Cost and Choice: An Inquiry in Economic Theory

James M. Buchanan.
Buchanan, James M.
(1919- )
CEE
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Editor/Trans.
First Pub. Date
1969
Publisher/Edition
Indianapolis, IN: Liberty Fund, Inc.
Pub. Date
1999
Comments
Foreword by Hartmut Kliemt.

Chapter 1Cost in Economic Theory

Classical Economics

6.1.0

If among a nation of hunters ... it usually costs twice the labour to kill a beaver which it costs to kill a deer, one beaver should naturally exchange for or be worth two deer.*4

6.1.1

The classical theory of exchange value is summarized in this statement. Adam Smith was not so careful as his modern counterpart who states his assumptions precisely, but perhaps this is why we still enjoy reading The Wealth of Nations. Normal or natural value in exchange is determined by the relative costs of production. This answers the central questions of classical economics.

6.1.2

Costs are calculated in units of resource input. "It usually costs" means that a specific resource outlay is required, an outlay that can be estimated in advance with some accuracy and measured ex post either by the resource owner or by an external observer who doubles as cost accountant. The relative costs of producing are objectively quantifiable, and no valuation process is necessary. Given a standard for measurement, relative costs can be computed like the relative weights of apples or potatoes. In Smith's elementary and conjectural model, the standard for measurement is a unit of homogeneous labor time. There are no nonlabor inputs (no other "negative goods"). The production functions for both deer and beaver are linear and homogeneous; that is to say, deer and beaver are available in unlimited supply at prevailing relative cost ratios.

6.1.3

Even in so simple a model, why should relative costs determine normal exchange values? They do so because hunters are assumed to be rational utility-maximizing individuals and because the positively valued "goods" and the negatively valued "bads" in their utility functions can be identified. If, for any reason, exchange values should settle in some ratio different from that of cost values, behavior will be modified. If the individual hunter knows that he is able, on an outlay of one day's labor, to kill two deer or one beaver, he will not choose to kill deer if the price of a beaver is three deer, even should he be a demander or final purchaser of deer alone. He can "produce" deer more cheaply through exchange under these circumstances. By devoting one day's time to killing a beaver and then exchanging this for deer, he ends up with three deer, not two. Since all hunters can be expected to behave in the same way, no deer will be produced until and unless the expected exchange value returns to equality with the cost ratio. Any divergence between expected exchange value and expected cost value in this model would reflect irrational behavior on the part of the hunters.

6.1.4

In this interpretation, the classical theory embodies the notion of opportunity cost. To the hunter at the point of an allocative decision, the cost of a beaver is two deer and the cost of a deer is one-half a beaver. At an expected exchange ratio of one for two, each prospective hunter must be on the margin of indifference. Physical production and production-through-exchange yield identical results. Labor time, the standard for measurement, is the common denominator in which the opportunity costs are computed.

6.1.5

Realized exchange value need not be equal to realized cost value in the elementary deer-beaver model or in the classical model generally. As interpreted here, there must be competitive indeterminacy in the allocation of resources to deer and beaver production. If, relative to prevailing demand patterns, a large number of hunters choose to produce beaver on a particular day, the price or market value of beaver will fall below cost. Or, alternatively, if the demand pattern shifts suddenly while the allocation of resources remains substantially unaltered, the same result can be forthcoming. Price, which is realized exchange value, can and will diverge from realized cost value. When this happens, however, some of the hunters will look back at the time of decision and conclude that mistakes were made.

6.1.6

There is one-way causality in this deer-beaver model. Relative costs determine normal exchange values. Implicitly, the theory assumes that predictions about production relations, the ratios of inputs to outputs, are considerably more accurate than predictions about demand patterns. This converts the theory into an elegant operational hypothesis. Objective, external measurements can be introduced which should yield predictions about normal exchange values. These predictions can be falsified.

6.1.7

The theory contains no prediction of normal exchange value when production is not possible, that is, when supply is fixed. Here normal exchange value, like realized exchange value in all cases, is set by the forces of demand. But to Adam Smith, this would not have embodied a predictive theory. No behavioral tendency can be introduced that relates the values of "goods" one to the other in terms of some objectively identifiable common denominator. For such fixed-supply goods, Smith would have said, simply, that no theory of value exists. Viewed in this context, J. S. Mill's infamous statement that nothing more could be said on the theory of value can be interpreted somewhat more sympathetically than modern economists have been wont to do.

6.1.8

Exchange value tends to equality with objectively measurable cost of production. This is a positive proposition and carries with it no normative content. Nothing is said or implied to the effect that market price should equal cost of production. In the direct sense, classical economics does not contain a normative theory of allocation. The equalization of return to similar units of resources tends to emerge from the basic postulate of rational behavior along with the implicit identification of "goods" and "bads" in the utility functions of individuals.

6.1.9

The interpretation sketched out above is unfair to those who have criticized classical economics. Confusions abound on many points of analysis. Once the extreme simplifications of Smith's homogeneous and single input model are dropped, the problems commence. The intricacies of classical reasoning are centered around the search for a comparable common denominator of value when inputs (negative goods) are heterogeneous. Ricardo's genius was not up to this challenge despite valiant efforts. Rent theory explained away, though somewhat unsatisfactorily, the return to land. But differing labor-capital ratios remained, and Marx's resort to "socially necessary" labor time was a retreat into the circularity that the whole classical theory was designed to circumvent.

6.1.10

Smith and, finally, Ricardo were both forced to rehabilitate the theory's pragmatic relevance at the expense of its elegance. Heterogeneous units of input were measured in the money prices established in factor markets. The cost of production for a good was computed in money. As an elementary explanation of the normal or natural exchange value for a specific good, the essential features of the deer-beaver model continue to hold. The normal exchange value of a pair of shoes tends to equal $10 if $10 is the money cost of producing shoes, the necessary outlay made to owners of all resource inputs. Unfortunately, the elegance and the objectivity of the deer-beaver world have disappeared in this more realistic cost-of-production model. The objective opportunity cost of a beaver in Smith's model is two deer because it "usually" takes twice as much physical input to produce a beaver. In the more realistic setting, what is the opportunity cost of a pair of shoes? Costs are measured in a numeraire and these reflect values of physical inputs. The direct connection between these inputs and alternative outputs is gone. To say that payments to resource owners amount to $10 does not, at least directly, imply that alternative products valued at $10 could be produced.

6.1.11

If costs are $10, the producer must expect a value of at least $10. The postulate of rational behavior along with the presumption that the numeraire is positively desired still implies that expected value be equal to or above costs. But what now determines costs? No longer is the theory simple enough to concentrate our attention on one moment of decision, one act of choice. Instead of this, we now must think of a chain of interlinked decisions over varying quantities of output, over separate time periods, and over many decision-makers. The producer, facing a near-certain outlay of $10, must expect a value in excess of this figure if he is to choose to produce. But resource owners, who are now conceptually separated from the producer-entrepreneur, must also make choices. Why does the unit mix of inputs sum to $10? Exchange values are established for resource units in markets, and each owner must be assumed to expect values in excess of costs when he makes a unit of resource available. But what are his costs? The classical economists were forced to discuss the costs of producing primary resources.

6.1.12

They thought themselves successful to an extent with respect to labor of an unskilled or common variety. In Smith's elementary model, the cost of a beaver is two deer, which for comparative purposes is measured as a day's labor, the time required to kill either of the alternatives open to the hunter. The cost of common labor time is the corn that is required to nourish the laborer and to allow him to reproduce his kind. Again, this corn can be measured in labor time required to produce the corn. But the analytical difference between these two statements is great, and in the latter we see a false extension of a basically correct theory of exchange value. The opportunity-cost theorem that is central to the deer-beaver model almost wholly disappears in the theory of wages for common labor. A day's labor time measures the cost of a beaver because it represents the genuine alternative product, two deer. A half-day's labor time presumably measures the cost of a laborer, not because it represents any genuine alternative product, but because it represents the outlay that is required to nourish him. The input-output relation has been subtly changed from that found in the simpler model. The labor input that measures the cost of a beaver is that required to produce an alternative, two deer. And no hunter would kill beaver unless the appropriate ratio of expected value holds. The outlay that may actually be required to kill a beaver is irrelevant to realized exchange value. By contrast, in the classical theory of wages, no consideration of the alternative to producing a laborer is included. Even the most sympathetic critic will find it difficult to read opportunity-cost thinking into the analysis.

6.1.13

It was perhaps in recognition of the difficulty here that both Smith and Ricardo shied away from rigorous analysis in discussing wages. A classical theory of sorts emerged which related wages to levels of subsistence. In this theory of wages based on Malthusian population principles, the cost theory of exchange value has lost almost all of its opportunity-cost moorings. Wages of common labor tend to subsistence levels, not because this is a predictable result of rational individual behavior, but because of the natural checks of famine and pestilence. The relationship between exchange value and individual choice behavior has been severed—and with it the essential logic of any cost-of-production theory. This classical theory of wages is almost devoid of behavioral content.

6.1.14

A source of some confusion that runs through and sometimes dominates classical discussion of cost has not been mentioned. This is the notion of pain cost, often called real cost. Not content with searching for a predictive theory of exchange value, the classical writers sought to "explain" the emergence of value in some basic philosophical sense. The toil and trouble, the physical pain, involved in working seemed to "justify" the payment of wages. Observation revealed that capital also received payment. Hence, the concept of abstinence developed by Senior seemed to place the capitalist alongside the wage-earner as a recipient of justifiable rewards. The importance of this real-cost doctrine in sowing confusion should not now be underestimated. Even today the theory of comparative advantage as taught by many sophisticated analysts contains its manifest nonsense, although fortunately little damage is done.*5

6.1.15

Cost does reflect pain or sacrifice; this is the elemental meaning of the word. But we must recognize the linguistic problem which confronts economists in the use of the word "cost" to refer to quite separate things. Any opportunity within the range of possibility that must be foregone in order to select a preferred but mutually excluding alternative reflects "costs" when it is "sacrificed." And its rejection must involve pain despite the fact that differentially greater pleasure is promised by the enjoyment of the mutually exclusive alternative. Cost and pain are far from being opposites, contrary to what loose discussion often seems to suggest; the concept of cost as pain or sacrifice is and must be central to the idea of opportunity cost. In certain aspects of the classical treatment, this pain-as-sacrifice concept was understood. As mentioned, the cost of capital accumulation was discussed in terms of "abstinence": by abstaining from consuming, capital is allowed to accumulate. Clearly, this involves opportunity-cost reasoning.

6.1.16

For the most part, however, the real-cost or pain-cost notion in classical economics refers to something quite different. Pain also arises when nothing is sacrificed in a behavioral context. Pain occurs when, as a result of a past chain of events, the utility of the individual is reduced without offsetting pleasures. The required outlay of labor may involve pain, something that can within limits be measured by sweat, muscle fatigue, and tears. The transfer of capital assets to meet a debt obligation, to pay taxes, or to pay tribute to a highwayman also involves pain, again something that can be proximately measured by a decrement in net worth on the individual's balance sheet. In this second sense, pain cost has no connection with deliberately sacrificed alternatives. The expectation of such pain may inform the comparison of alternative opportunities for choice, but the realization of such pain is irrelevant either in explaining or in justifying value. This vital distinction between the two separate notions of pain cost was not recognized by the classical economists or by many of their successors. The roots of many modern ambiguities lie in the classical failure to note this distinction, a failure that neoclassical economics did not remove satisfactorily.

Marginal-Utility Economics

6.1.17

A revolution in value theory took place after 1870. The classical cost-of-production theory was replaced by the marginal-utility theory, as the latter was variously developed by William Stanley Jevons, Karl Menger, and Leon Walras. These theorists were somewhat less obligated than their classical predecessors to define costs precisely for the simple reason that costs assumed much less importance for them in explaining exchange value. At least in the elementary stages of analysis, they seemed willing to accept classical definitions: Their quarrel with the classicists was not centered on the notion of cost. They considered their differences to be more profound. Regardless of the manner in which costs were defined, however, the marginal-utility theorists rejected classical analysis.

6.1.18

The development of a general theory of exchange value became a primary concern. Classical analysis was rejected because it contained two separate models, one for reproducible goods, another for goods in fixed supply. The solution was to claim generality for the single model of exchange value that the classical writers had reserved for the second category. Exchange value is, in all cases, said the marginal-utility theorists, determined by marginal utility, by demand. At the point of market exchange, all supplies are fixed. Hence, relative values or prices are set exclusively by relative marginal utilities.

6.1.19

If the revolution had amounted to nothing more than this, it would have scarcely warranted notice. The contribution of these theorists was not the mere substitution of a utility for a cost theory of exchange value. In the process of effecting this substitution, they were forced to develop the idea that values are set at the margin. In this manner, they were able to resolve the diamond-water paradox; value-in-use and value-in-exchange were no longer possibly contradictory. The economic calculus was born.

6.1.20

Nonetheless, there were losses in discarding the classical apparatus. In their search for a general theory, the marginal-utility economists largely abandoned a predictive theory of normal exchange value. They provided a satisfactory explanation of realized value; they did little toward developing analysis of expected or natural value. In the strict sense, theirs is a logical theory, not a scientific hypothesis capable of refutation. And as with all general theories, the marginal-utility theory explained too much.

6.1.21

The generality carried some secondary benefits, however, and a logical extension was the marginal-productivity theory of distribution. Since goods are valued in accordance with relative marginal utilities, resources should also be valued in accordance with the values of their final-product components. There was no call to go beyond the fixed supply of resources in a first approximation. For almost a century, the theory of population was dropped from the economist's kit of tools.

6.1.22

Marginal-utility economics is often called "subjective-value" economics, and the doctrinal revolution also carries this name. The classical cost-of-production theory was objective in the sense that external measurements of comparative costs were thought to provide predictions about normal exchange values of commodities. The replacement of this with a theory that explained relative exchange values by relative marginal utilities necessarily implies a loss of objective empirical content. Marginal utilities, however, were acknowledged to be dependent on quantities, and, for the whole group of demanders, on the supplies put on the market. Hence, even with a full knowledge of demand conditions, normal exchange values could not be predicted until and unless predictions were made about relative supplies. The cost or supply side of value had to be brought in. A one-sided explanation was no longer possible; demand-supply economics became a necessity.

6.1.23

Given a supply of a commodity, exchange value was determined by marginal utility, as worked out in a market interaction process. But utility is a subjective phenomenon, and it is not something that can be externally or objectively measured, as can classical cost-of-production. To understand this, let us think of a world of two commodities, each of which is in fixed supply, say, the world of bear and raccoon. Both are "goods," and each good is available in predictably fixed quantity in each period. If we know with accuracy the demand or marginal-utility schedules for all demanders, exchange value can be predicted. Note, however, that this prediction does not emerge as an outcome or result of a rational behavior postulate, at least in the same sense as the classical deer-beaver model. Suppose, given the fixed supplies along with the demand patterns, it is predicted that one bear will exchange for two raccoons. If realized values are observed to be different from those predicted, it is the result only of inaccuracy in the initial data upon which the predictions were made. No equilibrating mechanism is set in motion; there is no sense of error as there is in the deer-beaver model. No corrective process will emerge; values as realized are always "correct"; errors arise only in the data used by the observer. The marginal-utility theory in its elementary methodology here is akin to the simple Keynesian model of income determination. By contrast and despite its flaws, the classical cost-of-production theory is more closely analogous to the Swedish theory of income determination where expectations can explicitly enter the analysis.

6.1.24

To introduce elements of a predictive theory of exchange value required a return to quasi-classical analysis. Costs of production were acknowledged to influence exchange value through their effects on supply. And in discussing costs, the marginal-utility theorists could accept a money measure without ambiguity since they had no reason to search for a common denominator of physical resource inputs. The necessity of paying for inputs arises because these represent components of value in final products. This approach leads almost directly to opportunity-cost reasoning.

6.1.25

The value or price of resource units represents, especially for the Austrians (Menger, Böhm-Bawerk, Wieser), the value of product that might be produced by the same resource units in alternative uses or employments. This is the price that the user or employer of resources must advance in order to attract the resources away from such alternative opportunities. At the level of decision for the resource owner, the implicit opportunity-cost notion is identical to that which is present in Smith's deer-beaver model. To the Austrians, and notably to Wieser, rational behavior on the part of resource owners ensured the equalization of return in all employments.

6.1.26

Jevons was unique among the subjective-value theorists in his treatment of cost, and he is considerably more classical than Austrian. The cost of producing involves "pain," a concept that was almost entirely absent from Austrian discussion. This pain cost can be discussed in terms of marginal disutility. Jevons was thus the complete marginalist, and, for him, all choice reduced to a comparison of utilities and disutilities at the margin. He was able to resolve the diamond-water paradox through the use of, essentially, the classical apparatus. Because he did not sufficiently generalize the alternate-product conception, his cost theory was inferior to that of the Austrians or even to that which was implicit in Smith. Nonetheless, Jevons did concentrate attention on the act of economic choice, and this might have influenced Wicksteed in his major advances toward his wholly modern conception.

6.1.27

To the early Austrian theorists, costs of production are measured in money, and these reflect the value of output that might have been produced if the same resource inputs had been rationally applied in alternative employments. This is indeed an opportunity-cost notion, but it is subjective only in the sense that values of goods are set by their relative marginal utilities to demanders. Since these values are set in organized markets, they can be objectively measured.

The Marshallian Synthesis

6.1.28

Alfred Marshall thought that he had rewritten classical economics, incorporating, in the process, qualifications and criticisms which apparently were developed independently of the marginal-utility theorists, despite the similarities in treatment of particular elements. His period analysis provided a general model in which adjustment lags determined the relative explanatory power of marginal-utility and cost-of-production hypotheses. His sympathies were with the classicists, and he sensed the predictive advantages of the basic classical model.

6.1.29

Marshall was too sophisticated an analyst to overlook the simple idea of opportunity cost, but explicit statements of this notion cannot readily be found in his discussion. Cursory reading suggests that Marshall was willing to accept a naive classical version of real or pain cost arising from the exertions of the laborer and the abstinence of the capitalist. In part, his lack of precision in defining cost was due to his direct and pragmatic concern with explaining price formation. Marshall did not ask conceptual and definitional questions for their own sake, and he seemed willing to stop short of these inquiries once he had provided what he considered satisfactory answers to relevant practical questions.

6.1.30

For these purposes, money costs, as determined by prices set in factor markets, were sufficient. In the long period, after all adjustments are made and if other things do not change in the interim, prices tend to settle at the level of money costs when constant returns prevail. To Marshall this was a reasonably satisfactory statement, really all that could be expected from economics. His models, here as elsewhere, are often "fuzzy"—one feels deliberately so—not because he overlooked, but because he recognized the complexities involved in setting things all straight. Perhaps this is unduly sympathetic to Marshall, but one feels that, despite the ambiguities, he would never have made the blunders that his successors fell into over their failures to define costs properly.

Frank Knight and American Neoclassical Paradigms

6.1.31

In sharp contrast to Marshall stands Frank Knight whose "main concern is the correct definition of the problem...." He sensed the ambiguities that were present in the neoclassical, essentially Marshallian, treatment of cost. In a series of important papers written in the late 1920's and early 1930's, he established the conception of opportunity or alternate-product cost that became the paradigm for modern price theory, notably in its American-Chicago variant. Starting with Adam Smith's deer-beaver model, Knight demonstrated its inherent opportunity-cost content along the lines that I have sketched at the beginning of this chapter. "[T]he cost of beaver is deer and the cost of deer is beaver, and that is the only objective and scientific content in the cost notion."*6 The opportunity cost of a commodity is measured in units of alternate or displaced product and "all references either to 'sacrifice' or 'outlays' [should be] simply omitted."*7 "[C]ost must be measured in terms of products, and not of pains or outlays."*8

6.1.32

In this 1928 statement of what he considered to be the "correct" definition, Knight was following what he later acknowledged to be the standard Austrian position, especially that represented in Wieser. He also indicated in a later paper that this position was shared by Wicksteed. The cost of producing a unit of a commodity is simply measured by the alternative real product that might have been produced had the resource inputs used in production been rationally reallocated to other uses. The market value of these alternate products provides a common denominator for estimation, a value that is determined in the exchange process. Knight seems to have been correct in claiming this approach akin to that of Wieser who said: "Since each productive process diminishes this possession, it reduces utility—it costs, and it costs exactly as much as the value which the material and labor required would have produced if rationally applied."*9

6.1.33

Within a few years, however, Knight sensed that something was wrong with his straightforward alternate-product measure of opportunity cost. In papers published in 1934 and 1935, he tried to spell out his misgivings, but without great success.*10 He tried to modify the alternate-product definition of cost to take account of the differences in the irksomeness of different resource uses, especially with application to the allocation of labor. In an extremely complex argument, Knight claimed that to the extent that resource owners do not equalize pecuniary returns to resource units in all uses, the principle of alternate-product cost does not wholly apply. If the deer hunter accepts a relatively lower pecuniary reward for his more pleasant work, a dollar's worth of resource payment withdrawn from deer production and transferred to beaver production will increase "social" product by more than one dollar. Hence, the opportunity cost of the resultant increase in beaver production is more than the market value of the deer that the resource inputs might have produced before the transfer. Thus, the net change in irksomeness must also be acknowledged and counted.

6.1.34

This is surely a reasonable and fundamentally correct observation. It reflects, nonetheless, a notion of opportunity cost quite different from that which Knight had earlier advanced. The introduction of nonpecuniary advantages and disadvantages of resource uses severs the critically important link between the objectively measured market value of alternate product and the cost that enters into the subjective calculus of the decision-maker. This linkage is essential if the theory of value is to retain scientific content in any predictive sense. Without realizing it, Knight necessarily shifted from a positive model of behavior in which costs are objectively measurable into a logical model of choice in which costs are purely subjective. In the latter model, which has no predictive content, the market value of the displaced or alternate product has no direct relevance for the resource owner's decision. Hence, this value cannot in any way be considered as the measure of his cost. Properly interpreted, as Wicksteed came close to saying, the predicted or expected value of the alternate product at the moment of decision and as estimated by the chooser is the cost. And, under this definition, the nonmarket value of the alternate conditions of employment is included as an essential part of cost.

6.1.35

The initial position taken by Knight became the orthodox one, and it remains so in the major part of modern price theory. The opportunity-cost notion is central. "The cost of any alternative (simple or complex) chosen is the alternative that has to be given up; where there is no alternative to a given experience, no choice, there is no economic problem, and cost has no meaning."*11 "Economic cost, then, consists in the renunciation of some 'other' use of some resource or resource capacity in order to secure the benefit of the use to which it is actually devoted."*12 "The only general-cost theory which can be maintained will, after all, be that of alternative cost, best formulated as displaced product cost, but this must be stated subject to the qualification that it is true only 'in so far' as at equilibrium the indicated conditions obtain."*13

6.1.36

In the context of most theoretical discussion, these are perfectly acceptable and wholly correct statements. Cost is measured by the market value of displaced product. Cost is objective in that it can be estimated, at least in ex post terms, by external observers, despite the fact that market values are set, generally, by the subjective evaluations by many producers and consumers. Market prices measure collective evaluations at the margins of production, and prices are themselves objective.

6.1.37

These statements about cost are widely and uncritically accepted by most modern price theorists, most of whom fail to see that opportunity cost, so defined, has no connection with choice at all. It is precisely for this reason that the simple but subtle differences between this orthodoxy and the alternative London theory provide suitable subject matter for a small book.


Notes for this chapter


4.
Adam Smith, The Wealth of Nations (New York: Random House, Modern Library Edition, 1937), p. 47.

5.
Even in 1967 economists need to be warned of the fallacy. On this, see Royall Brandis, "The Myth of Absolute Advantage," American Economic Review, LVII (March 1967), 169-74.

6.
Frank H. Knight, "A Suggestion for Simplifying the Statement of the General Theory of Price," Journal of Political Economy, XXXVI (June 1928), 359.

7.
Ibid., 355.

8.
Ibid., 363.

9.
F. von Wieser, "The Theory of Value," Annals of The American Academy of Political and Social Science, II (March 1892), 618. See also F. von Wieser, Über den Ursprung und die Hauptgesetze des wirtschaftlichen Werthes (Wien, 1884), p. 100.

10.
Frank H. Knight, "The Common Sense of Political Economy (Wicksteed Reprinted)," Journal of Political Economy, XLII (October 1934), 660-73, reprinted in Frank H. Knight, On the History and Method of Economics (Chicago: University of Chicago Press, Phoenix Books, 1963), pp. 104-18. This is a review article of the two-volume edition of Wicksteed. Frank H. Knight, "Notes on Utility and Cost" (Mimeographed, University of Chicago, 1935). Published as two articles in German in Zeitschrift für Nationalökonomie (Vienna), Band VI, Heft 1, 3 (1935).

11.
Knight, "Notes on Utility and Cost," op. cit., p. 18.

12.
Ibid., p. 19.

13.
Knight, "The Common Sense of Political Economy (Wicksteed Reprinted)," op. cit., p. 116.

End of Notes


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