What Is Socially Necessary Abstract Labor Time?

To me, this is an easy question. SNALT, for a capitalist economy, is:

L = a₀ (I - A)^-1 y

The notation is from Luigi Pasinetti's Lectures on the Theory of Production. The idea can be empirically applied with data from national income and product accounts (NIPAs), using techniques explained in, for example, Ronald Miller and Peter Blair's Input-Output Analysis

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

Input	Iron Industry	Corn Industry
	I	II	III
Labor	a0,1 = 1	a0,2 = 5191/5770	a0,3 = (305/494) e(3/20) - σt
Type 1 Land	0	c1,2 = 1	0
Type 2 Land	0	0	c2,3 = e(3/20) - σt
Iron	a1,1 = 9/20	a1,2 = 1/40	a1,3 = (3/1976) e(3/20) - σt
Corn	a2,1 = 2	a2,2 = 1/10	a2,3 = (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

Technique	Type of Land
	Type 1	Type 2
Alpha	Fully farmed	Partially farmed
Beta	Partially farmed	Fully farmed
Gamma	Partially farmed	Farrow
Delta	Farrow	Partially 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:

(p_β a_1,1 + a_2,1)(1 + r) + w_β a_0,1 = p_β

(p_β a_1,2 + a_2,2)(1 + r) + w_β a_0,2 = 1

(p_β a_1,3 + a_2,3)(1 + r) + ρ₂ c_{2, 3} + w_β a_0,3 = 1

In these equations, p_β is the price of iron, w_β 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:

(p_γ a_1,1 + a_2,1)(1 + r) + w_γ a_0,1 = p_γ

(p_γ a_1,2 + a_2,2)(1 + r) + w_γ a_0,2 = 1

(p_γ a_1,3 + a_2,3)(1 + r) + w_γ a_0,3 > 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.

Some Difficulties In Reading Marx

"Let us take the process of circulation in a form under which it presents itself as a simple and direct exchange of commodities. This is always the case when two owners of commodities buy from each other, and on the settling day the amounts mutually owing are equal and cancel each other. The money in this case is money of account and serves to express the value of the commodities by their prices, but is not, itself, in the shape of hard cash, confronted with them. So far as regards use-values, it is clear that both parties may gain some advantage. Both part with goods that, as use-values, are of no service to them, and receive others that they can make use of. And there may also be a further gain. A, who sells wine and buys corn, possibly produces more wine, with given labour-time, than farmer B could, and B on the other hand, more corn than wine-grower A could. A, therefore, may get, for the same exchange-value, more corn, and B more wine, than each would respectively get without any exchange by producing his own corn and wine. With reference, therefore, to use-value, there is good ground for saying that 'exchange is a transaction by which both sides gain.'" -- Karl Marx, Capital, Chapter 5: Contradictions in the General Formula of Capital.

The sheer volume of his work makes Karl Marx difficult to read. Here I concentrate on the three volumes of Capital. In the Progress Publishers edition, they consist of 867 pages, 551 pages, and 948 pages, respectively. Is there a one-semester class in which students are expected to read all of that? If I were a scholar, I suppose I would be required to learn German. I suppose one ought to also look at the secondary and tertiary literature, maybe from one's home country. (The fact that I can assume such literature exists, wherever you are coming from, attests to Marx's importance.)

Another difficulty is in understanding why Marx chose his order of exposition. The introduction to the Grundrisse is an important text on method here, although I gather Marx came to think of the order in the main text of the Grundrisse as backwards. As I understand it, Marx works from higher levels of abstraction to lower levels, with the concrete being overdetermined, in some sense. These levels are supposed to be real abstractions. Reality is not generated out of thought, as in Hegel. One might respond to the first objection here by saying that the distinction between fixed and circulating capital is a volume 2 issue and can be ignored in volume 1. The overall arc is to consider production in volume 1, circulation combined rather mechanically with production in volume 2, and then the unity of production and circulation in volume 3. The decomposition of surplus value into profits, interest on monetary loans, rent, wages for non-productive workers (such as clerks hired by banks or lawyers) cannot be fully explained until volume 3. And Marx never even gets to taxes and the state. But, according to Lenin, I do not understand Capital since I have never read Hegel's Logic, as one might expect of one who has read some of Bertrand Russell.

Capital is an immanent critique, and it is not always easy to be sure of Marx's attitude to what he is writing about. I think that Marx does not expect prices of production to be proportional to labor values, for example. He says as much in a footnote at the end of chapter 5 in volume 1. It does not help the reader that he does not explain this as fully as he ever will until towards the start of volume 3, more than a thousand pages later. The distinction between labor values and exchange values seems to be of no matter in much of the middle of volume 2, where he explains how the wear and tear of long-lived machinery, the value of ancillaries such as fuel to keep machinery running and to light a factory, and raw materials that actually appear changed in the product all contribute to a commodity's value.

I think a major difficulty some have is that a Marx is developing a systems view or structural approach, in some sense. I find many bourgeois commentators seem to object to the labor theory of value based on the feelings of those engaged in a single transaction or on a supposed analysis of a single market. This standpoint seems besides the point to me. When, in volume 1, Marx talks about the capitalist, he is treating the capitalist as "capital personified", and so on.

I think a major point of Marx is why so many have necessary illusions about capitalism. So many perceive human social powers as relationships inherent in objects, such as money capital goods, or land, or inherent in institutions such as markets. This confusion and fetishism by apologists, at a superstructural level, has a role in keeping capitalism going. Marx tries to explain such illusions created by markets.

An issue I have is with readings of Marx as presenting a scientific, positivistic theory of prices and distribution. Even if not fully in the spirit of Marx, I find empirical work with input-output tables of interest, though.

Structural Dynamics With Extensive Rent

Figure 1: Variation in Switch Points with Time

1.0 Introduction

This post continues my effort to understand how fluke cases can partition parameter spaces in models of prices of production with extensive rent. Some background for this post is here, here, and here.

2.0 Technology

The technology is described by the coefficients of production in Table 1. I assume that requirements for use are such that they cannot be satisfied by cultivating only two types of land. After fully cultivating two types, the third type must also be introduced into cultivation, at least partially. See this post for a slightly longer description of the technology.

Table 1: The Coefficients of Production

Input	Iron Industry	Corn Industry
	I	II	III	IV
Labor	a0,1 = 1	a0,2 = 1/2	a0,3 = 3	a0,4(t) = 2.0743 e-0.03648 t
Type 1 Land	0	b1,2 = 1	0	0
Type 2 Land	0	0	b2,3 = 1	0
Type 3 Land	0	0	0	b3,4 = 1
Iron	a1,1 = 0	a1,2 = 1/2	a1,3 = 1/8	a1,4(t) = 0.3551 e-0.06337 t
Corn	a2,1 = 1/2	a2,2 = 0	a2,3 = 0	a2,4(t) = 0.3343 e-0.2906 t

Table 2 lists the techniques for this example. Feasiblity of a technique is determined by requirements for use. Only techniques in which all three types of land must be cultivated are listed.

Table 2: Techniques

Technique	Type of Land
	Type 1	Type 2	Type 3
Alpha	Fully farmed	Fully farmed	Partially farmed
Beta	Partially farmed	Fully farmed	Fully farmed

3.0 Prices

I consider the system of prices of production. Profits, rent and wages are paid out of the surplus at the end of the year. Each of the four processes contributes an equation to the system of price equations. A bushel corn is the numeraire, and rent must be zero on at least one type of land.

4.0 Variations of the Choice of Technique

In Figures 1 and 2, thin vertical lines partition time into numbered regions. In each numbered region, the variation of the cost-minimizing technique with distribution is qualitatively invariant. The partitions are labeled with a type of patterns of switch points. Region 8 is a narrow range of time between a pattern for the r-order of fertility, which occurs first, and a pattern over the axis for the rate of profits for the order of rentability.

Figure 2: Variation in Switch Points with Time (Start Enlarged)

The maximum rate of profit, the rate of profits at switch points, and the rate of profits for which the rent per acre for the two types of land that pay rent are equal are plotted as functions of time. Heavy solid lines, aside from the bound on the infeasible region, are for switch points on the inner frontier of the wage curves. These solid lines bound areas in which the cost-minimizing technique, Alpha, Beta, or Gamma, is as shown. Dashed lines are for switch points on the outer frontier. For each technique, the dashed lines bound areas in which the order of fertility does not change. Dotted lines are for the rate of profits at which rents are equal. The dashed lines bound areas, for each technique, in which the order of rentability is invariant. Table 3 summarizes how, within each numbered region, the choice of technique, the order of fertility, and the order of rentability vary with the rate of profits.

Table 3: Description of Partitions with Rate of Profits Given

Region	Range of r	Technique	Order of Fertility	Order of Rentability
1	0 < r < r1,2	Alpha	1 ,2, 3	ρ1 > ρ2 > 0. ρ3 = 0
	r1,2 < r < rmax, 3			ρ2 > ρ1 > 0. ρ3 = 0
2	0 < r < r1,2	Alpha	1 ,2, 3	ρ1 > ρ2 > 0. ρ3 = 0
	r1,2 < r < r*			ρ2 > ρ1 > 0. ρ3 = 0
	r* < r < rmax, 3		2, 1, 3
3	0 < r < r1,2	Alpha	1 ,2, 3	ρ1 > ρ2 > 0. ρ3 = 0
	r1,2 < r < r*			ρ2 > ρ1 > 0. ρ3 = 0
	r* < r < r**		2, 1, 3
	r** < r < rmax, 1	Beta	2, 3, 1	ρ2 > ρ3 > 0. ρ1 = 0
4	0 < r < r*	Gamma	1 ,3, 2	ρ1 > ρ3 > 0. ρ2 = 0
	r* < r < r1,2	Alpha	1 ,2, 3	ρ1 > ρ2 > 0. ρ3 = 0
	r1,2 < r < r**			ρ2 > ρ1 > 0. ρ3 = 0
	r** < r < r***		2, 1, 3
	r*** < r < rmax, 1	Beta	2, 3, 1	ρ2 > ρ3 > 0. ρ1 = 0
5	0 < r < r1,3	Gamma	1 ,3, 2	ρ1 > ρ3 > 0. ρ2 = 0
	r1,3 < r < r*			ρ3 > ρ1 > 0. ρ2 = 0
	r* < r < r**		3, 1, 2
	r** < r < r2,3	Beta	3 ,2, 1	ρ3 > ρ2 > 0. ρ1 = 0
	r2,3 < r < r***			ρ2 > ρ3 > 0. ρ1 = 0
	r*** < r < rmax, 1		2, 3, 1
6	0 < r < r1,3	Gamma	1 ,3, 2	ρ1 > ρ3 > 0. ρ2 = 0
	r1,3 < r < r*			ρ3 > ρ1 > 0. ρ2 = 0
	r* < r < r**		3, 1, 2
	r** < r < r2,3	Beta	3 ,2, 1	ρ3 > ρ2 > 0. ρ1 = 0
	r2,3 < r < rmax,1			ρ2 > ρ3 > 0. ρ1 = 0
7	0 < r < r*	Gamma	1 ,3, 2	ρ3 > ρ1 > 0. ρ2 = 0
	r* < r < r**		3, 1, 2
	r** < r < r2,3	Beta	3 ,2, 1	ρ3 > ρ2 > 0. ρ1 = 0
	r2,3 < r < rmax,1			ρ2 > ρ3 > 0. ρ1 = 0
8	0 < r < r*	Gamma	3, 1, 2	ρ3 > ρ1 > 0. ρ2 = 0
	r* < r < r2,3	Beta	3 ,2, 1	ρ3 > ρ2 > 0. ρ1 = 0
	r2,3 < r < rmax,1			ρ2 > ρ3 > 0. ρ1 = 0
9	0 < r < r*	Gamma	3, 1, 2	ρ3 > ρ1 > 0. ρ2 = 0
	r* < r < rmax,1	Beta	3 ,2, 1	ρ3 > ρ2 > 0. ρ1 = 0

Some of the fluke cases that partition the parameter space in this example arise in models with circulating capital alone and no scarce unproduced means of production. Some are specific to models with land. At the time for the partition between regions 2 and 3, a fluke switch point between the Alpha and Beta techniques occurs on the inner wage frontier with a wage of zero. This fluke is associated with the emergence of a range of low rates of profits in which the Beta technique is cost-minimizing. The partition between regions 3 and 4 is characterized by a fluke switch point on the inner wage frontier with a rate of profits of zero. This fluke is associated with the emergence of a range of high rate of profits in which the Gamma technique is cost-minimizing.

The three-technique pattern of switch points, defining the partition between regions 4 and 5, is probably the most visually noticeable fluke case depicted in Figure 8. For the coefficients of production at this instant in time, the wage curves for the Alpha, Beta, and Gamma techniques intersect at a single switch point. This switch point is simultaneously on the inner and the outer frontiers of the wage curves. For the rate of profits at which this switch point is defined, all three types of lands are equally fertile, and none of them pay any rent. None of the types of land need be fully cultivated prior to some other type being taken into cultivation.

Fluke cases I christen patterns for the r-order of fertility are specific to models of rent. A fluke switch point on the outer frontier also lies on the wage axis at the time which partitions regions 7 and 8. At this time, the range of the rate of profits in which the order of fertility is type 1, type 3, and type 2 land vanishes over the wage axis. The partition between regions 1 and 2 qualitatively resembles the partition between regions 5 and 6. A switch point on the outer frontier of wage curves occurs at a rate of profits of 100 percent, which is also the maximum rate of profits of the Beta technique. This fluke case is associated with a disappearance of a range of the rate of profits at which the order of fertility is type 2, type3, and type 1 lands.

A pattern for the w-order of fertility occurs at a time of approximately 6.0619. The switch point between type 1 and type 3 lands of the outer occurs at the same wage as the maximum wage for type 2 land. This switch point is associated with the disappearance of a range of wages at which the order of fertility is type 3, type 1, and type 2 lands, given the wage. Since the rate of profits is taken as given throughout this post, this example of a pattern for the w-order of fertility is merely an aside.

Patterns in the order of rentability are also specific to models with land. Consider the partition between regions 6 and 7. This is a pattern over the wage axis for the order of rentability. At a rate of profits of zero, the rent on an acre of type 1 and type 3 land is identical. This fluke case is associated with the disappearance of a range of the rate of profits in which the order of rentability is type 1, type 3, type 2 lands. The partition between regions 8 and 9 is a pattern over the axis for the rate of profits for the order of rentability.

5.0 Conclusions

This post continues to extend my research program. It illustrates that a divergence between the order of fertility and the order of rentability is not a fluke case.