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Competition and cooperation between public and private sectors in environmental maintenance Andrea Caravaggio Mauro Sodini University of Pisa September 4, 2019 Andrea Caravaggio (University of Pisa) Competition and cooperation September 4, 2019 1 / 17

Outline 1 Brief Literature review; 2 The John and Pecchenino’s model with environmental taxation; 3 The zero-maintenance case and the 1-D dynamic system; 4 The model with endogenous environmental impact; 5 Possible extensions and remarks. Andrea Caravaggio (University of Pisa) Competition and cooperation September 4, 2019 2 / 17

Literature review An overlapping generations model of growth and the environment (John and Pecchenino, 1994); Short-lived agents and the long-lived environment (John et al., 1995); Optimal tax schemes and the environmental externality (Ono, 1996); Environmental sustainability, nonlinear dynamics and chaos (Zhang, 1999); Multiple attractors and nonlinear dynamics in an overlapping generations model with environment (Naimzada and Sodini, 2010); Maladaptation and global indeterminacy (Antoci et al., 2019). Andrea Caravaggio (University of Pisa) Competition and cooperation September 4, 2019 3 / 17

The model (1) We consider an OLG framework (Diamond, 1965) in which: The utility function of the representative agent is given by: U ( c t +1 , E t +1 ) = ln( c t +1 ) + η ln( E t +1 ); where η > 0 is the given discount factor; The individual works only in young age, supplying one unit of labour and earning a real wage w t . She invests m A t for improvement in environmental quality and the government levies a tax on wage at the rate 0 < τ < 1. Then, the life-cycle budget constraints are: C t +1 = (1 + R t +1 − δ ) s t (1 − τ ) w t = s t + m A t C t +1 , m A t , s t ≥ 0 . Andrea Caravaggio (University of Pisa) Competition and cooperation September 4, 2019 4 / 17

The model (2) The environmental quality evolves according to John and Pecchenino (1994), E t +1 = (1 − b ) E t − β c t + γ ( m A t + τ w t ); where b ∈ (0 , 1) measures the autonomous decay of the environment while β, γ > 0 measure the negative impact of the consumption and the positive impact of the environmental improvement, respectively. A unique material good is produced by a representative firm using a Cobb-Douglas technology: Y t = F ( K t ) = Ak α t and the profit maximisation implies that w t = A (1 − α ) K α − 1 t r t = A α K α − 1 t Andrea Caravaggio (University of Pisa) Competition and cooperation September 4, 2019 5 / 17

Existence of a Zero-maintenance manifold The maximisation of the utility function under the intertemporal budget constraint allows to obtain the optimal mix of saving and maintenance private expenditure. In particular, A = (( α − 1)(( τ − 1) η + τ ) γ + βα ) Ak α t − k ( δ − 1) β + ( b − 1) E t m ∗ γ ( η + 1) where ∂ m ∗ ∂τ < 0, from which we obtain that A � E < � A > 0 m ∗ for E A = 0 m ∗ otherwise where E = (( α − 1)(( τ − 1) η + τ ) γ + β α ) AK t α − K t ( δ − 1) β � 1 − b Andrea Caravaggio (University of Pisa) Competition and cooperation September 4, 2019 6 / 17

The 1-D dynamical system in E t In the case of positive priva expenditure ( m ∗ A > 0), the relationship K t +1 = E t +1 ηγ follows by the market clearing condition K t +1 = s t . Then, similarly to Zhang (1999) we have to analyse the 1-D dynamic system � � E t � α � η E t +1 = ((1 − α ) γ − β ) γ A + ((1 − b ) ηγ + (1 − δ ) β ) E t η + 1 δ γ Andrea Caravaggio (University of Pisa) Competition and cooperation September 4, 2019 7 / 17

Stationary Points: Existence and stability By calculating the steady states of the system, we have � β ( δ − 1) − γ (1 + b η ) � 1 1 − α E ∗ 1 = E ∗ 2 = 0 ηγ, A ( βα − γ (1 − α )) Proposition A > 0 if and only if The map admits the fixed point E ∗ 1 with m ∗ βα β 1 − α < γ < (1 − α )( b δ (1 − τ ) − τ ) The fixed point E ∗ 1 is not stable for every configuration of parameters and the map may undergo a Flip bifurcation. Andrea Caravaggio (University of Pisa) Competition and cooperation September 4, 2019 8 / 17

Complex behaviour of the system for high levels of β According to Zhang (1999), complex dynamics may emerge as the environmental impact of consumption ( β ) increases. (a) (b) Figure: (a) α = 0 . 09 , τ = 0 . 4 , γ = 1 . 33 , A = 2 , η = 0 . 03 , b = 0 . 987 , δ = 0 . 03; (b) the graph performed for β = 1 . 86. Andrea Caravaggio (University of Pisa) Competition and cooperation September 4, 2019 9 / 17

The model with endogenous environmental impact of consumption We consider now that (i) the environmental impact of consumption is not constant and (ii) the public expenditure for environment does not enter directly the environment dynamics but it is employed in reducing the impact. Then, E t +1 = (1 − b ) E t − β t c t + γ m A t as in the previous specification, agents in economies with little capital or with high environmental quality may choose not to engage in environmental maintenance. Then, E < � � m ∗ A > 0 for E where � E = (( α − 1)(( τ − 1) η + τ ) γ + β t α ) AK t α − K t ( δ − 1) β t � . 1 − b Andrea Caravaggio (University of Pisa) Competition and cooperation September 4, 2019 10 / 17

Two different specifications for β t +1 We introduce the following specification for the dynamics of the environmental impact of consumption � � β 0 β t +1 = β t + − β t φ with i = 1 , 2 1 + φ i where φ 1 = τ w t and φ 2 = τ w t (alternative approaches) and β 0 > 0 is the Y t ”natural level” of environmental impact. Andrea Caravaggio (University of Pisa) Competition and cooperation September 4, 2019 11 / 17

The dynamic system with φ 1 By assuming to have positive private contribution m A t , the direct relation K t +1 = E t +1 continues to hold and then we have a 2-D dynamic system in ηγ E and β . In particular, we have:  � � � E t � α + E t ( δ − 1) β t   η E t +1 = A (( α − 1)( τ − 1) γ − β t α ) − ( b − 1) E t  η +1 η γ η γ � �   β 0 τ A ( E t η γ ) α ( α − 1) β t +1 = β t − η γ ) α ( α − 1)) − β t  (1 − τ A ( Et (1) Andrea Caravaggio (University of Pisa) Competition and cooperation September 4, 2019 12 / 17

The dynamic system with φ 1 (2) By analysing the system (1), we are not able to find fixed points in a closed form and (at this stage) we have proved the existence of couples ( E ∗ , β ∗ ) solving the system only numerically. We report the general form of the equilibrium: � β t ( δ − 1) − γ (1 + b η ) � 1 β 0 1 − α E ∗ = β ∗ = ηγ 1 − τ A ( Et A ( β t α − γ (1 − α )) η γ ) α ( α − 1) Andrea Caravaggio (University of Pisa) Competition and cooperation September 4, 2019 13 / 17

The dynamic system with φ 2  � � � E t � α + E t ( δ − 1) β t   η E t +1 = A (( α − 1)( τ − 1) γ − β t α ) − ( b − 1) E t  η +1 η γ η γ � �   β 0  β t +1 = β t − (1 − τ ( α − 1)) − β t τ ( α − 1) (2) Andrea Caravaggio (University of Pisa) Competition and cooperation September 4, 2019 14 / 17

The dynamic system with φ 2 (2) The dynamic system (2) is triangular and it is possible find fixed points in closed form � ( − 1 + τ ( α − 1)) ( b η + 1) γ 1 + ( δ − 1) B0 � 1 1 − α E ∗ = , 0 (( α − 1) ( τ − 1) ( α τ − τ − 1) γ 1 + B0 α ) A β 0 β ∗ = 1 + τ 1 − α ) Andrea Caravaggio (University of Pisa) Competition and cooperation September 4, 2019 15 / 17

Complex behaviour as productivity A varies Figure: (a) α = 0 . 16 , τ = 0 . 75 , γ = 1 . 33 , A = 2 , η = 0 . 034 , b = 0 . 987 , δ = 0 . 003 , β 0 = 0 . 001. we have that J 11 ( E ∗ , β t ) < − 1 for an appropriate parameters configuration, then Flip bifucations may occur. Andrea Caravaggio (University of Pisa) Competition and cooperation September 4, 2019 16 / 17

Thank you Andrea Caravaggio (University of Pisa) Competition and cooperation September 4, 2019 17 / 17