Optimization with Equality Constraints

Constrained Optimization

with Equality Constraints

$$\max\limits_{\substack{\mathbf x\\ \mathrm{subject\, to} \\ \\g_1(\mathbf x)=c_1\\[-6pt] \vdots \\[1.5pt] g_k(\mathbf x)=c_k }} f(\mathbf x)$$

$\max$ : maximize (the goal of the problem)
$f$ : the objective function (what we want to max.)
$\mathbf x=(x_1,\ldots,x_n)$ : vector of $n$ choice variables
$g_i$ : constraint function (there are $k\lt n$ of them)
$c_i$ : constraint bound constant
(there are $k$ of them)

Constrained Optimization with Equality Constraints

Conditions For A Solution

Given maximization problem define the Lagrangian,

$$ \mathcal{L}(\mathbf x, \boldsymbol \lambda)=f(\mathbf x)-\sum_{j=1}^{k}\lambda_{j}\cdot(g_{j}(\mathbf x)-c_{j})$$

Notice that the Lagrangian is a function of the choice variables $\mathbf x$ and also of the Lagrangian multipliers $\boldsymbol \lambda$

If $\mathbf x^\ast$ solves the maximization problem then there are Lagrange multipliers $\boldsymbol \lambda^\ast$ such that $$\nabla_{\mathbf x,\boldsymbol \lambda} \mathcal{L}=0,$$ where $\nabla_{\mathbf x} \mathcal{L}$ is the gradient of $\mathcal{L}$, the list of first partial derivatives of $\mathcal{L}$ with respect to each one of the variables: $$\diff{}{x_{i}} \mathcal{L}=0 \text{ for } i=1,\ldots,n \text{ and }\diff{}{\lambda_{j}} \mathcal{L}=0\text{ for }j=1,\ldots,k.$$

Warning! You must check if the solution of these equations corresponds to a maximum or if it corresponds to a minimum.

Constrained Optimization with Equality Constraints

Basic Recipe #1: Lagrangian Method Recipe

Set up the maximization problem (objective function, choice variables, constraints, etc...)
Write down the Lagrangian (objective function minus the difference of the LHS and RHS of the constraints weighted by the corresponding Lagrangian multiplier).
$\mathcal{L}(\mathbf x, \boldsymbol \lambda)=f(\mathbf x)-\sum_{j=1}^{k}\lambda_{j}\cdot(g_{j}(\mathbf x)-c_{j})$
Apply the basic recipe #0 to a problem in which: A) The objective function is the Lagrangian of this problem and, B) the choice variables are the choice variables of the original problem together with the Lagrange multipliers.
That is, find $(\mathbf{x},\boldsymbol{\lambda})$ that solves $\nabla_{(\mathbf{x},\boldsymbol{\lambda})}\mathcal{L}=\mathbf{0}$.
For each solution, check if it corresponds to a maximum or minimum.
Bonus/additional analysis (perform comparative statics, study how the solution depends on the exogenous variables (e.g., the constants in the constraints)).