14.1 Modeling Tool #3: Equilibrium

Up until now, we have been considering the problem of a single economic agent making a decision. In Part I, we considered the problem of someone in autarky — a single agent who didn’t interact with anyone else; their problem was how to allocate their resources to different productive ends. In Part II, we considered the problem of a utility-maximizing consumer, who has some money and uses it to buy goods at market prices; their problem was to choose what to buy (and therefore consume), given their income and the prices of goods. Then, in Part III, we considered the problem of a profit-maximizing firm, which made decisions about how much to produce, based on market prices for outputs and inputs. In each of these cases, a single agent had the ability to make a decision, and we were modeling how they would make that decision.

In a complex economy, though, people often say that we should “let the market decide” how much of a good the economy should produce. Unlike an individual agent, though, the market has no agency — it is not a single entity “deciding” anything at all. Rather, it represents the aggregate decisions billions of people and millions of firms. How can we analyze the “decision” of this decentralized mass of humanity?

It’s a bit of a misnomer to say that a market “decides” anything. Rather, the way economists see the market is as a system which converges to an equilibrium. There are many kinds of equilibria in economics, as different kinds of systems have different dynamics. Central to the notion of all economic equilibria is the mathematical notion of a “fixed point,” which can be most tersely summed up as \(x = f(x)\) where the variable $x$ describes a situation, and the function $f()$ describes the transition dynamics of how a system responds to that situation. For example, $x$ might be the number of firms in a market, and $f()$ would describe whether firms would enter or leave that market, depending on the number of firms already in it. If $\hat x$ is the equilibrium number of firms, then we might expect firms to enter the market if $x < \hat x$ and firms to leave the market if $x > \hat x$.

In the case of a market equilibrium, the situation $x$ may be considered a vector of all market prices, and the function $f()$ could describe how prices change given market conditions. The usual story told in Econ 1 to describe $f()$ in this case goes something like the following:

• If the price of a good is too high, supply will exceed demand (there will be a “surplus”). Sellers will start lowering their prices to clear out inventory.
• If the price of a good is too low, demand will exceed supply (there will be a “shortage”). Sellers will realize they can sell all their goods even if they raise their prices, so they’ll start doing that.
• At the equilibrium price, the quantity demanded will equal the quantity supplied, so there is no price pressure up or down.

This is generally illustrated using a graph like the following. Try dragging the line representing the price up and down:

Existence and uniqueness of equilibria

Two central questions about economic equilibria are whether they exist, and if so, whether they are unique. In the case of the market equilibrium above, an equilibrium exists if there is a price at which the quantity supplied is equal to the quantity demanded: that is, if the supply and demand curves ever cross. That equilibrium is unique if the two curves cross in exactly one place.

Examples of multiple equilibria are common in economics, but require moving beyond the assumptions of the traditional supply and demand model. For example, goods with “network externalities” have a benefits in proportion to the number of people who use them: e.g., fax machines are useful only if other people use fax machines. In the 1980’s, nearly everyone had a fax machine, so the demand for fax machines was high. Today, very few people use fax machines, so demand is low. Both cases are equilibria.

To model situations like this, think about an extension to the supply and demand model in which the demand for the good depends on the expectation of how many units are sold in equilibrium. Let’s call this number $\hat Q$, and assume that the demand for the good is increasing in $\hat Q$, so the more people expect others to buy the good, the more they want it themselves. You can see this situation in the diagram below:

If you increase or decrease $\hat Q$, you can see the demand shift in and out, and the equilibrium quantity respond accordingly. Notice that both $\hat Q = 10$ and $\hat Q = 60$ represent equilibria: if $\hat Q = 10$, demand is low, and the supply and demand curves cross at $Q = 10$. On the other hand, if $\hat Q = 60$, demand is high, and the equilibrium quantity is indeed 60. For every other value of $\hat Q$, the actual quantity and expected quantity are different; so those are the only two equilibria of the model.

Stability and instability of equilibria

Note that in this case, market forces cause the market price to converge to its equilibrium level. This results in a stable equilibrium: small shocks to supply or demand might result in a period of price adjustment, but the system will converge back to a new equilibrium price. There are also many equilibria which are unstable: for example, a fragile cease-fire between warring factions is an equilibrium if both sides hold their fire as long as the other side does; but it might be broken in an instant if a rogue soldier decides to discharge their weapon.

We can picture stable and unstable using the metaphor of a small ball in a semicircular bowl. If we place the ball inside the bowl and shake the bowl a bit, the ball will always return to the center; this is a stable equilibrium, and we have a good story for why the position of the ball will converge to its equilibrium. However, if we take the same bowl, turn it upside down, and carefully balance the ball on the top of it, we can see that even the slightest tremor will send the ball crashing off the bowl. In its position at the top of the bowl, the ball is still technically in equilibrium, as all the forces on it are balanced; but this equilibrium is clearly unstable.

In the model of network externalities above, we might assume that consumers update their beliefs about the number of units sold based on the number that has historically been sold. Try starting by setting $\hat Q$ equal to any number other than 0, 10, or 60, and moving the expectations toward the actual number. You can see that for any $\hat Q > 10$, the system converges to the equilibrium at $Q = 60$; and for any $\hat Q < 10$, the market collapses to $Q = 0$. Thus $Q = 0$ and $Q = 60$ are stable equilibria, while $Q = 10$ is an unstable equilibrium.