Essays on Continuous Time Games with Learning

ESSAYS ON CONTINUOUS-TIME GAMES WITH LEARNING Gonzalo S. Cisternas Leyton A DISSERTATION PRESENTED TO THE FACULTY OF PRINCETON UNIVERSITY IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY RECOMMENDED FOR ACCEPTANCE BY THE DEPARTMENT OF ECONOMICS Adviser: Yuliy Sannikov June 2013 c Copyright by Gonzalo S. Cisternas Leyton, 2013. All rights reserved. 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 em- ployers 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. Appli- cations 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 iii 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. iv Acknowledgements I would like to express my deepest gratitude to my advisor Yuliy Sannikov for his guidance, patience, support and invaluable insights through all these years. It has been an honor and a privilege to be his student, and a wonderful experience to learn from him. I would also like to express my sincere gratitude to Dilip Abreu and Stephen Morris for their tremendous support, thoughtful advice and wisdom. It has been a great pleasure to learn from them, and I am very thankful for all their attention. Thank you to my friends Nemanja Antic, Ben Brooks, Wei Cui, Josephine Duh, Leandro Gorno, Diogo Guillén,Constantinos Kalfarentzos, Rohit Lamba, Jay Lu, AndrésMaggi and Doron Ravid for allowing me to learn from them on a daily basis, and for making the Fisher Hall basement such a special place. Thank you to Markus Brunnermeier, Hector Chade, Sylvain Chassang, Eduardo Faingold, Henry Farber, Alex Mas and Bruno Strulovici for useful conversations about my work. Thank you to Andrea Nedic, for her infinite patience, constant support and loving care. Thank you to all my non-econ friends, for making Princeton feel like home. Special thanks to Kathleen DeGennaro and to Laura Hedden for administering the graduate program, and for their contributions to the department. Finally, thank you to my parents and grandparents, to whom I owe everything. v Contents Contents vi 1 Shock Persistence, Endogenous Skills and Career Concerns1 1.1 Introduction................................1 1.2 The Model.................................8 1.2.1 Output Technology, Skills Process and Human Capital....8 1.2.2 Connection with the Literature................. 14 1.2.3 Market Structure and Equilibrium Concept........... 16 1.3 Preliminary Results............................ 18 1.3.1 Learning: Filtering Equations.................. 19 1.3.2 Belief Distortion......................... 23 1.3.3 Problem Reduction: Main Lemma................ 27 1.4 Inefficiencies in Career Concerns Models................ 30 1.5 Human Capital Accumulation...................... 35 1.5.1 Weak Complementarity...................... 41 1.5.2 Strong Complementarity..................... 43 1.5.3 Predicted Path of Wages..................... 46 1.6 Conclusions................................ 48 1.7 Appendix A: Proofs............................ 50 1.8 Appendix B: General Human Capital Technologies.......... 57 vi 2 Two-Sided Learning and Moral Hazard 65 2.1 Introduction................................ 65 2.1.1 Literature............................. 70 2.2 Signal-Jamming Games: General Case................. 72 2.2.1 Set-up............................... 72 2.2.2 Learning and Belief Manipulation................ 76 2.2.3 Markov Perfect Equilibrium................... 83 2.2.4 Necessary Conditions for Markov Perfect Equilibria...... 85 2.2.5 The Incentives Equation: Examples............... 94 2.3 Signal-Jamming Games: Linear-Quadratic Environments....... 98 2.3.1 Existence Result......................... 99 2.3.2 The Structure of the Agent's Incentives............. 102 2.3.3 The Curvature Condition.................... 106 2.3.4 Connection with the Literature: Career Concerns....... 109 2.4 Investment Games............................ 110 2.4.1 Model and Learning Dynamics.................. 111 2.4.2 Necessary Conditions for Markov Perfect Equilibria...... 114 2.4.3 Application: Monetary Policy and Unobserved Inflation.... 117 2.5 Discussion................................. 124 2.5.1 Smoothness and Robustness................... 124 2.5.2 Computation: Off-Equilibrium Analysis............. 125 2.6 Conclusions................................ 126 2.7 Appendix................................. 127 3 Learning, Investment and Adjustment Costs 151 3.1 Introduction................................ 151 3.2 The Model................................. 156 3.3 Learning.................................. 158 vii 3.4 The Value of the Firm.......................... 160 3.5 Capital Dynamics and Optimal Investment Rule: The Time-Profile of Uncertainty................................ 162 3.6 First-Best: The Value of Information.................. 163 3.7 Conclusions................................ 165 3.8 Appendix................................. 166 Bibliography 176 viii Chapter 1 Shock Persistence, Endogenous Skills and Career Concerns 1.1 Introduction Environmental uncertainty plays an important role in the evolution of workers' perceived skills. While firms can influence an employee's productivity through tailored programs such as compensation schemes, on-the-job training or learning-by-doing, exogenous forces that affect the work environment can also have an important impact on performance. For instance, a worker's productivity may vary because unforeseen events force him to be assigned to a different task at which his productivity changes, or because tasks itself evolve due to technological progress. Moreover, these exogenous changes affect an employer's inference process about a worker's ability, as current performance can become a poor predictor of future one. Thus, in settings where wages are based on perceived skills, the degree of randomness of the environment is thus expected to influence the strategic behavior of a worker whose ultimate goal is to affect his future income stream by building a good reputation. 1 In this chapter I study how career concerns are shaped by the degree of randomness in the job environment. More specifically, I build on Holmstrom's (1982, 1999) seminal paper of career concerns in order to construct a continuous-time model of reputation that extends his work along two dimensions. First, I allow skills to be any process within the class of Gaussian diffusion (the continuous-time analog of an AR(1) process), with the persistence of shocks to productivity being the measure of environmental uncertainty. Second, I allow the worker to take actions that directly affect productivity. Since these actions have persistent effects on skills, they have the flavor of investments in human capital. The outcome is a very flexible and general framework that provides particularly clean insights on how belief-distortion mecha- nisms operate, on how wages and effort levels evolve over time, and on the extent to which reputation motives generate socially efficient outcomes. Traditional career concerns models have focused on issues such as the extent of markets' efficiency (Holmstrom's paper), on how different information structures and multitasking affect incentives (Dewatripont et al. (1999a,b)), on the interplay between implicit incentives and short-term contracts (Gibbons and Murphy (1992)) and even on herding behavior (Scharfstein and Stein (1990)). In this chapter, in turn, I provide a detailed analysis of how different degrees of environmental uncertainty influence the reputational motives faced by individuals in dynamic settings. It is widely understood that forward-looking agents evaluate not only the immediate reputational benefits from their actions, but also their long-run consequences. Such analysis is particularly

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