Exercise Set 8
Finding the Extremum of a Function

Objective function: z = xy

Constraint: x + y = 8

Note the following:
• The objective function, f(x,y), is xy.
• The choice variables are x and y.
• Rewrite the constraint in the form g(x,y) = c. Here, g(x,y) = x + y; c = 8.
• Denote the Lagrange multiplier by λ.

Step 1. Obtain the Lagrangian function, L(x,y,λ) = f(x,y) + λ[c - g(x,y)].

L = xy + λ(8 - x - y)

Step 2: Obtain the first-order derivatives of the Lagrangian function with respect to x , y and m respectively.

L₁ = y - λ

L₂ = x - λ

L₃ = 8 - x - y

Step 3: The first-order necessary conditions for an extremum are: L₁ = 0, L₂ = 0, L₃ = 0. Use the FOCs to solve for the critical values of x, y and λ.

x =4, y = 4, λ= 4

Step 4: Set up the bordered Hessian, H_bar. This is needed to verify second-order condition for a maximum or a minimum.

L₁₁ = 0, L₁₂ = 1

L₂₁ = 1, L₂₂ = 0

g₁ = 1, g₂ = 1

H_bar = ________________________

Step 5: Check the sign of the determinant of H_bar. If positive: maximum. If negative: minimum.

det(H_bar) = _____________

Solution:
The objective function is maximized at x = 4, y = 4.
The maximum value of the function is 16.

Question:
Suppose c increases to 25. Obtain the new critical values of x and y.