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: 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 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 output 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 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. 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: ∞ � β t u ( C t ) E 0 t =0 Final Good Production: 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 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 K ′ by a factor of h ( S ′ , γ ) = e − S ′ γ end-of-period capital stock ˜ to K ′ . That is, K ′ = h ( S ′ , γ ) ˜ K ′ , The approximating distribution of γ is π ( γ ) = λe − λγ . Let π ( γ ) be an alternative distribution, and m ( γ ) = ˆ π ( γ ) ˆ π ( γ ) be the likelihood ratio. The “distance” between ˆ π ( γ ) and π ( γ ) is measured by π ( γ ) , π ( γ )) ≡ � relative entropy ρ (ˆ [ m ( γ ) log m ( γ )] π ( γ )d γ . 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 + φ 0 E , respectively. The malevolent player chooses an alternative distribution ˆ π ( γ ) or, 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 m ( γ ) { u ( C ) min { C,E, ˜ K ′ ,S ′ } � � m ( γ ) V ( K ′ , S ′ ) + αm ( γ ) log m ( γ ) � + β π ( γ )d γ } s.t. K ′ ˜ F ( K, E ) − C = h ( S ′ , γ ) ˜ K ′ K ′ = S ′ = S + φ 0 E � 1 = m ( γ ) π ( γ )d γ 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 )d γ = 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 ′ ) m ( γ ) V ( K ′ , S ′ ) + αm ( γ ) log m ( γ ) � = min π ( γ )d γ m ( γ ) s.t. e − S ′ γ ˜ K ′ K ′ = � 1 = m ( γ ) π ( γ )d γ 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 ) ¯ αλ and H ( S ′ ; α, ¯ A A ) = α log(1 − ∆ S ′ ) is the robust where ∆ = 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 { log( C ) + β R ( V )( ˜ K ′ , S ′ ) } V ( K, S ) = max { C,E, ˜ K ′ ,S ′ } or equivalently, f ( S ) + ¯ A log( K ) + ¯ D C,E { log( C ) + β [ f ( S ′ ) + ¯ A log( ˜ K ′ ) + ¯ D + H ( S ′ ; α, ¯ = max A )] } s.t. K ′ ˜ F ( K, E ) − C = S ′ = S + φ 0 E 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: (1 − βθ ) K θ E ∗ ν = (1 − βθ ) K θ [ c E (1 − ∆ S )] ν C ∗ = E ∗ = c E (1 − ∆ S ) S ′∗ S + φ 0 c E (1 − ∆ S ) = π ∗ ( γ ) m ∗ ( γ ) π ( γ ) = λ ∗ e − λ ∗ γ ˆ = where λ ∗ = λ (1 − ∆ S ′∗ ) = λ (1 − ∆ φ 0 c E )(1 − ∆ S ) and ¯ ν (1 − β ) A c E = [ βα (1 − βθ )+ ν ] φ 0 ∆ and ∆ = αλ . Xin Li Borghan Narajabad Ted Loch-Temzelides Robust Optimal Taxation and Environmental Externalities
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