The Optimal Level of Experimentation

We assume that an impatient decision maker (DM) runs variable-size experiments at an increasing, strictly convex cost before choosing an irreversible action. We introduce and solve a tractable continuous time version of this problem --- a control of variance of a diffusion with uncertain mean. Assuming two states and two actions, we prove: (a) the optimal experimentation level rises in the Bellman value; and deduce testable implications, like (b) experimentation costs drift up; and (c) a more impatient decision maker may experiment more, given lump-sum final payoffs. We show that (a) and (b) are robust to finitely many states and actions, and we also extend an R&D interpretation of the model, where experimentation is monotonic not only in the value, but also in beliefs. Our intuition for our key monotonicity finding (a) is very economic. There are two decisions at each instant: stop or experiment, and then at what level n. The second choice equates the marginal costs and benefits of information: c'(n) = MB(n). In our diffusion setting, the marginal benefit of experimentation is constant, and so the total benefit TB is linear in the level: TB = n*MB = nc'(n). So the DM acts like a neoclassic competitive firm, producing information at an increasing marginal cost and selling it to himself at the fixed price c'(n). Since postponing the final decision entails a discounting cost, optimal stopping demands that the DM equate his producer surplus from experimentation nc'(n)-c(n) to the delay cost rV (given the interest rate r). Intuitively, the DM closes down his information firm (i.e. acts) when he cannot generate profits (producer surplus) to justify his capital rental (his deferred action). Since this surplus rises in quantity with convex costs, greater experimentation is needed to generate the higher surplus for a higher value V