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Lecture 4: Exchange Equilibrium


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Last time, we look at an individual agent’s optimization problem when they start with an endowment of goods, and can buy and sell those goods at market prices. In this lecture, we will examine market behavior in an exchange economy: that is, if an economy starts at some “endowment allocation” within the Edgeworth Box, and agents can buy or sell the goods (or otherwise trade with one another), how does the market arrive at an equilibrium? Essentially, we are going to develop our familiar model of supply and demand, but only involving exchanges of goods that already exist.

So, let’s go back to the Edgeworth Box and think about prices.

Mutual Gains from Trade

Last Thursday we established that some allocations in the Edgeworth Box are not Pareto efficient: there are other allocations which could make at least one agent better off, without making any agent worse off. In particular, this occurs when agents’ marginal rates of substitution at an allocation are different: that is, when the relative values they place on goods differ.

It follows that at such a point, there are potential mutual gains from trade: that is, the two agents can find another bundle which represents a Pareto improvement, and trade to it. The graph below shows Alison and Bob’s initial allocation $E$, both from the perspective of the Edgeworth Box and from their individual perspectives. Bundle $X$ represents a potential trade. If you drag bundle X around the box, you can see that some allocations (i.e. the ones in the “lens” of Pareto improvements) make both Alison and Bob better off. Others make only one of them better off, or even neither better off:

See interactive graph online here.

Such a bilateral trade is the most basic economic transaction, and can be used to analyze many situations. If both “goods” are goods, then it’s just describing trading one good for another. If good 1 is some good and good 2 is money, then it’s trading money for goods. And if good 1 is present consumption and good 2 is future consumption, then it’s one agent lending the other some money.

For each type of trade, there are various ways at which we can arrive at a new allocation:

While all of these are interesting, we’re going to focus (for now) on the last one: one in which there is a market price that all agents take as given, and each agent chooses how much to trade at that price. This is called a competitive equilibrium.

Competitive Equilibrium

In the model of competitive equilibrium in the Edgeworth Box, we’ll start from an assumption that both agents are price takers: that is, they believe that they can buy and sell goods from their endowment at given, market prices. In truth, this is a fanciful assumption for an economy of just two people; but in fact, as we’ll see, this model extends to an arbitrarily large number of agents. So let’s proceed using the assumption that there is a market price ratio that both agents take as given. We’ll call that price ratio $p = p_1/p_2$, since we know from our analysis in the past few chapters that it’s just the ratio (and not individual prices) which will determine behavior.

We can illustrate a price ratio in the Edgeworth Box as a line passing through the endowment with a (negative) slope equal to the price ratio. Note that this forms a mutual budget line: that is, it is the set of allocations to which you could trade from the endowment at market prices:


See interactive graph online here.


We’ve established that the potential for gains from trade in the Edgeworth Box exists when the two agents have different marginal rates of substitution at their endowment. By definition, this means that there is a range of price ratios between those two marginal rates of substitution; and since agents want to buy if their $MRS > p$ and sell if $MRS < p$, this means that in that range of price ratios, one of the agents will want to buy good 1 and sell good 2, and the other will want to sell good 1 and buy good 2.

Let’s look at little more at exactly who will buy, and who will sell:

The range of prices at which trade might occur are the price ratios that lie between their two MRS’s. In other words, the green area of Pareto improvements lies between the two dashed lines representing their MRS’s in the Edgeworth Box diagram below. The second graph shows a number line plotting different price ratios; the green area in this diagram represents the price ratios at which potential gains from trade might occur:


See interactive graph online here.


Now that we’ve established the range of prices at which trade might occur, let’s see what would actually happen at different price ratios.

Gross Demands in the Edgeworth Box

To start out, let’s look at where in the Edgeworth Box each agent would want to trade at different prices: that is, their gross demands. This is just applying everything we’ve done for the past few chapters: given and endowment and a price ratio, there is some ideal point that each agent would like to trade to. If we plot these two points in the Edgeworth Box, we can analyze the desired behavior at each price ratio:


See interactive graph online here.


Net Demands

Given each agent’s gross demands, we can derive their net demand for each good, which is the amount they wanted to end up with, minus their endowment. That is, if they start out with an endowment of $(e_1,e_2)$, and if they face price ratio $p = p_1/p_2$, their gross demands are represented by the $(x_1^\star(p),x_2^\star(p))$ to which they would optimally want to trade, and their net demands are

Since they want to buy one good and sell the other, one of these must be negative and the other one is positive: that is, at any price ratio, they would want to sell some of one of their goods and use the proceeds to buy the other. In particular:

Let’s just focus on the amount of good 1 each agent wants to trade.

See interactive graph online here.

When the price ratio is high, A wants to sell a lot of good 1, but B doesn’t want to buy very much. Likewise, when the price ratio is low, A doesn’t want to sell very much good 1, while B wants to buy a lot of it. In other words: we’ve developed a standard supply and demand model, using only endowments and preferences.

Interpreting the Equilibrium Price: Scarcity and Preferences

In the previous example, the two agents had Cobb-Douglas utility functions which put equal weight on each good, and there was twice as much good 1 as good 2; and the price ratio worked out so that the price of good 2 was twice that of good 1 (i.e. $p_1/p_2 = 1/2$). This is a feature of competitive equilibrium with Cobb-Douglas preferences, which can be used to illustrate some nice aspects of how the price ratio in competitive equilibrium relates to agents’ relative preferences for the two goods and the goods’ relative scarcity. To show why, we’ll have to do a bit of algebra.

Let’s go back to looking at both $p_1$ and $p_2$. Suppose each agent has normalized Cobb-Douglas preferences of the form \(u(x_1,x_2) = \alpha \ln x_1 + (1-\alpha) \ln x_2\) As we’ve shown many times, this means that each agent will spend fraction $\alpha$ of their income — in this case, the value of their endowment — on good 1, so their gross demand for good 1 will be \(x_1^* = \alpha \left ({p_1e_1 + p_2e_2 \over p_1}\right) = \alpha e_1 + \alpha e_2 \times {p_2 \over p_1}\) Subtracting $e_1$ from both sides, we can see that their net demand for good 1 is \(x_1^* - e_1 = -(1-\alpha)e_1 + \alpha e_2 \times {p_2 \over p_1}\) Now suppose that agent A’s value of $\alpha$ is $a$, while agent B’s value of $\alpha$ is $b$: \(\begin{aligned}u^A(x_1^A,x_2^A) &= a \ln x_1^A + (1-a) \ln x_2^A\\u^B(x_1^B,x_2^B) &= b \ln x_1^B + (1-b) \ln x_2^B\end{aligned}\) If we again assume that A is the supplier and B is the demander, A’s net supply and B’s net demand are given by \(\begin{aligned}s_1^A(p) &= (1-a)e_1^A - ae_2^A \times {p_2 \over p_1}\\d_1^B(p) &= -(1-b)e_1^B + be_2^B \times {p_2 \over p_1}\end{aligned}\) Equating these and solving for the price ratio $p_1/p_2$ gives us \(\begin{aligned} (1-a)e_1^A - ae_2^A \times {p_2 \over p_1} &= -(1-b)e_1^B + be_2^B \times {p_2 \over p_1}\\ (1-a)e_1^A + (1-b)e_1^B &= {p_2 \over p_1}\left(ae_2^A + be_2^B\right)\\ {p_1 \over p_2} &= {ae_2^A + be_2^B \over (1-a)e_1^A + (1-b)e_1^B} \end{aligned}\) OK, if you’ve tuned out because of all the algebra (full disclosure: I would!), just look at that last expression, which is our equilibrium price ratio in terms of four variables:

So how does the equilibrium price ratio change with each of these?

There are a few interesting special cases:

More generally, the equilibrium price ratio is jointly determined by the relative preferences for the two goods, and their relative scarcity. This result is also true for utility functions that aren’t Cobb-Douglas, but they’re not necessarily shown as clearly by the formula for the equilibrium price.

Application: Externalities and the Coase Theorem

Up until now, we’ve assumed \(u^A (x_1^A,x_2^A,x_1^B,x_2^B) = u^A (x_1^A,x_2^A)\) \(u^B (x_1^A,x_2^A,x_1^B,x_2^B) = u^B (x_1^B,x_2^B)\) That is, we’ve thought of utility being a function solely of one’s own consumption. Under this assumption (and other maintained assumptions, like smooth utility functions) we’ve shown that the set of Pareto efficient allocations is the one such that \(MRS^A (x_1^A,x_2^A) = MRS^B (x_1^B,x_2^B)\) What happens, though, if your consumption affects my enjoyment? It turns out we can use the Edgeworth Box to analyze situations like this as well.

Problem: Smoking Roommates

Suppose Chris and Ken are roommates. Chris is a non-smoker and doesn’t like smoke. Ken is a smoker and enjoys smoking quite a bit.

Let’s think of each of their utility as being a function of two things: the amount Ken smokes (good 1) and the amount of money they have (good 2). Let $s$ be the number of hours that Ken spends smoking, and let’s assume that Ken could spend up to 10 hours a day smoking: that is, $10 − s$ is the number of hours each day he doesn’t spend smoking. In a way, we can think of $x_1^K = s$ and $x_1^C = 10 − s$ — that is, Chris’s “consumption good” is the number of smoke free hours, and Ken’s “consumption good” is the number of hours spent smoking. (Assume that Ken would like to smoke for the full 10 hours.) Finally, let’s assume that Chris and Ken each have $€100$.

Just as we used an Edgeworth Box to analyze allocations of goods among people, we can use one to analyze possible outcomes in this model. The height of the box is the total amount of money Chris and Ken have ($€200$). The width of the box is the 10 hours in which Ken could smoke or not smoke:

See interactive graph online here.

Let’s suppose that both Chris and Ken have quasilinear preferences over smoking and money: in particular, Ken’s preferences are given by \(u^K(x_1^K,x_2^K) = k \ln x_1^K + x_2^K\) and \(u^C(x_1^C,x_2^C) = c \ln x_1^C + x_2^C\) Since money is “good 2” and smoke is “good 1,” this means that each person’s MRS is their willingness to pay for an hour of smoke (in Ken’s case) or an hour less of smoke (in Chris’s case): \(MRS^K = {k \over x_1^K}\) \(MRS^C = {c \over x_1^C}\) Now, we have that the amount of “good 1” that Ken has is the amount he smokes ($s$), and the amount of “good 1” that Chris has is the amount Ken doesn’t smoke ($10 - s$), so we along the contract curve, when these are equated, Ken’s marginal utility from the last hour he spends smoking is exactly equal to Chris’s marginal disutility from that hour. Furthermore, note that because we used quasilinear utility functions, the efficient amount of smoke doesn’t depend on money at all: it just depends on the amount of smoke! In particular, when these are equal, we have \(\begin{aligned} {k \over s} &= {c \over 10-s}\\ k(10-s) &= cs\\ 10k &= (k + c)s\\ s &= 10 \times {k \over k + c} \end{aligned}\) Note that $k$ and $c$ represent the intensity with which Ken and Chris think about smoking. If $k = c$, then the efficient quantity of smoke is 5; if $k > c$, it’s efficient for Ken to smoke more than 5 hours; and if $k < c$, it’s efficient for Ken to smoke less than 5 hours:

See interactive graph online here.

So, there is a single efficient quantity of smoke…how do we get there?

Property Rights and the Coase Theorem

In the Edgeworth Box, we generally start out with the notion of an “endowment.” As far as an endowment goes, the money part is easy: we’ve assumed each of them starts with €100, so $e_2^C=e_2^K = 100$. This means that we’re starting off with an endowment somewhere along the horizontal line in the diagram below:

See interactive graph online here.

But how can we think of an “endowment” of smoking time? One way of thinking of it is in terms of property rights. Suppose this is Ken’s house, and Chris is only paying rent; we might then assume that Ken has the right to smoke as much as he likes in his house. This would correspond to the point at the far right-hand end of the line in the box above. Likewise, if it’s Chris’s house, we might assume that he has the right to demand a smoke-free living environment; this would correspond to the point at the far left-hand end of the line. You could also imagine various different institutional arrangements which would give Ken the right to smoke some number $s$ hours per day. You can drag the point in the graph above left and right to assign different property rights.

Now, in general, most endowments won’t be Pareto efficient, unless they happen to lie directly on the contract curve. What then? Well, if we allow Chris and Ken to trade, everything we know from the arguments in the last chapter go through: they’ll keep trading until they end up on the contract curve, and we will end up at a Pareto efficient point. In fact, if we allow for there to be a market for the externality itself — that is, a market in which people can trade the right to smoke — then the competitive price in that market will converge to the price which lands you on the contract curve:

See interactive graph online here.

The Coase Theorem, named after Ronald Coase and based on his 1960 paper “The Problem of Social Costs,” essentially stipulates that as long as the costs of negotiation are sufficiently low, the problem of the inefficient externality can be solved by assigning property rights to the externality itself and allowing the interested parties to bargain with one another; and furthermore, that as long as everyone’s preferences are quasilinear, the efficient outcome (in this case, the quantity of smoke) will be the result regardless of the initial allocation of property rights. Of course, the property rights are themselves valuable: Ken end up better off if he starts out with the right to smoke, and Chris ends up better off if he starts out with the rights to clean air. But either way, creating a market for smoking means that in equilibrium, the amount of smoke is the Pareto efficient amount.

In addition to quasilinear preferences, the Coase theorem relies on some pretty strong assumptions about how the agents in the model come together. The last diagram shows a competitive equilibrium; but if we’re just talking about two roommates, the assumptions of competitive equilibrium (many buyers and sellers, etc.) won’t apply; so bargaining power can start to play a large role. All parties may not recognize the property rights of the others, or agree on a third party to impose those rights. And especially if the stakes are high, bargaining may not be costless; once the lawyers get involved, things can get expensive fast. Oh, and there could be asymmetric information, incentives to overstate the harm one feels from the externality, and so on. But the central Coasean idea is a powerful one: that the efficiency of markets can be brought to bear on things, like smoking, for which there is no existing market.

The Efficiency of Competitive Equilibrium and the Fundamental Theorems of Welfare Economics

This analysis of externalities points leads us to what are potentially the two most impactful “theorems” in all of economics, known as the “Fundamental Theorems of Welfare Economics.”

There is an elegant logical proof of the FWT, which goes like this: first consider a two-person, two-good model like we’ve been doing. Suppose the initial endowment is $E$, and $X$ is an equilibrium allocation. The budget line corresponding to the equilibrium connects $E$ and $X$ and divides the Edgeworth Box into to regions: those affordable to $A$ (below and to the left of the budget line) and those affordable to $B$ (above and to the right of the budget line). Now, because A chooses to trade to $X$, it must be the case that any allocation Y that A strictly prefers to X must be unaffordable to A – otherwise, A would have chosen Y over X. Visually, anything that A prefers to X must lie above and to the right of the budget line. By the same logic, any allocation Z that B strictly prefers to X must be unaffordable to B; i.e., lie below and to the left of the budget line. Since no allocation can simultaneously be in both regions, X must be Pareto efficient. (Things get a little more complicated with more goods and more people, but the logic remains the same.)

The SWT is a bit more controversial: it says that you don’t have to sacrifice efficiency for equity! You can just reallocate goods any way you want, and as long as you then let people trade, they’ll find their way to a Pareto efficient allocation. This is the intuition of the Coase Theorem: as long as (some) property rights are established, and you allow trade for an externality, you end up with an efficient quantity of the externality.

However, as is perfectly obvious from the various experiments in government appropriation of assets over the years, just knowing that the government might seize your property and reallocate it to someone else is distortionary and inefficient. In particular: we’ve just assumed that the goods in this model are “given,” and haven’t thought at all about how they came to be produced. If the government might take any goods you produce, your incentive to produce them diminishes considerably.

So: we need a model of where these goods come from! That’s what we’ll get to next week. :) See you then!

P.S. If you’re interested more on this, I would direct you to the extremely excellent Wikipedia article on the subject for more. Varian also has a great analysis in the Exchange chapter of his textbook.


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