1.1 Cast Away

In the movie “Cast Away,” a FedEx manager named Chuck Noland is stranded on an uninhabited island somewhere in the Pacific after a plane crash. All he has are the resources on the island — stones, branches, coconut trees, fish in the water — and a few FedEx packages from the plane. With these resources, and his own time, he figures out a way to survive, mainly on the fish he catches and the coconuts he’s able to break open.

As economists, we want to analyze the decisions made by billions of people all over the world, but at its core an economy is just like Chuck: it has resources, and technologies to produce goods from those resources. The fundamental problem we face is that of “scarcity and choice:” we have to choose how to use the scarce resources we have to produce the goods and services which we need to survive and thrive.

In the first part of this book, we’ll use the metaphor of Chuck’s problem – how to best allocate his time between producing fish and coconuts — to develop the fundamental modeling tool of constrained optimization that is at the core of many important economic models.

Technology and Production Functions

In Chapter 2 we will look at how economists model the technological processes by which resources are transformed into consumption goods, which we’ll call his production functions. We’ll assume that Chuck can’t catch fish or break open a coconut with his bare hands: he uses branches to make a fishing spear, and stone tools to break open the coconuts. We’ll therefore analyze how a multivariable production function can model how the quantity of an output (fish) depends on multiple inputs — for example, Chuck’s time (or “labor”) and and his available tools (or “capital”). We’ll then think about how we might model different kinds of production technologies using different functional forms: for example, what kind of mathematical function describes a production technology that uses labor and capital in conjunction with one another, and what kind of function describes a technology in which a good can be produced using either labor or capital. More than an attempt at realism — which, spoiler alert, it’s not going to be very realistic — what we’ll be developing here is the technique of modeling things in the real world using mathematical functions.

Resource Constraints and Production Possibilities

In Chapter 3 we will model the choice facing Chuck: how to allocate his resources across different possible uses. We’ll assume he has a certain amount of time to allocate either to producing fish or producing coconuts. Specifically, we’ll derive his production possibilities set from his production functions and his available resources.

This production possibilities set is our first example of a broader class of interest called a feasible set or choice set: that is, the set of options among which an economic agent is choosing. It’s essentially the “scarcity” half of “scarcity and choice.”

Preferences and the Utility Function

The “choice” half of “scarcity and choice” centers on the question of how Chuck should to spend his available resources. We will assume that Chuck has preferences over different possible combinations of fish and coconuts: he might prefer to have 10 coconuts and 10 fish to having 5 fish and 15 coconuts, for example. We will model his preferences by assuming that any possible combination of fish and coconuts would give Chuck some amount of “utility” or “payoff.”

We will model this almost like a production process, where just as a production function transforms resources into goods, Chuck’s utility function transforms goods into utility. This utility function will allow us to rank all of the possible combinations of fish and coconuts that Chuck can produce, in order to arrive at his optimal choice.

Constrained Optimization

Chapters 2-4 set up Chuck’s problem of scarcity and choice: the constraint is given by his available resources and technology, which together determine his choice set. His preferences allow him to rank all the elements of his choice set.

In Chapters 5 and 6 we go about solving this problem: how do we find Chuck’s optimal choice, or most preferred element of his choice set? More concretely, what allocation of Chuck’s resources would produce the combination of fish and coconuts which would give him the highest achievable utility?