Question 15.23: If a firm is attempting to maximize output Q = Q(x, y, z) su......
If a firm is attempting to maximize output Q = Q(x, y, z) subject to a budget of £5000 where the prices of the inputs x, y and z are £8, £12 and £6, respectively, what requirements are there for the relevant bordered Hessians to ensure that second-order conditions for optimization are met?
The Lagrange objective function will be
G = Q(x, y, z) + λ(5000 − 8x − 12y − 6z)
As there are three variables in the objective function and H_{B} is 4 × 4 then the second- order conditions for a maximum require that the determinant of the bordered Hessian of second-order partial derivatives \left|H_{B}\right| < 0. Therefore
\left|H_{B}\right| = \begin{vmatrix} Q_{xx}& Q_{xy} & Q_{xz}& −8 \\ Q_{yx}& Q_{yy}& Q _{yz} & −12 \\ Q_{zx}& Q_{zy}& Q_{zz}& −6 \\ −8 & −12& −6& 0 \end{vmatrix} \lt 0
As H_{B} is 4 × 4 the second-order conditions for a maximum also require that the determinant of the naturally ordered principal minor of H_{B} > 0. Thus, when the first row and column have been eliminated from H_{B} , this problem also requires that
\begin{vmatrix} Q_{yy}& Q _{yz} & −12 \\ Q_{zy}& Q_{zz}& −6 \\ −12& −6& 0 \end{vmatrix} \gt 0