Halfway down the Danube

Farm to Factory, or, Production junction, what's your function?

So far we've seen two examples of modeling centrally-planned economies. In the first, by the American game designer Chris Crawford, production depends only on the allocation of labor. In the second, by the Soviet economist G. A. Fel'dman, production depends only on the allocation of capital.

I want to pause here a moment to savor the irony.

However, their omissions aren't thoughtless ones. A turn in Crawford's game is twenty years, so Crawford made the assumption that any capital investment made in one turn would have depreciated into nothingness by the next. Fel'dman went in the opposite direction: his implicit assumption was of a near-infinite mass of surplus labor in the background that could be allocated wherever production required it.

(I should note that the leading capitalist economic growth model of the 1950s and 1960s, developed by Arthur Lewis, also made a similar assumption.)

Still, most current economists prefer to model economic production using both labor and capital as input variables. They usually use a standard formula called the Cobb-Douglas function, which looks like this:

output = residual * [labor ^ (1 - alpha)] * [capital ^ (alpha)]

where the variable alpha is the fraction of the economy's output that accrues to capital, and 1 - alpha is the fraction that accrues to labor. In advanced economies, this tends to be around 30% capital to 70% labor.

This formula has some nicely realistic properties. For instance, if you double both your labor and your capital -- basically cloning your workers and your factories -- you double your output. In the jargon, it has constant returns to scale.

Another nice thing is that this formula shows diminishing returns. Adding more capital or labor to a larger base gives you less output bang for the buck, which is also realistic.

Finally, see that term I called the residual? That's better known as total factor productivity, or TFP for short. It's the fudge factor that wraps up technological advances, increases in efficiency and organization, and that secret Chemical X into a nice numeric bundle. Total factor productivity is where the cutting edge of economic growth happens.

Notoriously, Soviet total factor productivity is calculated to have been stagnant or negative from 1970 on. This is usually thought to have been a major contributing factor to the demise of the Soviet Union. Allen, of course, has a contrary explanation.

Unfortunately, it requires a few more technical details.

The Cobb-Douglas production function is mathematically rather easy to manipulate, which is one of the reasons why economists like it so much.

An example: say you have a Cobb-Douglas economy running at a steady level, and you increase the amount of labor relative to capital by 1%. Then wages (the price of labor) relative to the price of capital (rents) will have to decrease by 1%. If you raise the relative amount of labor by 2%, relative wages will drop by 2%, and so on. On the other hand, if you drop the relative amount of labor to capital by 1% -- or raise the relative amount of capital to labor by 1%; same thing -- relative wages will rise by 1%.

Makes sense, right?

This isn't something deliberately built into the function. Rather, it's a mathematical consequence of the form of the equation. Technically speaking, the Cobb-Douglas function shows an elasticity of substitution between capital and labor exactly equal to 1.

But do real world economies do this? It makes intuitive sense, but following one's intuition in economics can put you on the fast track to economics hell.

Fortunately, it's been studied, and for advanced capitalist economies like Japan, the elasticity of substitution really is pretty close to one. So the Cobb-Douglas function is good enough for most purposes.

However, in 1970, Martin Weitzman showed that Soviet data better fit an economy with an elasticity of substitution of approximately 0.40. And in 1995, William Easterly and Stanley Fischer analyzed more recent data, and agreed.

What does that 0.40 mean? One way of looking at it is that in the Soviet Union, an increase in the capital-to-labor ratio had two and a half times more effect on relative wages than in Japan. Another way of looking at it is that an increase in relative wages in the Soviet Union had only 40% the effect on the capital-to-labor ratio as it did in Japan.

Which view is correct? Both of them.

As a result, the high growth and slowdown periods of the modernizing Soviet economy stand in extremely sharp contrast to each other, even compared to similar phases in other countries that modernized rapidly, like Japan. The growth period, from roughly 1928 to 1950, was a time of relative labor surplus, when the capital-to-labor ratio was low, and growth from capital intensification almost one-to-one; the slow period, from roughly 1965 to 1989, was a time of relative labor scarcity, when the capital-to-labor ratio was high, and growth from capital intensification nearly stagnant.

Or, to relate it back to the earlier Cobb-Douglas formula, it was as if the Soviet Union's alpha, the percentage of an economy's output accrued by its capital, dropped sharply as Soviet capital grew.

What made the Soviet Union so different from capitalist economies? Here's Easterly and Fischer's hypothesis:

The natural question to ask is why Soviet capital-labor substitution was more difficult than in Western market economies, and whether this difficulty was related to the Soviets' planned economic system. [... O]ne possible explanation for the Soviets' substitution problems would be that, under an autocratically directed economic system, they accumulated a narrow rather than a broad range of capital goods. Some forms of physical or human capital that were missing would have been market-oriented entrepreneurial skills, marketing and distributional skills, and information-intensive physical and human capital (because of the restrictions on information flows). It is more difficult to substitute more and more drill presses for a laborer than it is to substitute a drill press plus a computerized inventory and distribution system for a laborer. There is nothing that explicitly supports this conjecture in our results, but it is an interesting direction for further research.

Which brings us back to Allen's upshot: since total factor productivity is back-calculated from a Cobb-Douglas production function, and since the Soviet Union's economy did not fit a Cobb-Douglas production function, the apparent decline in TFP might simply represent the Soviet Union's extreme difficulty in substituting capital for labor as well as a modern market economy, and Soviet productivity might not have stagnated at all.

Except, of course, that Allen is contrary twice.

Posted by coyu at September 17, 2004 02:56 AM