Market For Numéraire Capital In General Equilibrium Models

"It is sometimes suggested that, because in the Walrasian model the existence of heterogeneous capital goods is recognized, Walras's analysis must be free of the error of employing the concept of quantities of 'capital in general'; but that is not so. In Walras's theory, the rate of interest is determined as the price which equates the quantity demanded and the quantity supplied of 'commodity E'. Commodity E represents the value of a certain set of producers' goods; ownership of commodity E is a title to the return on these items. Saving (demand for E), a function of the rate of interest, is represented as the purchase by rentiers of a quantity of commodity E (entitlement to the return on real assets of corresponding value); commodity E is issued by firms (supply of E) wishing to raise funds for the purchase of capital goods to the value of E issued; the quantity of E issued is also a function of the rate of interest. This is merely a version of the neoclassical productivity and thrift theory of interest." -- Roy H. Grieve, (2012). The Marginal Productivity Theory of the Price of Capital: An Historical Perspective on the Origins of the Codswallop, Real-World Economics Review, Issue no. 60.

I liked the recent article from which the above quote is taken. In this post, I make a few observations on the market for "capital in general". Grieve refers to, most specifically, the following passage:

"242. In order to efect the introduction of this new element rationally, we need only imagine a commodity (E) consisting of perpetual net income of which both the price p_e = 1/i and quantity demanded d_e are expressed in units of numéraire. i is the rate of perpetual net income. If the net income were not perpetual, its price would be p_e' < 1/i, which would [still] be a function of i.

A fairly exact counterpart of the ideal commodity (E) is to found in the perpetual net income whose variable rate i, once it has been determined for a certain period of time, serves as a basis for the computation of life insurance rates. Insurance companies are intermediaries between those who create savings, positive or negative, and the market for capital goods. Thus insurance companies require net income in order to pay death benefits and endowments to some; while they supply net income by paying annuities to others. If, all things considered, their reserves increase, the country is producing new capital goods; in the contrary case it is consuming existing capital goods.

In speaking here of the price of (E), I am simply reviving the old concept of number of years' purchase (twenty years' purchase, twenty-five years' purchase), which is the reciprocal of the more recent concept of rate (5 per cent = 1/20, 4 per cent = 1/25). I find it helpful to use these two concepts concurrently in developing a scientific theory of capital formation. Now, in light of these definitions, we may regard every member of an exchange economy as having, over a given period of time, a certain want for (E), that can be expressed by a function r = φ_e(q) which decreases as q increases, and as possessing a certain quantity of (E),

q_e = q_tp_t + ... + q_pπ_p + ... + q_kπ_k + q_k'π_k' + q_k''π_k'' + ...

which, within certain limits can be increased by demand, or decreased by offer, so that

φ_e(q_e + d_e) = p_eφ_a(d_a)

is the condition of maximum satisfaction... This condition combined with the equation of exchange

o_tp_t + ... + o_pp_p + ... + o_kp_k + o_k'p_k' + o_k''p_k'' + ...

= d_a + d_bp_b + d_cp_c + d_dp_d ... + d_ep_e

and with the other equations of maximum satisfaction, gives us the following [individual] demand for net income...:

d_e = f_e(p_t,...,p_p,...,p_k,p_k',p_k'',...,p_b, p_c, p_d,...,p_e)

The sum total of all individual demands for net income will be

D_e = F_e(p_t,...,p_p,...,p_k,p_k',p_k'',...,p_b, p_c, p_d,...,p_e)

This sum, D_e, is positive and equal to E_d for p_e = 0; and then it decreases as p_e increases while all other prices of services and products are assumed to be determined and constant, until it falls to zero for p_e = E_p; after which [as p_e increases still further], it becomes negative, first increasing and then decreasing (in absolute value) until it returns to zero again at p_e = ∞. Moreover, the algebraic sum of the individual excess of income over consumption will be

E = D_ep_e

= F_e(p_t,...,p_p,...,p_k,p_k',p_k'',...,p_b, p_c, p_d,...,p_e)p_e

= F_e(p_t,...,p_p,...,p_k,p_k',p_k'',...,p_b, p_c, p_d,...,i)

which is subtracted from income and added to capital ['au fonds'], thus constituting positive savings. As 1/i increases from zero to E_p, or alternatively as i decreases from ∞ to 1/E_p, E first increases from zero and decreases to zero again. Since we have chosen to put the offers of services on the left-hand side of the equation of equation when they are considered as positive quantities, and to put the demand for products on the right-hand side when they too are considered as positive quantities, we shall add the demand for new capital goods to the latter items and always assume it to be positive. In making this assumption we are restricting ourselves to a study of the production of new capital goods in a progressive society; and we are neglecting the study of consumption of existing capital goods in a retrogressive society.

If we let D_k, D_k', D_k'', ... designate the respective quantities of new capital goods (K), (K'), (K'') ... produced, we have the equation

D_kP_k + D_k'P_k' + D_k''P_k'' + ... = E."

-- Léon Walras, Element of Pure Economics or the Theory of Social Wealth (Translated by William Jaffé), Lesson 23, Section 242.

A lot of Walras is like these "thickets of algebra". (See the appendix below for some guidance to the notation taken from previous sections of the book.) Walras' book contains many concepts not explored by neoclassical economists until much later. For example, he is fairly explicit about the idea of human capital earlier in this chapter.

This section is part of a model of static equilibrium, not a steady state. Capital accumulation is going on at the instant in time for which the equilibrium is defined, but capital goods are not necessarily being produced in proportions to allow this equilibrium to be reproduced, either on the current or an expanded scale. The given data for this model include the initial quantities and distribution of land, population, and capital goods. At this point in his exposition of a succession of models, Walras has not yet introduced money into his exposition. Furthermore, he takes coefficients of production as givens. In later models, he relaxes both these assumptions. (For Walras, the values that variable coefficients take on is to be explained by the theory of marginal productivity.)

But what does it matter that Walras has a model in which a market for capital in general is included? For his model is logically overdetermined and therefore inconsistent, in general. The problem arises from given initial quantities of capital goods, the possibility that these capital goods must be themselves among the produced commodities, and an equilibrium condition that the same rate of profits must be earned for all commodities being produced.

A more apposite question might be how can a market for capital in general be included in the Arrow-Debreu model of intertemporal equilbria or in the Hicksian model of temporary equilibrium. I think the aggregation needed to specify the numéraire quantities of savings and investment at each time does not yield new independent equations in these models. On the other hand, agents must be willing to own all capital goods in equilibrium. And, some have claimed, that looking at savings and investment can tell us something about (in)stability in such models, either of tâtonnement or of equilibrium paths. Is this currently a live topic of debate?

But, in my opinion, none of these neoclassical models of General Equilibrium can possibly describe actually existing capitalist economies. They cannot possibly describe processes set in historical time, where the plans of separate agents do not exhibit a tendency to become coordinated. I find weird some of the defenses I have seen of General Equilibrium against Sraffians. Some have claimed that capital-reversing is neither necessary nor sufficient to demonstrate instability. One proves this by showing examples of instability without capital-reversing and of stability with capital-reversing. But, since the analysis demonstrates that instability can arise in general, no reason has been given to think actual capitalist economies will exhibit tendencies towards a General Equilibrium. Why study states in a model that will never be realized?

Appendix

• φ_e(q) is the marginal utility ("rareté") that a given individual obtains from the quantity q of perpetual net income.
• (A), (B), (C), ... denote different produced commodities.
• d_a denotes the quantity demanded by a given individual of the commodity (A).
• p_b denotes the (relative) price of the commodity (B) in terms of the numéraire (A).
• (T), (T'), (T''), ... denote different qualities of land.
• q_t denotes the (yearly) services provided by the (given) acres of land of the tth quality that a given individual owns.
• o_t denotes the (yearly) services offered by a given individual on the market for the acres of land of the tth quality that the individual owns.
• p_t denotes the rental price of an acre of the tth quality of land.
• (P), (P'), (P''), ... denote different qualities of labor.
• q_p denotes the (yearly) labor available of the pth quality from a given individual.
• o_p denotes the labor offered on the market by a given individual for the pth quality of labor.
• p_p denotes the (gross) wage for a person-year for the pth quality of labor.
• π_p denotes the wage for a person-year for the pth quality of labor, net of, for example, education expenses needed to maintain the human capital.
• (K), (K'), (K''), ... denote the (yearly) services of different kinds of capital goods.
• q_k denotes the (yearly) services available for the kth kind of capital goods owned by a given individual.
• o_k denotes the (yearly) services offered by a given individual on the market for the kth kind of capital good.
• p_k denotes the (gross) rental price for the services of the kth kind of capital good.
• π_k denotes the rental price for the services of the kth kind of capital good, net of depreciation and insurance charges.
• P_k denotes the price for a (physical) unit (not its services) of the kth kind of capital good.