5.1 Modeling Tool #1: Constrained Optimization

We are now ready to approach our first fundamental modeling tool of microeconomic analysis: constrained optimization. Before we think about Chuck’s problem in particular, let’s zoom out a little to think about the problem of constrained optimization more generally.

In microeconomics, we will generally be considering the problem of an “economic agent,” someone or something with agency to make a decision. In classical economic analysis agents were generally individual consumers (people buying things) and firms (businesses producing and selling things), but economists think of the decisions of lots of people and organizations, from family members dividing chores, to government officials trying to control a pandemic, to high school seniors deciding where to go to college.

Economists generally make three core assumptions about any decision being made:

1. We assume that the decision occurs within some choice space. This frames the problem: if there were no limits on what the agent could choose, what would the universe of options be? For example, in a model of where you choose to go to college, this could be the set of all colleges and universities in the world. In a model of shopping at a grocery store, this might be all the items for sale in the store.
2. We assume that the agent is usually constrained in their choice – for example, by available resources or technology – so that some elements of the choice space are available to them and others are not. We call the available options the feasible set. In a college choice model, this could be the set of colleges that accept you, along with the financial aid package they offer. In a model of shopping at a grocery store, this might be the set of all combinations of items which could be purchased with a given amount of money.
3. We assume that the agent has preferences over all elements of the choice space, which drive their decision. Importantly, we do not necessarily need to assume those preferences are rational, or smart, or motivated by good (or bad) intentions. An agent may be greedy or altruistic, vengeful or generous, patient or impatient – an economist doesn’t seek to tell people what they should prefer, only to understand what they do prefer.

Given the above assumptions – which are not necessarily as innocuous as they might seem at first blush! – economists assume that when people make decisions, they are choosing what from their perspective is their optimal choice (or one of several choices which are “tied for best”) from among the options in the feasible set. So, when we observe someone make a college choice, we assume that they chose what they considered was their best option from among the college offers they received; and when someone leaves a grocery store, they have filled their cart in the best way possible given their budget and their appetite.

It’s important to note that not all constrained optimization problems lend themselves to a mathematical solution: there’s no formula that you used to determine your best college choice! However, there is a wide class of economic problems that do lend themselves to mathematical analysis, and it’s those that we’ll spend the bulk of our time in this book modeling. To solve such a problem requires an understanding of two important functions:

• Scarcity: The feasible set will be a function of the scarce resources available to the agent. More resources will mean more feasible options to choose from.
• Choice: The optimal choice will be a function of the feasible set. Facing different feasible sets, the agent will make different choices.

Chuck’s constrained optimization problem

In the preceding chapters, we established the elements of a specific kind of choice: how to use limited resources and available production technologies to produce various goods. If we return to our “Cast Away” metaphor for the economy:

• We have defined Chuck’s choice space as all combinations of fish (good 1) and coconuts (good 2). This is shown as a Cartesian plane with quantity of fish ($x_1$) on the horizontal axis, and quantity of coconuts ($x_2$) on the vertical axis; we call this “good 1—good 2 space.”

• In Chapter 2, we described how Chuck’s available resources and production technologies together determine his feasible set:

  • The feasible set described mathematically by his resource constraint $L_1 + L_2 = \overline L$ and his production functions $f_1(L_1)$ and $f_2(L_2)$.

  • The feasible set is shown visually by the production possibilities frontier (PPF) in good 1—good 2 space.

  • The slope of the PPF is called the marginal rate of transformation (MRT), and represents the opportunity cost of an additional fish in terms of coconuts.

• In Chapter 3, we described Chuck’s preferences over bundles in the choice space:

  • Chuck’s preferences are described mathematically by his utility function $u(x_1,x_2)$.

  • They are shown visually by his indifference curves in good 1—good 2 space.

  • The slope of an indifference curve at point $X$ is called the marginal rate of substitution (MRS), and represents the number of coconuts Chuck would be willing to “pay” for an additional fish, if he were consuming at point $X$.

These two approaches have yielded the following two canonical diagrams. In the left-hand graph, we can see how the PPF divides the choice space into the feasible set (shaded) and the infeasible set; one point along the PPF is labeled $X$. In the right-hand diagram, we can see how an indifference curve divides the choices space into the set of bundles preferred to $X$ (shaded) and those dispreferred.

Chuck’s constrained optimization problem, therefore, is how he should divide his time between producing fish and coconuts in the way that provides him the most utility. To answer that, we bring these two graphs together, overlaying the PPF diagram with Chuck’s indifference map. Now, as we drag the (single) point X along the PPF, we can see that utility is maximized at the point (80, 36):

An astute student will notice that the optimal choice here involves a tangency condition: that is, at the optimum of (80, 36) the $MRS = MRT$. While that’s true in this case, there are lots of edge cases in which that won’t hold. We’re going to spend one lecture closely examining the “nice” case where the optimum is characterized by a tangency condition; and then one lecture examining these edge cases. But first, we’re going to look at the “gravitational forces” that “pull” an agent toward their optimal decision — forces which hold in all cases.