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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(ct+1, Et+1) = ln(ct+1) + η ln(Et+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 wt. She invests mA

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: Ct+1 = (1 + Rt+1 − δ)st (1 − τ)wt = st + mA

t

Ct+1, mA

t , st ≥ 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), Et+1 = (1 − b)Et − β ct + γ (mA

t + τ wt);

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: Yt = F(Kt) = Akα

t

and the profit maximisation implies that wt = A(1 − α)K α−1

t

rt = Aα K α−1

t

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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, m∗

A = ((α − 1)((τ − 1)η + τ)γ + βα)Akα t − k(δ − 1)β + (b − 1)Et

γ(η + 1) where ∂m∗

A

∂τ < 0, from which we obtain that

• m∗

A > 0

for E < E m∗

A = 0

• therwise

where

• E = ((α − 1)((τ − 1)η + τ)γ + β α)AKtα − Kt(δ − 1)β

1 − b

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The 1-D dynamical system in Et

In the case of positive priva expenditure (m∗

A > 0), the relationship

Kt+1 = Et+1 ηγ follows by the market clearing condition Kt+1 = st. Then, similarly to Zhang (1999) we have to analyse the 1-D dynamic system Et+1 = η η + 1

• ((1 − α)γ − β)γ A

Et δ γ α + ((1 − b)ηγ + (1 − δ)β)Et

• 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 E ∗

1 =

β(δ − 1) − γ(1 + bη) A(βα − γ(1 − α))

• 1

1−α

ηγ, E ∗

2 = 0

Proposition The map admits the fixed point E ∗

1 with m∗ A > 0 if and only if

βα 1 − α < γ < β (1 − α)(bδ(1 − τ) − τ) The fixed point E ∗

1 is not stable for every configuration of parameters and

the map may undergo a Flip bifurcation.

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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,

Et+1 = (1 − b)Et − βt ct + γ mA

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, m∗

A > 0

for E <

• E

where

• E = ((α−1)((τ−1)η+τ)γ+βt α)AKt α−Kt(δ−1)βt

1−b

.

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Two different specifications for βt+1

We introduce the following specification for the dynamics of the environmental impact of consumption βt+1 = βt +

• β0

1 + φi − βt

• φ

with i = 1, 2 where φ1 = τ wt and φ2 = τ wt

Yt

(alternative approaches) and β0 > 0 is the ”natural level” of environmental impact.

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The dynamic system with φ1

By assuming to have positive private contribution mA

t , the direct relation

Kt+1 = Et+1

ηγ

continues to hold and then we have a 2-D dynamic system in E and β. In particular, we have:        Et+1 =

η η+1

• A((α − 1)(τ − 1)γ − βt α)

Et

η γ

α + Et (δ−1)βt

η γ

− (b − 1)Et

• βt+1 = βt −
• β0

(1−τ A( Et

η γ )α(α−1)) − βt

• τ A( Et

η γ )α(α − 1)

(1)

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The dynamic system with φ1 (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: E ∗ = βt(δ − 1) − γ(1 + bη) A(βtα − γ(1 − α))

• 1

1−α

ηγ β∗ = β0 1 − τ A( Et

η γ )α(α − 1)

Andrea Caravaggio (University of Pisa) Competition and cooperation September 4, 2019 13 / 17

The dynamic system with φ2

       Et+1 =

η η+1

• A((α − 1)(τ − 1)γ − βt α)

Et

η γ

α + Et (δ−1)βt

η γ

− (b − 1)Et

• βt+1 = βt −
• β0

(1−τ (α−1)) − βt

• τ (α − 1)

(2)

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The dynamic system with φ2 (2)

The dynamic system (2) is triangular and it is possible find fixed points in closed form E ∗ = (−1 + τ (α − 1)) (bη + 1) γ1 + (δ − 1) B0 ((α − 1) (τ − 1) (α τ − τ − 1) γ1 + B0 α) A

• 1

1−α

, 0 β∗ = β0 1 + τ1 − α)

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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, β0 = 0.001.

we have that J11(E ∗, βt) < −1 for an appropriate parameters configuration, then Flip bifucations may occur.

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Thank you

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