However, due to a limited budget, we will have to choose the best combination that will maximize our budget and utility. This is called Constraint Maximization.

**Budget Constraints**

Typically, we can represent a constraint as:

Where P is the number of pizzas, and Pp is the price of each pizza.

Suppose that a pizza costs $20, and a movie costs $10, and we have $100 to spend, then we will have a graph as such:

We will be drawing $100 = $20P + $10M. The slope of this line is -Pm/Pp.

The price ratio is -0.5, which is the Marginal Rate of Transformation (MRT). It is the rate at which we can transform pizzas into movies.

The

**Opportunity Cost**of something is the value of foregone alternative. For example, if we forgo a pizza, then we are forgoing 2 movies. Or from the other perspective, a movie costs half a pizza. This is the Opportunity Cost of the situation. Opportunity costs only make sense if we have a limited budget.

Suppose that the price of pizza rose to $30. We will have the following graph now:

With the $100, we have things to choose from now. In other words, our

**Opportunity Set**has become more restricted.

We can also have a restricted Opportunity Set if our budget drops. Keeping the price constant as the first example, we will have the following graph if we drop our budget to $80. In this case, the slope remains the same.

We want to find out what is the furthest indifference curve given a budget constraint. An example when we combine the indifference curves and the budget constraint looks like this.

We can afford anything under our budget line. Notice that there are three points on our budget line itself. Points A and C are equivalent, but because we want the farthest indifference line (due to Non-Satiation), point D is actually the best we can get.

The best indifference curve we can get is one where the budget constraint line is tangent to the curve. In order to perform utility maximization, we need to get these things to be tangent by ensuring that the MRS is equal to the MRT. By setting MRS equal to MRT, we are setting marginal benefits equal to marginal costs.

**Deriving Marginal Utilities, MRS and MRT**

The marginal at each point is the partial derivative of the functional utility with respect to the item we are interested in. For example, if we are interested in the marginal utility function of pizza then we differentiate sqrt(PM) with respect to P.

Points A and C are both undesirable. It is clear that the MRT for the entire budget line is -0.5. Let's work out the MRS for all the points:

An MRS of -2 means that we are willing to trade 2 pizzas for a movie. However, the market MRT of -0.5 only requires us to trade half a pizza for a movie.

There are times where we will end up with negative quantities when we solve for the optimum point. It may be that we are looking at a problem with a corner solution.

Remember that MRS is the rate the consumer is willing to trade the y-axis for the x-axis. On the other hand, MRT is the actual market value ratio between the y-axis and the x-axis.

Note that B, D and E both satisfies that MRT equals MRS, but we want the farthest indifference point that is still within our budget. That is the basic goal of utility maximization.