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Chapter 12 Recursive competitive equilibria 12.1. Endogenous aggregate state variable For pure endowment stochastic economies, chapter 8 described two types of competitive equilibria, one in the style of Arrow and Debreu with markets that convene at time 0 and trade a complete set of history-contingent securities | Chapter 12 Recursive competitive equilibria 12.1. Endogenous aggregate state variable For pure endowment stochastic economies chapter 8 described two types of competitive equilibria one in the style of Arrow and Debreu with markets that convene at time 0 and trade a complete set of history-contingent securities another with markets that meet each period and trade a complete set of one-period ahead state-contingent securities called Arrow securities. Though their price systems and trading protocols differ both types of equilibria support identical equilibrium allocations. Chapter 8 described how to transform the Arrow-Debreu price system into one for pricing Arrow securities. The key step in transforming an equilibrium with time- 0 trading into one with sequential trading was to account for how individuals wealth evolve as time passes in a time-0 trading economy. In a time-0 trading economy individuals do not make any other trades than those executed in period 0 but the present value of those portfolios change as time passes and as uncertainty gets resolved. So in period t after some history s we used the Arrow-Debreu prices to compute the value of an individual s purchased claims to current and future goods net of his outstanding liabilities. We could then show that these wealth levels and the associated consumption choices could also be attained in a sequential-trading economy where there are only markets in one-period Arrow securities which reopen in each period. In chapter 8 we also demonstrated how to obtain a recursive formulation of the equilibrium with sequential trading. This required us to assume that individuals endowments were governed by a Markov process. Under that assumption we could identify a state vector in terms of which the Arrow securities could be cast. This aggregate state vector then became a component of the state vector for each individual s problem. This transformation of price systems is easy in the pure exchange economies of chapter 8 .
