Adapting to Rising Sea Levels: How Short-Term Responses Complement Long-Term Investment

Abstract

This paper presents a parsimonious model of a coastal locality’s adaptation to rising sea levels and uses the model to examine cost-minimizing policies involving two complementary approaches. One involves irreversible investment in sea walls and similar infrastructure. The other involves activities, such as beach scraping, that only provide temporary protection. Costs are minimized by delaying investment until the present value of the benefits from avoided inundation costs exceeds upfront investment costs by a margin that is economically significant. The premium, which can exceed 50% of investment costs, is higher when the sea level is rising more quickly. The ability to temporarily boost defenses is used aggressively: spending on temporary improvements immediately before investment is several times larger than its value immediately afterwards. Temporary improvements are made even when the marginal cost of increasing the effectiveness of defenses this way is significantly greater than the equivalent annual cost of permanently increasing effectiveness by investment.

Appendices

Appendices

Proofs

1.1 A.1 Proof of Proposition 1

Suppose that the local government invests immediately if and only if $z \ge {\hat{z}}$, for some constant ${\hat{z}}$. Further suppose that when $z = {\hat{z}}$, the local government increases the effectiveness of the defenses by ${\hat{i}}$, which resets the annual inundation cost to $\theta e^{-(x+{\hat{i}})}y = e^{- {\hat{i}}}{\hat{z}}$.

Let v(z) denote the present value of all future costs if the local government adopts this investment policy. If $z < {\hat{z}}$ then the local government waits, so that the present value satisfies

$$\begin{aligned} \rho v(z) = z + \frac{E[dv(z)]}{dt}. \end{aligned}$$

In the waiting region, z evolves according to the geometric Brownian motion

$$\begin{aligned} dz = (\mu +\delta ) z \, dt + \sigma z \, d\xi , \end{aligned}$$

which implies that v(z) satisfies the ordinary differential equation

$$\begin{aligned} 0 = \frac{1}{2} \sigma ^2 z^2 v''(z) + (\mu +\delta ) z v'(z) - \rho v(z) + z, \;\;\; 0< z < {\hat{z}}. \end{aligned}$$

The boundary conditions are $v(0)=0$ and $v({\hat{z}}) = f + c {\hat{i}} + v(e^{- {\hat{i}}} {\hat{z}})$. This system has solution

$$\begin{aligned} v(z) = \frac{z}{\rho -(\mu +\delta )} - A({\hat{i}},{\hat{z}}) z^{\beta }, \end{aligned}$$

where

$$\begin{aligned} A({\hat{i}},{\hat{z}}) = \frac{1}{{\hat{z}}^{\beta } (1 - e^{- \beta {\hat{i}}})} \left( \frac{(1-e^{- {\hat{i}}}){\hat{z}}}{\rho -(\mu +\delta )} - (f + c {\hat{i}}) \right) . \end{aligned}$$

As the solution for v(z) involves subtracting $A({\hat{i}},{\hat{z}}) z^{\beta }$, a cost-minimizing policy corresponds to values of ${\hat{i}}$ and ${\hat{z}}$ that maximize $A({\hat{i}},{\hat{z}})$.

Now consider the case where $z > {\hat{z}}$. If the local government increases the effectiveness of the defenses by i, the present value of costs equals $f + ci + v(e^{- i}z)$. The first-order condition for the cost-minimization problem is therefore $(e^{- i}z) v'(e^{- i}z)=c$. The form of this equation implies that for all values of z the local government should reset the annual inundation cost to the same level, which in this case equals $e^{- {\hat{i}}}{\hat{z}}$. That is, the solution of the first-order condition satisfies $e^{- i}z = e^{- {\hat{i}}} {\hat{z}}$, which implies that $i = {\hat{i}} + \log (z/{\hat{z}})$. The value function in this region is

$$\begin{aligned} v(z) = v(e^{- {\hat{i}}} {\hat{z}}) + f + c \left( {\hat{i}} + \log (z/{\hat{z}}) \right) = v({\hat{z}}) + c \log (z/{\hat{z}}), \end{aligned}$$

which can be written as

$$\begin{aligned} v(z) = v(e^{- {\hat{i}}} {\hat{z}}) + f + c \log \left( \frac{z}{e^{-{\hat{i}}}{\hat{z}}} \right) . \end{aligned}$$

1.2 Proof of Corollary 1

Immediately after investment, $\log z = \log {\hat{z}} - {\hat{i}}$. The next investment occurs when $\log z$ increases to $\log {\hat{z}}$; that is, as soon as $\log z$ has increased by ${\hat{i}}$. As $\log z$ follows arithmetic Brownian motion with drift $\mu + \delta - \frac{1}{2} \sigma ^2$ and volatility $\sigma$, it follows that the time between investments is distributed according to the inverse Gaussian distribution with mean ${\hat{i}}/(\mu + \delta - \frac{1}{2} \sigma ^2)$ and scaling parameter $( {\hat{i}}/\sigma )^2$ (Folks and Chhikara 1978).³⁶

1.3 Proof of Corollary 2

A policy is optimal if ${\hat{i}}$ and ${\hat{z}}$ maximize $A({\hat{i}},{\hat{z}})$. Differentiating A with respect to ${\hat{z}}$ and setting the result equal to zero shows that ${\hat{z}}$ satisfies equation (3). Using equation (3) to eliminate ${\hat{z}}$ from the expression for A shows that the optimal investment scale maximizes $A'$.

1.4 Proof of Corollary 3

Let the present value of future inundation costs equal w(z) if the annual inundation cost is currently equal to z. It follows that the present value of future inundation costs, measured immediately before investment, equals $w({\hat{z}})$ in the factual scenario and $e^{{\hat{i}}} w({\hat{z}})$ in the counterfactual scenario.³⁷ The benefit–cost ratio of investment therefore equals $BCR = (e^{{\hat{i}}} - 1) w({\hat{z}})/(f+c{\hat{i}})$.

An exact solution for w(z) can be obtained by dropping all investment-expenditure terms from the expression for v(z) in Proposition 1. This shows that

$$\begin{aligned} w({\hat{z}}) = \frac{{\hat{z}}}{\rho -(\mu +\delta )} \left( 1 - \frac{1-e^{-{\hat{i}}}}{1 - e^{-\beta {\hat{i}}}} \right) = \frac{{\hat{z}}}{\rho -(\mu +\delta )} \cdot \frac{e^{(\beta -1){\hat{i}}}-1}{e^{\beta {\hat{i}}}-1}, \end{aligned}$$

so that the benefit–cost ratio equals

$$\begin{aligned} BCR = \frac{{\hat{z}}}{\rho -(\mu +\delta )} \cdot \frac{e^{(\beta -1){\hat{i}}}-1}{e^{\beta {\hat{i}}}-1} \cdot \frac{e^{{\hat{i}}} - 1}{f+c{\hat{i}}}. \end{aligned}$$

(A.1)

This holds for all policy parameters ${\hat{i}}$ and ${\hat{z}}$. If the local government chooses the investment threshold in equation (3), equation (A.1) simplifies to equation (4).

The threshold for the benefit–cost ratio can be written in the form

$$\begin{aligned} BCR = \frac{\beta }{\beta -1} \left( 1-g(e^{{\hat{i}}}) \right) , \end{aligned}$$

where $g(u) = (u-1)/(u^\beta -1)$. Differentiating the expression for g(u) shows that $g'(u) = h(u)/(u^\beta -1)^2$, where $h(u) = u^\beta - 1 - \beta (u-1)u^{\beta -1}$. As $h(1)=0$ and $h'(u) = - \beta (\beta -1)(u-1)u^{\beta -2} < 0$ for all $u>1$, it follows that $h(u)<0$ for all $u>1$. This implies that $g'(u)<0$ for all $u>1$. Therefore $BCR$ is an increasing function of ${\hat{i}}$. L’Hôpital’s rule shows that $BCR =1$ when ${\hat{i}}=0$.

1.5 Proof of Lemma 1

Let $g(s) = e^{-s}z + \alpha s^2$, so that $g'(s) = -e^{-s}z + 2\alpha s$ and $g''(s) = e^{-s}z + 2\alpha > 0$. Solving the first-order condition $g'(s)=0$ for s shows that g(s) is maximized where s satisfies $s e^{s} = z/(2\alpha )$. This implies that $s^*(z) = W_0(z/(2\alpha ))$, which implies that

$$\begin{aligned} g(s^*(z)) = e^{-s^*(z)}z + \alpha (s^*(z))^2 = \alpha s^*(z) (2 + s^*(z)) = \alpha W_0\left( \frac{z}{2\alpha } \right) \left( 2 + W_0\left( \frac{z}{2\alpha } \right) \right) . \end{aligned}$$

1.6 Proof of Proposition 2

In the waiting region, the present value of future costs is

$$\begin{aligned} \left( e^{-s^*(z)}z + \alpha (s^*(z)^2) \right) dt + e^{-\rho \, dt} E[v(z+dz)], \end{aligned}$$

which can be written in the form

$$\begin{aligned} v(z) + \left( e^{-s^*(z)}z + \alpha (s^*(z))^2 + (\mu +\delta ) z v'(z) + \frac{1}{2} \sigma ^2 z^2 v''(z) - \rho v(z)\right) dt + o(dt). \end{aligned}$$

If the local government adopts the investment policy described by the constants ${\hat{i}}$ and ${\hat{z}}$, v(z) satisfies the ordinary differential equation

$$\begin{aligned} 0 = \frac{1}{2} \sigma ^2 z^2 v''(z) + (\mu +\delta ) z v'(z) - \rho v(z) + e^{-s^*(z)}z + \alpha (s^*(z))^2, \;\;\; 0< z < {\hat{z}}, \end{aligned}$$

together with the boundary conditions $v(0)=0$ and $v({\hat{z}}) = f + c {\hat{i}} + v(e^{- {\hat{i}}} {\hat{z}})$. This system has solution $v(z) = u(z) - B({\hat{i}},{\hat{z}}) z^{\beta }$, where

$$\begin{aligned} B({\hat{i}},{\hat{z}}) = \frac{1}{{\hat{z}}^{\beta } (1 - e^{- \beta {\hat{i}}})} \left( u({\hat{z}}) - u(e^{- {\hat{i}}} {\hat{z}}) - (f + c {\hat{i}}) \right) \end{aligned}$$

and u(z) is the particular solution that corresponds to the present value of future costs if the local government makes no future investments, while still adopting the cost-minimizing policy $s^*(z)$. As the solution for v(z) involves subtracting $B({\hat{i}},{\hat{z}}) z^{\beta }$, a cost-minimizing policy corresponds to values of ${\hat{i}}$ and ${\hat{z}}$ that maximize $B({\hat{i}},{\hat{z}})$.

B Miscellaneous Results

1.1 B.1 Steady-State Distribution

As long as z is away from the upper boundary at ${\hat{z}}$, it evolves according to $dz = (\mu +\delta ) z \, dt + \sigma z \, d\xi$, but as soon as it hits the upper boundary, z is reset to $e^{-{\hat{i}}}{\hat{z}}$. The Fokker–Planck equation for this process is

$$\begin{aligned} 0 = \frac{\partial ^{} p}{\partial t^{}} -\frac{1}{2} \sigma ^2 z^2 \frac{\partial ^{2} p}{\partial z^{2}} + (\mu +\delta -2 \sigma ^2) z \frac{\partial ^{} p}{\partial z^{}} + (\mu +\delta -\sigma ^2) p. \end{aligned}$$

I look for the stationary distribution; that is, I suppose that $\partial p / \partial t = 0$ and impose the boundary conditions $p(0)=0$ and $p({\hat{z}})=0$.

The differential equation and the left-hand boundary condition imply that $p(z) = C_1 z^{\gamma _1}$ in the region $0< z < e^{-{\hat{i}}}{\hat{z}}$, where $\gamma _1 = 2(\mu + \delta - \sigma ^2)/\sigma ^2$ and $C_1$ is a constant. The differential equation and the right-hand boundary condition imply that $p(z) = C_2 ( {\hat{z}}^{1+\gamma _1} /z - z^{\gamma _1})$ in the region where $e^{-{\hat{i}}}{\hat{z}}<z<{\hat{z}}$, where $C_2$ is a constant. The density function is continuous at the reset point, which implies that $C_1 = C_2 (e^{{\hat{i}}(1+\gamma _1)}-1)$. Finally, the area under the density function must equal one, which implies that

$$\begin{aligned} 1 = \frac{C_1 (e^{-{\hat{i}}}{\hat{z}})^{\gamma _1+1}}{\gamma _1+1} - C_2 \left( \frac{{\hat{z}}^{\gamma _1+1}}{\gamma _1+1} - {\hat{z}}^{\gamma _1+1} \log {\hat{z}} \right) + C_2 \left( \frac{(e^{-{\hat{i}}}{\hat{z}})^{\gamma _1+1}}{\gamma _1+1} - {\hat{z}}^{\gamma _1+1} \log (e^{-{\hat{i}}}{\hat{z}}) \right) . \end{aligned}$$

Substituting in the expression for $C_1$ and solving for $C_2$ shows that $C_2 = 1/({\hat{i}}{\hat{z}}^{\gamma _1+1})$.

Integrating zp(z) between 0 and ${\hat{z}}$, allowing for the piecewise nature of p(z), shows that

$$\begin{aligned} E[z] = \frac{\gamma _1+1}{\gamma _1+2} \cdot \frac{{\hat{z}}-e^{-{\hat{i}}}{\hat{z}}}{\log {\hat{z}} - \log (e^{-{\hat{i}}}{\hat{z}})} = \left( 1 - \frac{\sigma ^2}{2(\mu + \delta )} \right) \left( \frac{1-e^{-{\hat{i}}}}{{\hat{i}}}\right) {\hat{z}}. \end{aligned}$$

Likewise, integrating $(\log z) p(z)$ between 0 and ${\hat{z}}$ shows that

$$\begin{aligned} E[\log z] = \log {\hat{z}} - \frac{1}{2} \left( {\hat{i}} + \frac{1}{\frac{\mu + \delta }{\sigma ^2}-\frac{1}{2}}\right) . \end{aligned}$$

1.2 B.2 Model Calibration

In the model, the annual inundation cost equals $z = \theta e^{-x}y$, where $\theta$ is a positive constant, x measures the effectiveness of coastal defenses, and y equals local GDP. The expected growth rate in this variable, measured during a period when no investment occurs, equals

$$\begin{aligned} \frac{1}{z} \cdot \frac{E[dz]}{dt} = \mu + \delta . \end{aligned}$$

The parameter $\delta$ can therefore be calibrated using estimates of the growth rate in annual inundation costs, after controlling for local GDP growth.

Hallegatte et al. (2013) estimate average annual losses due to flooding for the world’s 136 largest coastal cities, including Miami. The authors consider 108 scenarios in total, comprising all combinations of three socio-economic scenarios, three climate-change scenarios, two subsidence scenarios, three defense-failure scenarios, and two protection-level scenarios. I focus on one of these, which assumes no socio-economic change, but allows for subsidence, maximum protection levels, and pessimistic defense-failure rates. This scenario assumes that the mean sea-level will rise by 10cm in 2030, 20cm in 2050, and 30cm in 2070. Without adaptation investment, the average annual loss is $672m in 2005, $1,256m in 2030, $2,350m in 2050, and $4,395m in 2070, implying average annual growth rates in losses of 2.53%, 2.82%, and 2.93% respectively.³⁸ I therefore set $\delta =0.028$.³⁹