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Continuous-Time Public Good Contribution under Uncertainty: A Stochastic Control Approach

  • Giorgio Ferrari
  • Frank Riedel
  • Jan-Henrik Steg

In this paper we study continuous-time stochastic control problems with both monotone and classical controls motivated by the so-called public good contribution problem. That is the problem of n economic agents aiming to maximize their expected utility allocating initial wealth over a given time period between private consumption and irreversible contributions to increase the level of some public good. We investigate the corresponding social planner problem and the case of strategic interaction between the agents, i.e. the public good contribution game. We show existence and uniqueness of the social planner's optimal policy, we characterize it by necessary and sufficient stochastic Kuhn-Tucker conditions and we provide its expression in terms of the unique optional solution of a stochastic backward equation. Similar stochastic first order conditions prove to be very useful for studying any Nash equilibria of the public good contribution game. In the symmetric case they allow us to prove (qualitative) uniqueness of the Nash equilibrium, which we again construct as the unique optional solution of a stochastic backward equation, although the latter is not related to a meaningful control problem. We finally also provide a detailed analysis of the so-called free rider effect.

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File URL: http://arxiv.org/pdf/1307.2849
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Paper provided by arXiv.org in its series Papers with number 1307.2849.

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Date of creation: Jul 2013
Date of revision: Jul 2015
Handle: RePEc:arx:papers:1307.2849
Contact details of provider: Web page: http://arxiv.org/

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  1. Lockwood, B. & Thomas, J.P., 1999. "Gradualism and Irreversibility," The Warwick Economics Research Paper Series (TWERPS) 525, University of Warwick, Department of Economics.
  2. Maria B. Chiarolla & Giorgio Ferrari & Frank Riedel, 2012. "Generalized Kuhn-Tucker Conditions for N-Firm Stochastic Irreversible Investment under Limited Resources," Center for Mathematical Economics Working Papers 463, Center for Mathematical Economics, Bielefeld University.
  3. Varian, Hal R., 1994. "Sequential contributions to public goods," Journal of Public Economics, Elsevier, vol. 53(2), pages 165-186, February.
  4. Theodore Groves & John Ledyard, 1976. "Optimal Allocation of Public Goods: A Solution to the 'Free Rider Problem'," Discussion Papers 144, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  5. Leslie M. Marx & Steven A. Matthews, 1997. "Dynamic Voluntary Contribution to a Public Project," Discussion Papers 1188, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  6. Fershtman, C. & Nitzan, S., 1988. "Dynamic Voluntary Provision Of Public Goods," Papers 21-88, Tel Aviv.
  7. Steg, Jan-Henrik, 2011. "Irreversible investment in oligopoly," Center for Mathematical Economics Working Papers 415, Center for Mathematical Economics, Bielefeld University.
  8. Marco Battaglini & Salvatore Nunnari & Thomas Palfrey, 2012. "The Free Rider Problem: a Dynamic Analysis," NBER Working Papers 17926, National Bureau of Economic Research, Inc.
  9. Y.M. Kabanov, 1999. "Hedging and liquidation under transaction costs in currency markets," Finance and Stochastics, Springer, vol. 3(2), pages 237-248.
  10. Kerry Back & Dirk Paulsen, 2009. "Open-Loop Equilibria and Perfect Competition in Option Exercise Games," Review of Financial Studies, Society for Financial Studies, vol. 22(11), pages 4531-4552, November.
  11. Parimal Bag & Santanu Roy, 2008. "On Sequential and Simultaneous Contributions under Incomplete Information," Departmental Working Papers 0805, Southern Methodist University, Department of Economics, revised Nov 2008.
  12. Jean-Jacques Laffont, 1988. "Fundamentals of Public Economics," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262121271, June.
  13. Giorgio Ferrari, 2012. "On an Integral Equation for the Free Boundary of Stochastic, Irreversible Investment Problems," Center for Mathematical Economics Working Papers 471, Center for Mathematical Economics, Bielefeld University.
  14. V. Filipe Martins-da-Rocha & Frank Riedel, 2008. "On Equilibrium Prices in Continuous Time," Papers 0802.3585, arXiv.org.
  15. V. Martins-da-Rocha & Frank Riedel, 2006. "Stochastic equilibria for economies under uncertainty with intertemporal substitution," Annals of Finance, Springer, vol. 2(1), pages 101-122, January.
  16. Bergstrom, Theodore & Blume, Lawrence & Varian, Hal, 1986. "On the private provision of public goods," Journal of Public Economics, Elsevier, vol. 29(1), pages 25-49, February.
  17. Palfrey, Thomas R. & Rosenthal, Howard, 1984. "Participation and the provision of discrete public goods: a strategic analysis," Journal of Public Economics, Elsevier, vol. 24(2), pages 171-193, July.
  18. Maria B. Chiarolla & Giorgio Ferrari, 2011. "Identifying the Free Boundary of a Stochastic, Irreversible Investment Problem via the Bank-El Karoui Representation Theorem," Papers 1108.4886, arXiv.org, revised Dec 2013.
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