7 - More Delays

Summary

Wait and see.

An equilibrium with delays of the game with periods does not converge, in general, to an equilibrium of the game with the same payoff and information structures but where time is continuous. In important cases, there is no equilibrium with continuous time. A waiting game takes place when the information is generated by the payoffs of actions, as when penguins watch for killer whales, oil drillers observe the results of exploration in neighboring patches, or agents receive gradually private information.

A model of economic growth or business cycles can be specified in periods or in continuous time. Its properties do not depend on that choice. In discrete time, quantities like consumption or output in a period have the same dimension as the length of the period, and if that length is vanishingly small, their ratios to the length are asymptotically equal to the flows of consumption and output in the continuous-time specification. This equivalence fails, generically, in models of social learning.

The essence of the equilibria with delay is the arbitrage between the opportunity cost of delay and the option value of delay, which is the expected value of undoing an investment after bad news. If the period shrinks, by arbitrage, the amount of information is smaller. However, this reduced information comes earlier. The overall effect is ambiguous, as shown in Section 7.1.1

When the period length is vanishingly short, the option value of delay shrinks to zero. However, the variance of the change of beliefs remains bounded below by a strictly positive number (Section 7.1.2).