Saturday, September 16, 2006

Contradiction In Arrow-Debreu Model Of Intertemporal Equilibrium?

This post looks at just one problem with the Arrow-Debreu model.

The data of the Arrow-Debreu model of intertemporal equilibrium include the initial quantities, that is, the endowments of all commodities. This specification includes the distribution of these endowments, a statement of who owns what. Some of these endowments are unproduced natural resources which yield services over future time periods in the model. Other endowments can be conceptualized as capital goods, as produced means of production.

In the model, markets clear once, at the beginning of time. The commodities traded in markets include promises to deliver specified commodities at specified dates in the future, perhaps contingent on the state of the world. In the full model, one can purchase now a contract to have delivered any specificied commodity at any date in the future. So market-clearing in this model implies that plans are pre-coordinated. No room exists for expectations about the future to be disappointed.

Where do the capital goods in existence at the beginning of time in the Arrow-Debreu model come from? Presumably, they were manufactured in a prior time period not formally modeled. Some entrepreneurs, with plans and expectations held before the start of time, were directing production toward the time in the model. The initial capital goods embody those plans and expectations. But, in the model, the relative quantities of those goods is arbitrary. For example, an endowment can be in excess supply, with an equilibrium price of zero. This arbitrariness seems to imply that those unmodeled past expectations and plans can be mistaken.

So the Arrow-Debreu model describes a world in which agents have made mistakes in the past and some plans made in the past have been toppled. But the plans and the expectations made now will be found correct forever in the future. It is a model in which agents must both have correct expectations and in which they can have mistaken expectations.

You can find in the literature comments on Joan Robinson's puzzlement at being given the Arrow-Debreu of intertemporal equilibrium as the answer to her question about what is the general neoclassical theory of value. She complained that she wanted to have a model that she could make stand up before she knocked it down. And she would ask, if the economy was not in equilibrium yesterday, why would you expect it to be in equilibrium tomorrow?

Suppose one models the economy as starting with an arbitrary set of capital goods, that is, with a short run model. Perhaps part of the data should then be the expectations, some of which are being disappointed, with which those capital goods were manufactured.
  • Gerard Debreu, Theory of Value: An Axiomatic Analysis of Economic Equilibrium, Yale University Press, 1959.

1 comment:

Blissex said...

«The data of the Arrow-Debreu model of intertemporal equilibrium include the initial quantities, that is, the endowments of all commodities. This specification includes the distribution of these endowments, a statement of who owns what. Some of these endowments are unproduced natural resources which yield services over future time periods in the model. Other endowments can be conceptualized as capital goods, as produced means of production. In the model, markets clear once, at the beginning of time.»

The purpose of this is to prove that the distribution of income for factors is uniquely determined by factor productivity, because that is the verity by which Economics gauges the validity of models.

There is no contradiction, because assumptions don't matter, only reaching the correct conclusion does.

Assuming initial endowments is absolutely vital to this purpose of the model.

The reason is that in order to prove the verity it is essential that factor demand must depend on their price instead of their profitability (because that involves the rate of interest).

That factor demand depends on price and not profitability is crucial to other tenets mainstream Economics, for example to prove Say's Law, without which it is phenomenally hard :-) to prove the existence and unicity of equilibrium; because if factor demand depends on profitability we get into Sraffian situations.

Unicity requires another assumption, convexity, which is assured only if demand (and supply) schedules are smoothly sloping downwards wrt to price.