Mathematical methods for economic theory

Martin J. Osborne

Contents Text Exercises

6.1.3 Exercises on optimization with an equality constraint: sufficient conditions for a local optimum for a function of two variables

For the problem
max_x,y x² + y² subject to x² + xy + y² = 3
find all the solutions of the first-order conditions and determine, if possible, whether each solution is a local maximizer or a local minimizer.
Solution
The bordered Hessian is

0 2x + y x + 2y

2x+y 2 − 2λ −λ

x + 2y −λ 2 − 2λ

.

The determinant of this matrix is −2[5x² + 8xy + 5y²] + 6λ[x² + xy + y²]. Evaluate this at the solutions of the first-order conditions (found in a previous problem):
- (1, 1, 2/3): determinant is −24 < 0, so (1, 1) is a local minimizer.
- (−1, −1, 2/3): determinant is −24 < 0, so (−1, −1) is a local minimizer.
- (√3, −√3, 2): determinant is 24 > 0, so (√3, −√3) is a local maximizer.
- (−√3, √3, 2): determinant is 24 > 0, so (−√3, √3) is a local maximizer.