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Renewable Resource Dynamics Under Correlated Environmental

Disturbances: The Case of Competitive Harvesting1

Michele Baggio

Dept. of Economics,

University of Connecticut

Lars J. Olson

Dept. of Agricultural and Resource Economics,

University of Maryland

Abstract

This paper examines long run dynamics of renewable resources under competitive harvest-

ing and autocorrelated growth shocks. We show that small perturbations in the temporal

correlation of disturbances may lead to regime shifts in long run dynamics. Using assump-

tions on the model primitives we characterize when a resource is sustainable in the long

run and we provide suﬃcient conditions for the existence of a unique limiting distribution

for the two–state Markov process. The usefulness of the results is demonstrated with two

classic models of renewable resource growth, the Beverton–Holt and Cushing models.

Keywords: Renewable resource dynamics, Autocorrelated shocks, Competitive harvesting

JEL: Q22, Q57

1Email: Michele Baggio, michele.baggio@uconn.edu; Lars J. Olson, ljolson@umd.edu. This research was

supported by the National Socio-Environmental Synthesis Center (SESYNC) under funding received from

the National Science Foundation DBI-1052875. We thank Erik Lichtenberg for valuable suggestions and

discussions. We also thank Doug Lipton, Ted McConnell, and Mathias Ruth.

July 7, 2017

1. Introduction

Scientists have long recognized that random environmental disturbances have important

consequences for renewable resource allocation (Ricker, 1958). In economic and ecolog-

ical models such disturbances are typically assumed to be identically and independently

distributed (i.i.d.) (Roughgarden, 1975; McGough et al, 2009). There is strong evidence,

however, that resource growth and productivity depend on persistent trends in the climate

and ocean environment. For instance, the interannual El Ni˜no Southern Oscillation, the

Paciﬁc decadal oscillation and the North Atlantic oscillation have been shown to aﬀect a

substantial portion of global ﬁsh production (see e.g., Hollowed et al, 2001; Beamish et al,

2004; Salinger, 2013). Serial correlation in environmental disturbances can lead to syn-

chronous trends in resource productivity (Lawton, 1988; Pimm and Redfearn, 1988) and a

persistent sequence of favorable or unfavorable conditions for population growth can result in

sustained shifts in population ﬂuctuations where the temporal correlation of environmental

disturbances exerts a strong inﬂuence on population variability (Wilmers et al, 2007).

The goal of this paper is to improve our understanding of how correlated environmental

disturbances impact the long run dynamics of renewable resources that are not eﬃciently

managed or where enforcement of property rights is weak. Many of the world’s renewable

resources fall in this category. Global ﬁshing eﬀort has grown 10-fold since 1950, increasing

in both intensity and geographic range (Watson et al, 2013), yet 23% of global catch is from

stocks that are not assessed (Costello et al, 2012) and the vast majority of countries have

received failing grades for compliance with the UN Code of Responsible Fisheries provisions

on management of stocks, ﬂeets, and gear (Pitcher et al, 2008, 2009). Widespread illegal

and unreported ﬁshing erodes the gains from those ﬁsheries that are managed (Agnew

et al, 2009). The extent to which competitive harvesting occurs under weak management is

evidenced by estimates that global ﬁshing rents are almost completely dissipated (Srinivasan

et al, 2012).

In this paper we examine the dynamics of resources stocks under competitive harvesting

and correlated environmental disturbances. The state evolves as a two-dimensional random

dynamic system (Bhattacharya and Majumdar, 2007; Stachurski, 2009). We characterize

1

when a resource stock in this system is sustainable in the long run and how resource sus-

tainability and the support of the marginal limiting distribution of stocks is aﬀected by

autocorrelation in environmental disturbances. We provide suﬃcient conditions for the ex-

istence of a unique limiting distribution for the two-state stock and disturbance Markov

process. When the invariant set has a strictly positive lower bound and if the elasticities

of the welfare, growth and disturbance transition equation satisfy a joint bound then an

”average contraction” condition holds and there exists a unique invariant distribution. The

usefulness of the results is demonstrated using two classic models of renewable resource

growth, the Beverton and Holt (1957) and Cushing (1971) models.

Our work extends the analysis of Mirman and Spulber (1984) who examine the mar-

ket for a renewable resource in the presence of i.i.d. environmental disturbances. They

focus mainly on the case where harvest costs are independent of the stock and equilibrium

harvests are constant. Our paper provides a more complete characterization of long-run dy-

namics for both stock-independent and stock-dependent welfare and for i.i.d. and correlated

disturbances. Our analysis also overlaps the literature on optimal intertemporal resource

allocation under uncertainty. Early work by Spulber (1982) extends Jaquette (1972) and

Reed (1974, 1979)’s (S,s) model of harvesting to the case where disturbances follow a Markov

process. With correlated shocks Spulber shows that the the optimal policy is non-stationary

in the sense that the critical (S,s) inventory levels depend on the most recent outcome of

the environmental disturbance. For the special case of a discrete logistic growth model and

a linear harvest cost function he establishes the convergence of stocks (and harvest) to a

unique invariant distribution and examines how this is aﬀected by economic and biological

parameters. Donaldson and Mehra (1983) extend the classic stochastic growth model of

Brock and Mirman (1972) to accomodate correlated productivity shocks. They examine

the characteristics of optimal consumption and investment policies and establish condi-

tions under which the capital stock converges to a unique invariant distribution. For the

class of CES/constant relative risk averse utility they show that consumption (or harvest)

increases(decreases) with the productivity shock if the preferences for intertemporal substi-

tution are inelastic(elastic). The countervailing incentives induced by a productivity shock

are highlighted by Hopenhayn and Prescott (1992) who indicate that ”a higher [productivity

2

shock] will lead to an increase in consumption in the present as well as in the future; but

since a higher [shock] also implies higher expected productivity of capital in the future, less

capital may be necessary to sustain the higher consumption.” Their paper provides suﬃcient

conditions for optimal investment to be monotone and for the existence of a limiting distri-

bution over capital stocks. In a dynamic optimization model with i.i.d. shocks, Nyarko and

Olson (1994) prove that resource stocks converge to a unique limiting distribution when

the welfare function satisﬁes a complementarity and a single-crossing condition, or when

production is suﬃciently variable.

Our approach to establishing conditions for the existence of a unique limiting distribution

relies on an average contraction theorem from the literature on iterated functions (Barnsley

and Elton, 1988; Diaconis and Freedman, 1999). Stachurski (2003) uses a similar result

due to Loskot and Rudnicki (1995) to prove a central limit theorem and shows how it can

be applied to the stochastic growth model with log utility and Cobb-Douglas production.

The average contraction theorem requires almost surely (a.s.) Lipschitz continuity of the

transition equations that govern the evolution of the state. In the stochastic growth model

with i.i.d shocks, Mitra and Privileggi (2009) provide conditions for Lipschitz continuity

of the capital stock transition equation when the elasticity of marginal product and the

elasticity of marginal utility are uniformly bounded. They also use a contraction condition

to establish convergence to a unique limiting distribution.

Singh et al (2006) analyze the optimal harvest of a renewable resource with costly adjust-

ment to capital when environmental disturbances are multiplicative and follow a Markov

process. Analogous to the result of Donaldson and Mehra (1983), Singh, Weninger and

Doyle suggest that if the consumption smoothing incentive is suﬃciently strong the opti-

mal harvest can vary positively with anticipated growth conditions. McGough et al (2009)

apply techniques developed in the real business cycle literature supplemented by numerical

simulations to analyze the optimal escapement policy for a resource characterized by a multi-

plicative, autocorrelated growth disturbance. Assuming interior solutions and a suﬃciently

small support for disturbances, the optimal policy can be approximated by log-linearizing

the Euler equation around the deterministic steady state. This enables the (approximately)

optimal escapement policy to be represented as a linear, covariance stationary process.

3

Using numerical simulations they ﬁnd that when disturbances are positively (negatively)

correlated escapement(harvest) should be increased following a positive disturbance to take

advantage of higher future (current) productivity.

2. The Model

In each period, t, the stock of a renewable natural resource, yt∈R+is harvested under

free-entry competition. Following the approach of Levhari et al (1981) it is convenient to

represent the competitive equilibrium harvest, ht,as the decentralized solution to:

max

0≤ht≤yt

W(ht, yt),(1)

where Wis a welfare function that reﬂects both the harvest costs that determine supply and

the consumer beneﬁts that deﬁne demand. Wtypically depends on ythrough the eﬀect of

the stock on harvest costs. The terms free-entry competition, equilibrium and open-access

are used interchangeably to refer to the representation in (1).

The welfare function is assumed to satisfy the following properties throughout the paper.

We refer to these collectively as assumption W. Subscripts denote partial derivatives.

•W-1. W(h, y) is strictly increasing in y.

•W-2. W(h, y) is strictly concave in h.

•W-3. W(h, y) is C2on R2

++.

•W-4. There exists some ye>0 such that Wh(0, y)>0 for all y > ye.

•W-5. Whh(h, y) + Why(h, y)≤0 for all 0 ≤h≤y.

The competitive harvest from yis H(y) = arg max W(h, y) subject to 0 ≤h≤y,

and escapement or investment is X(y) = y−H(y).Under W.1-W.3, H(y) and X(y)

are continuous functions. Whenever both harvest and escapement are strictly positive H

and Xare continuously diﬀerentiable and the implicit function theorem implies dH/dy =

4

−Why(h, y)/Whh(h, y) and dX/dy = (Whh(h, y) + Why(h, y))/Whh(h, y). Without W.4,

Wh<0 for all (h, y) and competitive harvesting is uneconomical. Hence, W.4 is a minimal

assumption for positive harvests to occur from some stock level. Under W.4, if a positive

harvest is economical from yethen a positive harvest is economical from larger stocks but

not necessarily from smaller stocks. We assume that yeis less than the maximum sustain-

able stock, deﬁned below in F.7. It is straight-forward to show that W.2-W.5 imply that

competitive escapement X(y) is continuous and increasing in y.

The resource grows according to a stochastic growth or stock-recruitment function de-

noted by yt+1 =F(xt, st+1) where the stock next period is a measurable function of es-

capement and a random environmental disturbance that aﬀects resource productivity. The

environmental disturbance sis assumed to belong to a compact interval S⊂[s, s] and to

evolve according to a measurable map gsuch that st+1 =g(st, εt+1),where εt

IID

∼φ, with

distribution φ∈P(Φ), Φ ∈ß(R),on an underlying probability space (Ω,z, ℘).

We assume Fsatisﬁes the following, referred to collectively as assumption F:

•F-1. F(x, s) is strictly increasing in xfor all s.

•F-2. F(x, s) is strictly concave in xfor all s.

•F-3. F(x, s) is C2on R2

+.

•F-4. Fx(0, s)>1 for all s.

•F-5. F(0, s) = 0, while F(x, s)>0 if x > 0 for all s.

•F-6. There exists an Asuch that F(A, s)≤Afor all s.

•F-7. F(x, s) is increasing in sfor all x.

Assumption F.4 ensures that the resource is biologically sustainable in the absence of

harvesting under every environmental disturbance. F.2 and F.6. imply the existence of a

maximum sustainable stock, A. Without loss of generality assume that y0∈(0, A].Deﬁne the

the period tstate zt= (xt, st).The state space, Z= [0, A]×S, is a complete, separable metric

space with metric, d. Let Zto be the product σ-algebra of Z. Together, the resource growth

5

function, competitive escapement, and the transition for environmental disturbances deﬁne

a two-state Markov process. Escapement follows xt+1 =ψ(xt, st, εt+1),with ψ(x, s, ε) =

X(F(x, g(s, ε))) and the state evolves according to the Markov transition rule:

zt+1 =

xt+1

st+1

=

ψ(xt, st, εt+1)

g(st, εt+1)

= Ψ(zt, εt+1).(2)

The growth function F, equilibrium escapement, X, and shock transition ginduce a

mapping P:Z×Z→[0,1] deﬁned by P(x, s;A×B) = φ({ε:X(F(x, s0)) ∈A, g(s, ε)∈B})

where A×B∈Z. That is, Pdeﬁnes the probability of moving from an initial state

z= (x, s) to z0= (x0, s0)∈A×B, where x0=X(F(x, s0)) and s0=g(s, ε).Let πt(z) be

a probability distribution on Z. Then Pmaps πtto a distribution over zt+1 according to

πt+1(A×B) = (πtP)(A×B) = Rπt(dz)P(z;A×B).For each tdeﬁne the t-period transition

kernel Pt(z;A×B) = RPt−1(z;A×B)P(z0;dz), P 1=P. The distribution π(z) is said to

be invariant or stationary if πP =π. If µP t→πas t→ ∞ for all µ∈P(Z) then πis a

unique invariant distribution and Pis globally stable.

3. Long Run Stock Dynamics

This section examines long run dynamics under competitive harvesting. The resource

is said to be sustainable from xif X(F(x, s)) ≥x. From W.5, F.1 and F.8 it follows

that if y0≥F(x, s) then xt≥xa.s. for all t. The weakest restriction that provides for

sustainability from some stocks is:

•A.1. There exists some x > 0 such that Wh(F(x, s)−x, F (x, s)) ≤0.

Recall that sis the worst environmental disturbance for growth. A.1 ensures that com-

petitive harvests are small enough to allow net growth in the stock under any environmental

disturbance when escapement is x.

Proposition 1. Under assumptions W, F, and A.1, if y0≥F(x, s) then xt≥x, a.s. for

all t.

6

Proof. First, suppose y0=F(x, s) Under A.1 the marginal returns from harvesting

F(x, s)−xare non-positive. Since Wis strictly concave, competitive harvests from y0are

less than or equal to F(x, s)−xand x0=F(x, s)−H(F(x, s)) ≥x, a.s. F.1 and F.7 imply

F(x0, s1)≥F(x, s) = y0.Since X(y) is increasing in yunder W.5, it follows that x1≥x0,

a.s. An induction argument completes the proof.

Proposition 1 allows that extinction may occur from small stocks and that the limiting

behavior of the distribution of resource stocks may be sensitive to initial conditions.

Under competitive harvesting, avoidance of extinction from any initial stock requires

harvest costs that depend on the stock. Otherwise the equilibrium harvest is constant

and immediate extinction results from initial stocks below this constant. The next results

characterize conditions that determine (a) when extinction is avoided from all x > 0, and

(b) when competitive harvesting from small stocks results in extinction.

•A.2. There exists a k > 0 such that Wh(F(x, s)−x, F (x, s)) <0 for all 0 < x < k.

Proposition 2. Assume W, F, and A.2. From any y0>0 there exists an x > 0 such

that xt≥xa.s. for all t.

Proof. Let F(x) = F(x, s).If A.2 holds and if y0< F (k) then analogous arguments to

the proof of Proposition 1 imply that xt≥F−1(y0) a.s. for all t. If y0≥F(k) then X(y)

increasing in yand A.2 imply that xt≥ka.s. for all t. Hence, xt≥min[F−1(y0), k] a.s. for

all t.

Assumption A.2 guarantees that as escapement increases away from zero, revenue from

harvesting net growth declines faster than the reduction in costs due to an increase in the

stock. Intuitively this will be true if it is not proﬁtable for competitive harvests to increase

in a direction (weakly) greater than net growth as the stock increases in the direction of

recruitment. When Wh(0, y)≤0 marginal returns from harvesting are negative and it is

not economical to harvest when stocks are very close to zero.

The next result characterizes when competitive harvesting drives the resource arbitrarily

close to extinction.

7

•A.3. There exists an ξ > 0 such that Wh(F(x, s)−x, F (x, s)) >0 for all 0 < x < ξ

Proposition 3. Assume W, F and A.3. If y0< F (ξ, s) then lim inftxt= 0,a.s. and

the resource becomes extinct or arbitrarily close to extinction.

Proof: Suppose the contrary. Then there exists some 0 < x0< ξ and a bx > 0 such that

lim inftxt=bx, a.s. For this to hold it must be that X(F(x, s)) ≥xfor some 0 < x < ξ.

This cannot hold under A.3 and W.2.

To illustrate Propositions 1-3, let W(h, y) = ph −αh2

2y,where p<αensures the entire

stock is not harvested at once. Under the worst environment, marginal returns from har-

vesting net growth are Wh(F(x, s)−x, F (x, s)) = p−α(F(x,s)−x)

F(x,s)=p−α+αx

F(x,s). Hence,

A.1 and A.2 are satisﬁed if Fx(0, s)>α

α−p,while A.3 holds if the inequality is reversed. For

this speciﬁcation the economic intuition behind the propositions is as follows. Equilibrium

harvests in the worst environment are H(F(x, s)) = pF (x,s)

α.The rate of increase in harvest

as xmoves away from zero is pFx(0,s)

α.To avoid extinction this needs to be less than the

net intrinsic growth rate Fx(0, s)−1.Hence, if Fx(0, s)>α

α−pthe stock is sustained under

competitive harvesting.

Application 1: The Beverton-Holt model with correlated disturbances. The

classic Beverton and Holt (1957) model is widely used to represent the growth of renew-

able resources and there is substantial evidence of autocorrelated disturbances for a large

number of ﬁsh species whose growth is ﬁt using the Beverton-Holt model (Thorson et al,

2014). The Beverton-Holt growth function can be expressed as F(x, s0) = s0x

(1+x·(s0−1)/A).

This speciﬁcation is convenient because s0=Fx(0, s0) and F(A, s) = Aso s0and Adenote

intrinsic growth and the maximum sustainable stock, respectively. We assume shocks are

correlated and follow st+1 =β+ρst+εt+1, with ε∈[ε, ε] and |ρ|<1.Deﬁne s(ρ) = β+ε

1−ρand

s(ρ) = β+ε

1−ρ.Assume s(ρ)≥s0≥s(ρ)>1.The bounds on s0also bound st.The inequality

s(ρ)>1 ensures Fx(0, s0)>1 for all possible realizations of s0, i.e., in the absence of har-

vesting the resource can biologically sustain itself under the worst environmental outcomes.

For purposes of illustration let competitive harvests be a piecewise linear function of the

stock H(y) = Max[0, M in[b+(1 −θ)y, y]]. This holds if W(h, y) = ph −αh2

yor if W(h, y) =

8

ph −(ech −1)e−ay,i.e. if ﬁshers are price takers with quadratic or exponential harvest costs

and demand is perfectly elastic. Under positive equilibrium harvests the Markov transition

function for the state is:

xt+1

st+1

=

max[0,−b+θA(β+ρst+εt+1)xt/(A+ (−1 + β+ρst+εt+1)xt)]

β+ρst+εt+1

(3)

First, suppose b= 0.If ρ < 1−θ(β+ε) then extinction occurs asymptotically from all

initial stocks. If 1 −θ(β+ε)<ρ<1−θ(β+ε) then the stock may become arbitrarily close

to extinction with positive probability. Finally, if 1 −θ(β+ε)< ρ, A.2 is satisﬁed and the

stock is bounded away from zero. The long run outcome for the stock clearly depends on

the degree of autocorrelation in environmental disturbances. Extinction is more(less) likely

under negatively(positively) correlated disturbances. When b > 0 immediate extinction oc-

curs from low stocks, even if the intrinsic biological growth rate is high. From high initial

stocks, the stock may converge to a limiting distribution bounded away from zero or it may

converge to extinction, depending on parameter values. Since ψx(0, s, ε) represents the net

growth rate of the harvested resource from low stocks it follows directly that, given s, posi-

tive autocorrelation enhances the prospects for conservation while negative autocorrelation

diminishes them.

To further illustrate, let W(h, y) = ph −(ech −1)e−ay .Note that W(0, y) = 0 for all y,

and Wsatisﬁes assumption W provided y > −1

aln(p

c) and c > a, which are necessary for

W.4 and W.5, respectively. For this speciﬁcation interior harvests satisfy H(y) = 1

c(ln(p

c) +

ay).Figures 1 and 2 depict how changes in ρcan aﬀect transition dynamics and long run

outcomes. Blue curves depict the Beverton-Holt growth function while red curves depict

the stock transition equation under competitive harvesting. Solid lines are the outcomes

under s(ρ) and dotted lines are the outcome under s(ρ).

In Figure 1, (β, ε, ε, A, p, c, a) = (1,1,5,100,0.04,0.04,0.02).Since p=c > a, A.2 is

satisﬁed and Proposition 2 applies if Fx(0, s(ρ)) = β+ε

1−ρ>c

c−a. In Figure 1a, autocorrelation

is high, ρ= 0.5, and extinction is avoided from low stocks. In Figure 1b, ρ= 0 and

shocks are not correlated. For this case A.3 holds, Proposition 3 applies, and the limiting

9

distribution has a support where that greatest lower bound on the resource stock is zero.

Figure 2 depicts a case where long run outcomes depend on both initial conditions

and the temporal correlation between environmental disturbances. Let (β, ε, ε, A, p, c, a) =

(3,1,3,100,1.21825,0.1,0.02). In this case p>cand A.3 holds. The entire stock is

harvested when stocks are low. When the autocorrelation in environmental disturbances

is high, A.1 is satisﬁed and the resource is sustainable from high initial stocks (Figure 2a,

ρ= 0.5). In Figure 2b, ρ= 0.2 and the harvested resource transition function is just

tangent to the 45 degree line at x= 25.From initial escapement below x= 25,the resource

is harvested to extinction. If the initial escapement is above 25 the resource converges

to a support bounded by [25,42.9729].The transition function tangency creates a regime

switching point where an inﬁnitesimal negative perturbation in autocorrelation causes A.1

to fail and leads the resource to be harvested to extinction from all initial stocks, as shown

in Figure 2c, where ρ= 0.

Given that Sis compact, the existence of an invariant distribution can be established

if the the Markov transition Ψ is either continuous (Stokey et al, 1989, Theorem 12.10)

or monotone (Hopenhayn and Prescott, 1992, Corollary 4). Our next results focus on

suﬃcient conditions for this invariant distribution to be unique. The lower and upper

bounds on the stochastic transition function for xtare deﬁned by Xm(x) = X(F(x, s)) and

XM(x) = X(F(x, s)); Xm(x) and XM(x) have at least one ﬁxed point, at the origin. Deﬁne

xm= min {x|Xm(x) = x},xM= max {x|XM(x) = x}, ym=F(xm, s) and yM=F(xM, s).

Observe that xm>0 under A.2. Without loss of generality assume xM< A. When A.2

holds, arguments similar to those in Nyarko and Olson (1991) can be used to show that the

set [0, xm)∪(xM, A] is transient and b

Z= [xm, xM]∪Sis an invariant set, although it need

not be the minimal invariant set.

To prove there is a unique invariant distribution requires some regularity on Ψ.Assume:

•A.4. Ψ(z, ε) is Lipschitz continuous ε-a.s. with Lipschitz constant

λ(ε) = sup

z6=z0∈

b

Z

d(Ψ(z, ε),Ψ(z0, ε))

d(z, z0).

Recall dis a metric on Z. Our results on the existence of a unique invariant distribution

rely on the following theorem.

10

Figure 1: Transition dynamics and long run outcomes as function of temporal correlation, (a) ρ= 0.5 and (b) ρ= 0. Blue

curves depict the Beverton-Holt growth function; Red curves depict the stock transition equation under competitive harvesting.

Solid lines denote the outcomes under s(ρ) and dotted lines are the outcome under s(ρ).

11

Figure 2: Transition dynamics and long run outcomes as function of initial conditions and temporal correlation ρ: (a) high

degree of autocorrelation; (b) medium degree of autocorrelation; (c) uncorrelated environmental disturbances. Blue curves

depict the Beverton-Holt growth function; Red curves depict the stock transition equation under competitive harvesting. Solid

lines denote the outcomes under s(ρ) and dotted lines are the outcome under s(ρ).

12

Theorem. Assume W, F, A.2 and A.4.

(a) (Barnsley and Elton, Theorem 1, 1988; Diaconis and Freedman, Theorem 1.1, 1999).

Suppose that Eφd(Ψ(z0, ε), z0)<∞for some z0∈b

Z, Eφλ(ε)<∞and Eφlog λ(ε)<0.

Then the Markov process induced by Ψ has a unique invariant distribution, π. For all tand

z∈b

Z, d(δzPt, π)≤KzRtfor some constants 0 < Kz<∞and 0 < R < 1,and δis the

indicator function.

(b) (Barnsley and Elton, Theorem 2, 1988; Steinsaltz, Proposition 2, 1999). Suppose for

some q > 0, Eφ[d(Ψ(z, ε), z)q]<∞and for some q > 0 sup Eφ

z6=z0∈

b

Zd(Ψ(z, ε),Ψ(z0, ε))

d(z, z0)q<1

then the Markov process induced by Ψ converges to a unique invariant distribution, π.

The main requirement of the theorem is that the Markov transition Ψ is contracting

on average. Part (b) weakens the contraction condition in (a) by moving the supremum

outside the expectation but it imposes a stronger moment condition involving the power

γrather than the logarithm. The theorem also requires Eφd(Ψ(z0, ε), z0)<∞.This holds

if average growth is bounded from some (x, s) pair. It is satisﬁed in our context as Ψ is

uniformly bounded on Zunder F.6 and the assumption that Sis compact. The condition

Eφλ(ε)<∞bounds (expected) marginal growth. In the diﬀerentiable version of our model

this translates to bounds on Xy, Fx, Fsand gson the subset deﬁned by b

Z. Local or n-step

versions of the average contraction conditions of the theorem are provided by Steinsaltz

(1999); Jarner and Tweedie (2001); Stenﬂo (2012) and Bhattacharya and Majumdar (2007,

Section 3.7). These are more general and allow the process to be non-contracting initially

as long as it eventually transitions to a set where the process is contracting on average.

Let dbe the sup norm. When Xand gare continuously diﬀerentiable the theorem can

be restated in terms of the primitives of our model as:

Proposition 4. Assume W, F, A.2 and A.4. Assume X(y) is C1on [ym, yM] and g(s, ε)

is C1in son [s, s].If

Eφsup

(x,s)∈

b

Z

[Xy(F(x, g(s, ε)))Fx(x, g(s, ε)), Xy(F(x, g(s, ε)))Fs(x, g(s, ε))|gs(s, ε)|,|gs(s, ε)|]<∞

and

Eφlog sup

(x,s)∈

b

Z

[Xy(F(x, g(s, ε)))Fx(x, g(s, ε)), Xy(F(x, g(s, ε)))Fs(x, g(s, ε))|gs(s, ε)|,|gs(s, ε)|]<0

,

13

then the conclusions of Theorem(a) hold and the Markov process induced by Ψ converges

to a unique invariant distribution.

To apply Proposition 4 to common speciﬁcations of harvesting renewable resources under

free-entry competition it is useful to consider a homeomorphism of the state space and

invoke the continuous mapping theorem. Deﬁne ext= log xt,est= log st, and let e

Z=

[ log xm,log xM]×[log s, log s].For any continuously diﬀerentiable function f(x), denote

the elasticity of fwith respect to xby ξf

x=fxx/f. When ψand gare C1the Lipschitz

constant for the transformed system is e

λ= sup[ξψ

x,|ξψ

s|,|ξg

s|] = sup[ξX

yξF

x, ξX

yξF

s0|ξg

s|,|ξg

s|].

To express ξX

yin terms of elasticities of the welfare function it is convenient to deﬁne

U(x, y) = W(y−x, y).When competitive equilibrium harvests are interior it follows that

ξX

y=−Uxyy/(Uxxx) = −ξUx

y/ξUx

x.

Proposition 5. Assume W, F, A.2, A.4 and s > 0. Assume X(y) is C1on [ym, yM] and

g(s, ε) is C1in son [s, s].Suppose

Eφsup

(ex,es)∈

e

Z

[sup[ξX

yξF

x, ξX

yξF

s0|ξg

s|,|ξg

s|]<∞and Eφlog sup

(ex,es)∈

e

Z

sup[ξX

yξF

x, ξX

yξF

s0|ξg

s|,|ξg

s|]<0.

Then the Markov process induced by Ψ converges to a unique invariant distribution.

Proof. A.2 ensures xm>0.Together with s > 0 this guarantees that the trans-

formed state (ext,est) is deﬁned on a bounded space, e

Z, and evolves according to ext+1 =

log ψ(eext, eest, εt+1) and est+1 = log g(eest, εt+1).In addition, the transformed system satisﬁes

Eφd(ez1(ez0, ε),ez0)<∞for some ez0∈e

Z. Strict monotonicity of the log transformation en-

sures a continuous inverse transformation from eztto zt.The continuous mapping theorem

(Mann and Wald, 1943) implies that if eztconverges to a unique invariant distribution, so

does zt.The proof is completed by applying the average contraction condition of Proposition

4 to the transformed system.

Proposition 5 requires that the Lipschitz constant associated with elasticities deﬁned

by the system be bounded and satisfy the average contraction condition. This Lipschitz

constant is a product of elasticities of marginal welfare, the elasticity of the resource growth

function, and the elasticity of the transition equation for environmental disturbances.

14

To conclude, we demonstrate the usefulness of Proposition 5 using two classic models of

renewable resources: the Beverton–Holt (1957) and Cushing (1971) growth functions. As

before, we assume linear and strictly interior escapement/investment X(y) = θy, 0 < θ < 1.

Recall, this can be generated by a welfare function that takes one of the functional forms

W(h, y) = ph −αh2

2ywith p < α, or W(h, y) = ph −(ech −1)e−ay with p=c > a, where θ=

α−p

αor θ=c−a

c,respectively. From this, we obtain ξψ

x(ε) = Fx(x, g(s, ε))x/F (x, g(s, ε))

and ξψ

s(ε) = Fs0(x, g(s, ε))gs(s, ε)s/F (x, g(s, ε)).

Application 1 (Beverton–Holt continued). The relevant elasticities for the Beverton–

Holt growth function are ξψ

x(ε) = A

A+ (g(s, ε)−1)xand ξψ

s(ε) = sgs(s, ε)(A−x)

g(s, ε)(A+ (g(s, ε)−1)x).

Since the resource is biologically sustainable in the absense of harvesting, g(s, ε)≥s > 1.

A.2 holds if θs > 1. This implies xm>0.It follows that ξψ

x(ε)<1 on b

Z. It only remains

to impose |ξg

s(ε)|<1.If this holds then |ξψ

s(ε)|<1 and the system converges to a unique

limiting distribution. The condition |ξg

s(ε)|<1 is satisﬁed for the disturbances in (3) with

|ρ|<1 and β+ε > 0.

Application 2. The Cushing (1971) model with correlated disturbances. The

resource growth function proposed by Cushing takes the form:

yt+1 =st+1 xb

t(4)

where 0 < b < 1 expresses density-dependence in recruitment. It is easy to check that (4)

satisﬁes assumptions F. We assume the transition function for environmental distubances

in given by:

st+1 =Bsρ

teεt+1 (5)

where |ρ|<1, and {εt}∞

t=1 is a sequence of i.i.d. random shocks taking values on a compact

set Σ = [ε, ε] with zero mean and variance σ2

ε.This can be expressed as the ﬁrst-order

autoregressive process: ln st+1 =β+ρln st+εt+1,where β= ln B. The Cushing model in

(4) and (5) is analogous to the speciﬁcation of production in the stochastic growth model

of Danthine et al (1983), with the distinction that harvesting in our model is the result

15

of free-entry competition with a stock-dependent welfare function, whereas in their model

consumption solves a stochastic dynamic programming problem with utility dependent only

on consumption.

Let s= exp β+ε

1−ρ>0 and s= exp β+ε

1−ρ.Straight-forward derivations show that xm=

(θs)1

1−b=θ1

1−bexp β+ε

(1−b)(1−ρ)>0 and xM= (θs)1

1−b=θ1

1−bexp β+ε

(1−b)(1−ρ)bound the sup-

port of the limiting marginal distribution for x. In addition,

max[ξX

yξF

x(ε), ξX

yξF

s0|ξg

s(ε)|,|ξg

s(ε)|] =

max[b, |ρ|]<1.Hence, Proposition 5 applies and the Markov process (xt, st) converges to a

unique invariant distribution on [xm, xM]×[s, s].

The impact of a change in the autocorrelation of environmental disturbances is imme-

diately apparent. An increase in ρshifts the support of the limiting distribution of shocks

upwards, and this shift is translated onto the support of the limiting distribution of re-

source stocks. Negatively correlated disturbances lead to smaller bounds on the support of

the limiting stock distribution, while positively correlated disturbances increase the bounds.

Following the approach in Danthine et al (1983) it can also be shown that the long run

expected value of the resource stock is:

lim

t→∞ E[xt] = θ1

1−bB1

(1−b)(1−ρ)lim

t→∞ E1

2! t−1

X

r=0

br

t−r−1

X

τ=0

ρt−r−1−τετ+1!2

+

1

4! t−1

X

r=0

br

t−r−1

X

τ=0

ρt−r−1−τετ+1!4

+. . .

(6)

which is increasing in ρ.

Remark: With minor modiﬁcations this analysis of the Cushing model can be extended

to any welfare function that generates an investment policy X(y) = θyν,ν≤1.

4. Concluding Remarks

In this paper we examined how correlated environmental disturbances impact the long

run dynamics of renewable resources that are not eﬃciently managed. Our contribution

provides a more complete characterization of long-run dynamics for both stock-independent

16

and stock-dependent welfare and for i.i.d. and correlated disturbances. In general, the

long run outcome for the stock depends on the degree of autocorrelation in environmental

disturbances and may be sensitive to initial conditions. Small perturbations in the temporal

correlation of disturbances may lead to regime shifts in long run dynamics. An average

contraction theorem is used to provide suﬃcient conditions for the existence of a unique

limiting distribution. We relate these conditions to the elasticity of marginal welfare, the

elasticity of resource growth and the elasticity of the disturbance transition. The usefulness

of the results is demonstrated using two classic models of renewable resource growth, the

Beverton–Holt and the Cushing models. Given the signiﬁcant inﬂuence that correlated

climate and weather disturbances have on renewable resources, characterizing their role in

determining resource dynamics under diﬀerent management regimes presents a worthwhile

area for future research.

17

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