It is also unscientific that, when such logical flaws are either pointed out by critics (as with Sraffa's critique in Chapter 6) or discovered by believers (as with the Sonnenschein-Mantel-Debreu conditions in Chapter 2), neocla.s.sical theory adopts 'ancillary a.s.sumptions' which are clearly absurd (such as the 'machines as putty' notions which were put forward during the debate with Sraffa and his supporters, or the SMD conditions used to save the theory of consumption, which amount to a.s.suming that all consumers have identical, income-independent tastes). This, to Lakatos, is the sign of a degenerative scientific research program which is preoccupied with adjusting its ancillary beliefs to defend its hard core, whereas a truly progressive program would be expanding the range of real-world phenomena which its theory explains. The school of economics which gives pride of place to the word 'rational' hardly displays rational behavior when its core beliefs are challenged. I expect that the new logical conundrums pointed out in this book will generate further displays of irrational behavior by conventional economists.3 Mathematics is therefore not the reason why conventional economics is so bad. Instead, bad economics is supported by bad mathematical practice. But this is only half of the story about how economics has abused mathematics. Economics has also accidentally inoculated itself against many of the advances of modern mathematics. One essential aspect of modern mathematics that economics has not realized is that mathematics today has a humility that economics lacks, because mathematicians have proved that mathematics has limits.

The limits to mathematics Economics remains perhaps the only area of applied mathematics that still believes in Laplace's dictum that, with an accurate enough model of the universe and accurate enough measurement today, the future course of the universe could be predicted.

For mathematicians, that dictum was dashed in 1899 by Poincare's proof of the existence of chaos. Poincare showed that not only was it impossible to derive a formula which could predict the future course of a dynamic model with three or more elements to it, but even any numerical approximation to this system would rapidly lose accuracy. The future could be predicted only if the present was known to infinite accuracy, and this was clearly impossible.

Today, mathematicians are quite comfortable with the proposition that most mathematical problems cannot be explicitly solved in a manner which yields the kind of didactic statements which economics makes as a matter of course such as 'perfect compet.i.tion gives superior welfare outcomes to monopoly,' 'free trade is superior to protection,' and so on. Such definitive p.r.o.nouncements are generally only possible when the problem is essentially the same as working out where two straight lines intersect. This cla.s.s of models is known as algebraic.

Some algebraic equations are rather difficult to solve because there is no standard formula to apply. But there are standard formulas available to solve systems of algebraic equations where all the equations are 'straight lines.' This is the type of mathematics which economic theory generally tries to apply to economic problems.

However, this body of mathematics is an appropriate model of only a tiny subset of real-world systems and that subset does not include true economic a.n.a.lysis.4 The more appropriate starting point for mathematical models of the economy is dynamic equations, in which the relationships between variables cannot be reduced to straight lines. These are known in mathematics as nonlinear differential equations. The vast majority of these cannot be solved, and once three or more such equations interact, they are impossible to solve.

Table 16.1 summarizes the situation. Economic theory attempts to a.n.a.lyze the economy using techniques appropriate to the upper left-hand part of Table 16.1 (with italic text), when in fact the appropriate methods are those in the lower right-hand part (with cells shaded gray).

Other developments, such as G.o.del's proof that a mathematical system cannot be self-contained so that it must take some axioms on faith and the proof that there were some mathematical problems which could not be solved, added to this realization by mathematicians and physicists that mathematics and science had innate limits. As a result, in place of Laplace's grand conceit, there is a humility to modern mathematics. The future cannot be known, mathematics cannot solve every problem, some things may not be knowable.

TABLE 16.1 The solvability of mathematical models (adapted from Costanza 1993) But these epiphanies pa.s.sed economists by: they continue to believe in a clockwork universe, in which a proper specification of the conditions of today could enable you to predict the future for all time. Nowhere is this vanity more obvious than in the school's defining works, Walras's Elements of Pure Economics and Debreu's Theory of Value.

Walras's arrogance towards those economists who would not practice mathematics is still the typical att.i.tude today held by economists towards those who criticize their use of mathematics: As for those economists who do not know any mathematics, who do not even know what is meant by mathematics and yet have taken the stand that mathematics cannot possibly serve to elucidate economic principles, let them go their way repeating that 'human liberty will never allow itself to be cast into equations' or that 'mathematics ignores frictions which are everything in social science' and other equally forceful and flowery phrases. They can never prevent the theory of the determination of prices under free compet.i.tion from becoming a mathematical theory. Hence, they will always have to face the alternative either of steering clear of this discipline and consequently elaborating a theory of applied economics without recourse to a theory of pure economics or of tackling the problems of pure economics without the necessary equipment, thus producing not only very bad pure economics but also very bad mathematics. (Walras 1954 [1874]) As this book has shown, it is neocla.s.sical economists who have been guilty of very bad mathematics. But just as important is the fact that they do not appreciate the limits to mathematics.

At least Walras could be forgiven for not being aware of Poincare's theorem of 1899 though he had sought out Poincare in a forlorn attempt to garner support for his mathematization of economics. Debreu's opus pre-dated the rediscovery of chaos by Lorenz, but that he could even conceive of modeling the economy as a system in which all production and exchange decisions were 'made now for the whole future,' and in which the theory of uncertainty was 'free from any probability concept and formally identical with the theory of certainty,' betrayed a profound lack of appreciation of the mathematics of his day (not to mention the real world).

The modern manifestation of this ignorance of the limits of mathematics is a widespread though not universal failure by economists to appreciate the importance of nonlinear a.n.a.lysis and chaos theory. If I had a cent for every time I heard an economist comment that 'chaos theory hasn't amounted to much' well, I wouldn't be wealthy, but I could afford an expensive meal or two.

Chaos theory has 'not amounted to much' in economics because its central tenets are ant.i.thetical to the economic obsession with equilibrium. In other sciences, chaos theory, complexity a.n.a.lysis and their close cousin evolutionary theory have had profound impacts. It shows how isolated economics has become from the scientific mainstream of the late twentieth and early twenty-first century that such ignorant views could be commonplace.

The recurring nightmare of straight lines Virtually every critique detailed in this book has led to the result that some relationship between phenomena that economics argued was curved had to instead be a straight line.

The economic theory of consumer behavior argued that a person's consumption of a commodity could change in any direction as his income rose: if it was a luxury, consumption would rise relative to other commodities; if a necessity, consumption could fall. Instead, the Sonnenschein-Mantel-Debreu conditions show that if the theory is to aggregate from the individual to the market demand curve, Engel curves must be straight lines.

The economic theory of production argues that output is subject to diminishing marginal returns, so that as the variable input rises, output rises less than proportionately the relationship is curved. Sraffa's critique shows that, in general, output should rise proportionately: the relationship should be a straight line.

The economic theory of compet.i.tion argues that perfect compet.i.tion is superior to monopoly. But the only conditions under which the comparison is watertight involve a straight-line relationship between inputs and outputs not the curved relationship a.s.serted by the concept of diminishing marginal productivity.

Why do straight lines haunt economic theory, when it is forced to be logical?

The answer to this dilemma has a lot to do with one of the basic notions of economics, the belief that society is no more than the sum of its parts. This a.s.serts that to work out the whole, all you have to do is add the parts up. This requires that the interactions between the parts are either zero or negligible. The only interaction that one variable can have with another is the one neocla.s.sical economists want to use, simple addition: your utility plus my utility equals social utility; your output plus my output equals industry output; and so on.

This categorically rules out interactions where one variable is multiplied by another (where both are likely to be large numbers). One obvious such interaction occurs in working out a firm's revenue. A firm's revenue equals the number of units it sells, times the sale price. Economics argues that the quant.i.ty a firm will sell is a function of price to invoke a higher supply from the firm, the price has to rise to offset the effect of diminishing marginal productivity.

If both the price and the quant.i.ty are treated as variables, then the firm's revenue is equal to one variable (price) times another (quant.i.ty). This can't be allowed if economists are to treat society as no more than the sum of its parts, so economists a.s.sume that the price a compet.i.tive firm faces is a constant. Then the firm's revenue equals a constant (price) times a variable (quant.i.ty).

However, neocla.s.sicals then want it both ways: they want price for the entire industry to be a variable, but price for the individual firm to be a constant, without the firms interacting in any way. This just can't happen mathematically as discussed in Chapter 4. So if they force this situation by making an invalid a.s.sumption, it inevitably means that something else that they want to treat as a variable has to instead be treated as a constant. Hence the recurrent nightmare of a straight line.

The future of mathematics in economics There is little doubt that the close identification of neocla.s.sical economics with mathematical a.n.a.lysis has given mathematics a bad name among critical economists, and members of the ordinary public who are critical of economics.

This is likely to lead to a backlash against mathematics in economics, if the discipline ever rids itself of its dominance by neocla.s.sical economics. This would be a great pity, since, as I hope this book has shown, properly used, mathematical reasoning debunks unsound economics. Furthermore, with its limitations fully appreciated, it and computer simulations can a.s.sist in the construction of sound alternative a.n.a.lyses. But if mathematics is avoided for its own sake, in reaction to how economics embraced it for its own sake, then the development of a meaningful economics will be stymied.

I now turn to some of the alternatives to conventional economics that do exist warts and all. We begin with the most radical alternative Marxian economics. You may, if you have typecast me as 'left-wing,' expect me to praise Marxian a.n.a.lysis. If so, you are in for a surprise.

17 | NOTHING TO LOSE BUT THEIR MINDS.

Why most Marxists are irrelevant, but most of Marx is not Marxian economics is clearly one of the alternatives to the neocla.s.sical way of 'thinking economically,' and by rights I should be discussing it in the next chapter, which looks at alternatives to conventional economics. However, in an ill.u.s.tration of the fact that conservative economists do not have a monopoly on unsound a.n.a.lysis, Marxian economics, as conventionally understood, is hobbled by a logical conundrum as significant as any of those afflicting neocla.s.sical economics.

This conundrum has split non-orthodox economists into two broad camps. One tiny group continues to work within what they see as the Marxian tradition, and spends most of its time trying to solve this conundrum. The vast majority largely ignore Marx and Marxian economics, and instead develop the schools of thought discussed in the next chapter.

I find this ironic, since if Marx's philosophy is properly understood, the conundrum disappears, and Marx provides an excellent basis from which to a.n.a.lyze capitalism though bereft of the revolutionary message that makes Marx both so appealing to his current followers, and anathema to so many others.

The kernel One defining belief in conventional Marxian economics is that labor is the only source of profit: while machines are necessary for production, labor alone generates profit for the capitalist. This proposition is a key part of the radical appeal of Marxism, since it argues that capitalist profit is based upon exploitation of the worker.

Marxists argue that labor is the only source of profit because it is the only commodity where one can distinguish between 'commodity' and 'commodity-power.' When any other commodity is sold, the purchaser takes it lock, stock and barrel. But with labor, the capitalist 'purchaser' does not own the worker. Instead, he pays a subsistence wage, which can be represented by a bundle of commodities; this is the cost of production of the ability to work, which Marxists describe as the commodity 'labor-power.' The capitalist then puts the laborer to work for the length of the working day, during which time the worker produces a different bundle of commodities that is worth more than his subsistence wage. The difference between the output of labor and the cost of maintaining labor-power is the source of profit.

Since no such distinction can be made for machinery, the capitalist 'gets what he paid for' and no more when he buys a machine, whereas with labor, he gets more than he paid for. Therefore machines transfer their value only to the product.

This proposition has been shown to lead to severe logical problems, so the vast majority of critical economists have in practice abandoned Marx's logic. However, a minority of economists continue to swear allegiance to what they perceive as Marx's method, and continue to strive to invent ways in which the proposition that labor is the only source of profit can be maintained.

The critiques which have been made of this notion on mathematical grounds are cogent, but have been challenged by Marxian economists on philosophical or methodological grounds.

However, there are philosophical reasons why the proposition that labor is the only source of profit are invalid, and these reasons were first discovered by Marx himself. Unfortunately, Marx failed to properly understand his own logic, and instead preserved a theory that he had in fact shown to be erroneous.

Once Marx's logic is properly applied, his economics becomes a powerful means of a.n.a.lyzing a market economy though not one which argues that capitalism must necessarily give way to socialism. Unfortunately, given the ideological role of Marxism today, I expect that Marxian economists will continue to cling to an interpretation of Marx that argues for capitalism's ultimate demise.

The roadmap In this chapter I explain the cla.s.sical economics concept of 'value,' and the manner in which Marx developed this into the labor theory of value. I ill.u.s.trate the logical problems with the proposition that labor is the only source of value. I then outline Marx's brilliant philosophical a.n.a.lysis of the commodity, and show that this a.n.a.lysis contradicts the labor theory of value by arguing that all inputs to production are potential sources of value.

Marxian economics and the economics of Marx If a nineteenth-century capitalist Machiavelli had wished to cripple the socialist intelligentsia of the twentieth century, he could have invented no more cogent weapon than the labor theory of value. Yet this theory was the invention, not of a defender of capitalism, but of its greatest critic: Karl Marx.

Marx used the labor theory of value to argue that capitalism harbored an internal contradiction, which would eventually lead to its downfall and replacement by socialism. However, Marx's logic in support of the labor theory of value had an internal contradiction that would invalidate Marx's critique of capitalism if it could not be resolved. Consequently, solving this enigma became the 'Holy Grail' for Marxist economists. Whereas nineteenth-century revolutionaries spent their time attempting to overthrow capitalism, twentieth-century revolutionaries spent theirs attempting to save the labor theory of value. Capitalism itself had no reason to fear them.

Despite valiant efforts, Marxist economists failed in their quest and they achieved little else. As a result, while Marx's thought still has considerable influence upon philosophers, historians, sociologists and left-wing political activists, at the beginning of the twenty-first century, Marx and Marxists are largely ignored by other economists.1 Most non-orthodox economists would acknowledge that Marx made major contributions to economic thought, but it seems that overall Samuelson was right: Marx was a 'minor Post-Ricardian' someone who took cla.s.sical economics slightly farther than had David Ricardo, but who ultimately led it into a dead end.

This conclusion is false. Properly understood, Marx's theory of value liberates cla.s.sical economics from its dependence on the labor theory of value, and makes it the basis for a deep and critical understanding of capitalism. But in a truly Machiavellian irony, the main factor obscuring this richer appreciation of Marx is the slavish devotion of Marxist economists to the labor theory of value.

To see why Marx's theory of value is not the labor theory of value, we have to first delve into the minds of the great cla.s.sical economists Adam Smith and David Ricardo.

Value a prelude The proposition that something is the source of value raises two questions: what is 'value' anyway, and why should any one thing be the source of it?

A generic definition of value one which encompa.s.ses the several schools of thought in economics which have used the term is that value is the innate worth of a commodity, which determines the normal ('equilibrium') ratio at which two commodities exchange. One essential corollary of this concept is that value is unrelated to the subjective valuation which purchasers put upon a product. In what follows, I'll use 'value' in this specific sense, not in any of its more colloquial senses.

The cla.s.sical economists also used the terms 'value in use' (or 'use-value') and 'value in exchange' (or 'exchange-value') to distinguish between two fundamental aspects of a commodity: its usefulness, and the effort involved in producing it. Value in use was an essential aspect of a commodity why buy something which is useless? but to the cla.s.sical economists, it played no role in determining price.

Their concept of usefulness was also objective, focusing upon the commodity's actual function rather than how it affected the user's feelings of well-being. The use-value of a chair was not how comfortable it made you feel, but that you could sit in it.

In contrast, the neocla.s.sical school argues that value, like beauty, is 'in the eye of the beholder' that utility is subjective, and that the price, even in equilibrium, has to reflect the subjective value put upon the product by both the buyer and the seller. Neocla.s.sical economics argues that the equilibrium ratio at which two products exchange is determined by the ratio of their marginal utilities to their marginal costs.

As we have seen in Chapters 3 and 4, there are serious problems with the economic theory of pricing. But it has some appeal in comparison to the cla.s.sical approach, since it seems reasonable to say that price should be determined both by the innate worth of a product, however that is defined, and by the buyer's subjective valuation of it.

The general cla.s.sical reply to this concept was that, sure, in the short run and out of equilibrium, that would be true. But the cla.s.sical school was more interested in 'long run' prices, and in the prices of things which could easily be reproduced.

In the long run, price would be determined by the value of the product, and not by the subjective valuations of the buyer or seller. For this reason, the cla.s.sical school tended to distinguish between price and value, and to use the former when they were talking about day-to-day sales, which could be at prices which were above or below long-run values.

As well as having some influence out of equilibrium, subjective utility was the only factor that could determine the value of rare objects. As Ricardo put it: There are some commodities, the value of which is determined by their scarcity alone. No labor can increase the quant.i.ty of such goods, and therefore their value cannot be lowered by an increased supply. Some rare statues and pictures, scarce books and coins, wines of a peculiar quality, which can be made only from grapes grown on a particular soil, of which there is a very limited quant.i.ty, are all of this description. Their value is wholly independent of the quant.i.ty of labor originally necessary to produce them, and varies with the varying wealth and inclinations of those who are desirous to possess them. (Ricardo 1817) Thus where scarcity was the rule, and the objects sold could not easily be reproduced, price was determined by the seller's and buyer's subjective utilities. But this minority of products was ignored by the cla.s.sical economists.

Marx gave an additional explanation of why, in a developed capitalist economy, the subjective valuations of both buyer and seller would be irrelevant to the price at which commodities exchanged.

This was the historical argument that, way back in time, humans lived in small and relatively isolated communities, and exchange between them was initially a rare and isolated event. At this stage, the objects being exchanged would be items that one community could produce but the other could not. As a result, one community would have no idea how much effort had gone into making the product, and the only basis for deciding how to exchange one product for another was the subjective valuation that each party put upon the products. As Marx put it: The exchange of commodities, therefore, first begins on the boundaries of such communities, at their points of contact with other similar communities, or with members of the latter. So soon, however, as products once become commodities in the external relations of a community, they also, by reaction, become so in its internal intercourse. The proportions in which they are exchangeable are at first quite a matter of chance. What makes them exchangeable is the mutual desire of their owners to alienate them. Meantime the need for foreign objects of utility gradually establishes itself. The constant repet.i.tion of exchange makes it a normal social act. In the course of time, therefore, some portion at least of the products of labor must be produced with a special view to exchange. From that moment the distinction becomes firmly established between the utility of an object for the purposes of consumption, and its utility for the purposes of exchange. Its use-value becomes distinguished from its exchange-value. On the other hand, the quant.i.tative proportion in which the articles are exchangeable, becomes dependent on their production itself. (Marx 1867) The most famous example of two products being exchanged on the basis of the perceived utility rather than their underlying value is the alleged exchange of the island of Manhattan for a bunch of beads.2 This price would never have been set if trade between the Dutch and the Indians had been a long-established practice, or if the Indians knew how little work it took to produce the beads.

In an advanced capitalist nation, factories churn out ma.s.s quant.i.ties of products specifically for exchange the seller has no interest in the products his factory produces. The sale price reflects the cost of production, and the subjective utility of the buyer and seller are irrelevant to the price.3 There is thus at least a prima facie plausibility to the argument that value alone determines the equilibrium ratio at which commodities are exchanged. The problem comes with the second question: what is the source of value?

Physiocrats The first economists to systematically consider this question4 answered that the source of all value was land.

The argument, in a nutsh.e.l.l, was that land existed before man did. Therefore man or more specifically, man's labor could not be the source of value. Instead, value came from the land as it absorbed the energy falling on it from the sun. Man's labor simply took the naturally generated wealth of the land and changed it into a different form. Land generated a surplus, or net product, and this enabled both growth and discretionary spending to occur.

Manufacturing, on the other hand, was 'sterile': it simply took whatever value the land had given, and transformed it into different commodities of an equivalent value. No formal proof was given of this latter proposition, beyond an appeal to observation: Maxims of Economic Government. I: Industrial work does not increase wealth. Agricultural work compensates for the costs involved, pays for the manual labor employed in cultivation, provides gains for the husbandmen, and, in addition, produces the revenue of landed property. Those who buy industrial goods pay the costs, the manual labor, and the gain accruing to the merchants; but these goods do not produce any revenue over and above this. Thus all the expenses involved in making industrial goods are simply drawn from the revenue of landed property no increase of wealth occurs in the production of industrial goods, since the value of these goods increases only by the cost of the subsistence which the workers consume. (Quesnay, cited in Meek 1972) Since land determined the value of commodities, and the price paid for something was normally equivalent to its value, the ratio between the prices of two commodities should be equivalent to the ratios of the land needed to produce them.

Smith (and Ricardo) The physiocratic answer to the source of value reflected the school's origins in overwhelmingly rural France. Adam Smith, a son of Scotland and neighbor to the 'nation of shopkeepers,' was strongly influenced by the physiocrats. But in The Wealth of Nations (which was published in the year in which the first steam engine was installed) Smith argued that labor was the source of value. In Smith's words: 'The annual labor of every nation is the fund which originally supplies it with all the necessaries and conveniences of life which it annually consumes, and which consist always either in the immediate produce of that labor, or in what is purchased with that produce from other nations' (Smith 1838 [1776]).

The growth of wealth was due to the division of labor, which increased because the expansion of industry allowed each job to be divided into ever smaller specialized sub-tasks. This allowed what we would today call economies of scale: an increase in the size of the market allowed each firm to make work more specialized, thus lowering production costs (his most famous example of this was of a pin factory; this pa.s.sage, which is better known than it is read, is reproduced on the web at Marx/More).5 Smith therefore had an explanation for the enormous growth in output which occurred during the Industrial Revolution. However, he had a dilemma: for reasons discussed below, Smith knew that, though labor was the source of value, it could not possibly determine price. Yet value was supposed to determine the ratio at which two commodities exchanged.

The dilemma arose because two commodities could exchange only on the basis of the amount of direct labor involved in their manufacture if only labor was required for their production. Smith gave the example of exchange in a primitive hunting society: In that early and rude state of society which precedes both the acc.u.mulation of stock and the appropriation of land, the proportion between the quant.i.ties of labor necessary for acquiring different objects seems to be the only circ.u.mstance which can afford any rule for exchanging them for one another. If among a nation of hunters, for example, it usually costs twice the labor to kill a beaver which it does to kill a deer, one beaver should naturally exchange for or be worth two deer. (Ibid.) However, once there had been an 'acc.u.mulation of stock' once a market economy had evolved then paying for the labor alone was not sufficient; the price had also to cover profit: As soon as stock has acc.u.mulated in the hands of particular persons, some of them will naturally employ it in setting to work industrious people, whom they will supply with materials and subsistence, in order to make a profit by the sale of their work, or by what their labor adds to the value of the materials. In exchanging the complete manufacture either for money, for labor, or for other goods, over and above what may be sufficient to pay the price of the materials, and the wages of the workmen, something must be given for the profits of the undertaker of the work who hazards his stock in this adventure. (Ibid.) So Smith was forced to concede that the price had to be high enough to pay for not just the hours of labor involved in making something, but also a profit. For example, if the deer hunter was an employee of a deer-hunting firm, then the price of the deer had to cover the hunter's labor, and also a profit margin for the firm.

The problem became more complicated still when land was involved. Now the price had to cover labor, profit, and rent. Smith's statement of this reveals that this 'father of economics' was rather more cynical and critical of market relations than some of his descendants: 'As soon as the land of any country has all become private property, the landlords, like all other men, love to reap where they never sowed, and demand a rent even for its natural produce' (ibid.).

In the end, Smith was reduced to an 'adding up' theory of prices: the price of a commodity represented in part payment for labor, in part payment for profit, and in part payment for rent. There was therefore no strict relationship between value and price.

Ricardo Though he paid homage to his predecessor, Ricardo was, to say the least, critical of Smith's treatment of the relationship between value and price. He began his Principles of Political Economy and Taxation (Ricardo 1817) with an emphatic statement of the belief he shared with Smith, that labor was the determinant of the value of a commodity: 'The value of a commodity, or the quant.i.ty of any other commodity for which it will exchange, depends on the relative quant.i.ty of labor which is necessary for its production' (ibid.). However, he was much more aware than Smith of the need for precise definitions, and of the difficulties in going from value to price.

Smith had used two measures of the amount of labor contained in a product: 'labor embodied' and 'labor commanded.' Labor embodied was the amount of direct labor time it actually took to make a commodity. Labor commanded, on the other hand, was the amount of labor-time you could buy using that commodity.

If, for example, it took one day for a laborer to make a chair, then the chair embodied one day's labor. However, that chair could well sell for an amount equivalent to two days' wages with the difference accounted for by profit and rent. The chair would therefore command two days' labor.

Ricardo argued that the former measure was far less volatile than the latter. He believed, in common with most cla.s.sical economists, that workers received a subsistence wage. Since this would always be equivalent to a fairly basic set of commodities so much food, clothing, and rental accommodation it would not change much from one year to the next. The latter measure, however, reflected the profit earned by selling the worker's output, and this would vary enormously over the trade cycle.

His solution for the value/price dilemma was that the price of a commodity included not just direct labor, but also the labor involved in producing any tools. Ricardo took up Smith's deer and beaver example and elaborated upon it. Even in Smith's example, some equipment had to be used to kill the game, and variations in the amount of time it took to make the equipment would affect the ratio in which deer and beavers were exchanged: Even in that early state to which Adam Smith refers, some capital, though possibly made and acc.u.mulated by the hunter himself, would be necessary to enable him to kill his game. Without some weapon, neither the beaver nor the deer could be destroyed, and therefore the value of these animals would be regulated, not solely by the time and labor necessary to their destruction, but also by the time and labor necessary for providing the hunter's capital, the weapon, by the aid of which their destruction was effected.

Suppose the weapon necessary to kill the beaver was constructed with much more labor than that necessary to kill the deer, on account of the greater difficulty of approaching near to the former animal, and the consequent necessity of its being more true to its mark; one beaver would naturally be of more value than two deer, and precisely for this reason, that more labor would, on the whole, be necessary to its destruction. (Ibid.) Thus the price of any commodity reflected the labor which had been involved in creating it, and the labor involved in creating any means of production used up in its manufacture. Ricardo gave many numerical examples in which the labor involved in producing the means of production simply reappeared in the product, whereas direct labor added additional value over and above its means of subsistence because of the difference between labor embodied (which equaled a subsistence wage) and labor commanded (which included a profit for the capitalist).6 However, Smith and Ricardo were both vague and inconsistent on key aspects of the theory of value.

Though he generally argued that labor was the source of value, on several occasions Smith counted the work of farm animals as labor.7 Though he failed to account for the role of machinery in the creation of value, he also argued that machines could produce more value than it took to produce them which would mean that machinery (and animals) would be a source of value, in addition to labor: 'The expense which is properly laid out upon a fixed capital of any kind, is always repaid with great profit, and increases the annual produce by a much greater value than that of the support which such improvements require' (Smith 1838 [1776]).

Ricardo more consistently implied that a machine added no more value to output than it lost in depreciation, but he also occasionally lapsed into completely ignoring the contribution of machinery to value.8 Marx's labor theory of value Where his forebears implied and were vague, Marx stated and was emphatic: labor was the only source of value, in the sense that it could add 'more value than it has itself' (Marx 1867). Marx called this difference between the value embodied in a worker and the value the worker added to production 'surplus value,' and saw it as the sole source of profit.

He was critical of Ricardo for not providing an explanation of why this difference existed in Ricardo's terms, for not having a systematic explanation of why labor embodied differed from labor commanded. As Marx put it: Ricardo starts out from the actual fact of capitalist production. The value of labor is smaller than the value of the product which it creates The excess of the value of the product over the value of the wages is the surplus-value For him, it is a fact, that the value of the product is greater than the value of the wages. How this fact arises, remains unclear. The total working-day is greater than that part of the working-day which is required for the production of wages. Why? That does not emerge. (Marx 1968 [1861]: Part II) The best that Ricardo could offer, Marx claimed, was that: [t]he value of labor is therefore determined by the means of subsistence which, in a given society, are traditionally necessary for the maintenance and reproduction of the laborers.

But why? By what law is the value of labor determined in this way?

Ricardo has in fact no answer, other than the law of supply and demand He determines value here, in one of the basic propositions of the whole system, by demand and supply as Say notes with malicious pleasure. (Ibid.) Similarly, Marx rejected Smith's musings on the productivity of machinery, and concurred with Ricardo that a machine only added as much value to output as it lost through depreciation: The maximum loss of value that they can suffer in the process, is plainly limited by the amount of the original value with which they came into the process, or in other words, by the labor-time necessary for their production. Therefore, the means of production can never add more value to the product than they themselves possess independently of the process in which they a.s.sist. However useful a given kind of raw material, or a machine, or other means of production may be, though it may cost 150, or, say 500 days' labor, yet it cannot, under any circ.u.mstances, add to the value of the product more than 150. (Marx 1867) Marx likewise concurred with Ricardo's definition of value, cited above, that it 'depends on the relative quant.i.ty of labor which is necessary for its production.' Value in turn determined the price at which commodities exchanged, with commodities of an equivalent value commodities containing an equivalent quant.i.ty of labor9 exchanging for the same price (in equilibrium).

This exchange of equivalents nonetheless still had to enable capitalists to make a profit, and Marx was disparaging of any explanation of profits which was based on 'buying cheap and selling dear': To explain, therefore, the general nature of profits, you must start from the theorem that, on the average, commodities are sold at their real values, and that profits are derived by selling them at their values, that is, in proportion to the quant.i.ty of labor realized in them. If you cannot explain profit upon this supposition, you cannot explain it at all. (Marx 1847) Marx gave two explanations for the origin of surplus value. One was a 'negative' proof, by a process of elimination based on the unique characteristics of labor. The other was a 'positive' proof, based on a general theory of commodities. Most Marxist economists are aware of only the negative proof.

The origin of surplus value (I) This was that labor was a unique commodity, in that what was sold was not actually the worker himself (which would of course be slavery), but his capacity to work, which Marx called labor-power. The value (or cost of production) of labor-power was the means of subsistence, since that is what it took to reproduce labor-power. It might take, say, six hours of labor to produce the goods which are needed to keep a worker alive for one day.

However, what the capitalist actually received from the worker, in return for paying for his labor-power, was not the worker's capacity to work (labor-power), but actual work itself. If the working day was twelve hours long (as it was in Marx's day), then the worker worked for twelve hours twice as long as it actually took to produce his value. The additional six hours of work was surplus labor, which accrued to the capitalist and was the basis of profit. As Marx put it: The laborer receives means of subsistence in exchange for his labor-power; the capitalist receives, in exchange for his means of subsistence, labor, the productive activity of the laborer, the creative force by which the worker not only replaces what he consumes, but also gives to the acc.u.mulated labor a greater value than it previously possessed. (Ibid.) This difference between labor and labor-power was unique to labor: there was no other commodity where 'commodity' and 'commodity-power' could be distinguished. Therefore other commodities used up in production simply transferred their value to the product, whereas labor was the source of additional value. Surplus value, when successfully converted into money by the sale of commodities produced by the worker, was in turn the source of profit.

The labor theory of value and the demise of capitalism This direct causal relationship between surplus value and profit meant there was also a direct causal relationship between what Marx called the rate of surplus-value and the rate of profit.

The rate of surplus value was the ratio of the surplus labor-time performed by a worker to the time needed to reproduce the value of labor-power. In our example above, this ratio is 1 to 1, or 100 percent: six hours of surplus labor to six hours of what Marx called necessary labor.

Marx defined the rate of profit as the ratio of surplus (which he denoted by the symbol s) to the sum of the inputs needed to generate the surplus. Two types of inputs were needed: necessary labor, and the means of production (depreciation of fixed capital plus raw materials, intermediate goods, etc.). Marx called necessary labor variable capital (for which he used the symbol v), because it could increase value, and he called the means of production used up constant capital (for which he used the symbol c), because it could not increase value.

Taking the example of weaving which Marx used extensively, during one working day a weaver might use 1,000 yards of yarn and wear out one spindle. The yarn might have taken twelve hours of (direct and indirect) labor to make, and the spindle the same. Thus the sum of the direct labor-time of the worker, plus the labor-time embodied in the yarn and the spindle, is thirty-six hours: twelve hours' labor by the weaver, twelve for the yarn, and twelve for the spindle. The ratio of the surplus to c + v is 6:30 for a rate of profit of 20 percent.

Marx a.s.sumed that the rate of surplus value the ratio of s to v was constant, both across industries and across time.10 Simultaneously, he argued that the compet.i.tive forces of capitalism would lead to capitalists replacing direct labor with machinery, so that for any given production process, c would get bigger with time. With s/v constant, this would decrease the ratio of s to the sum of c and v, thus reducing the rate of profit.

Capitalists would thus find that, regardless of their best efforts, the rate of profit was falling.11 Capitalists would respond to this by trying to drive down the wage rate, which would lead to revolt by the politically aware working cla.s.s, thus leading to a socialist revolution.12 Well, it was a nice theory. The problem was that, even if you accepted the premise that labor was the only source of value, the theory still had major logical problems. Chief among these was what became known as the transformation problem.

The transformation problem The transformation problem arises from the fact that capitalists are motivated not by the rate of surplus value, but by the rate of profit. If the rate of surplus value is constant across industries, and labor is the only source of surplus, then industries with a higher than average ratio of labor to capital should have a higher rate of profit. Yet if a capitalist economy is compet.i.tive, this situation cannot apply in equilibrium, because higher rates of profit in labor-intensive industries should lead to firms moving out of capital-intensive industries into labor-intensive ones, in search of a higher rate of profit.

Marx was not an equilibrium theorist, but this problem was serious because his description of equilibrium was inconsistent. Somehow, he had to reconcile a constant rate of surplus value across industries with at least a tendency towards uniform rates of profit.

Marx's solution was to argue that capitalism was effectively a joint enterprise, so that capitalists earned a profit which was proportional to their investment, regardless of whether they invested in a labor-intensive or capital-intensive industry: Thus, although in selling their commodities the capitalists of the various spheres of production recover the value of the capital consumed in their production, they do not secure the surplus-value, and consequently the profit, created in their own sphere by the production of these commodities So far as profits are concerned, the various capitalists are just so many stockholders in a stock company in which the shares of profit are uniformly divided per 100. (Marx 1894) He provided a numerical example (ibid.) that purported to show that this was feasible. He first provided a table (Table 17.1) showing the production of surplus value by a number of industries with differing ratios of variable to constant capital (in modern terms, varying labor-to-capital ratios).

In this 'value' table, a higher ratio of labor to capital is a.s.sociated with a higher rate of profit. Thus 'labor-intensive' industry III, with a labor-to-capital ratio of 2:3, earns the highest 'value' rate of profit of 40 percent, while 'capital-intensive' industry V, with a 1:20 ratio, makes a 'value' rate of profit of just 5 percent.

Then Marx provided a second table in which the same industries earned a uniform rate of profit, now in terms of price rather than value. In contrast to Table 17.1, now all industries earned the same rate of profit.

The numbers in this example appeared feasible. The sums are consistent: the sum of all prices in Table 17.2 equals the sum of the value created in Table 17.1; the sum of surplus value in Table 17.1 equals the sum of the differences between input costs (500) and the price of output in Table 17.2 (610). But this apparent consistency masks numerous internal inconsistencies. The best proof of this was provided by the Sraffian economist Ian Steedman (this next section is unavoidably technical, and can be skipped at first reading).

TABLE 17.1 Marx's unadjusted value creation table, with the rate of profit dependent upon the variable-to-constant ratio in each sector TABLE 17.2 Marx's profit distribution table, with the rate of profit now uniform across sector Marxist economics after Sraffa We have already seen in Chapter 6 the damage Sraffa's crucible did to the economic theory of price determination and income distribution. In an ill.u.s.tration of the comparatively non-ideological nature of Sraffian a.n.a.lysis, Steedman showed that Sraffa's method could equally well critique Marxian economics.

The basis of Sraffa's system is the acknowledgment that commodities are produced using other commodities and labor. Unlike conventional economics which has invented the fictional abstraction of 'factors of production' Marx's system is consistent with Sraffa's 'production of commodities by means of commodities' a.n.a.lysis (indeed, Marx's economics was a major inspiration for Sraffa).

Steedman began with an ill.u.s.trative numerical model of an economy with just three commodities: iron, corn and gold. Iron and labor were needed to produce all three commodities, but neither gold nor corn was needed to produce anything.13 Table 17.3 shows the quant.i.ties of inputs and outputs in Steedman's hypothetical economy.

TABLE 17.3 Steedman's hypothetical economy The numbers in this table represent arbitrary units: the iron units could be tons, the labor units hours, gold units ounces, and corn units bushels and any other set of arbitrary units would do as well. However, since each input is measured in a completely different unit, the numbers add up only down the columns: they don't add across the rows.

To a.n.a.lyze the labor theory of value, Steedman first had to convert these into the 'labor-value' units which Marx used. For simplicity, he set the labor-value of one unit ('hour') of labor at 1. Converted into value terms, Table 17.4 then says that it takes 28 times whatever the 'labor-value' of a ton of iron is, plus 56, to produce 56 times whatever the 'labor-value' of a ton of iron is. A bit of simple algebra shows that one ton of iron has a labor-value of 2.

TABLE 17.4 Steedman's physical table in Marx's value terms Similar calculations show that the labor-value of an ounce of gold is 1, and the labor-value of a bushel of corn is 4.

The next stage in the a.n.a.lysis is to work out the value of the commodity labor-power. It might appear that this has already been done didn't he set this equal to 1? No, because this represents the total amount of labor performed, and in Marx's theory, workers get paid less than this. They get paid, not for their contribution to output, but for the commodity labor-power, whose value is equal to the means of subsistence.

Steedman a.s.sumed that it took five bushels of corn to reproduce the labor used in this hypothetical economy. Therefore the total value of labor-power in the entire economy was equal to the labor-value of five bushels of wheat. Since a bushel of wheat has a labor-value of 4, this means that the value of labor-power across the whole economy was 20 (and therefore, one unit of labor had a labor-value of 1/4). The difference between this amount and the total labor performed 80 hours of labor, which we have set to equal 80 units of labor-value is surplus value. So v, in Marx's scheme, is 20, while s is 60, for a rate of surplus value of 300 percent.