Lecture 6: Competitive Equilibrium with Production and Trade

In the last lecture we established the basics of a production economy: each agent is endowed, not with goods, but with resources; and can transform those resources into goods via production functions.

We also saw that when they can sell those goods at market prices, they will choose the output combination that maximizes the market value of what they each produce. Furthermore, we saw that this will result in them producing a productively efficient point along their joint PPF – and even more, that they collectively choose the point along the joint PPF which has the highest market value.

Fundamentally, what this means is that we’ve established the supply choice of all agents in the economy. Today we’ll combine this analysis with the demand side of the equation, and solve for competitive equilibrium. What we’ll find is that at the equilibrium price ratio, the outcome is characterized by both productive and allocative efficiency.

Supply

Last time we established what each agent would do (i.e., how many fish and coconuts they would produce) at different prices. Now let’s look at how much fish they each choose to produce, as a function of the prices of fish and coconuts. We can then plot the global supply curve of fish, holding the price of coconuts constant (or equivalently, plot it as a function of the relative price of fish, $p_1/p_2$, which is all that matters). By the way, the production functions and optimal decisions are unchanged from lecture 5; review those for the basis of these calculations, I won’t take up space repeating them here.

For Alison and Bob, because their PPFs are curved (and smooth), each of them produces more and more fish as the price of fish rises. It may be shown that their supply functions are \(\begin{aligned} y_1^A(p_1,p_2) &= \frac{48p_1}{\sqrt{12p_1^2 + p_2^2} } & \hspace{0.25in} y_2^A(p_1,p_2) &= \frac{4p_2}{\sqrt{12p_1^2 + p_2^2} }\\ \\ y_1^B(p_1,p_2) &= \frac{48p_1}{\sqrt{12p_1^2 + 6p_2^2} } & \hspace{0.25in} y_2^B(p_1,p_2) &= \frac{24p_2}{\sqrt{12p_1^2 + 6p_2^2} } \end{aligned}\) and therefore the total amount produced of each good are the (rather nasty) sums of those two expressions. The important feature is that as $p_1$ increases, the total supply of good 1 smoothly increases as well, leading to an upward sloping supply curve:

Demand

Once each agent chooses what they’re going to produce, they then spend the money according to their utility functions.

How much money do they earn? Let’s use the function $m(p_1,p_2)$ to represent the market value of each agent’s production choice when they are maximizing that value: \(m(p_1,p_2) = p_1y_1^*(p_1,p_2) + p_2y_2^*(p_1,p_2)\) For example, the value of Alison’s production is \(\begin{aligned}m^A(p_1,p_2) &= p_1 \times \frac{48p_1}{\sqrt{12p_1^2 + p_2^2} } + p_2 \times \frac{4p_2}{\sqrt{12p_1^2 + p_2^2} }\\ &= \frac{48p_1^2 + 4p_2^2}{\sqrt{12p_1^2 + p_2^2} }\\ &= 4\sqrt{12p_1^2 + p_2^2}\end{aligned}\) We can think of this as the value of Alison’s endowment, when her endowment is determined by what she produces.

Let’s recall that her utility function (from lecture 2! remember that far back?) was the simple Cobb-Douglas $u(x_1,x_2) = x_1x_2$, meaning that she’ll spend half of her income on each good. Therefore her demand will be \(x_1^*(p_1,p_2) = {1 \over 2}{m(p_1,p_2) \over p_1} = \frac{4\sqrt{12p_1^2 + p_2^2} }{2p_1} = 2\sqrt{12 + \left({p_2 \over p_1}\right)^2}\) Again, don’t get lost in the math here (I won’t ever ask you to derive anything like it!) and stay focused on what’s going on: given prices $p_1$ and $p_2$, Alison is choosing the value-maximizing point along her PPF. That then determines a budget line, and she maximizes her utility subject to that budget line as usual. Visually, it looks like this:

Of course, Bob is doing the same thing; from his perspective, the problem looks like this:

Equilibrium

So what happens when we bring Alison and Bob’s supply curves together with their demands? This is the big, beautiful series of graphs. At low price ratios, they don’t produce much fish (because it’s not very valuable) but demand quite a bit (because it’s very cheap); hence there is excess demand, which would cause upward pressure on prices. This would cause both of them to produce more fish and fewer coconuts (moving to the right along the PPF). Eventually they arrive at a point where total supply exactly equals total demand.

Notice that in the Edgeworth box, we have disequilibrium at those low prices, and when we arrive at the competitive equilibrium, everything is beautiful and tangent and maximizing and efficient – the Invisible Hand has worked its magic.

Gains from Specialization and Trade

Let’s now close by thinking about Chuck and Wilson one last time. This model will probably be familiar to you from Econ 1, or even high school.

First, let’s think about what Chuck and Wilson would do if they couldn’t trade. Let’s assume each of them has the utility function \(u(x_1,x_2) = x_1x_2^2\) Since they have linear PPFs, the logic of the “Cobb-Douglas trick” applies here: they would consume at a point 1/3 of the way along the constraint. (In other words, just as they would spend 1/3 of their income on fish if this were a consumer problem, here they spend 1/3 of their labor producing fish.)

Of course, by definition, if Chuck can sell his output and then buy at any point along a budget line, he will be able to consume at a point outside his PPF. As we saw before, Chuck will maximize his value of production by specializing in one good or another. If Chuck has specialized in producing only fish, and produces $y_1 = 24$ fish, the monetary value of that decision is $24p_1$. Therefore, his budget line would be defined by the equation \(p_1x_1 + p_2x_2 = 24p_1\) On the other hand, if he has specialized in producing only coconuts, and produces $y_2 = 12$ coconuts, that would earn him $12p_2$, so his budget line would be \(p_1x_2 + p_2x_2 = 12p_2\) As long as he completely specializes in the more profitable good, his budget line will lie above his PPF: that is, he will be able to afford bundles with more of both fish and coconuts than if he had relied on only his own production! This makes him better off, because he can consume a bundle which gives him higher utility:

Wilson, of course, faces a similar problem. As we saw before, if the price ratio is in the range \({1 \over 2} < {p_1 \over p_2} < 1\) both Chuck and Wilson them will want to specialize in the good in which they have a comparative advantage:

But which of those prices will occur? Let’s assume both Chuck and Wilson have the utility function $u(x_1,x_2) = x_1x_2^2$, and think about their supply and demand for fish. For simplicity, let’s assume $p_2 = 1$, and calculate their demand for good 1 as a function of $p_1$.

When both Chuck and Wilson specialize, we’ve established that Chuck is producing $y_1^C = 24$ fish and Wilson is producing $y_2^W = 36$ coconuts. Therefore the monetary value of Chuck’s fish is $m^C = 24p_1$, while the monetary value of Wilson’s coconuts is $m^W = 36$. Using the Cobb-Douglas trick, we know that each of them will spend ${1 \over 3}$ of their money on fish. Therefore their (gross) demands for fish are \(\begin{aligned} x_1^C(p_1) &= {m^C \over 3p_1} = {24p_1 \over 3p_1} = 8\\ x_1^W(p_1) &= {m^W \over 3p_1} = {36 \over 3p_1} = {12 \over p_1} \end{aligned}\) Therefore, Chuck’s net supply curve is perfectly inelastic at $24 - 8 = 16$, and Wilson’s net demand curve is $d^W(p_1) = {12 \over p_1}$. Equating supply and demand gives us \(\begin{aligned} s^C(p_1) &= d^W(p_1)\\ 16 &= {12 \over p_1}\\ p_1^\star &= {12 \over 16} = 0.75 \end{aligned}\) And indeed, if we look in the Edgeworth Box, we can see that this is indeed the price ratio that causes each of them to want to trade to the same place in the Edgeworth Box:

So again, we have arrived at an equilibrium which is characterized by both productive efficiency (in a PPF diagram) and allocative efficiency (in an Edgeworth box).

End of the Neoclassical Model

So, that’s it. The past thirteen weeks has really been just one big model. Here it is in a nutshell.

The only things that truly matter in the world are real. They’re the planet we’re on, its people, its natural resources.

We have ways of transforming our resources (human and natural) into goods and services that make our lives better: food and clothing, phones and computers, fish and coconuts. Because the same resources can be used to produce different goods, and because resources are scarce, this means we face tradeoffs.

If there were only one person in the whole world, that person could make decisions however he or she wanted; but they would still face a tradeoff, and they would choose to produce the combination of goods such that their own willingness to trade one good for another (their MRS) reflected the opportunity cost they faced, based on the resources and technology available to them (their MRT). Put another way, they produce and consume goods such that the marginal benefit from consuming the last unit of each good exactly reflected the marginal cost of the resources used to produce it: \(MRS = MRT\) But there’s more than one person in the world. That opens up possibilities beyond what a single person can do, but it also introduces a coordination problem: how do billions of people coordinate on producing and trading millions of types of goods and services every day?

We have focused for thirteen weeks on how prices can solve this problem by serving as a coordination mechanism. If there is a price for everything, and if everyone is a price taker in all things, then prices convey information: if you can do something for cheaper than the market price, you can be better off. These price signals help you realize where your comparative advantage is, and therefore help you obtain the greater array of options (as shown by your budget set) so that you can live your best possible life.

In other words, when everyone sets their own personal marginal benefit or marginal cost equal to a common price – or their own personal MRT or MRS equal to a common price ratio – then we achieve both productive and allocative efficiency. When competitive markets are in equilibrium, there is no way to reallocate resources to produce more of one good without producing less of another; and there is no way to make one person better off without making someone else worse off.

Of course, that’s a lot of ifs. The big one is that everyone is a price taker: that no firm has market power, and no individual has negotiating power. That’s patently false. Another big one is that everyone has perfect information about all relevant aspects of a trade; that, too, is wrong. In other words, for the last seven weeks of the course, we’ll spend our time analyzing how and why this harmonious model of perfect competition breaks down.

But for now, enjoy the view from the mountaintop. And start studying for the first Checkpoint. :)

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