Total effort, competitive balance and the optimal contest success function

This paper shows that, in the constrained case, the value weighted contest success function (CSF) maximizes not only total effort, but every objective function which is increasing in total effort and competitive balance. In the unconstrained case, a first price all-pay auction with a reserve price is optimal only if the impact of competitive balance on the objective is small. Otherwise, a modified value weighted CSF is optimal.     

Contests where there is variation in the marginal productivity of effort

Summary. We provide a characterization of participants' behavior in a contest or tournament where the marginal productivity of effort varies across contestants and individual productivity is private information. We then consider the optimal design of such a contest. We first analyze contestant behavior for the usual type of contest, where the highest output wins. Abilities need not be independently distributed. We demonstrate that there is a unique symmetric equilibrium output function, that output is increasing in ability, and that marginal effort is increasing in ability, while effort decreases when the cost of effort increases. Next we consider the case where the highest output need not win, with independently distributed abilities. We analyze the contest designer's decisions in choosing contest rules optimal from her perspective. We show that the output produced, probability of winning, and contest designer's expected revenue are generally increasing in contestants' ability. We examine the relationship between the marginal cost of producing output and marginal utility per dollar of the net award for winning. Received: July 30, 1998; revised version: August 7, 2000     

Group contest success functions

This paper extends the axiomatic characterization of contest success functions of Skaperdas (Econ Theory 7:283–290, 1996) and Clark and Riis (Econ Theory 11:201–204, 1998) to contests between groups. Comments by Dan Kovenock significantly improved the paper and are gratefully acknowledged. I would also like to thank Pavlo Blavatskyy, Aron Kiss, Kai Konrad, Florian Morath, Dana Sisak, participants of the SFB/TR 15 meeting in Gummersbach 2004, the 2004 meeting of the Verein für Socialpolitik in Dresden, and two anonymous referees. Any errors are mine. Financial support from the Deutsche Forschungsgemeinschaft through SFB/TR 15 is gratefully acknowledged.     

Optimal contest design: volume and timing of rent seeking in contests

This paper undertakes a comparative analysis of rent-seeking contests in terms of the amount and the timing of effort they elicit from the participants. The optimal contest structure—the one that maximizes the discounted sum of efforts—is found to hinge on the degree of impatience of the contest organizers, the more patient of whom prefer longer contests consisting of pairwise matches among the contestants. Heterogeneity of the contestants' quality also turns out to play an important role in the comparison.     

Generalized contest success functions

The key element of models of contest is the contest success function (CSF) which specifies the winning probabilities of agents. The existing axiomatizations of CSFs assume that contestants can make only one type of investment. This paper generalizes these axiomatizations to the case where each agent can have multiple types of investments. This allows us to provide a unified framework to extend and interpret the results of Skaperdas (Econ Theory 7:283–290, 1996) and Clark and Riis (Econ Theory 11:201–204, 1998), and rationalize some seemingly ad hoc CSFs used by applied researchers. This paper has benefitted from the comments of Aicke Hinrichs, Rene Levinsky, and two anonymous referees. We are especially thankful to one of the referees for detailed comments which have helped improve every aspect of this paper.     

An investment contest to influence environmental policy

In an investment contest for environmental policy, polluters and victims of pollution invest in an increase of their marginal benefits of pollution and environmental quality, respectively. These investments influence time-consistent environmental policy. Investments will exceed their optimal level. The more victims there are, the lower aggregate investment, the lower payoffs for the victims and the higher payoffs for the polluters. The more polluters there are, the higher aggregate investment and the lower payoffs for polluters and victims. Asymmetries between polluter and victim result in less overinvestment and bias environmental policy in favour of the less productive side.     

Difference-form contest success functions and effort levels in contests

We examine players' equilibrium effort levels in a contest with difference-form contest success functions in which two players compete with each other to win a prize. We show the following. At the pure-strategy Nash equilibrium of a simultaneous-move game, and in the subgame-perfect equilibrium of a sequential-move game, only one of the players expends effort or neither player expends effort. If one player's composite strength is far greater than the other player's, only the player with greater composite strength expends effort whether they move simultaneously or sequentially. If the players' valuations for the prize and their marginal probabilities of winning at (0, 0) are sufficiently small, neither player expends effort whether they move simultaneously or sequentially.     

Cooperation by way of support in a rent seeking contest for a public good

This paper analyzes the efforts made to influence a decision that can have two outcomes. First, the agents on the same side designate an agent to play the contest for them. The inactive agents can then decide to support this active agent, i.e. pay a percentage of his effort. It is optimal to designate either the agent with the lowest stake (who will receive support from everyone else) or the agent with the highest stake (who will not receive support) as the active agent. We find that the side with the higher aggregate stake may have the lower success probability.     

On the existence of an implementable optimal income tax

Summary. This paper discusses the existence of an optimal income tax and distinguishes itself from the previous articles in two respects. In previous papers, the self selection condition was not necessarily consistent with the individual budget constraint. Furthermore, implementability in previous papers was implicit in individual ability, rather than individual income, as the basis of the tax function. We offer a different concept of the self selection conditions: Anti Normal Envy that is consistent with the individual budget constraint and that we show to be equivalent to the competitive equilibrium under a tax function based on income. We then establish the existence of an implementable optimal income tax.Received: 9 December 2000, Revised: 5 August 2003, JEL Classification Numbers: H21, C62, D59.Correspondence to: Jun IritaniThe authors are grateful to an anonymous referee of this journal whose suggestions were instrumental to our revision of the paper, and to Professors Takao Kataoka, Tomoyuki Kamo, Tetsuya Kishimoto, and Mototsugu Fukushige, and the members of the Kobe-Osaka Joint Seminar in Mathematical Economics, for their invaluable comments.     

Contest design and heterogeneity

I show that the effect of heterogeneity on contest investments depends on the structure of the competition, which implies that heterogeneity matters for optimal contest design. This insight helps to explain empirical evidence on the comparison of different contest structures.     

Some problems in modelling the optimal division of a ‘cake’ between an unknown number of participants

The trade-off between the survival time of a fixed population and its consumption rates, when resources are exhaustible, has been studied by Koopmans. A link is established between this model and the traditional neoclassical literature on optimum population. The model is embedded in a larger model in which limited control over population growth rates is feasible, but in which the initial population size acts as a constraint on development. The optimal depletion rate of the exhaustible resource and optimal survival times are then explored. This facilitates a sensitivity analysis of the optimal policy and a discussion of the robustness of the results of Koopmans.     

Computation of non-linear continuous optimal growth models: experiments with optimal control algorithms and computer programs

In this paper we discuss the use of optimal control methods for computing non-linear continuous optimal growth models. We have discussed various recently developed algorithms for computing optimal control, involving step-function approximations, Runge–Kutta solutions of differential equations, and we suggest that the discretization approach is preferable to methods which solve first-order optimality conditions. We have surveyed some powerful computer programs by : , and for computing such models numerically. These programs have no substantial optimal growth modelling applications yet, although they have numerous engineering and scientific applications. A computer program named by is developed in this study. Results are reported for computing the Kendrick–Taylor optimal growth model using and programs based on the discretization approach. References are made to the computational experiments with and . The results are used to compare and evaluate mathematical and economic properties, and computing criteria. While several computer packages are available for optimal control problems, they are not always suitable for particular classes of control problems, including some economic growth models. The -based and , however, offer good opportunities for computing continuous optimal growth models. It is argued in this paper, that optimal growth modellers may find that these recently developed algorithms and computer programs are relatively preferable for a large variety of optimal growth modelling studies.     

Identifying optimal contingent fiscal policies in a business cycle model

Optimal fiscal policy is indeterminate in a dynamic and stochastic environment. The complete characterization of the fiscal policy requires the use of identification constraints. In the literature either capital taxes or debt have been restricted to be not contingent on the state of nature. We propose a different type of identification constraints to have both policy variables state-contingent. Three alternative identification conditions are considered: (i) restrictions on the dynamic and stochastic behavior of the debt path; (ii) an exogenous debt path, and (iii) an exogenous belief function. The main result indicates that the optimal capital tax is zero and constant over the business cycle for any of the identification conditions used, suggesting that is optimal for the government to use debt return as a shock absorber, keeping capital taxes constant. The result is quite different from the previous literature, which obtains very volatile capital taxes. JEL Classification: E62, H21. We are grateful to Alfonso Novales, Víctor Ríos-Rull, Javier Vallés and two anonymous referees for helpful comments and suggestions. We acknowledge financial support from Spanish Ministerio de Ciencia y Tecnología (Ruiz and Pérez: BEC 2003-039; Manzano: BEC 2002-01995). Baltasar Manzano also acknowledges support from Xunta de Galicia (PGIDIT03PXIC30001PN, PGIDIT03CSO30001PR).     

On an optimal consumption problem for p-integrable consumption plans

A generalization is presented of the existence results for an optimal consumption problem of Aumann and Perles [4] and Cox and Huang [10]. In addition, we present a very general optimality principle. Received: July 14, 1999; revised version: September 9, 1999