On Walrasian Arrow-Debreu General Equilibrium and Rational Expectation (2): No-trade theorem and RBC/DSGE models
As in the previous story/post, when one says Walrasian analysis, it refers to Arrow-Debreu general equilibrium analysis, or shortening it, general equilibrium analysis. When I talk about DSGE models, I will refer to them separately — because while DSGE models are in some way rooted in general equilibrium anlaysis, they are not entirely rooted in general equilibrium analysis. This post intends to think about this.
It is well-known that if trade is ex-ante Pareto optimal, and human rationality and information structure is common knowledge (people have incomplete information of the world), then even if information changes, there cannot be any trade going on. This is what one calls as No-trade theorem, and for now I will not discuss more results on No-trade theorem. Suppose agents were previously in “ex-ante” general equilibrium. If information then changes, no trade will occur to change the allocation by No-trade theorem. (price vector changes, if the Walrasian market is to re-open after new information is revealed to some agents, however.)
It does seem that No-trade theorem restricts much about what general equilibrium markets. Once Arrow-Debreu trade occurs at time t=0, no trade is required, nor can occur. However, there are ways to get around this while maintaining the reasonable assumptions imposed by No-trade theorem, and of course those said in this post are not all of them.
If some to-be-realized states cannot be learned by agents, and agents cannot even know their existence is possible, then even after a trade already occurs, a new trade may occur. An obvious example is introduction of new products totally unanticipated by everyone.
Also, if agents can believe that they are not in state x even though they are in state x, then No-trade theorem can break down. This one will not be discussed.
Let me return to the question of some states that possibility of existence is not known. Essentially DSGE models re-open Walrasian markets at every period — reflecting the fact that there exists new products (even though models often have homogeneous goods, or representative goods). Since there are infinite periods in many DSGE models, this means that Walrasian markets are opened infinite times, while usual Arrow-Debreu general equilibrium analysis has finite number of goods. Here change in expectations has important consequences for each period, unlike in Arrow-Debreu general equilibrium, which lacks “business cycle.” It is obvious why DSGE models are used for business cycle analysis in this way. Essentially, modern DSGE models, that mostly can be said to be RBC variants, are incomplete market models.
However, one must be reminded of Maskin-Tirole critique (Maskin and Tirole (1999)), which roughly states that parties can write “contracts,” here including state-contigent trade promises, such that they know states even though they do not. Thus the above possibility of going around No-trade does pale somewhat. This, however, does not mean that the above possibility is wrong — rather, it is that necessity of deviations of incomplete market results from complete market results does not necessarily follow from unanticipated states/contingencies. We may still accept incomplete market as a feature of our reality and do analysis, at least in macro terms, and forget about theories of firms that investigate how firms arise.
Addendum: I may discuss Maskin-Tirole critique and some industrial organization/new institutional economics topics in a separate post.