The Hessian matrix was developed by Ludwig Otto Hesse (1811–1874), a German mathematician, though the term was first used by James Joseph Sylvester (1814–1897), an English mathematician who studied under Karl Gustav Jacob Jacobi (1804–1851). The Hessian is a square (the number of rows equal to the number of columns) and symmetric (if the rows are written as the columns, the same matrix is produced) matrix whose entries are second-order partial derivatives defined as . Since the Hessian is defined using second-order partial derivatives, the function must be continuous (smooth with no breaks or gaps) and differentiable (the derivative must exist at the point being evaluated).
The Hessian is used to characterize stationary or inflection points of a multivariable function, f (x 1, x 2), as maximums or minimums. The identification as a maximum or minimum requires knowledge about the leading principal minor, | Hk |—the determinant of the principal submatrix of order k. For a stationary point to be a maximum, the leading principal minors must alternate in sign with | H 1 | being negative. If all the leading principal minors are positive, then the stationary point is a minimum. Failure to satisfy either of these conditions, which includes a leading principal minor of value zero, means characterization of the stationary point is inconclusive. The drawback of the Hessian is that calculating all the leading principal minors becomes laborious for an n -variable function with n < 2.
The Hessian, as defined, is used to characterize stationary points of unconstrained optimization problems, which are drawn from the theory of the firm. Goods are produced using capital (K ) and labor (L ) with the following production function, f (L, K ). Firms must decide the optimal combination to maximize profit. Applying the Hessian to such a problem generates a condition for profit maximization. For | H 1 | to be negative, the marginal product of labor must be diminishing—additional labor beyond the optimal choice decreases productivity, therefore, decreasing profit. The positive requirement of | H 2 | means that the marginal product of capital must also be diminishing. Without knowledge of these conditions, changes to the firm’s production policy would result in inefficient use of resources and a decrease in social welfare. Therefore, the Hessian matrix is an important tool in the policy analysis of unconstrained choices.
SEE ALSO Inverse Matrix; Jacobian Matrix; Matrix Algebra
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Rhonda V. Sharpe
Idrissa A. Boly