y)

λ = -1/(3x-9)

3x-9 = -1/λ

x-3 = -1/(3λ)

x = -1/(3λ)+3

λ = -1/(2y)

y = -1/(2λ)

g(x,y) = 3x + 2y - 100

0 = 3[-1/(3λ)+3] + 2[-1/(2λ)] - 100

0 = -1/λ + 9 - 1/λ - 100

0 = -2/λ - 91

91 = -2/λ

λ = -2/91

Since we are just looking for the max amount of good y, we don't care about finding x:

y = -1/(2λ)

y = -1/[2(-2/91)]

y = -1/(-4/91)

y = 91/4

y = 22.75

Therefore, the consumer should purchase 22.75 of good y.

Hope this helped!