Lecture 6
UNCONSTRAINED OPTIMIZATION: THE MANY-VARIABLE CASE

1. Functions of two variables

• Consider the function z = f(x,y)

1.1 Find the extremum of the function

1. Obtain the first-order and second-order partial derivatives of the function:
   • f₁, f₂
   • f₁₁, f₁₂, f₂₁, f₂₂

2. Write down the first-order conditions for an extremum.
   • f₁(x,y) = 0     (1)
   • f₂(x,y) = 0     (2)

3. Solve (1) and (2) for x and y. Denote them x₀ and y₀.
   Note: There may be more than one set of (x,y) that satisfy equations (1) and (2).

4. The function attains an extremum value at (x₀,y₀).

5. Evaluate the second-order partial derivatives at (x₀,y₀).

6. Check the second-order conditions to verify whether the extremum is a maximum or a minimum:
   Maximum if: f₁₁ < 0, f₂₂ < 0, f₁₁f₁₂ > f₁₂²

   Minimum if: f₁₁ > 0, f₂₂ > 0, f₁₁f₁₂ > f₁₂²

1.2 The Hessian matrix, H

• The matrix whose elements are the second-order partial derivatives evaluated at (x₀,y₀):

		H 	=	(  f11	f12  )
				(  f21	f22  )	

• Obtained by deleting successive rows and columns of the Hessian determinant (the "all-but" rule):
  • H₁ = f₁₁ (delete all but the first row and column)
  • H₂ = f₁₁f₂₂ - f₁₂² (delete all but the first two rows and columns)

• Note: In the two-variable case, H₂ = det(H), the Hessian itself.

1.3 Second-order conditions revisited

• For a maximum: H₁ < 0, H₂ > 0
• For a minimum: H₁ > 0, H₂ > 0

2. General case

• Consider the function y = f(x₁,x₂,...,x_n)

2.2 First-order conditions (FOC)

• f₁ = 0, f₂ = 0, ..., f_n = 0

2.3 Second-order conditions (SOC)

Look at the signs of the principal minors.

• If they alternate in sign, the first being negative, the extremum is a maximum:
  H₁ < 0, H₂ > 0, H₃ < 0,...

• If all the n principal minors are positive, the extremum is a minimum:
  H₁ > 0, H₂ > 0, H₃ > 0,..., H_n > 0

3. Application

• Consider a firm that produces two goods. The demand for each, and the firm's cost function are given:
  • Demand for Good 1: Q₁ = 40 - 2P₁ - P₂
  • Demand for Good 2: Q₂ = 35 - P₁ - P₂
  • Cost function: C(Q₁,Q₂) = Q₁² + 2Q₂² + 10

• The firm wishes to maximize profits by choosing the optimal values of Q₁ and Q₂.

• Rewrite the demand functions as:
  • P₁ = f(Q₁,Q₂)
  • P₂ = g(Q₁,Q₂)

• Obtain the revenue functions:
  • R₁(Q₁,Q₂) = P₁Q₁
  • R₂(Q₁,Q₂) = P₂Q₂

• Obtain the profit function π(Q).

• Obtain the first-order and second-order derivatives of π(Q).

• Write down the FOC.

• Solve for the optimal values of the output levels.

• Check the SOC. [Use the Hessian.]

• Obtain the maximum value of π(Q).