Sunday, January 27, 2008

Therefore I Ain't Got No Money To Pay The Rent

1.0 Introduction
A vulgar belief is that competitive factor prices reward factor owners for the physical contributions of their services to output, in some sense. Although David Ricardo was not a vulgar economist - he is one of the paradigms for defining Classical economics - he made a similar mistake. In particular, Ricardo treated land as capable of being ordered from high rent to low rent based on physical data (fertility) alone. He only considered a special case where the order of rentability can be read off of physical data alone, rather than the more general case where the order of rentability can be altered by differences in distribution between wages and profits. Understanding the more general case seems appropriate for one who considers economics a social science.

This post demonstrates the limitations of the Ricardian doctrine. These limitations are shown by means of an example. Since so-called Neoclassical economics is often claimed to have been developed by extending the Ricardian theory of rent, the analysis presented here may be of slightly more contemporary interest as well.

2.0 The Technology
Consider a very simple economy that produces a single consumption good, corn, from inputs of labor and (seed) corn. Corn is grown on two grades of land. All production processes in this example require a year to complete. Only one production process is known for producing corn on each grade of land. These processes require the inputs shown in Table 1 to be available at the beginning of the year for each bushel corn produced and available at the end of the year.
Table 1: Inputs Required Per Bushel Corn Produced
Grade I LandGrade II Land
1 Acre1 Acre
1/4 Person-Years2/3 Person-Years
1/2 Bushels Corn1/3 Bushels Corn
Notice that the proportions of labor and seed corn differ on the two grades of land. This idea can be generalized to a model of the choice of technique in long period positions, where this model includes both extensive and intensive rent:
" is very well possible that a larger output per hectare is obtained by using less (or even nothing) of some input(s) and more of some other input(s) or some positive amount of some input(s) not used at all at the smaller level of production." -- Heinz D. Kurz and Neri Salvadori

Assume that the total corn output required in this economy can be grown on some combination of both grades of land, but exceeds the amount that can be grown on either grade alone. In other words, when enough corn is produced to satisfy this economy's requirement for use, some, but not all, of one grade of land will lie fallow. Thus, rent will almost always be paid on one grade of land, and the other is free (non-scarce).

Here's how Ricardo explains the emergence of rent:
"On the first settling of a country, in which there is an abundance of rich and fertile land, a very small proportion of which is required to be cultivated for the support of the actual population, or indeed can be cultivated with the capital which the population can command, there will be no rent; for no one would pay for the use of land, when there was an abundant quantity not yet appropriated, and, therefore, at the disposal of whosoever might choose to cultivate it...

If all land had the same properties, if it were unlimited in quantity, and uniform in quality, no charge could be made for its use, unless where it possessed peculiar advantages of situation. It is only, then, because land is not unlimited in quantity and uniform in quality, and because in the progress of population, land of an inferior quality, or less advantageously situated, is called into cultivation, that rent is ever paid for the use of it. When in the progress of society, land of the second degree of fertility is taken into cultivation, rent immediately commences on that of the first quality, and the amount of that rent will depend on the difference in the quality of these two portions of land...

Thus suppose land - No. 1, 2, 3, - to yield, with an equal employment of capital and labor, a net produce of 100, 90, and 80 quarters of corn. In a new country, where there is an abundance of fertile land compared with the population, and where therefore it is only necessary to cultivate No. 1, the whole net produce will belong to the cultivator, and will be the profits of the stock which he advances. As soon as population had so far increased as to make it necessary to cultivate No. 2, from which ninety quarters only can be obtained after supporting the labourers, rent would commence on No. 1; for either there must be two rates of profit on agricultural capital, or ten quarters, or the value of ten quarters must be withdrawn from the produce of No. 1, for some other purpose. Whether the proprietor of the land, or any other person, cultivated No. 1, those ten quarters would equally constitute rent; for the cultivator of No. 2 would get the same result with his capital, whether he cultivated No. 1, paying 10 quarters for rent, or continued to cultivate No. 2, paying no rent. In the same manner it might be shown that when No. 3 is brought into cultivation, the rent of No. 2 must be ten quarters, or the value of ten quarters, whilst the rent of No. 1 would rise to twenty quarters; for the cultivator of No. 3 would have have the same profits whether he paid twenty quarters for the rent of No. 1, ten quarters for the rent of No. 2, or cultivated No. 3 free of all rent." -- David Ricardo (1821)

3.0 Prices
Consider stationary prices where seed corn is paid for at the beginning of the year, and wages and rent are paid at the end of the year. Then prices must satisfy the following pair of equations:
( 1/2 )( 1 + r ) + ( 1/4 ) w + ρ(I) = 1
( 1/3 )( 1 + r ) + ( 2/3 ) w + ρ(II) = 1
  • Corn is the numeraire
  • w is the wage
  • r is the (common) rate of profits
  • ρ(I) and ρ(II) are the rents on the respective grades of land.
The condition that at least one grade of land be free is expressed by a third equation:
ρ(I) ρ(II) = 0
The last condition is that all rents must be nonnegative.

3.1 Case 1
Consider the case when wages are 1/2 bushel per person-year. The rate of profits is then 75%. The rent on grade II land is 1/12 bushel per acre, and grade I land is free.

3.2 Case 2
Consider the case when wages are 5/6 bushel per person-year. The rate of profits is 33.3%. The rent on grade I land is 1/8 bushel per acre, and grade II land is free.

4.0 Conclusions
A figure is useful for visualizing the analysis of rent. Figure 1 shows the wage-rate of profits curves for this example. One commodity is produced in the above example, and that commodity functions both as a capital good and as a consumption good. This leads to the wage-rate of profits curves being straight lines. In models with more goods, these curves, while still decreasing, can be of varying convexity. In models with more than two grades of land, more than two curves exist. The figure can be used to locate the grade of land at the extensive margin in working inward from the outer frontier. With a given real wage, one works from right to left. Can the grade of land associated with the rightmost curve satisfy the requirements for use? If so, all land is free, and that grade is used. Otherwise, look at the grade associated with the next curve inward. If both grades can satisfy the requirements for use, then the first grade pays rent, and the second grade is free. In the general case, the frontier associated with the extensive margin may be neither the outer nor the inner frontier formed by the curves on the graph. Furthermore, the order in which grades of land are introduced to satisfy higher requirements for use need not be the order from high-rent land to lower-rent land. If I fully understood my references, I probably could note other counterintuitive results.
Figure 1: Wage-Rate of Profits Frontier for Rent Analysis

Table 2: Summary
Wage Per Person-YearLocation of Extensive Margin
Between 0 and 2/3 Bushels Land I Non-Scarce, Land II Scarce
Between 2/3 and 1 BushelsLand I Scarce, Land II Non-Scarce
Table II summarizes the results from the example. The scarcity of land of the best grade is reflected in the use of both grades of land. But which land is best, and therefore scarce, varies with the distribution of income and the price system. Scarcity, then, generally cannot be simply read off of the technological data. It is determined with the price system.

Ricardo was wrong; the order of rentability may be altered by differences in distribution between wages and profits.

  • Christian Bidard, Prices, Reproduction, Scarcity, Cambridge University Press (2004).
  • Heinz D. Kurz and Neri Salvadori, "A Single Theory or Two? Walras's Critique of Ricardo".
  • Alberto Quadri-Curzio, "Rent, Income Distribution, and Orders of Efficiency and Rentability", in Essays on the Theory of Joint Production (ed. by Luigi L. Pasinetti), Columbia University Press (1980).
  • David Ricardo, On the Principles of Political Economy and Taxation, 3rd edition (1821).
  • Bertram Schefold, Mr. Sraffa on Joint Production and Other Essays, Unwin Hyman (1989).


YouNotSneaky! said...

"Only one production process is known for producing corn on each grade of land."

How crucial is this for the result? My sense is that this basically eliminates land as a non produced factor of production, leaving only labor, which lets the non-substitution theorem to go through, which the whole thing hinges upon.

Also - this one I'm not so sure about - it seems like the technology is constant returns (also needed for the non-substitution theorem) which makes this somewhat non-Ricardian since there the key is decreasing returns.

Robert Vienneau said...

Land is a case of joint production. In the example, the inputs to a production process are so much of a certain grade of land, labor, and (seed) corn. The outputs are corn and that same amount of the same grade of land. The so-called non-substitution theorem assumes no joint production. So, for this reason alone, that theorem does not apply.

In the example, production cannot be expanded indefinitely on a single grade of land. Eventually, both grades of land need to be used. And after a certain point, production cannot be expanded any more at all. I would say Constant Returns to Scale do not prevail in the example, and the example is in the spirit of the Ricardo quote I give.

If more than one process is known for a given grade of land, the possibility of an intensive margin arises. I don't fully understand an analysis that combines both intensive and extensive margins in a linear production model. But, in any case, I think the location of the margin is endogenous in the analysis of distribution.

YouNotSneaky! said...

Ah, I see. But then I'm not clear on something. Here land is essentially associated with a particular technique of production. In fact, you could pretty much say it is a technique of production, except for that its total stock is finite (which is where the sort of decreasing returns come in). So wouldn't that mean that all your standard linear production model of the sort you've been presenting involve joint production?

You got corn, labor and seed corn and two techniques of production. First one uses some ratio of seed corn and labor to produce corn and also a good called "the ability to use this technique again" (here relabeled "land")? The second one uses some other ratio of seed corn and labor to produce corn and also a good called "the ability to use this technique again".

This is sort of what I mean by that here land sort of disappears as a factor of production. I think for it to truly be a factor you do need that intensive margin.

At least as far as I'm understanding this. I could very well be misunderstanding it.

Robert Vienneau said...

YouNotSneaky misunderstands. The so-called non-substitution theorem is not like the Coase theorem. It has a statement to go along with the name.

YouNotSneaky! said...

The second comment wasn't about the non-substitution theorem. It was about joint production.