# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021179
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## 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 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 & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021179
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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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