Abstract
We derive a feedback equilibrium of an infinite-horizon dynamic Cournot game where production requires exploitation of a renewable mobile resource, such as migratory fish, wildlife, and groundwater. We study how a small increase in the resource mobility parameter (starting from a position of no resource mobility) impacts on the equilibrium and the associated consumer’s surplus, firms’ profits and social welfare. We show that consumer’s surplus and social welfare increase in the short run but decrease in the long run, while firms’ profits may either increase or decrease in the short run, depending on initial conditions, and increase in the long run. Over the entire planning horizon, both the discounted consumer’s surplus and the discounted social welfare decrease, whereas the discounted profits increase. This result remains valid also in the presence of a per unit tax on extraction.
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Notes
The concept of resource mobility in the context of international trade and specialization is discussed in Hummels et al. [31] and Krugman et al. [39], inter alia Krugman et al. [39], in particular, discuss the importance of resource mobility for market performance and economic growth. For a discussion on the implications of various aspects of resource mobility (including the movement of physical and intangible resources) for development see the World Investment Report 2013 by UNCTAD [50].
The assumption of resource homogeneity is relaxed in Colombo and Labrecciosa [19], who assume that, within the same species, there exist two varieties, one of which is of higher commercial value.
Ficsher and Mirman [26, 27] assume that each species is harvested by a single agent, and characterize and contrast cooperative and noncooperative strategies. In a model à la [26, 27, 43] and Rettieva [44] analyze the case of fish migration, whereas Breton et al. [14] analyze the case in which each species is harvested by a group of agents. These papers assume that the resource is directly consumed. On oligopoly exploitation in the presence of a two-species fish population (with ecological uncertainty) see Wang and Ewald [53], building on Jørgensen and Yeung [32].
Van der Ploeg and De Zeeuw [51] derive the optimal emission charge in a n -country differential game of international pollution control. Dockner and Long [24] study transboundary pollution in a two-player differential game, considering both linear and nonlinear feedback strategies. On transboundary pollution games, see also Benchekroun and Long [6], Jørgensen et al. [33], Kossioris et al. [37], Benchekroun and Chaudhuri [8], de Frutos and Martín-Herrían [21], de Frutos et al. [20], Boucekkine et al. [12, 13], and Yanase and Kamei [54], inter alia.
Kemp and Long [36] assume that n firms extract oil from a common pool and that oil is migratory (extraction in one location induces a flux).
We are grateful to an anonymous referee for suggesting this extension.
In a discrete-time setting, welfare analysis and consumer-surplus calculation are more challenging than in a continuous-time setting, requiring innovating special analytical tools.
Note that second order conditions are satisfied since the expression in curly brackets in (A.1) is concave in \(q_{i}\).
This can be easily verified by computing the eigenvalues of the Hessian matrix for each candidate.
References
Adams SB, Frissell CA, Rieman BE, Johnson SL (2004) The role of land use and climate change in the decline of native Pacific trout in the interior Columbia River Basin, USA. Landsc Ecol 19:513–530
Aarestrup K, Lucas MC, Olsen M (2019) The challenges and potential benefits of the use of environmental DNA (eDNA) for monitoring of aquatic populations. Genes 10:1–18
Başar T, Olsder GJ (1995) Dynamic noncooperative game theory. Academic Press, San Diego
Benchekroun H (2003) Unilateral production restrictions in a dynamic duopoly. J Econ Theory 111:214–239
Benchekroun H (2008) Comparative dynamics in a productive asset oligopoly. J Econ Theory 138:237–261
Benchekroun H, Long NV (1998) Efficiency inducing taxation for polluting oligopolists. J Public Econ 70:325–342
Benchekroun H, Long NV (2016) Status concern and the exploitation of common-pool renewable resources. Ecol Econ 125:70–82
Benchekroun H, Chaudhuri RA (2014) Transboundary pollution and clean technologies. Resour Energy Econ 36:601–619
Benchekroun H, Chaudhuri RA, Tasneem D (2020) On the impact of trade in a common property renewable resource oligopoly. J Environ Econ Manag 101:102304
Berndtsson JC, Bengtsson L (2015) Green roof performance towards management of runoff water quantity and quality: a review. Ecol Eng 77:156–167
Bolle F (1986) On the oligopolistic extraction of non-renewable common-pool resources. Economica 53:519–527
Boucekkine R, Fabbri G, Federico S, Gozzi F (2022) Managing spatial linkages and geographic heterogeneity in dynamic models with transboundary pollution. J Math Econ 98:102577
Boucekkine R, Fabbri G, Federico S, Gozzi F (2022) A dynamic theory of spatial externalities. Games Econ Behav 132:133–165
Breton M, Dahmouni I, Zaccour G (2019) Equilibria in a two-species fishery. Math Biosci 309:78–91
Colombo L, Labrecciosa P (2013) On the convergence to the Cournot equilibrium in a productive asset oligopoly. J Math Econ 49:441–445
Colombo L, Labrecciosa P (2015) On the Markovian efficiency of Bertrand and Cournot equilibria. J Econ Theory 155:332–358
Colombo L, Labrecciosa P (2018) Consumer surplus-enhancing cooperation in a natural resource oligopoly. J Environ Econ Manag 92:185–193
Colombo L, Labrecciosa P (2019) Stackelberg versus Cournot: a differential game approach. J Econ Dyna Control 101:239–261
Colombo L, Labrecciosa P (2022) Product quality differentiation in a renewable resource oligopoly. J Environ Econ Manag 111:102583
de Frutos J, López-Pérez PL, Martín-Herrán G (2021) Equilibrium strategies in a multiregional transboundary pollution differential game with spatially distributed controls. Automatica 125:109411
de Frutos J, Martín-Herrán G (2019) Spatial effects and strategic behavior in a multiregional transboundary pollution dynamic game. J Environ Econ Manag 97:182–207
Dockner E, Jørgensen S, Mehlmann A (1989) Noncooperative solutions for a differential game model of fishery. J Econ Dyn Control 13:1–20
Dockner E, Jørgensen S, Sorger Long NVG (2000) Differential games in economics and management science. Cambridge University Press, Cambridge
Dockner E, Long NV (1993) International pollution control: cooperative vs non cooperative strategies. J Environ Econ Manag 24:13–29
Dunham JB, Rieman BE (1999) Metapopulation structure of bull trout: influence of physical, biotic, and geometrical landscape characteristics. Ecol Appl 9:642–655
Fischer RD, Mirman LJ (1992) Strategic dynamic interaction: fish wars. J Econ Dyn Control 16:267–287
Fischer RD, Mirman LJ (1996) The compleat fish wars: biological and dynamic interactions. J Environ Econ Manag 30:34–42
Foster S, Garduño H, Evans R (2016) Groundwater in rural development: facing the challenges of supply and resource sustainability. Water Int 41:143–163
Haurie A, Krawczyzk JB, Zaccour G (2012) Games and dynamic games. World Scientific, Singapore
Hockley FA, Jones JI (2018) Ecological impacts of flow regulation and water abstraction on river ecosystems: a review. Freshw Biol 63:1090–1113
Hummels D, Ishii J, Yi KM (2001) The nature and growth of vertical specialization in world trade. J Int Econ 54:75–96
Jørgensen S, Yeung DK (1996) Stochastic differential game model of a common property fishery. J Optim Theory Appl 90:381–403
Jørgensen S, Martín-Herrán GG, Zaccour G (2010) Dynamic games in the economics and management of pollution. Environ Model Assess 15:433–467
Jun B, Vives X (2004) Strategic incentives in dynamic duopoly. J Econ Theory 116:249–281
Karp L (2017) Natural resource as capital. MIT Press, Cambridge
Kemp MC, Long NV (1980) Resource extraction under common access. In: Kemp MC, Long NV (eds) Exhaustible resources, optimality and trade. North Holland, Amsterdam, pp 128–133
Kossioris G, Plexousakis M, Xepapadeas A, de Zeeuw A (2011) On the optimal taxation of common-pool resources. J Econ Dyn Control 35:1868–1879
Koulovatianos C, Mirman LJ (2007) The effects of market structure on industry growth: Rivalrous non-excludable capital. J Econ Theory 133:199–218
Krugman P, Obstfeld M, Melitz M (2014) International economics: theory and policy, 10th ed. Pearson
Lal R (2015) Restoring soil quality to mitigate soil degradation. Sustainability 7:5875–5895
Long NV (2010) A survey of dynamic games in economics. World Scientific Publishing, Singapore
Long NV (2018) Resource economics. In: Başar T, Zaccour G (eds) Handbook of dynamic game theory. Springer, Cham
Mazalov VV, Rettieva AN (2010) Fish wars and cooperation maintenance. Ecol Model 221:1545–1553
Rettieva AN (2012) Stable coalition structure in bioresource management problem. Ecol Model 235:102–118
Roche KF, Budy P, Weber MA (2009) Effect of dams on fish assemblage structure in the upper Colorado River. Trans Am Fish Soc 138:570–585
Sandal LK, Steinshamn SI (2004) Dynamic Cournot-competitive harvesting of a common pool resource. J Econ Dyn Control 28:1781–1799
Schindler DE, Hilborn R (2015) Prediction, precaution, and policy under global change. Science 347:953–954
Silva AT, Lucas MC, Castro-Santos T, Katopodis C (2018) Assessing the impacts of barriers on fish passage and connectivity. Fish Fish 19:815–827
Thurow RF, Chandler GL, Rosgen DL (2003) Influences of human land use on stream habitat and salmonids in the lower Snake River Basin, northeastern Oregon. Trans Am Fish Soc 132:825–842
United Nations Conference on Trade and Development (UNCTAD) (2013) World Investment Report 2013: global value chains: Investment and Trade for Development, United Nations
Van der Ploeg F, De Zeuw AJ (1992) International aspects of pollution control. Environ Resour Econ 2:117–139
Vardar NB, Zaccour G (2020) Exploitation of a productive asset in the presence of strategic behavior and pollution externalities. Mathematics 8(10):1682. https://doi.org/10.3390/math8101682
Wang WK, Ewald CO (2010) A stochastic differential fishery game for a two species fish population with ecological interaction. J Econ Dyn Control 34:844–857
Yanase A, Kamei K (2022) Dynamic game of international pollution control with general oligopolistic equilibrium: Neary meets Dockner and Long. Dyn Games Appl 12:751–783
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We would like to thank two anonymous referees for insightful comments and suggestions. The usual disclaimer applies.
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In memory of Professor Long.
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Appendices
Appendix A: Proof of Proposition 1
Feedback equilibrium strategies must satisfy the following Hamilton–Jacobi–Bellman equations for the value functions \(V_{i}({\textbf{s}})\):
with \(i,j=1,2\), \(j\ne i\). The necessary (and sufficient) condition for an interior solution of the maximization of the right-hand side of (A.1) impliesFootnote 11:
Substitution of (A.2) into (A.1) gives a system of partial differential equations for \(V_{1}({\textbf{s}})\) and \(V_{2}({\textbf{s}})\). Given the linear-quadratic structure of the game, we guess value functions of the form:
where \(\kappa _{0},\kappa _{1},\kappa _{2},\kappa _{11},\kappa _{12},\kappa _{22}\) are coefficients to be identified. It follows that:
which implies the following linear strategy:
Consider the following solution:
and
By inserting (A.7), (A.10) and (A.11) into (A.5), we obtain \(q_{i}=\phi _{i}^{*}\), with \(\phi _{i}^{*}\) given in Proposition 1. It can be easily checked that for \(q_{i}=\phi _{i}^{*}\), the value functions \(V_{i}({\textbf{s}})\) with coefficients (A.6)–(A.11) satisfy the Hamilton–Jacobi–Bellman equations (A.1).
Appendix B: Proof of Corollary 1
Altogether, there exist six pairs of candidates for a feedback equilibrium, resulting from the standard application of the “undetermined coefficient technique,” with equilibrium strategies of the form \(\phi _{i}^{*}=\lambda _{0}s_{i}+\lambda _{1}s_{j}+\lambda _{2}\), \(i,j=1,2\), \(j\ne i\). Within the class of linear, symmetric, stationary strategies, the candidates for a feedback equilibrium are:
and
As usual in the literature, out of these six pairs, we select the one(s) inducing trajectories of the asset stocks that converge to globally asymptotically stable steady states. It can be easily checked that the only pair of strategies stabilizing the states for every possible initial conditions is (B.5), corresponding to \(\phi _{i}^{*}\) given in Proposition 1. Candidates (B.1) and (B.4) induce unstable trajectories, while candidates (B.2), (B.3) and (B.6) induce trajectories that converge to stationary points only for specific initial conditions. For candidates (B.2), (B.3) and (B.6), saddle path stability requires \(\alpha >r+\gamma \), \( \alpha >3r/2\), and \(\alpha >r\), respectively.Footnote 12 Note that candidate (B.1) corresponds to the open-loop (state-independent) solution. All of the other candidates are clearly state-dependent.
The steady-state levels of firm i’s asset stock associated with \((\phi _{1}^{*},\phi _{2}^{*})\) are given by:
with \(i=1,2\). From (B.7), we obtain:
where considering (B.5), \(\lambda _{0}^{*}\), \(\lambda _{1}^{*} \), and \(\lambda _{2}^{*}\) are the coefficient of \(s_{i}\), the coefficient of \(s_{j}\), and the constant, respectively. It is immediate to check that \(s_{\infty }\) given in (B.8) corresponds to \(s_{\infty }\) given in Corollary 1, and that \(s_{\infty }>0\) for \(\gamma \) sufficiently small. The (per firm) level of production associated with \(s_{\infty }\) is:
By computing the eigenvalues of the Hessian matrix, it can be verified that the steady state \(s_{\infty }\) is globally asymptotically stable if \(\alpha >r\) and \(\gamma <{\widehat{\gamma }}\), with \({\widehat{\gamma }}\)
or, equivalently, \(\alpha >{\widehat{\alpha }}\), with \({\widehat{\alpha }}\) given in Corollary 1. Global asymptotic stability implies that for any initial asset stocks \(s_{1,0}\), \(s_{2,0}\) (such that an interior solution exists) the couple of equilibrium strategies \((\phi _{1}^{*},\phi _{2}^{*})\) induces trajectories of the asset stocks that converge asymptotically to \(s_{\infty }\).
Appendix C: Proof of Proposition 2
As to (i), we have
which is decreasing in \(s_{0}\) and nil at \(s_{0}={\widetilde{s}}_{0}\), with
Hence,
However, since \({\widetilde{s}}_{0}>{\overline{s}}_{0}=(5\alpha -2r)/[6\alpha \left( 2\alpha -r\right) ]\), then
As to (ii), we have
which is concave in \(s_{0}\) and nil at \(s_{0}={\underline{s}}_{0,cs}\) and \( s_{0}={\overline{s}}_{0,cs}\), with
and
It follows that
As to (iii), we have
which is convex in \(s_{0}\) and nil at \(s_{0}={\underline{s}}_{0,\pi }\) and \( s_{0}={\overline{s}}_{0,\pi }\), with
and
It follows that
Finally, we have
which is convex in \(s_{0}\) and nil at \(s_{0}={\underline{s}}_{0,w}\) and \(s_{0}= {\overline{s}}_{0,w}\), with
and
Hence,
Appendix D: Proof of Proposition 3
As to (i), we have
since \(\alpha >r\) is required for the stability of the steady state.
As to (ii), we have
The steady-state price increases since p(Q) is decreasing in its argument.
As to (iii), we have
As to (iv), we have
Finally, we have
Appendix E: Proof of Proposition 4
As to (i), from (5), we have
since \(\alpha >r\) is required for the stability of the steady state, and the expression in square brackets is increasing in \(s_{0}\) and nil at
As to (ii), from (2), we have
As to (iii), from (6), we have
Appendix F: Proof of Proposition 5
As to (i), from (5), we have
since \(\alpha >r\) is required for the stability of the steady state, and the expression in square brackets is increasing in \(s_{0}\) and nil at
As to (ii), from (2), we have
As to (iii) and (iv), from (6) and (8), we have
with
Hence,
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Colombo, L., Labrecciosa, P. Resource Mobility and Market Performance. Dyn Games Appl 14, 78–96 (2024). https://doi.org/10.1007/s13235-023-00517-8
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DOI: https://doi.org/10.1007/s13235-023-00517-8
Keywords
- Resource mobility
- Spatially connected resources
- Social welfare
- Differential oligopoly games
- Feedback equilibrium