Essays on Continuous-Time Games with Learning

Abstract

This dissertation studies the impact of learning about unobserved payoff-relevant variables on economic decisions. In chapter 1, I study a labor market in which employers learn about a worker's unobserved skills by observing output. Skills evolve as a mean-reverting process with a trend that is potentially endogenous due to human capital accumulation. Output is additively separable in the worker's skills and in his hidden effort decision, and is also distorted by Brownian noise. Under general conditions, I show that there is an equilibrium in which effort is a deterministic function of time. This equilibrium is almost always inefficient.

In chapter 2, I study a class of continuous-time games in which one long-run agent and a population of small players learn about a hidden state from a public signal that is subject to Brownian shocks. The long-run agent can influence the small players' beliefs by affecting the signal or by affecting the hidden state itself, in both cases in an additively separable way. The impact of the small players' beliefs on the long-run agent's payoff is nonlinear. At a general level, I derive a necessary condition for Markov Perfect Equilibria in the form of an ordinary differential equation. In a subclass of games with linear-quadratic structure, I obtain closed-form solutions for global incentives through solving a new type of partial differential equation. Applications to procurement and monetary policy in the context of partial information are developed.

In chapter 3, joint with Yuliy Sannikov, a firm's earnings are driven by its stock of capital and by an underlying fundamental process. Earnings are not observable at the moment of investing in capital, thus making fundamentals unobserved. The manager learns about fundamentals by observing a signal which is distorted by Brownian noise. Investment is costly and subject to adjustment costs. We show that the sensitivity of investment to expected earnings increases as uncertainty decays over time if and only if earnings are a concave function of fundamentals. We also show that the firm's value is always below its corresponding value in the full-information benchmark.