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doi: 10.3934/dcdsb.2021179
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Environmental degradation and indeterminacy of equilibrium selection

Angelo Antoci 1, , Marcello Galeotti 2,3, and  Mauro Sodini 4,5,,

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 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.

Keywords: Economic growth model, environmental depletion, global analysis, global indeterminacy, poverty trap.
Mathematics Subject Classification: 34C60; 34C23; 49J15.
Citation: Angelo Antoci, Marcello Galeotti, Mauro Sodini. Environmental degradation and indeterminacy of equilibrium selection. Discrete & Continuous Dynamical Systems - B, 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.  Google Scholar

[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.   Google Scholar

[3]

A. Antoci, L. Gori, M. Sodini and E. Ticci, Maladaptation and global indeterminacy, Environment and Development Economics, 24 (2019), 643-659.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

J. Benhabib and R. E. Farmer, Indeterminacy and sunspots in macroeconomics, in Handbook of Macroeconomics, North-Holland, Amsterdam, (1999), 387–448. Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

J. M. Hartwick, Intergenerational equity and the investing of rents from exhaustible resources, The American Economic Review, 67 (1977), 972-974.   Google Scholar

[12]

J. M. Hartwick, Substitution among exhaustible resources and intergenerational equity, The Review of Economic Studies, 45 (1978), 347-354.  doi: 10.2307/2297349.  Google Scholar

[13]

P. Krugman, History versus expectations, Quarterly Journal of Economics, 106 (1991), 651-667.   Google Scholar

[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.   Google Scholar

[15]

K. Mino, Growth and Business Cycles with Equilibrium Indeterminacy, Advances in Japanese Business and Economics, Springer, Tokyo Japan, 2017.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

A. Skiba, Optimal growth with a convex-concave production function, Econometrica, 46 (1978), 527-540.  doi: 10.2307/1914229.  Google Scholar

[19]

R. M. Solow, Intergenerational equity and exhaustible resources, Review of Economic Studies: Symposium of the Economics of Exhaustible Resources, (1974), 29–46. Google Scholar

[20]

R. M. Solow, On the intergenerational allocation of natural resources, Scandinavian Journal of Economics, 88 (1986), 141-9.   Google Scholar

[21]

R. M. Solow, An almost practical step towards sustainability, Resources Policy, 16 (1993), 162-72.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.   Google Scholar

[3]

A. Antoci, L. Gori, M. Sodini and E. Ticci, Maladaptation and global indeterminacy, Environment and Development Economics, 24 (2019), 643-659.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

J. Benhabib and R. E. Farmer, Indeterminacy and sunspots in macroeconomics, in Handbook of Macroeconomics, North-Holland, Amsterdam, (1999), 387–448. Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

J. M. Hartwick, Intergenerational equity and the investing of rents from exhaustible resources, The American Economic Review, 67 (1977), 972-974.   Google Scholar

[12]

J. M. Hartwick, Substitution among exhaustible resources and intergenerational equity, The Review of Economic Studies, 45 (1978), 347-354.  doi: 10.2307/2297349.  Google Scholar

[13]

P. Krugman, History versus expectations, Quarterly Journal of Economics, 106 (1991), 651-667.   Google Scholar

[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.   Google Scholar

[15]

K. Mino, Growth and Business Cycles with Equilibrium Indeterminacy, Advances in Japanese Business and Economics, Springer, Tokyo Japan, 2017.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

A. Skiba, Optimal growth with a convex-concave production function, Econometrica, 46 (1978), 527-540.  doi: 10.2307/1914229.  Google Scholar

[19]

R. M. Solow, Intergenerational equity and exhaustible resources, Review of Economic Studies: Symposium of the Economics of Exhaustible Resources, (1974), 29–46. Google Scholar

[20]

R. M. Solow, On the intergenerational allocation of natural resources, Scandinavian Journal of Economics, 88 (1986), 141-9.   Google Scholar

[21]

R. M. Solow, An almost practical step towards sustainability, Resources Policy, 16 (1993), 162-72.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

Figure 1.  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} $.
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Figure 2.  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 $.
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Figure 3.  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 $.
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Figure 4.  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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