Unconstrained Optimization

$$\max_{\mathbf{x}} f(\mathbf{x})$$

• $\max$ : maximize (the goal of the agent's problem)
• $f$ : the objective function (what the agent wants to maximize, exogenous)
• $\mathbf{x} = (x_1, \ldots, x_n)$ : vector of $n$ choice variables (endogenous variable)
• We assume $f$ to be concave and differentiable with continuous derivatives.

Definition: A function $f$ is concave if the function evaluated at the weighted average of any two points is no less than the weighted average of the function evaluated at these two points. That is, for all $x$ and $y$ and for any $\alpha$ in $(0,1)$, we must have that
$f(\alpha x + (1-\alpha) x) \ge \alpha f(x) + (1-\alpha) f(y)$.

$$\max_{\mathbf{x}} f(\mathbf{x})$$

1. Obtain the gradient of the objective function,
   $\nabla f = \left(\dfrac{\partial f}{\partial x_1}, \ldots, \dfrac{\partial f}{\partial x_n}\right)$.
2. Solve the system of equations, the first-order conditions (FOC)
   $\nabla f = (0, \ldots, 0)$, for the choice variables $\mathbf{x} = (x_1, \ldots, x_n)$.
3. The solution(s) of the FOC are the candidate solutions for the maximization problem.
4. If there are multiple candidate solutions, evaluate the function at each candidate solution and pick those who yield the highest value.

Consider the consumer problem,

$$\max\limits_{\substack{x,y\\ \mathrm{subject\, to} \\ \\ p_x x \;+\; p_y y \;=\; i}} u(x,y)$$

A common trick is to use the equality constraints to eliminate variables from the problem,

$$\max\limits_{x} u\!\left(x,\frac{i-p_x x}{p_y}\right)$$

converting the constrained maximization problem into an unconstrained problem with fewer variables.

Let's take a look at the peer-effects model: individuals $1$ and $2$ choose actions $x$ and $y$. Individuals want everyone to choose their favorite actions $a_1$ and $a_2$ respectively, but if the peer chooses some different action, the individual also dislikes choosing actions far away from the peer's choice.

$$\begin{aligned} u_1(x,y) &= 1-(a_1-x)^2-\lambda\,(y-x)^2\\[2pt] u_2(x,y) &= 1-(a_2-y)^2-\lambda\,(x-y)^2 \end{aligned}$$

Assume individuals are able to contract on the choice of $x$ and $y$. What choices will then maximize their joint utility $u_1(x,y)+u_2(x,y)$?