## Abstract

We consider the prospects for sustainable growth using expected utility models of optimal investment under threat from natural disasters. Adoption of a continuous time, stochastic Ramsey growth model over an infinite time horizon permits the analysis of sustainability under uncertainty regarding adverse events, including both one-time and recurrent disasters. As appropriate to small economies, we consider adaptation to the risk of disaster. Natural disasters reduce capital stocks and disrupt the optimal consumption and felicity paths. While the time path of inter-temporal welfare might consequently shift downward, the path may still be non-decreasing over time, even without adding strong or weak sustainability constraints. Prudent disaster preparedness includes precautionary investment in productive capital, programs of adaptation to disaster risk, and avoiding distortionary policies undermining the prospects of optimality and sustainability.

### Similar content being viewed by others

## Notes

Our model employs continuous time stochastic optimal control, which we find more suitable for addressing the issues posed in the paper, as it affords clearer interpretation of results and avoids unnecessary computational complexity.

The model presented prominently features dynamic optimization to maximize inter-temporal welfare with inter-generational neutrality. Given that most economies operate along sub-optimal paths, well inside their production possibility frontiers due to myriad policy distortions, applying empirical data in attempt to validate or calibrate the model may be difficult to interpret.

This simplification seems very reasonable for a small economy. The size of a small economy’s capital stock (whether productive or natural capital) should not significantly influence the probability of a hurricane, tornado, cyclone, earthquake, tsunami, or even volcanic eruptions, to which small economies may be vulnerable. Setting P = P(K) for example, where K is the stock of total capital, would add unnecessary computational complexity to our model and render no additional insight regarding the prospect of sustaining inter-temporal welfare.

We present a model of optimal adaption, cautioning that results are highly dependent on poorly defined economic geography and key parameters, such as the productivity of adaptation, with values that remain largely undetermined.

See subsequent sections (Convergence to the Steady State with no Risk of Disaster; Paths of Convergence and Recovery Times) for discussions of convergence.

Following Weitzman (1976), K can be viewed as a composite of produced, natural, and human capital, aggregated from the shadow prices of the components.

This is easily generalized to the case where population grows at a constant rate (Solow 1956).

Without the Koopmans transformation, there can be many consumption paths for which the integral of U(C) is infinite (does not converge). By this means and by restricting attention to paths that are both feasible and “eligible” (undominated), Koopmans (1965) established that the transformed welfare function provides a complete ranking of those paths.

One alternative to ethical neutrality via the Koopmans transformation is the von Weizsäcker (1965) overtaking criterion. This criterion and modeling technique are used by Brock and Mirman (1973) for the case of stochastic optimal growth with no discounting. Another approach to modeling sustainable social preferences has been suggested by Chichilnisky (1996). In place of either the Koopmans or the von Weizsacker approach, she posits axioms specifying no “dictatorship” by either the present or the future.

The hazard rate in Tsur and Zemel (2006) is a function of the environmental stock. In the present treatment, the hazard rate is a parameter.

Treating damage fraction D

_{K}as a constant proportion of the capital stock is assumed to be a reasonable approach in the basic disaster model (without adaptation). We first regard all capital in the small economy as facing the same risk of natural disaster. Increasing the capital stock through the process of accumulation adds to the total stock of capital at risk to the same degree. We later endogenize D_{K}in a model of optimal investment in adaptation and derive a minimized damage fraction D*. We do not consider negative shocks to productivity, which van der Ploeg and de Zeeuw (2019) regard as regime shifting, and possibly non-recoverable, disasters. They address productivity shocks by permanently reducing total factor productivity A by an exogenous factor (1 – π) with (0 < π < 1) post disaster.The planner first solves the problem (equation (4)) assuming disaster risk, but as if actual occurrence were postponed indefinitely (the details of the optimal control solution for the planner’s problem are available in the appendix). Under that initial assumption, the extended Ramsey condition, equation (5), applies for all time t. Then, when an individual disaster does occur, say at time t = τ

_{1}, the planner adjusts according to a post-disaster numerical approximation of capital restoration (equation (7) and Fig. 1).There are opposing forces in precautionary investment. On the one hand, a greater hazard rate lowers the expected marginal product of new capital, tending to lower investment. On the other hand, there is a need for greater investment to increase consumption in the event of disaster. Specifically, for η > 1, precautionary investment increases with parametrically increasing P. That’s also the case in our present model as reflected in equation (5). For η < 1, there is less precautionary investment and more consumption now before disaster strikes. For η = 1 [U(C) = log C], the opposing forces are exactly offsetting; an increasing hazard rate has no effect on the level of precautionary investment. These results can be established using an adaptation of the two-period model in Gollier (2013) along with the more general felicity function \( \mathrm{U}\left(\mathrm{C}\left(\mathrm{t}\right)\right)=\frac{\left[\mathrm{C}{\left(\mathrm{t}\right)}^{\left(1-\upeta \right)}-1\right]}{1-\upeta} \) for η > 0.

See e.g. Friedman (1999), chapter 4, section 5 (The Fundamental Theorem of Calculus), page 127, Theorem 1: An indefinite integral of an integrable function is a continuous function.

Terada, S., Column: Fukushima thrives after recovering from 2011 earthquake, Honolulu Star Advertiser, 7/22/2019; and Davis, R., A Fukushima Ghost Town Seeks Rebirth Through Renewable Energy, Wall Street Journal, 7/12/2019).

Having accounted for intrinsic and amenity value of capital in the initial adaptation project at t = 0, the planner invokes the follow-on investment algorithm in terms of Q** for t > 0. This process eliminates the need to incorporate U(C, Q) rather than U(C) in the planner’s problem, equation (13).

Later in this section, we suggest an alternative dynamic equation for Q that might be used in a more complex, though less tractable model. Ideally, as a reward for greater computational complexity, such a model would permit investigation of research questions beyond the scope of this paper, including conditions under which adaptation capital and productive capital can serve as substitutes.

## References

Acemoglu D (2009) Modern economic growth. Princeton University Press, Princeton and Oxford

Adda J, Cooper R (2003) Dynamic economics. MIT Press, Cambridge and London

Anand S, Sen A (2000) Human development and economic sustainability. World Dev 21(12):2029–2049

Arrow K (1999) Discounting, morality, and gaming. In: Portney P, Weyant J (eds) Discounting and intergenerational equity. RFF Press, Washington, DC

Arrow K, Kurz M (1970) Public investment, the rate of return, and optimal fiscal policy. Johns Hopkins University Press, Baltimore and London

Arrow K, Dasgupta P, Goulder L, Daily G, Ehrlich P, Heal G, Levin S, Mäler GM, Schneider S, Starrett D, Walker B (2004) Are we consuming too much? J Econ Perspect 18(3):147–172

Arrow KJ, Dasgupta P, Goulder LH, Mumford KJ, Oleson K (2012) Sustainability and the measurement of wealth. Environ Dev Econ 17:317–353

Ayong Le Kama A (2001) Sustainable growth renewable resources, and pollution. J Econ Dyn Control 25(12):1911–1918

Barro R (2006) Rare disasters and asset markets in the twentieth century. Q J Econ 121(3):823–866

Barro R, Sala-i-Martin X (2004) Economic growth, 2nd edn. MIT Press, Cambridge

Ben-Shahar O, Logue K (2016) The perverse effects of subsidized weather insurance. Stanford Law Rev 68:571–626

Bretschger L, Karydas C (2018) Optimum growth and carbon policies with lags in the climate system. Environ Resource Econ 70(4):781–806

Brock W, Mirman L (1972) Optimal economic growth and uncertainty: the discounted case. J Econ Theory 4:479–513

Brock W, Mirman L (1973) Optimal economic growth and uncertainty: the no-discounting case. International Economic Review 14:497–513

Cavallo E, Noy I (2010) The Economics of natural disasters: a survey. Inter-American Development Bank, Working Paper No. IDB-WP-124

Cavallo E, Galiani S, Noy I, Pantano J (2010) Catastrophic natural disasters and economic growth. Inter-American Development Bank, Working Paper No. IDB-WP-183

Chichilnisky G (1996) An axiomatic approach to sustainable development. Soc Choice Welf 13:231–257

Chung JW (1994) Utility and production functions. Blackwell, Oxford/Cambridge

Clark WC (2007) Sustainability science: a room of its own. Proc Natl Acad Sci U S A 104(6):1737–1738

Cropper M (1976) Regulating activities with catastrophic environmental effects. Journal of Environmental Economics and Management 3:1–15

Dasgupta PS, Heal GM (1979) Economic theory and exhaustible resources. Cambridge University Press, Cambridge

de Hek P, Roy S (2001) On sustained growth under uncertainty. Int Econ Rev 42:801–813

Endress L, Roumasset J, Zhou T (2005) Sustainable growth with environmental spillovers. J Econ Behav Organ 58:527–547

Endress LH, Pongkijvorasin S, Roumasset J, Wada CA (2014) Intergenerational equity with individual impatience in a model of optimal and sustainable growth. Resour Energy Econ 36(2):620–635

Friedman A (1999) Advanced calculus. Dover Publications, Inc., New York

Gollier C (2001) The economics of time and risk. MIT Press, Cambridge

Gollier C (2013) Pricing the planet’s future. Princeton University Press, Princeton

Gollier C, Weitzman M (2010) How should the distant future be discounted when discount rates are uncertain? Econ Lett 107:350–353

Hallegatte S (2017) A normative exploration of the link between development, economic growth, and natural risk. Econ Disasters Clim Chang 1(1):5–31

Hallegatte S, Ghil M (2008) Natural disasters impacting a macroeconomic model with endogenous dynamics. Ecol Econ 68:582–592

Heal G (2000) Valuing the future: economic theory and sustainability. Columbia University Press, New York

Heal G (2001) Optimality or sustainability? Prepared for presentation at the European Association of Environmental and Resource Economists (EAERE) 2001 Conference, Southampton, England, June 2001

Ikefuji M, Horii R (2012) Natural disasters in a two-sector model of endogenous growth. J Public Econ 96:784–796

Judd K (1998) Numerical methods in economics. MIT Press, Cambridge

Koopmans TC (1965) On the concept of optimal economic growth. In: The econometric approach to development planning. Rand McNally, Chicago

Kousky C (2014) Informing climate adaptation: a review of the economic costs of natural disasters. Energy Econ 46:576–592

Laframboise N, Loko B (2012) Natural disasters: mitigating impact, managing risks. IMF Working Paper, WP/12/245

Lemoine D, Traeger C (2016) Ambiguous tipping points. J Econ Behav Organ 132:5–18

NOAA (2018) U.S. billion-dollar weather & climate disasters 1980–2018. https://www.ncdc.noaa.gov/billions/events.pdf

Noy I (2009) The macroeconomic consequences of disasters. J Dev Econ 88:221–231

Pindyck R, Wang N (2013) The economic and policy consequences of catastrophes. Am Econ J Policy 5(4):306–339

Polasky S, de Zeeuw A, Wagener F (2011) Optimal management with potential regime shifts. J Environ Econ Manage 62:229–240

Sargent T (1987) Dynamic macroeconomic theory. Harvard University Press, Cambridge

Scheffer M (2009) Critical transitions in nature and society. Princeton University Press, Princeton

Seierstad A (2009) Stochastic control in discrete and continuous time. Springer, Boston

Seierstad A, Sydsaeter K (1987) Optimal control theory with economic applications. Amsterdam, North-Holland

Shabnam N (2014) Natural disasters and economic growth: a review. Int J Disaster Risk Sci 5:157–163

Solow R (1956) A contribution to the theory of economic growth. Q J Econ 70(1):65–94

Solow R (1986) On the intergenerational allocation of natural resources. Scand J Econ 88(1):141–149

Solow R (1991) Sustainability: an economist’s perspective. Presented as the Eighteenth J. Seward Johnson Lecture to the Marine Policy Center, Woods Hole Oceanographic Institution, Woods Hole, Massachusetts. Published as Chapter 26. In: Stavins R (ed) Economics of the environment, 5th edn. W. W. Norton, New York

Stavins RM, Wagner AF, Wagner G (2003) Interpreting sustainability in economic terms: dynamic efficiency plus intergenerational equity. Economics Letters 79:339–343

Stokey N, Lucas R (1989) Recursive methods in economic dynamics. Harvard University Press, Cambridge

The Centre for Research on the Epidemiology of Disaster (CRED) report (2019), Natural Disasters 2018

The United Nations Office for Disaster Risk Reduction (2015) Disaster risk reduction and resilience in the 2030 agenda for sustainable development. http://www.unisdr.org/files/46052_disasterriskreductioninthe2030agend.pdf

Tsur Y, Zemel A (2006) Welfare measurement under threats of environmental catastrophes. Journal of Environmental Economics and Management 52:421–429

van der Ploeg F, de Zeeuw A (2019) Pricing carbon and adjusting capital to fend off climate catastrophes. Environ Resource Econ 72(1):29–50

Varian H (1992) Microeconomic analysis, 3rd edn. W.H. Norton & Company, New York and London

von Weizsäcker C (1965) Existence of optimal programs of accumulation for an infinite time horizon. Rev Econ Stud 32:85–104

Weitzman M (1976) On the welfare significance of national product in a dynamic economy. The Quarterly Journal of Economics 90(1):156–162

Weitzman M (2012) The Ramsey discounting formula for a hidden-state stochastic growth process. Environ Resource Econ 53(3):309–321

World Commission on Environment and Development (1987) Our common future. Oxford University Press, Oxford

World Vision (2018) 2010 Haiti earthquake: facts, FAQs, and how to help. https://www.worldvision.org/disaster-relief-news-stories/2010-haiti-earthquake-facts. Accessed 25 September 2018

## Author information

### Authors and Affiliations

### Corresponding author

## Additional information

### Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Appendix

### Appendix

### Extended Ramsey Equation for Sustainable Growth with Risk of Natural Disaster

We take ρ = 0, and the probability of risk is modeled as an exponential distribution with parameter P and density function Pe^{-Pt}. D_{K} is the fraction of capital stock damaged by the disaster. The planner’s problem is

subject to

The current value Hamiltonian corresponding to (A1) is

The standard first order conditions for the optimal control problem (A2) are

From (A3) and (A4) \( \dot{\lambda}={\dot{\mathrm{U}}}_{\mathrm{C}} \) and \( \dot{\lambda}=-\lambda \left[{\mathrm{F}}_{\mathrm{K}}-\left\{\updelta +\left[\mathrm{P}{\mathrm{e}}^{-\mathrm{Pt}}\left({\mathrm{D}}_{\mathrm{K}}\right)\right]\right\}\right] \). Equating expressions for \( \dot{\lambda} \) and rearranging yields:

where \( \mathrm{g}=\left(1/\mathrm{C}\right)\dot{\mathrm{C}} \). The extended Ramsey equation with disaster risk is then:

The second term on the RHS of (A6) represents the precautionary effect, inducing investment in precautionary capital.

As t → ∞ and the economy approaches the steady state, g → 0, and Pe^{−Pt} → 0. So in the steady state.

### Comparative Statics for Optimal Adaptation Capital Q*

Start with the result

for Q_{m} = A(K_{o})^{α}. It is sufficient to derive comparative statics in terms of logQ ∗ , given that dlogQ^{∗} = (1/Q^{∗})dQ^{∗}, with Q* > 0.

Using the second expression for Q*

Differentiation of this equation readily yields the results that comparative statics (in terms of both logQ* and Q*) are positive for P, D_{o} and K_{o}, but negative for θ.

The derivation of the comparative static for β proceeds as follows. Using the third expression for Q*,

Differentiating one term at a time leads to:

In the numerator, make the approximation log(1 + x) ≈ x with x = β − 1. Then.

Note that the quadratic term is zero for β = 1 but positive elsewhere. Hence J′(β) > 0.

Next,

If \( \left\{\frac{\mathrm{P}{\mathrm{D}}_{\mathrm{o}}{\left({\mathrm{K}}_{\mathrm{o}}\right)}^{1-\upalpha}}{\mathrm{A}}\right\}\ge 1 \), then L^{′}(β) ≥ 0. Otherwise L^{′}(β) < 0.

Finally, dlogQ^{∗}/dβ = J′(β) + L′(β). If L^{′}(β) ≥ 0, dlogQ^{∗}/dβ > 0; otherwise, the comparative static is ambiguous. For example, take α = 1/2, A = 1, P = D_{o} = 1/4. Then L^{′}(β) ≥ 0 for K_{o} ≥ 256; for A = 2, the condition requires K_{o} ≥ 1024.

### Solution of the Planner’s Problem for Optimal Adaptation at Time t = 0

Set U(C, Q) = [−C(t)^{−(η − 1)}] − [K_{0}PD(Q)]^{γ} with η > 1 and γ > 2. Then

The planner’s problem can be stated as:

subject to F(K_{0}) = C + {δ + PD(Q)}K_{0} + θQ where K_{0} is the initial capital stock and θ is the unit cost of adaptation capital. Form the Lagrangian:

The first order conditions are as follows:

Combining the first two FOCs yields

\( \mathrm{where}\ \frac{{\mathrm{U}}_2}{{\mathrm{U}}_1} \) is the negative of the marginal rate of substitution (MRS) between consumption and adaptation \( \left(\Delta \mathrm{C}/\Delta \mathrm{Q}<0\right) \) at time t = 0. For a solution to exist with\( \mathrm{where}\ \frac{{\mathrm{U}}_2}{{\mathrm{U}}_1}>0 \), we require that \( \left[{\mathrm{K}}_0\mathrm{PD}^{\prime}\left({\mathrm{Q}}^{\ast}\right)+\uptheta \right]>0 \). Using D(Q) = Do[1 - (Q^{β})/Qm] with 0 < β < 1 and Qm = [A(Ko)^{α}]/θ, it is easy to show that this condition is satisfied for a reasonable choice of parameters P, Do, α, and β (unit cost θ factors out).

Substituting full expressions for U_{1} and U_{2} in the term U_{2}/U_{1} and then combining with the first order condition ∂L/∂λ = 0 to substitute for C, leads to a challenging system of two nonlinear equations in the two variables C and Q. A numerical approach to the solution of this non-linear system will be required to yield the desired Q**.

One potential approach is the process of iteration, starting with the initial value Q* from the basic model and solving for C_{1} = F(Ko) - δKo - [KoPD(Q*)]. Then substituting for C_{1} in the system, numerically solve for Q_{1}. Now set C_{2} = F(Ko) - δKo - [KoPD(Q_{1})]. Continuing the process, generates a chain: Q* - > C_{1} - > Q_{1} - > C_{2} - > Q_{2} - > C_{3} - > Q_{3} - > ………

This process of iteration may lead to one of several outcomes:

- 1)
The process converges to a unique Q** with Q* < Q** < Qm.

- 2)
The process ultimately alternates between values Q^ and Q^^ with Q* < Q^ < Q^^ < Qm, in which case Q** may be approximated as [Q^^ - Q^]/2.

- 3)
The process diverges and fails to yield a numerical approximation for Q**, in which case other methods must be pursued.

For numerical approaches to solution, we refer to Judd (1998), especially chapters 5 (Nonlinear Equations) and 6 (Approximation Methods).

## Rights and permissions

## About this article

### Cite this article

Endress, L.H., Roumasset, J.A. & Wada, C.A. Do Natural Disasters Make Sustainable Growth Impossible?.
*EconDisCliCha* **4**, 319–345 (2020). https://doi.org/10.1007/s41885-019-00054-y

Received:

Accepted:

Published:

Issue Date:

DOI: https://doi.org/10.1007/s41885-019-00054-y