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Coping with Multiple Catastrophic Threats

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Abstract

We study intertemporal policies for dealing with multiple catastrophic threats with endogenous hazards, allowing, inter alia, for gradual mitigation efforts that accumulate to reduce occurrence risks. The long-run properties of the optimal policies are characterized in terms of the key parameters (damage, hazard sensitivity and natural degradation rate) associated with each type of catastrophic threat. Effects of background threats on the optimal response to a potential catastrophe are illustrated numerically.

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Notes

  1. More recent examples can be found in Polasky et al. (2011) and Tsur and Zemel (2014b).

  2. A case in point is the debate between advocates of an early vigorous climate policy (Stern 2007) and those supporting a more gradual approach (Nordhaus 2008).

  3. A similar question has been considered by MP in their extension dealing with partial alleviation.

  4. Note that with an iso-elastic utility, the specifications of modifying either consumption or utility by a constant factor are equivalent. To simplify, we assume that \(\eta =2\) and consider a stationary economy. The analysis extends in a straightforward manner to arbitrary \(\eta >1\) at the cost of somewhat more complicated algebra.

  5. The linear specification in (2.3) is made in the interest of simplicity and can be generalized without affecting the nature of our results. In particular, this specification allows us to avert a certain threat completely by reducing the relevant stock to \(P=0\). This possibility might appear unrealistic for some catastrophes, but can be plausible for others. For example, the risk of outbreak of a deadly disease can be eliminated once a cure or vaccine is discovered via sufficient R&D investments. It is easy to model catastrophes which cannot be averted by adding a constant term to (2.3) or by adjusting the parameters such that fixing the state at \(P=0\) involves an unacceptably low consumption rate.

  6. MP assume that \(\varphi \) is random, allowing for a different realization upon each occurrence. This distinction does not affect the model, when \(\varphi \) is interpreted as the expected value of the random factor, because the objective is specified as the expected welfare under a given policy, the damage enters linearly and the random damage and the time of its occurrence are assumed independent.

  7. A similar result is derived in MP (p. 2955) where equation (7) is analogous to our equation (2.10) above in that both equations require that the discount rate is large relative to the expected damage flow in order to avoid infinitely negative welfare. Indeed, this result is typical of models of recurrent events where discounting must offset the cumulative effect of an infinite series of damages [see Eq. (2.12) below]. The equations differ somewhat due to the zero growth rate and the explicit value \(\eta =2\) assumed in the present work. Observe further that (2.10) provides the steady state value only prior to the first occurrence. After n occurrences, the value is multiplied by \(\varphi ^n\) which diverges in the long run. This feature can be traced to the multiplicative damage postulated by MP and adopted also in the present work for ease of comparison. Alternatively, one can specify an additive damage (as in the recurrent events of Tsur and Zemel 1998, 2016a) and obtain a constant steady state value. This change will modify the algebraic details of the corresponding L-function, but hardly affect the nature of the solution.

  8. Noting (2.4), the expected payoff (2.6) can be expressed as \({\int _0^\infty f(P(t),q(t))e^{-\int _0^t[\rho +h(P(\tau ))]d\tau }dt}\). With \(\rho +h(P(t))\) representing the “effective discount rate,” f(P, q) can be interpreted as the “effective utility”.

  9. As in the case of an isolated catastrophe, f(P, q) is interpreted as the “effective utility” corresponding to the expected payoff (3.5).

  10. This simple assumption can hold for all P only under the linear hazard specification. When the derivative \(h_i'(\cdot )\) depends on the state \(P_i\), the damage parameters are equal only under specific arrangements of the states which yield equal marginal hazards. When the functions \(h_i(\cdot )\) are identical, this corresponds to the symmetric arrangement with all \(P_i\) equal.

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Authors and Affiliations

  1. Department of Environmental Economics and Management and the Center for Agricultural Economic Research, The Hebrew University of Jerusalem, POB 12, 7610001, Rehovot, Israel

    Yacov Tsur

  2. Department of Solar Energy and Environmental Physics, The Jacob Blaustein Institutes for Desert Research, Ben Gurion University of the Negev, 8499000, Sede Boker Campus, Israel

    Amos Zemel

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  1. Yacov Tsur
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  2. Amos Zemel
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Correspondence to Amos Zemel.

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Tsur, Y., Zemel, A. Coping with Multiple Catastrophic Threats. Environ Resource Econ 68, 175–196 (2017). https://doi.org/10.1007/s10640-017-0144-5

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