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Uncertainty over Uncertainty in Environmental Policy Adoption: Bayesian Learning of Unpredictable Socioeconomic Costs

Matteo Basei, Giorgio Ferrari, Neofytos Rodosthenous

Abstract

The socioeconomic impact of pollution naturally comes with uncertainty due to, e.g., current new technological developments in emissions' abatement or demographic changes. On top of that, the trend of the future costs of the environmental damage is unknown: Will global warming dominate or technological advancements prevail? The truth is that we do not know which scenario will be realised and the scientific debate is still open. This paper captures those two layers of uncertainty by developing a real-options-like model in which a decision maker aims at adopting a once-and-for-all costly reduction in the current emissions rate, when the stochastic dynamics of the socioeconomic costs of pollution are subject to Brownian shocks and the drift is an unobservable random variable. By keeping track of the actual evolution of the costs, the decision maker is able to learn the unknown drift and to form a posterior dynamic belief of its true value. The resulting decision maker's timing problem boils down to a truly two-dimensional optimal stopping problem which we address via probabilistic free-boundary methods and a state-space transformation. We completely characterise the solution by showing that the optimal timing for implementing the emissions reduction policy is the first time that the learning process has become ``decisive'' enough; that is, when it exceeds a time-dependent percentage. This is given in terms of an endogenously determined threshold function, which solves uniquely a nonlinear integral equation. We numerically illustrate our results, discuss the implications of the optimal policy and also perform comparative statics to understand the role of the relevant model's parameters in the optimal policy.

Uncertainty over Uncertainty in Environmental Policy Adoption: Bayesian Learning of Unpredictable Socioeconomic Costs

Abstract

The socioeconomic impact of pollution naturally comes with uncertainty due to, e.g., current new technological developments in emissions' abatement or demographic changes. On top of that, the trend of the future costs of the environmental damage is unknown: Will global warming dominate or technological advancements prevail? The truth is that we do not know which scenario will be realised and the scientific debate is still open. This paper captures those two layers of uncertainty by developing a real-options-like model in which a decision maker aims at adopting a once-and-for-all costly reduction in the current emissions rate, when the stochastic dynamics of the socioeconomic costs of pollution are subject to Brownian shocks and the drift is an unobservable random variable. By keeping track of the actual evolution of the costs, the decision maker is able to learn the unknown drift and to form a posterior dynamic belief of its true value. The resulting decision maker's timing problem boils down to a truly two-dimensional optimal stopping problem which we address via probabilistic free-boundary methods and a state-space transformation. We completely characterise the solution by showing that the optimal timing for implementing the emissions reduction policy is the first time that the learning process has become ``decisive'' enough; that is, when it exceeds a time-dependent percentage. This is given in terms of an endogenously determined threshold function, which solves uniquely a nonlinear integral equation. We numerically illustrate our results, discuss the implications of the optimal policy and also perform comparative statics to understand the role of the relevant model's parameters in the optimal policy.
Paper Structure (26 sections, 13 theorems, 110 equations, 6 figures)

This paper contains 26 sections, 13 theorems, 110 equations, 6 figures.

Table of Contents

  1. Introduction
  2. The optimal timing problem with uncertainty over uncertainty
  3. Markovian formulation of Problem \ref{['valf1']}
  4. The main result and its implications
  5. Decision making phases
  6. Sensitivity analysis: Optimal timing of environmental policy adoption
  7. Sensitivity analysis: Optimal socioeconomic costs
  8. Sensitivity analysis: Expected stock of pollutants until policy adoption
  9. On the proof of Theorem \ref{['thm:mainthm']}: Characterising the optimal emissions reduction time
  10. Reformulation of Problem \ref{['valf1']}
  11. Never adopt the policy
  12. Adopt the policy immediately
  13. Reformulation of Problem \ref{['partprob1']}: Dimensionality reduction
  14. Solution to the two-dimensional optimal stopping problem \ref{['valf2']}
  15. Well-posedness and initial properties of the value function $U$ defined in \ref{['valf2']}
  16. ...and 11 more sections

Key Result

Theorem 3.1

Assume $r>\alpha$, recall the Bayesian learning process $\Pi^{\pi}$ defined by eq:defPi and denote its transition density by $p_t(\pi,\pi')$, for $(\pi,\pi')\in (0,1)^2$, define the auxiliary "time-coordinate" process and introduce the continuous, nondecreasing function where Then, aiming at the minimisation of the overall socioeconomic costs of pollution in partprob1 for any $(z,\pi) \in \math

Figures (6)

  • Figure 1: A numerical calculation of the boundary function $z \mapsto c(z)$ solving \ref{['solc']} and dominating the threshold $z \mapsto m(z)$ defined in Theorem \ref{['thm:mainthm']}.
  • Figure 2: A numerical calculation of the boundary function $z \mapsto c(z)$ solving \ref{['solc']} with respect to different dissipation rates $\delta$ of the pollutant stock from the atmosphere, and the expected time to the policy adoption $\mathbb{E}[\tau^* | \tau^* < \infty]$ as a function of $\delta$.
  • Figure 3: A numerical calculation of the boundary function $z \mapsto c(z)$ solving \ref{['solc']} with respect to different sizes $\alpha$ (resp. $-\alpha$) of the average rate of increase (resp. decrease) of future socioeconomic costs of pollution, and the expected time to the policy adoption $\mathbb{E}[\tau^* | \tau^* < \infty]$ as a function of $\alpha$.
  • Figure 4: A numerical calculation of the boundary function $z \mapsto c(z)$ solving \ref{['solc']} with respect to different extents of volatility $\sigma$ in the socioeconomic costs of pollution, and the expected time to the policy adoption $\mathbb{E}[\tau^* | \tau^* < \infty]$ as a function of $\sigma$.
  • Figure 5: A numerical calculation of the optimal expected socioeconomic costs $x \mapsto V(x,p,\pi)$ solving \ref{['partprob1']} with respect to different: beliefs $\pi$ about an increasing average rate of future socioeconomic costs of pollution, dissipation rates $\delta$ of the pollutant stock from the atmosphere, extents of volatility $\sigma$ in the socioeconomic costs of pollution, and sizes $\alpha$ of the average rate of increase/decrease of future socioeconomic costs of pollution.
  • ...and 1 more figures

Theorems & Definitions (17)

  • Remark 2.1
  • Remark 2.2: Full information
  • Remark 2.3
  • Theorem 3.1
  • Remark 4.2
  • Lemma 4.3
  • Lemma 4.4
  • Proposition 4.5
  • Lemma 4.6
  • Proposition 4.7
  • ...and 7 more