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Repeated Experimentation to Learn About a Flow-Pollutant Threshold

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Abstract

We examine in discrete time the management of a flow pollutant that causes damage when it crosses a fixed but unknown threshold. The manager sequentially chooses a pollution level that allows learning about the threshold, thereby improving future decisions. If crossed, damage can be reversed at some cost. We analyze the conditions under which experimentation is optimal, and explore how experimentation depends on restoration costs, information about the threshold, and the discount rate. Our results suggest that the level of experimentation, defined as the difference between the optimal activity with and without learning, is non-monotonic in costs and decreasing in the discount rate. We identify two stopping boundaries for the experiment, depending on cost levels compared to the lower bound of the threshold’s interval. We show that when costs are high the stopping boundary under an infinite number of decisions is the same as when there are only two decision moments. A computational extension to more than two decisions suggests that an optimal sequence of experiments can cross the same threshold several times before experimentation ceases. These results shed light on a large class of environmental decision problems that has not been examined in the literature.

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Notes

  1. Remember that because we crossed the threshold we have \(\ell _{t+1}=\ell _t\).

  2. Recall that the second period of the two-period analysis represents an infinite horizon over which no further learning is allowed.

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Acknowledgments

Helpful comments on earlier versions of this article by Aart de Zeeuw, Amos Zemel, and an anonymous reviewer are gratefully acknowledged. Any remaining errors are the responsibility of the authors.

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Correspondence to Rolf Adriaan Groeneveld.

Appendix: Derivation of the Optimal Activity in \(t=T-1\)

Appendix: Derivation of the Optimal Activity in \(t=T-1\)

1.1 High Costs

If costs are ‘high’ in period \(T-1\) (\(c>1-\ell _{T-1}\)), then the present value of all expected future utility is as follows:

$$\begin{aligned}&V_{T-1}\left( x_{T-1},\ell _{T-1},u_{T-1}\right) \nonumber \\&\quad = {\left\{ \begin{array}{ll} x_{T-1} + \frac{\ell _{T-1}}{r} &{}\quad \mathrm{if }\,0 < x_{T-1} \le \ell _{T-1}\\ x_{T-1} + \frac{u_{T-1}-x_{T-1}}{u_{T-1}-\ell _{T-1}}\frac{x_{T-1}}{r} + \frac{x_{T-1}-\ell _{T-1}}{u_{T-1}-\ell _{T-1}}\left( \frac{\ell _{T-1}}{r}-c\right) &{}\quad \mathrm{if }\,\ell _{T-1} < x_{T-1} \le u_{T-1}\\ x_{T-1} + \frac{\ell _{T-1}}{r}-c &{}\quad \mathrm{if }\,u_{T-1} < x_{T-1} \end{array}\right. }\qquad \quad \end{aligned}$$
(21)

This utility function only has an interior solution for \(\ell _{T-1} < x_{T-1} \le u_{T-1}\). Setting the first derivative equal to zero and solving for \(x_{T-1}\) gives

$$\begin{aligned} x_{T-1}^H= \frac{u_{T-1}+\ell _{T-1}+r\left( u_{T-1}-\ell _{T-1}\right) -r c}{2}. \end{aligned}$$
(22)

This interior solution meets Condition (7b):

$$\begin{aligned} \frac{d^2 V_{T-1}}{dx^2_{T-1}}=\frac{-2}{r(u_{T-1}-\ell _{T-1})}. \end{aligned}$$
(23)

From Condition (7c) we derive the following two conditions for \(x_{T-1}^H\):

$$\begin{aligned} x_{T-1}^H&> \ell _{T-1} \Rightarrow u_{T-1} - \ell _{T-1} > \frac{r c}{1+r} \end{aligned}$$
(24a)
$$\begin{aligned} x_{T-1}^H&< u_{T-1} \Rightarrow u_{T-1} - \ell _{T-1} > -\frac{r c}{1-r}. \end{aligned}$$
(24b)

Finally, checking \(x_{T-1}^H\) with Condition (7d) gives

$$\begin{aligned} V_{T-1}^H\left( \ell _{T-1},u_{T-1}\right) >\ell _{T-1}+\frac{\ell _{T-1}}{r} \Rightarrow \frac{\left( r \left( u_{T-1}-c-\ell _{T-1} \right) +u_{T-1}-\ell _{T-1} \right) ^2}{4r(u_{T-1}-\ell _{T-1})} > 0,\nonumber \\ \end{aligned}$$
(25)

where \(V_{T-1}^H\left( \ell _{T-1},u_{T-1}\right) =V_{T-1}\left( x_{T-1}^H,\ell _{T-1},u_{T-1}\right) \). Conditions (24b) and (25) are trivial, so the only effective condition is Condition (24a).

1.2 Low Costs

If costs are ‘low’ in period \(T-1\) (\(c<1-\ell _{T-1}\)), then the present value of all expected future utility is as follows:

$$\begin{aligned}&V_{T-1}\left( x_{T-1},\ell _{T-1},u_{T-1}\right) \nonumber \\&\quad ={\left\{ \begin{array}{ll} x_{T-1} + \frac{1-c}{r} &{}\quad \mathrm{if }\,0 < x_{T-1} \le \ell _{T-1}\\ x_{T-1} + \frac{1-c}{r} - \frac{x_{T-1}-\ell _{T-1}}{u_{T-1}-\ell _{T-1}} c &{}\quad \mathrm{if }\,\ell _{T-1} < x_{T-1} \le 1-c\\ x_{T-1} + \frac{u_{T-1}-x_{T-1}}{u_{T-1}-\ell _{T-1}}\frac{x_{T-1}}{r} + \frac{x_{T-1}-\ell _{T-1}}{u_{T-1}-\ell _{T-1}}\left( \frac{1-c}{r}-c\right) &{}\quad \mathrm{if }\, 1-c < x_{T-1} \le u_{T-1}\\ x_{T-1} + \frac{1-c}{r}-c &{}\quad \mathrm{if }\,u_{T-1} < x_{T-1} \end{array}\right. }\qquad \quad \end{aligned}$$
(26)

Only one part of Eq. (26) has an interior solution: if \(1-c < x_{T-1} \le u_{T-1}\) we have

$$\begin{aligned} x_{T-1}^L = \frac{u_{T-1}+r\left( u_{T-1}-\ell _{T-1}\right) -c\left( 1+r\right) +1}{2} \end{aligned}$$
(27)

This interior solution meets Condition (7b):

$$\begin{aligned} \frac{d^2 V_{T-1}}{dx^2_{T-1}}=\frac{-2}{r(u_{T-1}-\ell _{T-1})} \end{aligned}$$
(28)

From Condition (7c) we can derive the following conditions:

$$\begin{aligned} x_{T-1}^L&> 1-c \Rightarrow u_{T-1} > \frac{1-c\left( 1-r\right) +r\ell _{T-1}}{1+r} \end{aligned}$$
(29a)
$$\begin{aligned} x_{T-1}^L&< u \Rightarrow u_{T-1} > \frac{1-c\left( 1+r\right) -r\ell _{T-1}}{1-r} \end{aligned}$$
(29b)

Finally, checking \(x_{T-1}^L\) with Condition (7d) gives

$$\begin{aligned}&V_{T-1}^L\left( \ell _{T-1},u_{T-1}\right) > \left( 1-c\right) \left( 1+\frac{1}{r}\right) \nonumber \\&\quad \quad \Rightarrow \begin{array}{l} u_{T-1} > \frac{1-c+r\left( \ell _{T-1}+c\right) +2\sqrt{rc(1-\ell _{T-1}-c)}}{1+r} \mathrm{or} \\ u_{T-1} < \frac{1-c+r\left( \ell _{T-1}+c\right) -2\sqrt{rc(1-\ell _{T-1}-c)}}{1+r} \end{array} \end{aligned}$$
(30)

where \(V_{T-1}^L\left( \ell _{T-1},u_{T-1}\right) =V_{T-1}\left( \ell _{T-1},u_{T-1},x_{T-1}^L\right) \). See Fig. 6 for a depiction of these conditions. Condition (29a) crosses Condition (30) at \([\ell _{T-1},u_{T-1}]=[1-c,(1-c+r)/(r+1)]\); Condition (29b) crosses Condition (30) at \([\ell _{T-1},u_{T-1}] = [1-((r+1)/r)c,1]\). Hence, Conditions (29a) and (29b) define the relevant segment of Condition (30), which is between \([\ell _{T-1},u_{T-1}]=[1-c,(1-c+r)/(r+1)]\) and \([\ell _{T-1},u_{T-1}] = [1-((r+1)/r)c,1]\). This is expressed by Condition (9b).

Fig. 6
figure 6

Conditions that determine the experimentation region in \(t=T-1\)

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Groeneveld, R.A., Springborn, M. & Costello, C. Repeated Experimentation to Learn About a Flow-Pollutant Threshold. Environ Resource Econ 58, 627–647 (2014). https://doi.org/10.1007/s10640-013-9713-4

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