On a special case of non-symmetric resource extraction games with unbounded payoffs
The game of resource extraction/capital accumulation is a stochastic infinitehorizon game, which models a joint utilization of a productive asset over time. The paper complements the available results on pure Markov perfect equilibrium existence in the non-symmetric game setting with an arbitrary number of agents. Moreover, we allow that the players have unbounded utilities and relax the assumption that the stochastic kernels of the transition probability must depend only on the amount of resource before consumption. This class of the game has not been examined beforehand. However, we could prove the Markov perfect equilibrium existence only in the specific case of interest. Namely, when the players have constant relative risk aversion (CRRA) power utilities and the transition law follows a geometric random walk in relation to the joint investment. The setup with the chosen characteristics is motivated by economic considerations, which makes it relevant to a certain range of real-word problems.
[1] Stokey, N.L., Lucas, R.E., & Prescott, E.C. (1989). Recursive Methods in Economic Dynamics. Harvard University Press.
[2] Ulus, A.Y. (2017). On discrete time infinite horizon optimal growth problem. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 8(1), 102–116.
[3] Levhari, D., & Mirman, L.J. (1980). The great fish war: An example using a dynamic Cournot-Nash solution. The Bell Journal of Economics, 11(1), 322–334.
[4] Sundaram, R.K. (1989). Perfect equilibrium in non-randomized strategies in a class of symmetric dynamic games. Journal of Economic Theory, 47(1), 153–177.
[5] G´orniewicz, O., & Wiszniewska-Matyszkiel, A. (2018). Verification and refinement of a two species fish wars model. Fisheries Research, 203, 22–34.
[6] Majumdar, M., & Sundaram, R. (1991). Symmetric stochastic games of resource extraction: The existence of non-randomized stationary equilibrium. In: Theory and Decision Library, Springer Netherlands, 175–190.
[7] Dutta, P.K., & Sundaram, R. (1992). Markovian equilibrium in a class of stochastic games: existence theorems for discounted and undiscounted models. Economic Theory, 2(2), 197– 214.
[8] Balbus, L., & Nowak, A.S. (2004). Construction of Nash equilibria in symmetric stochastic games of capital accumulation. Mathematical Methods of Operational Research, 60(2), 267–277.
[9] Ja´skiewicz, A., & Nowak, A.S. (2018). On symmetric stochastic games of resource extraction with weakly continuous transitions. TOP, 26(2), 239–256.
[10] Asienkiewicz, H., & Balbus, L. (2019). Existence of Nash equilibria in stochastic games of resource extraction with risk-sensitive players. TOP, 27(3), 502–518.
[11] Szajowski, P. (2006). Constructions of Nash equilibria in stochastic games of resource extraction with additive transition structure. Mathematical Methods of Operations Research, 63(2), 239–260.
[12] Balbus, L., & Nowak, A.S. (2008). Existence of perfect equilibria in a class of multigenerational stochastic games of capital accumulation. Automatica, 44(6), 1471–1479.
[13] He, W., & Sun, Y. (2017). Stationary markov perfect equilibria in discounted stochasticgames. Journal of Economic Theory, 169, 35– 61.
[14] Curtat, L.O. (1996). Markov equilibria of stochastic games with complementarities. Games and Economic Behavior, 17(2), 177–199.
[15] Balbus, L., Reffett, K., & Wo´zny, L. (2014). A constructive study of markov equilibria in stochastic games with strategic complementarities. Journal of Economic Theory, 150, 815–840.
[16] Nowak, A.S. (2007). On stochastic games in economics. Mathematical Methods of Operations Research, 66(3), 513–530.
[17] Amir, R. (1996). Continuous stochastic games of capital accumulation with convex transitions. Games and Economic Behavior, 15(2), 111–131.
[18] Ja´skiewicz, A., & Nowak, A.S. (2015). Stochastic games of resource extraction. Automatica, 54, 310–316.
[19] Friend, I., & Blume, M.E. (1975). The demand for risky assets. The American Economic Review, 65(5), 900–922.
[20] Szpiro, G.G. (1986). Relative risk aversion around the world. Economics Letters, 20(1), 19–21.
[21] Chiappori, P.A., & Paiella, M. (2011). Relative risk aversion is constant: Evidence from panel data. Journal of the European Economic Association, 9(6), 1021–1052.
[22] Bertsekas, D.P., & Shreve, S.E. (1978). Stochastic optimal control : the discrete time case. Academic Press New York.