Saturday, January 29, 2011

Entrepreneurial Profits In A Classical Model

I came upon this passage recently:
"Thus, in the short run, the entrepreneurs introducing new techniques would reap supernormal profits. [Suppose] there is continuous technical change in the system, which Marx assumes to be in the nature of capitalist competition and a requirement for the maintenance of the 'reserve army of labour', then there arises a permanent income category that cannot be accounted for by labour-time accounting... Though Pareto does not clearly separate such entrepreneurial income from returns to capital in general, or from the notion of productivity of capital, it is plain that he foreshadows the idea that was later developed by an explanation for positive profits in a capitalist economy." -- Ajit Sinha (Theories of Value from Adam Smith to Piero Sraffa, Routledge (2010) pp. 214-215.)
This suggests to me a puzzle: can I create a model in which entrepreneurs make profits even though the workers are paid what would be the entire net output if the technology in use during the period in which they are paid were to persist unchanged? This post demonstrates that Sinha's comment is well-founded.

2.0 A Model
2.1 The Technology
Consider a simple economy in which a single commodity, corn, is produced each year. Workers produce the annual output from inputs of (seed) corn and their labor. The technology is defined by:
  • a0(t): the labor (in person-years) needed as input per bushel corn produced in the tth year.
  • a1(t): the (seed) corn needed as input per bushel corn produced in the tth year.
The coefficients of production evolve as in the following two equations:
a0(t) = e-λ0t
a1(t) = c e-λ1t
where the positive constants λ0 and λ1 are the rate of decrease in the labor and (seed) corn inputs, respectively. I impose the condition that the quantity harvested must exceed the quantity of seed corn planted in the spring:
0 < c < 1

2.2 Quantity Flows
Let Q(t) be the bushels of corn produced during the tth year and available after the harvest at the end of year. Assume:
Q(t) = eλ0t
The labor employed each year is a0(t)Q(t), that is, one person-year.The seed corn, K(t), required for planting at the start of the tth year is:
K(t) = a1(t)Q(t) = c e-(λ1-λ0)t
The seed corn decreases each year if and only if the rate of decrease of the labor input per bushel corn produced exceeds the rate of decrease of the seed corn input per bushel corn produced:
λ1 < λ0

The surplus corn harvest, Y(t), over the seed corn planted at the start of the year is:
Y(t) = Q(t) - K(t) = eλ0t(1 - c e-λ1t)

2.3 Prices, Wages, And Distribution
Assume the labor hired during a given year is paid at the end of the year out of the harvest. The Sraffian price equations for this model are:
a1(t) + a1(t) w(t) = 1
where w(t) is the wage per person year, and I have taken a bushel of corn as the numeraire. It is easy to solve this equation to find that the wage is the net output produced by the person-year employed:
w(t) = Y(t)

If the seed corn required for a constant labor force declines year-by-year, this model provides a source of entrepreneurial profit:
π(t) = K(t + 1) - K(t) = c e-(λ1-λ0)t[e(λ0-λ1) - 1]
What happens if the condition on technological progress is not met? I haven’t worked out this case, but two possibilities seem to me to arise. In the first case, workers cannot consume the entire surplus each year. Perhaps, the capitalists obtain some accounting profits on their capital and they save those profits as additions to the seed corn each year. In the second case, the number of hours worked decline.


Ian Wright said...

I haven't yet read Ajit's book. But the notion that profit -- as a residual income -- cannot be accounted in terms of labor-time represents a misunderstanding of the aim of Marx's theory of value.

For example, the issue Ajit raises arises in more simpler cases, even before the existence of "profit on stock" or technical change.

Consider the case of a worker-only economy where -- say -- worker-owned firms earn "profits" from the out-of-equilibrium difference between their revenues and costs (e.g., due to scarcity pricing that rations goods). In this case the residual profit also bears no quantitative relation to labor time expended in production. And, of course, prices are not proportional to labor-values.

But Marx's theory of value does not require price-value proportionality; in fact -- just the opposite -- it requires dis-proportionality precisely because the mismatch between the labor-embodied in commodities and the labor-commanded by those commodities is the "transmission mechanism" by which the total social labor of society is allocated to social need. Rubin, the Bolshevik economist murdered by Stalin, is particularly strong on this point. Briefly, arbitrage will push the economy to reallocate labor from unprofitable to profitable ventures. Hypothetically, if this process has time to converge (i.e., assuming a constant demand and technology), then profits will fall to zero, and prices converge to natural prices proportional to labor-values.

What's my point? Marx understood very well that prices in general do not represent the labor-time embodied in commodities. They only do so in the very special case that -- unfortunately -- has become the only case considered by post-Sraffians, who in general seem either incapable or unwilling to develop dynamic models of the economy. Residual profit goes hand-in-hand with the necessary mismatch between labor-embodied and labor-commanded.

I'd suggest that only a very static a-causal methodological approach could entertain the idea that the existence of residual incomes, such as entrepreneurial profits, are somehow a counterexample to the labor theory of value.


Robert Vienneau said...

Ian, I think you might object to Sinha's reading of Marx. He accepts many of the traditional criticisms of the Marx's solution to the transformation problem and of Marx's law of the tendency of the rate of profit to fall.

I caution against grouping Sinha with undiferentiated post Sraffians. His take on Sraffa is opposed to seeing Sraffa's prices as gravitational attractors for market prices. He draws connections with Joan Robinson on Ricardo's corn model. I also see an affinity with Alessandro Roncaglia's idea that the given quantity flows in Sraffa's model are taken as a snapshot of the economy.

Emil Bakkum said...

I will not comment on the article itself. However, I find the idea of dynamic economics itself interesting. Technical progress implies that the technical coefficients itself change, and thus also the price equations. This poses a restriction in for instance the application of the Sraffa equations: after the introduction of an innovation, the prices need time to adapt.

This is even more the case with the Marxian equations. For instance TSSI assumes continuous price changes due to the increase in productivity. This would however again change the technical constants. Therefore TSSI seems restricted to the price adaptation AFTER the introduction of the innovation. And the introduction itself needs to be faster than the price adaptation (if I am correct).
Sincerely yours, Emil Bakkum
Utrecht - Netherlands

Emil Bakkum said...

Prof. Kliman informed me, that the equations can actually be written in terms of continuous time, e.g.,
(m(t) + d(m(t)/dt)(P(t) + dP(t)/dt) = (1/m(t))C(t) + L(t)
where P, C, and L are functions of I-O coefficients and per-unit prices.
He remarks, that in any case, the TSSI is an exegetical interpretation, not a model.
So TSSI is indeed totally dynamic.

Robert Vienneau said...

I recognize my assumption that ending and starting prices are the same in this sort of model is controversial.