Optimal Management of Environmental Externalities with Time Lags and Uncertainty

Abstract

Many environmental externalities occur with time lags that can range from a few days to several centuries in length, and many of these externalities are also subject to uncertainty. In this paper, we examine the key features of an optimal policy to manage environmental externalities that are both lagged and stochastic. We develop a two-period, two-polluter model and obtain closed-form solutions for optimal emissions levels under different combinations of damage functions and stochastic processes. These solutions show that it is not obvious whether greater control should be exerted on polluters that generate externalities with longer lags or on polluters that generate externalities with shorter lags. We find that the optimal ranking of polluters with respect to the length of the time lag associated with their externality will depend on (a) the discount rate, (b) conditional expectations of future states of the polluted resource, (c) persistence of the pollutant, and (d) initial conditions.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3

Notes

  1. 1.

    We are unable to derive closed-form solutions for optimal emissions levels when firms’ abatement costs functions are nonlinear. However, we are able to show that the relationship between key model parameters and optimal emissions levels are largely the same under linear or nonlinear cost functions. Please refer to “Optimal Emission Levels Under Generic Abatement Costs” section in “Appendix” for these derivations.

  2. 2.

    Because both the initial ambient contaminant concentration level (\(x_0\)) and the shock in time period 1 (\(\theta _1\)) are additive, it is possible to choose a different notation that represents the two initial conditions using only one variable. In fact, our analysis will show that changes in \(x_0\) and \(\theta _1\) have the same qualitative impact on optimal emissions. However, we have decided to retain the notation that separates the initial stock level from the initial shock in order to better conform to the literature, which often uses recursive formulations to characterize the evolution of stocks over time.

  3. 3.

    Note that in our model, positive (i.e. \({>}0\)) realizations of the shocks \(\theta _1\) and \(\theta _2\) are actually “bad” from society’s point of view because they lead to greater environmental damage.

  4. 4.

    In a model with more than two time periods, the conditional variance of the stochastic process in any given period s would be equal to:

    $$\begin{aligned} \text {Var} [\theta _s | \theta _1] = \sigma _\epsilon ^2 \sum _{i=0}^{s-1} \mu ^{2i}, \end{aligned}$$
    (10)

    which increases monotonically over time as it converges to the unconditional variance, \(\sigma _\epsilon ^2 / (1 - \mu ^2)\).

References

  1. Allen MR, Stott PA, Mitchell JFB, Schnur R, Delworth TL (2000) Quantifying the uncertainty in forecasts of anthropogenic climate change. Nature 407(6804):617–620

    Article  Google Scholar 

  2. Arthur WB, McNicoll G (1977) Optimal time paths with age-dependence: a theory of population policy. Rev Econ Stud 44(1):111–123

    Article  Google Scholar 

  3. Atkinson SE, Lewis DH (1974) A cost-effectiveness analysis of alternative air quality control strategies. J Environ Econ Manag 1(3):237–250

    Article  Google Scholar 

  4. Beck MB (1987) Water quality modeling: a review of the analysis of uncertainty. Water Resour Res 23(8):1393–1442

    Article  Google Scholar 

  5. Braga AL, Zanobetti A, Schwartz J (2001) The lag structure between particulate air pollution and respiratory and cardiovascular deaths in 10 US cities. J Occup Environ Med 43(11):927–933

    Article  Google Scholar 

  6. Brozović N, Schlenker W (2011) Optimal management of an ecosystem with an unknown threshold. Ecol Econ 70(4):627–640

    Article  Google Scholar 

  7. Camalier L, Cox W, Dolwick P (2007) The effects of meteorology on ozone in urban areas and their use in assessing ozone trends. Atmos Environ 41(33):7127–7137

    Article  Google Scholar 

  8. Caparrós A (2009) Delayed carbon sequestration and rising carbon prices. Clim Change 96(3):421–441

    Article  Google Scholar 

  9. Conrad JM, López A (2002) Stochastic water quality: timing and option value of treatment. Water Resour Res 38(5):2-1–2-7

    Article  Google Scholar 

  10. Conrad JM, Olson LJ (1992) The economics of a stock pollutant: Aldicarb on Long Island. Environ Resour Econ 2(3):245–258

    Article  Google Scholar 

  11. Ehrlich I, Becker GS (1972) Market insurance, self-insurance, and self-protection. J Polit Econ 80(4):623–648

    Article  Google Scholar 

  12. Essl F, Dullinger S, Rabitsch W, Hulme PE, Hülber K, Jaros̆ík V, Kleinbauer I, Krausmann F, Kühn I, Nentwig W, Vilà M, Genovesi P, Gherardi F, Desprez-Loustau M-L, Roques A, Pys̆ek P (2001) Socioeconomic legacy yields an invasion debt. Proc Natl Acad Sci USA 108(1):203–207

  13. Farrow RS, Schultz MT, Celikkol P, Van Houtven GL (2005) Pollution trading in water quality limited areas: use of benefits assessment and cost-effective trading ratios. Land Econ 81(2):191–205

    Article  Google Scholar 

  14. Fisher-Vanden K, Olmstead S (2013) Moving pollution trading from air to water: potential, problems, and prognosis. J Econ Perspect 27(1):147–171

    Article  Google Scholar 

  15. Fleming RA, Adams RM, Kim CS (1995) Regulating groundwater pollution: effects of geophysical response assumptions on economic efficiency. Water Resour Res 31(4):1069–1076

    Article  Google Scholar 

  16. Fox DG (1984) Uncertainty in air quality modeling. Bull Am Meteorol Soc 65(1):27–36

    Article  Google Scholar 

  17. Goering GE, Boyce JR (1999) Emissions taxation in durable goods oligopoly. J Ind Econ 47(1):125–143

    Article  Google Scholar 

  18. Hamilton SK (2012) Biogeochemical time lags may delay response of streams to ecological restoration. Freshw Biol 57(S1):43–57

    Article  Google Scholar 

  19. Harper CR, Zilberman D (1992) Pesticides and worker safety. Am J Agric Econ 74(1):68–78

    Article  Google Scholar 

  20. Horan RD (2001) Differences in social and public risk perceptions and conflicting impacts on point/nonpoint trading ratios. Am J Agric Econ 83(4):934–941

    Article  Google Scholar 

  21. Ibendahl G, Fleming RA (2007) Controlling aquifer nitrogen levels when fertilizing crops: a study of groundwater contamination and denitrification. Ecol Model 205(3–4):507–514

    Article  Google Scholar 

  22. Kamien MI, Muller E (1976) Optimal control with integral state equations. Rev Econ Stud 43(3):469–473

    Article  Google Scholar 

  23. Kim CS, Hostetler J, Amacher G (1993) The regulation of groundwater quality with delayed responses. Water Resour Res 29(5):1369–1377

    Article  Google Scholar 

  24. Krupnick AJ (1986) Costs of alternative policies for the control of nitrogen dioxide in Baltimore. J Environ Econ Manag 13(2):189–197

    Article  Google Scholar 

  25. Kuwayama Y, Brozović N (2013) The regulation of a spatially heterogeneous externality: tradable groundwater permits to protect streams. J Environ Econ Manag 66(2):364–382

    Article  Google Scholar 

  26. Laukkanen M, Huhtala A (2008) Optimal management of a eutrophied coastal ecosystem: balancing agricultural and municipal abatement measures. Environ Resour Econ 39(2):139–159

    Article  Google Scholar 

  27. Leland HE (1968) Saving and uncertainty: the precautionary demand for saving. Q J Econ 82(3):465–473

    Article  Google Scholar 

  28. Lieb CM (2004) The environmental Kuznetz curve and flow versus stock pollution: the neglect of future damages. Environ Resour Econ 29(4):483–506

    Article  Google Scholar 

  29. Malik AS, Letson D, Crutchfield SR (1993) Point/nonpoint source trading of pollution abatement: choosing the right trading ratio. Am J Agric Econ 75(4):959–967

    Article  Google Scholar 

  30. Meals DW, Dressing SA (2008) Lag time in water quality response to land treatment. Tetra Tech Inc, Fairfax

    Google Scholar 

  31. Meals DW, Dressing SA, Davenport TE (2010) Lag time in water quality response to best management practices: a review. J Environ Qual 39(1):85–96

    Article  Google Scholar 

  32. Meehl GA, Washington WM, Collins WD, Arblaster JM, Hu A, Buja LE, Strand WG, Teng H (2005) How much more global warming and sea level rise? Science 307(5716):1769–1772

    Article  Google Scholar 

  33. Muller E, Peles YC (1988) The dynamic adjustment of optimal durability and quality. Int J Ind Organ 6(4):499–507

    Article  Google Scholar 

  34. Muller E, Peles YC (1990) Optimal dynamic durability. J Econ Dyn Control 14(3–4):709–719

    Article  Google Scholar 

  35. Nerlove M, Arrow KJ (1962) Optimal advertising policy under dynamic conditions. Economica 29(114):129–142

    Article  Google Scholar 

  36. Newell RG, Pizer WA (2003) Regulating stock externalities under uncertainty. J Environ Econ Manag 45(2):416–432

    Article  Google Scholar 

  37. Nkonya EM, Featherstone AM (2000) Determining socially optimal nitrogen application rates using a delayed response model: the case of irrigated corn in Western Kansas. J Agric Resour Econ 25(2):453–467

    Google Scholar 

  38. Pindyck RS (2002) Optimal timing problems in environmental economics. J Econ Dyn Control 26(9–10):1677–1697

    Article  Google Scholar 

  39. Ragot L, Schubert K (2008) The optimal carbon sequestration in agricultural soils: do the dynamics of the physical process matter? J Econ Dyn Control 32(12):3847–3865

    Article  Google Scholar 

  40. Runkel M (2003) Product durability and extended producer responsibility in solid waste management. Environ Resour Econ 24(2):161–182

    Article  Google Scholar 

  41. Sandmo A (1970) The effect of uncertainty on saving decisions. Rev Econ Stud 37(3):353–360

    Article  Google Scholar 

  42. Schwartz J (2000) The distributed lag between air pollution and daily deaths. Epidemiology 11(3):320–326

    Article  Google Scholar 

  43. Shogren JF, Crocker TD (1991) Risk, self-protection, and ex ante economic value. J Environ Econ Manag 20(1):1–15

    Article  Google Scholar 

  44. Shogren JF, Crocker TD (1999) Risk and its consequences. J Environ Econ Manag 37(1):44–51

    Article  Google Scholar 

  45. Shortle JS (1987) Allocative implications of comparisons between the marginal costs of point and nonpoint source pollution abatement. Northeastern J Agric Resour Econ 16(1):17–23

    Google Scholar 

  46. Solomon S, Plattner G-K, Knutti R, Friedlingstein P (2009) Irreversible climate change due to carbon dioxide emissions. Proc Natl Acad Sci USA 106(6):1704–1709

    Article  Google Scholar 

  47. UNEP (2011) Integrated Assessment of Black Carbon and Tropospheric Ozone. United Nations Environment Programme and World Meteorological Organization, Nairobi, Kenya, and Geneva, Switzerland

  48. van Imhoff E (1989) Optimal investment in human capital under conditions of nonstable population. J Hum Resour 24(3):414–432

    Article  Google Scholar 

  49. Weitzman ML (1974) Prices vs. quantities. Rev Econ Stud 41(4):477–491

    Article  Google Scholar 

  50. Weitzman ML (2010) What is the “damages function” for global warming—and what difference might it make? Clim Change Econ 1(1):57–69

    Article  Google Scholar 

  51. Wigley TML (2005) The climate change commitment. Science 307(5716):1766–1769

    Article  Google Scholar 

  52. Yadav SN (1997) Dynamic optimization of nitrogen use when groundwater contamination is internalized at the standard in the long run. Am J Agric Econ 79(3):931–945

    Article  Google Scholar 

  53. Zemel A (2012) Precaution under mixed uncertainty: implications for environmental management. Resour Energy Econ 34(2):188–197

    Article  Google Scholar 

Download references

Acknowledgments

The authors wish to thank Amy Ando, John Braden, Dallas Burtraw, Ximing Cai, Sahan Dissanayake, Dave Finnoff, Iddo Kan, Alan Krupnick, Chuck Mason, Ariel Ortiz-Bobea, Juan Robalino, Jay Shogren, Fred Sterbenz, Tom Tietenberg, Al Valocchi, Klaas van ’t Veld, Guillermo Vuletin, Quinn Weninger, and three anonymous referees for valuable comments and suggestions. This project was completed with financial support from the National Science Foundation, EAR 0709735.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Yusuke Kuwayama.

Appendix: Mathematical Derivations

Appendix: Mathematical Derivations

Optimal Emission Levels Under Uncertainty Type 1 and Quadratic Damages

Given the form of the abatement cost function in (1), the quadratic form of the damage function in (2), and the stochastic process described in Sect. 3.3.1, the social planner’s objective is to:

$$\begin{aligned}&\min _{e_A,e_B} \quad \phi (\bar{e} - e_A) + \phi (\bar{e} - e_B) + \left[ \alpha (e_A + \theta _1) + \frac{\gamma }{2} (e_A + \theta _1)^2 \right] \nonumber \\&\quad +\,\mathbb {E} \left\{ \left. \beta \left[ \alpha (e_B + \theta _2) + \frac{\gamma }{2} (e_B + \theta _2)^2 \right] \right| \theta _1 \right\} . \end{aligned}$$
(21)

We can make use of the fact that if a random variable y has mean \(\mathbb {E}[y]\) and variance \(\sigma _y^2\), then \(\mathbb {E}(\alpha y + \frac{\gamma }{2} y^2) = \alpha \mathbb {E}[y] + \frac{\gamma }{2} [\sigma _y^2 + (\mathbb {E}[y])^2 ]\). Since \(\mathbb {E}[\theta _2 | \theta _1] = \mu \theta _1\) and \( Var [\theta _2 | \theta _1] = \sigma _\epsilon ^2\) by definition, the social planner’s objective can be rewritten as:

$$\begin{aligned}&\min _{e_A,e_B} \quad \phi (\bar{e} - e_A) + \phi (\bar{e} - e_B) + \left[ \alpha (e_A + \theta _1) + \frac{\gamma }{2} (e_A + \theta _1)^2 \right] \nonumber \\&\quad +\, \beta \left\{ \alpha (e_B + \mu \theta _1) + \frac{\gamma }{2} \left[ \sigma _\epsilon ^2 + (e_B + \mu \theta _1)^2 \right] \right\} . \end{aligned}$$
(22)

The first-order conditions for an interior solution to this minimization problem are:

$$\begin{aligned}&-\phi + \gamma \left( e_A^*+ \theta _1\right) + \alpha = 0,\end{aligned}$$
(23)
$$\begin{aligned}&-\phi + \beta \left[ \gamma (e_B^*+ \mu \theta _1) + \alpha \right] = 0. \end{aligned}$$
(24)

Equation (23) can be rewritten to obtain the closed-form solution for Firm A’s optimal emissions:

$$\begin{aligned} e_A^*= \frac{\phi - \alpha }{\gamma } - \theta _1. \end{aligned}$$
(25)

Equation (24) can be rearranged to obtain the closed-form solution for Firm B’s optimal emissions:

$$\begin{aligned} e_B^*= \frac{1}{\gamma } \left( \frac{\phi }{\beta } - \alpha \right) - \mu \theta _1. \end{aligned}$$
(26)

Optimal Emission Levels Under Uncertainty Type 1 and Exponential Damages

Given the abatement cost function in (1), the exponential form of the damage function in (3), and the stochastic process described in Sect. 3.3.1, the social planner’s objective is to:

$$\begin{aligned}&\min _{e_A,e_B} \quad \phi (\bar{e} - e_A) + \phi (\bar{e} - e_B) + \left[ \exp (\psi (e_A + \theta _1)) - 1 \right] \nonumber \\&\quad +\, \mathbb {E}\left\{ \left. \beta \left[ \exp (\psi (e_B + \theta _2)) - 1 \right] \right| \theta _1 \right\} . \end{aligned}$$
(27)

We can make use of the fact that if a random variable y is normally distributed with mean \(\mathbb {E}[y]\) and variance \(\sigma ^2_y\), then \(\mathbb {E}[\exp (y)] = \exp (\mathbb {E}[y] + \frac{\sigma ^2_y}{2})\). Since \(\mathbb {E}[\theta _2 | \theta _1] = \mu \theta _1\) and \(\text {Var} [\theta _2 | \theta _1] = \sigma _\epsilon ^2\) by definition, the social planner’s objective can be rewritten as:

$$\begin{aligned}&\min _{e_A,e_B} \quad \phi (\bar{e} - e_A) + \phi (\bar{e} - e_B) + \left[ \exp (\psi (e_A + \theta _1)) - 1 \right] \nonumber \\&\quad +\,\beta \left[ \exp \left( \psi \left( e_B + \mu \theta _1\right) + \frac{\psi \sigma _\epsilon ^2}{2} \right) - 1 \right] . \end{aligned}$$
(28)

The first-order conditions for an interior solution to this minimization problem are:

$$\begin{aligned}&-\phi + \psi \exp \left( \psi \left( e_A^*+ \theta _1\right) \right) = 0,\end{aligned}$$
(29)
$$\begin{aligned}&-\phi + \psi \beta \exp \left( \psi \left( e_B^*+ \mu \theta _1\right) + \frac{\psi ^2 \sigma _\epsilon ^2}{2} \right) = 0. \end{aligned}$$
(30)

Taking the natural logarithm of both sides of Eqs. (29) and (30) rearrangement yields the closed-form solution for Firms A and B optimal emissions:

$$\begin{aligned} e_A^*= & {} \frac{1}{\psi } \ln \left( \frac{\phi }{\psi } \right) - \theta _1,\end{aligned}$$
(31)
$$\begin{aligned} e_B^*= & {} \frac{1}{\psi } \ln \left( \frac{\phi }{\psi \beta } \right) - \mu \theta _1 - \frac{\psi \sigma _\epsilon ^2}{2}. \end{aligned}$$
(32)

Optimal Emission Levels Under Uncertainty Type 2 and Quadratic Damages

Given the form of the abatement cost function in (1) and the quadratic form of the damage function in (2), the social planner’s objective is to:

$$\begin{aligned}&\min _{e_A,e_B} \quad \phi (\bar{e} - e_A) + \phi (\bar{e} - e_B) + \left[ \alpha (\rho x_0 + e_A + \theta _1) + \frac{\gamma }{2} (\rho x_0 + e_A + \theta _1)^2 \right] \nonumber \\&\quad +\, \mathbb {E} \left\{ \left. \beta \left[ \alpha \left( \rho ^2 x_0 \,{+}\, \rho e_A \,{+}\, \rho \theta _1 \,{+}\, e_B \,{+}\, \theta _2\right) \,{+}\, \frac{\gamma }{2} \left( \rho ^2 x_0 \,{+}\, \rho e_A \,{+}\, \rho \theta _1 \,{+}\, e_B \,{+}\, \theta _2\right) ^2 \right| \theta _1 \right] \right\} .\nonumber \\ \end{aligned}$$
(33)

We can make use of the fact that if a random variable y has mean \(\mathbb {E}[y]\) and variance \(\sigma _y^2\), then \(\mathbb {E}(\alpha y + \frac{\gamma }{2} y^2) = \alpha \mathbb {E}[y] + \frac{\gamma }{2} [\sigma _y^2 + (\mathbb {E}[y])^2 ]\). Since \(\mathbb {E}[\theta _t] = 0\) and \( Var [\theta _t] = \sigma _\theta ^2\) by definition, the objective function can be rewritten as:

$$\begin{aligned}&\min _{e_A,e_B} \quad \phi (\bar{e} - e_A) + \phi (\bar{e} - e_B) + \left[ \alpha (\rho x_0 + e_A + \theta _1) + \frac{\gamma }{2} (\rho x_0 + e_A + \theta _1)^2 \right] \nonumber \\&\quad +\,\beta \left\{ \alpha \left( \rho ^2 x_0 + \rho e_A + \rho \theta _1 + e_B\right) + \frac{\gamma }{2} \left[ \sigma _\theta ^2 + \left( \rho ^2 x_0 + \rho e_A + \rho \theta _1 + e_B\right) ^2 \right] \right\} .\qquad \quad \end{aligned}$$
(34)

The first-order conditions for an interior solution to this minimization problem are:

$$\begin{aligned}&- \phi + \alpha + \gamma \left( \rho x_0 + e_A^*+ \theta _1\right) + \beta \left[ \alpha \rho + \gamma \left( \rho ^2 x_0 + \rho e_A^*+ \rho \theta _1 + e_B^*\right) \rho \right] = 0,\qquad \quad \end{aligned}$$
(35)
$$\begin{aligned}&- \phi + \beta \left[ \alpha +\gamma \left( \rho ^2 x_0 + \rho e_A^*+ \rho \theta _1 + e_B^*\right) \right] = 0. \end{aligned}$$
(36)

Solving this system of two equations and two unknowns yields closed-form solutions for optimal emissions by Firms A and B:

$$\begin{aligned} e_A^*= & {} \frac{\phi - \phi \rho - \alpha }{\gamma } - \rho x_0 - \theta _1,\end{aligned}$$
(37)
$$\begin{aligned} e_B^*= & {} \frac{\phi - \alpha \beta - \beta \rho (\phi - \phi \rho - \alpha )}{\gamma \beta }. \end{aligned}$$
(38)

Optimal Emission Levels Under Uncertainty Type 2 and Exponential Damages

Given the form of the abatement cost function in (1) and the form of the damage function in (3), the social planner’s objective is to:

$$\begin{aligned}&\min _{e_A,e_B} \quad \phi (\bar{e} - e_A) + \phi (\bar{e} - e_B) + \left[ \exp \left( \psi \left( \rho x_0 + e_A + \theta _1\right) \right) - 1 \right] \nonumber \\&\quad +\,\mathbb {E} \left\{ \left. \beta \left[ \exp \left( \psi \left( \rho ^2 x_0 + \rho e_A + \rho \theta _1 + e_B + \theta _2\right) \right) -1 \right] \right| \theta _1 \right\} . \end{aligned}$$
(39)

We can make use of the fact that if a random variable y is normally distributed with mean \(\mathbb {E}[y]\) and variance \(\sigma ^2_y\), then \(\mathbb {E}[\exp (y)] = \exp (\mathbb {E}[y] + \frac{\sigma ^2_y}{2})\). Since \(\mathbb {E}[\theta _t] = 0\) and \( Var [\theta _t] = \sigma _\theta ^2\) by definition, the objective function can be rewritten as:

$$\begin{aligned}&\min _{e_A,e_B} \quad \phi (\bar{e} - e_A) + \phi (\bar{e} - e_B) + \left[ \exp \left( \psi (\rho x_0 + e_A + \theta _1)\right) - 1 \right] \nonumber \\&\quad +\,\beta \left[ \exp \left( \psi (\rho ^2 x_0 + \rho e_A + \rho \theta _1 + e_B) + \frac{\psi ^2 \sigma _\theta ^2}{2} \right) - 1 \right] . \end{aligned}$$
(40)

The first-order conditions for an interior solution to this minimization problem are:

$$\begin{aligned}&- \phi + \psi \exp \left( \psi \left( \rho x_0 + e_A^*+ \theta _1\right) \right) \nonumber \\&\quad +\,\beta \psi \rho \exp \left( \psi \left( \rho ^2 x_0 + \rho e_A^*+ \rho \theta _1 + e_B^*\right) + \frac{\psi ^2 \sigma _\theta ^2}{2} \right) = 0, \end{aligned}$$
(41)
$$\begin{aligned} - \phi + \beta \psi \exp \left( \psi \left( \rho ^2 x_0 + \rho e_A^*+ \rho \theta _1 + e_B^*\right) + \frac{\psi ^2 \sigma _\theta ^2}{2} \right) = 0. \end{aligned}$$
(42)

Solving this system of two equations and two unknowns yields closed-form solutions for optimal emissions by Firms A and B:

$$\begin{aligned} e_A^*= & {} \frac{1}{\psi } \ln \left( \frac{\phi - \phi \rho }{\psi } \right) - \rho x_0 - \theta _1,\end{aligned}$$
(43)
$$\begin{aligned} e_B^*= & {} \frac{1}{\psi } \left[ \ln \left( \frac{\phi }{\beta \psi } \right) - \rho \ln \left( \frac{\phi - \phi \rho }{\psi } \right) \right] - \frac{\psi \sigma _\theta ^2}{2}. \end{aligned}$$
(44)

Optimal Emission Levels Under Generic Abatement Costs

In this appendix, we derive relationships between key model parameters and optimal emissions levels using generic abatement cost functions and an exponential damage function.

Generic Abatement Costs, Exponential Damages, and Uncertainty Type 1

Under uncertainty type 1, the social planner’s objective is to:

$$\begin{aligned}&\min _{e_A,e_B} \quad C(\bar{e} - e_A) + C(\bar{e} - e_B) + \exp (\psi (e_A + \theta _1)) - 1 \nonumber \\&\quad +\, \mathbb {E} \left\{ \beta \left[ \exp (\psi (e_B + \theta _2)) - 1 \right] | \theta _1 \right\} . \end{aligned}$$
(45)

As in “Optimal Emission Levels Under Uncertainty Type 1 and Exponential Damages” section in “Appendix”, we can rewrite the social planner’s objective as:

$$\begin{aligned}&\min _{e_A,e_B} \quad C(\bar{e} - e_A) + C(\bar{e} - e_B) + \exp (\psi (e_A + \theta _1)) - 1 \nonumber \\&\quad +\,\beta \left[ \exp \left( \psi (e_B + \mu \theta _1) + \frac{\psi \sigma _\epsilon ^2}{2} \right) - 1 \right] . \end{aligned}$$
(46)

The first-order conditions for an interior solution to this minimization problem are:

$$\begin{aligned}&-C'\left( \bar{e} - e_A^*\right) + \psi \exp \left( \psi \left( e_A^*+ \theta _1\right) \right) = 0,\end{aligned}$$
(47)
$$\begin{aligned}&-C'\left( \bar{e} - e_B^*\right) + \beta \psi \exp \left( \psi \left( e_B^*+ \mu \theta _1\right) + \frac{\psi ^2 \sigma _\epsilon ^2}{2} \right) = 0. \end{aligned}$$
(48)

In order to examine how \(e_A^*\) and \(e_B^*\) vary with \(\beta \), we take the total derivative of the first-order conditions with respect to \(\beta \) and rearrange to get:

$$\begin{aligned}&C''\left( \bar{e} - e_A^*\right) \frac{de_A^*}{d \beta } + \psi ^2 \exp \left( \psi \left( e_A^*+ \theta _1\right) \right) \frac{de_A^*}{d \beta } = 0,\end{aligned}$$
(49)
$$\begin{aligned}&C''\left( \bar{e} - e_B^*\right) \frac{de_B^*}{d \beta } + \beta \psi ^2 \exp \left( \psi \left( e_B^*+ \mu \theta _1\right) + \frac{\psi ^2 \sigma _\epsilon ^2}{2} \right) \frac{de_B^*}{d \beta }\nonumber \\&\quad +\,\psi \exp \left( \psi \left( e_B^*+ \mu \theta _1\right) + \frac{\psi ^2 \sigma _\epsilon ^2}{2} \right) = 0. \end{aligned}$$
(50)

Equations (49) and (50) can be rearranged to obtain expressions for the change in optimal emissions for Firms A and B in response to a change in \(\beta \):

$$\begin{aligned}&\frac{de_A^*}{d \beta } = 0,\end{aligned}$$
(51)
$$\begin{aligned}&\frac{de_B^*}{d \beta } = - \frac{\psi \exp \left( \psi \left( e_B^*+ \mu \theta _1\right) + \frac{\psi ^2 \sigma _\epsilon ^2}{2} \right) }{C''\left( \bar{e} - e_B^*\right) + \beta \psi ^2 \exp \left( \psi \left( e_B^*+ \mu \theta _1\right) + \frac{\psi ^2 \sigma _\epsilon ^2}{2} \right) }. \end{aligned}$$
(52)

Assuming that \(C''(\cdot ) > 0\), Eq. (52) implies that \(\frac{de_B^*}{d \beta } < 0\). A similar procedure yields expressions for the change in optimal emissions in response to changes in other modeling parameters, \(\theta _1, \mu \), and \(\sigma _\epsilon ^2\):

$$\begin{aligned} \frac{de_A^*}{d \theta _1}= & {} - \frac{\psi ^2 \exp \left( \psi \left( e_A^*+ \theta _1\right) \right) }{C''\left( \bar{e} - e_A^*\right) + \psi ^2 \exp \left( \psi \left( e_A^*+ \theta _1\right) \right) } < 0,\end{aligned}$$
(53)
$$\begin{aligned} \frac{de_B^*}{d \theta _1}= & {} - \frac{\beta \mu \psi ^2 \exp \left( \psi \left( e_B^*+ \mu \theta _1\right) + \frac{\psi ^2 \sigma _\epsilon ^2}{2} \right) }{C''\left( \bar{e} - e_B^*\right) + \beta \psi ^2 \exp \left( \psi \left( e_B^*+ \mu \theta _1\right) + \frac{\psi ^2 \sigma _\epsilon ^2}{2} \right) } < 0,\end{aligned}$$
(54)
$$\begin{aligned} \frac{de_A^*}{d \mu }= & {} 0,\end{aligned}$$
(55)
$$\begin{aligned} \frac{de_B^*}{d \mu }= & {} - \frac{\beta \psi ^2 \theta _1 \exp \left( \psi \left( e_B^*+ \mu \theta _1\right) + \frac{\psi ^2 \sigma _\epsilon ^2}{2} \right) }{C''\left( \bar{e} - e_B^*\right) + \beta \psi ^2 \exp \left( \psi \left( e_B^*+ \mu \theta _1\right) + \frac{\psi ^2 \sigma _\epsilon ^2}{2} \right) } \lesseqgtr 0,\end{aligned}$$
(56)
$$\begin{aligned} \frac{de_A^*}{d \sigma _\epsilon ^2}= & {} 0,\end{aligned}$$
(57)
$$\begin{aligned} \frac{de_B^*}{d \sigma _\epsilon ^2}= & {} - \frac{\beta \psi ^3 \exp \left( \psi \left( e_B^*+ \mu \theta _1\right) + \frac{\psi ^2 \sigma _\epsilon ^2}{2} \right) }{2 \left[ C''\left( \bar{e} - e_B^*\right) + \beta \psi ^2 \exp \left( \psi \left( e_B^*+ \mu \theta _1\right) + \frac{\psi ^2 \sigma _\epsilon ^2}{2} \right) \right] } < 0. \end{aligned}$$
(58)

Generic Abatement Costs, Exponential Damages, and Uncertainty Type 2

Under uncertainty type 2, the social planner’s objective is to:

$$\begin{aligned}&\min _{e_A,e_B} \quad C(\bar{e} - e_A) + C(\bar{e} - e_B) + \exp (\psi (\rho x_0 + e_A + \theta _1)) - 1 \nonumber \\&\quad +\,\mathbb {E} \left\{ \beta \left. \left[ \exp \left( \psi \left( \rho ^2 x_0 + \rho e_A + \rho \theta _1 + e_B + \theta _2\right) \right) \right] - 1 \right| \theta _1 \right\} . \end{aligned}$$
(59)

As in “Optimal Emission Levels Under Uncertainty Type 2 and Exponential Damages” section in “Appendix”, we can rewrite the social planner’s objective as:

$$\begin{aligned}&\min _{e_A,e_B} \quad C(\bar{e} - e_A) + C(\bar{e} - e_B) + \exp (\psi (\rho x_0 + e_A + \theta _1)) - 1 \nonumber \\&\quad +\,\beta \left[ \exp \left( \psi \left( \rho ^2 x_0 + \rho e_A + \rho \theta _1 + e_B\right) + \frac{\psi ^2 \sigma _\theta ^2}{2} \right) - 1 \right] . \end{aligned}$$
(60)

The first-order conditions for an interior solution to this minimization problem are:

$$\begin{aligned}&- C'\left( \bar{e} - e_A^*\right) + \psi \exp \left( \psi \left( \rho x_0 + e_A^*+ \theta _1\right) \right) \nonumber \\&\quad +\, \beta \rho \psi \exp \left( \psi \left( \rho ^2 x_0 + \rho e_A^*+ \rho \theta _1 + e_B^*\right) + \frac{\psi ^2 \sigma _\theta ^2}{2} \right) = 0,\end{aligned}$$
(61)
$$\begin{aligned}&- C'\left( \bar{e} - e_B^*\right) + \beta \psi \exp \left( \psi \left( \rho ^2 x_0 + \rho e_A^*+ \rho \theta _1 + e_B^*\right) + \frac{\psi ^2 \sigma _\theta ^2}{2} \right) = 0. \end{aligned}$$
(62)

In order to examine how \(e_A^*\) and \(e_B^*\) vary with \(\beta \), we take the total derivative of the first-order conditions with respect to \(\beta \) and rearrange to get:

$$\begin{aligned}&\left[ C''\left( \bar{e} - e_A^*\right) + \psi ^2 \exp (\varvec{\Gamma }_1) + \beta \rho ^2 \psi ^2 \exp (\varvec{\Gamma }_2) \right] \frac{de_A^*}{d \beta }+\, \beta \rho \psi ^2 \exp (\varvec{\Gamma }_2) \frac{de_B^*}{d \beta }\nonumber \\&\quad = - \psi \rho \exp (\varvec{\Gamma }_2),\end{aligned}$$
(63)
$$\begin{aligned}&\beta \rho \psi ^2 \exp (\varvec{\Gamma }_2) \frac{de_A^*}{d \beta } + \left[ C''\left( \bar{e} - e_B^*\right) +\,\beta \psi ^2 \exp (\varvec{\Gamma }_2) \right] \frac{de_B^*}{d \beta } = - \psi \exp (\varvec{\Gamma }_2), \end{aligned}$$
(64)

where

$$\begin{aligned} \varvec{\Gamma }_1= & {} \psi \left( \rho x_0 + e_A^*+ \theta _1\right) ,\end{aligned}$$
(65)
$$\begin{aligned} \varvec{\Gamma }_2= & {} \psi \left( \rho ^2 x_0 + \rho e_A^*+ \rho \theta _1 + e_B^*\right) + \frac{\psi ^2 \sigma _\theta ^2}{2}. \end{aligned}$$
(66)

Equations (63) and (64) can be rewritten in matrix form:

$$\begin{aligned}&\left[ \begin{array}{c@{\quad }c} C''\left( \bar{e} - e_A^*\right) + \psi ^2 \exp (\varvec{\Gamma }_1) + \beta \rho ^2 \psi ^2 \exp (\varvec{\Gamma }_2) &{} \beta \rho \psi ^2 \exp (\varvec{\Gamma }_2) \\ \beta \rho \psi ^2 \exp (\varvec{\Gamma }_2) &{} C''\left( \bar{e} - e_B^*\right) + \beta \psi ^2 \exp (\varvec{\Gamma }_2) \end{array} \right] \nonumber \\&\quad \times \, \left[ \begin{array}{c} \frac{de_A^*}{d \beta } \\ \frac{de_B^*}{d \beta } \end{array} \right] = \left[ \begin{array}{c} - \psi \rho \exp (\varvec{\Gamma }_2) \\ - \psi \exp (\varvec{\Gamma }_2) \end{array} \right] . \end{aligned}$$

We define the following three matrices:

$$\begin{aligned} \mathbf {G}= & {} \left[ \begin{array}{c@{\quad }c} C''\left( \bar{e} - e_A^*\right) + \psi ^2 \exp (\varvec{\Gamma }_1) + \beta \rho ^2 \psi ^2 \exp (\varvec{\Gamma }_2) &{} \beta \rho \psi ^2 \exp (\varvec{\Gamma }_2) \\ \beta \rho \psi ^2 \exp (\varvec{\Gamma }_2) &{} C''\left( \bar{e} - e_B^*\right) + \beta \psi ^2 \exp (\varvec{\Gamma }_2) \end{array} \right] ,\\ \mathbf {G_1}= & {} \left[ \begin{array}{c@{\quad }c} - \psi \rho \exp (\varvec{\Gamma }_2) &{} \beta \rho \psi ^2 \exp (\varvec{\Gamma }_2) \\ - \psi \exp (\varvec{\Gamma }_2) &{} C''\left( \bar{e} - e_B^*\right) + \beta \psi ^2 \exp (\varvec{\Gamma }_2) \end{array} \right] ,\\ \mathbf {G_2}= & {} \left[ \begin{array}{c@{\quad }c} C''\left( \bar{e} - e_A^*\right) + \psi ^2 \exp (\varvec{\Gamma }_1) + \beta \rho ^2 \psi ^2 \exp (\varvec{\Gamma }_2) &{} - \psi \rho \exp (\varvec{\Gamma }_2) \\ \beta \rho \psi ^2 \exp (\varvec{\Gamma }_2) &{} - \psi \exp (\varvec{\Gamma }_2) \end{array} \right] . \end{aligned}$$

Assuming that \(C''(\cdot ) > 0\), we can determine whether the determinants of the above three matrices are positive or negative:

$$\begin{aligned} \det (\mathbf {G})= & {} C''\left( \bar{e} - e_A^*\right) C''\left( \bar{e} - e_B^*\right) + \psi ^2 C''\left( \bar{e} - e_B^*\right) \exp (\varvec{\Gamma }_1) \nonumber \\&\quad +\,\beta \rho ^2 \phi ^2 C''\left( \bar{e} - e_B^*\right) \exp (\varvec{\Gamma }_2) +\,\beta \psi ^2 C''\left( \bar{e} - e_A^*\right) \exp (\varvec{\Gamma }_2)\nonumber \\&\quad +\,\beta \psi ^4 \exp (\varvec{\Gamma }_1) \exp (\varvec{\Gamma }_2) > 0,\end{aligned}$$
(67)
$$\begin{aligned} \det (\mathbf {G_1})= & {} - \rho \psi C''\left( \bar{e} - e_B^*\right) \exp (\varvec{\Gamma }_2) < 0,\end{aligned}$$
(68)
$$\begin{aligned} \det (\mathbf {G_2})= & {} - \psi \exp (\varvec{\Gamma }_2) [C''\left( \bar{e} - e_A^*\right) + \psi ^2 \exp (\varvec{\Gamma }_1)] < 0. \end{aligned}$$
(69)

By Cramer’s rule, it follows that:

$$\begin{aligned} \frac{de_A^*}{d \beta }= & {} \frac{\det (\mathbf {G_1})}{\det (\mathbf {G})} < 0,\end{aligned}$$
(70)
$$\begin{aligned} \frac{de_B^*}{d \beta }= & {} \frac{\det (\mathbf {G_2})}{\det (\mathbf {G})} < 0, \end{aligned}$$
(71)

A similar procedure yields expressions for the change in optimal emissions in response to changes in other modeling parameters, \(x_0, \theta _1, \rho \), and \(\sigma _\theta ^2\):

$$\begin{aligned} \frac{de_A^*}{d x_0}< 0, \ \frac{de_B^*}{d x_0}< 0, \ \frac{de_A^*}{d \theta _1}< 0, \ \frac{de_B^*}{d \theta _1}< 0, \ \frac{de_A^*}{d \rho } \lesseqgtr 0, \ \frac{de_B^*}{d \rho } \lesseqgtr 0, \ \frac{de_A^*}{d \sigma _\theta ^2}< 0, \ \frac{de_B^*}{d \sigma _\theta ^2} < 0.\nonumber \\ \end{aligned}$$
(72)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kuwayama, Y., Brozović, N. Optimal Management of Environmental Externalities with Time Lags and Uncertainty. Environ Resource Econ 68, 473–499 (2017). https://doi.org/10.1007/s10640-016-0026-2

Download citation

Keywords

  • Environmental externalities
  • Time lags
  • Uncertainty
  • Persistence

JEL Classification

  • H23
  • Q5
  • D9
  • Q2