Development Level and Optimal Mixture of Adaptation and Mitigation Investments∗ Wataru Nozawa† Tetsuya Tamaki‡ Shunsuke Managi§ 744 Motooka, Nishi-ku, Fukuoka 8190395 Japan School of Engineering, Kyushu University September 12, 2016

Abstract Determining the optimal combination of mitigation and adaptation investments is an important topic in policy making to combat climate change. Some analytical results on the relationship between the optimal ratio of adaptation to mitigation and development level have been reported in the literature. In this paper, we examine this relationship in greater detail using a simple model with general return functional forms and analytically show that the relationship can take various forms. The insights obtained in the simple model are useful to understand more complicated models.

1

Introduction

Adaptation, the importance of which was recognized later than that of mitigation, has acquired a prominent place in policy making on climate change. The Conference of the Parties 21st Session to the UN Convention on Climate Change (COP21), which was held in 2015, recognized an increasing role for adaptation and emphasizes the benefits of cost reductions for adaptation efforts. New York City’s Flexible Adaptation Pathways and the Climate Smart Adaptation of the Queensland Climate Change Centre of Excellence are famous examples of adaptation efforts at the regional level. The growing attention on the topic has demanded studies on various topics concerning adaptation. de Bruin et al. (2009) and de Bruin (2011) add an adaptation decision to the DICE model, which is an Integrated Assessment Model (IAM), and examined how adaptation interacts with mitigation decisions in balancing between climate change costs and economic costs. ? focus on the fiscal revenue effects of mitigation and adaptation options and discuss implications in the context of optimal taxation. Because resources are limited, how to allocate them between mitigation and adaptation investments is an important problem. A notable feature of the problem is that an increase in a mitigation ∗ This research was supported by the Environment Research and Technology Development Fund (S-14) of the Ministry of the Environment, Japan, and Specially Promoted Research through a Grant-in-Aid (Scientific Research 26000001) from the Japanese Ministry of Education, Culture, Sports, Science and Technology (MEXT). † E-mail: [email protected]; Phone and Fax: +81928023429; Corresponding author ‡ E-mail: [email protected] § E-mail: [email protected]

1

measure decreases the necessity of adaptation, and vice versa. Some analytical results on efficient divisions have been reported in the literature; Ingham et al. (2013) show that, in a variety of simple mitigation-adaptation investment problems, the two measures are substitutes. Zemel (2015) focuses more on the dynamic aspect of the problem and relates the optimal timing of beginning investments in adaptation measures to the amount of existing pollution stocks. Br´echet et al. (2013) explore the relationship between the level of production efficiency and the efficient share of adaptation relative to mitigation. They report an analytical result that an economy with very low productivity should engage in mitigation only and that the optimal ratio of adaptation investment to mitigation investment increases as productivity increases up to a threshold level, beyond which the ratio decreases under a specific set of functional forms. The relationship is explored through an IAM in Bosello et al. (2010), who explore the difference between OECD and non-OECD countries in the composition of adaptation types, namely, anticipatory, reactive, and innovation, based on a calibrated result in an AD-WITCH model. In this paper, we examine this relationship in greater detail. First, we formulate a simple static model of mitigation and adaptation investments with general return functional forms in Section 2. The model includes a parameter capturing the amount of available resources for mitigation and adaptation, which we interpret as the level of development. The objective function is assumed to take a multiplicative form to capture the feature of the mitigation-adaptation investment problem whereby an increase in a given mitigation measure decreases the necessity of adaptation, and vice versa. The multiplicative form can endogenously arise in some specific contexts, as we discuss in Section 4, and should be understood as a crude way to incorporate the feature into the model in other contexts. Our main result is reported in Section 3. After showing that an equation determines the optimal ratio, using the equation, we analytically prove that the relationship can exhibit various patterns: any functions in a wide class can yield the optimal ratio if we appropriately select a pair of return functions. Note that all functional forms for mitigation and adaptation returns satisfy reasonable assumptions, in that they are monotonic and have diminishing marginal returns. To demonstrate the breadth of the range of potential functions, we present several examples. The examples demonstrate that the optimal ratio can be (i) constant, (ii) 0 (meaning no adaptation) for low productivity and increasing for high productivity, (iii) infinity (meaning no mitigation) for low productivity and decreasing for high productivity, (iv) increasing for low productivity and decreasing for high productivity, (v) decreasing for low productivity and increasing for high productivity, (vi) monotonically increasing, and (vii) monotonically decreasing. Although our model is static and extremely simple, the insight from the examples is useful to understand more complicated models, such as that of Br´echet et al. (2013). In Section 4, we show the optimal ratio in the model of Br´echet et al. (2013) is determined by an equation similar to that in our simple model. Then, we provide examples in which various patterns that are similar to the examples in our model arise. We also relate our result to Zemel (2015). In his model, the multiplicative form arises endogenously and, therefore, offers an interpretation of the multiplicative form of utility functions. We show that the optimal ratio is characterized similarly in Zemel (2015). Our result re-emphasizes the importance of collecting empirical evaluations of costs and benefits of mitigation and adaptation measures, which is far from new and repeatedly highlighted in the literature (e.g., IPCC (2007)). Unless we know the form of mitigation and adaptation return functions, we cannot determine the optimal division of resources. Our result indicates that it is unlikely that calibration can solve the problem, unless a quite wide class of return functions,

2

which have many parameters, is employed. Calibrating such parameters would also require further empirical evidence. As discussed in Agrawala et al. (2011), empirical work is at present limited to specific sectors and areas. Pindyck (2013) criticizes IAM models for various arbitrary in modeling decisions, including the choice of functional forms. This paper addresses a similar problem in mitigation-adaptation investment and describes how functional forms can affect the outcomes of models with more technical details in this specific context.

2

Model

The model is static. There is a exogenous resource Y > 0 that can be used for mitigation B ≥ 0 and adaptation D ≥ 0: B + D ≤ Y. (1) Allocations are evaluated by −η(D)g(B),

(2)

where η : [0, ∞) → (0, ∞) and g : [0, ∞) → (0, ∞) are both strictly decreasing, meaning that mitigation and adaptation are intrinsically beneficial. Although this model abstracts from why mitigation and adaptation are beneficial, it is nevertheless useful for understanding more structured models, such as that of Br´echet et al. (2013). The multiplicative form is a simple but crude way to capture an important property of the mitigation-adaptation investment problem: an increase in mitigation investment decreases the necessity of adaptation investment, and vice versa. Even with such a simplifying assumption, the relationship between the optimal ratio and the amount of resources exhibits various patterns, as we show later. The planner’s problem is to maximize (2) subject to (1), B ≥ 0, and D ≥ 0. We assume that η and g are continuously differentiable and strictly log-convex: log η(D) and log g(B) are strictly convex. This is a sufficient condition for the strict concavity of the objective function and simplifies the analysis.

3

Solution

Because the objective is strictly concave and the constraints are all linear, the optimal solution (B ∗ , D∗ ) is characterized by the KKT condition −η(D∗ )g 0 (B ∗ ) − λ + µ = 0, −η 0 (D∗ )g(B ∗ ) − λ + ξ = 0, λ[Y − B ∗ − D∗ ] = 0, µB ∗ = 0, ξD∗ = 0, and the constraints. Note that the resource constraint is satisfied at the equality because the objective is strictly increasing. The following lemma summarizes the characterization of the solution.

3

Lemma 1. (B ∗ , D∗ ) = (Y, 0) if and only if −

η 0 (0) g 0 (Y ) ≤− . η(0) g(Y )

−

η 0 (Y ) g 0 (0) ≥− . η(Y ) g(0)

(B ∗ , D∗ ) = (0, Y ) if and only if

(B ∗ , D∗ ) is interior and characterized by η 0 (D∗ ) g 0 (B ∗ ) = − , η(D∗ ) g(B ∗ ) B ∗ + D∗ = Y.

−

if and only if −

(3) (4)

η 0 (0) g 0 (Y ) η 0 (Y ) g 0 (0) >− and − <− . η(0) g(Y ) η(Y ) g(0)

The proof is standard, and hence, it is omitted. Define Fg : [0, ∞) → R and Fη : [0, ∞) → R by g 0 (B) , g(B) η 0 (D) . Fη (D) = − η(D) Fg (B) = −

Then, by the strict log-convexity, their inverses are well defined. When the solution is interior, the optimal adaptation D∗ can be found by solving D∗ + Fg−1 (Fη (D∗ )) = Y. Because, again, by strict log-convexity, Fg−1 ◦ Fη is strictly increasing, the equation shows that D∗ is increasing in Y . In the following, we denote the optimal adaptation by D∗ (Y ), emphasizing that it is a function of the amount of resources Y . Our main interest concerns how the optimal ratio of adaptation to mitigation R(Y ), which can be expressed as D∗ (Y ) R(Y ) = −1 , Fg (Fη (D∗ (Y ))) moves as the amount of resource Y increases. An explicit solution is often unavailable for R(Y ). We can nevertheless investigate the shape of R(Y ) in the following way. Define ˆ R(D) =

D

. Fg−1 (Fη (D))

If D∗ is differentiable, ˆ ∗ (Y )) dD∗ (Y ) dR(Y ) dR(D = . dY dD dY ˆ Because D∗ (Y ) is strictly increasing, the sign of dR/dY is the same as that of dR/dD. Therefore, ˆ we can investigate the slope of R by investigating the slope of R. The following proposition, which is our main result, states that the optimal ratio, as a function of the amount of resources, can exhibit various patterns. 4

Proposition 1. For any continuously differentiable function rˆ : (0, ∞) → (0, ∞) such that ∞ and for any D ∈ (0, ∞) such that rˆ0 (D) 1 < rˆ(D) D

´∞ 0

rˆ(D) D dD

holds, there exists a pair of continuously differentiable, strictly decreasing, and strictly log-convex functions (g, η) such that D rˆ(D) = −1 Fg (Fη (D)) holds. Proof. The proof is constructive. Take an arbitrary function rˆ : (0, ∞) → (0, ∞) that satisfies the inequality condition. Set g(B) = 1/B; then, it clearly is continuously differentiable, strictly decreasing, and strictly log-convex, and Fg (B) = Fg−1 (B) = 1/B. Set  ˆ η(D) = exp −

D

 rˆ(y) dy . y

0

Then, it is clearly continuously differentiable and strictly decreasing. Strict log-convexity is proven as follows:  ˆ D  d2 rˆ(y) d2 log η(D) = − dy . dD2 dD2 y  0  d rˆ(D) = − dD D 1 0 = − 2 (ˆ r (D)D − rˆ(D)) > 0. D The last inequality directly follows from the inequality condition. Because Fη (D) =

rˆ(D) , D

the existence of a pair (g, η) of our interest is proved. A natural question is whether the class of functions rˆ that satisfies the inequality condition is wide or narrow. First, note that any decreasing and integrable functions are included in the class. The following example shows that the class contains a quite complicated function, although it is unrealistic. Example 1. The return functions are given by g(B) = B −1 , η(D) = D

−2

 ˆ exp − 0

D

 sin2 y dy . y2

Then, g and η are continuously differentiable, strictly decreasing, and strictly log-convex. We have ˆ R(D) ≡

D D sin2 D = = + 2.  h 2 i−1 D Fg−1 (Fη (D)) 1 sin D +2 D

D

5

<

3.5

optimal ratio; D/B

3 2.5 2 1.5 1 0.5 0

0

2

4

6

8

10

resource; Y

Figure 1: Optimal ratio R(Y ) in Example 1 This function exhibits repetitive oscillation with decreasing amplitude, and thus the optimal ratio R(Y ) also exhibits such a pattern. Figure 1 displays the result of a numerical computation, which shows the actual shape of R(Y ). The next example is more standard in economic models and used in Br´echet et al. (2013). Example 2 (Ratio increasing for low level and decreasing for high level). The return functions are given by g(B) = B −1 , η(D) = η + (¯ η − η)e−aD , where η > 0, η¯ > 0, and a > 0 are parameters that satisfy η < η¯. This pair of functions are taken from Br´echet et al. (2013). The function g and η satisfy all the assumptions. By lemma 1, as long η+(¯ η −η)

as 0 ≤ Y ≤ a(¯η−η) , the optimal adaptation is at the boundary D = 0, and for Y > solution is interior and characterized by ∗

"

a(¯ η − η)e−aD D∗ + η + (¯ η − η)e−aD∗

η+(¯ η −η) a(¯ η −η) ,

the

#−1 = Y.

To our knowledge, this equation does not allow any explicit solution, and we cannot obtain any explicit solution for R(Y ). In this example, ˆ R(D) = and

a(¯ η − η)e−aD D, η + (¯ η − η)e−aD

a(¯ η − η)e−aD d ˆ R(D) = [η + (¯ η − η)e−aD − Daη]. dD [η + (¯ η − η)e−aD ]2

6

(5)

optimal ratio; D/B

1.2 1 0.8 0.6 0.4 0.2 0

0

5

10 resource; Y

15

20

Figure 2: Optimal ratio in Example 2

optimal ratio; D/B

15 2 2 2 2

10

= 10!5 = 10!3 = 10!1 = 10

5

0

0

5

10 resource; Y

15

20

Figure 3: Optimal ratio function for different values of η d ˆ d ˆ This function is strictly decreasing, dD R(0) > 0, and limD→∞ dD R(D) < 0. Therefore, we can tell that R is increasing up to some level Y¯ and decreasing beyond that level. Figure 2 depicts the function R(Y ) when η = 10, η¯ = 100, and a = 1. Equation (5) shows that the assumption that η > 0 is necessary for the optimal ratio function to be hump shaped. If η were equal to 0, then d ˆ dD R(D) would be always positive. This observation is intuitive. The assumption that η > 0 means that even an infinite amount of investment in adaptation cannot avoid all damage, as discussed in Br´echet et al. (2013). In other words, even when substantial resources are available, there is nevertheless a considerable role left for the other measure, namely, mitigation. Figure 3 depicts the optimal ratio functions for various values of η, holding the other parameter values the same as above. We can see that the peek moves to the right as η approaches 0.

For other examples, see the Appendix, which shows various patterns of the change in the optimal ratio caused by productivity change: (i) constant, (ii) 0 (which means that the solution is at the boundary D = 0) for low productivity and increasing for high productivity, (iii) infinity (meaning that the solution is at the boundary B = 0) for low productivity and decreasing for high productivity, (iv) increasing for low productivity and decreasing for high productivity, (v) decreasing for low productivity and increasing for high productivity, (vi) monotonically increasing, and (vii) monotonically decreasing. 7

What does the result mean? The most important implication is that the assumptions, that returns are increasing and have diminishing marginal returns and that an increase in mitigation investment decreases the necessity of adaptation investment, and vice versa, do not say much about how the optimal investment ratio is related to the available resources. In other words, the relationship depends on the details of the shapes of the return functions, for which there is little empirical evidence. This discussion is applicable to more complicated models. If a model has similar multiplicative forms, an equation similar to Equation (3) characterizes the optimal investment ratio, as we show in the next section.

4

Relating the result to more complicated models

The model in this paper is extremely simple, but the insights are useful to understand more complicated models. To demonstrate this point, first, we show the optimal ratio is characterized by a similar equation to that in the model of Br´echet et al. (2013). Then, we provide examples in which various patterns that are similar to the examples in Section A arise. We also show that the same equation characterizes the optimal ratio in Zemel (2015), noting how the multiplicative form arises endogenously.

4.1 4.1.1

Br´ echet et al. (2013) Model

Time is continuous and indexed by t. The horizon is infinite. There is a consumption good that can be converted into production capital or adaptation capital. At each moment t, the planner observes production capital K(t), adaptation capital D(t), and the pollution stock P (t). Production capital K(t) produces AK(t)α units of the consumption good as output, where A > 0 is a productivity parameter. In this model, because production is endogenous, the parameter A is the exogenous parameter that determines resource availability. We interpret A as the level of development. The planner must determine how to use the output. It can be used for production capital investment IK (t), adaptation capital investment ID (t), mitigation effort B(t) ≥ 0, or consumption C(t) ≥ 0: AK(t)α = IK (t) + ID (t) + B(t) + C(t).

(6)

The evolution of production and adaptation capital is standard: ˙ K(t) = IK (t) − δK K(t), ˙ D(t) = ID (t) − δD D(t),

(7) (8)

where K(0) and D(0) are given and δK ≥ 0 and δD ≥ 0 are depreciation parameters. The law of motion of the pollution stock is given by P˙ (t) = AK(t)α g(B) − δP P (t),

(9)

where P (0) is given, δP ≥ 0 is a depreciation parameter, and g : [0, ∞) → [0, ∞) is a decreasing function. Here, g(B) determines the rate of pollution emission from production activity. The more mitigation effort B is made, the less pollution is emitted. The planner evaluates allocations by ˆ ∞ e−ρt U (C(t), P (t), D(t))dt (10) 0

8

where ρ > 0 is discount factor, U is given by U (C, P, D) = log C − η(D)v(P ), where η : [0, ∞) → [0, ∞) and v : [0, ∞) → [0, ∞) is a strictly decreasing function. An increase in the pollution stock P reduces utility, while an increase in adaptation capital D attenuates the utility reduction caused by pollution. The planner’s problem is to maximize (10) subject to (6), (7), (8), and (9). 4.1.2

Solution

The discounted-value Hamiltonian for the planner’s problem is ˆ = [log C − η(D)v(P )] + λ1 [AK α − IK − ID − B − C] H +λ2 [IK − δK K] + λ3 [ID − δD D] +λ4 [AK α g(B) − δP P ], where λi , i = 1, 2, 3, 4, are the costate variables. The first-order conditions are C −1 − λ1 = 0, −λ1 + λ4 AK α g 0 (B) = 0, −λ1 + λ2 = 0, −λ1 + λ3 = 0, λ1 αAK

α−1

− λ2 δK + λ4 αAK α−1 g(B) = ρλ2 − λ˙ 2 , −η 0 (D)v(P ) − λ3 δD = ρλ3 − λ˙ 3 , 0

−η(D)v (P ) − λ4 δP

= ρλ4 − λ˙ 4 .

(11) (12)

Eliminating λs, IK , and ID yields AK α = δK K + K˙ + δD D + D˙ + B + C, C˙ g(B) = ρ+ , αAK α−1 − δK + αK −1 0 g (B) C α P˙ + δP P = AK g(B), C˙ −Cη 0 (D)v(P ) − δD = ρ + , C " # 00 (B) ˙ ˙ δ 1 C K g P = ρ+ +α + 0 B˙ . −Cη(D)v 0 (P ) − AK α g 0 (B) AK α g 0 (B) C K g (B) As in Br´echet et al. (2013), we restrict our attention to steady states. Setting all variables to be constant over time, we have AK α = δK K + δD D + B + C, g(B) αAK α−1 − δK + αK −1 0 = ρ, g (B) δP P = AK α g(B), −Cη 0 (D)v(P ) − δD = ρ, ρ δP = . −Cη(D)v 0 (P ) − α 0 α AK g (B) AK g 0 (B) 9

The three equations at the bottom of the system yield − where ε =

ρ+δD ρ+δP δP .

η 0 (D) g 0 (B) v 0 (P ) = −ε , η(D) g(B) v(P )

(13)

This equation is very similar to Equation (3). The difference is that the 0

(P ) equation above has two additional terms, the coefficient ε and the ratio vv(P ) . The former appears because mitigation B is modeled as a flow, while adaptation D is modeled as a stock, and the latter appears because mitigation B is modeled to affect utility only indirectly, through reducing the pollution stock P .

4.1.3

Examples

We compute the optimal ratio numerically for three chosen examples, which have counterparts in examples in Section A. In all of the examples, the function v is fixed as v(P ) =

P2 , 2

and the economic parameters are fixed as follows: α = 0.66, ρ = 0.01, δP = 0.0002, γ = 0.16. Example 3 (Constant ratio). The return functions are given by g(B) = B −θ , η(D) = D−γ , where θ > 0 and γ > 0 are parameters. Figure 4 depicts the ratio for θ = 1 and γ = 1.

optimal ratio; D/B

2 1.5 1 0.5 0

0

5

10 productivity; A

15

20

Figure 4: Optimal ratio in Example 3

Example 4 (Ratio increasing for low level and decreasing for high level). The return functions are given by g(B) = B −1 , η(D) = η + (¯ η − η)e−aD , 10

where η > 0, η¯ > 0, and a > 0 are parameters that satisfy η < η¯. This is the case analyzed in Br´echet et al. (2013). Figure 5 depicts the ratio that is numerically calculated for η = 0.0025, η¯ = 0.003, and a = 0.005.

optimal ratio; D/B

200 150 100 50 0

0

5

10 productivity; A

15

20

Figure 5: Optimal ratio in Example 4

Example 5 (Ratio decreasing for low level and increasing for high level). The return functions are given by g(B) = g + (¯ g − g)e−bB , η(D) = D−1 , here g > 0, g¯ > 0, and b > 0 are parameters that satisfy g < g¯. Figure 6 depicts the ratio that is numerically calculated for g = 0.00005, g¯ = 0.1, and b = 0.001.

optimal ratio; D/B

35 30 25 20 15 10 5 0

0

5

10 productivity; A

15

20

Figure 6: Optimal ratio in Example 5

4.2

Zemel (2015)

In this model, there is no parameter that determines the amount of available resources and mitigation is modeled more implicitly than in the models that we have seen above. Although these 11

features make it difficult to compare the model of Zemel (2015) with those above, the optimal mitigation and adaptation are linked by a similar equation to that in the simple model and that of Br´echet et al. (2013). Furthermore, we will see how the multiplicative form arises endogenously in this model. 4.2.1

Model

Time is continuous and indexed by t. The horizon is infinite. At each moment t, the planner ¯ A unit determines the production activity level Y (t) ≥ 0 and adaptation investment I(t) ∈ [0, I]. of production activity yields a unit of consumption good and a unit of emissions, which contributes to increasing the pollution stock: P˙ (t) = Y (t) − δP P (t), (14) where δP > 0 denotes the natural pollution decay rate. The adaptation investment increases adaptation capital D(t), which evolves as ˙ D(t) = I(t) − δD D(t),

(15)

where δD > 0 denotes the depreciation rate, at a unit of utility cost. For a pair (Y (t), I(t)), the planner obtains a flow utility u(Y (t)) − I(t). The pollution stock determines the risk of the occurrence of a disastrous event, which yields damage η(D(t)) in terms of utility, where η is a function that satisfies η > 0, η 0 < 0, and η 00 < 0. The survival probability of the disastrous event S(t) evolves according to ˙ S(t) = −v(P (t))S(t), (16) where v is a function that satisfies v(0) = 0, v 0 > 0, and v 00 ≥ 0. Let T denote the random time at which the disastrous event occurs. The optimization problem can be described through the principle of optimality: ˆ T  V (P (s), D(s)) = max Es e−ρt [u(Y (t)) − I(t)]dt + e−ρT (V (P (T ), D(T ))) , (17) s

¯ where Es is subject to (14), (15), (16), P (0) = P0 , S(0) = 1, D(0) = 0, Y (t) ≥ 0, and I(t) ∈ [0, I], a expectation operator conditional on information up to s, and ρ > 0 is a discount factor. 4.2.2

Solution

Assuming that the solution is interior,1 the Hamilton-Jacobi (HJ) equation can be written as ρV = max {u(Y ) − I + VP [Y − δP P ] + VD [I − δD D] + v[−η]} . Y,I

(18)

The last term is important. It appears here as an instantaneous change in value caused by the disastrous event, which occurs at rate v(P ) and yields damage η(D). The multiplicative form arises due to the expected value calculation, to which multiplication is intrinsic. Note also that if we set the utility as u(Y ) − I − η(D)v(P ) 1

Boundary solutions are also characterized similarly. See Proposition 1 in Zemel (2015).

12

in the first place, we would obtain exactly the same HJ equation. Necessary conditions for (interior) optimality are u0 (Y ) + VP

= 0,

−1 + VD = 0. Defining λ3 = VD and λ4 = VP , we have −η 0 (D)v(P ) − λ3 δD = ρλ3 − λ˙ 3 , −η(D)v 0 (P ) − λ4 δP = ρλ4 − λ˙ 4 . Note that these equations are the same as those in (11) and (12). Again, restricting our attention to steady states, we have 1 v 0 (P ) η 0 (D) = −ˆ ε 0 , − η(D) u (Y ) v(P ) which is similar to Equations (3) and (13). The coefficient εˆ appears because the two measures are 1 modeled as stocks, and the term u0 (Y ) appears because mitigation and adaptation have different impacts on utility: the shadow value of the former is u0 (Y ), while that of the latter is 1.

5

Conclusion

We have explored the optimal mitigation-adaptation ratio in a simple model and proposed an equation that links the optimal investments in the two measures. Using the equation, we have demonstrated that the relationship between the optimal ratio and amount of resources available can exhibit various patterns, according to the functional forms assumed for the returns to mitigation and adaptation. In other words, the basic assumptions in a mitigation-adaptation investment problem that returns are diminishing and that an increase in investment in one measure reduces the necessity of the other leave a high degree of freedom in the relationship. We have demonstrated that this insight is useful for understanding more complicated models, such as Br´echet et al. (2013) and Zemel (2015), by showing that similar equations characterize the optimal ratio and displaying a variety of patterns of the relationship in our examples. As noted in the model section, the multiplicative form of the utility function is an important assumption of our analysis. One possible interpretation is that, in the context in which it is reasonable to assume that mitigation determines the probability of the occurrence of a harmful event and adaptation determines the damage of it, the multiplicative form is simply a reduced form of the expected value calculation. In other contexts, it should be interpreted as simply a crude means of capturing the substitutability between mitigation and adaptation. We leave analyses with general dependance to future work. Our result implies that it is necessary to know the forms of the mitigation and adaptation return functions. This is a question that should be answered empirically. As discussed in Agrawala et al. (2011), we have only limited results to answer the question.

References Shardul Agrawala, Francesco Bosello, Carlo Carraro, Enrica De Cian, Elisa Lanzi, et al. Adapting to climate change: costs, benefits, and modelling approaches. International Review of Environmental and Resource Economics, 5(3):245–284, 2011. 13

Francesco Bosello, Carlo Carraro, and Enrica De Cian. Climate policy and the optimal balance between mitigation, adaptation and unavoided damage. Climate Change Economics, 1(02):71–92, 2010. Thierry Br´echet, Natali Hritonenko, and Yuri Yatsenko. Adaptation and mitigation in long-term climate policy. Environmental and Resource Economics, 55(2):217–243, 2013. Kelly C de Bruin, Rob B Dellink, and Richard SJ Tol. Ad-dice: an implementation of adaptation in the dice model. Climatic Change, 95(1-2):63–81, 2009. Kelly Chloe de Bruin. Distinguishing between proactive (stock) and reactive (flow) adaptation. Available at SSRN 1854285, 2011. Alan Ingham, Jie Ma, and Alistair M Ulph. Can adaptation and mitigation be complements? Climatic change, 120(1-2):39–53, 2013. IPCC. Climate Change 2007:Fourth Assesment report of the IPCC. Cambridge University Press, 2007. Robert S Pindyck. Climate change policy: What do the models tell us? Journal of Economic Literature, 51(3):860–872, 2013. Amos Zemel. Adaptation, mitigation and risk: An analytic approach. Journal of Economic Dynamics and Control, 51:133–147, 2015.

A

Examples

Example 6 (Constant ratio). The return functions are given by g(B) = B −θ , η(D) = D−γ , where θ > 0 and γ > 0 are parameters. These functions are strictly decreasing, continuously differentiable, and strictly log-convex, and thus satisfy the assumptions. The functions Fg and Fη are expressed as Fg (B) = θB −1 , Fη (D) = γD−1 . By solving D∗ + θ[γD∗−1 ]−1 = Y, we obtain explicit solutions D∗ (Y ) =

γ θ+γ Y

R(Y ) ≡

(and B ∗ (Y ) =

θ θ+γ Y

). The optimal ratio is

D∗ (Y ) γ = . −1 ∗ θ Fg (Fη (D (Y )))

Figure 7 depicts the function R(Y ) when θ = γ = 1. 14

optimal ratio; D/B

2 1.5 1 0.5 0

0

5

10 resource; Y

15

20

Figure 7: Optimal ratio in Example 6 Example 7 (Increasing ratio with boundary solution). The return functions are given by g(B) = B −θ , η(D) = e−aD , where θ > 0 and a > 0 are parameters. The function g satisfies all of the assumptions, but η satisfies only continuous differentiability and strict decreasingness. It is not strictly log-convex but only weakly log-convex. The KKT condition still characterizes the optimal solution and lemma 1 holds. Because g 0 (B) g(B) η 0 (D) − η(D) −

= θB −1 , = a,

as long as 0 ≤ Y ≤ θ/a, the inequality −

η 0 (0) g 0 (Y ) ≤− η(0) g(Y )

holds and the optimal adaptation is at the boundary D = 0. For Y > θ/a, the solution is interior, and by solving D∗ + θa−1 = Y we obtain explicit solutions D∗ = Y − θa−1 and B ∗ = θa−1 . Therefore, the ratio is given by ( 0 for Y ∈ (0, θ/a], R(Y ) = Y −θ/a for Y ∈ (θ/a, ∞). θ/a Figure 8 depicts the function R(Y ) when θ = 5 and a = 1. Example 8 (Decreasing ratio with boundary solution). This is simply a symmetric version of Example 7.The return functions are given by g(B) = e−bB , η(D) = D−γ , 15

optimal ratio; D/B

3 2.5 2 1.5 1 0.5 0

0

5

10 resource; Y

15

20

Figure 8: Optimal ratio in Example 7

optimal ratio; D/B

12 10 8 6 4 2 0

0

5

10 resource; Y

15

20

Figure 9: Optimal ratio in Example 8 where b > 0 and γ > 0 are parameters. The function η satisfies all of the assumptions, but g satisfies only continuous differentiability and strict decreasingness. It is not strictly log-convex but only weakly log-convex. As long as 0 ≤ Y ≤ γ/b, the inequality γY −1 ≡ −

η 0 (Y ) g 0 (0) ≥− ≡b η(Y ) g(0)

holds and the optimal mitigation is at the boundary B = 0. For Y > γ/b, the solution is interior, and by solving γ/b + B ∗ = Y we obtain explicit solutions D∗ = γ/b and B ∗ = Y − γ/b. Therefore, the ratio is given by ( ∞ for Y ∈ (0, γ/b], R(Y ) = γ/b for Y ∈ (γ/b, ∞). Y −γ/b Figure 9 depicts the function R(Y ) when γ = 5 and b = 1.

16

optimal ratio; D/B

5 4 3 2 1 0

0

5

10 resource; Y

15

20

Figure 10: Optimal ratio in Example 9 Example 9 (Ratio decreasing for low level and increasing for high level). This is a symmetric version of Example 2. The return functions are given by g(B) = g + (¯ g − g)e−bB , η(D) = D−1 , here g > 0, g¯ > 0, and b > 0 are parameters that satisfy g < g¯. The functions g and η satisfy all of the assumptions. As long as 0 ≤ Y ≤ b(¯gg¯−g) , the inequality −

η 0 (Y ) g 0 (0) ≥− η(Y ) g(0)

holds and the optimal adaptation is at the boundary B = 0. For Y > and characterized by   (¯ g − g) b − D∗−1 1 = Y. D∗ + log b g D∗−1

g¯ b(¯ g −g) ,

the solution is interior

To the best of our knowledge, this equation does not allow any explicit solution, and we investigate the shape of R(Y ) by investigating the shape of   (¯ g − g) b − D∗−1 D 1 ˆ R(D) ≡ −1 = D log b g D∗−1 Fg (Fη (D)) and

  g¯ − g b − D−1 d ˆ 1 1 R(D) = log + . −1 dD b g D b − D−1

d ˆ d ˆ This function is strictly increasing for D = 1/b, limD→1/b dD R(D) = −∞, and limD→∞ dD R(D) = ¯ ∞. Therefore, we can tell that R is increasing up to some level Y , and decreasing beyond that level. Figure 10 depicts the function R(Y ) when g = 10, g¯ = 100, and b = 1.

Example 10 (Ratio monotone without boundary solution). The return functions are given by g(B) = exp(−bB π ), η(D) = exp(−aDσ ), 17

where π ∈ (0, 1) and σ ∈ (0, 1) are parameters. The functions g and η satisfy all of the assumptions. Because η 0 (0) g 0 (0) − =− = ∞, η(0) g(0) the solution is always interior. The solution is characterized by D∗ +

hσ π

D∗σ−1

i

1 π−1

= Y.

To the best of our knowledge, this equation does not allow any explicit solution, and we investigate the shape of R(Y ) by investigating the shape of ˆ R(D) ≡ Because

hσ i− 1 hσ i 1 1−σ D π−1 1−π σ−1 = D D = D1− 1−π . −1 π π Fg (Fη (D))  hσ i 1  1−σ d ˆ 1−σ 1−π R(D) = 1− D− 1−π , dD π 1−π

the ratio is globally   increasing constant   decreasing

if σ > π, if σ = π, if σ < π.

Figure 11 depicts the ratio R(Y ) when a = b = 1, π = 0.5, and σ = 0.25, and Figure 12 depicts the ratio R(Y ) when a = b = 1, π = 0.25, and σ = 0.5.

optimal ratio; D/B

3 2.5 2 1.5 1 0.5 0

0

5

10 resource; Y

15

20

Figure 11: Optimal ratio in Example 10, when a = b = 1, π = 0.5, and σ = 0.25

18

optimal ratio; D/B

8 6 4 2 0

0

5

10 resource; Y

15

20

Figure 12: Optimal ratio in Example 10, when a = b = 1, π = 0.25, and σ = 0.5

19