Saturday, June 04, 2011

Play It Cool, Daddy-O

1.0 Introduction
"Is Von Neumann Square?" is one of my favorite titles for an article in economics1. This post is about a case in which Von Neumann is more hep2.

Sraffa’s book presents a succession of models in which, after the second chapter, the system of price equations have one degree of freedom. This is usually taken to be a trade-off between wages and the rate of profits. Once the distribution of the surplus product is exogenously specified, prices are determined.

Some, such as Michael Mandler and Paul Samuelson, have criticized Sraffian economics on the basis that this number of degrees of freedom is arbitrary. Cases can arise in which the system of price equations has either more or less than one degree of freedom. This post illustrates a case in which more than one degree of freedom exists.

2.0 The Example
2.1 Technology and Quantity Flows
Consider a very simple economy in which laborers produce corn from seed corn on lands of definite types. Two types of land are available. Assume that this economy has 100 acres of land of each type available. Two Constant-Returns-to-Scale processes are known for producing corn. As shown in Table 1, each process requires inputs of a single type of land, as well as labor and seed corn. The technology is such that the order in which types of land will be rented can be read off directly from the technology. As I have previously pointed out, this is not a general property in long period models analyzing rent. I think this special case property, however, is not what drives the existence of possibly more than one degree of freedom.
Table 1: The Technology
α
Process
β
Process
Labor1 person-year1 person-year
Type I Land1 acre0 acre
Type II Land0 acre1 acre
Corn1/5 bushels1/4 bushels
Outputs1 bushel corn1 bushel corn

Under the assumptions, anywhere from zero to 200 bushels of corn can be produced as gross output in this economy. Assume that the gross output of this economy is 100 bushels of corn. Then cost-minimizing firms will cultivate all of type I land, and all of type II land will lie fallow.

2.2 The Price System
For stationary-state prices, no process can earn pure economic prices. This condition imposes the following inequalities:
(1/5)(1 + r) + ρ1 + w ≥ 1
(1/4)(1 + r) + ρ2 + w ≥ 1
  • w is the wage (bushels per person-year), paid at the end of the year
  • r is the rate of profits
  • ρ1 is the rent (bushels per acre) on type I land, paid at the end of the year
  • ρ2 is the rent (bushels per acre) on type II land, paid at the end of the year
An equality applies for any process in use.

Land of a given type can be modeled, in an alternative specification of the technology, as jointly produced at the end of the period from the inputs of labor, seed corn, and that type of land3. As long as less than 200 bushels of corn are produced, at least one type of land will pay no rent:
ρ1 ρ2 = 0

2.2.1 First Special Case
Consider what would happen if the gross output was infinitesimally less. Both types of land would be in excess supply. The rent on both would be zero:
ρ1 = ρ2 = 0
The solution in this case is:
0 ≤ r ≤ 4
w = (1/5)(4 - r)
Only type I land is cultivated. The number of processes in use is equal to the number of produced commodities, that is produced goods with a positive price. The system of price equations has one degree of freedom.
(1/5)(1 + r) + ρ1 + w = 1
(1/4)(1 + r) + ρ2 + w > 1

2.2.2 Second Special Case
Consider, however, what would happen if the gross output was infinitesimally more. Both types of land would be cultivated. Type I land would not be able to produce all the output quantity needed for the requirements for use, and it would have a positive rent:
ρ1 > 0
Type II land would be in excess supply, and it would have a rent of zero.
ρ2 = 0
The price system becomes:
(1/5)(1 + r) + ρ1 + w = 1
(1/4)(1 + r) + ρ2 + w = 1
The solution is:
0 ≤ r ≤ 3
w = (1/4)(3 - r)
ρ1 = (1/20)(1 + r)
Two produced commodities with positive prices exist: corn and the first type of land. And two processes are activated.

2.2.3 The Case With Two Degrees of Freedom
But type II land does not need to be cultivated in the case under consideration. Thus, the costs of cultivating type II land can exceed the revenues, and the rent on Type I land is not determined by the price equations. Only the first equation in the system of equations for prices need obtain.
(1/5)(1 + r) + ρ1 + w = 1
The second process is still characterized by an inequality:
(1/4)(1 + r) + ρ2 + w ≥ 1
This system has the solution:
0 ≤ r ≤ 4
0 ≤ ρ1 ≤ (1/20)(1 + r)
ρ2 = 0
w = (1/5)(4 - 5ρ1 - r)
Figure 1 illustrates one projection of this solution into two dimensions. The lines closer to the origin are drawn for a higher rent on the first type of land.
Figure 1: Variation in the Wage-Rate of Profits Frontier with Rent

3.0 Conclusions
I am loath to argue that the extra degree of freedom in this example is negligible since it arises only for a knife-edge. If the quantity produced is a hair larger or a hair smaller, the input-output matrices for commodities with positive prices are square. But in a larger model, the quantity produced is a choice variable. I also don't see why Sraffian models must not have more than one degree of freedom.

Footnotes
1 If I’ve actually read this article, it must have been in a reprint in some collection.

2 Some posts take me a while to write. I began this one the day after Arthur Laurents died.

3 I don't here show the derivation of rent from such a model.

References
  • Christian Bidard (1986) "Is Von Neumann Square?" Journal of Economics, V. 46: pp. 407-419.
  • Michael Mandler (20xx) "Sraffian Economics (new developments)" New Palgrave, 2nd edition.

1 comment:

Lord Keynes said...

Sorry, this is not a response to your post here, but a follow up on your comment on ABCT on my blog.

Jesús Huerta de Soto (Money, Bank Credit and Economic Cycles, Ludwig von Mises Institute, Auburn, Ala, 2006, pp. 406–408) makes comments trying to defend ABCT from charges that is does not work when credit goes to consumers:

“We are now able to identify the modifications, if any, to be made to our analysis when, as in modern economies, a significant portion of the credit expansion banks bring about without the support of voluntary saving takes the form of consumer credit. This analysis is of great theoretical and practical importance, since it has at times been argued that, to the extent credit expansion initially falls on consumption and not on investment, the economic effects which trigger a recession would not necessarily appear. Nevertheless this opinion is erroneous for reasons this section will explain. It is first necessary to point out that most consumer credit is extended by banks to households for the purchase of durable consumer goods. We have already established that durable consumer goods are actually true capital goods which permit the rendering of direct consumer services over a very prolonged period of time. Therefore from an economic standpoint, the granting of loans to finance durable consumer goods is indistinguishable from the direct granting of loans to the capital-intensive stages furthest from consumption. In fact an easing of credit terms and a decline in interest rates will provoke, among other effects, an increase in the quantity, quality and duration of so-called “durable consumer goods,” which will simultaneously require a widening and lengthening of the productive stages involved, especially those furthest from consumption” (Huerta de Soto 2006: 406; this can also be read here).

Do you have any response to this?
How can loans to consume "durable consumer goods" be "indistinguishable from the direct granting of loans to the capital-intensive stages furthest from consumption"??