2.4 Summary

The General Case

A more general Lagrangian, with $n$ choice variables, $N_E$ equality constraints, and $N_I$ inequality constraints, will look like this: \(\mathcal{L} = u(x) + \sum_{j=1}^{N_E} \lambda_j h_j(x) + \sum_{k=1}^{N_I} \mu_k g_k(x)\)

where each $h_j(x) = 0$ is one of the $N_E$ equality constraints, with multiplier $\lambda_j$, and each $g_k(x) \geq 0$ is one of the $N_I$ inequality constraints, with multiplier $\mu_k$.

The first-order conditions consist of one equation for each of the $n$ choice variables, \(\frac{\partial \mathcal{L}}{\partial x_i} = \frac{\partial u}{\partial x_i} + \sum_{j=1}^{N_E} \lambda_j \frac{\partial h_j}{\partial x_i} + \sum_{k=1}^{N_I} \mu_k \frac{\partial g_k}{\partial x_i} = 0 \quad \text{for } i = 1, \dots, n\) and one equation for each of the $N_E$ equality multipliers, \(\frac{\partial \mathcal{L}}{\partial \lambda_j} = h_j(x) = 0 \quad \text{for } j = 1, \dots, N_E\) As before, we do not differentiate with respect to the $\mu_k$; instead each of the $N_I$ inequality constraints contributes a complementary slackness condition, \(\mu_k \, g_k(x) = 0, \quad \mu_k \geq 0 \quad \text{for } k = 1, \dots, N_I\)