Constrained optimisation problem / firm profit function

Question

A firm uses two inputs x and y, and their profit function is P(x,y)=2xy+2x+3y. Input x costs $2 each and y costs $1 each and they are constrained to spend a total of $100 on inputs.

If the firm wants to maximise profit, they should use _____ of input x, _____ of input y. In addition, the shadow price will be ______.

Round your answer to one demical place only where necessary.

Answer 1

Consider using Lagrange Multipliers

Profit Function: P(x,y) = 2xy+2x+3y

Constraint Function: g(x,y) = 2x+y-100

Lagrangean Function: L(x,y,λ) = P(x,y) + λg(x,y) = (2xy+2x+3y) + λ(2x+y-100)

Find critical points

∂L/∂x = 2y+2+2λ

∂L/∂y = 2x+3+λ

2y+2+2λ = 0 --> λ = -y-1

2x+3+λ = 0 --> λ = -2x-3

λ = -y-1 --> y = -λ-1

λ = -2x-3 --> x = (-1/2)(λ+3)

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

0 = 2(-1/2)(λ+3)+(-λ-1)-100

0 = -λ-3-λ-1-100

0 = -2λ-104

2λ = -104

λ = -52

x = (-1/2)(λ+3)

x = (-1/2)(-52+3)

x = (-1/2)(-49)

x = 49/2 = 24.5

y = -λ-1

y = -(-52)-1

y = 52-1 = 51

The shadow price measures the increase in profit that would result from a one-unit increase in the budget for input y, while holding all other inputs and output constant:

Shadow Price = ∂P/∂y = 2x+3 = 2(24.5)+3 = 49+3 = 52

If the firm wants to maximize profit, they should use 24.5 of input x, 51 of input y. In addition, the shadow price will be $52 (per unit of input y).

Hope this helped!