22.6 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:

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:

• If $MRS^A < MRS^B$ at the initial endowment, then B is more willing to give up good 2 to get more good 1 than A is; so the “lens” of potential improvements lies above and to the left of the endowment point, and potential trades involve A trading some good 1 for some of B’s good 2.
• If $MRS^A > MRS^B$ at the initial endowment, then A is more willing to give up good 2 to get more good 1 than B is; so the “lens” of potential improvements lies below and to the right of the endowment point, and potential trades involve A trading some good 2 for some of B’s good 1.

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:

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:

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

• Net demand for good 1: $x_1^\star(p) - e_1$
• Net demand for good 2: $x_2^\star(p) - e_2$

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:

• If $MRS(e_1,e_2) > p$, the agent would want to buy good 1 and sell good 2; so we could define \(\begin{aligned}d_1(p) &= x_1^\star(p) - e_1 \\ s_2(p) &= e_2 - x_2^\star(p)\end{aligned}\)
• If $MRS(e_1,e_2) < p$, the agent would want to sell good 1 and buy good 2; so we could define \(\begin{aligned}s_1(p) &= e_1 - x_1^\star(p) \\ d_2(p) &= x_2^\star(p) - e_2\end{aligned}\)
• If $MRS(e_1,e_2) = p$, the agent would want to stay at their endowment; so at this price ratio, $d_1(p) = s_1(p) = 0$. In other words, this price ratio represents the intercept of both their net demand and net supply curves.

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

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.

Note: everything below this is newly written and may contain typos. Please let me know if you catch any!

Walras’ Law

OK, we’ve solved for the equilibrium price ratio by setting the net demand for good 1 equal to the net supply of good 1…but what about the market for good 2? Is it possible for the market for good 1 to be in equilibrium, but the market for good 2 to not be in equilibrium? No — and we can see why, both intuitively and algebraically.

• Intuitively, at the equilibrium price, both A and B want to trade to the same point in the Edgeworth Box. That is, they want to trade a particular amount of good 1 for a particular amount of good 2. Since they agree on the amount of good 2, the market for good 2 must also be in equilibrium.
• Mathematically, we can prove that market 2 must clear by looking at the market value of the two goods traded. Let’s return to $p_1$ being the price of good 1, and $p_2$ being the price of good 2. For agent A, the amount of money they get from selling good 1 must equal the amount they spend buying good 2: \(p_1(e_1^A - x_1^A) = p_2(x_2^A - e_1^A)\) Likewise, for agent B, the amount of money they get from selling good 2 must equal the amount they spend buying good 1: \(p_1(x_1^B - e_1^B) = p_2(e_2^B - x_2^B)\)If the market for good 1 is in equilibrium, it means that the left-hand sides of these two equations must be equal:\(\begin{aligned}\overbrace{e_1^A - x_1^A}^\text{A's net supply of good 1} &= \overbrace{x_1^B - e_1^B}^\text{B's net demand for good 1}\\p_1(e_1^A - x_1^A) &= p_1(x_1^B - e_1^B)\end{aligned}\)and so the right-hand sides must also be equal:\(\begin{aligned}p_2(x_2^A - e_2^A) &= p_2(e_2^B - x_2^B)\\ \underbrace{x_2^A - e_2^A}_\text{A's net demand for good 2} &= \underbrace{e_2^B - x_2^B}_\text{B's net supply of good 2}\end{aligned}\)

In fact, it can be shown that in the more general case of many different goods, if all markets except one are clearing, the last one must clear as well. This is called Walras’ Law and is worth looking up if you’re interested.

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:

• Agent A’s preferences put a weight of $a$ on good 1 and $1 -a$ on good 2
• Agent B’s preferences put a weight of $b$ on good 1 and $1 -b$ on good 2
• $e_1^A$ and $e_1^B$ are the endowments of good 1
• $e_2^A$ and $e_2^B$ are the endowments of good 2

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

• An increase in $a$ or $b$ will increase $p_1/p_2$. In other words, the more agents like good 1 relative to good 2, the higher its relative price will be.
• An increase in $e_1^A$ or $e_1^B$ will decrease $p_1/p_2$. In other words, the less scarce good 1 is, the lower its relative price will be.
• An increase in $e_2^A$ or $e_2^B$ will increase $p_1/p_2$. In other words, an increase in the amount of good 2 makes good 1 relatively more scarce, and therefore increases the price of good 1 relative to good 2 (or decreases the price of good 2 relative to good 1)

There are a few interesting special cases:

• Equal preferences: As the example we used to derive equilibrium above, if all agents put an equal weight on each good, so $a = b = 1-a = 1-b = {1 \over 2}$, the equilibrium price is determined entirely by the relative amounts of the two goods (i.e. the dimensions of the Edgeworth Box): \({p_1 \over p_2} = {e_2^A + e_2^B \over e_1^A + e_1^B} = \frac{\text{Total amount of good 2}}{\text{Total amount of good 1}}\)
• Equal endowments: Suppose $e_1^A = e_2^A = e_1^B = e_2^B$, so the Edgeworth Box is a square, and the original allocation is in its center. In this case, the equilibrium price is determined entirely by the relative preferences of the agents: \({p_1 \over p_2} = {a + b \over (1-a) + (1-b)} = \frac{\text{Sum of preference weights on good 1}}{\text{Sum of preference weights on good 2}}\)

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.