Robust Optimal Taxation and Environmental Externalities Xin Li - - PowerPoint PPT Presentation

Robust Optimal Taxation and Environmental Externalities

Robust Optimal Taxation and Environmental Externalities

Xin Li Borghan Narajabad Ted Loch-Temzelides

Department of Economics Rice University

Xin Li Borghan Narajabad Ted Loch-Temzelides Robust Optimal Taxation and Environmental Externalities

Robust Optimal Taxation and Environmental Externalities Outline

Outline

1 Introduction 2 One-Energy-Sector Model 3 Complete Model 4 Numerical Results

Xin Li Borghan Narajabad Ted Loch-Temzelides Robust Optimal Taxation and Environmental Externalities

Robust Optimal Taxation and Environmental Externalities Introduction

Outline

1 Introduction 2 One-Energy-Sector Model 3 Complete Model 4 Numerical Results

Xin Li Borghan Narajabad Ted Loch-Temzelides Robust Optimal Taxation and Environmental Externalities

Robust Optimal Taxation and Environmental Externalities Introduction

Research Question

Environmental Externality: CO2 Emission → Global Temperature → output. Model Uncertainty: Our knowledge is limited regarding how CO2 emission affects future global mean temperature and

• utput.

Robust Control: The fear of catastrophic events alters the “optimal” path of energy extraction and thereby production and consumption. We want to find out: Robust vesion of optimal energy extraction How to achieve it in a decentralized economy

Xin Li Borghan Narajabad Ted Loch-Temzelides Robust Optimal Taxation and Environmental Externalities

Robust Optimal Taxation and Environmental Externalities Introduction

Literature Review

Golosov, Hassler, Krusell, and Tsyvinski (GHKT, 2012):

• 1. Assume that the mapping from carbon concentration to
• utput damages is subject to a risk γ: y = e−γS ˜

y (the true distribution of γ is known).

• 2. Obtain analytical expressions for carbon externality.
• 3. Assess optimal use of energy based on average performance

(w.r.t. the know distribution)

• 4. Optimal taxation: A Pigouvian tax on carbon emission

with a rate equal to the marginal externality of carbon.

Xin Li Borghan Narajabad Ted Loch-Temzelides Robust Optimal Taxation and Environmental Externalities

Robust Optimal Taxation and Environmental Externalities Introduction

Literature Review

Hansen and Sargent (2002, 2008) — Robust Control:

• 1. Decision maker does not know the true distribution of γ,

but has an approximating one in his mind.

• 2. Only concerns about social (individual) walfare in the worst

case scenario.

• 3. Chooses the worst case distribution of γ around a

neighborhood centered at the approximating one.

• 4. Uses “entropy” as a measure of the deviation from the

approximating model.

• 5. Linear-Quadratic-Gaussian world.

Xin Li Borghan Narajabad Ted Loch-Temzelides Robust Optimal Taxation and Environmental Externalities

Robust Optimal Taxation and Environmental Externalities Introduction

Our Contribution

To the Literature of Environmental Externality:

• 1. Incorporate model uncertainty into GHKT (2012) and

characterize optimal allocation and tax as functions of the size

• f model uncertainty.
• 2. The concern of robustness matters both qualitatively and

qnantatively. To the Literature of Robust Control:

• 1. Study a dynamic model outside the LQG world.
• 2. Develop a simple method which divide the model into two

parts and solve them separately.

Xin Li Borghan Narajabad Ted Loch-Temzelides Robust Optimal Taxation and Environmental Externalities

Robust Optimal Taxation and Environmental Externalities One-Energy-Sector Model

Outline

1 Introduction 2 One-Energy-Sector Model 3 Complete Model 4 Numerical Results

Xin Li Borghan Narajabad Ted Loch-Temzelides Robust Optimal Taxation and Environmental Externalities

Robust Optimal Taxation and Environmental Externalities One-Energy-Sector Model

Basic Setup

Preference: E0

∞

• t=0

βtu(Ct) Final Good Production: F(K, E) = KθEν, θ + ν ≤ 1 where E denotes energy input measured in its carbon content. In addition, the extraction cost of E is zero. Capital Accumulation (with 100 percent depreciation): ˜ K′ = F(K, E) + (1 − δ)K − C Law of motion of atmospheric carbon concentration: S′ = S + φ0E

Xin Li Borghan Narajabad Ted Loch-Temzelides Robust Optimal Taxation and Environmental Externalities

Robust Optimal Taxation and Environmental Externalities One-Energy-Sector Model

Externality and Uncertainty

(elaborate) Introducing a stochastic variable γ, which reduces the end-of-period capital stock ˜ K′ by a factor of h(S′, γ) = e−S′γ to K′. That is, K′ = h(S′, γ) ˜ K′, The approximating distribution of γ is π(γ) = λe−λγ. Let ˆ π(γ) be an alternative distribution, and m(γ) = ˆ

π(γ) π(γ) be the

likelihood ratio. The “distance” between ˆ π(γ) and π(γ) is measured by relative entropy ρ(ˆ π(γ), π(γ)) ≡

• [m(γ) log m(γ)]π(γ)dγ.

Xin Li Borghan Narajabad Ted Loch-Temzelides Robust Optimal Taxation and Environmental Externalities

Robust Optimal Taxation and Environmental Externalities One-Energy-Sector Model

A Zero-Sum Dynamic Game

We first focus on a robust version of social planner’s problem and then show that the robust social optimal allocation is implemented via a Pigouvian tax in a decentralized economy. Two-person zero-sum dynamic game: In any period, The beginning-of-the-period capital K is given. The planner chooses E, production takes place and the planner chooses C. ˜ K′ and S′ evolve according to ˜ K′ = F(K, E) + (1 − δ)K − C and S′ = S + φ0E, respectively. The malevolent player chooses an alternative distribution ˆ π(γ)

• r, equivalently, m(γ), to minimize welfare. Accordingly, the

next period K′ = h(S′, γ) ˜ K′.

Xin Li Borghan Narajabad Ted Loch-Temzelides Robust Optimal Taxation and Environmental Externalities

Robust Optimal Taxation and Environmental Externalities One-Energy-Sector Model

A Zero-Sum Dynamic Game (Continued)

(elaborate) V (K, S) = max

{C,E, ˜ K′,S′}

min

m(γ){u(C)

+β m(γ)V (K′, S′) + αm(γ) log m(γ)

• π(γ)dγ}

s.t. ˜ K′ = F(K, E) − C K′ = h(S′, γ) ˜ K′ S′ = S + φ0E 1 =

• m(γ)π(γ)dγ

Xin Li Borghan Narajabad Ted Loch-Temzelides Robust Optimal Taxation and Environmental Externalities

Robust Optimal Taxation and Environmental Externalities One-Energy-Sector Model

A Zero-Sum Dynamic Game (Continued)

• αm(γ) log m(γ)π(γ)dγ = αρ(ˆ

π(γ), π(γ)): Deviation from the approximating distribution will be penalized by adding αρ(ˆ π(γ), π(γ)) to the objective function. A greater α means a greater penalty associated with the deviation of γ from its approximating distribution, thus, a less concern about robustness.

Xin Li Borghan Narajabad Ted Loch-Temzelides Robust Optimal Taxation and Environmental Externalities

Robust Optimal Taxation and Environmental Externalities One-Energy-Sector Model

Robustness (the inner minimization problem)

(Skip) Define the robustness part of the problem by R(V )( ˜ K′, S′) = min

m(γ)

m(γ)V (K′, S′) + αm(γ) log m(γ)

• π(γ)dγ

s.t. K′ = e−S′γ ˜ K′ 1 =

• m(γ)π(γ)dγ

Xin Li Borghan Narajabad Ted Loch-Temzelides Robust Optimal Taxation and Environmental Externalities

Robust Optimal Taxation and Environmental Externalities One-Energy-Sector Model

Robustness (Continued)

(Skip) Suppose V (·) takes the form V (K′, S′) = f(S′) + ¯ A log(K′) + ¯ D, and subsititute it into the above problem, we obtain: m∗(γ) = (1 − ∆S′)e∆λS′γ and R(V )( ˜ K′, S′) = f(S′) + ¯ A log( ˜ K′) + ¯ D + H(S′; α, ¯ A) where ∆ =

¯ A αλ and H(S′; α, ¯

A) = α log(1 − ∆S′) is the robust version externality of carbon.

Xin Li Borghan Narajabad Ted Loch-Temzelides Robust Optimal Taxation and Environmental Externalities

Robust Optimal Taxation and Environmental Externalities One-Energy-Sector Model

Optimal Choice (the outer maximization problem)

(Skip) With the above results, the maximization problem can be written as V (K, S) = max

{C,E, ˜ K′,S′}

{log(C) + βR(V )( ˜ K′, S′)}

• r equivalently,

f(S) + ¯ A log(K) + ¯ D = max

C,E {log(C) + β[f(S′) + ¯

A log( ˜ K′) + ¯ D + H(S′; α, ¯ A)]} s.t. ˜ K′ = F(K, E) − C S′ = S + φ0E

Xin Li Borghan Narajabad Ted Loch-Temzelides Robust Optimal Taxation and Environmental Externalities

Robust Optimal Taxation and Environmental Externalities One-Energy-Sector Model

Equilibrium

Proposition 1: The two-person zero-sum dynamic game described by admits a unique feedback (Markov perfect) equilibrium, in which the equilibrium strategies are given by: C∗ = (1 − βθ)KθE∗ν = (1 − βθ)Kθ[cE(1 − ∆S)]ν E∗ = cE(1 − ∆S) S′∗ = S + φ0cE(1 − ∆S) ˆ π∗(γ) = m∗(γ)π(γ) = λ∗e−λ∗γ where λ∗ = λ(1 − ∆S′∗) = λ(1 − ∆φ0cE)(1 − ∆S) and cE =

ν(1−β) [βα(1−βθ)+ν]φ0∆ and ∆ = ¯ A αλ.

Xin Li Borghan Narajabad Ted Loch-Temzelides Robust Optimal Taxation and Environmental Externalities

Robust Optimal Taxation and Environmental Externalities One-Energy-Sector Model

Remarks

V (K, S) is increasing in K, decreasing in S, and jointly concave in K and S. A greater concern of robustness (α ↓) is going to lower E∗, C∗ and S′∗. The worst case distribution is also exponential, ˆ π∗(γ) = λ∗e−λ∗γ. Since λ∗ < λ, the worst case mean of γ, (λ∗)−1, is strictly greatly than the approximating mean, λ−1.

Xin Li Borghan Narajabad Ted Loch-Temzelides Robust Optimal Taxation and Environmental Externalities

Robust Optimal Taxation and Environmental Externalities One-Energy-Sector Model

Remarks (Continued)

The marginal externality of Carbon emission, measured in the unit of utility and evaluated at E∗, is given by λs = −β ∂V (K′, S′) ∂E |K′∗,S′∗ = ν cE(1 − βθ)(1 − ∆S) = ν (1 − βθ)E∗ . The marginal externality of Carbon emission, measured in the unit of comsuption goods and evaluated at E∗, is given by Λs = λs u′(C∗

t ) = νKθE∗ν−1.

The marginal externality of Carbon emission, measured as a percentage of output and evaluated at E∗, is given by ˆ Λs = Λs F(K, E) = ν E∗ .

Xin Li Borghan Narajabad Ted Loch-Temzelides Robust Optimal Taxation and Environmental Externalities

Robust Optimal Taxation and Environmental Externalities One-Energy-Sector Model

Decentralization

We now turn to the decentralized problem. Suppose the government imposes a percentage tax τt on emissions, Et.

V (k, K, S) = max

c,˜ k′ min ˆ π(γ)

• u(c) + βˆ

Eγ

• V (k′, K′, S′) + α log

ˆ π(γ) π(γ)

• s.t.

c + ˜ k′ = r(K, S)k + τ(K, S)E(τ) + πprofit ˜ K′ = G(K, S) k′ = k′−γS′˜ k′ K′ = K′−γS′ ˜ K′ S′ = S + φ0E(S)

Xin Li Borghan Narajabad Ted Loch-Temzelides Robust Optimal Taxation and Environmental Externalities

Robust Optimal Taxation and Environmental Externalities One-Energy-Sector Model

Decentralization (Continued)

Proposition 2 Suppose the government sets E = cE(1 − ∆S) or, equivalently, τt = Λs, and the tax proceeds are rebated lump-sum to the representative consumer. Then the competitive equilibrium allocation coincides with the solution to the robust planner’s

• problem. That is, c∗ = C∗ = (1 − βθ)Kθ[cE(1 − ∆S)]ν.

Xin Li Borghan Narajabad Ted Loch-Temzelides Robust Optimal Taxation and Environmental Externalities

Robust Optimal Taxation and Environmental Externalities Complete Model

Outline

1 Introduction 2 One-Energy-Sector Model 3 Complete Model 4 Numerical Results

Xin Li Borghan Narajabad Ted Loch-Temzelides Robust Optimal Taxation and Environmental Externalities

Robust Optimal Taxation and Environmental Externalities Complete Model

Complete Model

In this section, we extend the analytical one-sector model as follows: Three Energy Sectors:

• 1. The oil sector produces oil (E1) at zero cost but is subject

to a resource feasibility constraint, R0 > 0.

• 2. The coal and the green energy sector use linear

technologies Ei = AiNi; i = 2, 3. The stock of coal is assumed to be infinity.

• 3. A2N2 and A3N3 grow at a rate of two percent per year.

Xin Li Borghan Narajabad Ted Loch-Temzelides Robust Optimal Taxation and Environmental Externalities

Robust Optimal Taxation and Environmental Externalities Complete Model

Complete Model

The composite energy is produced by E = (κ1Eρ

1 + κ2Eρ 2 + κ3Eρ 3)1/ρ.

As in GHTK, suppose S = P + T where P and T are the permanent and temporary components of S, respectively. P ′ = P + φL(E1 + E2) and T ′ = (1 − φ)T + (1 − φL)φ0(E1 + E2).

Xin Li Borghan Narajabad Ted Loch-Temzelides Robust Optimal Taxation and Environmental Externalities

Robust Optimal Taxation and Environmental Externalities Complete Model

Complete Model

Now assume that the approximating distribution of γ is normal with mean ¯ γ and variance σ2; i.e., π ∼ N(¯ γ, σ2). It follows from the inner-minimization problem that ˆ π∗(γ) ∼ N(¯ γ + ¯ Aσ2 α S′2, σ2) H(S′; α, ¯ A) = −(¯ γ + ¯ Aσ2 2α S′) ¯ AS′

Xin Li Borghan Narajabad Ted Loch-Temzelides Robust Optimal Taxation and Environmental Externalities

Robust Optimal Taxation and Environmental Externalities Complete Model

Social Planner

f(N, P, T, R) = max

E1,E2,E3,E,P ′,T ′,S′,R′

{ 1 1 − βθ log[(1 − E2 A2N − E3 A3N )1−θ−νEν] +β[f(N′, P ′, T ′, R′) + H(S′; α, ¯ A)]} s.t. E = (κ1Eρ

1 + κ2Eρ 2 + κ3Eρ 3)1/ρ

N′ = (1 + g)N R′ = R − E1 ≥ 0 P ′ = P + φL(E1 + E2) T ′ = (1 − φ)T + (1 − φL)φ0(E1 + E2) S′ = P ′ + T ′

Xin Li Borghan Narajabad Ted Loch-Temzelides Robust Optimal Taxation and Environmental Externalities

Robust Optimal Taxation and Environmental Externalities Complete Model

FONC and Externalities

(skip) We calculate the marginal externalities caused by P and T respectively, and show that the externality caused by Carbon emission is a weighted sum of the two. ˆ ΛP = −(1 − βθ) ∂f ∂P = θ¯ γ

+∞

• j=1

βj 1 − ∆St+j ˆ ΛT = −(1 − βθ) ∂f ∂T = θ¯ γ

+∞

• j=1

[β(1 − φ)]j 1 − ∆St+j ˆ ΛS = φLˆ ΛP + (1 − φL)φ0 1 − φ ˆ ΛT lim

α→+∞

ˆ ΛS

t

= θ¯ γ φLβ 1 − β + (1 − φL)φ0β 1 − (1 − φ)β

• Xin Li

Borghan Narajabad Ted Loch-Temzelides Robust Optimal Taxation and Environmental Externalities

Robust Optimal Taxation and Environmental Externalities Complete Model

FONC and Externalities

(skip) FONC’s: νκ1 E1−ρ

1

Eρ − ˆ ΛS = β

• νκ1

(E′

1)1−ρ(E′)ρ − (ˆ

ΛS)′

• ,

(∂E1) νκ2 E1−ρ

2

Eρ − ˆ ΛS = 1 − θ − ν A2N0 , (∂E2) νκ3 E1−ρ

3

Eρ = 1 − θ − ν A3N0 . (∂E3)

Xin Li Borghan Narajabad Ted Loch-Temzelides Robust Optimal Taxation and Environmental Externalities

Robust Optimal Taxation and Environmental Externalities Numerical Results

Outline

1 Introduction 2 One-Energy-Sector Model 3 Complete Model 4 Numerical Results

Xin Li Borghan Narajabad Ted Loch-Temzelides Robust Optimal Taxation and Environmental Externalities

Robust Optimal Taxation and Environmental Externalities Numerical Results

Calibration

Table: Calibration Summary

Parameter φ φL φ0 θ Value 0.0228 0.2 0.393 0.3 Parameter ν β ρ 1 + g Value 0.04 0.98510

• 0.058

1.0210 Parameter P0 T0 R0 κ1 Value 103 699 800 0.5008 Parameter κ2 A2,0 A3,0 λ−1 Value 0.08916 7,693 1,311 2.38×10−5

Xin Li Borghan Narajabad Ted Loch-Temzelides Robust Optimal Taxation and Environmental Externalities

Robust Optimal Taxation and Environmental Externalities Numerical Results

Optimal Energy Path when R0 = 253.8GtC (The same as in GHKT)

Figure: Optimal Use of Energy when R0 = 253.8

Xin Li Borghan Narajabad Ted Loch-Temzelides Robust Optimal Taxation and Environmental Externalities

Robust Optimal Taxation and Environmental Externalities Numerical Results

Optimal Energy Path when R0 = 8000GtC (Including Methane)

Figure: Optimal Use of Energy when R0 = 8000

Xin Li Borghan Narajabad Ted Loch-Temzelides Robust Optimal Taxation and Environmental Externalities

Robust Optimal Taxation and Environmental Externalities Numerical Results

Optimal Energy Path when R0 = ∞

Figure: Optimal Use of Energy when R0 = ∞

Xin Li Borghan Narajabad Ted Loch-Temzelides Robust Optimal Taxation and Environmental Externalities

Robust Optimal Taxation and Environmental Externalities Numerical Results

More Graphs for R0 = ∞

Figure: Global Average Temperature when R0 = ∞

Xin Li Borghan Narajabad Ted Loch-Temzelides Robust Optimal Taxation and Environmental Externalities

Robust Optimal Taxation and Environmental Externalities Numerical Results

More Graphs for R0 = ∞ (Continued)

Figure: The Path of Output and Capital when R0 = ∞

Xin Li Borghan Narajabad Ted Loch-Temzelides Robust Optimal Taxation and Environmental Externalities

Robust Optimal Taxation and Environmental Externalities Numerical Results

Thank You!

Xin Li Borghan Narajabad Ted Loch-Temzelides Robust Optimal Taxation and Environmental Externalities