"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 Schumpeter...as 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.
a0(t) = e-λ0t
a1(t) = c e-λ1twhere 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) = eλ0tThe 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)tThe 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) = 1where 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.