UBC Economics Math Camps
A function of \(n\) variables has \(n^2\) possible second derivatives: \(\frac{\partial^2 f(x)}{\partial x_i\partial x_j} = \frac{\partial \left( \frac{\partial f(x)}{\partial x_i} \right) }{\partial x_j}\)
Hessian matrix collects them: \(H(x) = \left[ \frac{\partial^2 f(x)}{\partial x_i\partial x_j} \right]_{i,j}\)
A function \(f\) is concave if, and only if, \(H(x)\) is negative semi-definite for all values \(x\)
If \(H(\cdot )\) is negative definite then \(f\) is strictly concave.
The first-order conditions are:
\[ \begin{aligned} \Lambda_{x} &= f_{x} (x^*,y^*) - \lambda^* g_{x}(x^*, y^*) = 0.\\ \Lambda_{y} &= f_{y} (x^*,y^*) - \lambda^* g_{y}(x^*, y^*) = 0.\\ \Lambda_{\lambda} &= c - g(x^*, y^*) = 0. \end{aligned} \]
These are the necessary conditions for the optimum. To derive sufficient conditions, we define a bordered Hessian:
\[ H = \begin{bmatrix} 0 & g_{x} & g_{y} \\ g_{x} & f_{xx}-\lambda g_{xx} & f_{xy}-\lambda g_{xy} \\ g_{y} & f_{xy}-\lambda g_{xy} & f_{yy}-\lambda g_{yy} \end{bmatrix} \]
Bordered Hessian augments the regular Hessian matrix of the objective function with the gradients of the constraint functions.
\[ \frac{f_{x} (x^*, y^*)}{f_{y} (x^*, y^*)} = \frac{g_{x} (x^*, y^*)}{g_{y} (x^*, y^*)} \]
This, together with the constraint, \(c − g(x^∗, y^∗) = 0\), yields two equations in the two unknowns, \((x^∗, y^∗)\).
Variable \(\lambda^*\) can then be solved residually from one of the first-order conditions above.
The first-order conditions and second-order conditions in this constrained problem are closely related to the corresponding conditions in the unconstrained problem studied earlier.
To show this, notice that the constraint \(g(x, y) = c\) can be re-arranged to write \(y\) as a function of \(x\), say
\[ y=h(x) \]
Then, by substituting it into the objective function, the objective function becomes
\[ f(x, h(x)) \]
The original problem with two choice variables and a constraint has been reduced to a one-variable unconstrained problem
Maximizing \(f (x, h(x))\) with respect to \(x\), the first-order condition is
\[ 0 = f_{x} + h' f_{y} \]
where \(h' = \frac{dy}{dx} = - \frac{g_{x}}{g_{y}}\)
\[ 0 = f_{x} - \frac{g_{x}}{g_{y}} f_{y} \]
\[ \frac{f_{x}}{f_{y}} = \frac{g_{x}}{g_{y}}, \]
which is the same condition obtained under the Lagrangian. Together with the constraint \(g(x,y)=c\) it can be solved for \((x^*, y^*)\).
The Lagrange multiplier measures the marginal impact on the value of the objective due to changing the constraint “a little” (marginally).
To see this, consider the change in the objective function \(f\) when the constraint \(c\) changes: \(df/dc\).
\[ \frac{df}{dc} = \frac{\partial f}{\partial x} \frac{\partial x}{\partial c} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial c} \]
\[ \begin{aligned} \frac{\partial f}{\partial x} = \lambda \frac{\partial g}{\partial x} = \lambda g_x \\ \frac{\partial f}{\partial y} = \lambda \frac{\partial g}{\partial y} = \lambda g_y \end{aligned} \]
\[ \frac{dg}{dc} = g_x \frac{\partial x}{\partial c} + g_y \frac{\partial y}{\partial c} = 1 \]
\[ \frac{df}{dc} = \lambda \]