Private Benefits,Warm Glow,and Reputation in the Free and Open Source Software Production Model

A great deal of production and consumption behavior takes place in the context of social organizations that seem to fall outside of the traditional paradigm of profit/utility maximization. These organizations are voluntary in nature and rely on contributions from members to achieve their objectives. Examples include the Linux operating system and other FOSS projects, political movements, churches and religious groups, Habitat for Humanity, and similar charitable organizations. In this paper, we consider a world containing agents with heterogeneous abilities who may voluntarily choose to make effort contributions to one or more different public projects. Agents are motivated by a desire to be seen as significant contributors to important and valuable projects, the warm glow from the act of contributing, and a desire to directly enjoy the benefits of projects when complete. We find that contributions from others can be either strategic complements or substitutes. We show that Nash equilibria exist and study how agents’ abilities and project quality affect the equilibrium levels of contributions.     

Asymmetric treatment of identical agents in teams

We investigate when identical agents will be treated asymmetrically in a simple team setting. Asymmetric treatment is optimal when the agents’ individual contributions to team performance are strategic complements. Symmetric treatment of identical agents is optimal when the agents’ contributions are strategic substitutes or when they are independent.     

COLLUSION,FLUCTUATING DEMAND,AND PRICE RIGIDITY*

We study an infinitely repeated Bertrand game in which an i.i.d. demand shock occurs in each period. Each firm receives a private signal about the demand shock at the beginning of each period. At the end of each period, all information but the private signals becomes public. We consider the optimal symmetric perfect public equilibrium (SPPE) mainly for patient firms. We show that price rigidity arises in the optimal SPPE if the accuracy of the private signals is low. We also study the implications of more firms and firms' impatience on collusive pricing.     

Implications of Economic Interdependence for Endogenous Wage Indexation Decisions

This paper shows how economic interdependence affects the indexation decisions of atomistic wage setters in an environment in which monetary authorities do not observe stochastic disturbances before making their policy choices. If stochastic disturbances are common across countries, interdependence has no effect on equilibrium indexation choices in identical countries. However, if disturbances are country specific, numerical simulations show that interdependence is likely to reduce equilibrium indexation choices relative to a small open economy. We also show that indexation choices may be either strategic complements or strategic substitutes, but that strategic complementarity becomes more likely as the degree of interdependence rises.     

Inflation, Welfare, and Public Goods

In this article we study the welfare effects of monetary policy in a simple overlapping generation economy in which agents voluntarily contribute to a public good. Inflation has two effects at equilibrium: it increases voluntary contributions and it misallocates private consumption across time. We show that the aggregate effect is welfare-improving for "not too large" inflation rates. Moreover, there exists an optimal inflation rate.     

Herding with collective preferences

This paper studies a simple model of observational learning where agents care not only about the information of others but also about their actions. We show that despite complex strategic considerations that arise from forward-looking incentives, herd behavior can arise in equilibrium. The model encompasses applications such as sequential elections, public good contributions, and leadership charitable giving.     

Uncertainty, co-ordination and path dependence

Our objective is to assess whether dynamics hinder or assist co-ordination in a game with strategic complementarities. We study two dynamic aspects: different agents make decisions at different points in time; and extra information about a payoff-relevant state of nature becomes available over time. We find that the dynamic resolution of information matters most for uniqueness of equilibrium. This is demonstrated by showing that the condition for uniqueness is weaker when learning occurs. We also analyse how successfully agents co-ordinate when there is a unique equilibrium. Finally, we show that path dependence occurs: the order in which signals arrive matters, as well as the total amount of information received.     

Better-reply dynamics and global convergence to Nash equilibrium in aggregative games

We consider n-person games with quasi-concave payoffs that depend on a player's own action and the sum of all players' actions. We show that a discrete-time, stochastic process in which players move towards better replies—the better-reply dynamics—converges globally to a Nash equilibrium if actions are either strategic substitutes or strategic complements for all players around each Nash equilibrium that is asymptotically stable under a deterministic, adjusted best-reply dynamics. We present an example of a 2-person game with a unique equilibrium where the derivatives of the best-reply functions have different signs and the better-reply dynamics does not converge.     

Cross‐border Lobbying in Preferential Trading Agreements: Implications for External Tariffs and Welfare

This paper examines the effect of cross‐border lobbying on domestic lobbying and on external tariffs in both Customs Union (CU) and Free Trade Area (FTA). We do so by developing a two‐stage game which endogenizes the tariff formation function in a political economic model of the directly unproductive rent‐seeking activities type. We find that cross‐border lobbying un‐ambiguously increases both domestic lobbying and the equilibrium common external tariffs in a CU. The same result also holds for FTA provided tariffs for the member governments are strategic complements. We also develop a specific oligopolistic model of FTA and show that tariffs are indeed strategic complements in such a model.     

Endogenous Timing in Strategic Environmental Policymaking

In this paper, we endogenize the timing of policymaking in a simple two-country model of strategic environmental policy. We consider a timing game in which two policymakers non-cooperatively decide their preferred sequence of moves before setting emission tax rates. We show that whether the policymakers implement emission tax policies simultaneously or sequentially crucially depends on the magnitude of environmental damages. When the damages are insignificant, the tax rates are strategic substitutes, and the simultaneous-move policymaking emerges in equilibrium. In contrast, when the damages are significant, the tax rates are strategic complements, and sequential-move policymaking emerges. We also extend the model by allowing for differences in the vulnerability to environmental damages between countries. When the differences are large, the unique equilibrium of the game is the situation where the less vulnerable country acts as a leader. In the case where multiple equilibrium emerges, the risk-dominant equilibrium is also that where the less vulnerable country leads.     

Competitive bargaining equilibrium

In a simple exchange economy we propose a bargaining procedure that leads to a Walrasian outcome as the agents become increasingly patient. The competitive outcome therefore obtains even if agents have market power and are not price-takers. Moreover, where in other bargaining protocols the final outcome depends on bargaining power or relative impatience, the outcome here is determinate and depends only on preferences and endowments. Our bargaining procedure involves bargaining over prices and maximum quantity constraints, and it guarantees convergence to a Walrasian outcome for any standard exchange economy. In contrast, without quantity constraints we show that equilibrium is generically inefficient.     

Second Mover Disadvantages in a Three-Player Stackelberg Game with Private Information

I consider a three-player Stackelberg. game where each player has its private signal concerning stochastic demand. I show that at a perfectly revealing equilibrium, the second mover earns the lowest and the third mover the highest expected profit of the three. I characterize this result by the strategic substitutes or complements relationships among three firms at the equilibrium. Journal of Economic Literature Classification Numbers: C72, D82, and L13.     

Dynamic coordination with timing frictions: Theory and applications

We present a general framework of dynamic coordination with timing frictions. A continuum of agents receive random chances to choose between two actions and remain locked in the selected action until their next opportunity to reoptimize. The instantaneous utility from each action depends on an exogenous fundamental that moves stochastically and on the mass of agents currently playing each action. Agents' decisions are strategic complements and history matters. We review some key theoretical results and show a general method to solve the social planner's problem. We then review applications of this framework to different economic problems: network externalities, statistical discrimination, and business cycles. The positive implications of these models are very similar, but the social planner's solution points to very different results for efficiency in each case. Last, we review extensions of the framework that allow for endogenous hazard rates and ex ante heterogeneous agents.     

Strategic Complements, Substitutes, and Ambiguity: The Implications for Public Goods

We examine the effect of ambiguity in symmetric games with aggregate externalities. We find that ambiguity will increase/decrease the equilibrium strategy in games with strategic complements/substitutes and positive externalities. These effects are reversed in games with negative externalities. We consider some economic applications of these results to Cournot oligopoly, bargaining, macroeconomic coordination, and voluntary donations to a public good. In particular we show that ambiguity may reduce free-riding. Comparative statics analysis shows that increases in uncertainty will increase donations, to a public good. Journal of Economic Literature C72, D81, H41.     

Fiat money and the value of binding portfolio constraints

We establish necessary and sufficient conditions for the individual optimality of a consumption-portfolio plan in an infinite horizon economy where agents are uniformly impatient and fiat money is the only asset available for intertemporal transfers of wealth. Next, we show that fiat money has a positive equilibrium price if and only if for some agent the zero short sale constraint is binding and has a positive shadow price (now or in the future). As there is always an agent that is long, it follows that marginal rates of intertemporal substitution never coincide across agents. That is, monetary equilibria are never full Pareto efficient. We also give a counter-example illustrating the occurrence of monetary bubbles under incomplete markets in the absence of uniform impatience.     

A global game with strategic substitutes and complements: Comment

In Karp et al. (2007), an argument is made to show existence of Bayesian–Nash equilibrium in global games that may include both strategic substitutes and complements. This note documents a gap in the proof of that statement.     

Farsighted network formation

This paper studies a model of dynamic network formation when individuals are farsighted: players evaluate the desirability of a “current” move in terms of its consequences on the entire discounted stream of payoffs. We define a concept of equilibrium which takes into account farsighted behavior of agents and allows for limited cooperation amongst agents. We show that an equilibrium process of network formation exists. We also show that there are valuation structures in which no equilibrium strategy profile can sustain efficient networks. We then provide sufficient conditions under which the equilibrium process will yield efficient outcomes.     

Strategic complementarities and search market equilibrium

In this paper, we apply supermodular game theory to the equilibrium search literature with sequential search. We identify necessary and sufficient conditions for the pricing game to exhibit strategic complementarities and prove existence of equilibrium. We then show that price dispersion is inherently incompatible with strategic complementarities in the sense that the Diamond Paradox obtains when firms are identical and is robust within the class of search cost densities that are small near zero and support strategic complementarities. We also show that a major criticism of the literature, that agents act as if they know the distribution of prices, can be justified in the sense of convergent best response dynamics.     

Over- and under-investment according to different benchmarks

In a two-stage oligopoly, with investment in the first stage and quantity or price competition in the second stage, there is a “Common Wisdom” Theorem which states that we find over-investment if the goods are substitutes and competition is in strategic substitutes, or if goods are complements and competition is in strategic complements, and that we find under-investment if we have complements and strategic substitutes or substitutes and strategic complements. The existing literature, however, lacks a proof of this theorem and, in particular, it lacks a systematic comparison of the different benchmarks for over- and under-investment. A “naive” benchmark is the cost efficient investment with respect to the subgame perfect (closed loop) equilibrium quantities. Alternative benchmarks (which are more often proposed) are the open loop equilibrium investment or the welfare maximizing investment. The chosen benchmark is critical because the Common Wisdom Theorem applies (under certain conventional conditions) only for the naive benchmark. The other two benchmarks give rise to subcases.     

Coordination and learning in games with strategic substitutes and complements

This paper experimentally compares the impact of the presence of strategic substitutes (GSS) and complements (GSC) on players’ ability to successfully play equilibrium strategies. By exploiting a simple property of the ordering on strategy spaces, our design allows us to isolate these effects by avoiding other confounding factors that are present in more complex settings, such as market games. We find that the presence of strategic complementarities significantly improves the rate of Nash play, but that this effect is driven mainly by early rounds of play. This suggests that GSS may be more difficult to learn initially, but that given sufficient time, the theoretically supported globally stable equilibrium offers a good prediction in both settings. We also show that increasing the degree of substitutability or complementarity does not significantly improve the rate of Nash play in either setting, which builds on the findings of previous studies.