constrained optimisation problem

Question

What is the shadow price in the following constrained optimisation problem?

Max y = 3x₁x₂ + x₂²

s.t. 2x₁ + x₂ = 100

Answer 1

Let's rewrite each function into something more understandable and use Lagrange Multipliers:

Max: M(x,y) = 3xy + y²

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

Lagrangian Function: L(x,y,λ) = M(x,y) + λg(x,y) = (3xy+y²) + λ(2x+y-100)

Find critical points

∂L/∂x = 3y+2λ

∂L/∂y = 3x+2y+λ

3y+2λ = 0 --> y = -2λ/3

3x+2y+λ = 0

3x+2(-2λ/3)+λ = 0

3x-4λ/3+λ = 0

3x-λ/3 = 0

3x = λ/3

x = λ/9

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

0 = 2(λ/9)+(-2λ/3)-100

0 = 2λ/9-2λ/3-100

0 = -4λ/9-100

4λ/9 = -100

λ = -900/4

λ = -225

x = λ/9 = -225/9 = -25

y = -2λ/3 = -2(-225)/3 = 450/3 = 150

Shadow Price = ∂M/∂y = 3x+2y = 3(-25)+2(150) = -75+300 = $225 (per unit of input y)

Hope this helped!