# American Institute of Mathematical Sciences

November  2021, 26(11): 5755-5767. doi: 10.3934/dcdsb.2021179

## Environmental degradation and indeterminacy of equilibrium selection

 1 Department of Economic and Business Sciences, University of Sassari, Sassari, Italy 2 Department of Statistics, Informatics and Applications, University of Florence, Italy 3 INdAM (National Institute of High Mathematics), Group of Analysis and Probability, Rome, Italy 4 Department of Law, University of Naples, Federico II, Naples, Italy 5 Department of Finance, Faculty of Economics, Technical University of Ostrava, Ostrava, Czech Republic

* Corresponding author: mauro.sodini@unipi.it

Received  December 2020 Revised  May 2021 Published  November 2021 Early access  August 2021

This paper analyzes an intertemporal optimization problem in which agents derive utility from three goods: leisure, a public environmental good and the consumption of a produced good. The global analysis of the dynamic system generated by the optimization problem shows that global indeterminacy may arise: given the initial values of the state variables, the economy may converge to different steady states, by choosing different initial values of the control variable.

Citation: Angelo Antoci, Marcello Galeotti, Mauro Sodini. Environmental degradation and indeterminacy of equilibrium selection. Discrete and Continuous Dynamical Systems - B, 2021, 26 (11) : 5755-5767. doi: 10.3934/dcdsb.2021179
##### References:
 [1] A. Antoci, M. Galeotti and P. Russu, Poverty trap and global indeterminacy in a growth model with open-access natural resources, Journal of Economic Theory, 146 (2011), 569-591.  doi: 10.1016/j.jet.2010.12.003. [2] A. Antoci and S. Borghesi, Preserving or escaping? On the welfare effects of environmental self-protective choices, Journal of Socio-Economics, 41 (2012), 248-254. [3] A. Antoci, L. Gori, M. Sodini and E. Ticci, Maladaptation and global indeterminacy, Environment and Development Economics, 24 (2019), 643-659. [4] A. Antoci, S. Borghesi, M. Galeotti and P. Russu, Living in an uncertain world: Environment substitution, local and global indeterminacy, Journal of Economic Dynamics and Control, 126 (2021), 103929. doi: 10.1016/j.jedc.2020.103929. [5] G. Bella, P. Mattana and B. Venturi, Shilnikov chaos in the Lucas model of endogenous growth, Journal of Economic Theory, 172 (2017), 451-477.  doi: 10.1016/j.jet.2017.09.010. [6] G. Bella and P. Mattana, Global indeterminacy and equilibrium selection in a model with depletion of non-renewable resources, Decisions Economics Finance, 41 (2018), 187-202.  doi: 10.1007/s10203-018-0218-z. [7] J. Benhabib and R. E. Farmer, Indeterminacy and sunspots in macroeconomics, in Handbook of Macroeconomics, North-Holland, Amsterdam, (1999), 387–448. [8] A. Caravaggio and M. Sodini, Nonlinear dynamics in coevolution of economic and environmental systems, Frontiers in Applied Mathematics and Statistics, 4 (2018), 1-17. [9] O. A. Carboni and P. Russu, Linear production function, externalities and indeterminacy in a capital-resource growth model, Journal of Mathematical Economics, 49 (2013), 422-428.  doi: 10.1016/j.jmateco.2013.04.002. [10] E. Fernández, R. Pérez and J. Ruiz, The environmental Kuznets curve and equilibrium indeterminacy, Journal of Economic Dynamics and Control, 36 (2012), 1700-1717.  doi: 10.1016/j.jedc.2012.05.004. [11] J. M. Hartwick, Intergenerational equity and the investing of rents from exhaustible resources, The American Economic Review, 67 (1977), 972-974. [12] J. M. Hartwick, Substitution among exhaustible resources and intergenerational equity, The Review of Economic Studies, 45 (1978), 347-354.  doi: 10.2307/2297349. [13] P. Krugman, History versus expectations, Quarterly Journal of Economics, 106 (1991), 651-667. [14] P. Mattana, K. Nishimura and T. Shigoka, Homoclinic bifurcation and global indeterminacy of equilibrium in a two-sector endogenous growth model, International Journal of Economic Theory, 5 (2009), 25-47. [15] K. Mino, Growth and Business Cycles with Equilibrium Indeterminacy, Advances in Japanese Business and Economics, Springer, Tokyo Japan, 2017. [16] P. Russu, Mathematical analysis of an economic growth model with perfect-substitution technologies, Nonlinear Analysis: Modelling and Control, 25 (2020), 84-107.  doi: 10.15388/namc.2020.25.15733. [17] S. P. Sethi, Nearest feasible paths in optimal control problems: Theory, examples and counterexamples, Journal of Optimization Theory and Applications, 23 (1977), 563-579.  doi: 10.1007/BF00933297. [18] A. Skiba, Optimal growth with a convex-concave production function, Econometrica, 46 (1978), 527-540.  doi: 10.2307/1914229. [19] R. M. Solow, Intergenerational equity and exhaustible resources, Review of Economic Studies: Symposium of the Economics of Exhaustible Resources, (1974), 29–46. [20] R. M. Solow, On the intergenerational allocation of natural resources, Scandinavian Journal of Economics, 88 (1986), 141-9. [21] R. M. Solow, An almost practical step towards sustainability, Resources Policy, 16 (1993), 162-72. [22] F. Wirl, Stability and limit cycles in one-dimensional dynamic optimizations of competitive agents with a market externality, Journal of Evolutionary Economics, 7 (1997), 73-89. [23] A. Yanase, Impatience, pollution, and indeterminacy, Journal of Economic Dynamics and Control, 35 (2011), 1789-1799.  doi: 10.1016/j.jedc.2011.06.010.

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##### References:
 [1] A. Antoci, M. Galeotti and P. Russu, Poverty trap and global indeterminacy in a growth model with open-access natural resources, Journal of Economic Theory, 146 (2011), 569-591.  doi: 10.1016/j.jet.2010.12.003. [2] A. Antoci and S. Borghesi, Preserving or escaping? On the welfare effects of environmental self-protective choices, Journal of Socio-Economics, 41 (2012), 248-254. [3] A. Antoci, L. Gori, M. Sodini and E. Ticci, Maladaptation and global indeterminacy, Environment and Development Economics, 24 (2019), 643-659. [4] A. Antoci, S. Borghesi, M. Galeotti and P. Russu, Living in an uncertain world: Environment substitution, local and global indeterminacy, Journal of Economic Dynamics and Control, 126 (2021), 103929. doi: 10.1016/j.jedc.2020.103929. [5] G. Bella, P. Mattana and B. Venturi, Shilnikov chaos in the Lucas model of endogenous growth, Journal of Economic Theory, 172 (2017), 451-477.  doi: 10.1016/j.jet.2017.09.010. [6] G. Bella and P. Mattana, Global indeterminacy and equilibrium selection in a model with depletion of non-renewable resources, Decisions Economics Finance, 41 (2018), 187-202.  doi: 10.1007/s10203-018-0218-z. [7] J. Benhabib and R. E. Farmer, Indeterminacy and sunspots in macroeconomics, in Handbook of Macroeconomics, North-Holland, Amsterdam, (1999), 387–448. [8] A. Caravaggio and M. Sodini, Nonlinear dynamics in coevolution of economic and environmental systems, Frontiers in Applied Mathematics and Statistics, 4 (2018), 1-17. [9] O. A. Carboni and P. Russu, Linear production function, externalities and indeterminacy in a capital-resource growth model, Journal of Mathematical Economics, 49 (2013), 422-428.  doi: 10.1016/j.jmateco.2013.04.002. [10] E. Fernández, R. Pérez and J. Ruiz, The environmental Kuznets curve and equilibrium indeterminacy, Journal of Economic Dynamics and Control, 36 (2012), 1700-1717.  doi: 10.1016/j.jedc.2012.05.004. [11] J. M. Hartwick, Intergenerational equity and the investing of rents from exhaustible resources, The American Economic Review, 67 (1977), 972-974. [12] J. M. Hartwick, Substitution among exhaustible resources and intergenerational equity, The Review of Economic Studies, 45 (1978), 347-354.  doi: 10.2307/2297349. [13] P. Krugman, History versus expectations, Quarterly Journal of Economics, 106 (1991), 651-667. [14] P. Mattana, K. Nishimura and T. Shigoka, Homoclinic bifurcation and global indeterminacy of equilibrium in a two-sector endogenous growth model, International Journal of Economic Theory, 5 (2009), 25-47. [15] K. Mino, Growth and Business Cycles with Equilibrium Indeterminacy, Advances in Japanese Business and Economics, Springer, Tokyo Japan, 2017. [16] P. Russu, Mathematical analysis of an economic growth model with perfect-substitution technologies, Nonlinear Analysis: Modelling and Control, 25 (2020), 84-107.  doi: 10.15388/namc.2020.25.15733. [17] S. P. Sethi, Nearest feasible paths in optimal control problems: Theory, examples and counterexamples, Journal of Optimization Theory and Applications, 23 (1977), 563-579.  doi: 10.1007/BF00933297. [18] A. Skiba, Optimal growth with a convex-concave production function, Econometrica, 46 (1978), 527-540.  doi: 10.2307/1914229. [19] R. M. Solow, Intergenerational equity and exhaustible resources, Review of Economic Studies: Symposium of the Economics of Exhaustible Resources, (1974), 29–46. [20] R. M. Solow, On the intergenerational allocation of natural resources, Scandinavian Journal of Economics, 88 (1986), 141-9. [21] R. M. Solow, An almost practical step towards sustainability, Resources Policy, 16 (1993), 162-72. [22] F. Wirl, Stability and limit cycles in one-dimensional dynamic optimizations of competitive agents with a market externality, Journal of Evolutionary Economics, 7 (1997), 73-89. [23] A. Yanase, Impatience, pollution, and indeterminacy, Journal of Economic Dynamics and Control, 35 (2011), 1789-1799.  doi: 10.1016/j.jedc.2011.06.010.
Global indeterminacy scenario. Parameter values: $\overline{E} = 1.3$; $\alpha = 0.3$, $\beta = 24.84$, $\gamma = 1.3$, $\delta = 1.06,$ $\varepsilon = 0.4,$ $\rho = 0.04$. Initial values of the state variables $E(0) = 1.04025820702807;$ $K(0) = 5.99124478008392$. For $L(0) = 0.425697415099600$ the trajectory evolves on the stable manifold of $P_{2}$ converging to the saddle $P_{2}$. For a larger value of $L(0)$ (in figure $L(0) = 0.681115864159360$) trajectories converge to the sink $P_{1}$. For lower values of $L(0)$ (in figure $L(0) = 0.383127673589640$) trajectories reach the plane $K = 0$ in finite time. The dashed curve shows the branch of the unstable manifold of $P_{2}$ converging to the sink $P_{1}$.
Hopf bifurcation and global indeterminacy. The parameter values are the same as in the previous figure except for $\delta = 1.035.$ Trajectories converging to the limit cycle $\Gamma$ surrounding $P_{1}$ coexist with trajectories on the stable manifold of the saddle $P_{2}$. Initial values of the state variables for the trajectories in figure are $E(0) = 2.20617886672602,$ $K(0) = 5.35230152562421$. The red trajectory converging to the saddle $P_{2}$ starts with $L(0) = 0.605972067919661$, the one converging to $\Gamma$ with $L(0) = 0.666569274711627$.
Evolution of the attractor of the system. The parameter values are the same as in the previous figures except for $\delta$. Starting from $\delta = \delta_{1}>1.0374$ for which $P_{1} = \left( 6.225232325,0.3491884150,0.35\right)$ is a sink (coordinates of the stationary states are independent of $\delta$), if we let $\delta$ decrease, $P_{1}$ undergoes a supercritical Hopf bifurcation ($\delta\simeq1.0374$). Letting $\delta$ decrease further we observe an expansion of the attractive limit cycle surrounding $P_{1}$. The labels near the closed curves indicate the values of $\delta$ in the simulation: $\delta_{2} = 1.035,$ $\delta _{3} = 1.03,$ $\delta_{4} = 1.028,$ $\delta_{5} = 1.027$.
Case $\delta<1.$ The parameter values are the same as in the previous figures except for $\delta = 0.95$.No attractor exists$.$ The phasespace is projected on the plane $K,E$. Three trajectories are depicted starting from the same initial values of the state variables $K,$ $E$ ($K(0) =$ $11.8279$, $E(0) = 0.6949$), but with different initial values of the jumping variable $L.$ The dashed one reaches (in finite time) the plane $K = 0$ ($L(0) = L_{1} = 0.2800$); the dash-dot one reaches (in finite time) the plane $E = 0$ ($L(0) = L_{2} = 0.4550$); the solid one tends (in infinite time) to the saddle point $P_{2}$ ($L(0) = L_{3} = 0.3585$).
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