Discussion Paper Series No. 178
Sustainability or ruin of a common resource
economy with random jump
FUJISAKI Masatoshi

University of Hyogo

KATAYAMA Seiichi

Kobe University

OHTA Hiroshi

Kobe University

December

2005

Discussion papers are a series of manuscripts in their draft form.
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indicated by the author. For that reason Discussion Papers may not bequoted, reproduced or distributed without the written consent of theauthor.

Sustainability or ruin of a common resource
economy with random jump
FUJISAKI Masatoshi, University of Hyogo,
KATAYAMA Seiichi, Kobe University, RIEBOHTA Hiroshi, Kobe University, GSICS
2005.11.22.
Abstract
We consider a common resource economy in which agents exploit the common resource, and use it to produce goods and consume the goods produced.
Also the agents can invest in private and productive capital. The resource extracted from the common resource is non-renewable and the common pool isunder unceratinty in the sense that it could have a sudden increase or decreasein the course of extraction. In this model we shall explore the probability ofsustainability or ruin of the economy. We compare and discuss the optimal
extraction rules and the probabilities under cooperative and non-cooperative
regimes.

Keywords: common property resource, private capital accumulation, pure jumpprocess, exhaustion probability, HJB (Hamilton-Jacobi-Bellman) equation
JEL Classification: Q32

Corresponding author: Seiichi Katayama, RIEB, Kobe University, 2 Rokkodai,
Nada-ku, Kobe, Japan, Fax: +81 78 803 7012
E-mail:katayama@rieb.kobe-u.ac.jp10Introduction

On the plannet Earth the amount of natural resources is limited and their endowments are unevenly located. However, resources held by some limited number ofcountries are of free access under suitable economic conditions. From the consumers’
point of view natural resource is nothing but a common property.
It is well known that when agents have free access to a common property resource,
they tend to over-exploit the resource stock. The size of the resource stock thus
declines at a much faster rate than the socially optimal rate. In most of the modelsof common property resources, the typical assumptions are that there are no otherstocks than resources, such as private capital, and that the extracted resources are
directly consumed .
To shed light on the alternative capital asset than resources, Long and Katayama(2002) have presented a model of exploitation of a common property resource, when
agents can also invest in private and productive capital. The resource extracted
from a common pool is non-renewable in the model. They investigate the patternof consumption and capital accumulation. In the present paper, we try to extend
their result to the case where a common pool is under uncertainty in the sense thatit could have a sudden increase or decrease in the course of extraction.
The extension is quite natural when we see the present state of international
crude oil market. Some oil producing countries have encountered the technologicaldifficulties of extraction and /or social hazards. Also the past history of oil showsthat unexpected exploration of new oil reserves occasionally happened. Even so, weknow that the total reserve in the earth is limited, and it is expected that the resource
is finally exhaustible. As a remedy, people can accumulate capital for substituting
the exhaustible resource and prolong the ultimate depletion of resource stock.
To take into account this consideration to our discussion, we present a modelof uncertainty on the available reserve of resource and build in a physical capitalas substitute for the exhaustible resource. Speciffically, we build a model with arandom jump in the stock of the resource. Fujisaki, Katayama and Ohta (2005)
have already discussed whether there is an optimal solution to this type of model.
One of the main issues in the economy of this sort is whehter it could be sustainable or reaching to the ruin situation where it would have hit the minimum
boundary of the resource reserve and the economy would cease to be active. It isinteresting to see how high such probability is and which factor affects the probability. If the economy could be in destruction, which probability of ruin is higher
under coopreative regime or under non-cooperative regime? Thus, we derive andcompare the probabilities of ruin in both regimes. As far as the authors have known,
there is no existing literature discussing sustainability and ruin of resource economy
in probability sense as in this paper. The pattern of the resource extraction would
be affected by the uncertainty. We obtain the optimal resource extraction rule, andask how this rule is affected by the jump process in resource reserve and between
the co-operative and non-cooperative regimes.
The structure of the paper is as follows. The next section develops the model.
In the section 2 the cooperative outcome is stated. In the section 3 exhaustion andsustainable probabilities are derived. The non-cooperative outcome follows in thesection 4, and the comparison with cooperative outcome is made in terms of the ruin
2

probability and the extarction rates in the section 5. The final section concludes
the paper.

1

The Model

The basic framework of the model is same as Long and Katayama (2002).
There are n identical agents having common access to a stock of nonrenewable
natural resource, denoted by S(t). Each agent i also owns a private capital stock
Ki (t). Agent i extracts the amount Ri (t) of the common resource stock (i=1,...,n).
n
X
Extraction is costless. Total extraction in the economy at time t is R(t) =Ri (t),
i=1

and the reserve depletes according toṠ(t) = −R(t)if it is not subject to any uncertainty.
However, the reserve may be augmented or damaged several times in the finite
horizon and the reserve size is affected by those jumps in magnitude. Let us consider
a probability space (Ω, F, P, F = {Ft }) satisfying the usual conditions and the jumpprocess is given on this basis. The jump process takes the form dJ(t), and theresource stock is governed by the following stochastic differential equation (in short,
SDE)dS(t) = dJ(t) − R(t)dt.
The stok level at time t is
Z t
(1.1)

R(s)ds,S(t) = S0 + J(t) −0where J(t) is a pure jump process given by
Z tZ
S(s−) · zN (ds, dz).
(1.2)
J(t) =
0R\{0}N (·) is a Poisson Random measure with intensity
Z
E[N (A)] = λ σ(z)dtdz,
AA ∈ B(R+ × R), a Borel set and λ > 0. λσ(z)dz is called Lévy measure and weassume thatZ ∞|z|2 ∧ 1 σ(z)dz < ∞.
−∞The Lévy measure expresses the possible jump size and λ is the average number ofjumps to occur during unit time interval. This rather technical assumption exculudes
such cases that the jumps occur intensively in any finite intervals and/or the sizes
are extremely large.
Assume that the extracted resource cannot be directly consumed. Instead, agent
i uses Ri as an input, which, in combination with his labor input and his privately
3owned capital stock Ki yields an output Yi of final good. For simplicity we choose ameasurement unit by fixing the labor input to unity. Therfore, agent i’s production
function is
Yi = Ri1−β Kiα ,where 0 < α, β < 1. Agent i consumes Ci (t), and the remaining quantity is invested
to accumulate his physical capital. The rate of accumulation of the privately owned
capital stock is thusdKi (t) = (Ri1−β Kiα − Ci )dt.

(1.3)

Each individual utility is increasing in consumption Ci (t):Ui = (1 − γ)−1 Ci1−γ ,where 0 < γ < 1. Each agent wishes to maximize the integral of the stream of
discounted utility
Z
∞

max
0

(1 − γ)−1 Ci1−γ e−ρt dt

subject to (1.1) and (1.3), and the initial conditions
S(0) = S0 ,Ki (0) = Ki0 .
γ is the elasticity of marginal utility, and for mathematical simplicity as in Long
and Katayama (2002) assume that γ = α.

2The cooperative outcome

If the agents cooperate each other, they will collectively seek to maximize the samelevel of their welfare. They will choose the rate of extraction per agent Rh andconsumption per agent Ch to maximize
Z ∞max(1 − α)−1 Ch1−α e−ρt dt
0subject to dKh (t) = (Rh (t)1−β Kh (t)α − Ch (t))dt,
(2.1)



dS(t) = dJ(t) − nRh (t)dt,

and the boundary conditions³
P

S(0) = S > a > 0, Kh (0) = Kh ≥ 0.
´
lim S (t) < ∞ = 1, lim Kh (t) ≥ 0.

t→∞

t→∞

For the resource economy to be analytically meaningful, the boundary condition in
the future is introduced so that the resource stock in any event should be finite.
4

Define the value function V (S, Kh ) for this maximization problem by

(2.2)

Z τ
V (S, Kh ) = max E[ (1 − α)−1 Ch1−α e−ρt dt + g(S(τ ), Kh (τ ))e−ρτ ],
Ch ,Rh ≥00where ρ > 0 is a discount factor, g is a given function which represents the evaluationof remaining stocks, 0 < α, β < 1. τ is the first exit time from D,
τ = inf{t > 0; (S(t), Kh (t)) ∈/ D}, = ∞ if {·} = φ,where D = {(S, K); S > a, K > 0} and a is a positive constant. Here it is assumed
that the whole system collapses when the economy reaches to τ , and that it isnecessary for the economy to keep the minimum sustainable level a of the resource
stock. This assumption is innnocuous.
It is known (see e.g. Kushner and Dupuis (2001)) that this optimization problem
is equivalent to the following Hamilton-Jacobi-Bellman equation;
(2.3)





(1 − α)−1 Ch1−α + VKh (Rh1−β Khα − Ch )




Z
ρV (S, Kh ) = max
Ch ,Rh >0  +λ {V (S + Sz, Kh ) − V (S, Kh )} σ(z)dz + VS (−nRh ) Rfor (S, Kh ) ∈ D, andV (S, Kh ) = g(S, Kh ) for (S, Kh ) ∈ ∂D,
where VKh = ∂V /∂Kh and VS = ∂V /∂S.
The first order condition for maximization areCh−α − VKh = 0(1 − β)Rh−β Khα VKh − nVS = 0

They turn to be
Ch = (VKh )−1/α
and

½
Rh =

nVS
(1 − β)VKh Khα¾−1/βSubstituting these conditions into Eq.(2.3), we obtain the partial differential equation for V (S, Kh ). As Long and Katayama (2002) have indicated, the solution to thedifferential equation is not simple. Instead of solving it directly we take the same
solution as the one used by them. Assume that the partial differential equation has
a simple solution
V (S, Kh ) = AKh1−α + BS 1−β ,5where A and B are positive constants to be determined. Then Eq.(2.3) becomes
ρ(AKh1−α + BS 1−β ) = (1 − α)−1 (VKh )1−1/α + VKh {Rh1−β Khα − (VKh )−1/α }Z−nRh VS + λ {V (S + Sz, Kh ) − V (S, Kh )}σ(z)dzα{(1 − α)AKh−α }1−1/α + Rh1−β Khα {(1 − α)AKh−α }
1−α
Z
−β
1−β
{(1 + z)1−β − 1}σ(z)dz−nRh (1 − β)BS + λBS== α(1 − α)−1/α A1−1/α Kh1−α + {β(nB)1−1/β [(1 − α)A]1/β + λBd(β, σ)}S 1−β ,where d(β, σ) is given by the following formula.
Zd(β, σ) = {(1 + z)1−β − 1}σ(z)dz.
For this equation to hold for all Kh > 0 and S > 0, it is necessary that the following
conditions are satisfied; ρA = α(1 − α)−1/α A1−1/αρB = β(nB)1−1/β [(1 − α)A]1/β + λBd(β, σ).

Assume that(A.1)

δ ≡ ρ − λd(β, σ) > 0.

Therefore, we deduce

µ ¶α
α
A=
(1 − α)−1 ,
ρ

and

µ ¶β
µ ¶α
β
α
β−1
B=
n
.
δ
ρ

From these results, we can obtain the following.
Proposition 2.1 Assume that g in (2.2) is of the formg(S, K) = AK 1−α + BS 1−β ,where A and B are defined by the above formulas, then the optimal consumption Ch
and the optimal resource extraction Rh are given by

(2.4)


³ρ´

Kh
C
=

h


α
µ ¶µ ¶
δ
S Rh =
βn6respectively. Moreover, the corresponding optimal trajectory of two stocks (Kh , S)
are given by the following differential equations;

(µ
)
¶1−β
³ρ´

δS(t)


dKh (t) =Kh (t)α −Kh (t) dt



nβ
α
(2.5)


µ ¶


δ


S(t)dt dS(t) = dJ(t) −βwith initial (boundary) conditions Kh (0) = K0 > 0 and S(0) = S0 > a > 0.
The phase diagram for this system in the (Kh , S) plane is almost the same asLong and Katayama (2002) or Fujisaki, Ibuki and Katayama (2004). In fact, it isclear that if the jumps occur in the negative direction which could happen in suchcase as sudden destruction of the resource stock, then both the resource stock S andcapital stock Kh will eventually converges to the boundary (0, a).
Remark 2.1 It is easy to see that the assumption (A.1) is satisfied if we choose ρand λ so that d(β, σ) < ρ/λ. The economic meaning is that the average size and thenumber of jumps are restricted not to be so large.
Remark 2.2 The optimal consumption rule (2.4) is independent of the stock of
the resource, and as (Long and Katayama(2002)) shows, the extraction by agent hdepends only on the resource stock per head, S/n. Notice that Long and Katayama(2002) has derived the optimal extraction function as Rh = [ρ/β] × [S/n] in theabsence of jumps in resource size. Our result is obtained by replacing ρ with δ =ρ − λd(β, σ). If d(β, σ) ≥ 0, then the optimal extraction is revealed to be lessunder jumps than without them for the same level of resource stock. Moreover as
δ is decreasing in λ, the optimal extraction decreases as the resource jumps more
frequently. It is the way for economic agents to react more cautiously to cope withthe uncertainty. However, the more cautious behavior is not applied to consumption,
since the optimal level Ch is shown to be the same as in Long and Katayama (2002).

3

Sustainablility and ruin probabilities

In this paragraph, we will calculate the exhaustion probability of the stock
process. Using the optimal resource extraction rules (2.4) and optimal trajectories(2.5), the stock process is given by the following formula;
Z t
(3.1)
S(t) = S0 + J(t) −ηS(s)ds,0

where J is a pure jump process given by (1.2) and we put η = δ/β. Note that
(3.1) is written in terms of the following stochastic differential equation, so calledstochastic exponential;(3.2)

dS(t) = S(t−)dY (t), S(0) = S0
7

where Y is Lévy process given by
Z tZ
Y (t) =0zN (ds, dz) − ηt.
R\{0}We will need the following assumptions.
(A.2)

inf{∆Y (t), t > 0} > −1 (a.s.),

where ∆Y (t) ≡ Y (t) − Y (t−). Otherwise, S(t) may be negative (see Remark 3.1(1)). As it can be seen from (3.2), (A.2) implies that the jump size should not be toolarge for S(t) ≥ 0. In the following, assume that the support of the measure σ(z)dz
is included in the interval (−1, ∞) (cf. Remark 2.1). Then it is well known (seeApplebaum (2004)) that the unique solution of Eq.(3.2) is represented as follows;
Q
S(t) = S0 eY (t) s≤t [1 + ∆Y (s)]e−∆Y (s)
(3.3)
RtR
= S0 exp{−ηt + 0 {z6=0,z>−1} log(1 + z)N (ds, dz)}.
Note that if S0 > a > 0, then S(t) > 0 for all t > 0.
Put S̃(t) = log S(t) then
Z tZ
log(1 + z)N (ds, dz)}.
S̃(t) = log S0 + {−ηt +0{z6=0,z>−1}

Defineτa = inf{t > 0; S(t) ≤ a}, = ∞ if {·} = φ,thenτa = inf{t > 0; S̃(t) ≤ log a}.
τa is the time for the economy to reach break down limit.
3.1 Sustainable caseWe are going to investigate under what condition the system is sustainable in thesense that the system does not ruin. For ease of computations, suppose that {νi , i <∞} and {κi , i < ∞} are mutually independent sequences of iid random variables
and they are point masses of the Poisson random measure N (ds, dz). Furthermore
we assume that ν0 = 0, the τn = νn+1 − νn are exponentially distributed with mean
1/λ, and the κn has distribution σ(z)dz. Roughly speaking, the νn and κn mean
the jump times and jump sizes of the process respectively (see Gaier, Grandis andSchachermayer (2003), pp.28∼30). Note that (A.2) means that for any n, κn > −1.
Then we have a lemma (c.f. Long and katayama (2002), Boikov (2003), and Fujisaki,
Katayama and Ohta (2005)).
Lemma 3.1 Let k > 0, then it follows thatE[e−kS̃(t) ] = exp{−kc + t{kη − λ(1 − Φ(k))},
where

Z
−k

Φ(k) = E[(1 + κi ) ] =8(1 + z)−k σ(z)dz.

ProofE[e−kS̃(t)Z tZ] = E[exp{−k(c − ηt +log(1 + z)N (ds, dz))}0−k(c−ηt)=e{z6=0,z>−1}

Z tZ
E[exp{−klog(1 + z)N (ds, dz)}]0−k(c−ηt)=e{z6=0,z>−1}E[exp{−kN (t)Xlog(1 + κn )} ≡ I1 × I2 ,n=0

where c = log S0 and for each t, N (t) denotes the Poisson random measure withintensity λt, the number of jump times νn which is less than t.
We will calculate only the expectation part I2 . Since both sequences {κn } and{νn } are mutually independent iid sequences,
I2 =

∞
X

E[exp{−k

=

( l
)−k
Y
E[
(1 + κi )] × P (N (t) = l)
i=1

l=0

=

log(1 + κn )}; N (t) = l]

n=0

l=0
∞
X

l
X

l
∞ Y
X

E[(1 + κi )−k ] × P (N (t) = l).

l=0 i=1

Since N (t) is Poisson process with intensity λt, it follows that
P (N (t) = l) =Then we deduce
I2 =

∞
X

(Φ(k))l

l=0(λt)l −λte .
l!
(λt)l −λte ,l!

= e−λt(1−Φ(k)) ,from which the assertion follows immediately. (c.q.f.d.)Assume that there exists a k ∗ > 0 such that(A.3)

k ∗ η − λ(1 − Φ(k ∗ )) = 0,then we have the following.
∗

Lemma 3.2 M ∗ (t) ≡ e−k S̃(t) is a martingale if and only if (A.3) holds. Moreover,∗is also martingale.
it can be shown that Mta = Mt∧τa9

Proof We have to show that for any t > s, E[M ∗ (t)|Fs ] = M ∗ (s), where Ft is thefiltration generated by {Nt }. But this follows from the fact that M ∗ (t) is stationary
process with independent increments and (A.3). Since t ∧ τa is bounded stopping
time, the last assertion is obvious. (c.q.f.d.)Therefore,∗∗∗
E[Mta ] = E[Mt∧τ] = E[e−k S̃(t∧τa ) ] = e−k cafor all t > 0. Since S̃(τa ) ≤ log a on {τa ≤ t}, we deduce that∗

P (τa ≤ t) ≤ e−k (log a0 −log a)
for all t > 0. Note that c = log S0 = log a0 > log a.
An intuitive meaning of (A.3) is that average number of jumps and the sizes are
small (see Remark 3.1(2)). We can summarize as follows.
Proposition 3.1.1 Assume (A.3), then∗
(1) P (τa < ∞) = supt P (τa ≤ t) ≤ e−k (log a0 −log a) < 1. Therefore, in this case,the probability that exhaustion does not occur is strictly positive, in other words, theruin probability < 1.
(2) P (τa ≤ t) → 0 as a0 → ∞(3) P (τa ≤ t) → 0 as a → 0.
The economic meaning of this proposition is that as far as average number ofjumps and the sizes are small, the ruin probability of the economy within limited
time is less than one. And the probability goes to zero when the initial resource
stock is large or the minimum stock requirement is small.
3.2 Ruin case
Next we study the case when the stock process is exhaustible eventually withprobability 1. Due to (3.3), S̃(t) is written as follows:˜S̃(t) = S̃0 − ηt + J(t),
where
˜ =
J(t)

Z tZ
log(1 + z)N (ds, dz),
0

R\{0}

For any f ∈ C 1 , due to Ito’s formula,Z t∧τaf (S̃(t ∧ τa )) − f (S̃0 ) =(−η)∂x f (S̃(s))ds0

Z t∧τa Z+[f (S̃(s−) + log(1 + z)) − f (S̃(s−))]N (ds, dz)
0

R\{0}

10

Taking the mathematical expectation, we get
E[f (S̃(t ∧ τa ) − f (S̃0 )]Z t∧τaZ t∧τa Z= E[[f (S̃(s−) + log(1 + z)) − f (S̃(s−))]λdsσ(z)dz(−η)∂x f (S̃(s−))ds +00

Z t∧τa= E[
Lf (S̃(s−))ds]
0

where

Z
Lf (x) = (−η)∂x f (x) + λ[f (x + log(1 + x)) − f (x)]σ(x)dx.

Assume that there exists a nondecreasing function f ∈ C 1 such that
(A.4)

Lf (x) ≤ −1

for all x ≥ S̃0 . Then one deduce
Z t∧τaE[f (S̃t∧τa ) − f (S̃0 )] = E[
Lf (S̃(s))ds] ≤ −E[t ∧ τa ],0or equivalently,
E[t ∧ τa ] ≤ E[f (S̃0 ) − f (S̃t∧τa )].
Note that the right side is bounded because
sup E[f (S̃0 ) − f (S̃t∧τa )] ≤ E[f (S̃0 ) − f (log a)],0≤t<∞

and we have the following.
Proposition 3.2.1 Assume (A.4), then E[τa ] is bounded. Moreover,P (τa ≥ t) ≤ κ/t → 0
as t → ∞, where κ = f (S̃0 ) − E[f (S̃(τa )]. Therefore, in this case, the exhaustion
occurs with probability 1.
It shoud be noticed that (A.3) and (A.4) are mutually inconsistent. Therefore
under (A.4), it is not the case that average number of jumps and the size are small.
The probability for the economy to survive eventually goes to zero under the assumption (A.4).
Remark 3.1 (1) The assumption (A.2) can be dropped and we can extend to thecase where ∆Y (t) is arbitrary. But in this case S(t) may take negative values sothat it is not good for our case.
(2) The assumption (A.3) is equivalent to the following equation with respect tok > 0,(3.4)

1 − Φ(k) = kξ,
11where ξ = η/λ = δ/βλ. Since Φ(k) ≥ 0, kξ ≤ 1. Note also that it is necessarythat Φ(k) ≤ 1 because kξ > 0. Since Φ(0) = 1, k = 0 is trivial solution of Eq.(3.4).
Assume that Φ0 (0) < 0 and choose ξ such that −Φ0 (0) > ξ. then it is easy to seethat we can find a nontrivial solution k > 0 of Eq.(3.4). This is equivalent to thefollowing inequality:Z(3.5)
log(1 + z)σ(z)dz > 0.
If the jump sizes are always positive, i.e. the density σ(z) has compact support in(0, ∞), then the above inequality is clearly satisfied, so that there is a positive solution of Eq.(3.4). However, if there arises negative jumps, then it depends on thesizes of negative jumps. Some examples are shown in the appendix in which there
exists solution k > 0.
(3) Put f (x) = cx where c is a positive constant. Then the assumption (A.4) means
that
Z
(3.6)

c{−η + λlog(1 + z)σ(z)dz} ≤ −1.

Suppose thatZ
(3.7)

−η + λlog(1 + z)σ(z)dz < 0,then we can choose c > 0 so that (3.6) holds. It is obvious that this is valid if only
negative jumps occur. Note also that (3.7) implies that E[S̃(t)] ≤ log a0 for all t andalso E[S̃(t ∧ τa )] ≤ log a0 . Moreover, ifZlog(1 + z)σ(z)dz < 0,then (A.4) is clearly satisfied (cf.(3.5)). For example, put σ(z) be uniformly distributed such that σ(z) = 1 on (−1/2, 1/2), then (A.4) is fulfilled.

4

Non-cooperative outcome

In this section we turn to the non-cooperative scenario following Long and Katayama(2002). It should be noticed that for the existence of a Markov perfect equilibrium
every agent has common knowledge on a pure jump process J(t), and take it into
consideration to set her optimization problem. Following the same way as Long
and Katayam (2002), it turns out that there exists a Markov perfect equilibrium inwhich all agents play linear strategies of the form
Rj = µj (S) = rS
andCj = θj (Kj ) = cKj
12

Therefore the resource stock S and capital stock Ki for each agent i are givenby the following (cf.(2.1)):
 dKi (t) = (Ri1−β Kiα − Ci )dt,
(4.1)
dS(t) = −(Ri + r(n − 1)S)dt + dJ(t),where r is a positive constant, J(t) is given by (1.2) and the boundary conditions
are the same as the cooperative case in §2. Moreover, suppose that each agent i’svalue function is given by the following.
Z τi
(4.2)
V (S, Ki ) = max E[(1 − α)−1 Ci1−α e−ρt dt + g(S(τi ), Ki (τi ))e−ρτi ],Ci ,Ri ≥00where τi is the first exit time from D,
τi = inf{t > 0; (S(t), Ki (t)) ∈/ D},
D = {(S, K); S > a, K > 0} and a is a positive constant. By using the same method
as Long and Kayama (2002) and §2, we have the following result. Suppose that β
and the number of agents n satisfies that
(A.5)

1 − n(1 − β) > 0 or equivalently, n < 1/(1 − β).

Proposition 4.1 Assume (A.5) and that g in (4.2) is given of the formg(S, Ki ) = ÂKi1−α + B̂S 1−β ,where Â and B̂ are positive constants. Assume also that the system is in a symmetricequilibrium, i.e.
#1/β"Â(1 − α).
r=B̂Then the optimal consumption Ci and the optimal resource extraction Ri are givenbyρCi = × Ki



α
(4.3)

δ


× S,
 Ri =
1 − n(1 − β)respectively. Moreover, the corresponding optimal trajectory of two stocks (Ki , S)
are given by the following differential equations;

)
(µ
¶1−β
nρo

δS


Kiα −
Ki dt
dKi (t) =
1 − n(1 − β)
α
(4.4)


½




 dS(t) = dJ(t) −nδ
1 − n(1 − β)¾Sdtwith initial (boundary) conditions Ki (0) = K0 > 0 and S(0) = S0 > a > 0.
13

If 1 < n < 1/(1 − β), then the extraction rate in the cooperative scenario is lessthan the non-cooperative one. It is also shown that each agent’s welfare under the
non-cooperative scenario is worse than the cooperative one.
Next it is interesting to calculate the ruin probability under the non-cooperative
scenario. Let η 0 = nδ/{1 − n(1 − β)}, then η < η 0 if 1 < n < 1/(1 − β), whereη = δ/β. Then optimal stock process S 0 (t) is written as follows;
Z t
0
(4.5)
S (t) = S0 + J(t) −η 0 S 0 (s)ds,
0or equivalently,
dS 0 (t) = S 0 (t−)dY 0 (t),Z tZ
0
Y (t) =zN (ds, dz) − η 0 t.
0

R\{0}

Put S̃ 0 (t) = log S 0 (t) and τa0 = inf{t > 0; S 0 (t) ≤ a}, then τa0 = inf{t > 0; S̃ 0 (t) ≤log a}. Since η < η 0 , log S 0 (t) ≤ log S(t) or equivalently, S 0 (t) ≤ S(t). This implies
that τa0 ≤ τa so that P (τa < ∞) ≤ P (τa0 < ∞). In other words, the ruin probability
under non-cooperativer regime within finite time is not smaller than the one under
the cooperative regime. More precisely, by using similar arguments as in §3, we canobtain the following.
Proposition 4.2 Assume that there exists a k 0∗ such that
(A.6)

k 0∗ η 0 − λ(1 − Φ(k 0∗ )) = 0,then the assertion of Theorem 3.1 still holds for τa0 and k 0∗ .
As for the probability P (τa0 > t), we can proceed the same way as §3.

5Comparison of extraction patterns and consumption

In this section we compare various extraction patterns, and discuss the role ofcooperation regime and uncertainty.
The optimal resource extraction rule under cooperation without jumps:
RhC =

ρS
βn

and that of under non-cooperation without jumps:
RhN =

ρS
1 − n (1 − β)by Long and Katayama (2002) .

14

With jump process they are given by (2, 4) and (4.3). Denoting RhCJ for thecooperation and RhN J for the non-cooperation under the regimes with jump process,we have derived
δS
RhCJ =βn
RhN J =

δS
1 − n (1 − β)It is easily seen that
RhCJ < RhC < RhN , RhCJ < RhN J < RhN ,The ordering of RhN J and RhC depends on the relative magnitude of effects ofcooperation and uncertainty. However, we can say that agents are most conservative under cooperative regime with uncertainty and least conservative under noncooperative regime without uncertainty in terms of the resource extraction. It should
be noticed that the optimal consumption rule is the same under cooperative andnon-cooperative regimes as given (2.4) and (4.3) . Further, it is known from Long
and Katayama (2002) that the optimal level Ch is the same as under certainty. The
response for economic agents are free from uncertainty and socio-economic relationships among agents in their consumption behavior.

6

Concluding remarks

We have studied a differential game model of exploitation of a common property
resource under unceratinty, when agents can also invest in private and productive
capital. The resource extracted from a common pool is non-renewable, and thecommon pool is under uncertainty in the sense that it could have a sudden increaseor decrease in the course of extraction. The stochastic reserve process is expressedby Lévy process.
We derived the optimal consumption and resource extraction rule and the sustainability and ruin probabilities of the stock process. One of the interesting result isthat as far as average number of jumps and the sizes are small, the ruin probabilityof the economy within limited time is less than one. And the probability goes tozero when the initial stock is large or the minimum stock requirement is small.
The extraction rate in the cooperative regime is less than the non-cooperative
regime under unceratinty. Finally it is interesting to find that ruin probability
within finite time horizon under non-cooperative regime is not smaller than the one
under cooperation.

7

Appendix

Two examples are shown here in which there exists solution k > 0 in Eq.(3.4).
Example 1z < −1/2
 0
2z + 1 −1/2 ≤ z < 0
σ(z) =
 −(4/3)ze0≤z15

It is easily seen that Φ(0) = 1, Φ(k) is strictly decreasing for 0 < k ≤ 2 and then it
is increasing for 2 < k and limk→∞ Φ(k) = ∞. Therefore, if we choose ξ so smallthat ξ < −Φ0 (0) ≈ 0.3178, then there exists a solution k ∗ > 0 of Eq.(3.4).
Example 2


0



2z + 1σ(x) =−(2/3)z + 10

z < −1/2−1/2 ≤ z < 00 ≤ z < 3/2
3/2 ≤ zIf we choose ξ so small that ξ < −Φ0 (0) ≈ 0.2356, then there exists a solution k ∗ > 0
of Eq.(3.4).

16

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2004.
[2] Boikov,A.V. The Cramér-Lundberg model with stochastic premium process, Theory Probab. Appl. vol.47, No.3, pp.489-493, 2003.
[3] Fujisaki, M., Ibuki, K. and Katayama, S. Renewable common property resourceand private capital accumulation Shōdai-ronshū, vol 55, No.5, pp.1-6, Kobe University of Commerce. 2004,[4] Fujisaki, M., Katayama, S. and Ohta, H. Common property resource and private
capital accumulation with random jump. In: C. Deissenberg and R. Hartl (eds),
Optimal Control and Dynamic Games, pp.77-84, Springer, Dordrecht,2005.
[5] Gaier,J., Grandis,P and Schachermayer,W. Asymptotic ruin probabilities andoptimal investment, The Annales of Applied Probability, vol.13, no.3, pp.10541076, 2003.
[6] Kushner, H.J. and Dupuis, P. Numerical Methods for Stochastic Control Problems in Continuous Time, Springer, 2001.
[7] Long,N.V. and Katayama,S. Common property resource and private capital accumulation. In: G.Zaccour (eds), Optimal Control and Differential Games, pp.193209, Kluwer Academic Publishers, Boston/Dordrecht/London,2002.
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