¹ A. Wald, “Some Systems of Equations in Mathematical Economics,” Econometrica, October, 1951 (the results summarized in that paper were first published in German in 1935–36); K. Arrow and G. Debreu, “Existence of an Equilibrium for a Competitive Economy,” Econometrica, July, 1954; R. Dorfman, P. A. Samuelson, R. M. Solow, Linear Programming and Economic Analysis, New York, 1958, pp. 351–75.

² References will be to W. Jaffé’s English edition, New York, 1959 (translated from the 4ème édition définitive, Paris, 1926).

³ The quantities of services supplied for productive use O_t…O_p…O_k… differ from the quantities available in the economy Q_t…Q_p…Q_k … because a demand for direct use is assumed to exist (compare Elements, p. 238).

⁴ For eq. (1) (2) (3) (4) see, respectively, eq. (2)(4)(3)(1) at pp. 239–40, Elements.

⁵ See the references given above, p. 367, n. 1.

⁶ See below, pp. 379–80.

⁷ Compare the four papers delivered at the Société Vaudoise des Sciences Naturelles in 1875–76 (Théorie Mathématique de la Richesse Sociale, Paris, 1877, later reprinted with the same title in Lausanne, 1883).

⁸ Up to the third edition of the Elements, Walras limited his capital theory to fixed capitals by assuming that “consumers’ goods and raw materials are consumed as soon as they are produced and are not stocked in advance.” Only labour, land and fixed capital need then appear among the productive services (Elements, 3^rd ed., Paris, 1896, par. 175). This uneasy assumption is abandoned in the fourth edition where, in Part VI, a “theory of circulation” appears in which circulating capitals are treated in a way strictly analogous to fixed capitals. To avoid unnecessary complications, that theory will not be explicitly considered in the text, although the applicability to it of the conclusions we shall reach will then be indicated (below, p. 376, n. 25).

⁹ Elements, pp. 268–69. Walras's calculation of the amortization quota as a constant fraction of the price of the new capital good ignores the interest obtainable on the amortization quota during the time between the moment it is received and the moment of replacing the capital good. This, by giving rise to a “net price” of the service independent of the age of the capital good, brings to the paradox that the price of an old capital good (i.e., the price on which yearly net income at the rate i obtains) is the same as for a new capital good of that kind (compare e.g., Elements, p. 303). This conclusion would be acceptable only if amortization took the form of a “maintenance” by which the fixed capital could be made to last forever with constant efficiency. In the general case the correct relationship for the rate of net income on a fixed capital good with constant efficiency for a life of t years (where 1/t corresponds to Walras's s amortization coefficient) is of the type:

The discussion in this paper would need no modification, if the above relationship is substituted for the one given by Walras.

¹⁰ Elements, pp. 271–72.

¹¹ Elements, p. 271.

¹² Elements, pp. 272–76. Walras's text is ambiguous on whether F_e expresses gross savings (with D_k, D_k … constituting the entire output of capital goods for the period) or net savings (with capital outputs including, besides D_k, D_k′ … capital replacements, cQ_k, c′Q_k′ …, where Q_k, Q_k′ … are the stocks of K, K′ … already in existence). Starting from the fourth edition of the Elements, F_e is obtained in the following way. A commodity (E) is introduced consisting of perpetual future income, which can be bought at the price . Individual demands d_e for (E) are taken to be determined by the conditions of consumers’ equilibrium including those relating to a “marginal utility” function of (E):

where q_e is the perpetual future income the individual already possesses, i.e., his present net income. The expression (d_ep_e) must then give the net savings of the individual during the period considered, for d_e = 0 means that the individual is keeping his perpetual future income q_e constant and must therefore be replacing his capital stocks q_k, q_{k ′} … as they wear out. Since F_e is the algebraic sum of the individual functions giving (d_ep_e), F_e should give aggregate net savings. At pp. 275–76 Walras seems to interpret it that way. However, if F_e is net savings, then two things should follow: (a) the individual budget equations should include a replacement expenditure (cq_kP_k + c′q_k′P_k′ + …) not included by Walras; he instead generally refers to F_e as the “excess of income over consumption,” an expression he had before defined as gross savings (pp. 272–73); (b) total current outputs of capital goods should be (D_k+ cQ_k), (D_k′ + c′Q_k′) … and these terms should appear on the right hand of equations (3), where Walras has only D_k, D_k′ … Also, in the editions of the Elements preceding the fourth, F_e clearly expressed gross savings. For a consistent exposition of Walras's theory we have opted for F_e as gross savings (W. Jaffé interprets F_e as net savings, Elements, pp. 531–32; G.J. Stigler, as gross savings, Production and Distribution Theories, New York, 1941, p. 241).

¹³ Elements, p. 276.

¹⁴ Gross savings (equal to zero) and demands for consumers’ goods are both assumed to be independent of the rate of net income, so that i does not appear in equations (1) and (4). Equations (1) – (4) are then sufficient to determine the (2m + 2n – 1) unknowns appearing in them, as equations of pure exchange and production.

¹⁵ Each D_k may take any value between zero and the one which would absorb total gross savings for the period. The “initial” values of R′, R″ … can therefore be taken as any set of rational numbers between zero and infinity. The infinite value of one ratio would mean assuming D_k = 0 and would involve infinite values of all other ratios. In that case the ratios can be defined by using at denominator one of the D_k's larger than zero.

¹⁶ System (P) can be envisaged as consisting of equations of exchange and production, with (A) as an equation giving the demand for a special composite commodity, consisting of K, K′, K″ … in undefined proportions, and with the one equation (B) giving the costs conditions for the additional price (the rate of net income) introduced with the demand for the composite commodity. The indefiniteness of the proportions in which the ℓ new capital goods are demanded emerges in the degrees of freedom in system (P), where the (2m + 2n + l) unknowns of the complete system appear in (2m + 2n + 1) independent equations. When such proportions are defined by arbitrary values of R′, R″ …, system (P) will admit solution as any system of equations of exchange and production.

The possibility of multiple economically meaningful solutions of (P) for a given set of ratios R′, R″ …, will not be considered in the text, since it would complicate the discussion without affecting its upshot.

¹⁷ If the given production coefficients for K, K′ … satisfy the following series of equalities:

(the ratios for services not used for the production of any capital good can be ignored), the unknowns D_k, D_k′ … in equations (3) and (A) could be reduced to one unknown (a “capital output” using productive services in the uniform proportions in which they are used for the several capital goods). System (P) would then be sufficient to determine that output and all other unknowns, independently of R′, R″, R‴…

¹⁸ If, for instance, we increased R′ = while keeping constant R″, R‴ … we should compare the ratios between each production coefficient of K′ and the coefficient, for the same service, of an output X consisting of K, K″, K‴ … in the proportions defined by R″, R‴ … The increase of R′ will result in a rise of the price of any service T relative to the price of any service T′, when (variations of the prices of services not used for the production of any capital good are of little interest, as will become clear). In cases of multiple solutions for system (P), comparison of certain solutions might give price variations for some services, in the other direction, but such exceptions need not be considered in our discussion (compare p. 372, n. 16 (above)). A more important consideration is that a change in any R would somewhat affect the amount of gross savings by affecting the prices and rate of net income on the left of equation (A). To that extent we shall have a rise or fall in the relative prices of services used in different proportion in consumers’ goods processes than in capital goods processes. These effects can be safely ignored in our discussion since they are as likely to widen as to narrow the disparities between rates of return, which are our sole concern here.

¹⁹ It is to be remembered that capital outputs cannot affect capital stocks until a period subsequent to the one considered in the equations.

²⁰ Two points should be mentioned for the sake of completeness. The first is that the limit to the adjustments in costs of production would not be effective if we could assume that the costs of production of all capital goods tended to zero with their current outputs; that would be the case if all services used for producing each capital good were specific to that process and their supply prices tended to zero with the quantities supplied.

The second regards an equality of the rates of return obtained by changes in the “initial” values of R′, R″ intended to act on the relative prices of capital services rather than on their relative costs. Such changes should consist of increasing the relative outputs of the capital goods using as inputs relatively more of the capital goods with lower “initial” rates of return. However, even if the relevant production coefficients made these adjustments possible, a solution of the complete system thus obtained, and not obtainable also by the process described in the text, would be bound to be a position of unstable equilibrium. After a disturbance, market forces would move the economy away from that position and not back to it.

²¹ A striking instance of this deficiency of Walras’ s procedure would occur for an economy which happened to be potentially stationary, in the sense that gross savings would exactly pay for the replacement in kind of the capital goods and for replacement during the period. Assuming that Walras's equations for that situation had a solution, the resulting D_k, D_k′ … would not in general be the physical replacements of the several stocks; the economy would not be stationary, since some stocks would rise and some would fall. If, on the other hand, we stipulated the D_k, D_k′ … to be the physical replacements, the equations would not, in general, admit solution. This dilemma may help to explain why Walras avoided the case of a stationary economy, with the unconvincing argument that to have a supply and demand of capital goods, it is necessary to substitute to the conception of a stationary economy that of a progressive economy” (Elements, collation note(n), pp. 586–87). But, we may ask, what about the demand of capital goods for replacements? G. Barone took up the point in answer to a criticism by Wicksell, and held that, in Walras's system, the prices of capital goods and the rate of net income could be determined also in a stationary economy through the re-investment of amortization quotas (“Sopra un libro del Wicksell,” Giornale degli Economisti, Nov., 1895, p. 538; a view analogous to that of Barone is held by Hicks, “Léon Walras,” Econometrica, 1939, p. 396; by G. J. Stigler, History of the Theories of Production and Distribution, New York, 1949, p. 259; by W. Jaffé, “Léon Walras's Theory of Capital Accumulation,” in the symposium, Studies in Mathematical Economics, Chicago, 1942, p. 42). Barone, however, seems to overlook the dilemma already pointed out. Walras, on the other hand, kept to the assumption of a “progressive” economy also in the 4th ed. of the Elements (1900), although the justification for the assumption is there such as not to exclude the possibility of determining the rate of net income, even if the economy happened to be potentially stationary (Elements, p. 269).

²² With relative capital outputs R′, R″ … determined by additional conditions, the independent equations in the system would exceed by the number of unknowns. Given some hypothesis as to the distribution among individuals of variable capital stocks, the system could determine, given the stock of one kind of capital good, the other stocks compatible with equilibrium. That would define the only configurations of stocks for which the system would admit solution, where R′, R″ are determined by additional conditions.

²³ The way in which elements (a) and (b) affect the “ranges” mentioned in the text can be thus described. A given change in relative capital outputs, such as to preserve unchanged the total value of capital outputs at the previous prices, would cause in the capital goods sector of the economy as a whole a “deficit” of some kinds of services and a “surplus” of others. These “deficits” and “surpluses” will be smaller the closer the proportions in which the services are used to produce the several capital goods (they would all be zero if the proportions were the same). Given, on the other hand, the size of the “deficits” and “surpluses” of the capital goods sector, the rise in the prices of “deficit” services relative to the prices of “surplus” services will be smaller the more substitutable for each other consumers’ goods using in different proportions the two kinds of services, and the more positively elastic the supplies for productive use of the two groups of services; both of these effects amount to a substitution between the two groups of services in the system. Thus the change of relative costs of capital goods for given changes of their relative outputs will tend to be the smaller the closer the proportions between services in the capital processes, and the higher the substitutability in the system between services used in different proportions in the capital processes. The smaller such changes in relative costs of capital goods, the smaller (other things being equal) the “ranges” spoken of in the text.

²⁴ The smaller the amount of gross savings the smaller the size of the changes in the outputs of each kind of capital good possible in the system, and therefore (other things being equal) the smaller both the changes of relative costs of capital goods obtainable by changes in R′, R″ …, and the “ranges” spoken of in the text. In the extreme case of zero gross savings, the “ranges” narrow down to single points.

²⁵ The inconsistency discussed in this section with reference to the theory of fixed capital of Part V of the Elements is equally present in the strictly analogous treatment of circulating capital first given by Walras in Part VI of the 4th edition of the Elements. Services of “availability” are there defined for stocks of “raw materials” and finished goods, possessed in given quantities by individuals in the economy. Additional equations analogous to equations (3) and (4) give demand and supply for the service of availability of the stock of each kind of circulating capital. Additional equations analogous to (B) state the equilibrium condition of a uniform net rate of return over costs for circulating capitals, equal to the net rate of return on fixed capitals (op. cit., p. 323). These latter equations will generally be inconsistent with the remainder of the system for the reasons discussed in this section.

²⁶ Elements, pp. 282 ff. It is to be stressed that Walras explicitly assumes all the data not to be affected by the course of the tâtonnements. From this it appears that the tâtonnements cannot be correctly interpreted as an attempt by Walras to develop a “dynamic” theory of the path by which the economy reaches equilibrium, for, if they were intended to be that, the constancy of such data as the stocks of capital goods or commodities carried over from stage to stage of the tâtonnements, would void such theory of all its content. It seems more correct to interpret Walras's tâtonnements purely as a device for an analysis, however imperfect, of the existence of a stable solution for the system. This purely analytical role of the tâtonnements becomes clear in the fourth edition of the Elements, where the device of “tickets” allows all contracts to be provisional, and no exchange to take place until the equilibrium system of prices is obtained (Elements, p. 262 and pp. 282–3).

²⁷ Op. cit., 3rd ed., par. 253. The passage can be found in all earlier versions of the theory, starting from the paper delivered in Ju1y, 1876, at the Société Vaudoise des Sciences Naturelles (compare Théorie Mathématique de la Richesse Sociale, Paris, 1877, pp. 568–69). Another passage of similar content remains in Lesson 23 of the 4th ed. (op. cit., p. 271), in striking contradiction with the revised argument developed in Lesson 25 of that edition. Also in “The Geometrical Theory of the Determination of Prices,” Am. Ac. of Pol. and Social Sciences Proceedings, Vol. 63 (1892–93).

²⁸ Elements, p. 293.

²⁹ Walras thought that the erroneous proposition quoted in our text “followed quite explicitly from the laws of variation of the prices. of services [already] established” (Elements, p. 291). If we turn to Lesson 22 of the 3rd ed. for such “laws of variation”, we find: “If, all other things being equal, the quantity of one of these [productive] services in the hands of one or more holders increases or decreases, the price of this … service will decrease or increase (Elements, 3^rd ed., par. 223 our italics).

³⁰ Up to the third edition of the Elements the tâtonnements admitted production of both consumers goods and capital goods at prices and quantities other than the equilibrium ones. This should not have caused any confusion, since any effects of such capital outputs on relative capital stocks in the course of the tâtonnements was explicitly ruled out by the assumption that “at each successive stage of the tâtonnements the entrepreneurs will find in the economy landowners, workers and capitalists possessing the same quantities of services …” (Elements, 3rd ed., par. 247). An ambiguous passage occurs, however, a few lines later where Walras writes that, once the equilibrium quantities and prices have been reached, “production … can continue but with the changes entailed by the existence of new capital goods.” It may also be significant that in the fourth edition, where the newly introduced device of “tickets” allows no production or exchange to take place during the tâtonnements the above ambiguous passage is substituted by the following: “Although the economy is becoming progressive, it remains static because of the fact that the new capital goods play no part in the economy until later in a period subsequent to the one under consideration” (Elements, p. 283).

³¹ Elements, p. 292.

³² Elements, p. 293, where Walras allows that a decrease of the output of K′ would lower the costs of production of K, whose output is assumed constant.

³³ Elememts, pp. 292–93.

³⁴ Elements, p. 308.

³⁵ See the Appendix to this paper.

³⁶ Elements, p. 294, our italics.

³⁷ A thing should be noted. The capitalization of the prices of services on the left member of equations (B) involves an amortization (compare above, p. 369 n. 9). For the existing capital goods whose price falls below what corresponds to their costs of production, that amortization would be insufficient for their replacement in kind.

³⁸ A. Wald had avoided the difficulty by assuming that the demand price of a consumer good becomes infinitely great as the rate of output tends to zero (see condition (5) in “On Some Systems of Equations of Mathematical Economics,” Econometrica, 1951, p. 373–74). In the later demonstrations of the existence of solutions for the equations of production and exchange, that unwarranted assumption was eliminated, and the possibility of demand prices below costs of production was allowed for consumers’ goods, provided their outputs be zero (see the references given above, p. 367 n. 1).

A second modification of the equations of production and exchange is required to ensure solution. As Walras had noticed (Elements, p. 250) the possibility exists that the supply of some productive services should exceed demand even at zero prices. This possibility can be taken care of by substituting the sign ≤ for the sign = in equations (3), with the additional condition that if the sign > holds, the price of that service be zero (compare Dorfman, Samuelson, Solow, Linear Programming and Economic Analysis, p. 360, and the literature there quoted). This only entails that some of the agents of production appearing in the system are free.

³⁹ It is the peculiarity of consumers’ goods as such that stocks of them need not exist in the system. If, on the other hand, stocks of consumers’ goods are introduced, difficulties arise for a notion of static equilibrium, whether the stocks be of the kinds of consumers’ goods whose output is zero, or not.

⁴⁰ It might be argued that Walras's original notion too is one of temporary equilibrium, since it allows for net accumulation. However, if we could assume the “correct” proportions among the capital stocks, it could be granted that the growth of the several stocks during the time required for tendencies to equilibrium to assert themselves, would be sufficiently small, compared with existing stocks, to be safely ignored. To abstract from it would then be no less legitimate than to abstract from changes in population when taking as given the supply of labor. When, instead, the given physical composition of capital is not the “correct” one, the concentration of gross accumulation on few stocks results in positive and negative changes in them which can no longer be ignored as small. Further, the transfers of fixed capitals between firms and lines of production, necessary to achieve equilibrium in production, are then found to involve, by themselves, changes of the stocks, as will be indicated in the text.

⁴¹ The condition of a uniform price for each kind of service includes the condition of equality between costs and prices of products. In Walras's system, entrepreneurs supply a particular kind of productive service.

⁴² As for circulating capitals, it is obvious that these stocks, taken by Walras as data of his system (compare above, p. 376, n. 25), cannot be assumed constant during the adjustments to equilibrium in production and exchange; in step with such adjustments the stocks would tend to change so as to be available in the “correct” quantities.

⁴³ If that dynamic analysis were defended on the ground that the errors would cancel out in the end, since the system would tend, with time, to the same “stationary” state (a state with zero net savings) whatever the particular path of accumulation, then the claims to fruitfulness of that analysis could be no more than those of a static theory of value and distribution limited, by its choice of data, to the analysis and comparison of “stationary” states. In a theory of “stationary” states the difficulties raised by the notion of given capital supplies would of course be eliminated, since capital supplies, in whatever form, could no longer be part of the data. But with it, so much else is eliminated that little of interest would remain.

We might note here a misapprehension for which it is sometimes implied that the theories of Böhm Bawerk and Wicksell were intended for “stationary” situations as here defined. (Compare J. R. Hicks, Value and Capital, London, 1946, pp. 117–19.) In fact their theories were no more intended for stationary states than Walras's theory, as is clearly shown by their use of a “quantity” of capital (Bőhm Bawerk's subsistence fund) as one of the data. The implicit justification of their static hypothesis appears to have been not the stationary state, but rather the consideration that, at a general theoretical level, the effects of accumulation could be safely ignored, as sufficiently small, both with regard to changes of capital quantity during the time necessary for the tendency to equilibrium to assert itself and with regard to engendered expectations of future changes. Nor, if the notion of a “subsistence fund” held, would that justification of a static theory appear any weaker than the analogous ones relating to changes in population, or consumers’ tastes, or technical knowledge. The “subsistence fund” was not, in fact, a consistent conception of capital. But then, it seems, the difficulty with regard to capital lay in that notion, and not in the static character of the theories.

⁴⁴ For a concise demonstration of the impossibility of a consistent measurement of a “quantity” of capital and average period of production, compare the recent book by P. Sraffa, Production of Commodities by Means of Commodities, Cambridge, 1960, p. 38.

⁴⁵ The equations of general equilibrium including capital formation are formulated in the note to par. 135 (Cours, Vol. I, Paris, 1896, pp. 59–61, n. 1).

⁴⁶ Replacement items appear in the demands of services arising from production (compare Pareto's equations (6), Cours, p. 60, n. 1).

⁴⁷ Cours., pp. 60–61, n. 1.

⁴⁸ Manuel, Paris, 1909, pp. 610–13. No appreciable difference can be noted in this respect between the earlier Italian edition (Milano, 1906) and the revised French edition.

⁴⁹ Manuel, p. 361, our Italics.