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Updated, 19 April 2004
ECON/SØK500 Natural resource economics, Spring 2003
Olav Bjerkholt:
Lecture notes on the Theory of Nonrenewable Resources
7. Renewable resources: Optimal taxation under an open-access regime
Perman et al. (2003) Chapter 17 Renewable resources cover a wide range of renewable
resource problems. As we can give these problems quite limited attention within our
course, we concentrate on the analytics of a stylized approach to the renewable resource
problem, interpretable as the harvesting of a non-migratory species of fish at a particular
location or even as a freshwater stock in a pond or reservoir.
We use control theory to study the optimal harvest, i.e. optimal from a social efficiency
point of view. We confront this with the harvest implied by "open access", as discussed
below. Finally, we try to derive taxation rules that might lead an open access regime
towards the optimal solution.
The renewable resource problem differs from the nonrenewable resource problem,
naturally in the law of motion of the resource stock. While the nonrenewable resource
stock in a non-stochastic setting, is changes as we have seen, by the stock being
diminished over time by the amount of resource depleted (harvested) from it, i.e.
St   Rt .
For the renewable resource problem there is another element in the law of motion of the
resource stock, namely the endogenous growth of the stock, endogenous because it is
modeled as dependent upon the resource stock. We can write it as
(7.1)
St  G ( St )  Rt
(In this lecture note the analytics is slightly different from Perman et al. (2003), and there
are minor differences in notation. One difference is the use of R instead of H for the
harvest (depletion) of the stock, consistent with the notation used for nonrenewable
resources. The use of H for harvest is common in this literature, but should be avoided as
H also is used for the Hamiltonian. One of the Perman et al. (2003) authors may also
have tried to pursue this idea, see the unnumbered equation (iii) just below (17.35) on
p.580!)
What kind of function is G(S)? This is discussed at some length in Perman et al. (2003),
section 17.1, particularly Box 17.1. For our purpose here we may as well assume the
simplest version, the logistic form given in (17.3) as
(7.2)
G ( St )  g (1 
St
S MAX
) St
2
This function implies that the growth rate of S (with no harvesting) decreases from g at
S=0 to a growth rate of zero at S=SMAX. The absolute growth increases from zero at S=0
to a maximum of 1/2 gSMAX at S=1/2 SMAX and then declines and reaches zero at S=SMAX..
This logistics function, even in a more complicated version, is still a gross simplification
of the actual dynamics of endogenous growth of fish and other biological species. The
function does not pay attention to the full population dynamics which requires
representation of the size of the different cohorts, fertility patterns etc. Neither does it pay
attention to stochastic elements in the reproductive patterns, whether endogenous or
caused by external factors. Even less does it pay attention to the interrelation between the
species under consideration and other species. We leave, however, these aspects aside.
Just by looking at a graphical representation of the growth function G(S), one realizes
that this renewable resource has a infinite number of steady states, i.e. an infinite number
of combinations of stock size and harvest size that will leave the stock unchanged. This is
also too obvious to be constructed as an important insight. We also can draw the
conclusion that there is one stock size that allows a maximal size of the steady state
harvest, called the maximum sustainable yield, RMSY, with corresponding stock size SMSY.
We can add to this that for all other yields, that is R between zero and RMSY, there are two
different stock sizes that can yield this harvest in a steady state.
To harvest the renewable resource requires expenditure of effort which again requires
costs. Just as in the nonrenewable resource theory we can assume a cost function as a
function of harvest as a function representing the minimum cost to give rise to the given
harvest, b  b( Rt ) . But even more than for the nonrenewable resource there are reasons
for including the stock level in the cost function, i.e. , with the usual properties:
bR  0, bRR  0, bS  0, bRS  0 . The cost of catching an amount of a non-migratory fish
can reasonably be assumed to be lower the denser the fish stock is. The same will hold
for other populations to which this theory may apply.
We can then proceed to formulate the optimal control problem for harvesting the
renewable resource when the price of harvested resource is given as p and the rate of
discount as r.

(7.3)
Max  [ pRt  b( Rt , St )]e  rt dt
{ Rt }
subject to (7.1) and S 0  S .
0
The Hamiltonian in current values is then
(7.4)
H C ( Rt , St ,t )  pRt  b( Rt , St )  t  (G(St )  Rt )
where  t is the shadow price of the resource stock. The necessary condition for a
maximum is
(7.5)
p  bR ( Rt , St )  t  0
The shadow price must obey
(7.6)
t  rt  t  G ( St )  bS ( Rt , St )
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We see from (7.5) that the shadow price here as in the nonrenewable case has an
interpretation as resource rent, the net of price over marginal cost. The three equations
(7.1), (7.5) and (7.6) which jointly determines the optimal path of Rt , St and t are not by
immediate inspection easy to see through. We shall try to make it more transparent by
investigating whether there are stationary states solutions, i.e. solutions such that St  S *
for all t, and as a consequence also constant Rt and  t . It then follows that Rt  G( St ) and
t  0 .Using this we can rewrite (7.6) with substitution from (7.5) and write it in three
alternative ways
(7.7)
r  G( S * )  bS (G ( S * ), S * ) /( p  bR (G ( S * ), S * )
(7.8)
r  ( p  bR (G ( S * ), S * )  p  G ( S * )  (bR (G ( S * ), S * )  G ( S * )  bS (G ( S * ), S * ))
(7.9)
( p  bR (G( S * ), S * ) 
p  G( S * ) (bR (G( S * ), S * )  G( S * )  bS (G ( S * ), S * ))

r
r
(7.7) looks like a Hotelling rule(!), but perhaps difficult to interpret as it stands here. We
have in fact shown by this reasoning that there are stationary solutions, which should not
surprise for this kind of problem. From (7.7) we see that there is a correspondence
between r and stationary values of S, say S*=S*(r), thus for given r there is one stationary
solution. (Try to figure out whether a higher r corresponds to a lower or higher stationary
value of S?) If the start-out value of the stock S0 is different from S*(r), we would expect
the optimal path to lead from S0 to - or approaching - S*(r). This can be shown.
The other two alternative formulations, (7.8) and (7.9), are easier to interpret. We could
have simplified the expressions by noting that bR (G ( S * ), S * )  G( S * )  bS (G ( S * ), S * ) is the
total marginal cost of b(G(S),S), i.e. the marginal cost w.r.t S.
(7.10)
db(G ( S ), S )
 bR (G ( S * ), S * )  G( S * )  bS (G ( S * ), S * )
dS
Before we interpret these solutions, let us try to reason along another track. If we think only about
stationary solutions, then everything is the same in every period. Let us simply ignore the time
dimension and search for the stationary stock value which gives the highest surplus, calculated
simply as the revenue over cost, in other words
Max[ pG ( S )  b(G ( S ), S )]
(7.11)
S
The first-order condition is
(7.12)
pG( S ) 
db
 bR (G ( S ), S )  G( S )  bS (G ( S ), S )
dS
We can rewrite this as
(7.13)
G( S # )  bS (G ( S # ), S # ) /( p  bR (G ( S # ), S # )  0
Perman et al. calls this optimal value S# for the “static provate-prperty steady state”
(p.573). By comparison with (7.7) we find that the value of S in this static optimization
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problem is the same as the value we found in the optimal control problem for the
stationary state corresponding to r=0. What is the explanation for this?
Let us first look at a graphical representation (without graphs!). With S measured on the
X-axis, we can draw the pG(S), which of course is a second degree polynomial curve, just
as (7.2), only multiplied with p. It starts in origo, rises to a maximum as discussed above,
and reaches zero again at S=SMAX.
What does the cost function b(G(S),S) look like over the relevant interval. Let us think of
the cost function as proportional in R, i.e. b(R,S)=c(S)R. Consider the point S=SMAX . At
this point the cost function must arguably be zero, as the catch is zero. When S decreases
from SMAX, the catch increases from zero and the cost increases because the catch
increases. At the same time the decrease in S works through the second argument to
increase the costs. Hence, we can draw the cost function leftwards from SMAX as
increasing, although we may have difficulty saying more about its shape without more
information about the technology. The cost function increases as S becomes smaller and
cuts through the pG(S) curve at some point S**.
We can then from (7.12) conclude that the solution point S# can be found just where the
tangents of pG(S) and the cost function are parallel (assuming that the shape of the cost
function is such that this point actually maximizes the vertical difference between the two
curves).
The point S# seems to be an optimal point for a steady state-solution as the profit is higher
that in any other point and the same profit will be realized every year. Why then did we
find above that proper optimization over time gives something else. Let us look at (7.8)
and (7.9) again. Let us think of S# as the starting point. The left-hand side of (7.8)
expresses the annual gain from catching an infinitesimal amount of S, thus moving to the
left on the pG(S) curve, and the right hand side expresses how much we give up in annual
revenue. As the right-hand side is zero for S=S# , we must move further to the left to
achieve the equality expressed by (7.8), and further the higher is r. So what it means is
that catching some of the stock now outweighs the future losses of a lower permanent
revenue.
(7.9) expresses the same idea only put in present-value terms. The left-hand side is the
present value of catching a unit of the stock today, the left-hand side expresses the cost in
terms of the present value of the permanent income from a changed steady state.
Thus we have found that dynamic optimization results in steady state solution which is
S=S# for r=0 and for higher r we find a new steady state S* to the left of S#. What if the
discount rate gets very high, let us say an infinite discount rate? The answer is easy, in
this case future income has no present value. Hence, the optimal action must be to catch
immediately all stock down to S** and continue in the steady state which gives no profit!
The dynamic solution we have found to the optimal exploitation of a biological stock
(fish) under the stated conditions, is by Perman et al.(2003) called the “present-valuemaximizing fishery model” (p.574) and as “private-property” optimum. We might as well
call it the (social) planning solution. The solution pays full attention to the connection
between stock, steady-state harvest and cost. Private agents would do that only if they
owned the stock (and were fully informed). We can regard the solution we have found as
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competitive solution if the agents have enforceable property rights to the stock. One way
of thinking about this is that the entire fishery could be divided up in separate privately
owned stocks. In practice this is not easily possible for many of the stocks to which this
model could be relevant. We can, alternatively, regard the solution as the outcome of an
institutional system which simulates the private property solution, either a voluntary
cooperation or a system of tradable quotas or by taxation.
This points to the central element the fishery problem, the externality of the stock. If we
had the open access system, i.e. free entry of agents and no property rights or alternative
systems, no agent would have any incentive to pay attention to the dynamic effects of
stock changes. Under open-access conditions we have the following problem:
(7.14)
Max[ pR  b( R, S )]
R
with first order condition
p  bR  0
(7.15)
If we have the cost function suggested above with cost proportional with the harvest level
for any given size of the stock, we see that (7.15) implies
(7.16)
pR  bR  R  b( R, S )
Hence, we end up in S** as defined above, where revenue equal costs. Open-access can
only give a short term profit as the stock is harvested down to the S** level. After that
there is no profit. Open access thus gives the same solution as present-value maximizing
with infinite discount rate!
Can the externality of the open access fishery be corrected by means of taxation? Let us
try. We introduce a tax t on the harvest. The agents will now maximize
(7.17)
Max[( p  t ) R  b( R, S )]
R
with first order condition
(7.18)
p  bR  t
By comparison with (7.9) we see that we can actually get to the optimal solution S* by
setting that tax t to
d ( pG ( S ) db
t [
 ]S  S *
(7.19)
dS
dS
It seems very simple in practice, it may not seem so simple in practice, for a number of
reasons. Neither has it been used much in real life.
The stock externality is the essence of this problem, but there could also be another kind
of externality that makes the open access fishery different from the optimal harvesting,
namely inefficiency in the harvesting. The efficiency of each agent may depend upon
how many others are fishing, in short, fishing vessels may get in the way of each other.
Also this externality may be corrected by taxation, at least theoretically. To discuss it, it
is better to use another parametrization of the problem that the one used above. Cf
Perman et al. (2003), p.563. We introduce explicitly a variable representing the “effort”,
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E, which could be fishing vessels or some other instrumental measure of the effort. The
cost can then be represented simply as
b  qE
(7.20)
where q is the cost of each effort unit. The harvest is a function of effort and stock,
R=F(E,S). Perman et al. (2003 suggest that this function could be
R  e E  S
(7.21)
where e is a coefficient. We can then reformulate the dynamic optimization problem as

(7.22)
Max  [ p  F ( Et , St )  q  Et ]e  rt dt
{ Et }
0
The problem is equivalent to the one we have solved above. We leave it as an exercise to
work out the further details of the optimization problem.
Within this problem we could formulate the externality of the effort by a slight
reformulation of the production function. We could let the catch function for agent i be
Ri=F(Ei, Ê, S), where Ei is the effort of agent i and Ê be the total effort with a negative
effect on Ri. This externality could be corrected by a tax on q.