The 2 × 2 Exchange Economy

In this chapter the focus shifts from the individual agent to the market. Specifically we examine the allocation that results if all economic agents are price takers and prices adjust until markets clear. Rather than attempt to bring firms and consumers into the analysis all at once, we focus here on equilibrium in which there is no production. Consumers have endowments of commodities that they may exchange. As we see later, the ideas developed here extend very directly to economies with production.

Even though this chapter focuses on equilibrium outcomes, it is helpful to keep in mind a possible adjustment process that might lead to equilibrium. Suppose that there is an auctioneer who calls out prices for each of the commodities. Consumers and firms respond with the demands that they would make at these prices. The auctioneer lowers prices in markets where there is excess supply and raises them in markets where there is excess demand. At a price-taking (Walrasian) equilibrium, all markets clear.

Strategic Equilibrium

In all the previous chapters, the primary focus was on resource allocation via a Walrasian equilibrium (WE). In a WE allocation all players are price takers so there are no strategic issues. However, the price-taking assumption only makes sense if there are a sufficiently large number of competing players. As we have seen, if a production set exhibits increasing returns to scale, one firm can produce at a lower cost than two or more firms so there is a natural monopoly. When a commodity is sold by one firm there is a strategic issue. Instead of being a price taker, the firm is a price setter, choosing a pricing strategy to maximize the firm's payoff. But suppose that production sets in an industry exhibit increasing returns to scale at low outputs and decreasing returns at outputs that are a significant fraction (but less than 50%) of market demand. Then average cost is minimized, with a few firms producing near the average cost-minimizing output. Now strategic issues become much more subtle because a change in the production plan of one firm affects the sales of that firm's competitors. This typically causes a reaction by each competitor. Making a good choice then requires all players to forecast the actions of their competitors. Using the language of social competition (sports, card games, etc.) any such strategic competition is called a game and the participants in the game are called players.

To begin, consider the following simple economic game. There are two players. Player i, i = 1, 2 is the manager of firm i. Each player submits the price of the firm's product for the next week to be posted on the web. To keep things simple, each player sets a high price pH or a low price pL. Let Ai be the set of possible actions; then Ai = {pH, pL}.

Supporting Prices

The pursuit of self-interest is central to economics. Thus a deep understanding of the theory of maximization is essential to effective theorizing. In particular, the theory of constrained maximization is so crucial that we explore it in this first chapter. In contrast to a purely mathematical exposition, the emphasis here is on prices.

Our first topic is the role of supporting prices. We explore the issue of how prices can be used to provide incentives for an economic agent to make a desired choice. In addition to being of direct importance, supporting prices are central to the theory of constrained maximization. Section 1.2 explores this point, emphasizing the intuition behind the formal mathematics. Section 1.3 then examines how the maximized payoff of an agent is affected by a change in the environment (a parametric change.) Section 1.4 contains a formal proof of the necessary conditions for a constrained maximum. An example showing how to apply the necessary conditions is presented in Section 1.5.

What Is a Firm?

In this chapter the focus switches to the transformation of commodities by firms. Within a firm, raw materials and other commodity inputs are processed by labor and managerial inputs to produce goods and services. These outputs may be for consumption (final products) or for sale as inputs to other firms (intermediate products). The amount of output that can be produced depends on the technology (machinery, buildings, etc.) held by the firm.

This is relatively straightforward. Consider, for example, a newsprint manufacturer. It transforms the primary raw materials of lumber, energy, and labor into giant rolls of paper ready for delivery to daily newspapers, using an array of machines. However, from a broader perspective, the machines are also inputs. In addition to purchasing labor inputs and raw materials, the firm can purchase additional capital equipment (for the same plant or to build a new plant) and so alter the set of available outputs. From this perspective, the technology of a firm is a set of blueprints for the transformation of commodities.