Session 9: Dynamic Games, Contracts, and Markets

I argue that in long-term consumer–producer relationships, menus of contracts can be advantageously replaced by a single generous contract such that, at any point in time, the sum of a consumer’s transfers is equal to the sum of transfers they would have made under the contract that is the best for them in hindsight. Such generous long-term contracts can increase skeptical consumers’ demand for complex and higher-powered contracts while approximately implementing the same outcomes as the underlying menu evaluated by a rational decision maker. Applications include voluntary load shedding in retail electricity markets and cost sharing in health insurance.

This paper develops dual recursive methods to study dynamic games with imperfect monitoring. It begins by defining two kinds of perfect public equilibrium that exploit communication differently, then formulates sequence problems that realize their welfare as linear programs. Their value functions are equated to maximal fixed points of corresponding functional equations defined by convex pro-grams, with implied dual functional equations amenable to computation. In games with frequent actions, this yields welfare optimality (HJB) equations that relate the curvature of welfare to welfare itself. I apply these results to the repeated Prisoners’ Dilemma, optimal dynamic contracting and approximate equilibrium.

This paper studies optimal information disclosure in dynamic economies with income risk where an incumbent firm knows more than the market about a consumer’s persistent type. When the incumbent can commit to long-term contracts, optimal disclosure prescribes no information revelation. Without commitment, no cross-subsidization is feasible for any disclosure policy due to adverse selection. Partial disclosure is typically optimal, and it implements intertemporal consumption smoothing. Lastly, we show that banning long-term relationships can be beneficial to consumers. We apply these findings to lending markets, offering a new perspective on policies like open banking and consumer data ownership.

This paper presents a dynamic model of firm financing where firms use financial slack to reduce rent extraction by financiers with bargaining power. Financing is lumpy because it is optimal to bargain infrequently. Moreover, firms may finance ‘early’ before exhausting internal funds to bargain when their outside options are better. Financing rents are thus endogenous to firms’ dynamic financing strategy. Firms with good financing alternatives raise financing early to reduce rents, whereas firms lacking such alternatives raise financing after exhausting funds to avoid frequently paying endogenously large rents. Investment irreversibility increases financing rents, and disproportionately so for less productive firms.

Bank supervisors employ stress tests and disclose valuable information about banks to the stakeholders. Such tests are also conducted frequently. A credible stress test design must incorporate the stakeholders’ strategic responses to test results and ensure that a bank that passes a stress test does not subsequently fail, even under the most adverse equilibrium. To study credible stress tests, we analyze bank run as a canonical regime change game under incomplete information and asynchronous moves. We find that a higher testing frequency reduces the need for stricter test stringency to maintain credibility. We then characterize the optimal credible stress test policy and demonstrate its dependence on model primitives.

In canonical models with wealth inequality, it is undesirable to tax capital: redistribution can be achieved through consumption taxation without any distortions. These models assume that consumption is observable to the government. However, in practice, business owners can disguise their consumption as a business expense to avoid or evade consumption taxes. How do such imperfections in the enforceability of consumption taxes affect optimal tax policy? We approach this question from a dynamic mechanism design perspective in general equilibrium. The key trade-off that the planner faces is that capital taxation can provide more revenue than consumption taxation, but lowers wages over time. We analytically solve for the long run state of the economy under the optimal dynamic mechanism. The optimal allocation requires sustained intertemporal distortions—as entailed, for example, by sustained capital taxation. The optimal policy can be implemented by a combination of time-varying linear taxes/subsidies on wealth, capital income, and consumption.

We study a standard binary social learning model where agents - information is serially correlated -  it is generated by a Markov process. There is a unique equilibrium in which a herd, sometimes incorrect, always forms. In the long run, does greater persistence increase the likelihood that an incorrect herd forms? In the medium run (prior to the formation of a herd), does a greater similarity information - higher persistence - lead to a greater similarity of actions? The answer to both questions is no.

Financial markets (and more generally the real economy) display a wide range of important nonlinearities. This paper focuses on stock returns, which are skewed left – generating crashes – and whose volatility moves over time, is itself skewed, is strongly related to the level of prices, and displays long memory. This paper shows that such behavior is almost inevitable when prices are formed by investors acquiring information about the true, but latent, value of stocks. It studies a general model of filtering in which agents receive signals about the fundamental value of the stock market and dynamically update their beliefs (potentially with biases). When those beliefs are non-normal and investors believe crashes can happen, prices generically display the range of nonlinearities observed in the data. While the model does not explain where crashes come from, it shows that investors believing that prices can crash is sufficient to generate the rich higher-order dynamics observed empirically. In a simple calibration with iid shocks to fundamentals, the model fits well quantitatively, and regression-based tests support the model’s mechanism.

We analyze a reputational bargaining game in which a central player negotiates simultaneously with two peripheral players. Each player is either rational or a commitment type who never concedes and insists on a fixed share, and concessions are publicly observed. The central player’s type is global, so actions in one dispute update beliefs in the other and generate reputational spillovers. The game admits a unique equilibrium, enabling a sharp comparison with the bilateral benchmark of Abreu and Gul (2000). Spillovers are payoff-relevant if and only if a peripheral is uniquely the most reputable player initially. In that case, spillovers overturn the bilateral prediction that toughness pays: the central player is never strictly better off and can be strictly worse off; the strongest peripheral loses; and the weakest peripheral can benefit, especially when the center’s higher-stakes dispute is with the other peripheral.

We study the role of timestamps in a dynamic disclosure game. At a random date, a sender privately obtains one piece of hard evidence about a hidden binary state evolving as a continuous-time Markov chain and can disclose it at any later date. When evidence carries a timestamp, the unique equilibrium features immediate disclosure of good evidence and disclosure of bad evidence after a deterministic, timestamp-dependent delay. Without timestamps, unless the prior is low, equilibrium must feature delayed disclosure of good evidence; under some conditions, bad evidence is never disclosed. We construct an equilibrium in which good evidence is initially delayed, then stochastically released, and eventually disclosed immediately. Timestamps prevent pretending old good evidence is fresh and allow proving bad evidence is old, thereby accelerating disclosure of good evidence and facilitating disclosure of bad evidence.

We study continuous-time persuasion where a sender controls both how informative a signal is over time and when to stop providing information to a receiver. Given an exogenous signal process, the sender can both garble the evolving signal path and delay the receiver’s decision at a convex, increasing cost of time. We show that, although both instruments are available, any optimal persuasion scheme is fully transparent: the sender keeps the signal fully informative and persuades solely by choosing when to stop.

We study a continuous-time game of bad-news learning between two symmetrically uninformed players: a decision-maker who samples to learn a binary state and a manipulator who can adaptively delay the arrival of breakdown signals at a cost in the low state. Equilibria with manipulation have a simple structure: along the no-breakdown history, the decision-maker stops deterministically, and the manipulator randomizes between never manipulating and manipulating at full intensity from a deterministic time until the decision-maker stops. These equilibria exist on a nondegenerate interval of priors; for any fixed prior in this interval, manipulation starts at the same time and the decision maker stops at the same time, so multiplicity is confined to the mixing probability. A Wald equilibrium without manipulation also exists on this interval, and the players rank Wald and manipulation equilibria in opposite order. Manipulation can flip the comparison of sampling procedures: in equilibrium, static sampling can outperform sequential sampling, and a hybrid protocol can outperform both.

We consider a dynamic adverse selection model where privately informed sellers of divisible assets can choose how much of their asset to sell at each point in time to competitive buyers. With commitment, delay and lower quantities are equivalent ways to signal higher quality. Only the discounted quantity traded is pinned down in equilibrium. With spot contracts and observable past trades, there is a unique and fully separating path of trades in equilibrium. Irrespective of the horizmon and the frequency of trades, the same welfare is attained by each seller type as in the commitment case. When trades can take place continuously over time, each type trades all of its assets at a unique point in time. Thus, only delay is used to signal higher quality. When past trades are not observable, the equilibrium only coincides with the one with public histories when trading can take place continuously over time.

Motivated by concerns that AI-driven entry-level automation may deprive new generations of valuable work experience, this paper studies how technological change affects the intergenerational transmission of tacit knowledge—practical, hard-to-codify skills acquired through workplace interaction. I develop a task-based overlapping-generations model in which novices acquire tacit knowledge by working alongside experts. Knowledge-transfer contracts are incomplete because tacit knowledge is embodied and non-verifiable. In equilibrium, endogenous growth arises because only the most knowledgeable experts manage production and transmit their expertise to multiple novices, diffusing best practices. I show that improvements in entry-level automation increase output upon adoption but can reduce growth and welfare, even without reducing entry-level employment. This occurs when such improvements reallocate novices away from the most productive experts, weakening the diffusion of best practices. By contrast, technological improvements that increase the span of control of the most productive experts—such as those that create new labor-intensive tasks—strengthen knowledge transmission and raise growth.

This paper develops a theory of employer competition over hiring standards in labor markets where employers rely on third-party certification to screen applicants. A revenue-maximizing certifier sells tests to an applicant, who possesses imperfect private information about his ability and seeks to persuade employers to offer him employment. The certifier faces a joint screening and information design problem in designing a test allocation. The distortions from screening reduce the overall informativeness of the test allocation, steering the applicant supply to- wards less selective employers. This incentivizes the more selective employers to lower their standards, intensifying employer competition.

People make correct inferences from \textit{observed} events but make mistakes when reasoning about hypothetical events. I study the implications in competitive markets. I define Dynamic Cursed Expectations (DCE), where agents treat events occurring at different future periods as independent. I build on it to define Dynamic Cursed Expectations Equilibrium (DCEE) and prove its existence for financial economies in which traders have CARA preferences, access to riskless storage, and signals that inform only about the next period's dividends. I apply DCEE to an asset pricing model and show that in DCEE agents underestimate the variance of risky assets, which under risk aversion leads to overvaluation relative to the Rational Expectations Equilibrium benchmark.

We study misspecified Bayesian learning in principal-agent relationships, where an agent is assessed by an evaluator and rewarded by the market. The agent’s outcome depends on their innate ability, costly effort—whose effectiveness is governed by a productivity parameter—and noise. The market infers the agent’s ability from observed outcomes and rewards them accordingly. The evaluator conducts costly assessments to reduce outcome noise, which shape the market’s inferences and provide implicit incentives for effort.

Society—including the evaluator and the market—holds dogmatic, inaccurate beliefs about ability, which distort learning about effort productivity and effort choice. This, in turn, shapes the evaluator’s choice of assessment. We describe a feedback loop linking misspecified ability, biased learning about ef-fort, and distorted assessment. We characterize outcomes that arise in stable steady states and analyze their robust comparative statics and learning foundations. Applications to education and labor market reveal how stereotypes can reinforce across domains—sometimes disguised as narrowing or even reversals of outcome gaps—and how policy interventions targeting assessment can help.

This paper analyzes a dynamic interaction between a fully rational, privately informed sender and a boundedly rational, uninformed receiver with memory constraints. The sender controls the flow of information, while the receiver designs a decision-making protocol that uses a finite state space to learn and to provide incentives. We characterize optimal protocols and quantify the scope for manipulation and the incentive cost of guarding against it. We show that distinctive behavioral patterns that might otherwise appear erratic or psychologically driven—such as information disengagement, opinion polarization conditional on the same information, and indecision near the decision point—emerge as systematic equilibrium responses to asymmetric rationality and information. The model provides an expressive framework for procedural rationality in strategic settings.