Robust Optimal Taxation and Environmental Externalities

Transcription

1 Robust Optimal Taxation and Environmental Externalities Xin Li Borghan Narajabad Ted Loch-Temzelides Department of Economics Rice University

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

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

4 Introduction Research Question Environmental Externality: CO 2 Emission Global Temperature output. Model Uncertainty: Our knowledge is limited regarding how CO 2 emission affects future global mean temperature and output. 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

5 Introduction Literature Review Golosov, Hassler, Krusell, and Tsyvinski (GHKT, 2012): 1. Assume that the mapping from carbon concentration to output damages is subject to a risk γ: y = e γs ỹ (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.

6 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.

7 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 of 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.

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

9 One-Energy-Sector Model Basic Setup Preference: Final Good Production: E 0 β t u(c t ) t=0 F (K, E) = K θ E ν, 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 + φ 0 E

10 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 ˆπ(γ) be an alternative distribution, and m(γ) = ˆπ(γ) π(γ) be the likelihood ratio. The distance between ˆπ(γ) and π(γ) is measured by relative entropy ρ(ˆπ(γ), π(γ)) [m(γ) log m(γ)]π(γ)dγ.

11 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 + φ 0 E, respectively. The malevolent player chooses an alternative distribution ˆπ(γ) or, equivalently, m(γ), to minimize welfare. Accordingly, the next period K = h(s, γ) K.

12 One-Energy-Sector Model A Zero-Sum Dynamic Game (Continued) (elaborate) V (K, S) = max {C,E, K,S } min {u(c) m(γ) +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 + φ 0 E 1 = m(γ)π(γ)dγ

13 One-Energy-Sector Model A Zero-Sum Dynamic Game (Continued) αm(γ) log m(γ)π(γ)dγ = αρ(ˆπ(γ), π(γ)): Deviation from the approximating distribution will be penalized by adding αρ(ˆπ(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.

14 One-Energy-Sector Model Robustness (the inner minimization problem) (Skip) Define the robustness part of the problem by [m(γ)v R(V )( K, S ) = min (K, S ) + αm(γ) log m(γ) ] π(γ)dγ m(γ) s.t. K = e S γ K 1 = m(γ)π(γ)dγ

15 One-Energy-Sector Model Robustness (Continued) (Skip) Suppose V ( ) takes the form V (K, S ) = f(s ) + Ā log(k ) + D, and subsititute it into the above problem, we obtain: and m (γ) = (1 S )e λs γ R(V )( K, S ) = f(s ) + Ā log( K ) + D + H(S ; α, Ā) where = Ā αλ and H(S ; α, Ā) = α log(1 S ) is the robust version externality of carbon.

16 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 )} or equivalently, f(s) + Ā log(k) + D = max C,E β[f(s ) + Ā log( K ) + D + H(S ; α, Ā)]} s.t. K = F (K, E) C S = S + φ 0 E

17 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)k θ E ν = (1 βθ)k θ [c E (1 S)] ν E = c E (1 S) S = S + φ 0 c E (1 S) ˆπ (γ) = m (γ)π(γ) = λ e λ γ where λ = λ(1 S ) = λ(1 φ 0 c E )(1 S) and ν(1 β) c E = [βα(1 βθ)+ν]φ 0 and = Ā αλ.

18 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, ˆπ (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.

19 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 λ s = β V (K, S ) K E,S = ν c E (1 βθ)(1 S) = The marginal externality of Carbon emission, measured in the unit of comsuption goods and evaluated at E, is given by Λ s = λs u (C t ) = νkθ E ν 1. The marginal externality of Carbon emission, measured as a percentage of output and evaluated at E, is given by ˆΛ s = Λ s F (K, E) = ν E. ν (1 βθ)e.

20 One-Energy-Sector Model Decentralization We now turn to the decentralized problem. Suppose the government imposes a percentage tax τ t on emissions, E t. [ {u(c) + βêγ V (k, K, S ) + α log V (k, K, S) = max min c, k ˆπ(γ) s.t. c + k = r(k, S)k + τ(k, S)E(τ) + π profit K = G(K, S) k = k γs k K = K γs K S = S + φ 0E(S) ( ˆπ(γ) π(γ) )]}

21 One-Energy-Sector Model Decentralization (Continued) Proposition 2 Suppose the government sets E = c E (1 S) or, equivalently, τ t = Λ s, 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)k θ [c E (1 S)] ν.

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

23 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 (E 1 ) at zero cost but is subject to a resource feasibility constraint, R 0 > The coal and the green energy sector use linear technologies E i = A i N i ; i = 2, 3. The stock of coal is assumed to be infinity. 3. A 2 N 2 and A 3 N 3 grow at a rate of two percent per year.

24 Complete Model Complete Model The composite energy is produced by E = (κ 1 E ρ 1 + κ 2E ρ 2 + κ 3E ρ 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 + φ L (E 1 + E 2 ) and T = (1 φ)t + (1 φ L )φ 0 (E 1 + E 2 ).

25 Complete Model Complete Model Now assume that the approximating distribution of γ is normal with mean γ and variance σ 2 ; i.e., π N ( γ, σ 2 ). It follows from the inner-minimization problem that H(S ; α, ˆπ (γ) N ( γ + Āσ2 α S 2, σ 2 ) Āσ2 Ā) = ( γ + S )ĀS 2α

26 Complete Model Social Planner f(n, P, T, R) = max E 1,E 2,E 3,E,P,T,S,R s.t. 1 { 1 βθ log[(1 E 2 A 2 N E 3 A 3 N )1 θ ν E ν ] +β[f(n, P, T, R ) + H(S ; α, Ā)]} E = (κ 1 E ρ 1 + κ 2E ρ 2 + κ 3E ρ 3 )1/ρ N = (1 + g)n R = R E 1 0 P = P + φ L (E 1 + E 2 ) T = (1 φ)t + (1 φ L )φ 0 (E 1 + E 2 ) S = P + T

27 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. ˆΛ P = (1 βθ) f P = θ γ + j=1 ˆΛ T = (1 βθ) f T = θ γ + j=1 ˆΛ S = φ ˆΛP L + (1 φ L)φ 0 ˆΛ T 1 φ [ lim ˆΛ S φl β α + t = θ γ 1 β + (1 φ ] L)φ 0 β 1 (1 φ)β β j 1 S t+j [β(1 φ)] j 1 S t+j

28 Complete Model FONC and Externalities (skip) FONC s: νκ 1 E 1 ρ 1 E ˆΛ S ρ = β νκ 2 E 1 ρ [ ] νκ 1 (E 1 )1 ρ (E ) ρ (ˆΛ S ), ( E 1 ) 2 E ˆΛ S = 1 θ ν, ρ A 2 N 0 ( E 2 ) νκ 3 E 1 ρ = 1 θ ν. 3 E ρ A 3 N 0 ( E 3 )

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

30 Numerical Results Calibration Table: Calibration Summary Parameter φ φ L φ 0 θ Value Parameter ν β ρ 1 + g Value Parameter P 0 T 0 R 0 κ 1 Value Parameter κ 2 A 2,0 A 3,0 λ 1 Value ,693 1,

31 Numerical Results Optimal Energy Path when R 0 = 253.8GtC (The same as in GHKT) Figure: Optimal Use of Energy when R 0 = 253.8

32 Numerical Results Optimal Energy Path when R 0 = 8000GtC (Including Methane) Figure: Optimal Use of Energy when R 0 = 8000

33 Numerical Results Optimal Energy Path when R 0 = Figure: Optimal Use of Energy when R 0 =

34 Numerical Results More Graphs for R 0 = Figure: Global Average Temperature when R 0 =

35 Numerical Results More Graphs for R 0 = (Continued) Figure: The Path of Output and Capital when R 0 =

36 Numerical Results Thank You!