Sunday, February 22, 2009

Another Reswitching Example

1.0 Introduction
I have previously presented examples of reswitching and capital reversing. These phenomena were discovered as part of the Cambridge Capital Controversy, and you will occasionally see commentators bring up the CCC on various blogs. This post presents another example. This example has a different structure than my favorite models. In this example, not all commodities are produced with the aid of commodities. Some capital goods not explicitly shown are produced directly with unassisted labor.

2.0 Some Theory And An Example
Consider a firm choosing among a set of techniques for producing a given commodity. Each technique, as in Table 1, exhibits Constant Returns to Scale (CRS) and can be represented as a series of dated labor inputs. The first element in this series is the amount of labor that must be applied in the technique during the current production cycle to produce one unit of the commodity. The second element is the amount of labor that must be applied one cycle ago to produce the capital goods required for use by the first labor input in the series. And so on. This representation of a technique is known as a flow-input, point-output model.

Table 1: The Technology
Labor Hired for Each Technique
033 Person-Years0 Person-Years
10 Person-Years52 Person-Years
220 Person-Years0 Person-Years

Consider the the cost of producing a unit of output with a given technique, where the calculation of the cost is performed at the end of the year in which the output becomes available. The cost is the sum of the cost of the labor inputs over previous years, with the cost of each labor input including an interest charge:
L0 w + L1 w (1 + r) + L2 w (1 + r)2 + ... + Ln w (1 + r)n + ...
where w is the wage for a unit of labor, r is the rate of profits, and (L0, L1, L2, ...) is the series of dated labor inputs representing the technique. The wage is paid at the end of the year for the labor expended during that year.

The best rate of profits (also known as the interest rate) available to the firm should be used in finding the cost of producing a unit of output with each technique. Since the only use of financial capital available to the firm here is to produce with one or another of the techniques of production, the best rate of profits is the rate of profits obtainable with cost-minimizing techniques. The following algorithm traverses the techniques to find the rate of profits associated with cost-minimizing techniques:
  1. Choose a technique.
  2. For the given wage w, find the rate of profits r that, when used to cost up the labor inputs, equates that cost to the price of a unit of the produced commodity.
  3. Cost up the labor inuts for all techniques with this wage and rate of profits.
  4. Choose a technique with the minimum of the costs found in Step 3.
  5. If the technique for which the rate of profits was obtained in Step 2 can be selected in Step 4, stop. You have found a cost-minimizing technique and the rate of profits.
  6. Else, go to Step 2 and repeat with the newly selected technique.

Figures 1, 2, and 3 illustrate the application of the algorithm to the example. Figure 1 shows the wage-rate of profits curves associated with each technique. In flow-input, point-output models, these curves never cross the axis for the rate of profits. Instead, the curves approach it asymptotically. Points at which such curves intersect are called switch points. The "perverse" or "paradoxical" switch point is at a wage of 1/78 units of output per person year and a rate of profits of 50%. The higher-wage switch point is at a wage of 5/286 units of output per person-year and a rate of profits of 10%. A switch point is called "paradoxical" merely because it illustrates behavior contradicting mistaken and outdated neoclassical beliefs. The algorithm can be considerably simplified:

Theorem: Consider a technology specified as a choice among techniques, where each technique is represented by a flow-input, point-output model. Let the outer envelope of the wage-rate of profits curves for each technique be constructed by finding the maximum rate of profits for each wage over all wage-rate of profits curves. For each point on the outer envelope, a technique in which the technique’s corresponding wage-rate of profits curve contains that point is selected by the above algorithm as a cost-minimizing technique at that wage.

Figure 1: Rate of Profits by Technique

The theorem is easily applied to the example. By the theorem, the alpha technique is cost-minimizing for wages between the switch points. The beta technique is cost-minimizing for wages below 1/78 units of output per person-year and between wages of 5/286 and the maximum wage of 1/52 units of output per person-year.

Now to show that the algorithm yields the same answers. Suppose one starts by choosing the alpha technique at Step 1. The rate of profits sought in Step 2 is read off the wage-rate of profits curve for the alpha technique in Figure 1, given the wage. The costs found in Step 3 are shown in Figure 2. Figure 1 shows that the alpha technique is, indeed, cost minimizing for intermediate wages, that is, between the switch points. If the wage is either too low or too high, the beta technique is selected in Step 4, as shown in Figure 2. According to Step 6, the flow of control in the algorithm then goes to Step 2 with the beta technique selected. Figure 3 shows the costs of the techniques using the rate of profits for the beta technique. Here too, the algorithm concludes with the same answer as suggested by the theorem.

Figure 2: Cost of Techniques at Alpha Rate of Profits

Figure 3: Cost of Techniques at Beta Rate of Profit

3.0 Conclusion
Once the cost-minimizing technique has been determined, one can consider how much labor firms will want to hire per unit output at any given wage. Figure 4, which is based on the assumption of a stationary-state output, illustrates this calculation for the example. If a firm is operating the alpha technique, for example, to produce an unchanged output over the next three years, workers producing consumption goods available at the end of the current year, producing capital goods to be used to produce consumption goods available at the end of the next year, and producing capital goods to be used to produce capital goods available at the end of the next year will all be working side-by-side. That is, 53 workers will be employed per unit output under the alpha technique for a stationary state. As the graph shows, around the switch point at the lower wage, a higher wage is associated with a cost-minimizing technique in which firms want to employ more workers for a given stationary-state output.

Figure 4: Labor Intensity of Cheapest Technique

So much for the theory that wages and employment are determined by the intersection of well-behaved supply and demand curves in the labor market.


Chris said...

This helped me a lot. It reminds me the examples that demonstrate why net present value is always a better measure than internal rate of return. A little googling reveals I'm not the first to see the connection:

Anybody who payed attention in econ 101 knows that labor markets aren't simple markets. I'm new to the PK world, so I'm not exactly clear on what the implications of this are, or what's really at stake with the CCC. Is there a policy or social subtext to this? Thanks.

YouNotSneaky! said...

Your Figure 4 is a labor demand curve. With a bit of weirdness in it, but it's still a labor demand curve.

Robert Vienneau said...


Thanks for the link. The Internal Rates of Return (IRRs) for the difference in labor inputs for the two techniques in the example are the switch points. Stiglitz pointed this out in 1974.

An IRR is calculated in Step 2 of the algorithm, and Net Present Values (NPVs) are found, more or less, in Step 3. I thought it necessary to present the algorithm for those that know your point about NPV.

Mainstream economists, such as Bliss, Burmeister, Hahn, Samuelson, and Solow, have acknowledged the correctness of such examples. Nevertheless, I expect mainstream economists obtaining PhDs. recently from the "best" schools to never have heard of reswitching or the Cambridge Capital Controversy. On the other hand, many heterodox economists will accept that the CCC demonstrated decades ago that much of mainstream teaching is simply incorrect. This is a puzzle. I don't see why only leftists should be able to do arithmetic.

These points are quite debatable. Cohen and Harcourt (2003) is an interesting recent survey on topic.

walt said...

It's not a simple case of an inability to do arithmetic. The two sides don't share enough to successfully communicate. Part of the gulf is that the mainstream position has changed since the CCC.

From the current mainstream viewpoint, your example has missing markets. The rate of profits would be determined by consumers' decision to consume or invest. The wage rate would be determined by workers' labor-leisure trade-off. If the model has multiple solutions, then it has multiple solutions. Models that have more than a finite number of solutions are regarded as pathological (and there's some mathematical argument to that effect).

The idea that you can regard the economy as having a stock of homogeneous capital is regarded as an empirical fact, rather than a logical necessity.

Robert Vienneau said...

I don't see the post as presenting a closed model of the economy. I don't see why economists cannot discuss an open model.

Furthermore, I am aware that one can close the the model with agents maximizing utility, both across time and by trading off leisure and the commodities purchased out of wages. Stephen Marglin has shown that some other closures are weakly empirically superior.

I cannot cite any empirical evidence that the economy has a capital stock homogeneous with a single consumption good - at least, empirical evidence in which that hypothesis has passed potentially falsifying tests. (Anybody trying to construct such a test should be aware of Franklin Fisher's simulation results and Anwar Shaikh's humbug production function.)

walt said...

I'm sorry that I didn't respond to this until it was ready to scroll off the page. I'm not an economist (though I have taken a few classes) -- I would just like to see you make strong arguments instead of weak ones. I've read your blog for two years now, without ever understanding what was at stake in the capital controversy. I had to do a bunch of outside reading to figure out what it was about. Part of what made it confusing is that the two sides didn't even agree on what it was about, let alone the conclusion. From the mainstream point of view, it's an argument over whether a certain model is a good one. From the other side, somehow the whole issue indicts the whole moral basis for capitalism.

The neoclassical synthesis of Solow and Samuelson has been replaced with a much more right-wing flavor of economics, which grounds the justification of the superiority of capitalism on completely different grounds, ones that are immune to the reswitching argument. If you believe the general equilibrium story, then it doesn't matter if there are a hundred thousand different kinds of capital goods. If you don't believe the general equilibrium story, then you're better off attacking that.

If it just bugs you that so many people use such a stupid model of the aggregate economy, then go ahead and attack that stupid model. But that doesn't get you to concluding that all of mainstream economics is vacuous.

Robert Vienneau said...

Walt, I'm not exactly fast in posting either.

I'm not hurt at hearing that one might not find my writings on the CCC crystal clear. I suspect Walt can say the same about whatever else he's read too. I agree that consensus does not exist on what the CCC is about.

The Cambridge-Italian school does not use the CCC exclusively to attack an aggregated macroeconomic model. They have used it to raise stability issues about GE models at least since Garegnani's reply to Bliss' comment on Garegnani (1970).

I have difficultly in view GE models as a foundation for right wing nonsense on laissez-faire. I think mainstream economists who specialize in studying GE would agree with me. Maybe that's more true of an older generation (e.g., Debreu and Hahn).

Anyways, the focus of my post here is neither GE nor aggregate macroeconomics. It is closer to part of a partial equilibrium model of a labor market.