by Carrie
Sun Nov 13th, 2005 at 12:12:30 PM EST
As a mathematical physicist I am bound to take a peculiar approach to economic issues, so I hope you will forgive me if this diary strikes you as, well, peculiar.
On the other hand, we have had a number of interesting comment threads about free trade and comparative advantage, the nontrivial true result of economic theory. A number of mathematically inclined types around here, including me, can't seem to let an economics discussion go by without taking a swipe at economists for mathematical naïveté or physics envy, and we keep hinting at parallels between economics and our own pet theories. It's about time we stuck our neck out. Maybe we'll get those promised diaries about economics and thermodynamics, economics and dynamical systems, or economics and Darwinian evolution.
So let me sum up the gist of my argument so you can jump directly to the comments after this if you wish. To make things easy to calculate, easy to visualize and easy to reason with, economic models are often formulated in terms of linear equations (if this is twice as big then that will be twice as big, too!). The problem is, to justify linearity one would have to impose a long list of implausible conditions on the economic system at hand. Ceteris paribus does look like a weak excuse. Comparative advantage is indeed a robust result which does not depend on all these extraneous hypotheses, but it is often presented (as it was originally by Ricardo) with the help of a linear model.
Thinking about this linear model I realized that division of labour can be described by a linear model without stretching reality too much, so I start from there. I imagine an economy consisting of just two people producing two goods. A simple linear diagram illustrates why specialization helps, and why it is possible that some tasks won't be carried out by those most capable of carrying them out. The result, though, is that the production line of more than one person is not linear but convex, and this is what makes division of labour possible.
I then look at comparative advantage, which is the generalization of division of labour from two people to two countries. Now, the discussion of division of labour makes it clear that a linear model is no longer adequate to describe the production of either economy, so I give a diagram full of nonlinear production curves.
A feature of both analyses is that, starting from a situation where both people (countries) share work (production) equally across all tasks (goods), a fairly large reassignment of tasks is needed to achieve a modest improvement in efficiency. If we take into account how reluctant people are to change careers, or how long it takes to convert an entire industry to another purpose, it is easy to see that the optimal solutions on paper will usually be implemented at a large human cost, if at all. This gives an explanation, on paper for the widespread resistance to liberalization.
Division of Labour
Alice (A) and Bob (B) work together. They need to perform two tasks, X and Y. They have an egalitarian agreement whereby each tries to spend the same amount of time on each of the two tasks. In the diagram below, the line C represents the possible amounts of X and Y that they can obtain in this way. The dashed line to the point C represents one possible choice.
An alternative arrangement will be a
Pareto improvement if it increases the amount of both X and Y that is accomplished. This is impossible staying on the line C. All the points on C are
indifferent. The dotted lines from C delimit the directions of Pareto improvements.
A Pareto improvement can be achieved (along the bold line) by having Bob specialize completely in Y and Alice partially in X. The bold arrows represent the necessary adjustments. They are quite drastic even for a modest improvement in overall performance. The direction of the adjustments leading to a Pareto improvement is determined by the slopes of the lines A and B.
The convex broken line D is the "production possibility frontier" of Alice and Bob together. It would only be a straight line if the lines A and B were perfectly parallel.
Comparative Advantage
Here is an analogous diagram illustrating comparative advantage. We now have, say, Australia and Belgium, producing two commodities X and Y. Each country has a
convex, nonlinear production curve due to division of labour within it.
The curve C can be obtained by assuming that Australia and Belgium produce the goods X and Y in the same ratio. The curve D, which is Pareto-optimal, has the property that the vertices of the parallelogram on the curves A, B, and D correspond to where all three curves have the same
slope. This common slope is the
relative price of the goods X and Y assuming a given ratio of production of X to Y, and that the productive resources of Australia and Belgium are fully employed. Comparative advantage is advantage in terms of this relative price. In the linear model the slopes are all independent of what the two economies are doing, and so a lot of the richness of possible behaviours is lost. Does linearity throw the baby out with the bathwater?
Once again the gain in productivity from C to D is much smaller than the necessary adjustments in the employment of resources within Australia and Belgium.
These are the generic features of the diagram. There are a lot of features that are not generic. The way I've drawn the curves, neither country is more efficient than the other across the board, but Australia has a larger aggregate production on account of its larger size. The most efficient producers of X are in Australia, and the most efficient producers of Y are in Belgium. Basically, the linear model I drew above can be fully specified by three numbers, but the convex model for the second diagram has a whole lot more features that affect the answers to questions such as: can free trade between A and B wipe out the entire production of X or Y in one of the two countries? This is determined by the slopes of the curves A and B at the endpoints. The diagram will be qualitatively different if the curves A and B intersect. In that case, if we imagine that the economies of A and B "evolve in time" by sliding continuously on the production curves, there may be a suboptimal equilibrium where A and B are both poised at the crossing point. And many more nongeneric features.
Conclusion
This should come as no surprise, but as soon as the model got a little more realistic it also got a little unwieldly. The linear model can be dissected
ad nauseam in all generality, but the convex one cannot, or not so easily. The fact of comparative advantage reduces (mathematically) to the fact that free trade partly combines the smaller economies into a single one. This means that the possible arrangements of the whole economy are more numerous than the arrangements where each subeconomy must satisfy all its own needs. The underlying mathematical fact is that optimizing over a larger set of possibilities improves the optimum.
But this whole analysis is static. We can introduce bastard dynamics by imagining that the economies of A and B are like beads sliding on wires (the production curves) but we are basically unable to study questions of time scales. We also forget that in order for the beads to slide on the wires a large number of resources need to be transferred from one industry to another. In a real economy all these things take time, and we are talking about real people. Also, all the responses are delayed with respect to the signals that cause them, and the responses tend to overshoot. When an economist draws a diagram like this, he is behaving like a Platonic Socialist master planner even if he's modelling a free market: he has access to perfect information about the system and can assume that resources are allocated to their optimal uses as needed. But even this Platonic allusion is misleading: it is the ideal economic model that is a shadow of the real world.
Update [2005-11-15 19:41:3 by Migeru]: By popular demand... here are some equations. And then after that I really stick my neck out and make some outlandish claims about economics based on a drawing on a piece of paper. So don't hit me too hard.
Gold for Soybeans
I wanted to call this "oil for food", but it would have made the argument a little convoluted, as well as requiring me to put money on an oil standard.
There is a lot of information that can be obtained from the production curve of an economy. As I have mentioned above, its slope at any given point is the relative price of the commodities X and Y. Suppose that Y is a foodstuff (say, soybeans), and that X is something the people use as money (say, gold). At any given point of the production the slope of the normal (M) is the (marginal) number of units of gold per unit of soybeans, i.e., the price of food. If we plot the rate of production of Y against the slope M we obtain the supply curve of soybeans:
If you want some equations, suppose the production curve is defined by some equation of the form
U(x,y) = 0
then, the ratio of partial derivatives
M = Uy/Ux
is the ratio of the price of Y to the price of X, that is, the price of Y if X is used as money. Then, the quantity
G = X + (Uy/Ux)Y
is the value of X + Y measured in units of X, that is, the GDP. The value G' is the GDP if you used Y as currency. For those of you Thermodynamics buffs, G as a function G(M) is the Legendre transform of Y(X).
Now I am ready to justify why the production curve of an economy cannot be linear. First of all, the GDP and the price of food in that economy would be fixed and immutable (assuming the economy is fuctioning efficiently). But, worst of all, the supply of food could take any value but its price would be fixed.
Nonconvexity and catastrophes
I really don't want to write any more equations: they're ugly. Besides, if you're a thermodynamics buff you can take what I said earlier about U(x,y) and run with it, and if not it's just Greek to you. Instead I'll draw another couple of pretty pictures.
So, why should the actual production curve of an economy (as opposed to the ideal production possibility frontier) be convex anyway? All it needs to be is Pareto-indifferent, but that only means that as you increase the production of X you must decrease the production of Y. The bold curve below on the left is Pareto-indifferent but not convex. What's wrong with that?
Well, what's wrong is that the portion of the curve below the thin straight line is not convex, and it corresponds to a bizarre situation in which decreasing the price of Y increases the supply (see on the right). This is an unstable situation. The market will never do that. In fact, what will happen is that, as the price M increases, the supply of Y increases slowly and then, at some point, as increasing demand tries to push the price of Y a little higher a huge irreversible rearrangement of productive resources (from X to Y) is triggered, causing a supply glut together with a collapse of the price (along the dashed arrow in the diagram on the right). If the system starts out in the high-supply (of Y), high-price regime and the demand starts to sag, driving the price and supply down, again a point will be reached where a large shock in the economy shifts production from Y to X, with a drop in Y production and a price increase (along the other dashed arrow).
Now, the whole economy is likely to rearrange itself in such a way that the production curve changes. Such a large amount of resources change use so quickly that all bets are off. The system will likely settle somewhere between the thin solid line (the convex envelope of the production curve) and the curved dashed line (the production possibility frontier). This is, I suppose, the creative destruction of Schumpeter.
How can something like this come to pass? Starting from a stable, convex production curve, as time goes by innovation at different rates in different sectors of the economy (and even in different parts of the same sector) causes the production curve to push outwards unevenly, and it may cease to be convex. Now, if the demand does not change much the new instability in the system may not be triggered for a long time. So, the production curve of the economy will naturally develop "kinks" which will be relaxed in more or less catastrophic events. The demand changes triggering these events need not be very large, and the root causes of the catastrophe will not be apparent.
By the way, in the 1970's Catastrophe Theory was popularized by René Thom. The S-shaped curved on the right is called a "Fold Catastrophe". The Thermodynamics buffs will realize this is pretty much analogous to the Van der Waals model of the gas/liquid phase transition in imperfect gases.
To link back to the discussion of free trade, opening an economy to free trade will expose its economic sectors to large changes in price and demand (how large? look at the first two diagrams I drew for this diary above). If the economy is efficient and in a stable "convex" state, nothing much will happen, but if the economy has been closed on itself for a long time, and has had little mobility of labour and capital among its different sectors, its production curve will likely be far from the PPF as well as rather bumpy, and a few of these "catastrophes" will be triggered. The economy will settle down to a more productive and stable state (hopefully), but not without a lot more pain than the first part of this diary hinted at.
