The Second Order Conditions for Multiple Choice Variables

Note: Here, I will present solve problems typical of those offered in a mathematical economics or advanced microeconomics course. The problems were authored by Dr. Charles N. Steele and are reprinted with his generous permission. The solutions to the problems are my own work and not necessarily the only way to solve the problems.

Problem: Explain the second order conditions in the case 1, 2, and n > 2 choice variables for a maximum or minimum in an unconstrained choice problem. What is the relationship with concavity or convexity of the objective function?

Solution:

Second order conditions for 1 choice variable, where z = F(x):

Maximum: d²z < 0

Minimum: d²z > 0

Second order conditions for 2 choice variables, where z = F(x, y):

Maximum: d²z < 0 iff F_xx< 0, F_yy< 0, and F_xxF_yy > (F_xy)²

Minimum: d²z > 0 iff F_xx > 0, F_yy > 0, and F_xxF_yy > (F_xy)²

Second order conditions for n choice variables, where z = F(x₁, x₂, ..., x_n):

Definition: A matrix is positive definite if all of its eigenvalues are greater than 0. Equivalently, a matrix is positive definite if all upper left corners of the matrix have positive determinants.

A matrix is negative definite if all of its eigenvalues are less than 0.

Let x₁, x₂, ..., x_n be the choice variables under consideration. And let all second partial derivatives of the function F exist with respect to the choice variables. Then the Hessian of F is the matrix of second partial derivatives defined as follows

H(F)_ij(x) =

[F_{(x_1)(x_1)} F_{(x_1)(x_2)} ... F_{(x_1)(x_n)}]

[F_{(x_2)x_1)} F_{(x_2)(x_2)} ... F_{(x_2) (x_n)}]

... ... ... ...

[F_{(x_n) (x_1)} F_{(x_n) (x_2)} ... F_{(x_n) (x_n)}], where F_{(x_i) (x_j)} is the partial derivative of F with respect to x_i and x_j.

If at a point K, the Hessian of F is negative definite (i.e., d²z < 0), then F has a local maximum at K.

If at a point K, the Hessian of F is positive definite (i.e., d²z > 0), then F has a local minimum at K.

If the Hessian of F has both positive and negative eigenvalues at point K, then K is a saddle point for F.

Relationship between second order conditions and concavity or convexity of the objective function:

If the objective function is always concave, then any extreme point will be a maximum.

If the objective function is always convex, then any extreme point will be a minimum.

See Mr. Stolyarov's complete list of Mathematical Economics Problems and Solutions.