Walras's Progressive Theory of Capital

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Léon Walras, in Part 5 of his Elements of Pure Economics (1874) presented a theory of capital and interest as the next-to-last step towards a complete general equilibrium model. Walras's theory of capital was among the most innovative in his time and rather confusing. The following is a simplistic elucidation of Walras's theory, and is far from controversial. A more careful and deeper analysis is contained in Jaff・(1942), van Daal and Jolink (1993), Kompas (1992) and Walker (1996). See also our discussion of the Walras-Cassel system.

Walras divided between the stock of capital and the flows of "services" of capital (i.e. the income flows arising from that stock). Capital, he argued, was not "used up" in the production process, but rather provided "services" and it was these services that entered the production process. Indeed, in this way, Walras considered labor and land to be capital as well because they provided "services" to production and were themselves (usually) not used up. It was only in later editions of the Elements that he considered the issue of "circulating" capital explicitly.

In Walras's system, then, given a capital stock, we obtain a given flow amount of capital services. This can be considered the "supply" of capital services. The "demand" for capital services is merely the consumers' demand for an amorphous commodity he called "savings in general" which arise from consumers' desires to transfer some amount of current purchasing power to demand future goods and services. Capital services, then, are the mechanism which makes this transfer possible. The price of capital services is determined by their demand (i.e. savings) and supply (i.e. existing capital services)

However, a few problems soon emerge. Firstly, capital goods have the unique feature that, unlike labor and land, they can themselves be "produced" and thus are not "endowed". As the amount of capital services available depends on the amount of capital stock, then by increasing capital stock, we can increase capital services artificially - thus, there is really no "endowment" of capital services either.

Furthermore, one needs to be able to "price" capital stock. As a produced commodity, it comes under the rule of cost-of-production pricing: "[c]apital goods proper are artificial capital goods: they are products and their prices are subject to the law of cost of production" (Walras, 1874: p.271). However, at the same time, Walras proposed that the price of capital stock is equated to the capitalized net income of capital services, which in turn depends on the price of capital services. This lands Walras's analysis in something of a knot for the price of capital seems to come now under two laws of pricing: by supply and demand (as other factors of production); by cost of production (as other produced goods). The link between these two was never really satisfactorily answered by Walras, even with the introduction of "time" (which enables us to freeze the supply of capital stock temporarily) or the introduction of circulating capital and cost-of-production inequalities in later work.

Walras's theory of capital was developed for a "progressive" or growing economy. To get there, it might be useful to first start from a static state and formulate the general equilibrium equations for his capital model, and then consider the modifications that have to be made to incorporate the progressive case.

Let us assume two produced commodities (consumer goods and capital goods) and two factors (capital and labor). Let X_c be the output of consumer goods and X_k be the output of capital goods. Both use capital and labor, let K_c and L_c be the amounts of capital and labor used in the production of X_c amount of consumption goods and let K_k and L_k be the amounts of capital and labor used in the production of X_k amount of labor goods. We can consider this in two general production functions X_c = F_c(K_c, L_c) and X_k = F_k(K_k, L_k), but we shall give it a precise form, as Walras (1874) did, by assuming fixed unit input coefficients. Define a_LC = L_c/X_c, a_KC = K_c/X_c and a_LK = L_k/X_k, a_KK = K_k/X_k as the unit input coefficients (for mnemonic purposes, read the subscript in the order "input-output" so that the coefficient a_ij denotes the amount of factor i needed to produce a single unit of output j). Thus, the production conditions become:

X_c = L_c + K_c = a_LCX_c + a_KCX_c

X_k = L_k + K_k = a_LKX_k + a_KKX_k

as the production requirements for consumer and capital goods respectively. Thus:

L_c + L_k = a_LCX_c + a_LKX_k

K_c + K_k = a_KCX_c + a_KKX_k

are the total demands for labor and capital respectively. Note that factor demand can be imputed via technology from the amount of outputs produced (X_c, X_k). Now, if we assume a fixed supply of labor and capital, L and K (thus a static economy) then it must be that:

L_c + L_k ｣ L

K_c + K_k ｣ K

i.e. labor and capital demanded must not exceed labor or capital supplied. If we were to assume market clearing in the factor markets, then these inequalities would become equalities, so that:

L_c + L_k = L

K_c + K_k = K

or, expressing this in terms of unit-input coefficients:

a_LCX_c + a_LKX_k = L

a_KCX_c + a_KKX_k = K

If we were to let x｢ = [X_c X_k]｢ and v｢ = [L K]｢ and B be a 2 ｴ 2 matrix:

a_LC a_LK
B =
a_KC a_KK

then we can rewrite the entire system as Bx = v. The implied generalization to n goods and m factors can be immediately noted by letting B be an (m ｴ n) matrix, x be a (n ｴ 1) vector and v a (m ｴ 1) vector. However, for our purposes, let us continue with the simple two-sector, two-factor model.

Let us now assume the Walrasian type of perfect competition where entrepreneurs "break even" in every commodity, i.e. no net profits, so total revenue must equal total costs. Total revenues are merely the price of the commodity multiplied by the amount produced, whereas total costs are the wages and rental rates paid to the labor and capital employed in the production of the goods, i.e.

p_cX_c = wL_c + rK_c

p_kX_k = wL_k + rK_k

where (p_c, p_k) are the prices of consumer and capital goods respectively while (w, r) are the wage and rental rates per unit of labor and capital respectively. Substituting for L_c, K_c, L_k and K_k, we obtain:

p_cX_c = wa_LCX_c + ra_KCX_c

p_kX_k = wa_LKX_k + ra_KKX_k

or simply, dividing through by outputs X_c and X_k respectively:

p_c = wa_LC + ra_KC

p_k = wa_LK + ra_KK

so, letting p｢ = [p_c p_k]｢ , w｢ = [w r]｢ and letting A be a 2 x 2 matrix:

a_LC a_KC
A =
a_LK a_KK

we can rewrite the entire system as p = Aw. Thus, again generalizing to n goods and m factors, we obtain a (n ｴ 1) vector p, a (n ｴ m) matrix A and a (m ｴ 1) vector w. Notice that the transpose of matrix A is merely the matrix B we derived earlier, A｢ = B. Thus, we could write our earlier equation as v = A｢ x.

We have yet to tackle the issue of the demand for consumer goods and capital goods. For the sake of argument, let D_c be the demand for consumer goods and D_k be the demand for capital goods. Then, in market clearing:

D_c = X_c

D_k = X_k

However, note that we already have a problem: namely, who demands capital goods? In other models, we can obtain the demand for capital as factors by summing input requirements, K_c + K_k. Is there any relationship between D_k and (K_c + K_k)? This is where the difficulty comes in. If we supposed X_k was a consumer good, then there would be no problem as D_k would, like D_c, be determined by consumer demands and would have nothing to do with factor demands. However, as capital is a produced good, then there ought to be a relationship between the demand for capital output and the demand for capital as a factor.

Walras's resolution was to divide the two: D_k is the demand for new capital goods, whereas K_c + K_k is the demand for capital services on existing capital goods. Hence, the output of new capital goods, X_k, does not feed into production contemporaneously as it only yields capital services in the future. Thus, we see immediately the intimate interrelationship between time and capital in Walras's theory.

But the question remains hanging: what determines the demand for new capital goods, D_k? Walras proposed that this is merely savings. Thus, savings by households translates into a demand for new capital goods. In this simple two-sector world, then, households either consume or save - or, equivalently, demand consumer goods D_c and new capital goods, D_k. Thus, in aggregate budget constraint faced by households implies that:

p_cD_c + p_kD_k ｣ Y

where total value of demand for consumer and new capital goods on the left must be less than or equal to total income Y. This is merely total factor income when factor markets clear, i.e. Y = rK + wL, or:

Y = w(a_LCX_c + a_LKX_k) + r(a_KCX_c + a_KKX_k)

then the Walras's Law constraint can be rewritten:

p_cD_c + p_kD_k = w(a_LCX_c + a_LKX_k) + r(a_KCX_c + a_KKX_k)

As this arises from the households' decisions, we must still ask what are determinants of D_c and D_k. Essentially, preferences (utility functions), commodity prices (p_c, p_k), factor returns (w, r) and endowments of factors (in total equal to L and K, which we can reduce to Y) will determine D_c and D_k. If we aggregated household demands, we would obtain aggregate demand functions of the form:

D_c = D_c(p_c, p_k, w, r, Y)

D_k = D_k(p_c, p_k, w, r, Y)

for consumption goods and capital goods accordingly. However, households obtain no utility from capital goods as they cannot be consumed. Thus, D_k, rightly, are seen as "savings" and so can be considered as demanded in order to be able to claim future output.

Walras did not spend much time discussing the desire or demand for future goods, preferring to just subsume it all under "savings". As a result, a few more words on this matter are warranted. We have used r to denote the return to capital services. Walras defined "net income" to a unit of a capital good to be r - (m + v)p_k where m is the depreciation rate and v is the insurance rate and p_k, of course, is the price of the capital good. Ignoring insurance (setting v = 0), then the "net income" of a unit of capital good would be simply r. The ratio of net income to the price of existing capital Walras called the "rate of net income" and denoted it by:

i = (r - m p_k)/p_k = r/p_k - m

Thus, we can express the price for the capital good as:

p_k = r/(i - m )

Now, Walras actually went further and claimed that, in equilibrium, ratio of net income to the price of the capital good would be equal for all capital goods. So, in general, if we have m capital goods, then i = r₁/p₁ = r₂/p₂ = ... = r_m/p_m. Thus, all capital goods, in equilibrium, have the same rate of net income.

Now, Walras allowed for two types of savings: savings "in kind", in which households save entire vectors of capital goods and receive a stream of various net returns, or savings in a numeraire commodity. Let us consider the latter case where agents save "purchasing power" in the form of a homogeneous substance he called "perpetual net income" and denoted E. We can think of it, heuristically, as an asset which pays a unit of the numeraire good every period forever. Thus, the price of a unit of E is p_e = 1/i, i.e. the value of a unit of E is the perpetually discounted unit of account. But i, as we noted earlier, was merely r/p_k. Thus, the households' consumption and savings decisions are reduced to considering merely i instead of r and p_k (or, when we have multiple capital goods, a whole array of r_i and p_i).

As noted earlier, households save and thus purchase E in order to be able to claim future output. In turn, the assets E are sold by firms to households in order to enable them to purchase the new capital goods. Thus, firms sell asset E to households and purchase the newly produced capital goods X_k. Thus, p_kD_k = E/i and the Walras's Law constraint becomes:

p_cD_c + E/i = w(a_LCX_c + a_LKX_k) + r(a_KCX_c + a_KKX_k)

Households, thus, maximize utility subject to a constraint similar to this one (just add the subscript for each individual household). Note that, now, having subsumed r and p_k into i, then the aggregate demand functions become:

D_c = D_c(p_c, w, i, Y)

E = E(p_c, w, i, Y)

where D_c is the demand for consumption goods and E is merely "savings". As noted earlier, Walras, himself, did not explicitly consider future "utility" from future consumption - thus he allows E to represent it.

One last condition is necessary. To tie up our entire model in a simple fashion, let us eliminate time by assuming a stationary state condition so that new capital goods are demanded merely to replace depreciated capital, i.e. D_k = m K. In this manner, we can eliminate the issue of "expectations" of different future capital requirements and thus determine a single level of K instead of a moving path of K. Consequently, let us recapitulate the equations and unknowns in Walras's capital model.

where we thus have twelve types of equations. The types of unknowns are D_c, E, Y, X_c, X_k, p_c, p_k, w, r, i, D_k and K, which are also twelve. By Walras's Law, one of the equations is redundant and, setting a numeraire good, p_c = 1, one of the prices can be eliminated, thus we have a system of eleven types of equations and eleven types of unknowns. It is important to note that K is one of the unknowns, thus we do not assume it is an exogenously endowed factor but a producible means of production, i.e. "capital" properly speaking.

How is this generalized to multiple commodities and multiple factors? Let there be n produced consumption goods, k produced factors (i.e. capital goods) and m non-produced factors (i.e. labor, land, etc.). In this case, the letters on the right hand side above denote the number of equations of each type. Thus, reading from the top, we have n of the D_c equations, one E equation, one Y equation, m non-produced factor market clearing conditions, k produced factor market clearing conditions, n consumption goods market equilibrium conditions, k capital goods market equilibrium conditions, n price-cost equalities for consumption goods, k price-cost equalities for capital goods, one definition of savings E, k net income equations and k stationary state conditions. Thus, we have (3 + 3n + m + 5k) equations.

The generalization is straightforward enough, only the following might merit noticing. Firstly, we only have one equation for the earlier E/i = p_kD_k. This is because, with multiple goods, p_kD_k now becomes the sum of such terms over all capital goods, i.e. E/i = p₁D₁ + p₂D₂ + .... p_kD_k, so that we still only have one definition of E. In contrast, we have k conditions i = r/p_k - m , because, as noted earlier, it is supposed to hold for every capital good and thus every p_k.

The number of unknowns are as follows: we have n consumption demands (n times D_c), one savings (E), one total income (Y), n outputs of consumption goods (n times X_c), k outputs of capital goods (k times X_k), n consumption good prices (p_c), k capital goods prices (p_k), m factor returns on non-producible factors (m times w), k factor returns on producible factors (k times r), one rate of net income (i), k capital demands (k times D_k) and k capital stocks (k times K). Thus the total number of unknowns is (3 + 3n + m + 5k).

Thus, even with multiple goods and factors, the number of equations is equal to the number of unknowns. We can eliminate one of the equations by Walras's Law and one of the unknowns by the numeraire, and thus end up with (2 + 3n + m + 5k) equations and unknowns.

One bizarre notion must be underlined: the stationary state condition. Knut Wicksell (1893) pointed out that the interest rate was indeterminate in Walras's model. However, as Enrico Barone (1895) noted and we can see above clearly, the stationary condition allows for determination. However, by effectively reducing the demand for new capital goods to replacement capital, we have thus reduced the determination of the interest rate to the market for depreciation. But suppose capital did not depreciate. Then Wicksell's critique is legimitate: there is no determinate rate of interest.

To reincorporate the determinateness of interest in an economy without depreciation, we must have a "progressive" or "growing" economy with net savings and thus replace the stationary state condition above with some other, more dynamic, condition. This would require capital demands to be dependent on future demands, etc., which brings in expectations and thus might imply the concept of a temporary sequential equilibrium in a growing economy. This seems pretty much what Walras himself had in mind.

This point is interesting because it brings forth a notion that was quite unique to Walras. After Walras, the Neoclassical theories of capital (e.g. J.B. Clark, 1899) seemed to be able to determine interest rates in static economies. But in Walras, the rate of interest is only determinate in a "progressive" economy. If one thinks of the work of Schumpeter (1911) or von Neumann (1937), where stationary economies are indeed associated with zero interest rates, one can immediately realize their affinity to Walras's notion (see our discussion of steady-state growth and also the Walras-Cassel system.)

Selected References

E. Barone (1895) "Sopra un Libro del Wicksell", Giornale degli Economisti, Vol. 11, p.524-39.

J.v. Daal and A. Jolink (1993) The Equilibrium Economics of Léon Walras. London: Routledge.

W. Jaff・(1942) "Léon Walras' Theory of Capital Accumulation", in O. Lange, F. McIntyre and O. Yntema, editors, Studies in Mathematical Economics and Econometrics. Chicago: University of Chicago Press.

T. Kompas (1992) Studies in the History of Long-Run Equilibrium Theory. Manchester, UK: Manchester University Press.

D.A. Walker (1996) Walras's Market Models. Cambridge, UK: Cambridge University Press.

L. Walras (1874) Elements of Pure Economics: Or the theory of social wealth. 1954 translation of 1926 edition, Homewood, Ill.: Richard Irwin.

K. Wicksell (1893) Value, Capital and Rent. 1970 reprint of 1954 edition, New York: Augustus M. Kelley.

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