## Constrained Optimization

Moneyball offers a unique way for students to familiarize themselves with the concepts of indifference curves and budget constraints. A budget constraint is frequently depicted as a straight line showing the amount of Good X and Good Y that can be purchased at a particular income. Indifference curves are a more abstract representation of the various combinations of Good X and Good Y that make a consumer/producer equally well off.

For a student with little experience thinking in the abstract, this may be a difficult introduction to a topic that is absolutely essential to intermediate microeconomics. Using concepts from Moneyball enables instructors to engage their students in problems based on budget constraints and indifference curves, and helps them answer questions based on those curves.

Some relevant questions that might be asked are:

• How much talent can a team afford given their payroll constraints?

• How can a General Manager maximize utility (winning percentages, attendance, etc) by evaluating which input (either player or skill type) can be purchased in order to maximize utility for the price (training or payroll cost)?

If an instructor wanted to use actual data, there is ample data online to answer these questions. An instructor could quickly compile a spreadsheet of salaries and performance metrics from websites such as Baseball Reference.

For a more general approach, an instructor could start by playing the trailer to Moneyball to set the tone for the examples that will be used:

Afterward, an instructor could then present a budget constraint like the one below, but labeling the axes with valuable traits in baseball. We have labeled our budget constraint with On Base Percentage (OBP) and Slugging (SLG):

After drawing the graph, the instructor can discuss the Moneyball approach to valuing those items. If most teams do not value OBP as much as SLG, it becomes much less costly to purchase additional units of OBP relative to SLG, allowing teams that are willing to purchase OBP to get more units for the same cost than a team looking to purchase SLG.

For teaching indifference curves, a similar application can be presented, and students can discuss how hard it is to find more and more players that specialize in a given statistic or to find players that are truly exceptional in a given statistic (both variations on the concept of consuming a higher quantity of one good or another).

The instructor can then illustrate that the cheapest way to consume on a given indifference curve is typically to find a blend of the available goods, since diminishing marginal utility means that each additional unit of a good is worth less to consumers than the unit before. Teams, therefore, have an incentive to avoid overspecializing, just like consumers have an incentive to eat a diet that is not exclusively made up of ramen noodles.