Friday, June 20, 2014

A Sophisticated Neoclassical Response To Cambridge Capital Controversies

A Reproduction Scheme (Based on Biddard 1990)
1.0 Introduction

One way of reading the Cambridge Capital Controversy (CCC) is an internal exploration of and debate about neoclassical price theory1. Both sides agreed to concentrate on the case of perfect competition, with no principal agent problems, no asymmetric information, etc. The Cambridge-Italian critics thought themselves to have demonstrated that neoclassical economists could not consistently with their theory claim that equilibrium prices were indices of relative scarcity. Such a claim is not well-founded in the theory, and economists should turn away from biotechnological determinism and turn toward developing price theories in which class power matters.

In this post, I want to outline the sophisticated neoclassical response, in the 1970s, to the Cambridge-Italian critics.

2.0 General Equilibrium Models of Intertemporal and Temporary Equilibrium

This neoclassical response asserted that price theory was best expressed in terms of General Equilibrium Theory (GET). Capital theory involves production over time. Models of intertemporal and temporary equilibrium have been developed in GET. And these models, it is claimed, are both logically consistent and unaffected by Cambridge-Italian criticism2.

2.1 The Arrow-Debreu Model of Intertemporal Equilibrium

The Arrow-Debreu model is a model of disaggregated individuals interacting solely through a single centralized market in existence at the beginning of time. Commodities are distinguished by their physical characteristics, where they become available, when they become available, and the state of the world in which they become available. The givens in the Arrow-Debreu model consist of, roughly:

  • Tastes: Each agent can choose the more preferred consumption plan, when presented with any pair of such plans. A consumption plan specifies what commodities the agent consumes at each point in space and time in each possible state. It also specifies what inputs that the agent controls are supplied as inputs into the firms at each point in space and time and for each state.
  • Technology: What commodities can be produced for supply for consumption from each possible list of inputs is known by everybody.
  • Endowments: A list of commodities available at the beginning of time is given. The specification of endowments includes who owns what. Likewise, the ownership (shares) of the given finite number of firms is known.

Equilibrium is achieved in some sort of no-time before the beginning of time. The centralized market opens, and the auctioneer informs all agents of the prices of all commodities. These commodities include, for example, a contract to buy an umbrella in New York City on 20 June 2015, given it is raining. The agents inform the auctioneer of which contracts they are willing to enter and offer at the given prices. The auctioneer adjusts prices, depending on mismatches between offers and demands, until all markets clear. No transactions are allowed to take place until equilibrium is achieved3.

In an equilibrium, the plans of all agents are pre-reconciled. For any commodity with a positive (spot or forward) price, the quantity supplied equals the quantity demanded. Some goods are not commodities in equilibrium; they have a price of zero. The supply of these goods exceeds the demand.

Once equilibrium has been established, all agents make their promised trades and carry out their plans throughout time.

2.2 The GET Model of Temporary Equilibrium

The idea that all plans are only made in one big market transaction at one instant of time is hard to swallow. This constraint is relaxed in models of temporary equilibrium, as developed by, for example, J. R. Hicks (1946).

In Hicks' model, a market opens at the start of each successive week. If I recall correctly, with a single exception, all markets are spot. That is, supplies and demands are contracted each Monday to be delivered or taken during the following week, but not for later weeks. If an agent plans, for example, to hire labor two weeks hence, they must agree on the wage on the Monday at the start of that week. No market exists in which a worker can be hired for a period of more than one week.

In this model, the plans of agents only need to be consistent in equilibrium for a single week. Supply and demand of both factors of production and commodities for consumption match in the spot market. But these supplies and demands may be based on plans that entail inconsistencies in future weeks. When agents see spot markets failing to clear, say, next week, they revise their plans. And, once again, these revisions and the haggling in the market are assumed to bring about instantaneous market clearing.

The single exception to the requirement that all markets be spot is a market for something like a bond or an annuity. Forward markets exist for all time periods in which one can offer to pay a unit of the numeraire commodity at the start of this week for delivery of a given quantity of the numeraire quantity at the start of, say, the next week. So a whole complex of interest rates exist for the numeraire. An individual's expectations include expectations of movements of future spot prices of all commodities. That is, each individual has expectations for not only future movements in, say, one week interest rates for the numeraire commodity but beliefs about own-rates of interest for all commodities. Nothing in the model brings about consistency among agents of these expectations and beliefs about the future.

This model continues to impose the requirement that no false trading occur. When the markets open on Monday, no transactions can take place until a market-clearing set of prices is found. Only after equilibrium has been achieved do commodities change hands. And production proceeds in each week during the period in which markets are closed.

These models impose very few restrictions on equilibrium, even with specific assumptions on expectations. Perhaps models of temporary equilibrium are better, within mainstream economics, than the Arrow-Debreu model of exploring the formation of expectations.

3.0 Acceptance of Cambridge-Italian Criticisms

These models of general equilibrium may be internally consistent. Both sides of the CCC, however, came to recognize they did not support the beliefs about causal properties still relied on to this day in mainstream applied theory. The faulty and unfounded idea is something like this: compare two equilibria, in which the exogenous (given) data is identical, except the quantity of some given endowment varies across the two cases. Then, if one abstracts from violations of the assumptions of pure competition and is ignorant of price theory, one might expect the price of that endowment to be higher in the case where it is more scarce. Similarly, such an ignoramus would expect the price of commodities produced more intensively with the more scarce endowment to be higher. Despite the short run nature of the models outlined above, such beliefs are unfounded in GET. Here is Christopher Bliss forthrightly acknowledging such:

"Even people who have made no study of economic theory are familiar with the idea that when something is more plentiful its price will be lower, and introductory courses on economic theory reinforce this common presumption with various examples. However, there is no support from the theory of general equilibrium for the proposition that an input to production will be cheaper in an economy where more of it is available." -- Christopher Bliss (1975).
3.1 The Meaning of Quantities, Prices, and Commodities in GET

One issue with GET is remaining clear on the meaning of prices, commodities, and endowments. As with Bliss, I have stated the claim about endowments, prices, and scarcity indices in a timeless, static equilibrium. However, the two disaggregated models outlined above are set in logical time. For example, consider the endowment of a natural resource, such as oil. Only the endowment at the beginning of time is taken as data; the quantities of oil available at the start of the second, third, etc., weeks are different commodities. And these quantities are found by solving the model; they are not taken as given. Likewise, a grade of gasoline produced from oil is a different commodity, depending on which week in which it is produced. Christian Bidard is clear about this distinction between commodities that may be physically identical, but available at different times:

"Intertemporal general equilibrium prices associated with a finite or infinite path of consumption and accumulation have no general properties: the reason being that goods called corn at date t, iron at date t, corn at date t + 1, iron at date t + 1 are formally considered as four distinct commodities of an atemporal economy." -- Christian Bidard (1990).

You might find in GET a statement about the price of one commodity and the quantity of one commodity that goes into the production of that commodity at some more-or-less distant time beforehand. But this is not a statement about the whole vector of prices of physically identical commodities distinguished by the time of their availability. In general, GET theory does not provide simple intuitive propositions about prices and quantities of multiple commodities.

3.2 Steady States

The possibility of a third model might be thought to provide a work-around. Consider the limit, as time increases without bound in the models, of relative quantities and relative prices all referring to that single point in time at infinity. This is a model of a steady-state4. Does this background help explain the following from Frank Hahn?:

"It is possible that the outputs produced in an Arrow-Debreu economy in the far distant future are independent of its initial endowments. That would mean that in such an economy the relative scarcities prevailing now would have no influence on the relative prices and rentals in the distant future. This should be enough to persuade the critics that the theory is not committed to a relative scarcity theory of distribution, though they seem to believe it is and that often motivates them in their attacks." -- Frank Hahn (1981).

Christopher Bidard also considers such a limiting process:

"The subscripts refer to the date, the horizontal arrow indicates production by means of inputs and labor and the data in italics are neoclassical theory calls the 'endowments' of the economy. The interesting feature of this scheme is that all inputs at, except for the very initial ones a0, are obtained as the result of previous production. If we admit that the influence of the primitive inputs a0 vanishes in the long run, we have a pure reproduction process (a mechanical endogenization of labour, identified with a given wage basket, would be useful for a comparison with von Neumann's theory, but the operation is not necessary here)." -- Christian Bidard (1990).

In a simple model of a steady state, one might as well drop time indices off state variables for relative quantities and relative prices. They are invariant over time in such a model. But even so, one cannot apply theorems of static equilibrium to such a model. The logic of steady states is not one of allocating scarce resources. Inputs into production, insofar as they are produced, are not exogenous givens. Rather, they are found as part of the model solution. And the kind of relationships that Bliss says above are without foundation in GET are equally unfounded in models of steady-states.

4.0 Does a Logically Consistent Vulgar Neoclassical Theory Exist?

I consider the beliefs that economics is solely about the allocation of scarce resources and that equilibrium prices are indices of relative scarcity to be vulgar neoclassical doctrines. Beliefs about properties of equilibrium, if you are interested in mathematically formalized Neoclassical models, should be logical conclusions derived from assumptions. And those assumptions should be on the primitives of the model. For example, one might have some assumptions about the tastes, production functions, or patterns of initial endowments.

Here Edwin Burmeister states that no such vulgar neoclassical theory is known to exist, albeit in the context of the analysis of an aggregate market for capital:

"Imposing some set of conditions on the technology ... should be sufficient to assure that real Wicksell effects are always negative. Such conditions would be of interest - especially if they could be empirically tested - since they would validate the qualitative conclusions derived from the one-good models often used in macroeconomics without any theoretical justification for ignoring aggregation problems. Moreover, Burmeister ... has proved that a negative real Wicksell effect is a necessary and sufficient condition for the existence of an index of capital ..., and a neoclassical aggregate production function defined across steady state equilibria such that (i) [consumption per head is a function of the index of capital per head], (ii) [the equilibrium interest rate is equal to the marginal product of the index of capital], and (iii) [This index of capital exhibits declining marginal returns]. Unfortunately, no set of such sufficient conditions is known, but the literature on capital aggregation suggests that they would impose severe restrictions on the technology." -- Edwin Burmeister (1987).

Economists who hold fast to vulgar neoclassical economics may present formal models. But often their mathematics is imprecise, disguising muddle and confusion. It is not a matter of having unrealistic assumpions; it is a matter of having assumptions that do not imply one's conclusions.

5.0 Conclusion

As far as I can tell, most economists do not learn the literature of their subject. Not only do most economists stay ignorant of the view of the English side of the CCC. They also remain ignorant of the most sophisticated neoclassical response. Thus, they ask to see irrelevant empirical evidence5 for the existence of reswitching, and do not acknowledge their applied stories6 are without foundation in rigorous neoclassical price theory.

  1. For the purposes of this post, I bracket out the Sraffian reinterpretation of classical and Marxist economics, the Sraffian claim to have reconstructed a viable alternative price theory, and arguments over the compatibility of this theory with the economics of Keynes.
  2. It is not clear that these models cannot be attacked by Sraffians. Do they contain or do they need to contain a market for income in general at each point of time, as in Walras's work?
  3. Why this haggling over prices would ever approach equilibrium is unclear in theory.
  4. The Turnpike Theorem asserts that intertemporal equilibrium paths starting from appropriately selected initial conditions will spend most of their time around a steady state, even if such a steady state is not the final destination of such a path at a final given final time at which the model ends. On the other hand, the Sonnenschein-Debreu-Mantel theorem suggests any dynamics is possible. Thus, intertemporal paths may have no tendency to approach such steady states, even if they have a local saddle-point stability.
  5. I am not sure that all those who ask for empirical evidence of reswitching are clear what they are asking, even though I often use "rewitching" as a synecdoche myself. Anyways empirical evidence exists for Sraffa effects. If Sraffa effects were empirically unlikely, one would be faced with the (unmet) theoretical challenge of outlining a theory to explain this supposed unlikeliness.
  6. For example, on the supposed decreasing employment effects, under perfect competition, of a minimum wage above the (what is that) equilibrium real wage.
  • Christian Bidard (1990). From Arrow-Debreu to Sraffa, Political Economy: Studies in the Surplus Approach, V. 6: pp. 125-138.
  • Christopher J. Bliss (1975). Capital Theory and the Distribution of Income, Amsterdam: North Holland Press.
  • Edwin Burmeister (1980). Capital Theory and Dynamics, Cambridge: Cambridge University Press.
  • Edwin Burmeister (1987). Wicksell Effects, The New Palgrave: A Dictionary of Economics (ed. by J. Eatwell, M. Milgate, and P. Newman).
  • Frank Hahn (1981). General Equilibrium Theory, in The Crisis in Economic Theory (ed. by. D. Bell and I. Kristol), Basic Books.
  • Frank Hahn (1982). The Neo-Ricardians, Cambridge Journal of Economics, V. 6: pp. 353-374.
  • J. R. Hicks (1946). Value and Capital: An Inquiry into Some Fundamental Principles of Economic Theory, 2nd edition, Oxford: Oxford University Press.
  • J. M. Grandmont (1977). Temporary General Equilibrium Theory, Econometrica, V. 45, N. 3: pp. 535-572.
  • Fabio Petri (2004). General Equilibrium, Capital and Macroeconomics: A Key to Recent Controversies in Equilibrium Theory, Edward Elgar.

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