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Social Choice and Welfare

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01-08-2014 | Original Paper

A Clarke tax tâtonnement that converges to the Lindahl allocation

Author: Matt Van Essen

Published in: Social Choice and Welfare | Issue 2/2014

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Abstract

This paper uses the Clarke mechanism to construct an incentive compatible t âtonnement process which converges to the Lindahl allocation of a stylized public good economy when consumers have quadratic preferences. We show truth-telling to be an ex-post perfect equilibrium in the infinite horizon game induced by the tâtonnement.

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See Lindahl (1919) or see Malinvaud (1971) for a discussion of Lindahl’s t âtonnement.

This has created a literature trying to eliminate this distortion. Green and Laffont (1979), Rob (1982), and Sinn (1986) have all shown that the Clarke tax vanishes as the economy grows leaving an approximate, non-Lindahl, Pareto optimal allocation. Walker (1980) proves that there is no dominant strategy mechanism that generally achieves Pareto efficiency in a public good economy.

A mechanism is stable if it induces a game whose equilibrium of interest is locally stable under a specified learning algorithm. Common examples of learning algorithms are myopic best reply and fictitious play. See, for example, Healy and Mathevet (2012) for a general discussion of stability in mechanism design.

Iterative adjustment concepts have also been applied in the regulation of natural monopoly literature. See Finsinger and Vogelsang (1981) or Cox and Issac (1987).

These prices are called Lindahl prices. See Foley (1970) for a rigorous definition of the Lindahl equilibrium.

We assume each consumer has enough of the private good to cover their Lindahl tax.

See Fudenberg and Tirole (1991). The one shot deviation property can be applied because: (1) The message space is a finite discrete set in each period, thus per round payoffs are bounded; (2), future payoffs are discounted, thus the game satisfies the “continuity at infinity” requirement.

In this case, \(r\) is only restricted to be between \(0\) and \(1\).

These values were computed using a MATLAB program, the code for this program is available upon request.

Chen Y (2002) A family of supermodular nash mechanisms implementing Lindahl allocations. Econ Theor 19:773–790CrossRef

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Clarke E (2000) Demand revelation and the provision of public goods. Excel Press, Lincoln

Cox J, Issac M (1987) Mechanisms for incentive regulation: theory and experiment. RAND J Econ 18(3):348–359CrossRef

Dreze JH, de la Poussin V (1971) A t âtonnement process for public goods. Rev Econ Stud 38(2):133–150CrossRef

Finsinger J, Vogelsang I (1981) Alternative institutional frameworks for price incentive mechanisms. Kyklos 34(3):388–404CrossRef

Foley D (1970) Lindahl’s solution and the core of an economy with public goods. Econometrica 38:66–72CrossRef

Fudenberg D, Tirole J (1991) Game theory. MIT Press, Cambridge

Green J, Laffont J (1979) Incentives in public decision making. Studies in Public Economics, North-Holland

Healy P, Mathevet L (2012) Designing stable mechanisms for economic environments. Theor Econ 7:609–661CrossRef

Hurwicz L (1979) Outcome functions yielding Walrasian and Lindahl allocations at nash equilibrium points. Rev Econ Stud 46:217–224CrossRef

Kim T (1987) Stability problems in the implementation of Lindahl allocations doctoral Thesis. University of Minnesota

Kim T (1993) A stable nash mechanism implementing Lindahl allocations for quasi-linear environments. J Math Econ 22:359–371CrossRef

Lindahl E (1919) Just taxation: a positive solution. In: Musgrave R, Peacock A (eds) English translation of portion of Die Gerechtigkeit der Besteuerung, in classics in the theory of public finance. MacMillan, New York

Malinvaud E (1971) A planning approach to the public good problem. Swed J Econ 73:96–112CrossRef

Milgrom P (2004) Putting auction theory to work. Cambridge University Press, CambridgeCrossRef

Pivato M (2011) A fair pivotal mechanism for nonpecuniary public goods. Trent University, New York

Rob R (1982) Asymptotic efficiency of the demand revealing mechanism. J Econo Theory 28:207–220CrossRef

Sinn H (1986) Sufficient conditions for a vanishing Clarke tax—a note. J Econ 46(1):65–72CrossRef

Tideman T, Tullock G (1976) A new and superior process for making social choices. J Political Econ 84(6):1145–1159CrossRef

Van Essen M, Lazzati N, Walker M (2012) Out-of-equilibrium performance of three Lindahl mechanisms: an experiment. Games Econ Behav 74:366–381CrossRef

Van Essen M (2013) A simple supermodular mechanism that implements Lindahl allocations. J Public Econ Theory 15:363–377CrossRef

Van Essen M (2012a) Information complexity, punishment, and stability in two nash efficient Lindahl mechanisms. Rev Econ Design 16:15–40CrossRef

Van Essen M (2012b) A note on the stablity of Chen’s Lindahl mechanism. Soc Choice Welf 38:365–370CrossRef

Walker M (1980) On the non-existence of a dominant strategy mechanism for making optimal public decisions. Econometrica 48:1521–1540CrossRef

Walker M (1981) A simple incentive compatible scheme for attaining Lindahl allocations. Econometrica 49:65–71CrossRef

Title: A Clarke tax tâtonnement that converges to the Lindahl allocation
Author: Matt Van Essen
Publication date: 01-08-2014
Publisher: Springer Berlin Heidelberg
Published in: Social Choice and Welfare / Issue 2/2014
Print ISSN: 0176-1714
Electronic ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-013-0781-8

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