Jacobian Matrix

BIBLIOGRAPHY

The Jacobian matrix was developed by Carl Gustav Jacob Jacobi (1804–1851), a German Jewish mathematician. The Jacobian is a matrix whose entries are first-order partial derivatives defind as where the function is given by m real-valued component functions, y₁(x₁, …,x_n), …,y_m (x₁, …,x_n), continuous (smooth with no breaks or gaps) and differentiable (the derivative must exist at the point being evaluated). If m = n, then the Jacobian matrix is a square matrix. This matrix is denoted by J_F (x ₁, … , x_n).

The interesting concept about the Jacobian is its determinant: Jacobian determinant, |J|. The analysis of the |J| permits one to characterize the behavior of the function around a given point, which has uses in the social sciences.

First, |J| is used to test functional dependence, linear and nonlinear, of a set of equations. If |J| = 0, the equations are functionally dependent. If |J| > 0, the equations are functionally independent. Note |J| does not determine the functional relationship, linear or nonlinear. Second, if |J| at a given point is different from zero, the function is invertible near that point, that is, an inverse function exists. Then the Jacobian determinant in conjunction with the implicit function theorem can be used to identify changes in an endogenous variable, which may be a choice or optimization variable, as an exogenous variable changes.

Unlike the Hessian matrix, the Jacobian can be used to analyze constrained-optimization problems. However, like the Hessian, calculating the |J| becomes laborious as the dimensions of the matrix increase. In addition, the Jacobian is difficult to use with a nonlinear optimization problem, which produces a Jacobian matrix with elements that may not be constant.

SEE ALSO Inverse Matrix; Matrix Algebra

BIBLIOGRAPHY

Chiang, Alpha C., and Kevin Wainwright. 2005. Fundamental Methods of Mathematical Economics. 4th ed. Boston: McGraw-Hill/Irwin.

Dowling, Edward T. 2001. Schaum’s Outline of Theory and Problems of Introduction to Mathematical Economics. 3rd ed. Boston: McGraw-Hill.

Fuente, Angel de la. 2000. Mathematical Methods and Models for Economists. New York: Cambridge University Press.

Jehle, Geoffrey A., and Philip J. Reny. 2001. Advanced Microeconomic Theory. 2nd ed. Boston: Addison-Wesley.

MacTutor History of Mathematics Archive. University of St. Andrews, Scotland. http://www-groups.dcs.st-and.ac.uk/~history/.

Silbergberg, Eugene, and Wing Suen. 2001. The Structure of Economics: A Mathematical Analysis. 3rd ed. Boston: McGraw-Hill/Irwin.

Simon, P. Carl, and Lawrence Blume. 1994. Mathematics for Economists. New York: Norton.

Varian, Hal R. 1992. Microeconomic Analysis. 3rd ed. New York: Norton.

Rhonda V. Sharpe

Idrissa A. Boly