The optimal management of renewable resources under the risk of potential regime shift

Journal of Economic Dynamics & Control 40 (2014) 195–212

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The optimal management of renewable resources under the risk of potential regime shift Bijie Ren n, Stephen Polasky Department of Applied Economics, University of Minnesota, 231 Ruttan Hall, 1994 Buford Avenue, St. Paul, MN 55108, United States

a r t i c l e i n f o

abstract

Article history: Received 28 January 2013 Received in revised form 13 November 2013 Accepted 5 January 2014 Available online 14 January 2014

Complex dynamic systems can undergo changes in feedbacks between system components causing a rapid and persistent shift in system behavior (“regime shifts”), and potentially reduce welfare from declining provision of important ecosystem services. In this paper, we provide an analytical condition that determinfilefs whether the threat of a potential regime shift causes management to be more aggressive or more precautionary. In numerical simulations we find that aggressive management can occur for reasonable parameter values, which is counter prior results that the potential for harmful regime shift always leads to precautionary management. & 2014 Elsevier B.V. All rights reserved.

JEL classification: O44 Q01 Q20 Q57 Keywords: Optimal management Uncertainty Growth Renewable resources Regime shift

1. Introduction Complex systems, such as ecosystems or market economies, are characterized by interactions among multiple components (e.g., interactions among species or among multiple producers and consumers). Such complex systems can undergo changes in feedbacks between system components causing rapid and persistent shifts in system behavior. Examples of such regime shifts in ecological systems include shifts in coral reefs between coral-dominated and algaldominated systems (Hughes et al., 2003), and in shallow lakes between oligotrophic and eutrophic conditions (Scheffer, 2004). In financial markets, where the expectations of other traders influence returns, shifts in investor sentiment lead to shifts between bull and bear markets (Scharfstein and Stein, 1990; Banerjee, 1992). Similarly, changes in expectations can lead to shifts between multiple potential equilibria in the entire economy (Azariadis, 1981; Cass and Shell, 1983). Regime shifts can also occur in social systems (e.g., consumer fads) and in political systems with changes in governments (literally regime shifts). Regime shifts are characterized not only by relatively rapid shifts in system behavior but also by hysteresis. Once a regime shift has occurred it may be difficult or impossible to reverse the process and recover the original regime.

n

Corresponding author. Tel.: þ1 612 360 7903. E-mail addresses: [email protected] (B. Ren), [email protected] (S. Polasky).

0165-1889/$ - see front matter & 2014 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jedc.2014.01.006

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In ecological systems, an increase in phosphorus inputs into shallow lakes can trigger a shift from oligotrophic to eutrophic conditions but it may take a far greater reduction in phosphorus inputs over an extended period of time to shift the lake back from eutrophic to oligotrophic conditions (Scheffer et al., 2001). Similarly, it may take prolonged good or bad economic news before a sufficient number of people shift expectations and generate a new equilibrium. A regime shift may have either positive or negative effects on welfare. Of greatest concern for management are cases where a regime shift causes a decline in welfare. For example, a shift from coral-dominated to algal-dominated reef may reduce tourism and fish harvests, and a shift from a bull to a bear market can reduce wealth and potentially trigger a financial crisis. In this paper we analyze the case where a potential regime shift results in lower welfare. Early research on regime shifts in resource economics focused on catastrophic collapse with zero utility post regime shift. Cropper (1976) analyzed potential system collapse triggered when the state variable exceeds an uncertain threshold. Reed (1987, 1988) used the Pontryagin maximum principle to transform Cropper0 s stochastic problem into a deterministic problem to find an analytical solution. With an exogenous probability of catastrophic collapse, optimal environmental management is more aggressive compared to the case with no threat of collapse. The potential to lose the resource in the future gives more incentive to use the resource in the current period rather than save for the future (“use it or lose it”). When the risk of collapse is endogenous, optimal management is contingent on the shape of the hazard function and the magnitude of disutility induced by the collapse and can be either aggressive or precautionary (Clarke and Reed, 1994). More recent research has analyzed the effect of regime shift that reduces stock but not to zero, changes the growth dynamics of the stock, or shifts preferences. This literature has generally found that the threat of regime shift causes optimal management to be more precautionary. Tsur and Zemel (1998) studied a model of pollution in which “reversible events” lead to a reduction in post event welfare. They conclude that “reversible events always imply more conservation” (p. 968), which is equivalent in our terms to the threat of potential regime shift always leading to precautionary management. This result follows from the assumption that the hazard rate and the penalty inflicted by the regime shift are non-decreasing functions of the pollution level. Polasky et al. (2011) developed unambiguous results for a potential regime shift that reduces the natural growth of a renewable resource. With a linear benefit function, they found that a potential regime shift has no effect on the optimal management when the risk is exogenous, and induces precautionary management when the risk is endogenous. Precautionary management is also found by de Zeeuw and Zemel (2012) in the context of pollution control where a regime shift causes a structural change in preferences but does not affect the pollution decay rate. Numerical simulations by Gjerde et al. (1999) and Keller et al. (2004) in a potential regime shift for climate change also suggest precautionary management. This paper builds from Polasky et al. (2011) in which a regime shift reduces the natural growth of a renewable resource. We use a general utility function rather than a linear function as in Polasky et al. (2011). We use dynamic programming methods to evaluate the changes in value of harvesting the resource caused by biophysical change in the resource growth function. This change in value is captured by a damage function based on the value functions before and after the regime shift. As found in the early literature (Clarke and Reed, 1994; Tsur and Zemel, 1998), the shape of the damage function is crucial to determine how a potential regime shift influences optimal management. In contrast with the previous research in which the shape of the damage function is given by assumptions (Clarke and Reed, 1994; Tsur and Zemel, 1998), we use general forms of utility function, natural growth function and hazard function to study the shape of the damage function analytically. We show that damage function can be either increasing or decreasing in resource stock. Therefore, the threat of potential regime shift can make the management more aggressive (when the damage function increases in stock) or more precautionary (when the damage function decreases in stock) compared to management without threat of regime shift. We find that a potential regime shift affects optimal management through multiple effects, which have not been fully captured by the existing literature. If lowering exploitation of the resource lowers the risk of a regime shift, the resource manager will take precautionary measures and lower exploitation compared to a case without potential regime shift. We call this effect the “risk reduction effect.” The risk reduction effect occurs when the probability of a regime shift is endogenous. The risk reduction effect is the only effect that occurs when the utility function is linear (Polasky et al., 2011). However, there are two additional effects from a potential regime shift with general forms of utility function and resource growth function. First, a regime shift will cause future resource availability to be lower, which lowers post-regime shift harvest and raises post-regime shift marginal utility of harvest. A forward looking manager will take this into account by reducing the initial exploitation rate thereby saving more stock to increase future harvests (“consumption smoothing effect”). Second, a regime shift reduces the resource growth rate and makes saving the resource a poorer investment because it will have a lower rate of return. A forward looking manager will take this effect into account by increasing the current exploitation rate and saving less stock for the future (“investment effect”). The investment effect causes optimal management to be more aggressive, but the risk reduction effect and the consumption smoothing effect cause optimal management to be more precautionary. We provide a condition under which the investment effect outweighs the other two effects, leading to more aggressive management in the face of a potential regime shift compared to the case without threat of potential regime shift. With aggressive management a potential regime shift will increase current resource exploitation and reduce resource stocks as compared to the case with no potential regime shift. This result is surprising in light of the previous literature in which the potential for a regime shift was thought to cause optimal management to be more precautionary.

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However, similar results to ours can be found in the consumption-saving problem from standard growth models. An exogenous shock to the general economic condition that reduces the return on capital would lead to lower future income and encourage higher saving to smooth consumption over time. On the other hand, this shock would also cause the return on saving to be lower and discourage saving. Thus the overall effect of a negative economic shock depends on the relative magnitude of these two effects, which are sometimes called income effect and substitution/price effect. In our model of renewable resource management, we find that aggressive management can occur for reasonable parameter values. We simulate the effect of a potential regime shift on either the carrying capacity or the intrinsic growth rate of the natural growth of a renewable resource. Our simulation shows that a potential regime shift is likely to cause aggressive management if risk endogeneity is suitably small and the elasticity of intertemporal substitution (EIS) is suitably large. Our simulation results also show that the relationship between the effect of a potential regime shift and relative risk aversion can be non-monotonic. It is well known that environmental uncertainty can have complicated effect on the optimal management (Pindyck, 1984). Holding the relative risk aversion constant, an increase in the variance of a stochastic shock can induce a non-monotonic response to the hazard (Saphores, 2003; Brozovic and Schlenker, 2011; Zemel, 2011). We also find that increasing risk aversion can have non-monotonic impacts on optimal management. In the next section we build intuition by solving for a two-period model where the potential regime shift may occur after the harvest decision is made in the initial period. In Section 3 we consider a discrete time infinite horizon model with a nonzero probability of regime shift in each period. Section 4 illustrates the main results using a numerical simulation. Section 5 contains concluding comments. 2. A two-period example We start with a two-period model that is relatively simple to analyze but illustrates most of the important results contained in the infinite-horizon model analyzed in Section 3. Consider the optimal management problem for a renewable resource stock whose harvest supplies the only consumption good in the economy. The growth of the renewable resource is characterized by a production function f : S  X-S, where s A S  R þ represents resource stock size, and x A X  R þ represents an aspect of the environmental quality. We assume that both S and X are nonempty, convex and compact. We can write the production function as f ¼ ϕ þ s, where ϕ represents natural growth. For example, ϕ ¼ gsð1  s=KÞ is the standard logistic natural growth function. In this case, x could represent the intrinsic growth rate g or the carrying capacity K. We assume that f satisfies the following assumptions: Assumption 1a. f ðs; xÞ is twice continuously differentiable on S  X; Assumption 1b. f 1 ðs; xÞ 4 0 and f 11 ðs; xÞ o0;1 Assumption 1c. f 2 ðs; xÞ 40 and f 12 ðs; xÞ 4 0; Assumption 1d. For all x A X, f ð0; xÞ ¼ 0, and there exists carrying capacity K such that s r f ðs; xÞ r K for all 0 r s rK, and f ðs; xÞ os for all s 4K. A regime shift is introduced as a decrease of x by Δx 4 0. Then by Assumption 1c, for any given stock level s, a regime shift would reduce the resource production f ðs; xÞ and the gross return rate Rðs; xÞ ¼ f 1 ðs; xÞ. Given initial resource stock s0, the stock after growth is f ðs0 ; xÞ in the initial period t ¼0. The resource manager chooses harvest, h0 A ½0; f ðs0 ; xÞ, which is nonstorable and consumed immediately to generate utility according to the function u : R þ þ -R. We assume that u satisfies the following assumptions: Assumption 2a. u(h) is twice continuously differentiable on R þ þ ; Assumption 2b. u0 ðhÞ 40 and u″ðhÞ o 0; Assumption 2c. limh-0 u0 ðhÞ ¼ þ 1 and limh- þ 1 u0 ðhÞ ¼ 0; Assumption 2d. hu″ðhÞ=u0 ðhÞ ¼ γ 40 (constant relative risk aversion). After h0 is chosen, the resource stock in the beginning of period t¼1 is s1 ¼ f ðs0 ; xÞ  h0 . We assume that there is a probability of regime shift at t¼1 yielding either normal growth, f ðs1 ; xÞ, or reduced growth f ðs1 ; x  ΔxÞ. The probability of regime shift is determined by a hazard function λ : S-½0; 1, which satisfies the following assumptions: Assumption 3a. λðsÞ is twice continuously differentiable on S; Assumption 3b. λ0 ðsÞ r0. 1 Throughout this paper we will use subscript i to denote the partial derivative of a function to its ith argument, and subscript ij to denote the mixed second order partial derivatives to its ith and jth arguments.

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Table 1 Two-period model. t¼ 0

t¼ 1

Initial stock

Stock after growth

Harvest

Initial stock

Stock after growth¼ harvest

Probability

s0

f ðs0 ; xÞ

h0

s1 ¼ f ðs0 ; xÞ  h0

f ðs1 ; xÞ ¼ h1 f ðs1 ; x  ΔxÞ ¼ h 1

1  λðs1 Þ: no shift λðs1 Þ: shift occurs

Because t¼1 is the last period, with a strictly increasing utility function the resource manager would harvest the entire resource stock after growth, h1 ¼ f ðs1 ; xÞ for normal growth, h 1 ¼ f ðs1 ; x  ΔxÞ for reduced growth. The two-period model is summarized in Table 1. The renewable resource management problem considered here has the same structure as the standard growth model, where consumption and capital stock correspond to h and s in our model. A regime shift of the renewable resource growth in our model is similar to a decline in productivity of capital in the standard growth model. The resource manager maximizes the net present value of utility from harvest where period 1 values are discounted by β. Then define X ¼ fðx; ΔxÞ A R2þ j x A X; x  Δx A Xg, for any ðx; ΔxÞ A X and s0 A S, the resource manager solves the following optimization problem: wðs0 Þ ¼ maxfuðh0 Þ þ β½ð1 λðs1 ÞÞuðh1 Þ þ λðs1 Þuðh 1 Þg by choosing h0. This problem is equivalent to wðs0 Þ ¼ maxfuðf ðs0 ; xÞ s1 Þ þ βuðf ðs1 ; xÞ  dðs1 ÞÞg

ð1Þ

s1

where dðs1 Þ ¼ λðs1 Þ½uðf ðs1 ; xÞÞ  uðf ðs1 ; x  ΔxÞÞ:

ð2Þ

Note that d defined by (2) represents the damage to the net present value caused by the risk of potential regime shift. RS Solving (1) the optimal resource stock at the beginning of period 1 in the case with a potential regime shift, s1 , satisfies the following first order condition: 0

RS 0 RS RS u0 ðf ðs0 ; xÞ sRS 1 Þ ¼ βf 1 ðs1 ; xÞu ðf ðs1 ; xÞÞ  βd ðs1 Þ:

ð3Þ n

If there is no risk of potential regime shift, or λðÞ ¼ 0, the optimal resource stock s1 in the risk-free case would satisfy u0 ðf ðs0 ; xÞ sn1 Þ ¼ βf 1 ðsn1 ; xÞu0 ðf ðsn1 ; xÞÞ:

ð4Þ RS s1

n

with s1 . Notice that the left hand side of Without specific functional forms of f, u and λ we cannot explicitly compare (4) is strictly increasing in s1 and the right hand side of (4) is strictly decreasing in s1, thus the optimal solution sn1 is uniquely 0 determined. On the other hand, by Assumptions 1–3, we can see that d ðsÞ in (3) converges uniformly to 0 on S as RS 0 n Δx converges to 0.2 Therefore, for any s1 satisfying (3),3 we know limΔx-0 sRS 1 ¼ s1 , because limΔx-0 d ðsÞ ¼ 0 and the RS n convergence does not depend on s. Based on this result, we can discuss the sign of s1  s1 at the margin for small Δx. n RS n RS n We call the management aggressive, unchanged, or precautionary if sRS 1 os1 , s1 ¼ s1 , or s1 4 s1 . By (3) and (4) these 0 0 RS 0 RS situations correspond to d ðsRS Þ 40, d ðs Þ ¼ 0 or d ðs Þ o0, as summarized in Table 2. Intuitively, if the damage of a 1 1 1 0 potential regime shift is increasing in the resource stock ðd ðsÞ 4 0Þ, a resource manager would increase the harvest and save less stock (aggressive management) to reduce the damage. In the appendix we show that lim

Δx-0

0

n  ∂hn1

n  ∂hn1

n  ∂Rn d ðsRS 1 Þ þ λ sn1 Rn u″ h1 þ λ sn1 u0 h1 ¼ λ0 sn1 u0 h1 ∂x ∂x ∂x Δx

ð5Þ 0

n

where h1 ¼ f ðsn1 ; xÞ and Rn ¼ Rðsn1 ; xÞ ¼ f 1 ðsn1 ; xÞ. Then the management is aggressive at the margin, or limΔx-0 d ðsRS 1 Þ=Δx 40, if and only if the following condition is true: γξh |{z}

þ

Consumption smoothing

θξh |{z}

o

Risk reduction

ξR |{z}

ð6Þ

Investment

where n

ξh ¼

∂h1 x ; ∂x hn1

ξR ¼

∂Rn x ; ∂x Rn

n

θ¼ 

λ0 ðsn1 Þh1 : λðsn1 ÞRn

2 Because f ðs; xÞ and f 1 ðs; xÞ are continuous and strictly increasing in x, and u and u0 are continuous and strictly monotonic, then by Dini0 s theorem the convergence of uðf ðs; x  ΔxÞÞ-uðf ðs; xÞÞ, u0 ðf ðs; x  ΔxÞÞ-u0 ðf ðs; xÞÞ and f 1 ðs; x  ΔxÞ-f 1 ðs; xÞ are all uniform on compact set S as x  Δx-x. 3 The arguments here are not sufficient to establish the uniqueness of the solutions to (3).

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Table 2 Optimal management in the two-period model. 0

Aggressive

n sRS 1 o s1

d ðsRS 1 Þ40

Unchanged

n sRS 1 ¼ s1

d ðsRS 1 Þ¼0

Precautionary

n sRS 1 4 s1

d ðsRS 1 Þo0

0 0

If the risk is exogenous ðλ0 ¼ 0Þ, then θ ¼ 0, and the optimal management strategy is determined by the relative magnitudes of ξR and γξh. ξR is the elasticity of gross return rate (Rn) with respect to x. The regime shift that occurs will reduce the gross return rate, thus the resource stock will become a poorer investment after the regime shift. The resource manager will take this effect into account by increasing the harvest in the initial period and save less resource stock for the n future (the “investment effect”). ξh is the elasticity of optimal harvest ðh1 Þ with respect to x. The regime shift that occurs will also cause lower resource availability, which lowers post-regime shift harvest and raises post-regime shift marginal utility of harvest. The resource manager will take this effect into account by reducing the harvest in the initial period thereby saving more stock to smooth harvests across time. The elasticity of intertemporal substitution (EIS), 1/γ, measures the optimal intertemporal adjustment of consumption in response to the fluctuation of the gross return rate. Thus γ serves as a unit adjustment factor such that γξh and ξR are comparable. We call γξh the “consumption smoothing effect”. Lower EIS (1/γ) implies that the fluctuation of the gross return rate has less impact on consumption. When the risk is endogenous ðλ0 o 0Þ, then θ 4 0. By lowering harvest the resource manager can lower the risk of regime shift, and thus will tend to make optimal management less aggressive. We call this effect the “risk reduction effect.” With a positive risk reduction effect the aggressive management condition (6) is less likely to hold, and the optimal management is more likely to be precautionary. RS In this two-period model we focused on the optimal resource stock s1 under the risk of potential regime shift. We found RS that s1 can be either greater than, equal to, or lower than the risk-free optimal resource stock sn1 , depending on relative magnitudes of consumption smoothing effect, investment effect, and risk reduction effect. In the next section, we will extend the two-period model to an infinite horizon model, and discuss how a potential regime shift will affect the resource stock in the steady state. 3. The infinite horizon model We begin by briefly reviewing the classical renewable resource management problem in a risk-free environment. Then we introduce the risk of potential regime shift and show what impact it has on optimal management at the margin. We derive a condition under which the pre-shift optimal management is aggressive compared with the risk-free situation. 3.1. Classical renewable resource management problem Given the initial resource stock, s0, the resource manager chooses the optimal plan of harvests ðht Þt Z 0 to solve the following optimization problem: 1

Vðs0 Þ ¼ max ∑ βt uðht Þ

ð7Þ

t¼0

subject to st þ 1 ¼ f ðst ; xÞ  ht

ð8Þ

and nonnegativity constraints. An equivalent dynamic programming problem is characterized by the following Bellman equation: Vðst ; xÞ ¼

max

fuðf ðst ; xÞ  st þ 1 Þ þ βVðst þ 1 ; xÞg

st þ 1 A Γðst ; xÞ

ð9Þ

where Γðst ; xÞ ¼ fst þ 1 A S j 0 rst þ 1 r f ðst ; xÞg:

ð10Þ

Here we treat x as another state variable. The optimal policy for the Bellman equation is defined as gðs; xÞ ¼ fyA Γðs; xÞj Vðs; xÞ ¼ uðf ðs; xÞ yÞ þβVðy; xÞg:

ð11Þ

4

The first order condition of (9) is

u0 ðf ðs; xÞ  gðs; xÞÞ ¼ βV 1 ðgðs; xÞ; xÞ:

ð12Þ

4 We assume that Vðs; xÞ is continuous and strictly increasing in both arguments, and strictly concave and continuously differentiable in s; and gðs; xÞ is a continuous single-valued function (see Stokey et al., 1989, Section 5.1).

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Thus gðs; xÞ is chosen such that the marginal value of harvest in the current period and shadow value of the resource stock in the next period are equalized. The envelope condition of (9) is V 1 ðs; xÞ ¼ u0 ðf ðs; xÞ gðs; xÞÞf 1 ðs; xÞ:

ð13Þ

From (12) and (13) we can solve the Euler equation u0 ðht Þ ¼ βf 1 ðst þ 1 ; xÞu0 ðht þ 1 Þ

ð14Þ t 0

The solution of (9) is characterized by the Euler equation (14) and a transversality condition limt-1 β u ðht Þst þ 1 ¼ 0. The Euler equation (14) shows that the present value of marginal utility is equalized across periods. The Euler equation (14) together with the growth equation (8) compose a two-dimensional dynamic system ðst ; ht Þt Z 0 , of which a nontrivial n steady state ðsn ; h Þ is uniquely determined by f 1 ðsn ; xÞ ¼ 1=β

ð15Þ

n

h ¼ f ðsn ; xÞ  sn :

ð16Þ n

Throughout this paper we assume that s A S exists. 3.2. Renewable resource management with a potential regime shift Now we introduce a potential regime shift. As in the two-period model, the regime shift is characterized by a decrease of x by Δx 40. In the infinite horizon model, we assume that the regime shift can happen at any time t 40 with probability λðst Þ, and is irreversible. Given ðst ; x; ΔxÞ A S  X , the Bellman equation under the risk of potential regime shift is Wðst ; x; ΔxÞ ¼

max

fuðf ðst ; xÞ  st þ 1 Þ

st þ 1 A Γðst ; xÞ

þ βð1  λðst þ 1 ÞÞWðst þ 1 ; x; ΔxÞ þ βλðst þ 1 ÞVðst þ 1 ; x ΔxÞg

ð17Þ

where Γðst ; xÞ is given by (10). The optimal policy is defined as qðs; x; ΔxÞ ¼ fy A Γðs; xÞj Wðs; x; ΔxÞ ¼ uðf ðs; xÞ  yÞ þ βð1 λðyÞÞWðy; x; ΔxÞ þβλðyÞVðy; x  ΔxÞg

ð18Þ

Intuitively, the net present value derived from the renewable resource under the risk, W, depends on the resource stock s, the level of resource production x, and the magnitude of the potential regime shift Δx. The following result shows that regime shift has a negative impact on the net present value of resource stock. Proposition 1. For any ðs; x; ΔxÞ A S  X , Vðs; x  ΔxÞ oWðs; x; ΔxÞ o Vðs; xÞ: Proof. See the Appendix A. The Bellman equation (17) can be rewritten as Wðst ; x; ΔxÞ ¼

max

fuðf ðst ; xÞ  st þ 1 Þ þ βWðst þ 1 ; x; ΔxÞ

st þ 1 A Γðst ; xÞ

 βDðst þ 1 ; x; ΔxÞg

ð19Þ

where Dðst þ 1 ; x; ΔxÞ ¼ λðst þ 1 Þ½Wðst þ 1 ; x; ΔxÞ  Vðst þ 1 ; x  ΔxÞ:

ð20Þ

By Proposition 1 we know that Dðs; x; ΔxÞ 4 0 for any ðs; x; ΔxÞ A S  X . Thus, similar to the two-period model, the risk of potential regime shift reduces net present value by an amount βD. In the appendix we show that Wð; x; ΔxÞ is continuously differentiable under fairly general conditions for the hazard function λ. Then solving (19) the first order condition is u0 ðf ðs; xÞ  qðs; x; ΔxÞÞ ¼ βW 1 ðqðs; x; ΔxÞ; x; ΔxÞ  βD1 ðqðs; x; ΔxÞ; x; ΔxÞ

ð21Þ

and the envelope condition is W 1 ðs; x; ΔxÞ ¼ u0 ðf ðs; xÞ qðs; x; ΔxÞÞf 1 ðs; xÞ:

ð22Þ

Then by (21) and (22), for any ðx; ΔxÞ A X , a steady state resource stock sRS under the risk of potential regime shift, if it exists, is determined by
 1 D1 ðsRS ; x; ΔxÞ f 1 sRS ; x ¼ þ 0 β u ðf ðsRS ; xÞ sRS Þ

ð23Þ

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201

3.3. Effect of potential regime shift on optimal management We are interested in how sRS determined by (23) compares to sn determined by (15). Proposition 2. For any ðx; ΔxÞ A X , if sRS exists, then sRS ⋛sn 3 D1 ðsRS ; x; ΔxÞ⋚0: Proof. Because u0 4 0, by (15) and (23) the desired result follows immediately from the concavity of f in s.

□

Proposition 2 shows that the effect of the risk of a potential regime shift is determined by the properties of the damage function D defined by (20). If the damage of a potential regime shift is decreasing in the resource stock (D1 o 0), a resource manager would reduce the harvest to increase the resource stock in the steady state (precautionary management). This result is found in prior paper (Tsur and Zemel, 1998). However, if the damage of a potential regime shift is increasing in the resource stock ðD1 4 0Þ, a resource manager would maintain lower resource stock in the steady state (aggressive management) to reduce the damage. We now decompose D1 to provide more intuition on when D1 4 0, and then derive a condition under which optimal management is aggressive. From (20) D1 ðs; x; ΔxÞ ¼ λ0 ðsÞ½Wðs; x; ΔxÞ  Vðs; x ΔxÞ þ λðsÞ½W 1 ðs; x; ΔxÞ V 1 ðs; x  ΔxÞ:

ð24Þ

We know from Proposition 1 that Wðs; x; ΔxÞ 4 Vðs; x  ΔxÞ. With endogenous risk ðλ0 o0Þ, the first term on the right hand side of (24) is negative. Endogenous risk gives the resource manager an incentive to increase the resource stock in order to lower the probability of regime shift, the “risk reduction effect.” The second term on the right hand side of (24) depends on the difference between shadow prices of resource stocks in the pre-shift risky and post-shift risk-free environments. In Polasky et al. (2011) the shadow value of stock is always equal to prices, which is constant. Therefore, shadow values are equal before and after the regime shift ðW 1 ¼ V 1 Þ, which leaves only the first term. This is why in their model there no effect of the potential regime shift when risk is exogenous ðλ0 ¼ 0Þ and management is always precautionary when risk is endogenous ðλ0 o 0Þ. In our model, the shadow values differ pre- vs. post-regime shift. By the envelope conditions of W and V, we know the shadow values are W 1 ðs; x; ΔxÞ ¼ u0 ðhw Þf 1 ðs; xÞ and V 1 ðs; x  ΔxÞ ¼ u0 ðhv Þf 1 ðs; x  ΔxÞ, where hw ¼ f ðs; xÞ  qðs; x; ΔxÞ and hv ¼ f ðs; x  ΔxÞ  gðs; x  ΔxÞ are the harvests in corresponding cases. In our model, a regime shift has opposite effects on marginal utility u0 and gross return rate f1. As in the two-period model, the regime shift will cause lower resource availability, f ðs; xÞ 4 f ðs; x ΔxÞ, thus tending to reduce harvests after the regime shift and raise post-shift marginal utility. With the potential for regime shift, the resource manager will reduce current harvest in order to have more stock in the future when marginal utility is higher (the “consumption smoothing effect”). However, because a regime shift also results in a lower gross return rate to resource stock, f 1 ðs; xÞ 4 f 1 ðs; x ΔxÞ, the resource manager tends to increase current harvest in order to reduce the potential loss of value of the resource stock caused by the regime shift (the “investment effect”). Depending on which of these effects dominates, we can have either W 1 ðs; x; ΔxÞ 4V 1 ðs; x  ΔxÞ or the reverse. In the following discussion we provide a condition under which the investment effect will outweigh the risk reduction and consumption smoothing effects so that the pre-shift management is aggressive ðsRS o sn Þ. In the risk-free case, steady state resource stock sn is sn ðxÞ ¼ fyA S j y ¼ gðy; xÞg and by (15) we know that sn ðxÞ is a single-valued continuously differentiable function. Similarly, under the risk of potential regime shift, a steady state resource stock sRS, if it exists, satisfies sRS ðs; ΔxÞ ¼ fy A S j y ¼ qðy; x; ΔxÞg: In general the existence and uniqueness of sRS is not guaranteed by (23). The following proposition shows that the correspondence sRS ðx; ΔxÞ is nonempty when Δx is sufficiently small, and converges to sn when Δx converges to 0. Lemma 3. For any x A X, there exists δ 40 such that for any jΔxjo δ, sRS ðx; ΔxÞ is nonempty and lim sRS ðx; ΔxÞ ¼ sn ðxÞ:

Δx-0

Proof. See the Appendix A. n

As Lemma 3 shows, the effect of a potential regime shift on steady state resource stock is continuous. Let h ¼ f ðsn ; xÞ  sn denote the steady state harvest and Rn ¼ f 1 ðsn ; xÞ denote the steady state gross return rate in the risk-free case, the following proposition provides a condition under which the optimal management is aggressive under the risk of a potential regime shift.

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Proposition 4 (Aggressive management condition). There exists δ 40 such that Δx A ð0; δÞ implies sRS o sn , if the following condition is true: γξnh |{z}

Θξnh |{z}

þ

Consumption smoothing

o

Risk reduction

Rn n ξ R  κ fflR} |fflfflfflffl ffl{zfflfflfflffl

ð25Þ

n

Investment

where ξnh ¼

n

∂h x ; ∂x hn

ξnR ¼

∂Rn x ; ∂x Rn

n

Θ¼ 

λ0 ðsn Þh

λðsn ÞðRn  1 þ λðsn ÞÞ n

and κ A ð0; 1Þ is the eigenvalue of dynamic system (8) and (14) close to the risk-free steady state ðsn ; h Þ, given as follows: κ ¼ 1 þ 0:5ð1=β 1 βΩ 

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1=β 1 βΩÞ2  4βΩÞ

where n

Ω ¼ h f 11 ðsn ; xÞ=γ o 0:

Proof. See the Appendix A. The right hand side of (25) is the investment effect, which is analogous to the right hand side in condition (6) for the twoperiod model. In the infinite horizon model there is an additional term Rn =ðRn  κÞ. Because κ is the eigenvalue of dynamic system (8) and (14) close to the risk-free steady state, in the optimal solution the evolution of the resource stock is approximated by st þ 1  sn ¼ κðst  sn Þ. This means that the effect through the adjustment of resource stock (the investment effect) accumulates over an infinite future. In the two-period model, the entire resource will be harvested at t¼1. If we consider the two-period harvesting plan as the solution of a dynamic system, the resource stock converges to 0 instantaneously after t¼1 without any preservation for t Z2, or equivalently κ ¼ 0. Thus the right hand side of (25) reduces to ξnR as (6) in the two-period model. The left hand side of (25) consists of the consumption smoothing effect and the risk reduction effect. When the risk is exogenous ðλ0 ¼ 0Þ, then Θ ¼ 0. Thus, similar to the two-period model, with exogenous risk, the consumption smoothing effect γξnh is the only effect that provides an incentive to be more precautionary in harvesting the resource stock. When the risk is endogenous ðλ0 o0Þ, the resource manager tends to be more precautionary because of the risk reduction effect Θξnh . It is also interesting to note how optimal management changes with γ. For sufficiently small γ and exogenous risk ðλ0 ¼ 0Þ, n optimal management will be aggressive. For the investment effect ξnR Rn =ðRn  κÞ, notice that steady state ðsn ; h Þ is determined n n by (15) and (16), thus R and ξR are independent of γ. But we can show that 0

1

dκ βB 1 þβ  β2 Ω C ¼ @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1A 40 dΩ 2 ð1 þβ  β2 ΩÞ2 4β and lim κ ¼ 0;

Ω-  1

lim κ ¼ 1:

Ω-0

Because Ωo 0, we find that κ is increasing in γ and κ A ð0; 1Þ. Therefore, γ affects the investment effect only through κ, and the investment effect is bounded below by ξnR when γ is small. As γ increases, the investment effect also increases, but because κ o1, the investment effect is bounded above by ξnR Rn =ðRn 1Þ. For the consumption smoothing effect γξnh , because ξnh is independent of γ, γξnh would be very small if γ is close to 0. Thus, with exogenous risk, sufficiently small γ implies aggressive management because the investment effect is bounded below.5 However, when the risk is endogenous, the optimal management could be precautionary even when γ is close to 0. Notice that both Θ and ξnh are independent of γ. Thus, if  λ0 is suitably large, the risk reduction effect Θξnh would dominate the investment effect which is bounded above. The consumption smoothing effect γξnh increases proportionally with γ, but the investment effect is bounded above. Thus, even with exogenous risk, optimal management will be precautionary when γ is sufficiently large, because the consumption smoothing effect will dominate the investment effect. 5 This does not mean that with exogenous risk and γ ¼0 the optimal management is aggressive, because in this case κ is not well-defined. Polasky et al. (2011) showed that when the utility function is linear (γ¼ 0) and the risk is exogenous that a potential regime shift has no effect on optimal management.

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4. Examples: the potential regime shift on carrying capacity and intrinsic growth rate In this section we illustrate the main model results using a standard constant relative risk aversion utility function uðhÞ ¼

1γ

h 1γ

and a resource production function f ¼ ϕ þs where  s ϕ ¼ gs 1  K is the standard logistic natural growth function with carrying capacity K and intrinsic growth rate g. A regime shift can be characterized by either a reduction of K by ΔK, or a reduction of g by Δg. By Eq. (15), the risk-free steady state resource stock is  βð1 þgÞ 1 : ð26Þ sn ¼ K 2βg We assume βð1 þ gÞ 4 1 to ensure that a strictly positive risk-free steady state resource stock sn exists. We use the following hazard function: λðsÞ ¼

2λ : 1 þ expðηðs=sn  1ÞÞ

Clearly λðsn Þ ¼ λ and λ0 ðsn Þ ¼  0:5λη=sn . Given λ A ð0; 1Þ, the risk is exogenous when η ¼ 0 and endogenous when η 40. We are interested in characterizing how the steady state resource stock is affected by the regime shift. Based on (26), the n steady state harvest is h ¼ f ðsn ; xÞ  sn and the steady state gross return rate is Rn ¼ f 1 ðsn ; xÞ. Using definitions given in (25) and evaluating a potential regime shift in carrying capacity K or intrinsic growth rate g, we find that ðξnR =ξnh Þjx ¼ K ¼ 1 β þ βg

ð27Þ

ðξnR =ξnh Þjx ¼ g ¼ 1  β:

ð28Þ

Because βg 4 0, the investment effect is relatively stronger for the regime shift on K. Thus, it is more likely that a regime shift will cause aggressive management when carrying capacity K is reduced compared to the case with a reduction in intrinsic growth rate g. We set discount factor β ¼0.98. The estimates of EIS (1/γ) vary considerably in recent literature, and we consider γ in the range ð0; 3Þ.6 We normalize carrying capacity at K ¼100, and set the intrinsic growth rate g at 0.05, 0.2, or 0.5.7 To characterize the probability of a potential regime shift, we set λ ¼ 0:1, so when s ¼ sn the probability of regime shift is 10%. The risk endogeneity value, η, is set at 0, 1, 2, 3, or 4. These parameter values are summarized in Table 3. 4.1. The effect of potential regime shift on steady state We assume that the regime shift will cause a 5% reduction on either K or g. We numerically solve the Bellman equations (9) and (17).8 Fig. 1 shows simulation results for a potential regime shift on g, and Fig. 2 shows simulation results for a potential regime shift on K. Both figures plot the percentage change of steady state stock ðsn  sRS Þ=sn % against γ, for η ¼0, 1, 2, 3 and 4 and g ¼0.05, 0.2 and 0.5. Values above 0 indicate aggressive management while values below 0 indicate precautionary management. The simulations show that higher values of η (greater endogeneity of risk) shift the curve downward, making optimal management more precautionary. As predicted by (27) and (28), the regime shift on K causes aggressive management more often than regime shift on g (compare Fig. 2 with Fig. 1). This result occurs because, ceteris paribus, the regime shift on K has a relatively stronger investment effect. For the regime shift on g, Fig. 1 shows that optimal management is mostly precautionary. However, when the intrinsic growth rate is very low (g¼0.05), aggressive management is optimal with exogenous risk and when γ is lower than approximately 0.4. 6 Mulligan (2002) used U.S. national account data and estimated that EIS ranges from 0.49 to 2.05 with different specifications. Vissing-Jorgensen (2002) estimates EIS using micro data from the U.S. Consumer Expenditure Survey (CEX), and found that EIS is around 0.3–0.4 for stockholders and 0.8–1 for bondholders. Gruber (2006) also used the CEX data but considered the variation across individuals in the capital income tax rate. His estimates of EIS vary from 1.54 to 2.36. 7 The intrinsic growth rate g of marine fish species varies from 0.025 to 0.75 (Clark et al., 2010). Musick (1999) suggested that g Z 0:5 implies that a fish species has high resilience to fishing pressure, and g r 0:05 implies low resilience to fishing pressure. 8 To numerically solve the Bellman equations, we use the COMPECON toolbox developed by Miranda and Fackler (2004). The solver “dpsolve” is modified to incorporate state-dependent discount rate. The codes are available upon request.

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Table 3 Key parameter values. Preference

Resource production

Hazard function

β

γ

K

g

λ

η

0.98

0–3

100

0.05, 0.2, 0.5

0.1

0, 1, 2, 3, 4

Fig. 1. Change of the steady state resource stock with a potential regime shift on g. From top to bottom: η ¼ 0, 1, 2, 3, 4.

Fig. 2. Change of the steady state resource stock with a potential regime shift on K. From top to bottom: η ¼0, 1, 2, 3, 4.

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For the regime shift on K, Fig. 2 shows that aggressive management is possible for many reasonable combinations of γ, g and η. For γ o 2, the simulations show that optimal management is aggressive with exogenous risk and for combinations of low value of g and η. However, when γ is sufficiently high, optimal management is always precautionary. Figs. 1 and 2 also show that the effect of a potential regime shift can change non-monotonically as γ increases. When γ is very small, the consumption smoothing effect is close to 0. Thus, if the risk reduction effect is stronger than the investment effect, the optimal management is precautionary. As γ increases, both the consumption smoothing effect and the investment effect increase, but the risk reduction effect is independent of γ. Because the investment effect increases relatively faster than the consumption smoothing effect in the beginning, optimal management becomes more aggressive as γ increases. However, the investment effect is bounded above but the consumption smoothing effect increases proportionally in γ. Thus, when γ is sufficiently large, the consumption smoothing effect will become dominant and optimal management will become more precautionary as γ increases. 4.2. The effect of potential regime shift on transition path We also simulate the optimal transition path of stock and harvest ðst ; ht Þ under the risk of a potential regime shift, and compare it with the optimal path in the risk-free case. We assume that the initial resource stock equals the risk-free steady state sn. In our simulation the resource manager knows the hazard function, but does not know the exact time of the regime shift. To illustrate the effect of a potential regime shift, we present a transition path for the case where the regime shift occurs at t¼60. Because the date of the regime shift is unknown to the resource manager, the optimal transition path before the regime shift is independent of the time of its occurrence. In the simulation we set g ¼0.05, γ¼1 and η¼1. The values of the other parameters are the same as given previously. With these parameter values, a regime shift on g causes precautionary management and a regime shift on K causes aggressive management. The transition paths on stock and harvest with a potential regime shift on intrinsic growth rate g are plotted in Fig. 3. If the resource manager is ignorant of the risk and follows the “risk-free” path (asterisks), the resource stock would stay at sn n and harvest would equal h ¼ f ðsn ; xÞ  sn in each period. At t¼60, the intrinsic growth rate falls so that the new risk-free steady state resource stock and harvest are lower (illustrated by the horizontal dotted line after t ¼60). After the regime shift, the optimal management would follow the standard risk-free path that converges to the lower steady state asymptotically. From the transition path of ht we can see that if the manager is ignorant of the regime shift and follows the “risk-free” path, ht would jump downward at t¼60 in order to get on the path that converges to the lower steady state. In contrast, a resource manager who anticipates a potential regime shift that will result in lower stock growth and lower future harvests has an incentive to build up the resource stock before the regime shift. Precautionary management results in a smoother harvest transition path at t¼60, higher harvests and stock levels following the regime shift. The transition paths on stock and harvest with a potential regime shift on carrying capacity K are plotted in Fig. 4. The investment effect is relatively more important in the regime shift on K than on g. Because of the decline in the gross return

Fig. 3. Transition paths of ðst ; ht Þ under the risk of a potential regime shift on g.

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Fig. 4. Transition paths of ðst ; ht Þ under the risk of a potential regime shift on K.

of the stock post regime shift, the optimal management plan following a regime shift is to quickly reduce stock level towards the new lower steady state stock level. There are relatively large losses from having stock levels that differ from the postregime desired steady state stock. In the case when the resource manager is ignorant of the potential regime shift, the harvest level post-regime shift would actually jump up to a higher level in order to more quickly draw down stock towards the new optimal steady state level. A resource manager that anticipates a potential regime shift will draw down stock prior to a regime shift (aggressive management). Therefore, the gap between the stock size when the regime shift occurs and the desired post-regime shift steady state stock will be smaller. The upward jump in harvest rates will be smaller resulting in a smoother harvest transition path when a regime shift occurs. Were the time of the regime shift known ahead of time, the resource stock would be managed in a way to eliminate the jump in harvest and have a completely smooth transition path. Without prior knowledge of the date when a potential regime shift will occur, aggressive management will reduce but not eliminate the upward jump in harvest when the regime shift occurs. With endogenous risk, changes in stock size will change the probability of regime shift. Fig. 5 shows the hazard rate of the potential regime shift through time for t o 60 as a function of the stock size ðλðst ÞÞ for the cases shown in Figs. 3 and 4. If the resource manager is ignorant of the potential for regime shift and follows the “risk-free” path, the hazard rate would be constant at λðsn Þ ¼ λ ¼ 0:1. If the regime shift is on the carrying capacity K, the resource manager who is aware of this risk would harvest the resource aggressively, and this leads to higher hazard rate. On the other hand, when the regime shift is on the intrinsic growth rate g, the optimal management is precautionary and this would reduce the hazard rate along the transition path.

5. Conclusion In this paper we showed that the risk of a potential regime shift could cause the optimal management of renewable resources to be precautionary, unchanged, or aggressive as compared to the risk-free case without regime shift. Our results contrast with those of other recent papers on regime shifts in systems dynamics that show that a regime shift will cause management to be more precautionary (Polasky et al., 2011; de Zeeuw and Zemel, 2012). Our model includes non-linear benefits functions (varying marginal benefits) and a regime shift on resource growth function, which introduce additional effects into the model. Our model shows that the optimal management depends on the relative magnitudes of three effects: the risk reduction effect, the consumption smoothing effect, and the investment effect. When the risk of a regime shift declines with an increase in resource stock, the resource manager will have an incentive to lower harvest and increase stock thereby reducing the risk of regime shift (risk reduction effect). A regime shift results in lower resource availability and thus reduces harvest and raises the marginal utility. A forward-looking resource manager will take this effect into account by reducing initial harvest thereby saving more stock to smooth harvest rates through time (consumption smoothing effect). However, a regime shift also lowers the gross return rate thus saving the resource stock becomes a poorer investment. A forward looking resource manager will take this into account by increasing initial harvest and saving less stock for the future (the investment effect). Both the risk reduction effect and the consumption smoothing effect cause management to be

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Fig. 5. Hazard rate of a potential regime shift.

more precautionary. The investment effect has the opposite impact and causes management to be more aggressive. We characterize conditions under which the investment effect outweighs the other two effects so that a potential regime shift will cause management to be more aggressive overall as compared to the risk-free case. In a numerical simulation with constant relative risk aversion utility and logistic growth, we showed that a potential regime shift can generate more aggressive management for reasonable parameter values. In particular, we showed that when the regime shift reduces the carrying capacity of the renewable resource then the optimal management tends to be more aggressive with a potential regime shift. In contrast, management tends to be more precautionary with a reduction in the intrinsic growth rate. While the model incorporates a number of effects, we readily acknowledge that there are additional interesting issues that we have not considered. For example, we considered a regime shift that affects the growth function but not one that directly affects the utility function, or has a simultaneous effect on both. We have relied on a relatively simple hazard function rather than explicitly considering the underlying mechanisms that may govern regime shifts. There may be additional management approaches besides simply managing the level of stock that can affect the likelihood of regime shift, such as reduction of pollution or investments in environmental improvements. In addition, it may be possible to learn about the probability of crossing a threshold or to get early warning signals of impending regime shift (Scheffer et al., 2009; Biggs et al., 2009). Finally, we assume that there is a single potential regime shift and that this regime shift is irreversible. In reality there may be many potential regimes with the potential to flip back and forth among regimes. Considerations of this sort would complicate the analysis but would not necessarily add insight into optimal management. Appendix A Derivation of Eqs. (5) and (6). By (2) we can show that 0

0 RS RS RS d ðsRS 1 Þ ¼ λ ðs1 Þ½uðf ðs1 ; xÞÞ  uðf ðs1 ; x  ΔxÞÞ 0 RS RS þλðsRS 1 Þ½u ðf ðs1 ; xÞÞf 1 ðs1 ; xÞ RS u0 ðf ðsRS 1 ; x  ΔxÞÞf 1 ðs1 ; x  ΔxÞ RS RS ¼ λ0 ðsRS 1 Þ½uðf ðs1 ; xÞÞ  uðf ðs1 ; x ΔxÞÞ RS 0 RS 0 RS þλðsRS 1 Þf 1 ðs1 ; xÞ½u ðf ðs1 ; xÞÞ  u ðf ðs1 ; x  ΔxÞÞ 0 RS RS RS þλðsRS 1 Þu ðf ðs1 ; x  ΔxÞÞ½f 1 ðs1 ; xÞ  f 1 ðs1 ; x  ΔxÞ: n Because limΔx-0 sRS 1 ¼ s1 , for the three terms on the right hand side, we have that

RS

 ∂f ðsn1 ; xÞ uðf ðsRS 1 ; xÞÞ  uðf ðs1 ; x  ΔxÞÞ ¼ u0 f sn1 ; x Δx-0 ∂x Δx
n  ∂hn1 0 ; ¼ u h1 ∂x

lim

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B. Ren, S. Polasky / Journal of Economic Dynamics & Control 40 (2014) 195–212 0 RS

 ∂f ðsn1 ; xÞ u0 ðf ðsRS 1 ; xÞÞ u ðf ðs1 ; x  ΔxÞÞ ¼ u″ f sn1 ; x ∂x Δx
n  ∂hn1 ; ¼ u″ h1 ∂x

lim

Δx-0

lim

Δx-0

RS f 1 ðsRS ∂Rn 1 ; xÞ f 1 ðs1 ; x  ΔxÞ : ¼ ∂x Δx

Then (5) is solved as 0

n  ∂hn1

n  ∂hn1

n  ∂Rn d ðsRS 1 Þ þ λ sn1 Rn u″ h1 þ λ sn1 u0 h1 : ¼ λ0 sn1 u0 h1 Δx-0 Δx ∂x ∂x ∂x

lim

From (5) we can rearrange terms to find that 0 u0 ðh1 Þλðsn1 ÞRn d ðsRS 1 Þ ¼ Δx-0 Δx x n

lim

n

n λ0 ðsn1 Þh1 x ∂hn1 x ∂h x ∂Rn γ n 1 þ n n n n R ∂x λðs1 ÞR h1 ∂x h1 ∂x

!

n

¼

u0 ðh1 Þλðsn1 ÞRn ð θξh  γξh þξR Þ: x 0

n

Because u0 ðh1 Þλðsn1 ÞRn =x 40, aggressive management ðd 4 0Þ implies γξh þ θξh o ξR . Proof of Proposition 1. We first prove the second inequality of this proposition. Define an operator T as ðTwÞðs; x; ΔxÞ ¼ max fuðf ðs; xÞ yÞ y A Γðs; xÞ

þ βð1 λðyÞÞwðy; x; ΔxÞ þβλðyÞVðy; x  ΔxÞg:

ðA:1Þ

As noted in the classical model, V is continuous and strictly increasing in both arguments. Then following standard arguments in literature (Theorem 4.6 of Stokey et al., 1989), T has a unique fixed point Wðs; x; ΔxÞ ¼ lim ðT n wÞðs; x; ΔxÞ

ðA:2Þ

n-1

and Wðs; x; ΔxÞ is continuous, strictly increasing in s and x, and strictly decreasing in Δx, as inherited from u and V. Also by (9), we can verify that ðTVÞðs; x  ΔxÞ ¼ Vðs; x  ΔxÞ given Δx ¼ 0. Because the fixed point is unique for any ðs; x; ΔxÞ A S  X , we know that Wðs; x; 0Þ ¼ Vðs; xÞ. Then Wðs; x; ΔxÞ oVðs; xÞ is established because Wðs; x; ΔxÞ is continuous and strictly decreasing in Δx. To prove the first inequality of this proposition, define another operator L as ðLwÞðs; x; ΔxÞ ¼

max

y A Γðs; x  ΔÞ

fuðf ðs; x  ΔxÞ  yÞ

þ βð1 λðyÞÞwðy; x; ΔxÞ þβλðyÞVðy; x  ΔxÞg:

ðA:3Þ

The standard arguments from Stokey et al. (1989) ensure that L has a unique fixed point, and according to (9) we can verify that Vðs; x  ΔxÞ ¼ ðLVÞðs; x  ΔxÞ for any ðs; x; ΔxÞ A S  X , or equivalently Vðs; x  ΔxÞ ¼ lim ðLn wÞðs; x; ΔxÞ:

ðA:4Þ

n-1

By (10) Γðs; x  ΔxÞ  Γðs; xÞ, then (A.1) and (A.3) imply that for any w it is true that Tw 4 Lw. Thus lim ðT n wÞðs; x; ΔxÞ 4 lim ðLn wÞðs; x; ΔxÞ

n-1

n-1

and the desired result follows from (A.2) and (A.4).

□

Continuous differentiability of Wð; x; ΔxÞ. Although the continuity and monotonicity of W are established in the proof of Proposition 1 with standard assumptions, to further establish the concavity and differentiability of Wð; x; ΔxÞ, we need stronger assumption on the hazard function λ. Assumption 3c. Given any twice differentiable function w which is weakly concave on S, jλ0 ðsÞj þ jλ″ðsÞj is bounded above by a function M w ðsÞ : S-R þ , such that wλ ðsÞ ¼ ð1  λðsÞÞwðsÞ þ λðsÞVðs; x  ΔxÞ is strictly concave in s for arbitrary ðx; ΔxÞ A X . The intuition behind Assumption 3c is as the following. Let V 0 and V″ denote the first and second derivative9 of V with respect to s. For any s A S, we know w″λ o0 if and only if ð1  λÞw″ þ λV″ o2λ0 ðw0 V 0 Þ þ λ″ðw  VÞ: Given that w and V are twice differentiable on compact set S, we know that w V and w0  V 0 are bounded. Because w″ r0, and V″ o0 as noted in the classical model, the last equation holds when Mw is sufficiently small. When w is close to V and w0 is close to V 0 , then the function Mw can be large. 9

Santos (1991, 1992) established the C2 differentiability of the value function under standard assumptions which are satisfied in our model.

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209

With Assumption 3c, we know that the weak concavity of any function wð; x; ΔxÞ on compact set S is inherited by ðTwÞð; x; ΔxÞ (defined in the proof of Proposition 1) as strict concavity, thus the fixed point Wð; x; ΔxÞ is strictly concave. Then, for any ðx; ΔxÞ A X consider an arbitrary s0 in the interior of S, because u satisfies the Inada condition, we know qðs0 ; x; ΔxÞ A intΓðs0 ; xÞ. Thus according to the envelope theorem of Benveniste and Scheinkman (1979) (or Stokey et al., 1989, Theorem 4.11), Wð; x; ΔxÞ is continuously differentiable on the interior of S. Proof of Lemma 3. We first show that the policy function qðs; x; ΔxÞ converges to gðs; xÞ independent of s as Δx converges to 0. Then use this result to prove the convergence of sRS to sn. From the proof of Proposition 1 we know Wðs; x; 0Þ ¼ Vðs; xÞ, then according to (11) and (18) qðs; x; 0Þ ¼ gðs; xÞ. Thus we only need to show qðs; x; ΔxÞ-qðs; x; 0Þ uniformly. Because Vðs; xÞ is continuous on S  X and strictly increasing in x, by Dini0 s theorem Vð; x  ΔxÞ-Vð; xÞ uniformly as Δx-0, thus Proposition 1 ensures Wð; x; ΔxÞ-Vð; xÞ uniformly. With the uniform convergence of value functions, we can use Stokey et al. (1989, Theorem 3.8) to establish the uniform convergence of policy function and only check the assumptions of the theorem. Consider a sequence fxn g which converges from below to x. Define functions ωn and ω on S  Γðs; xÞ as ωn ðs; yÞ ¼ uðf ðs; xÞ  yÞ þ βð1  λðyÞÞWðy; x; x  xn Þ þ βλðyÞVðy; xn Þ; ωðs; yÞ ¼ uðf ðs; xÞ  yÞ þ βð1 λðyÞÞWðy; x; 0Þ þ βλðyÞVðy; xÞ: We know ωn ðs; yÞ-ωðs; yÞ uniformly because of the uniform convergence of value functions. By definition (18) qðs; x; x xn Þ ¼ fy A Γðs; xÞjωn ðs; yÞ ¼ Wðs; x; x  xn Þg; qðs; x; 0Þ ¼ fy A Γðs; xÞjωðs; yÞ ¼ Wðs; x; 0Þg: Then by Stokey et al. (1989, Theorem 3.8), qðs; x; x  xn Þ-qðs; x; 0Þ uniformly if Γðs; xÞ is nonempty, compact, convex and continuous at any ðs; xÞ A S  X, and ωn ðs; yÞ and ωðs; yÞ are continuous on S  Γðs; xÞ and strictly concave in y. The assumptions on Γ are satisfied by definition (10). The continuity of ωn and ω is ensured by the continuity of W and V. And Assumption 3c guarantees that ωn and ω are strictly concave in y. Thus the desired assumptions are satisfied, and the uniform convergence of policy functions is established. Now we prove the convergence of sRS to sn. First notice that (12) ensures that gðs; xÞ is strictly increasing in s. This implies that the convergence to sn from any s0 A S is monotonic. Then consider arbitrary ɛ 4 0 and Nɛ ðsn Þ ¼ ðsn ɛ; sn þ ɛÞ, for any sL and sH in Nɛ ðsn Þ and sL o sn o sH , we have sL o gðsL ; xÞ osn ogðsH ; xÞ osH : For all s2 = N ɛ ðsn Þ, because qðs; x; ΔxÞ-gðs; xÞ uniformly as Δx-0, we can pick some δ1 4 0 such that for any jΔxj oδ1 , jgðs; xÞ  qðs; x; ΔxÞj o min fjgðsL ; xÞ sL j; jgðsH ; xÞ  sH jg. Then the following is true jqðs; x; ΔxÞ  sj

Z jgðs; xÞ  sj  jgðs; xÞ qðs; x; ΔxÞj Z minfjgðsL ; xÞ  sL j; jgðsH ; xÞ sH jg jgðs; xÞ  qðs; x; ΔxÞj

40

Thus for all s2 = N ɛ ðsn Þ, there exists δ1 40 such that jΔxjo δ1 implies jqðs; x; ΔxÞ  sj 4 0. On the other hand, we can pick δ2 4 0 such that for any jΔxjo δ2 , the following is true: sL o qðsL ; x; ΔxÞ o sn o qðsH ; x; ΔxÞ osH : Thus qðsL ; x; ΔxÞ  sL 4 0 4 qðsH ; x; ΔxÞ sH : Assumption 3c ensures that Wð; x; ΔxÞ is strictly concave. Then applying the theorem of the maximum we know qðs; x; ΔxÞ s is continuous in s. Then there exists some sRS A ðsL ; sH Þ  Nɛ ðsn Þ such that qðsRS ; x; ΔxÞ sRS ¼ 0. Therefore, given x A X, for any ɛ 4 0, there exists some δ ¼ min fδ1 ; δ2 g 4 0 such that jΔxj oδ implies sRS ðx; ΔxÞ A N ɛ ðsn ðxÞÞ and there is no s2 = Nɛ ðsn ðxÞÞ satisfies qðs; x; ΔxÞ ¼ s. This establishes the convergence of sRS ðx; ΔxÞ to sn ðxÞ. □ Proof of Proposition 4. We want to find the condition under which for any x A X it is true that limΔx-0 ðsn ðxÞ sRS ðx; ΔxÞÞ=Δx 40. By Lemma 2, this statement is equivalent to lim

Δx-0

D1 ðsRS ðx; ΔxÞ; x; ΔxÞ 40 Δx

or
 WðsRS ; x; ΔxÞ  VðsRS ; x  ΔxÞ λ0 sn lim Δx Δx-0
n W 1 ðsRS ; x; ΔxÞ V 1 ðsRS ; x  ΔxÞ 4 0: þ λ s lim Δx Δx-0

ðA:5Þ

To simplify the notation, we use sn to denote sn ðxÞ and sRS to denote sRS ðx; ΔxÞ unless otherwise noted. The rest of this proof is organized in steps.

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Step 1: We first consider the first term on the left hand side of (A.5). By (19) Wð; x; ΔxÞ evaluated at steady state sRS is
 uðf ðsRS ; xÞ sRS Þ  βDðsRS ; x; ΔxÞ : W sRS ; x; Δx ¼ 1 β Let ΔsRS ¼ gðsRS ; x  ΔxÞ  sRS , by Lemma 3 we know limΔx-0 ΔsRS ¼ 0. Then use condition (12) we can show that

 V sRS þΔsRS ; x Δx V sRS ; x  Δx
RS 
 ¼ V 1 s þ ΔsRS ; x  Δx ΔsRS þ o ΔsRS
 u0 ðf ðsRS ; x  ΔxÞ  gðsRS ; x  ΔxÞÞΔsRS þ o ΔsRS : ¼ β Also by (9),
 V sRS ; x Δx uðf ðsRS ; x ΔxÞ  gðsRS ; x ΔxÞÞ 1 β β½VðsRS þ ΔsRS ; x  ΔxÞ VðsRS ; x  ΔxÞ þ 1β uðf ðsRS ; x ΔxÞ  gðsRS ; x ΔxÞÞ ¼ 1 β
 u0 ðf ðsRS ; x  ΔxÞ  gðsRS ; x  ΔxÞÞΔsRS þ o ΔsRS : þ 1β ¼

Then WðsRS ; x; ΔxÞ VðsRS ; x  ΔxÞ Δx-0 Δx uðf ðsRS ; xÞ sRS Þ  uðf ðsRS ; x ΔxÞ  gðsRS ; x ΔxÞÞ ¼ lim Δx Δx-0 u0 ðf ðsRS ; x  ΔxÞ  gðsRS ; x  ΔxÞÞΔsRS  lim Δx-0 Δx DðsRS ; x; ΔxÞ :  β lim Δx Δx-0

ð1  βÞ lim

ðA:6Þ

We calculate each term on the right hand side of (A.6) separately lim

Δx-0

lim

Δx-0

lim

Δx-0

uðf ðsRS ; xÞ  sRS Þ  uðf ðsRS ; x  ΔxÞ gðsRS ; x  ΔxÞÞ Δx 
n 
n ΔsRS 0 ; f 2 s ; x þ lim ¼u h Δx-0 Δx u0 ðf ðsRS ; x  ΔxÞ  gðsRS ; x  ΔxÞÞΔsRS Δx
n ΔsRS 0 ; ¼ u h lim Δx-0 Δx
 DðsRS ; x; ΔxÞ WðsRS ; x; ΔxÞ VðsRS ; x  ΔxÞ ¼ λ sn lim : Δx-0 Δx Δx

Substituting last three equations into (A.6) and rearranging terms n

lim

Δx-0

WðsRS ; x; ΔxÞ  VðsRS ; x  ΔxÞ u0 ðh Þf 2 ðsn ; xÞ ¼ : Δx 1  β þ βλðsn Þ

ðA:7Þ

Step 2: Now we consider the second term on the left hand side of (A.5). By (22) W 1 ðsRS ; x; ΔxÞ ¼ u0 ðf ðsRS ; xÞ sRS Þf 1 ðsRS ; xÞ: Also by (13) V 1 ðsRS ; x  ΔxÞ ¼ u0 ðf ðsRS ; x ΔxÞ  gðsRS ; x ΔxÞÞf 1 ðsRS ; x ΔxÞ: Then lim

Δx-0

W 1 ðsRS ; x; ΔxÞ  V 1 ðsRS ; x  ΔxÞ Δx 

n

n
n  ΔsRS 0 : f 2 sn ; x þ lim ¼ u h f 12 s ; x þ f 1 sn ; x u″ h Δx-0 Δx

ðA:8Þ

B. Ren, S. Polasky / Journal of Economic Dynamics & Control 40 (2014) 195–212

211

Eq. (15) implies that sn is continuously differentiable in x, and by Assumption 1b and 1c dsn f ðsn ; xÞ ¼  12 n 40: f 11 ðs ; xÞ dx

ðA:9Þ

Let κðx ΔxÞ be defined with x replaced by x  Δx. The differentiability of sn in x and C2 differentiability of f ensure that limΔx-0 κðx  ΔxÞ ¼ κðxÞ ¼ κ. Consider the risk-free steady state sn evaluated at x  Δx and the risk-free policy function g evaluated at ðsRS ðx; ΔxÞ; x ΔxÞ, we have gðsRS ðs; ΔxÞ; x  ΔxÞ sn ðx ΔxÞ ¼ κðx  ΔxÞðsRS ðs; ΔxÞ sn ðx ΔxÞÞ: Then lim

Δx-0

ΔsRS Δx gðsRS ðs; ΔxÞ; x ΔxÞ  sRS ðs; ΔxÞ Δx ð1  κðx  ΔxÞÞðsn ðx  ΔxÞ  sRS ðx; ΔxÞÞ ¼ lim Δx-0 Δx  sn ðxÞ sRS ðx  ΔxÞ sn ðxÞ  sn ðx ΔxÞ  lim ¼ ð1  κÞ lim Δx-0 Δx-0 Δx Δx  sn ðxÞ  sRS ðx ΔxÞ dsn  : ¼ ð1  κÞ lim Δx Δx-0 dx ¼ lim

Δx-0

n

0

Then substituting the last equation into (A.8) and using (15), (A.9), Ω ¼ h f 11 ðsn ; xÞ=γ and γ ¼  hu″ðhÞ=h ðhÞ, we can show that lim

Δx-0

W 1 ðsRS ; x; ΔxÞ  V 1 ðsRS ; x  ΔxÞ Δx n 
n  u″ðh Þ dsn sn  sRS f 2 s ; x  ð1  κ βΩÞ þ ð1  κÞ lim : ¼ β dx Δx-0 Δx

ðA:10Þ

Step 3: By (15) and (23) lim

Δx-0

f 1 ðsn ; xÞ  f 1 ðsRS ; xÞ 1 D1 ðsRS ; x; ΔxÞ ¼ : lim n Δx Δx u0 ðh Þ Δx-0

Then
 sn sRS f 11 sn ; x lim Δx-0 Δx λ0 ðsn Þ WðsRS ; x; ΔxÞ  VðsRS ; x ΔxÞ lim ¼ n 0 Δx u ðh Þ Δx-0 

λðsn Þ W 1 ðsRS ; x; ΔxÞ  V 1 ðsRS ; x  ΔxÞ : lim n 0 Δx u ðh Þ Δx-0

Substituting (A.7) and (A.10) into the last equation we have  
 βΩ sn sRS Θ dsn  1 κ  f sn ; x  ð1  κ  βΩÞ lim ¼ 1 þ λðsn Þ Δx-0 Δx γ 2 dx where n

Θ¼ 


 1 Rn ¼ f 1 sn ; x ¼ : β

λ0 ðsn Þh

λðsn ÞðRn  1 þ λðsn ÞÞ

Because Ω o0 and 0 o κ o1, then 1  κ βΩ=λðsn Þ 4 0. If a regime shift causes aggressive pre-shift management at the margin, it is true that limΔx-0 ðsn  sRS Þ=Δx 40, or equivalently 
 Θ dsn f sn ; x o ð1  κ  βΩÞ : 1þ γ 2 dx n

Step 4: Linearizing (8) and (14) around ðsn ; h Þ, the system dynamics are determined by the eigenvalues and eigenvectors of matrix ½ð1=β; 1Þ; ðΩ; 1  βΩÞ, and the eigenvalues κ satisfies ð1=β κÞð1  βΩ κÞ þ Ω ¼ 0 n

ðA:11Þ n

Then by (A.9), (A.11) and Ω ¼ h f 11 ðs ; xÞ=γ, we find that n

ð1 κ  βΩÞ

dsn h f 12 ðsn ; xÞ ¼ : dx ðRn  κÞγ

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Then we know ðγ þΘÞ

f 2 ðsn ; xÞ Rn f 12 ðsn ; xÞ o n : n Rn R κ h

Define n

∂h x ∂ðf ðsn ; xÞ sn Þ x f 2 ðsn ; xÞx n ¼ n ¼ n ∂x ∂x h h h n n n ∂R x ∂f ðs ; xÞ x f ðs ; xÞx ξnR ¼ ¼ 1 ¼ 12 n ∂x ∂x hn Rn R ξnh ¼

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