Friday, July 23, 2021

Extensive Rent For A Reswitching Example

Figure 1: Wage Curves and Rent
1.0 Introduction

I might as well illustrate an example with extensive rent and reswitching. I find it incredible that the agents in these sorts of models understand the implications of, say, a variation of the distribution of income for their self-interests. Nevertheless, I try to note the consequences of variation in the distribution of income and perturbations of model parameters on prices of production. And I do not worry too much about disequilibria.

2.0 Technology and Requirements for Use

Consider a capitalist economy in which two commodities, iron and corn, are produced. One process is known for producing iron. In the iron industry, workers use inputs of iron and corn to produce an output of iron. The output of the iron industry is one ton with the inputs shown in Table 1. Two processes are known for producing corn. Each corn-producing process operates on a specific type of land. The coefficients of production shown in Table 1 are for an output of one bushel corn. These processes can be thought of as examples of joint production. Their outputs are corn and the same quantity of land used as input, unchanged by the production process. Presumably, some of the labor in these processes is used to maintain the land in a given state. For this post, I assume σt is 17/100.

Table 1: The Coefficients of Production
InputIron IndustryCorn Industry
Labora0,1 = 1a0,2 = 5191/5770a0,3 = (305/494) e(3/20) - σt
Type 1 Land0c1,2 = 10
Type 2 Land00c2,3 = e(3/20) - σt
Irona1,1 = 9/20a1,2 = 1/40a1,3 = (3/1976) e(3/20) - σt
Corna2,1 = 2a2,2 = 1/10a2,3 = (229/494) e(3/20) - σt

The specification of technology is completed by noting the values of parameters for the quantities available of non-produced means of production. For this numerical example, let there be 100 acres of type 1 land and 100 acres of type 2 land. The iron-producing process and each corn-producing process exhibits constant returns to scale, up to the limits imposed by the endowments of land.

I consider stationary states with a net output consisting solely of corn. A bushel corn is the numeraire. Any one of four techniques can be used to produce corn, depending on the requirements for use. The process for producing iron is part of each technique. Table 2 specifies which types of land are fully or partially farmed in each technique. In the Alpha and Beta techniques, both types of land are cultivated, with one type only partially farmed. In the remaining two techniques, one type of land is left totally farrow. Which techniques are feasible depends on the endowments of the land and on the requirements for use.

Table 2: Techniques
TechniqueType of Land
Type 1Type 2
AlphaFully farmedPartially farmed
BetaPartially farmedFully farmed
GammaPartially farmedFarrow
DeltaFarrowPartially farmed

Suppose requirements for use, that is, net output of corn, exceed 55.112 bushels and fall below 80.90. Delta is not feasible. Beta and Gamma are feasible. With Alpha, corn is in excess supply.

2.0 Prices of Production

I have asserted above that only the Beta and Gamma techniques are feasible, given technology, endowments, and requirements for use. A system of prices of production is associated with each technique. For Beta, type 2 land pays a rent. For Gamma, neither type of land pays a rent.

3.1 Prices for Beta

Suppose managers of firms have adopted the Beta technique. Prices of production satisfy the following system of three equations:

(pβ a1,1 + a2,1)(1 + r) + wβ a0,1 = pβ

(pβ a1,2 + a2,2)(1 + r) + wβ a0,2 = 1

(pβ a1,3 + a2,3)(1 + r) + ρ2 c2, 3 + wβ a0,3 = 1

In these equations, pβ is the price of iron, wβ is the wage, ρ2 is the rent per acre for type 2 land, and r is the given rate of profits. The left-hand side (LHS) of each equation is the cost of operating the corresponding process at a unit level. Costs include the cost of previously produced commodities used as raw material or ancillary inputs, the going rate of profits on these costs, rent, and wages. Since type 1 land is not fully cultivated, it obtains no rent. The right-hand side (RHS) is the revenue obtained from the corresponding process.

For prices of production, costs do not exceed revenue for any operated process. Furthermore, supernormal profits cannot be made in any prices.

3.2 Prices for Gamma

Now suppose instead that the Gamma technique is adopted by managers. Prices of production, in analogous notation, must satisfy the following system of equalities and inequalities:

(pγ a1,1 + a2,1)(1 + r) + wγ a0,1 = pγ

(pγ a1,2 + a2,2)(1 + r) + wγ a0,2 = 1

(pγ a1,3 + a2,3)(1 + r) + wγ a0,3 > 1

3.3 The Choice of Technique

Which system of equations and inequalities prevails for a given rate of profits. The analysis of the choice of technique, in models of extensive rent, can still be based on wage curves. In both the Beta and the Gamma techniques, the first two equations for prices of production are in three variables: the price of iron, the wage, and the rate of profits. Thus, one can solve for the wage as a function of the rate of profits. This is the curve labeled 'Type 1 Land' in the left panel in Figure 1 above.

For the Beta technique, one can solve the last equation for the rent on type 2 land, given the solution from the first two equations. This decomposition of the equations shows that land is a non-basic commodity, in Sraffa's terminology. Hence, a tax on land will not affect the price of iron.

The wage curve for type 2 land can be found from the system of equalities and inequalities for the Delta technique. This wage curve is also shown in Figure 1.

Consider the outer frontier of the wage curves in Figure 1. If requirements for use can satisfied by only cultivating that type of land, then the cost-minimizing technique at a given rate of profits is the corresponding technique. That is, Gamma is cost-minimizing for rates of profits between the switch points.

If the technique for the wage curve on the frontier is not feasible, the corresponding type of land will be fully cultivated. To find the cost-minimizing technique drop down to next wage curve at the given rate of profits. In this example, the cost-minimizing technique corresponds to the wage curve on the inner frontier of the wage curves. So Beta is cost-minimizing at low and high rates of profits. The same rate of profits is made in operating both type 1 and type 2 land, and type 2 land pays a rent.

Whether or not type 2 land is introduced into cultivation alongside partial cultivation of type 1 land depends on the rate of profits. When type 2 land is fully cultivated, less of type 1 land is farmed.

4.0 Conclusion

Type 1 land is partially farmed. Whether or not type 2 land is fully farmed or left farrow depends on distribution. For high and low rates of profits (or low and high wages), type 2 land is fully farmed and owners of type 1 land receive a rent. For intermediate rates of profits (or wages), type 2 land is left farrow, and no land receives a rent.

Employment is greater under Gamma than when the Beta technique is adopted. Thus, around the switch point at the lower wage, an increased wage is associated with each worker benefitting and employment being increased. Owners of type 2 land have a stake in how the social question is being decided among workers and capitalists.


Blissex said...

«I find it incredible that the agents in these sorts of models understand the implications of, say, a variation of the distribution of income for their self-interests.»

I don't find it incredible unless the differences are too modest, because the above embodies an engineer's view of a deliberative design, but in the political economy "the markets", or rather the collective "tatonnement" of market participants can mean that their blind collection exploration of the optimization landscape leads to emergent behaviour "as if" deliberative design process had been used; for example if one looks at "the markets" as a Montecarlo exploration of the optimization landscape, those who *purely by chance* choose to aim for a different distribution of income do get better profits and thus can outcompete the others.

«Nevertheless, I try to note the consequences of variation in the distribution of income»

More generally in your investigation of patterns of "anomalies" of the optimization landscape, did you bother looking at Thom's "catastrophe theory" and in particular at the most elementary "catastrophe", saddle"/"negative curvature"? In simple models perhaps something as simple as "saddle" is most common.

«and perturbations of model parameters on prices of production»

Obligatory XKCD cartoon:

Blissex said...

«"tatonnement" of market participants can mean that their blind collection exploration of the optimization landscape leads to emergent behaviour "as if" deliberative design process had been used»

That is quite similar to the argument used in evolutional biology to "explain" the many and sometimes quite weird still functional designs of eyes. That and path dependency which is a very big deal in biology and political economy (in biology paths dependency is mostly embedded in DNA, in political economy mostly in the present capital endowment, therefore the desperate "leets"/"putty" definition of "capital" by "sell-side" Economists and the existence of wicksellian etc. "anomalies" in sraffian models).

Blissex said...

«Whether or not type 2 land is introduced into cultivation alongside partial cultivation of type 1 land depends on the rate of profits.»

It looks like that again I cannot hold myself back from commenting here, fortunately pixels or character codes are inexpensive and nobody is forced to read what I write.

The sraffian/wicksellian anomalies here are of a somewhat less interesting sort: type 1 and type 2 land are different capital goods, and the Cambridges Capital Controversy showed that "neoclassical" models can have multiple consumer goods, but they break if there is more than one capital good. These anomalies are static, and don't presuppose path dependency, it just "happens" that one type of land is better if the rate of profits is this or that. Our blogger is interested in finding out a pattern in the shapes of these landscapes, and that is though interesting.

But sraffian/wicksellian anomalies that depend on the rate of interest are far more interesting, because the rate of interest is the same as the rate of profit only in useless models, and the rate of interest introduces path dependency and time. First two introductory concepts:

* Buying and selling in most Economics models as perfectly symmetric, transactions are exchanges: buying a pizza paying $10 can be also described as selling $10 and being paid with a pizza. But there is an intrinsic asymmetry in most transactions: the buyer becomes less liquid, and the seller becomes more liquid. After that transaction the wealth of wither side has not changed, the buyer owned a $10 banknote and now owns a $10 pizza, the seller vice-versa. Buying means going short liquidity and long something else (like eating), and vice-versa for selling. That is really really really important (and related to having a balance sheet view of the political economy as in H. Minsky and M. Pettis).

* J. M. Keynes pointed out that the interest rate is not the price of money, but the price of liquidity (even if money usually is liquid), one of the greatest insights in political economy studies ever, because there is "liquidity preference" as by remaining more liquid an investor can widen their choice of investment across time: they can wait for better investment opportunities, while by committing to one they give up that option. Investing is buying less liquid assets by paying with more liquid assets, on the expectation that the less liquid assets will give better income. That (expected) difference in between less and more liquid assets is what "justifies" interest rates, not the absolute rate of profit delayed consumption. The interest rate is the price of liquidity optionality.

If we introduce interest rates in the model we necessarily introduce path dependency and time as investment decisions are both time dependent and are not exactly reversible: having bought a new factory with money for $10m, selling the factory to regain liquidity usually does not work 100% because of liquidity preference. Investing is as a rule easier than liquidating an investment because of liquidity preference (even if during a famine a $10 slice of bread is more liquid than the $10 banknote).

The related sraffian/wicksellian anomalies then are dynamic anomalies. That is also why in "neoclassical" model there can be only one capital good which is also perfectly liquid (the "leets" or "putty" concept of capital introduced by J. B. Clark).

The asymmetries introduced by liquidity also cause a problem for the "labor theory of value", as "value" does not depend only on the labor content of goods, but also on the cost of the interest rates embedded in the capital goods used in production. The theory of K. Marx is not affected because in it "value" is by definition only the "labor of free employees".

Blissex said...

«If we introduce interest rates in the model we necessarily introduce path dependency and time as investment decisions are both time dependent and are not exactly reversible»

There is a subtle point here about which I have mixed feelings...

Liquidity is a form of optionality, that is an abstract quality. There is another relevant and related quality in political economies, which is promptness (or lateness): suppose that the a haircut takes a standard half hour, but it can be done tomorrow or next week: some people will pay extra for it to be done tomorrow, because promptness has some utility to them. Like liquidity, promptness is an immaterial quality that nonetheless can be given a price.

My feeling is that liquidity and promptness are qualities like beauty or fertility, which also affect prices, as more fertile land or a more beautiful car will have a higher price than otherwise, and like the premium for fertility or beauty the price of liquidity or promptness is a rent.

But I also feel that liquidity and promptness are different qualities from beauty or fertility, because they can be "financialised", that is securitised, and traded sort of independently from the goods they refer to, while fertility and beauty are intrinsic to a particular plot land or particular car, and that this is because both liquidity and promptness are about timing and optionality, which are context dependent.

For example the main source of liquid assets are banks (their liabilities and cheques drawn on them) and the treasury of the government (bills), if they are "bankable" (a misnomer), and because those assets are nearly immaterial "writings" they are almost pure forms of liquidity itself, which cannot be done with the fertility of land or the beauty of cars.

Anonymous said...

This example reminds me of the previous set of exercises around two-four technique patterns and if augmentation from two to three types of land can reproduce those results.

Blissex said...

«The sraffian/wicksellian anomalies here are of a somewhat less interesting sort:»

Just remembered a particularly relevant passage:
«Thus, many years ago, Robinson (1953, p. 590) put back on the agenda what we now call path-dependent equilibria:
“the very process of moving has an effect upon the destination of the movement, so that there is no such thing as a position of long-run equilibrium which exists independently of the course which the economy is following at a particular date.”
The title of her 1975 paper, “The Unimportance of Reswitching” (Robinson, 1975a), reflected her belief that while reswitching and capital-reversing were problematic for neoclassical capital theory, her methodological critique was far more important.»

But I still thinking that investigating the patterns of "anomalies" might well give good insights not just as to the limitations of the neoclassical model, but on how to replace it with a more general one.

Robert Vienneau said...

I have an extension with three types of land in which I get three-technique patterns of switch points.

Looking at fluke cases can occupy me for some time. I have decided that looking at how market price dynamics are qualitatively altered by variations in the analysis of the choice of technique in the system of prices of production is beyond me. I hope that these analyses, however, interest those concentrating on historical time.