Markov Equilibria Research Articles

The dynamics of distributive politics

We study dynamic committee bargaining over an infinite horizon with discounting. In each period, a committee proposal is generated by a random recognition rule, the committee chooses between the proposal and a status quo by majority rule, and the voting outcome in period t becomes the status quo in period t + 1. We study symmetric Markov equilibria of the resulting game and conduct an experiment to test hypotheses generated by the theory for pure distributional (divide-the-dollar) environments. In particular, we investigate the effects of concavity in the utility functions, the existence of a Condorcet winning alternative, and the discount factor (committee “impatience”). We report several new findings. Voting behavior is selfish and myopic. Status quo outcomes have great inertia. There are strong treatment effects that are in the direction predicted by the Markov equilibrium. We find significant evidence of concave utility functions.

Sequential auctions with randomly arriving buyers

We analyze a dynamic market in which buyers compete in a sequence of private-value auctions for differentiated goods. New buyers and new objects may arrive at random times. Since objects are imperfect substitutes, buyersʼ values are not persistent. Instead, each buyerʼs private value for a new object is a new independent draw from the same distribution. We consider the use of second-price auctions for selling these objects, and show that there exists a unique symmetric Markov equilibrium in this market. In equilibrium, buyers shade their bids down by their continuation value, which is the (endogenous) option value of participating in future auctions. We characterize this option value and show that it depends not only on the number of buyers currently present on the market and the distribution of their values, but also on anticipated market dynamics.

Heterogeneous discounting in economic problems

SUMMARYThere is a growing literature considering deviations from standard discounting. In this paper, we analyze a continuous‐time model in finite horizon in which the agent discounts the instantaneous utility function and the final function at constant but different instantaneous discount rates of time preference. Within this context we can model problems in which, when the time t approaches to the final time, the valuation of the final function increases compared with the previous valuations in a way that cannot be explained by using a constant or a variable discount rate. Despite its simplicity, the differential time preference suffices to obtain time‐inconsistent results if standard optimal control theory is applied, for the same reason as in problems with non‐constant discounting or hyperbolic preferences. We derive a dynamic programming equation whose solutions are time‐consistent Markov equilibria. We solve a simple model and discuss some extensions of the model. Copyright © 2011 John Wiley & Sons, Ltd.

Theory of Hot Brownian Motion

The erratic motion of suspended colloidal particles due to thermal fluctuations of the solvent molecules is well known as Brownian motion and has long found its theoretical explanation as an equilibrium phenomenon. In this work, we abandon the homogeneity of temperature and viscosity and address so-called hot Brownian motion [Rings et al., Phys. Rev. Lett., 2010, 105 (9), 090604] of (laser-)heated nanoparticles. We develop an effective Markovian equilibrium description for this non-equilibrium phenomenon by deriving the appropriate effective temperature and viscosity parameters from a generalized theory of fluctuating hydrodynamics. On the basis of our systematic mathematical formulation we propose a numerically accurate simplified model that yields analytical results.

Repeated bargaining with reference-dependent preferences

We consider a two-player bargaining model in which one or both players have reference-dependent preferences, but are otherwise perfectly rational. Our behavioural assumption is that players with reference-dependent preferences prefer impasse to consuming strictly less than their current reference points. Reference points adjust each period according to some exogenously specified law of motion. When reference points do not adjust following disagreement, we show that disagreement does not arise in equilibrium, but they do influence the division of the pie. In contrast, when reference points adjust downwards following disagreement, disagreements arise and players may try to manipulate the reference point of their opponent. When reference points adjust downwards following a rejection, for a particular Markov equilibrium, we show that the set of feasible allocations can be divided into agreement and disagreement regions. In particular, there are thresholds such that if one (or more) player’s reference point is above the threshold, disagreement necessarily arises.

Numerical Simulation of the Overlapping Generations Models with Indeterminant Equilibria

In this paper we explorer the computation and simulation of stochastic overlapping generation (OLG) models. To do so we compute all Markovian equilibria adopting a recently developed numerical algorithm. Among the models we studied, the inde- terminacy in deterministic OLG model results in many di erent equilibrium paths corresponding to the initial condition that all asymptotically converge to the same steady state. The introduction of uncertainty brings at least two type of indetermi- nacies. First, there exists numerous selections of transition and policy functions from the equilibrium set. Each of them generates a competitive equilibrium sequence with distinct statistics and quite sizable di erence in welfare. Second, the possibility of switching between these transition and policy functions will support a continuum of competitive equilibrium paths starting with the same initial conditions.

Social security as Markov equilibrium in OLG models: A note

I refine and extend the Markov perfect equilibrium of the social security policy game in Forni (2005) for the special case of logarithmic utility. Under the restriction that the policy function be continuous, instead of differentiable, the equilibrium is globally well defined and its dynamics always stable.

Political intergenerational risk sharing

In a stochastic two-period OLG model, featuring an aggregate shock to the economy, ex-ante optimality requires intergenerational risk sharing. We compare the level of intergenerational risk sharing chosen by a benevolent government and by an office-seeking politician. In our political system, the transfer of resources across generations is determined as a Markov equilibrium of a probabilistic voting game. Low realized returns on the risky asset induce politicians to compensate the old through a PAYG system. This political system typically generates an intergenerational risk sharing scheme that is (i) larger, (ii) more persistent, and (iii) less responsive to the realization of the shock than the social optimum. This is because the current politician anticipates her transfers to the elderly to be compensated by future politicians through offsetting transfers, and hence overspends.

Demographics and the Politics of Capital Taxation in a Life-Cycle Economy

This article studies the effects of demographics on the mix of tax rates on labor and capital. It uses a quantitative general-equilibrium, overlapping-generations model where tax rates are voted without past commitments in every period and characterized as a Markov equilibrium. In the United States, the younger voting-age population in 1990 compared to 1965 accounts for the observed decline in the relative capital tax rate between those two years. A younger population raises the net return to capital, leads voters to increase their savings, and results in a preference for lower taxes on capital. Conversely, aging might increase capital taxation. (JEL E13, H24, H25, J11)

Stock market insider trading in continuous time with imperfect dynamic information

In this paper, I study the equilibrium pricing of asset shares in the presence of dynamic private information. The market consists of a risk-neutral informed agent who observes the firm value, noise traders and competitive market makers who set share prices using the total order flow as a noisy signal of the insider's information. I provide a characterization of all optimal strategies and prove the existence of both Markovian and non-Markovian equilibria by deriving closed-form solutions for the optimal order process of the informed trader and the optimal pricing rule of the market maker. The consideration of non-Markovian equilibrium is relevant since the market maker might decide to re-weight past information after receiving a new signal. Also, I show that (1) there is a unique Markovian equilibrium price process that allows the insider to trade undetected and (2) the presence of an insider increases the market's informational efficiency, in particular, for times close to dividend payment.

On Optimal and Neutrally Stable Population Equilibrium in Voluntary Partnership Prisoner's Dilemma Games

In a voluntary partnership prisoner's dilemma, equilibrium discounted payoff at the origin of a partnership also serves as the worst feasible threat with which to sustain cooperation. In a population model, we introduce the Markov strategy and characterize neutrally stable equilibrium. We construct simple Markov strategies that achieve the highest discounted payoff at the origin among all subgame perfect equilibrium distributions that have either eternal cooperation or eternal (matched) alternation on their equilibrium paths. Partnerships past the first strangers’ phase never break up, and potential punishments are all performed within the partnership. With this feature, our optimal Markov equilibria prove to be neutrally stable, while many commonly known ones in the literature are not.

Investment dynamics in electricity markets

We investigate the incentives for investments in capacity in a simple strategic dynamic model with random demand growth. We construct non-collusive Markovian equilibria where the firms’ decisions depend on the current capacity stock only. The firms maintain small reserve margins and high market prices, and extract large rents. In some equilibria, rationing occurs with positive probability, so the market mechanism does not ensure ‘security of supply’. Usually, the price cap reflects the value of lost energy or lost load (VOLL) that consumers place on severely reducing consumption on short notice. Our analysis identifies a minimum price cap, unrelated to the VOLL, that allows the firms to recoup their investment and production costs in equilibrium. However, raising the price cap above this minimum increases market prices and reduces consumer surplus, without affecting the level of investment.

A Foundation for Markoiv Equilibria in Infinite Horizon Perfect Information Games

We study perfect information games with an infinite horizon played by an arbitrary number of players. This class of games includes infinitely repeated perfect information games, repeated games with asynchronous moves, games with long and short run players, games with overlapping generations of players, and canonical non-cooperative models of bargaining. We consider two restrictions on equilibria. An equilibrium is purifiable if close by behavior is consistent with equilibrium when agents' payoffs at each node are perturbed additively and independently. An equilibrium has bounded recall if there exists K such that at most one player's strategy depends on what happened more than K periods earlier. We show that only Markov equilibria have bounded memory and are purifiable. Thus if a game has at most one long-run player, all purifiable equilibria are Markov.

A Dynamic Model of the Limit Order Book

of a Markov equilibrium in which the bid and ask prices depend only on the numbers of buy and sell orders in the book, and which can be characterized in closed-form in several cases of interest. My model generates empirically verified implications for the shape of the limit-order book and the dynamics of prices and trades. In particular, I show that buy and sell orders can cluster away from the bid-ask spread, thus generating a hump-shaped limit-order book. Also, following a market buy order, both the ask and bid prices increase, with the ask increasing more than the bid—hence the spread widens.

A dynamic model for advertising and pricing competition between national and store brands

Abstract We study the relationship between the pricing and advertising decisions in a channel where a national brand is competing with a private label. We consider a differential game that incorporates the carryover effects of brand advertising over time for both the manufacturer and the retailer and we account for the complementary and competitive roles of advertising. Analysis of the obtained equilibrium Markov strategies shows that the relationship between advertising and pricing decisions in the channel depends mainly on the nature of the advertising effects. In particular, the manufacturer reacts to higher competitive retailer’s advertising levels by offering price concessions and limiting his advertising expenditures. The retailer’s optimal reaction to competitive advertising effects in the channel depends on two factors: (1) the price competition level between the store and the national brands and (2) the strength of the competitive advertising effects. For example, in case of intense price competition between the two brands combined with a strong manufacturer’s competitive advertising effect, the retailer should lower both the store and the national brands’ prices as a reaction to higher manufacturer’s advertising levels. For the retailer, the main advantage from boosting his competitive advertising investments seems to be driven by increased revenues from the private label. The retailer should however limit his investments in advertising if the latter generates considerable competitive effects on the national brand’s sales.

Migration-Regime Liberalization and Social Security: Political-Economy Effect

The pay-as-you-go social security system, burdened by dwindling labour force, can benefit from immigrants, with birth rates that exceed the native-born birth rates. Thus, the social security system effectively provides an incentive to liberalize migration policy through a political-economy mechanism. The paper examines a dynamic political-economy mechanism through which the social security system influences the young decisive voter's attitudes in favour of a more liberal immigration regime. A Markov equilibrium with social security consists of a more liberal migration policy, than a corresponding equilibrium with no social security.

Intergenerational Long Term Effects of Preschool - Structural Estimates from a Discrete Dynamic Programming Model

This paper formulates a structural dynamic programming model of preschool investment choices of altruistic parents and then empirically estimates the structural parameters of the model using the NLSY79 data. The paper finds that preschool investment significantly boosts cognitive and non-cognitive skills, which enhance earnings and school outcomes. It also finds that a standard Mincer earnings function, by omitting measures of non-cognitive skills on the right hand side, overestimates the rate of return to schooling. From the estimated equilibrium Markov process, the paper studies the nature of within generation earnings distribution and intergenerational earnings and schooling mobility. The paper finds that a tax financed free preschool program for the children of poor socioeconomic status generates positive net gains to the society in terms of average earnings and higher intergenerational earnings and schooling mobility.

Endogenous Network Dynamics

In all social and economic interactions, individuals or coalitions choose not only with whom to interact but how to interact, and over time both the structure (the “with whom”) and the strategy (“the how”) of interactions change. Our objectives here are to model the structure and strategy of interactions prevailing at any point in time as a directed network and to address the following open question in the theory of social and economic network formation: given the rules of network and coalition formation, the preferences of individuals over networks, the strategic behavior of coalitions in forming networks, and the trembles of nature, what network and coalitional dynamics are likely to emerge and persist. Our main contributions are (i) to formulate the problem of network and coalition formation as a dynamic, stochastic game, (ii) to show that this game possesses a stationary correlated equilibrium (in network and coalition formation strategies), (iii) to show that, together with the trembles of nature, this stationary correlated equilibrium determines an equilibrium Markov process of network and coalition formation, and (iv) to show that this endogenous process possesses a finite, nonempty set of ergodic measures, and generates a finite, disjoint collection of nonempty subsets of networks and coalitions, each constituting a basin of attraction. We also extend to the setting of endogenous Markov dynamics the notions of pairwise stability (Jackson-Wolinsky, 1996), strong stability (Jacksonvan den Nouweland, 2005), and Nash stability (Bala-Goyal, 2000), and we show that in order for any network-coalition pair to persist and be stable (pairwise, strong, or Nash) it is necessary and sufficient that the pair reside in one of finitely many basins of attraction. The results we obtain here for endogenous network dynamics and stochastic basins of attraction are the dynamic analogs of our earlier results on endogenous network formation and strategic basins of attraction in static, abstract games of network formation (Page and Wooders, 2008), and build on the seminal contributions of Jackson and Watts (2002), Konishi and Ray (2003), and Dutta, Ghosal, and Ray (2005).

A General Theory (and Some Evidence) of Expectation Traps in Monetary Policy

I show that multiple equilibria are a general property of economies under full monetary policy discretion. Three simple conditions are sufficient to rule out, generically, a unique equilibrium in a static economy. The key departure from Barro and Gordon (1983) is to consider bounded welfare costs of inflation. I also show that in a two Markov equilibrium economy the inflation response to certain perturbations is, generically, qualitatively different in each equilibrium. Finally, I discuss some evidence on inflation dynamics that supports the hypothesis that U.S. monetary policy was caught in an expectation trap during the high inflation episode of the 1970s.

Time-Consistent Public Policy

In this paper we study how a benevolent government that cannot commit to future policy should trade off the costs and benefits of public expenditure. We characterize and solve for Markov-perfect equilibria of the dynamic game between successive governments. The characterization consists of an inter-temporal first-order condition (a “generalized Euler equation”) for the government, and we use it both to gain insight into the nature of the equilibrium and as a basis for computations. For a calibrated economy, we find that when the only tax base available to the government is capital income—an inelastic source of funds at any point in time—the government still refrains from taxing at confiscatory rates. We also find that when the only tax base is labour income the Markov equilibrium features less public expenditure and lower tax rates than the Ramsey equilibrium.