Lecture 19: Partial Equilibrium

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Modeling Tool #3: Equilibrium

Up until now, we have been considering the problem of a single economic agent making a decision. In the first four weeks of this course, analyzed 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, last week, we considered the problem of a competitive 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:

See interactive graph online here.

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:

See interactive graph online here.

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.

See interactive graph online here.

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.

Perfect Competition

While there are many equilibrium concepts in economics, the one we’re going to concentrate on here is the model of perfectly competitive markets. Such a market is characterized by four key assumptions:

Lots of buyers and sellers: no one agent has any market power
A homogoeneous good: sellers are not differentiated by quality or any other metric; everyone is selling an identical product
Perfect information: all agents know the price being asked by all sellers
Free entry and exit: there are no barriers to entry, so firms can enter (or leave) the market at any time

The first three assumptions imply that all buyers and sellers are price takers. Importantly, this does not mean that any agent is forced to accept a price. Rather, it means:

All sellers believe that there is a market price at which they can sell as much of their goods as they would like. This price then becomes the optimal price for them to charge: if they charged more, they would sell no goods (because perfect information and homogeneity mean that consumers would just buy at the market price from one of their competitors), while if they charged less, they would earn less revenue for any given quantity sold for no reason.
All buyers know that sellers can sell as much of their product as they like at the market price, so there is no sense in trying to negotiate lower prices.

In fact, we have already analyzed the behavior of buyers and sellers under these assumptions. In lecture 8 we derived the demand function for an individual who took prices as given, and in the last lecture we derived the supply function for an individual firm that took prices as given. In this chapter we’ll bring consumers and firms together and analyze market equilibrium. In order to do that, though, we need to move from these individual demand and supply curves to market demand and supply curves.

Market demand

In lecture 8, we analyzed the problem of a utility-maximizing consumer who could buy goods at constant, given prices. Specifically, we assumed that if they had devoted a budget of $m$ dollars to buying two goods (goods 1 and 2), and faced prices $p_1$ and $p_2$ for those goods, that their utility-maximizing behavior could be described by the demand functions $x_1^\star(p_1,p_2,m)$ and $x_2^\star(p_1,p_2,m)$. This told us their optimal bundle as a function of all prices and incomes.

Let’s zero in on the demand for a particular good; we might write this individual demand as $d(p)$, where $p$ is the market price of the good, and $d$ is the quantity demanded by the individual. To analyze market demand for this good, we need to aggregate the quantity demanded by all individuals, to determine the market demand curve $D(p)$. To do this, we’re going to sum the demand of different individuals. For example, if we have $N_C$ consumers, we can number them consumer 1, 2, 3, etc., up to $N_C$. Then, if $d^1(p)$ is (Note: Notation alert: we’re using the superscripts here to denote which consumer we’re talking about; these aren’t exponents.) the demand function for individual 1, and $d^2(p)$ is the demand for individual 2, and so forth, we can write the overall market demand $D(p)$ as $D(p) = d^1(p) + d^2(p) + d^3(p) + \cdots + d^{N_C}(p)$ or, more succinctly, $D(p) = \sum_{i=1}^{N_C}d^i(p)$ which we can read as “the total quantity of a good demanded at price $p$ is the sum, for each $i$ from 1 to $N_C$, of the quantity demanded by each individual $i$ at that price.”

Now, as the above setup implies, economists understand that people are different. However, modeling or predicting the preferences of individuals can be hard or even impossible, let alone getting data on billions of individuals’ preferences. For that reason (among others), economists sometimes model market demand “as if” there are a lot of identical individuals, each of whom has the same “average-ish” preferences leading to the individual demand function $d(p)$. In this case the total market demand is just the number of consumers times the amount demanded (on average) by each consumer: $D(p) = \sum_{i = 1}^{N_C}d^i(p) = N_Cd(p)$ For example, let’s suppose that in some market, there are $N_C$ consumers who each have preferences that may be represented by the Cobb-Douglas utility function $u(x_1,x_2) = \alpha \ln x_1 + (1 - \alpha)\ln x_2$ We showed in Chapter 8 that this would imply that each individual would spend fraction $\alpha$ of their income on good 1, leading to the individual demand function $d(p) = {\alpha m \over p}$ The following diagrams show this situation. The diagram on the left shows the demand curve for each individual; the diagram on the right shows the market demand.

See interactive graph online here.

Try changing the preferences parameter $\alpha$, the income $m$, and the number of consumers to see how the two graphs are affected. Which factors affect both the amount demanded by an individual, and the market? Which only affect market demand, and not the individual demand?

Market supply

On Monday we analyzed the problem of a profit-maximizing firm that bought resources (labor and capital) and used those resources to produce goods, which it then sold. We defined a perfectly competitive firm as one which was a price taker in both input and output markets: specifically, we assumed that they could buy any quantity of labor and capital at prices $w$ and $r$ respectively, and they could sell any amount of output at price $p$.

Just as we wrote the quantity of a good demanded by an individual consumer as $d(p)$ and the market demand as $D(p)$, let’s write the amount of a good supplied by a firm as $s(p)$ and the market supply as $S(p)$. And just as we aggregated the individual demands of all consumers in a market to get $D(p)$, we need to aggregate the quantity supplied by all firms to determine the market supply curve $S(p)$. Thus, if we have $N_F$ firms, and $s^j(p)$ is the supply function for firm $j$, we can write the overall market supply $S(p)$ as $S(p) = s^1(p) + s^2(p) + s^3( p) + \cdots + s^{N_F}(p)$ or, more succinctly, $S(p) = \sum_ {j=1}^{N_F}s^j(p)$ which we can read as “the total quantity of a good supplied at price $p$ is the sum, for each $j$ from 1 to $N_F$, of the quantity supplied by each firm $j$ at that price.”

Just as we did with demand, let’s look at an example in which there are $N_F$ identical firms, just to keep the math simple. In this case, if each firm has the individual supply function $s(p)$, the total market supply is just the number of firms times the amount supplied by each firm: $S(p) = \sum_{i = j}^{N_F}s^j(p) = N_Fs(p)$ For example, let’s suppose that in some market, there are $N_F$ firms who each have access to the Cobb-Douglas production function $f(L,K) = \sqrt{LK}$ We showed in Chapter 13 that, if capital is fixed at $\overline K$, the short-run supply function for such a firm is $s(p) = {\overline Kp \over 2w}$ Therefore the total amount supplied by the market is $S(p) = N_Fs(p) = {N_F \overline Kp \over 2w}$ The following diagrams show this situation. The diagram on the left shows the supply curve for each firm; the diagram on the right shows the market supply.

See interactive graph online here.

Again, you can play with the sliders to see how the individual supply curve and market supply curves are affected.

Partial equilibrium

We’re now — finally! — ready to bring market demand and market supply together.

Market equilibrium occurs when the price is such that the market quantity demanded equals the market quantity supplied.

Formally, a price $p^\star$ is an equilibrium price in a market if:

Consumer optimization: each consumer $i$ is consuming the quantity $x^i(p^\star)$ that solves their utility maximization problem.
Firm optimization: each firm $j$ is producing the quantity $q^j(p^\star)$ that solves its profit maximization problem.
Market clearing: the total quantity demanded by all consumers equals the total quantity supplied by all firms: $D(p) = S(p)$.

For example, in the past two sections, we derived the supply and demand functions $\begin{aligned}S(p) &= N_F \times {\overline K p \over 2w}\\ D(p) &= N_C \times {\alpha m \over p}\end{aligned}$ If we set these two equal to one another, we get $\begin{aligned}S(p) &= D(p)\\ N_F \times {\overline K p \over 2w} &= N_C \times {\alpha m \over p}\\ p^2 &= {N_C \over N_F} \times {2 \alpha m w \over \overline K}\\ p^\star &= \sqrt{\frac{N_C}{N_F} \times {2 \alpha m w \over \overline K}}\end{aligned}$ Plugging this back into either the demand or supply function gives us the quantity in the market will be $Q^\star = S(p^\star) = D(p^\star) = \sqrt{N_FN_C\overline K \alpha m \over 2w}$ This looks like a lot of variables! But in fact, what we’re seeing is that we can do all our usual Econ 1 comparative statics just from these expressions:

If more consumers enter the market $(\uparrow N_C)$, or if consumers desire this good more $(\uparrow \alpha)$, or if consumers get more money $(\uparrow m)$ the demand curve shifts to the right; the equilibrium price and quantity both rise.
If more firms enter the market $(\uparrow N_F)$ or if firms invest more capital in producing this good $(\uparrow \overline K)$, the supply curve shifts to the right; the equilibrium price falls, and the equilibrium quantity rises.
If the wage rate increases $(\uparrow w)$, the supply curve shifts to the left; the equilibrium price rises, and the equilibrium quantity falls.

You can see how each of these effects (and their opposites) play out in the following diagrams. The middle diagram shows market supply and demand; the left diagram shows individual demand, and the right diagram shows individual supply.

See interactive graph online here.

Try changing the parameters, and see what happens to the supply or demand curves.

Equilibrium with different consumers and firms

In the previous example, we analyzed a situation in which there are a lot of identical consumers and firms. Such a model is called a “representative agent” model, and economists often use this model for situations in which it is reasonable to talk about “average preferences” or “average production technologies.” However, the world is a gloriously varied place, so we’d like to also be able to analyze situations in which consumers and firms aren’t all the same.

To do this, let’s work through an example in which we have two consumers with different preferences, and two firms with different levels of capital. In particular, we’ll consider a market for sandwiches in which there are two consumers (Adam and Eve) and two firms (Subway and Togo). Even though there are a small number of consumers and firms, we’re going to model this “as if” all agents are price takers.

Demand

Our two consumers, Adam and Eve, have preferences over sandwiches (good 1) and money spent on other goods (good 2) $\begin{aligned} u^A(x_1,x_2) &= 8 \ln x_1 + x_2\\ u^E(x_1,x_2) &= 4 \ln x_1 + x_2 \end{aligned}$ Let’s assume that sandwiches are a relatively small part of each of their budget, so we can ignore corner solutions in which Adam or Eve spends their entire budget on sandwiches. Furthermore, since good 2 “money spent on other things,” the price of good 2 is just 1; so let’s write the price of a sandwich as $p$. Finally, let’s write $A$ for the quantity of sandwiches Adam consumes, and $E$ for the quantity of sandwiches Eve consumes. Therefore, Adam’s $MRS = 8/A$ and Eve’s $MRS = 4/E$.

Following our procedure from Part II, Adam and Eve will each set their MRS equal to the price ratio (in this case, $p$): $\text{Adam's optimization (tangency) condition: }{8 \over A} = p$ $\text{Eve's optimization (tangency) condition: }{4 \over E} = p$ Solving for the quantity demanded in each case, Adam’s demand for sandwiches will be $d^A(p) = {8 \over p}$ and Eve’s will be $d^E(p) = {4 \over p}$ The overall demand in this “market” is therefore $\begin{aligned} D(p) &= d^A(p) + d^E(p)\\ &= {8 \over p} + {4 \over p}\\ &= {12 \over p} \end{aligned}$ Visually, we can understand this as the horizontal summation of their demand curves. In the following diagram, drag the price up and down to see how many sandwiches each of them demands at that price, and therefore what the total quantity demanded in the market is:

See interactive graph online here.

Supply

Suppose our two firms, Subway and Togo’s, each have the sandwich production function $f(L,K) = \sqrt{LK}$ Furthermore, in the short run, suppose Subway has $\overline K^S = 4$ units of capital, Togo’s has $\overline K^T = 2$ unit of capital, the price of labor is $w = 4$, and the price of capital is $r = 2$.

Suppose $S$ is the quantity of sandwiches produced by Subway, and $T$ is the quantity of sandwiches produced by Togo’s. Using our methods from Unit III, we can derive the cost functions for Subway and Togo’s as $c^S(S) = S^2 + 8 \Rightarrow MC^S(S) = 2S$ $c^T(T) = 2T^2 + 4 \Rightarrow MC^T(T) = 4T$ Since each firm will set $p = MC$, their supply functions will be $s^S(p) = {p \over 2}$ $s^T(p) = {p \over 4}$ The overall supply in this “market” is therefore $\begin{aligned} S(p) &= s^S(p) + s^T(p)\\ &= {p \over 2} + {p \over 4}\\ &= {3p \over 4} \end{aligned}$ Again, visually this is the horizontal summation of the two supply curves:

See interactive graph online here.

Equilibrium price, quantity, and allocation

Setting market demand equal to market supply gives us the market equilibrium price: $\begin{aligned} S(p) &= D(p)\\ {3p \over 4} &= {12 \over p}\\ p^2 &= 16 \\ p^\star &= 4\end{aligned}$ At a price of $p = 4$:

Adam demands $8/4 = 2$ sandwiches
Eve demands $4/4 = 1$ sandwich
Subway produces $4/2 = 2$ sandwiches
Togo’s produces $2/2 = 1$ sandwich

Note that competitive markets have both determined the overall quantity of sandwiches (three) and also who gets sandwiches (Adam gets more than Eve) and who produces sandwiches (Subway makes more than Togo’s). This allocation comes directly from the utility and production functions: Adam likes sandwiches more than Eve does, and Subway has more capital than Togo’s (and correspondingly lower marginal costs).

An important question, by the way, is is this the “right” overall quantity and allocation of sandwiches? We’ll answer that on Friday…

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