Understanding the Components of Optimization [closed]

Question

What are policy function, objective function, constraint function, choice variables and parameters within the context of a utility maximization problem?

I don't understand which each of these are and I was not able to find out from resources online. Your help will be much appreciated.

Answer 1

I will give a general answer (for the specification of utility maximization), and I will give a more specific example of Cobb-Douglas utility: $$U(x_i,y_i)=x_i^{\alpha_i}y_i^{1-\alpha_i}$$

$\textbf{Objective function:}$ In the utility maximization problem, the objective function is going to be the utility function. This is the function which you are trying to maximize. Therefore the objective function for agent $i$ in our example is: $$U(x_i,y_i)=x_i^{\alpha_i}y_i^{1-\alpha_i}$$

$\textbf{Constraint function:}$ The constraint function is the function that limits the agent's ability to consume. If these functions were not in place, any agent with a strictly increasing utility function would choose to consume infinite goods. The two most common types of constraints are resource constraints and budget constraints. Resource constraints are just a maximum amount of a resource available (which makes sense because we know that resources are finite). A budget constraint is basically a constraint that says that you cannot consume more goods than you can buy. Examples of the two are below:

Resource constraint: $$\sum_{i=1}^nx_i=A,\quad \sum_{i=1}^ny_i=B$$ where $A,B<\infty$

Budget constraint: $$P_xx_i+P_yy_i= Z_i$$ where $Z_i$ is income of agent $i$ and $Z_i<\infty$

$\textbf{Choice variables:}$ These are the variables that each agent chooses to maximize utility. In our example the choice variables for agent $i$ are $x_i$ and $y_i$.

$\textbf{Parameters:}$ Parameters are also called state variables. These are the variables which the agent has no control over. In our example the parameters are $\alpha_i,B,A,Z_i, P_y,P_x$.

$\textbf{Policy function:}$ The policy function is also called the decision rule. This is what you get from solving the maximization problem. In the context of utility maximization problem, the policy function is the Marshallian demand (how much you want of each good). The policy function gives you your choice variables as functions of your state variables/parameters.