The economic insurance value of ecosystem resilience

Ecological Economics 101 (2014) 21–32

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Ecological Economics journal homepage: www.elsevier.com/locate/ecolecon

Analysis

The economic insurance value of ecosystem resilience Stefan Baumgärtner ⁎, Sebastian Strunz Department of Sustainability Science, Leuphana University of Lüneburg, Germany Department of Economics, Leuphana University of Lüneburg, Germany

a r t i c l e

i n f o

Article history: Received 3 July 2009 Received in revised form 21 December 2013 Accepted 16 February 2014 Available online 7 March 2014 JEL classification: Q57 Q56 D81 G22

a b s t r a c t Ecosystem resilience, i.e. an ecosystem's ability to maintain its basic functions and controls under disturbances, is often interpreted as insurance: by decreasing the probability of future drops in the provision of ecosystem services, resilience insures risk-averse ecosystem users against potential welfare losses. Using a general and stringent definition of “insurance” and a simple ecological–economic model, we derive the (marginal) economic insurance value of ecosystem resilience and study how it depends on ecosystem properties, economic context, and the ecosystem user's risk preferences. We show that (i) the insurance value of resilience is negative (positive) for low (high) levels of resilience, (ii) it increases with the level of resilience, and (iii) it is one additive component of the (overall always positive) economic value of resilience. © 2014 Elsevier B.V. All rights reserved.

Keywords: Ecosystem Economic value Insurance Resilience Risk Risk preferences

1. Introduction Ecosystems that are used and managed for the ecosystem services they provide may exhibit multiple stability domains (“basins of attraction”) that differ in fundamental system structure and behavior. As a result of exogenous natural disturbances or human management, a system may flip from one stability domain into another one with different basic functions and controls (Holling, 1973; Levin et al., 1998; Scheffer et al., 2001). As a consequence, also the level, composition and quality of ecosystem services may abruptly and irreversibly change. Examples encompass a diverse set of ecosystem types that are highly relevant for economic use, such as boreal forests, semi-arid rangelands, wetlands, shallow lakes, coral reefs, and high-seas fisheries (Gunderson and Pritchard, 2002). The term “resilience” has been used to denote an ecosystem's ability to maintain its basic functions and controls under disturbances (Carpenter et al., 2001; Holling, 1973). The economic relevance of

⁎ Corresponding author at: Sustainability Economics Group, Leuphana University of Lüneburg, P.O. Box 2440, D-21314 Lüneburg, Germany. Tel.: + 49 4131 677 2600; fax: +49 4131 677 1381. E-mail address: [email protected] (S. Baumgärtner). URL: http://www.leuphana.de (S. Baumgärtner).

http://dx.doi.org/10.1016/j.ecolecon.2014.02.012 0921-8009/© 2014 Elsevier B.V. All rights reserved.

ecosystem resilience is obvious, as a system flip may entail huge welfare losses. 1 For example, a combination of drought, fire and ill-adapted livestock grazing management in sub-Saharan Africa, central Asia and Australia have lead to severe degradation and desertification of semi-arid rangelands, which provide subsistence livelihood for more than 1 billion people worldwide. Once degraded, these grassland ecosystems cannot be used as pasture anymore (Perrings and Stern, 2000; Perrings and Walker, 1995). In Africa alone, almost 75% of semi-arid regions are threatened by degradation and desertification ([UNO] United Nations Organisation, 2002). Worldwide, the income loss associated with desertification of agricultural land is estimated to some 42 billion US dollars per year ([UNCCD] Secretariat of the United Nations Convention to Combat Desertification, 2005). An ecosystem's resilience in a given stability domain can be measured by the probability that exogenous perturbations make the system flip into another stability domain. Therefore, enhancing the resilience of a particular (desired) domain reduces the likelihood of a flip into another

1 Accordingly, some have included a reference to the provision of desired ecosystem services right into the definition of ecosystem resilience, e.g. as the capacity of an ecosystem “to maintain desired ecosystem services in the face of a fluctuating environment and human use” (Brand and Jax, 2007: 3, referring to Folke et al., 2002).

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(less desired) domain. It is for this reason that ecosystem resilience has been referred to as “insurance”, e.g. in the following manner: “Resilience can be regarded as an insurance against flips of the system into different basins of stability.” [Mäler (2008: 17)]

“[R]esilience […] provides us with a kind of insurance against reaching a non-desired state.” [Mäler and Li (2010: 708) and Mäler et al. (2009: 48)]

“The link between biodiversity, ecosystem resilience and insurance should now be transparent. […] It follows that the value of biodiversity conservation lies in the value of that protection: the insurance it offers against catastrophic change.” [Perrings (1995: 72)]

“The resilience of the ecological system provides ‘insurance’ within which managers can affordably fail and learn while applying policies and practices.” [Holling et al. (2002: 415)] So far in the resilience literature, the term “insurance” is employed in a rather metaphoric manner — as a metaphor for “keeping an ecosystem in a desirable domain”. It is used to convey the message that resilience is a desirable property of some ecosystem since it helps to prevent potential catastrophic and irreversible reductions in ecosystem service flows. While ecosystem resilience obviously and undoubtedly includes an insurance aspect, no explicit attempt has been made so far to use a clearly defined concept of “insurance” from the established literature on insurance and financial economics. As a result, it remains unclear what exactly constitutes the insurance value – understood in a rigorous economic sense – of ecosystem resilience, how it depends on ecosystem properties, economic context, and the ecosystem user's risk preferences, and how it relates to the all-encompassing economic value of ecosystem resilience. In order to conceptually determine and to empirically capture the economic value of ecosystem resilience, Mäler and his co-authors use the shadow price of resilience as a measure of its economic value (Mäler, 2008; Mäler and Li, 2010; Mäler et al., 2007; Walker et al., 2010). They calculate the present discounted value of future improvements in expected welfare from ecosystem services, where these future improvements accrue from reduced risks of a system flip due to a unit increase in the initial level of resilience. While this approach establishes an adequate measure of the marginal economic value of resilience in an explicitly dynamic setting, it has two shortcomings in view of understanding the insurance function of resilience. First, in the explicit modeling it focuses on the dynamics of ecological–economic systems and pays less attention to the insurance aspect. In particular, none of the papers cited above analyzes how the economic value of resilience depends on the type or degree of ecosystem users' risk aversion. Second, the shadow price of resilience is the economic value of resilience which includes, but is larger than its insurance value. As resilience also has economic value beyond its insurance value, it remains to be clarified what fraction of the economic value of resilience is due to its insurance function, and how this insurance function relates to the other value-constituting functions to make the overall economic value of resilience. In this paper, we aim to close these gaps and to provide some conceptual clarification. Any idea of “insurance” fundamentally refers to a combination of three elements: (i) the objective characteristics of risk in terms of different possible states of nature, (ii) the decision maker's subjective risk preferences over these states, and (iii) a mechanism that allows mitigation of (i) in view of (ii). We believe that the ongoing

discussion of resilience as an insurance could be clarified and fruitfully advanced if reference to these three elements was made explicitly and rigorously, and we propose an analytical framework for that purpose. For clarity, we perform an atemporal analysis that focuses on the riskand-insurance aspect, but neglects explicit dynamics of the ecological– economic system. Our analysis therefore complements the strand of literature cited above that focuses explicitly on dynamics but is less explicit on the risk-and-insurance aspect. We adopt a clear and generally accepted definition of “insurance” from the risk and finance literature, according to which insurance is an action or institution that mitigates the influence of uncertainty on a person's well-being (McCall, 1987). Based on this definition, we conceptualize resilience's (marginal) economic insurance value2 as the value of one very specific function of resilience: to reduce an ecosystem user's income risk from using ecosystem services under uncertainty. We also analyze how exactly the insurance value of ecosystem resilience depends on ecosystem properties, economic context, and on the ecosystem user's risk preferences. Our analysis yields several interesting results. First, the insurance value of resilience is negative for low levels of resilience and positive for high levels of resilience. That is, ecosystem resilience actually functions as an insurance only at sufficiently high levels of resilience; it does not function as an insurance at low levels of resilience. Second, the (marginal) insurance value of resilience increases with the level of resilience — for some ecosystem types even monotonically. This is in contrast to normal economic goods, the (marginal) value of which decreases with their quantity. Third, the insurance value of resilience is one additive component of its economic value. That is, the economic value of resilience is larger than just its insurance value. While the latter may be negative, the economic value of resilience is always positive. The paper is organized as follows. In Section 2, we present a stylized model of an ecological–economic system that describes how different degrees of ecosystem resilience are related to different system outcomes, and how this contributes to an ecosystem user's well-being under uncertainty. In Section 3, we clarify what exactly we mean by “insurance value”. In Section 4, we present our results about the insurance value and the economic value of ecosystem resilience, with all proofs and formal derivations contained in Appendix A. In Section 5, we discuss these findings and draw conclusions. 2. Model Consider an ecosystem that potentially exhibits two different stability domains with respective levels of ecosystem services-production. One domain is characterized by a high level of ecosystem service provision and corresponding net income yH N 0, the other domain is characterized by a low level of ecosystem service provision and corresponding net income yL N 0, with yL b yH so that Δy :¼ yH −yL N0

ð1Þ

is the potential income loss when the system flips from the highproduction into the low-production stability domain. Initially, the ecosystem is in the high-production stability domain. In this domain, an exogenous stochastic disturbance threatens to trigger a flip into the low-production stability domain. Such a flip may occur with probability p with 0 ≤ p ≤ 1. Conversely, the ecosystem stays in the high-production domain with probability 1 − p. In line with Holling's (1973) notion of resilience as the maximum amount of disturbance a system can absorb in a given stability domain while still remaining in that stability domain, we define and measure 2 As it should be clear by now that we are concerned with the economic insurance value of resilience here, as opposed to the so-called “ecological insurance hypothesis” of biodiversity (e.g. Gonzalez et al., 2009; Yachi and Loreau, 1999), we drop the adjective “economic” from here on.

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23

resilience as a continuous variable R ∈ [0,1] that determines the probability of the system flipping from the high-production into the lowproduction stability domain through a continuously differentiable and strictly monotonic function: ′

′

p ¼ pðRÞ with p ðRÞ≤0 for all R and p ðRÞb0 for all R∈ð0; 1Þ ð2Þ

and

pð0Þ ¼ 1; pð1Þ ¼ 0:

ð3Þ

In words, the higher the ecosystem's resilience in the highproduction domain, the lower the probability that it flips into the lowproduction domain due to exogenous disturbance; without resilience it flips for sure; and with maximum resilience it will certainly not flip.3 For future reference, we define R through pðR Þ ¼ 1=2 as the level of resilience at which the probability of a system flip exactly equals the probability of the system not flipping. This is the level of resilience at which the future state of the system is maximally uncertain. In order to give more ecological structure to our ecosystem model Eqs. (2)–(3), in some parts of our analysis we assume the following more specific model about the relationship between the level of resilience R and the flip probability p (cf. Fig. 1): pðRÞ ¼ 1−R

1−σ

with −∞bσb þ 1:

ð4Þ

This model has the fundamental resilience-defining properties Eqs. (2) and (3). In addition, it has the analytically handsome property that p′(⋅) is a constant-elasticity function of R, where the parameter σ is the (constant) elasticity of p′(⋅),4 i.e. σ specifies by how much (in percent) the flip probability's slope increases when the level of resilience increases by 1%. For short, we will refer to σ as “the ecosystem's elasticity”. As σ may be positive or negative, one has5 8 9
if and only if

8 9
Lacking ecological evidence or a plausible guess on the value of σ, we will study the full range of theoretically possible values of σ. Notwithstanding this generality, the case of σ = 0 has an epistemically outstanding importance. For, one may argue that one can meaningfully define and measure the variable “resilience” only through, and not independently of, the observable variable “flip probability”.6 Such an epistemic equivalence between the variable R and the observable p is exactly what is expressed by σ = 0. In this case, Eq. (4) reduces to a linear negative relationship, p(R) = 1 − R, so that resilience is measured directly in units of reduced flip-probability. As this case has an epistemically outstanding importance, we will treat the case of σ = 0 as the preeminent case, and discuss the cases of σ b 0 and σ N 0 against it. With the ecosystem model Eqs. (2) and (3) – or, more specifically, model (4) – income is a binary random variable and the ecosystem user faces an income lottery {yL,yH; p(R),(1 − p(R))}. That is, given 3 While Assumption (2) is generic and crucially drives our results, Assumption (3) only serves to obtain simple expressions of results. It may be weakened (p(0) = pmax and p(1) = pmin with 1 ≥ pmax N pmin ≥ 0) to increase realism, but at the price of more cumbersome derivations and expressions. 4 Note that Eq. (4) implies −p″(R)R/p′(R) = σ. 5 For σ = 0, p″(R) = 0 holds also for R = 0 and R = 1. Yet, for σ b 0, one has p″(0) = 0, and for σ → 1, one has p″(1) → 0. 6 If the system's state space was one-dimensional, one could indeed meaningfully define and measure the system's resilience (sensu Holling, 1973) independently of the system's flip probability, namely as the “distance” in state space – measured in units of the single state variable – between the current system state and the threshold between stability domains (e.g. Walker et al., 2010). However, if the system is characterized by more than one state variable, the “distance” in state space is not uniquely defined but requires some metric which is not naturally given. Then, the system's resilience in a given stability domain cannot be measured through some distance in state space, but only through the observable consequence in terms of flip probability.

Fig. 1. Flip probability p as a function of resilience R for different signs of the ecosystem elasticity σ, according to the constant-elasticity model, p(R) = 1 − R1 − σ with −∞ b σ b +1 (Eq. (4)).

that the system is initially in the high-production stability domain and is characterized by a level R of resilience, the system will provide net income yL with probability p(R) and net income yH with probability 1 − p(R). Obviously, with changing level of resilience R the statistical distribution of income will also change. As in our simple analytical framework only the level of resilience R may vary, R uniquely characterizes the income lottery. One may thus speak of “the income lottery R”. We assume that the ecosystem user only cares about (uncertain) income, and not directly about the underlying state of the system in terms of resilience. The ecosystem user's preferences over income lotteries are represented by a von Neumann–Morgenstern expected utility function U ¼ ER ½uðyÞ

with

′

″

u ðyÞN0 and u ðyÞb0 for all y;

ð5Þ

where ER is the expectancy operator based on the probabilities of lottery R,y ∈ {yL,yH} is actually realized net income, and u(y) is a continuous and differentiable Bernoulli utility function which is assumed to be increasing and strictly concave, i.e. the ecosystem user is non-satiated and risk averse.7 In order to study in the most simple way how the insurance value of resilience depends on the ecosystem user's degree of risk aversion, we assume that the ecosystem user is characterized by constant absolute risk aversion in the sense of Arrow (1971) and Pratt (1964), i.e. −u″(y)/u′(y) ≡ const., so that the Bernoulli utility u(y) function is −ρy

uðyÞ ¼ −e

with

ρN0;

ð6Þ

where the parameter ρ measures the ecosystem user's risk aversion. 3. Conceptual Clarification: Insurance Value Before we derive results about the insurance value of ecosystem resilience in the next section, in this section we provide exact definitions of the terms “insurance”, “insurance value” and “economic value”. Adopting a very general and widely accepted definition, insurance may be defined in the following way (cf. McCall, 1987).

7 While risk aversion is a natural and standard assumption for households (Besley, 1995; Dasgupta, 1993: Chapter 8), it appears as an induced property in the behavior of companies which are fundamentally risk neutral but act as if they were risk averse when facing e.g. external financing constraints or bankruptcy costs (Caillaud et al., 2000; Mayers and Smith, 1990).

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Definition 1. Insurance is an action or institution that mitigates the influence of uncertainty on a person's well-being or on a firm's profitability. In the concrete setting described in the previous section, the term “insurance” takes on a more concrete meaning. As a person's (here: the ecosystem user's) well-being is determined by a preference relation over income lotteries, insurance is about the mitigation of income uncertainty, and the person's risk preferences specify what changes in the income lottery actually constitute a “mitigation”. Thereby, uncertainty exists due the existence of many potential future states of nature (here: high and low ecosystem-service production), in each of which the state-specific income is known (yH and yL) and the probability of which is also known (1 − p(R) and p(R)). That is, uncertainty comes in the form of risk in the sense of Knight (1921). In this more concrete understanding of the term, insurance may come in many forms. One example is the classic insurance contract that an insuree signs with an insurance company under private law, and which specifies that the insuree pays an insurance premium to the insurance company in all states of nature and in exchange obtains from the insurance company an indemnification payment if and only if one particular unfavorable state of nature should occur. Another example is so-called “self-protection” (Ehrlich and Becker, 1972), which means that a person undertakes some action that reduces the probability by which an unfavorable – in terms of net income – state of nature occurs. In this terminology, an increase in the ecosystem's resilience by the ecosystem user may be interpreted as insurance because it is an action that may provide self-protection in terms of net income obtained from the ecosystem.8 In order to precisely define and measure the insurance value of some act of self-protection (here: an increase in the ecosystem's resilience), we follow Baumgärtner (2007: 103–104). One standard method of how to value, in units of income, the riskiness of an income lottery to a decision maker is to calculate the risk premium Π of the lottery, which is defined by (e.g. Kreps, 1990 and Varian, 1992: 181)9 uðER ½y−Π ðRÞÞ ¼ ER ½uðyÞ:

ð7Þ

In words, the risk premium Π(R) is the amount of money that leaves a decision maker equally well-off, in terms of utility, between the two situations of (i) receiving for sure the expected pay-off from the income lottery R, ER[y], minus the risk premium Π(R), and (ii) facing the risky income lottery R with random pay-off y.10 In the model employed here, the risk premium as defined by Eq. (7) uniquely exists because, by assumption, income y is taken from a compact set (i.e. an interval of the positive real numbers) and u is continuous and strictly increasing (Kreps, 1990: 84). The defining Eq. (7) implies that if the decision maker is risk averse, i.e. the Bernoulli utility function u is strictly concave, the risk premium Π(R) is strictly positive. And obviously, the risk premium Π(R) depends on the level of resilience R, because R determines the probabilities underlying the expectancy operators in the defining Eq. (7). The insurance value of resilience can now be defined based on the risk premium of the income lottery R as follows.

8 Yet another form of insurance is so-called “self-insurance” (Ehrlich and Becker, 1972), which means that a person undertakes some action that reduces the size of the net income loss in case the unfavorable state of nature should occur. As an example in our setting, one may think of the ecosystem user making an investment into adaptation to the unfavorable (that is, low-production) state of nature, thus reducing the potential income loss. 9 By Eq. (7), ER[y] − Π(R) is the certainty equivalent of lottery R, as it yields exactly the same expected utility as the risky lottery, ER[u(y)]. 10 The risk premium is, thus, the maximum amount of money that a decision maker would be willing to pay for getting the expected pay-off from the income lottery for sure instead of facing the risky income lottery with random pay-off.

Definition 2. The insurance value V of resilience is given by the change of the risk premium Π(R) of the income lottery R due to a marginal change in the level of resilience R:

IðRÞ :¼ −

dΠ ðRÞ : dR

ð8Þ

Thus, the insurance value of ecosystem resilience is the marginal value, measured in units of income, of its function to reduce the risk premium of the ecosystem user's income risk from using ecosystem services under uncertainty. In general, it depends on the existing level of resilience R. The minus sign in the defining Eq. (8) serves to express a reduction of the risk premium as a positive value. As it is apparent already from Definition 2 (and as it will become more explicit in the following section), the insurance value of ecosystem resilience has, in general, an objective and a subjective dimension. The objective dimension is captured by the ecosystem's sensitivity of the flip probability p(R) to changes in the level of resilience, σ; the subjective dimension is captured by the ecosystem user's degree of risk aversion, ρ. If the flip probability would not vary with the level of resilience (i.e. p′(R) ≡ 0), or if the ecosystem user was risk-neutral (i.e. ρ = 0), the risk premium Π of income lottery R (Eq. (7)) is zero for all R, thus yielding a vanishing insurance value of resilience. The insurance value of resilience is only one part of resilience's economic value, namely the value of its function to reduce the risk premium of the ecosystem user's income risk from using ecosystem services under uncertainty. Beyond its insurance value, resilience also has economic value in its function to increase the ecosystem user's expected income from ecosystem services. Fig. 2 illustrates these two functions of resilience — to increase the ecosystem user's expected income and to potentially reduce the risk premium of the ecosystem user's income lottery. The figure depicts the Bernoulli utility function u(y) as a function of income y,11 and a situation of R ¼ R so that pðR Þ ¼ 0:5 ¼ 1−pðR Þ, that is, the low and the high-income state occur with equal probability. Expected income in this situation is ER[y], expected utility is ER[u(y)], and the risk premium, according to the defining Eq. (7), is Π(R). A strictly positive risk premium only emerges for a strictly concave utility function u(y), that is, for risk-aversion. According to Assumption (2), an increase ΔR in the level of resilience decreases the probability p(R) of low income yL . In Fig. 2, ΔR is such that the probability of low income decreases to p ðR þ ΔRÞ ¼ 0:25 and the probability of high income correspondingly increases to 1−pðR þ ΔRÞ ¼ 0:75 . The increase ΔR in resilience, thus, increases the expected income to ER + ΔR[y], increases expected utility to ER + ΔR[u], and decreases the risk premium of the ecosystem user's income risk lottery to Π(R + ΔR). The latter is the insurance effect of resilience. It only occurs under risk aversion, that is, for a strictly concave utility function u(y); and its extent depends on the curvature of the utility function, that is, the degree of risk aversion. In contrast, the extent of the increase in expected income is independent of the degree of risk-aversion. It would also emerge under riskneutrality, that is, for a linear utility curve. Both functions of resilience – to increase the ecosystem user's expected income and to potentially reduce the risk premium of the ecosystem user's income lottery – together make the economic value of resilience, which is due to the overall increase in expected utility. In order to characterize the insurance value of resilience as a fraction of its economic value we adopt the following general and widely accepted definition of economic value under uncertainty (e.g. Freeman, 2003: Chap. 8).

11 Note that the figure does not exactly show the constant-absolute-risk-aversion utility Function (6), but some monotonic transformation.

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25

risik premium

350

risk premium insurance value economic value expected value

30

300 250

20

200

15

150

10

100 50

5 0 0.0 -5

resilience 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 1.0 -50 -100

-10 Fig. 2. An increase ΔR in the level of resilience decreases the probability p(R) of low income yL. It, thus, increases the expected income E[y] and decreases the risk premium Π(R). Overall, expected utility E[u] increases. (Parameter values: R ¼ R so that pðR Þ ¼ 0:5; ΔR such that pðR þ ΔRÞ ¼ 0:25.)

Definition 3. The economic value V of resilience is given by the maximum ex-ante willingness to pay W per unit for a marginal increase of ΔR in the level of resilience R: W ðΔRÞ ; V ðRÞ :¼ lim ΔR ΔR→0

ð9Þ

Fig. 3. Risk premium (orange curve), insurance value (green curve), expected value (vertical distance between green and blue curves) and economic value (blue curve) as a function of resilience for the case of intermediate ecosystem elasticity 0≤ σ ≤ σ . The dashed e (Parameter values: σ = line marks the maximum-income-risk level of resilience R ¼ R. 0, Δy = 110, ρ = 0.017).

value of ecosystem resilience. The risk premium of the income lottery R is given by  i 1 h ρΔy Π ðRÞ ¼ −pðRÞΔy þ ln 1 þ pðRÞ e −1 ; ρ

ð11Þ

which implies

where W is defined through ER ½uðyÞ ¼ ERþΔR ½uðy−W ðΔRÞÞ:

insurance value , economic value

35

25

ð10Þ

In words, we measure the economic value of a change ΔR in resilience as the ecosystem user's maximum willingness to pay ex ante, i.e. independently of what state of nature actually turns out, for that change.12 The maximum willingness to pay W for the increase ΔR in resilience is the amount of money that leaves an individual indifferent, in terms of expected utility, between the two situations of (i) resting in the original position with resilience R and (ii) paying the amount W and getting into a situation with resilience R + ΔR.13 As value is typically expressed as a per-unit quantity characterizing a marginal change, we divide W by ΔR and let ΔR → 0 to obtain the marginal economic value of resilience. In the simple model studied here, with no other constraints or alternative options for action in place, the economic value of resilience as defined by Definition 3, evaluated at the socially optimal level of resilience, is exactly its shadow price (as measured e.g. by Mäler et al., 2007; Mäler and Li, 2010; Walker et al., 2010). 4. Results Using the concepts defined in Section 3, we can make statements about the model described in Section 2, and, thus, about the insurance

Π ð0Þ ¼ Π ð1Þ ¼ 0 Π ðRÞN0

ð12Þ

and

for all R∈ð0; 1Þ:

ð13Þ

Proof. See Appendix A 1. □ This means that the risk premium of income lottery R is strictly positive at all levels of resilience in between the minimum level of 0 and the maximum level of 1, and is zero only at the extreme levels of 0 and 1. That is, income is risky at all levels of resilience in between 0 and 1; and only at the extreme levels of 0 and 1 does the income risk vanish, as in the case R = 0 the system will flip into the low-productivity domain with income yL for certain, and at R = 1 the system will remain in the high-productivity domain with income yL for certain. As a consequence of Results (12) and (13), the risk premium varies with the level of resilience in a non-monotonic way (Figs. 3 and 4, orange line). This establishes the following properties of the insurance value of resilience. Proposition 1. The insurance value of resilience, I(R), is given by ( ′

IðRÞ ¼ p ðRÞ Δy−

) 1 eρΔy −1  ρΔy  ; ρ 1 þ pðRÞ e −1

ð14Þ

which has the following properties. 12

This ex-ante, i.e. state-independent, willingness to pay is often called the “option price of a risk reduction” (e.g. Freeman, 2003: 213). 13 In the language of welfare measurement, willingness to pay (WTP), as defined through Eq. (10), is the Hicksian compensating variation for a finite change of ΔR in the level of resilience (Freeman, 2003: Chap. 3; Hicks, 1943). Alternatively, one could also use the Hicksian equivalent variation to measure the monetary value of the welfare change associated with a finite change of ΔR in the level of resilience, that is, the minimum ex-ante, i.e. state-independent, amount of monetary compensation to the individual (“willingness to accept”, WTA) that leaves the individual indifferent between the two situations of (i) resting in the original position with resilience R and receiving a monetary payment of WTA and (ii) getting into a situation with resilience R + ΔR. In general, WTP and WTA will differ. However, for continuously differentiable probability and utility functions, Eqs. (2) and (5), and for the marginal changes studied here, i.e. for ΔR → 0, WTP and WTA coincide, so that the value of V(R) does not depend on whether WTP or WTA is used in the defining Eq. (9).

(i) For all R ∈ (0,1)14 8 9
where

e :¼ p−1 R

8 9
ð15Þ



ð16Þ

14 For σ = 0, the statement about the sign of I(R) holds also for R = 0 and R = 1. Yet, for σ b 0, one has I(0) = 0; for σ → 1, one has I(1) → 0.

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risk premium

500

20

400

15

300

10

200 100

5

resilience 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 1.0 -100 -200

-10

600

insurance value economic value

25

500

20

400

15

300

10

200

5

100

0 0.0 -5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

resilience

0.9

0 1.0 -100

insurance value , total value

risik premium

25

0 0.0 -5

30

600

economic value

risik premium

30

700

35

700

risk premium insurance value

insurance value , total value

35

-200

-10

Fig. 4. Risk premium (orange curve), insurance value (green curve), expected value (vertical distance between green and blue curves) and economic value (blue curve) as a function of resilience for the two extreme cases of negative ecosystem elasticity (σ b 0, left) and very large positive ecosystem elasticity (σ Nσ, right). The dashed line marks the maximum-income-risk e (Parameter values, left: σ = −0.88; right: σ = 0.92; both: Δy = 110, ρ = 0.017). level of resilience R ¼ R.

e so that RbRb1

and

ð17Þ

e e e dR dR dR N0; N0; b0: dρ dΔy dσ

ð18Þ

Ið0ÞbIð1Þ:

In particular, it is strictly monotonically increasing depending on ecosystem elasticity: there exists σ with 0 bσ ≤1 and lim

σ ¼ 1;

σ ¼ 0;

ð20Þ

if σb0 if and only if if σNσ

0 ≤σ ≤σ ; ð21Þ

lim

ρΔy→0

so that

′

I ðRÞN0

ð25Þ

σ→−∞

e . e is given by Eq. (16) and RbRbR where R

8 e e > < for RNR for all R∈ð0; 1Þ > : e e for RbR

  e e e e ¼ 0 for σ b 0, and through R e e is defined through Π ″ R where R n o e ″ e for σ N 0. eNR min RjΠ ðRÞ ¼ 0 for σNσ , so that R b b (iii) For all R ∈ (0,1) 8 9 b dI ðRÞ < = ¼ 0 dρ : ; N

8 9
and

8 9 b dI ðRÞ < = ¼ 0 dΔy : ; N

8 9
and

8 9 b> > > > > > > ¼> = < dI ðRÞ N 0 > dσ > > ¼> > > > ; : > b

8 0 Rb 0R > > > > R ¼ R < 0 0 RbRbR for > ′ > > > : R ¼ R′ RNR

lim IððRÞ ¼ 0;

ð22Þ

lim IðRÞ ¼ 0;

ð23Þ

ρ→0

Δy→0

′

Proof. See Appendix A 2. □ ð19Þ

ρΔy→þ∞

lim I ðRÞ ¼ lim IðRÞ ¼ 0;

σ→1

0

(ii) The insurance value is globally increasing with resilience,

dσ N0; dðρΔyÞ

and

ð24Þ

Result (15) states that the insurance value of resilience may be negative or positive, depending on the level of resilience R: there is a unique e (defined by Eq. (16)), such that the insurance value of threshold value R e and strictly positive for all RNR e resilience is strictly negative for all RbR e ¼ 0:647 in Fig. 3, R e ¼ 0:794 in Fig. 4 (Figs. 3 and 4, green line; R e ¼ 0:004 in Fig. 4 right). left, R The economic intuition behind this result is as follows. At the resile the risk premium Π(R) (Eq. (11)) is maximal, that is, the ience level R monetary value of the income risk is maximal (Figs. 3 and 4, orange e a marginal increase in resilience increases the line). Hence, for RbR risk premium, which makes the insurance value of resilience negative, e a marginal increase in resilience reduces the risk premium, and for RNR which makes the insurance value positive. e (Eq. (16)), is strictly This maximum-income-risk level of resilience, R in between R and 1, where RN0 denotes the level of resilience at which the probability of a system flip exactly equals the probability of the system not flipping (Result (17)).15 So, the maximum-income-risk level of e is always strictly larger than the level of resilience R at which resilience R the future state of the system is maximally uncertain. Also, the range  i e of low levels of resilience for which the insurance value of resil0; R ience is strictly negative, is non-empty. e inFurthermore, the maximum-income-risk level of resilience R creases with the degree of risk aversion ρ and the potential income loss Δy, it decreases with the ecosystem's elasticity σ (Result (18)). Result (19) states that the insurance value of ecosystem resilience globally increases with the level of resilience: it is strictly higher for the maximum level of resilience than for zero resilience. Result (21) states that there exists a domain of (positive) values of ecosystem elasticity, 0 ≤σ ≤σ , including the preeminent case of σ = 0, for which the insurance value of ecosystem resilience increases even strictly monotonically with the level of resilience, and it does so over the entire range of resilience (Fig. 3, green line). The underlying reason is that, here, the risk premium is strictly concave over the entire range of 15 Note that R, which is defined through pðR Þ ¼ 1=2, will be greater than (equal to, smaller than) 1/2 for σ b 0 (= 0, N 0).

S. Baumgärtner, S. Strunz / Ecological Economics 101 (2014) 21–32

resilience (Fig. 3, orange line). This domain of ecosystem elasticities is bounded from below by zero, and from above by some maximal value σ , which has the properties stated in Result (20): it increases with the risk-aversion-weighted potential income loss, ρΔy, and for ρΔy going to infinity (zero) approaches the maximal ecosystem elasticity of one (zero). Only as ecosystem elasticity σ turns negative or exceeds the threshold value σ, it may happen that the insurance value locally decreases.16 For negative ecosystem elasticity, σ b 0, it may be that the (negative) e e insurance value locally decreases at levels of resilience smaller than R (Fig. 4 left, green line); and for very large positive ecosystem elasticity, σ Nσ , it may be that the (positive) insurance value locally decreases at e e (Fig. 4 right, green line). The underlylevels of resilience greater than R ing reason is that, here, the risk premium is locally strictly convex (Fig. 4 left and right, orange line). Over the full range of R, the insurance value is then non-monotonic (Fig. 4 left and right, green line), a possibility already hypothesized by Walker et al. (2010). In economic terms, an increasing insurance value means that the higher the level of resilience, the more valuable – as an insurance – is a marginal increase in resilience. This is unusual and in contrast to normal economic goods, the marginal value of which decreases with their amount: normally, the more abundant a good, the less valuable the next marginal unit. Technically, the increasing marginal value of resilience comes about as the objective function, expected utility Eq. (5), when expressed as a function of the level of resilience, is non-concave in R. Result (22) states how the ecosystem user's degree of risk-aversion affects the insurance value. If the ecosystem user was risk neutral (ρ = 0), the insurance value would vanish for all levels of resilience R. With increasing risk-aversion, the insurance value increases for e where it is positive, and decreases for low levels high levels of RNR, e of RbR, where it is negative. The underlying economic reason is that for a risk-neutral individual, the risk premium would be 0 for all R. The more risk-averse the ecosystem user is, the larger the perceived riskiness of the income lottery R and the larger the associated risk premium (Eq. (11)), hence, the steeper the curve for I (Fig. 3, green line). Similarly, for the potential income loss Δy (Result (23)). For equal income levels in both stability domains, which means no income loss in case of a system flip (Δy = 0), the risk premium (Eq. (11)) would be zero over the whole range of R and the insurance value would vanish for all levels of resilience R. With increasing potential income loss Δy, the risk premium increases and the I-curve gets steeper. Hence, the ine and increases for RNR. e surance value decreases for RbR e Also, R shifts to the right with both increasing risk-aversion ρ and increasing potential income loss Δy. For very high values of ρ or Δy the Ie since the insurance value curve appears to be sharply bended around R, e rises faster with ρ or Δy in the range of RNR than it increases in the range e of RbR. Result (24) states that the insurance value decreases with increasing e and ecosystem elasticity for low and high levels of resilience, Rb0 RbR e , and increases with increasing ecosystem elasticity in RNR′ NR between, ′R b R b R′. This can be seen from comparing the green lines in Figs. 4 (left), 3 and 4 (right), as σ increases in this order.

16 A parameter value of σ b 0 in Function (4) implies a relationship between p and R such that the first marginal units of resilience starting from R = 0 do not have any significant impact on the reduction of the flip probability p. Only increases in resilience close to the maximum level of R = 1 do significantly lower the flip probability p. For such ecosystems, the insurance value of resilience decreases for small levels of resilience and increases for high levels of resilience close to R = 1 (Fig. 4 left, green line). Conversely, a parameter value of σ close to its maximum value of 1 means that the first marginal unit of resilience has a huge impact on the reduction of the flip probability p, whereas all later units of resilience only have a negligible effect. Under such circumstances, the insurance value of resilience steeply increases in the vicinity of R = 0 from a negative value to its maximum (positive) value and then decreases with R (Fig. 4 right, green line).

27

The underlying economic reason is that the risk premium (Eq. (11)) increases (decreases) with the ecosystem's elasticity for levels of resile ience below (above) the maximum-income-risk level of resilience, R. That is, in the range of resilience where the riskiness of income ine an increase in ecosyscreases (decreases) with resilience, i.e. for RbðNÞR, tem elasticity increases (decreases) the riskiness of income. This can be seen from comparing the orange lines in Figs. 4 (left), 3 and 4 (right), as σ increases in this order. Ecosystem elasticity thus has the very same ambivalent role as ecosystem resilience for the riskiness of income. Result (25) states that the insurance value vanishes as the ecosystem's elasticity approaches either its maximum or its minimum value. This can also be seen from comparing the green lines in Figs. 4 (left and right) and 3. The underlying economic reason is again in the properties of the risk premium (Eq. (11)), which vanishes as the ecosystem's elasticity approaches either its maximum or its minimum value. In either limiting case, according to model (4), the flip probability p(R) does not depend on the level of resilience any more, except for the extreme levels of R = 0 (for σ → 1) or R = 1 (for σ → − ∞) where it jumps from one to zero or from zero to one, respectively. As a result, the risk premium is non-vanishing only at R = 0 (for σ → 1) or R = 1 (for σ → − ∞), but vanishes for all other levels of resilience.17 Having discussed the properties of the insurance value of resilience, we now turn to discussing how the insurance value of ecosystem resilience relates to its economic value. Proposition 2. The economic value of resilience, V(R), is given by ′

V ðRÞ ¼ −p ðRÞ

ρΔy

1 e −1  ; ρ 1 þ pðRÞ eρΔy −1

ð26Þ

which has the following properties: V ðRÞ ≡ −p ðRÞΔy þ I ðRÞ ;

′

ð27Þ

V ðRÞ ≥0

for all R :

ð28Þ

Proof. See Appendix A 3. From Eq. (27) it becomes obvious that the (marginal) economic value of resilience, V(R), is the sum of two components: the expected increase in income due to a marginal increase in resilience, −p′(R)Δy, which is always positive,18 and the insurance value of a marginal increase in resilience, I(R), which may be negative or positive (cf. Proposition 1). This reflects the fact that an increase in ecosystem resilience has two effects on the ecosystem user's uncertain income (cf. the discussion in Section 3 and the illustration in Fig. 2): (i) it raises the expected income; (ii) it may raise or lower the riskiness of income, i.e. deviations from expected income. Thus, the economic value of resilience is more than its insurance value, or, put the other way round, the insurance value is a value component over and above the expected value of resilience. Figs. 3 and 4 show the economic value as a function of resilience (blue line). In the figures, the expected value of resilience, −p′(R)Δy, is the vertical difference between the curves for I (green) and V (blue). Whereas the insurance value I(R) of resilience may be positive or negative, depending on the level of resilience R, the expected value of resilience, − p′(R)Δy, is positive at all levels of resilience R.19 As a 17 Note that an overall vanishing risk premium, except for either R = 0 (for σ → 1) or R = 1 (for σ → − ∞) is compatible with the statement that the risk premium ine because R e decreases with σ (Result (18)). creases with σ for RbR, 18 By Assumption (2), p′(R) b 0 for all R ∈ (0,1). 19 Note that for σ = 0, one has p′(R) = −1 = const., so that the expected value of resilience does not depend on the level of resilience. That is, the vertical difference between the curves for I (green) and V (blue) in Fig. 3 is constant and simply equal to Δy.

28

S. Baumgärtner, S. Strunz / Ecological Economics 101 (2014) 21–32

e where the insurance value is negative, the ecoconsequence, for RbR nomic value of resilience is smaller than its expected value. Yet, at all levels of resilience the economic value is positive (Result (28)). That means, even if the insurance value should be negative, the expected value of resilience is large enough to offset this negative effect on the economic value. 5. Discussion and Conclusion In this paper we have provided a conceptual clarification of the insurance value of ecosystem resilience. We have adopted a general and widely accepted definition of insurance as mitigation of the influence of uncertainty on a person's well-being (McCall, 1987), and of insurance value as a reduction in the risk premium of the person's income risk lottery (Baumgärtner, 2007). That way, we have clearly distinguished the insurance value of ecosystem resilience, which is due to its function to reduce the riskiness of income (“risk mitigation”), from other components of its economic value, which are due to resilience's function to raise the expected income from ecosystem services. This distinction is important for economic analyses of decision-making problems because the two functions may have different substitutes or complements. Our analysis has yielded several important results. First, the insurance value of resilience is negative for low levels of resilience and positive for high levels of resilience. That is, ecosystem resilience actually functions as an insurance, i.e. it reduces the riskiness of income from ecosystem services, only at sufficiently high levels of resilience; it does not function as an insurance but – just on the contrary – increases the riskiness of income at low levels of resilience. Second, the (marginal) insurance value as well as the (marginal) economic value of resilience increase globally with the level of resilience — for some ecosystem types (namely those with moderately positive elasticity) even monotonically: the higher the level of resilience, the more valuable is another unit of resilience. This is in contrast to normal economic goods, the (marginal) value of which decreases with their quantity. As unusual as this increasing-returns property may be for normal economic goods, it is not implausible and also known from other goods which are of systemic importance and thus give rise to a non-concavity in the objective function, such as information (Radner and Stiglitz, 1984) and biodiversity conservation (Hunter, 2009). The management consequences for such non-convex ecological–economic systems are discussed e.g. by Dasgupta and Mäler (2003). Third, the insurance value of resilience is one additive component of its economic value. The other component is the rise in expected income due to a higher level of resilience. So, the insurance value of resilience, which is due to its risk-mitigation function, is a value component over and above the change in the expected value of the income lottery. While the former may be positive or negative, the latter is always positive, and the economic value of resilience is always positive. One reason for distinguishing between the two value components of ecosystem resilience, and for studying the insurance value separately, might be that in an encompassing management-and-decision context the different functions of resilience may have different substitutes or complements. For example, in many rural areas of developing countries there is no substitute for agro-ecosystem resilience in enhancing the mean level of farming income, but there is now more and more financial insurance available that serves as a substitute for resilience's function to mitigate income risks (Baumgärtner and Quaas, 2009; Quaas and Baumgärtner, 2008). While we have made one specific assumption about risk preferences, i.e. constant absolute risk aversion, actually all of our results qualitatively hold more generally for all risk preferences satisfying the vonNeumann–Morgenstern axioms. These axioms, including continuity and context-independence, appear plausible for standard small-risk situations. But one may doubt that they adequately describe people's risk preferences when it comes to catastrophic (i.e. discontinuous) risk that

irreversibly threatens the subsistence level of income, as it is the case for many threats to the resilience of life-supporting ecosystems. For such risks, it may be interesting to study how resilience provides insurance under, e.g., safety-first preferences (Kataoka, 1963; Roy, 1952; Telser, 1955). One general lesson from our analysis for further discussions of resilience as an insurance is that the concept of insurance fundamentally refers to both the objective characteristics of risk in terms of different possible states of nature and people's subjective risk preferences over these states. In particular, explicit reference to people's risk preferences is needed to meaningfully discuss insurance, to specify the insurance value of resilience, and to meaningfully distinguish the insurance value from other components of the economic value of ecosystem resilience. This is critical for adequately taking into account the insurance function of resilience in the sustainable management of ecosystems and their services for human well-being (Bateman et al., 2011: Sec. 8.2; Brand, 2009). Acknowledgments We are grateful to Geir Asheim, Fridolin Brand, Philippe Delacote, Moritz Drupp, Eli Fenichel, Michael Hanemann, Karl-Göran Mäler, Charles Perrings, Martin Quaas, Kerry Smith, Brian Walker, participants at the conferences Resilience 2011, SURED 2010, BIOECON 2010, EAERE 2009, ESEE 2009, and German Economic Association 2009, and – last, but not the least – three anonymous referees for helpful discussion and comments. Financial support from the German Federal Ministry of Education and Research under grant 01LL0911E is gratefully acknowledged. Appendix A Throughout the Appendix, we denote the risk-aversion-weighted income loss by λ := ρΔy. A 1. Proof of Results (11)–(13) Explicating the general definition of the risk premium (Eq. (7)) by the CARA-utility Function (6) yields −ρ½ð1−pðRÞÞyH þpðRÞyL −Π ðRÞ

−e

¼ −ð1−pðRÞÞe

−ρyH

−ρyL

−pðRÞe

;

ðA:29Þ

which can be rearranged into ρΠðRÞ

e

¼

ð1−pðRÞÞe−ρyH þ pðRÞe−ρyL : e−ρ½ð1−pðRÞÞyH þpðRÞyL 

ðA:30Þ

Using Δy = yH − yL, Eq. (A.30) can be solved for Π(R), which leads to Result (11). Inserting p(0) = 1 or p(1) = 0 into Eq. (11) immediately yields Π = 0 (Result (12)). Strict positivity of Π(R) for all R ∈ (0,1) (Result (13)) can be demonstrated as follows. Note that pðRÞλ

1−pðRÞNe

λ

−pðRÞe

ðA:31Þ

because the right hand side of this inequality approaches 1 − p(R) as λ → 0 and strictly monotonically decreases with λ, i   d h pðRÞλ λ pðRÞλ λ e −pðRÞe ¼ pðRÞ e −e b0; dλ since λ N 0 and R ∈ (0,1), i.e. 0 b p(R) b 1. Inequality Eq. (A.31) can be rearranged pðRÞλ

1−pðRÞNe

λ

−pðRÞe

ðA:32Þ

S. Baumgärtner, S. Strunz / Ecological Economics 101 (2014) 21–32

  λ pðRÞλ 1 þ pðRÞ e −1 Ne

ðA:33Þ

h  i λ ln 1 þ pðRÞ e −1 NpðRÞλ

ðA:34Þ

29

so that Eq. (A.41) can be rewritten as   e ≡ F ðλÞ and p R

e ≡ p−1 ð F ðλÞÞ: R

ðA:43Þ

Note that h  i λ −pðRÞλ þ ln 1 þ pðRÞ e −1 N0;

ðA:35Þ

λ

λ

e −1−λ e −1 1 1   ¼ lim ¼ lim ¼ ; λ→0 ð1 þ λÞeλ −1 λ→0 2 þ λ 2 λ eλ −1 ^ ðapply lʼHopitalʼs rule twiceÞ

lim F ðλÞ ¼ lim

λ→0

λ→0

which yields, after dividing by ρ N 0, Result (13). A 2. Proof of Proposition 1

lim F ðλÞ ¼ lim

Differentiating the risk premium Π(R) (Result (11)) with respect to R yields ′

Π ðRÞ ¼ −

( ) p′ ðRÞ eλ −1  λ  : λ− ρ 1 þ pðRÞ e −1

ðA:36Þ

Substituting λ = ρΔy, and taking into account the minus sign from the defining Eq. (8) immediately yields Result (14).

λ→þ∞ ′

λ→þ∞

1 1 − lim ¼ 0; λ λ→þ∞ eλ −1

8 9
lim λ−

R→0

λ

R→1

ðA:39Þ

With Π′(R) being continuous on (0,1), limR → 0Π′(R) N 0 and limR → 1Π′(R) b 0, according to the  intermediate-value theorem, e ¼ 0. It is determined from e ð0; 1Þ with Π ′ R there exists R∈ Eq. (A.36) and p′(R) ≠ 0 for all R ∈ (0,1) (Assumption (2)) through λ−

λ

e −1    ¼ 0; e eλ −1 1þp R

−2 ¼

∞ X λn n¼1

F ðλÞN0

For R → 1, and again using ex N 1 + x for all x ≠ 0, one has eλ −1 eλ −1 λ  λ  ¼ λ− ¼ λ−e þ 1 1þ0 1 þ pðRÞ e −1 bλ−1−λ þ 1 ¼ 0:

−λ

ðA:37Þ

eλ −1 eλ −1 −λ  λ  ¼ λ−   ¼ λ−1 þ e 1 þ pðRÞ e −1 1 þ eλ −1 Nλ−1 þ 1−λ ¼ 0:

ðA:40Þ

n!

þ

for all λ

1 1 − ; λ eλ −1

¼1þ

∞ X λn n¼1

n!

;

(follows immediately from Eq. (A.44) to (A.46)). Since by Assumption (2), p(R) is continuous and strictly monotonic, i.e. p′(R) ≠ 0 for all R ∈ (0,1), and F(λ) is continuous and strictly monotonically decreasing between 1/2 and 0 (Properties (A.64)–(A.66)), Eq. (A.43) can be uniquely solved e (Result (16)). Pulling all this information together, it folfor R lows that Result (A.37) holds. Result (15) then follows immediately from Definition (8).   e ¼ F ðλÞN0 Next, from Eq. (A.47) it follows immediately that p R e for all λ. for all λ, which implies, with p′(R) b 0 for all R, that Rb1 On the other hand, from Eqs. (A.44) and (A.46) one has that   e ¼ F ðλÞb1=2 for all λ, F(λ) b 1/2 for all λ N 0. Hence, p R e for all which implies, with p′(R) b 0 for all R ∈ (0,1), that RNR λ. This establishes Result (17). With Eq. (A.43), Assumption (2) (p′(R) b 0 for all R ∈ (0,1)) and Property (A.46), it follows that e dR 1 ′ ¼   F ðλÞN0: dλ p′ R e

ðA:48Þ

e From that, with λ ≡ ρΔy it follows immediately that dR=dρN0 e e can and dR=dΔyN0 (Result (18)). Using Eqs. (4) and (A.43), R be rewritten as ðA:49Þ

from which it follows that ðA:41Þ

e In order to study the existence, uniqueness, and properties of R introduce F ðλÞ :¼

n!

ðA:47Þ

1 1−σ

  1 e ¼ − 1 : p R λ eλ −1

∞ X λn

∞ ∞ X ð−λÞn n¼2m X 2λ2m ¼ ð n! 2mÞ! n¼1 m¼1

e ¼ p−1 ð F ðλÞÞ ¼ ½1− F ðλÞ ; R

which can be rearranged into

ðA:46Þ

n¼0

λ

ðA:38Þ

lim λ−

for all λN0

ðas; through Taylor series expansion; e ¼

and therefore e þ e

First, note that with Assumption (2), Π′(R) Eq. (A.36) is continuous on (0,1). Also by Assumption (2), p′(R) ≠ 0 for all R ∈ (0,1). Hence, the sign of Π′(R) is determined by the sign of the term in braces. For R → 0, using ex N 1 + x for all x ≠ 0, one has

ðA:45Þ

λ

1 e þ 2 λ2 eλ −1 1 1 ¼ λ − b0 e þ e−λ −2 λ2

F ðλÞ ¼ −

ad (i) To start with, we show that for all R ∈ (0,1)20 8 9
ðA:44Þ

ðA:42Þ

20 For σ = 0, the statement about the sign of Π′(R) holds also for R = 0 and R = 1. Yet, for σ b 0, one has Π′(0) = 0; for σ → 1, one has Π′(1) → 0.

e dR ¼ ½1−F ðλÞ dσ

1 1−σ

ln½1−F ðλÞ

1 2

ð1−σ Þ b0 ;

ðA:50Þ

since 0 b F(λ) b 1/2 (from Eqs. (A.44) to (A.47)) and σ b 1 (by Assumption (4)) imply that the first and third factors are strictly positive and the second is strictly negative. ad (ii) Under Assumption (4) one has pðRÞ ¼ 1−R

1−σ

ðA:51Þ

30

S. Baumgärtner, S. Strunz / Ecological Economics 101 (2014) 21–32 ′

−σ

p ðRÞ ¼ −ð1−σ ÞR

so that

ðA:52Þ

8 e e > < for RNR Π ðRÞb0 for all R∈ð0; 1Þ > : e e for RbR

if σ b0 if and only if if σ Nσ

″

″

p ðRÞ ¼ σ ð1−σ ÞR

−σ −1

:

ðA:53Þ

Result (19) can be demonstrated by noting that Result (14) implies  1 −λ λ−1 þ e ρ  1 ′ λ ¼ p ð1Þ λ−e þ 1 ; ρ ′

Ið0Þ ¼ p ð0Þ

and

Ið1Þ ðA:54Þ

where −λ

λ−1 þ e

N0 and

λ

λ−e þ 1b0;

0≤σ ≤σ ;

ðA:60Þ   e e e is defined through Π ″ R e ¼ 0 for σ b 0, and through where R n o e e e ¼ min RjΠ ″ ðRÞ ¼ 0 for σ Nσ , so that R eNR e for σ N 0. R b b Result (20) is Result (A.59), and Result (21) follows immediately from Definition (8) and Result (A.60). ad (iii) We first show that the risk premium Π(R) (Eq. (11)) has the following properties:For all R ∈ (0,1)

ðA:55Þ

dΠ ðRÞ N0 dρ

and

dΠ ðRÞ N0 dΔy

and

lim Π ðRÞ ¼ 0;

ðA:61Þ

lim Π ðRÞ ¼ 0;

ðA:62Þ

ρ→0

since ex N 1 + x for all x N 0. Under Assumption (4), one has Eq. (A.52), so that 9 8 ′ ′ > = < p ð0Þ ¼ 0 and p ð1Þb0 > ′ ′ p ð0Þb0 and p ð1Þb0 > > : ′ ′ 21 ; p ð0Þb0 and p ð1Þ ≤0

if

8 9 < σ b0 = σ ¼0 : : ; σ N0

ðA:56Þ

Combining Eqs. (A.54)–(A.56), one has 8 9 < I ð0Þ ¼ 0 and Ið1ÞN0 = if Ið0Þb0 and I ð1ÞN0 : ; Ið0Þb0 and I ð1Þ≥0

8 9 < σ b0 = σ ¼0 ; : ; σ N0

ðA:57Þ

Δy→0

8 9 8 9 N
ðA:63Þ

and lim Π ðRÞ ¼ lim Π ðRÞ ¼ 0:

ðA:64Þ

σ →−∞

σ→1

which means that, in any case, I(0) b I(1), which is Result (19). Differentiating Eq. (A.36) with respect to R yields: ( " # " #2 ) 1 eλ −1 eλ −1 ″ ′  λ  þ p ðRÞ  λ  : p ðRÞ λ− Π ðRÞ ¼ − ρ 1 þ pðRÞ e −1 1 þ pðRÞ e −1 ″

ðA:58Þ

Proof. The first derivative of Π(R) (Eq. (11)) with respect to ρ is o dΠ ðRÞ 1 n ′ ¼ 2 λG ðλÞ−GðλÞ with GðλÞ : dρ ρh  i λ

¼ ln 1 þ pðRÞ e −1

Inserting Eqs. (A.51)–(A.53) into Eq. (A.58) yields an explicit equation for Π″(R) in the elementary parameters of the model, σ, ρ and Δy. We have conducted systematic numerical simulations of this equation with Microsoft EXCEL. Starting with very large negative values of σ, we increased the value of σ until its maximum of + 1. The simulations showed that there is a range of values for σ for which Π″(R) is strictly negative for all R. The lower bound of this range is 0, its upper bound, denoted by σ, close to +1. Below this range, i.e. for strictly negative values of σ, we found that Π″(R) N 0 for small values of R and Π″(R) b 0 for large values of R. We found that the value of R e e is smaller than R. e at which Π″(R) changes its sign, denoted by R, Increasing σ (at strictly negative values of σ) shrinks the range of R for which Π″(R) N 0. At σ = 0, Π″(R) becomes strictly negative for all R. Only for σ Nσ does Π″(R) become positive again for some values of R, this time for large values of R, and we found the value of R at which Π″(R) changes its sign from nege e is greater than R. e To determine the value of ative to positive, R, σ , we varied the risk-aversion-weighted potential income loss, λ = ρΔy, from 0 to + ∞, observing a continuous and strictly monotonic increase of σ from 0 to +1. For example, with parameter values of ρ = 0.017 and Δy = 110, which were used to produce Figs. 3 and 4, σ ¼ 0:89. The findings from these simulations can be summarized and formally expressed as follows: There exists σ with 0bσ ≤1 and dσ N0; dðρΔyÞ

lim

ρΔy→þ∞

σ ¼ 1;

lim

ρΔy→0

σ ¼ 0;

ðA:59Þ

:

ðA:65Þ

Note that G(λ) is strictly monotonically increasing and strictly convex in λ for all R ∈ (0,1), as ′

G ðλÞ ¼

″

λ

pðRÞe   N0; 1 þ pðRÞ eλ −1

G ðλÞ ¼ h

ðA:66Þ

pðRÞð1−pðRÞÞeλ  i2 λ 1 þ pðRÞ e −1 N0 ;

ðA:67Þ

so that G′(λ) N G(λ)/λ for all λ N 0. Therefore the term in braces in Eq. (A.65) is strictly positive, and dΠ(R)/dρ N 0 for all λ N 0. This establishes Result (A.61a). Result (A.61b) follows from Eq. (11) as follows:

lim Π ðRÞ ¼ −pðRÞΔy þ lim

ρ→0

h  i ln 1 þ pðRÞ eρΔy −1

ρ→0

^ pitalʼs rule using lʼHo

ρΔy

pðRÞΔy e   ρ→0 1 þ pðRÞ eρΔy −1

¼ −pðRÞΔy þ lim

¼ −pðRÞΔy þ pðRÞΔy ¼ 0:

ρ

ðA:68Þ

ðA:69Þ

ðA:70Þ

S. Baumgärtner, S. Strunz / Ecological Economics 101 (2014) 21–32

That the risk premium increases with Δy can be seen from the first derivative of Π(R) with respect to Δy: " # dΠ ðRÞ eλ  λ  −1 ¼ pðRÞ dΔy 1 þ pðRÞ e −1 ðA:71Þ  1 −1 : ¼ pðRÞ pðRÞ þ ð1−pðRÞÞe−λ As e−λ b 1 for λ N 0, and 0 b p(R) b 1 for R ∈ (0,1), the denominator in the fraction is strictly smaller than 1, so that the term in brackets is strictly positive and the whole expression is strictly positive, which yields Result (A.62a). Setting Δy = 0 in Expression (11) for Π(R) obviously yields Π(R) ≡ 0 (Result (A.62b)). From Result (11) it follows that ( ) dΠ ðRÞ 1 eλ −1 dpðRÞ  λ  ¼− λ− : dσ ρ dσ 1 þ pðRÞ e −1

ðA:72Þ

From Eq. (A.36) it is apparent that ( λ−

eλ −1   1 þ pðRÞ eλ −1

) ¼ −ρ

Π ′ ðRÞ ; p′ ðRÞ

ðA:73Þ

so that Eq. (A.72) becomes

Solving for W(ΔR), using Δy = yH − yL and λ ≡ ρΔy, yields

W ðΔRÞ ¼

ð1−pðRÞÞ þ pðRÞeλ 1 ln : ρ ð1−pðR þ ΔRÞÞ þ pðR þ ΔRÞeλ

ðA:80Þ

Using Eq. (A.80) in Definition 3 and applying l'Hôpital's rule, one has ð1−pðRÞÞþpðRÞeλ

V ðRÞ ¼

lnð1−pðRþΔRÞÞþpðRþΔRÞe 1 lim ρ ΔR→0 ΔR

λ

1 ð1−pðR þ ΔRÞÞ þ pðR þ ΔRÞeλ ¼ lim ρ ΔR→0 ð1−pðRÞÞ þ#pðRÞeλ " ρΔy d 1−pðRÞ þ pðRÞe  dΔR 1−pðR þ ΔRÞ þ pðR þ ΔRÞeλ

ðA:81Þ

ðA:82Þ

" # λ 1 d 1−pðRÞ þ pðRÞe lim ¼ ρ ΔR→0 dΔR 1−pðR þ ΔRÞ þ pðR þ ΔRÞeλ

ðA:83Þ

h ih i 1−pðRÞ þ pðRÞeλ −p′ ðR þ ΔRÞ þ p′ ðR þ ΔRÞeλ 1 ¼ − lim h i λ 2 ρ ΔR→0 1−pðR þ ΔRÞ þ pðR þ ΔRÞe

ðA:84Þ

¼−

dΠ ðRÞ Π ′ ðRÞ dpðRÞ ¼ ′ : dσ p ðRÞ dσ

31

p′ ðRÞ eλ −1  : ρ 1 þ pðRÞ eλ −1

ðA:85Þ

ðA:74Þ

As p′(R) b 0 for all R ∈ (0,1), and, with Assumption (4), dp(R)/dσ b 0 for all R ∈ (0,1), the sign of dΠ(R)/dσ is determined by the sign of Π′(R). With Result (A.37), Result (A.63) then follows immediately. Result (A.64) follows from Result (11) and noting that model (4) implies

ad (i) Rearranging Result (26), and using Result (14), immediately yields Result (27). ad (ii) Expression (A.85) for V is non-negative for all R, since −p′(R) is non-negative and the term (eλ − 1) is strictly positive for all R. Hence, Result (28) holds.

lim pðRÞ ¼ 0 as well as

References

lim

σ→−∞

σ →1

pðRÞ ¼ 1

for all R∈ð0; 1Þ : ðA:75Þ

Now, Results (22), (23), and (24) follow from Definition (8), the fact that the function Π(R) is continuous and differentiable, and Results (13), (A.61), (A.62), and (A.63). In, addition, systematic numerical simulations of Eq. (14), using model (4), for all −∞ b σ b 1 and for various ρ,Δy N 0 have been employed to demonstrate Result (24). Result (25) follows from Definition (8), the fact that the function Π(R) is continuous and differentiable, and Result (A.64). A 3. Proof of Proposition 2 Explicating the general definition of the ecosystem user's willingness to pay (Eq. (10)) by the CARA-utility Function (6) yields −ρyH

−ð1−pðRÞÞe

−ρyL

−pðRÞe

ðA:76Þ

h i −ρðyL −W ðΔRÞÞ −ρðyH −W ðΔRÞÞ þ ð1−pðR þ ΔRÞÞe ¼ − pðR þ ΔRÞe ¼ −e

ρW ðΔRÞ

−ρyL

−ρyH 

ðA:77Þ

:

ðA:78Þ

−ð1−pðRÞÞe−ρyH −pðRÞe−ρyL : ½pðR þ ΔRÞe−ρyL þ ð1−pðR þ ΔRÞÞe−ρyH 

ðA:79Þ

pðR þ ΔRÞe

þ ð1−pðR þ ΔRÞÞe

Rearranging leads to −e

ρW ðΔRÞ

¼

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