Adapting to rates versus amounts of climate change: A case of adaptation to sea level rise Supplementary Information

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1 Adapting to rates versus amounts of climate change: A case of adaptation to sea level rise Supplementary Information Soheil Shayegh, Juan Moreno-Cruz, Ken Caldeira We formulate a dynamic model to solve the problem of finding an optimal investment strategy, in which a single decision maker such as a private investor or local government aims to maximize the net present value of a unit investment in the coastal zone. We first introduce the notation for formulating this problem and then look at different adaptation strategies and compare their outcomes. We end with a numerical example. S Model Formulation S. Variables B x,t : Infrastructure capital L x,t : Land attractiveness R x,t : Revenue from a unit investment W x,t : Net present value of unit investment D h : Cost of building and maintaining the dike S.2 Parameters δ: Infrastructure depreciation rate θ: Coefficient of land attractiveness r: Discount rate R : Constant revenue v: Speed of sea level rise h: Effective height of the dike s: Slope of the shore α, β: coefficients of cost function

2 S.3 Equations Land attractiveness L x = e θx Infrastructure capital dynamics B x,t t = δb x,t B x,t = e δt 2 Revenue from a unit investment R x,t = B x,t L x R 3 Cost of building a dike D h = α + β h 4 2

3 S2 Assessing investment strategies First we analyze the case where there is no protection against the sea level rise. In this case the only adaptation measure will be to retreat and invest in areas far from the sea. For a given zone that lies in distance x from the sea, the land attractiveness is increasing with the rate of sea level rise L x,t = e θx vt/s. For simplicity, later we assume s =. Therefore the revenue from a unit investment is R x,t = e δt L x,t R. The total elapsed time before this location is flooded can be calculated as t max = x/v. The net present value of unit investment then can be expressed as: W x = = = tmax=xs/v xs/v xs/v e rt R x,t = xs/v e rt e δt e θx vt/s R e θv/s r δt e θx R xs/v = e θx R e θv/s r δt e rt e δt L x,t R θv/s r δx = e θx R e v/s θv/s r δ R e θx = e r+δ θv/s x v/s 5 r + δ θv/s This equation gives the necessary condition for feasibility of the solution when r + δ θv/s θ r + δ v/s 6 In order to find the critical location for the unit investment where the total revenue is maximum, we take the derivative of W x with respect to x: W x x r + δ r+δ = v/s e v/s x = θe θx x = ln r+δ θv/s θ r+δ θv/s W = R r+δ θv/s θv/s r+δ θv/s r + δ 7 In the limits when θ r+δ, we have v/s x v/s. The general shape of W r+δ x function is shown in Figure S. 3

4 Figure S: Net present value of unit investment W x as a function of location x. 4

5 S2. No adaptation We can think of no adaptation case as a case with no sea level rise. In that sense it will be the limiting case where v in Equation 7. In this case the optimal location x. In this case the investment is only made at the closest place to the current shoreline which will be lost due to the upcoming sea level rise. Figure S2: In the case of no adaptation the net present value of unit investment goes to zero as the sea level rises. S2.2 Adaptation to amounts of change In this case a buffer zone is created near the shoreline. The sea is expected to rise only up to the limits of this zone and therefore, the new investments are assumed to be protected beyond the buffer zone. Lets denote by x the depth of the buffer zone. In this case, the investors assume that the land attractiveness remains unchanged and therefore their perceived net present value NPV of unit investment can be written as: W x = = R e θx r + δ e rt e δt e θx R = e θx R e r+δt for x x 8 From the equation above it is apparent that W is decreasing in x and therefore, the optimal investment happens at x = x. The maximum perceived NPV in this case will be W x = R e θ x. This location is considered to have the most land attractiveness because r+δ investment is prohibited within the buffer zone. However, in reality the sea level continues to rise until it surpasses the buffer zone at time t =. So the real net present value W x v/s 5

6 comes from investing right at the border of the buffer zone where we have: W x = R e θ x e r+δ θv/s x v/s 9 r + δ θv/s We can compare this case with the optimal case presented in Equation 7. if the buffer zone ends before the optimal location x, that is x > x, there will be a missed opportunity since the optimal location for the unit investment would still be feasible but the investment is made closer to the shoreline instead of at its optimal location. On the other hand, If x x the best place to investment will be at the edge of the buffer zone that is farther from the shoreline and therefore generates less NPV. 6

7 a x x: If the buffer zone ends before the optimal location x, the NPV of unit investment at the edge of the buffer zone is less than the optimal NPV at x. b x < x: If the buffer zone goes beyond the optimal location x, the best location to make a unit investment will be at the edge of the buffer zone at x that generates less NPV than investing at x. Figure S3: Adapting to the amount of change 7

8 S2.3 Adaptation to rates of change In the this case a unit investment can be made at the optimal location x as it is shown in Figure S4. The maximum achievable net present value from a unit investment is W. Figure S4: Adapting to the rate of change. The optimal location for unit investment is at x. S2.4 Adaptation with protection To account for the dike case, we assume that only one dike can be built in the current location of the shoreline. Further, we assume that the cost of building and maintaining the dike is a linear function of its height as expressed in Equation 4. We can consider dike as an option that delays the sea level rise for a finite amount of time. This means the dike can be built only in a predefined height. When the sea level reaches this height h after a certain time period defined as h/v, we will retreat inland and invest similar to the case with no dike. Therefore, for any given height of h we need to calculate and compare two cases. The first is the case of investment beyond the protected zone. The protection option provides a temporary hold to the sea level rise and therefore, keeps the land attractiveness fixed until the sea level reaches the height of the dike. Once the water surpassed the dike, it will approach the investment location with horizontal speed of v/s. The attractiveness increases as the distance from the shoreline decreases at a constant rate. The second option is to invest within the protected zone right behind the dike where it has the highest land attractiveness. This investment lasts shorter compared to the previous case. After the investment is flooded, the optimal investment will follow the optimal investment discussed in Section S2.3. 8

9 In order to be able to compare the net present value of these two scenarios and decide about the optimal approach, we need to have the same time scale for both scenarios. Since the fist option investing beyond the protected zone last longer, we pick its time scale and adopted for the second scenario.. Investing beyond the protected zone W x>h/s,h = t=h/v t= e rt e δt e θx R + = R e θx [ e r+δh/v r + δ t=xs/v t=h/v e rt e δt e θx vt/s R + e r+δ θv/sh/v e r+δ θv/sxs/v r + δ θv/s [ e = R e θx r+δh/v + e r+δ θv/sh/v r + δ r + δ θv/s }{{} A h [ ] = R e θx A h e Bxs/v B e r+δ θv/sxs/v r + δ θv/s }{{} B ] ] The optimal location of the unit investment is a value of x that maximizes W : x = argmaxw x>h/s,h = v/sb ln W x, h = R A h θa h θ/b + s/v θ/b θa h θ/b + s/v θ/b + s/v 2. Investing behind the dike W 2 x=,h = t=h/v t= e rt e δt e θx R = R e r+δh/v 2 r + δ Total net present value of unit investment, therefore, can be found by comparing this two options. In long run and with sequential investing, the net present value is found by adding optimal investment at each time step and applying the discount factor to it. As discussed in Section S2, after the last piece of dike goes under water, the optimal strategy will be the same as the case with no protection. S2.5 Net NPV from protection Equations and 2 provide a framework for calculating the optimal NPV of a unit investment as a function of the dike s height. To calculate the total NPV, we calculate the NPV from a unit investment through time while the sea level is rising and the effective height of the dike is decreasing. Furthermore, we can consider some initial infrastructure by the 9

10 shoreline. In this case, there is an economic incentive for protecting these assets against the sea level rise. Such economic incentive appears as an additional term in calculating the total NPV. Therefore, for any given dike with the initial height h, the total NPV can be written as: NP V dike = max Wx >h/s,h, W 2 x =,h + + r max Wx >h v/s,h v, W 2 x =,h v + + r max W 2 x >h 2v/s,h 2v, W 2 x =,h 2v r max W h v x >v/s,v, W 2 x =,v + R B r + δ [ e r+δh/v ] + Wno dike 3 +r where R B r+δ [ e r+δh/v ] is the NPV of preexisting infrastructure calculated from Equation 2 by considering the initial infrastructure B instead of a unit investment. In this case, at each time step a unit investment is made to recover the loss from capital depreciation. Therefore, the steady-state infrastructure will be B = /δ. We can compare NP V dike with NP V no dike and calculate the difference as the net NPV from protection. Similar to the case above, NP V no dike can be written as a convolution of investments for any given time duration that is translated from a given height of the dike. Investment in optimal location x generates Wno t dike after t time steps: Wno t dike = = = t t t e rt R x,t = t e rt e δt e θx vt/s R e θv/s r δt e θx R t = e θx R e θv/s r δt R e θv/s r δt = e θx θv/s r δ θv/s θv/s r δ θv/s = R r+δ θv/s r δ e rt e δt L x,tr e θv/s r δt 4

11 The NPV of investment in the no-dike case for a time equivalent to a given dike s height can be calculated from this equation: NP V no dike = { h/v + + t= { } + r W t t no dike + r h x s v x s/v t= if h x s + r W t t no dike + h x s v i= } W + r i if h>x s +r τ W no dike 5 where W is the optimal NPV of a unit investment from Equation 7. Since the last argument in the above equation is independent of the dike s height h, we can ignore it when calculating the net NPV as a difference between two cases of NP V dike and NP V no dike. Alternatively, we can estimate the height of the dike using a simple approximation. If the dike is built right where the current shoreline is and the property to be protected is right behind the dike, the optimal height of the dike delays the sea-level rise until the investment is fully depreciated. Therefore, the time needed for the sea-level to reach the top of the dike should be the same as the time needed to depreciate the investment: h v = r + δ h = v r + δ 6

12 S3 Numerical example Here we look at the case with a set of parameters as following: δ =.25 yr : Infrastructure depreciation rate θ =. m : Coefficient of land attractiveness r =.5 yr : Discount rate R = $: Constant revenue v = cm yr : Speed of sea level rise s = cm km : Slope of the shore In this case condition 6 holds as θ =. and r+δ v/s =.75. The optimal values are calculated as x = 3 m and W = $9.78 Red dashed line in Figure S5. The net present value of unit investment in locations closer to or farther from the shoreline is less than this optimal value. For example, the NPV drops by 4% to $8.36 for the location at half the distance from the shoreline at x = 55 m Dark green dashed line in Figure S5. Similarly, the NPV for the location two times farther from the shoreline at x = 62 m decreases by 7% to $8.3 Light green dashed line in Figure S5. Figure S5: Net present value of unit investment W x as a function of location x. The optimal location for unit investment is at x = 3 m. Investments in the areas closer to the sea x < 3 m will disappear before generating the maximum revenue. On the other hand. Investing in farther locations x > 3 m is suboptimal too since the investment will depreciate faster than the rate of increase in land attractiveness. 2

13 If we assume a buffer zone of x = 5 m length, the NPV of unit investment at the edge of the buffer zone can be calculated as $8.97 using Equation 9. This indicates about 8% reduction in every dollar investment compared to investment in the optimal location. Similarly, if we build a dike of.5 m height that protects an area of 5 m length, the NPV of unit investment behind the dike can be obtained from the Equation 2 that is equal to $3.2. This means a 33% increase in NPV due to the protection. S3. Assessment of protection benefit Figure S6 shows the net NPV accumulation in addition to the protection cost curve as functions of the protection height. If the dike is relatively cheap, the optimal height of the dike can be found from the point where there is the largest gap between the revenue from the investment and the cost of the dike line C-D. We assume the parameters of the cost to be α = $2 and β = $8 cm. For this set of parameters, the optimal height of the dike is 2 cm that can protect 2 m of the shoreline for 2 years. If a dike shorter than that represented by point A is considered, the cost of dike is larger than the NPV of protected investment and therefore building the dike is not justified. Similarly, if the protection period is longer than depreciation and discounting period i.e. dike is higher than that represented by point B building a high dike is not justified. Moreover, building a dike is not a viable economic option if the dike is too expensive i.e. the cost curve does not intersect the NPV curve. Figure S6: The feasible height of the dike can be obtained from the intersection of the dike s cost and NPV of unit investment. Building a shorter dike than that represented by point A or higher dike than that represented by point B is not economically feasible. The optimal height of the dike is where the gap between the two graphs is maximum C-D. 3

14 S3.2 Impact of rate of sea level rise and discount rate on optimal solution Increase in the rate of sea level rise v is our equations motivates investment in the areas farther from the shoreline as presented in Equation 7. Figure S7 shows the impact of change in the rate of sea level rise on the optimal location for unit investment. In the base case where the rate of sea level rise is. cm yr, the optimal location for unit investment is at 3 m distance from the shoreline.if the rate of sea level rise increase to 2 cm yr the optimal location moves to 48 m distance from the shoreline. On the other hand, if the rate of sea level rise decreases to.5 cm yr, the optimal location moves closer to the shoreline and will be at 93 m distance from the shoreline. Figure S7: The optimal location for unit investment changes as the rate of sea level rise changes. The optimal location for the base case is located at 3 m distance from the shoreline blue line. Higher rates of sea level rises green lone induce investment in the areas farther from the shoreline. In contrast, lower rates of sea level rise orange line makes the area near the shoreline more attractive for investment. We can investigate the impact of change in discount rate on the optimal location for unit investment in a similar fashion. Contrary to the previous case, higher rate of discounting encourages the investments in the areas closer to the shoreline. Lower discount rates on the other hand, provides more incentive to invest farther from the shoreline. Figure S8 demonstrates the change in the optimal location of unit investment compared to the base case. 4

15 Figure S8: The optimal location for unit investment changes as discount rate changes. The optimal location for the base case is located at 3 m distance from the shoreline blue line. Higher discount rates green lone induce investment in the areas closer to the shoreline. In contrast, lower discount rates orange line makes the area farther from the shoreline more attractive for investment. 5