Renewable Resources

Department of Agricultural and Resource Economics EEP 101

University of California at Berkeley David Zilberman

Spring Semester, 1999

Chapter #11

Renewable Resources

Renewable resources are resources that grow. Unlike the case of nonrenewable

resources in which a fixed stock is depleted over time, renewable resources are

natural resources that can reproduce, grow, and die.

Economically important renewable resources include:

• Forests

• Fisheries

• Grasslands (used for grazing)

• Underground water (groundwater) fed by rainfall and surface water percolation

Issues for analysis:

• Growth functions and equations of motion for renewable resource systems.

• Steady-state behavior of renewable resource systems.

• Open access, inefficient market outcomes, and policy corrections.

• Dynamic behavior of renewable resource systems.

A steady-state is a permanent level of stock that is maintained throughout time. An

Equation of Motion is a formula that defines what happens to the stock over

time. For example, when S

denotes the resource inventory at period t:

• If

S St t+ −1

> 0, then the resource stock is growing over time

• If

S St t+ −1

< 0, then the resource stock is shrinking over time

• If

S St t+ −1

= 0, then the resource stock is in a steady state ( S

= S

t+1

= S

t+2

….)

Note: For clarity, we will consider only one stock variable (for example, only one species

of fish) in each of our renewable resource models. More complex models consider

multiple stock variables. Models with multiple stock variables may be required if, for

example, two species interact in a predator-prey relationship.

Also, we will also use a single indicator variable for the resource, which is Stock

• Lots of times having only one indicator variable of a species is not good

For example, when the renewable resource is trout in a lake:

• Need to know the number of fish (Stock)

• Also need to know the number of juvenile fish (Cohorts)

Yet, the use of a single indicator variable allows us to formulate tractable models that give

us information about how we should manage the resource. These models can get quite

complicated with multiple stock variables and more than one indicator variable.

For now, let us focus our analysis on biological renewable resources. We will consider

non-biological renewable resources (e.g., groundwater) later in the course.

Economic models of biological resources combine biological models and economic models;

for this reason, such combined models are called bioeconomic models. Shaefer's

(1954) Biomass Model is a classic example of a biological model used in bioeconmic

models. The following discussion of growth functions, steady-state, carrying capacity and

equations of motion is based on Shaefer's model.

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A Biological Model of A Fishery

Growth Functions

Let St represent the stock of a renewable resource at time t. For example, let St represent

the biomass of a fish population at time t.

Let g(At, St) represent the growth function of the stock during time period t, where

growth is a function of the level of the stock at the beginning of period t, St, and an

exogenous parameter At, which represents factors other than stock which might affect

growth in period t. This simple growth function neglects other biologically-important

determinents of stock growth (e.g.'s: age distribution, sex distribution, size distribution),

but its simplicity allows us to concentrate on the dynamic interaction of economics and

biology in renewable resource models. More complex models take additional factors into

account.

Carrying Capacity and MSS

Notice that growth is influenced by the level of the stock St, which represents fish biomass

in our example. If the level of the stock is low, Births > Deaths of fish so that the fish

population grows. Yet, the growth of the fish population will be low, because few fish

exist to reproduce. If the stock is high, food scarcity, predation, territorialism, and/or an

increased incidence of disease may limit the growth of fish stock. If the stock is very high,

growth may be negative for a period of time, that is Births < Deaths, until the stock level

falls to the carrying capacity of the environment.

The carrying capacity of an environment is the maximum level of stock that the

environment can sustain indefinitely. Carrying capacity is also called the

maximum sustainable stock. At the maximum sustainable level, Births = Deaths, and

the growth rate of the stock is zero (i.e., the fish population is constant, or in a steady-

state). Without human intervention, the stock of fish will reach the carrying capacity of the

lake and the population of fish will stay at the maximum sustainable stock indefinitely.

Steady-States

At the carrying capacity, the growth of the fish population is zero and the stock is in a

steady-state. A steady-state is a situation in which the level of the stock is constant over

time. There can be more than one steady-state. For example, a stock level of zero is also a

steady-state, because at zero the stock level cannot rise (i.e., there are no fish to reproduce)

and the stock level cannot fall below zero stock. [Note that although the growth of the

stock can be negative, the level of the stock cannot fall below zero.]

-3-

The preceding discussion is often illustrated by using a graph like the one presented in

Figure 9.1 below. For now, let's assume for simplicity that A

does not influence growth,

so we can drop A

from the growth function.

Figure 9.1

Fisheries growth function g(S

St = stock of renewable resource at time t

g(St) = biological growth of renewable resource at time t

There are two steady-states in Figure 9.1: the origin and S

mss

. To the right of the origin, the

growth of the stock at first increases with the level of the stock (e.g., point S1), but then

food scarcity, disease, etc. cause growth to decline with further stock increases (e.g., point

S2). Eventually, carrying capacity S

mss

is reached and growth falls to zero. To the right of

Smss (e.g., point S3), the stock is at an unsustainably high level, and growth is negative

(e.g., g(S3) ). To the right of S

mss

, stock will fall until carrying capacity is reached.

Equation of Motion

If we were to harvest an amount greater than the growth of the resource in a period,

i.e., if X

> g(S

), then the stock would be lower in the following period. Depending on the

level of the stock, such a harvest could either decrease growth (e.g., if the level of the stock

were S1) or increase growth (e.g., if the level of the stock were S2).

We can express the relationships between stock level, growth and harvest in an equation

of motion for the renewable resource:

t+1

= S

+ g(S

) - X

which states, in words, that "stock next period equals stock this period plus growth this

period minus harvest this period." [Aside: In this formulation of the equation of motion,

we are assuming that growth occurs before harvest in each time period.]

-4-

We can rearrange the equation of motion as follows:

t+1

- S

= g(S

) - X

In this form we see that the equation of motion says that "the change in the stock equals

growth minus harvest."

If we are in steady-state, then we know that the change in the stock is zero:

t+1

- S

= 0 = g(S

) - X

or, rearranging,

g(S

) = X

Hence, in steady-state, growth equals harvest. Of course, if we are in steady-state and

harvest equals zero, then growth g(S

) must equal zero, which is consistent with our earlier

discussion about steady-states.

Sustainable Yields and MSY

We have not yet talked about the optimal harvest of fish. A sustainable yield of fish refers

to the level of harvest which will result in a steady-state fish population. Every point on the

curve above represents a sustainable yield. It is possible, of course, to harvest fish off the

curve. Say we harvest a larger amount of fish than the growth of the fish stock, i.e., at a

point above the curve; what happens then is that the overall stock decreases by the amount

of the additional harvest and the system is not in a steady-state. The optimal policy, as

we will show, is to maintain a steady-state, or sustainable, level of harvest

through time.

Notice that if an amount of the stock X

is harvested each period such that X

= g(S

), then

the stock will remain at level S

. In this case the owner would be "harvesting only the

growth" and leaving the stock at its original level each period. Because the owner could

then harvest X

=g(S

) indefinitely, such a harvest level is termed a sustainable yield.

There are many possible sustainable yields. In fact, each point on the g(S

) curve to the left

of S

mss

could be a sustainable yield if harvest were carried out such that X

= g(St). Note

that the level of sustainable yield depends on the level of the stock St. That is, every

sustainable yield is associated with a steady-state stock of fish.

The highest possible sustainable yield is called the maximum sustainable yield,

denoted X

msy

. In Figure 9.1, X

msy

= g(S

msy

). The level of the stock associated with X

msy

called "the stock level that supports maximum sustainable yield," denoted S

msy

in Figure 9.1.

When Biologists were first analyzing Fishery Dynamics, they noted that the largest level of

sustainable harvest would be at the level X

msy

. At last, they thought, we do not need

economists to tell us what the optimal harvest level is for fish! And the reasoning was that

we would obviously be better off at the steady-state level associated with the highest annual

harvest of fish. Needless to say, this reasoning was wrong.

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Optimal Fish Harvest in Steady-State

with Interest Rate of Zero

Before we develop a full, dynamic, economic model of a fishery, let's examine an example

of fish harvest in steady-state. In this example, we will maximize the benefits minus the

costs of fish harvest subject to the constraint that we maintain a steady-state. Note that this

may not be socially optimal, because we still have not shown that it is socially optimal to be

in a steady-state or described the conditions under which this premise holds. Thus, the

problem we are about to solve is a second-best problem. Nonetheless, it is an important

second-best problem because policy-makers and their constituents often want steady-state

solutions because of the economic stability and predictability that they provide. Steady-

state solutions are important to policy-makers because steady-state solutions are sustainable

over the long run, i.e., the fish species (and the fisherpersons) don't go extinct!

In steady-state, the fish stock is not changing, thus harvest must equal growth:

g(A

) = X

When we examine a fishery in steady-state, the steady-state equation above serves as a

constraint. Let's assume that our objective is to maximize the total benefits, B(X

), derived

from fish harvest minus the total cost associated with fish harvest. Suppose that the total

cost of fish harvest is C(X

, S

• C(X

, S

) depends on the level of fish harvest, X

, and on the stock of fish, S

(this

is because the fewer the number of fish, the more expensive it becomes to locate

and catch them in the lake).

> 0 (the more you fish, the more it costs; i.e., positive MC)

< 0 (the cost of fishing declines as the level of stock increases)

Our choice variables are:

• the level of harvest X

, and

• the stock of fish S

Note: We indirectly control S

by changing X

. To keep the example simple, let's assume a

zero interest rate, so that we do not need to discriminate between different periods in time.

Our problem is now:

max B(X

) - C(X

)

subject to: g(A

, S

) = X

THE GREAT THING ABOUT STEADY-STATE: Because we will be in steady-

state, the fish stock and the harvest level will not be changing, hence the (second-best)

levels of X

and S

will be the same for every time period. Thus, we only need to examine

one of the time periods. So, the dynamic problem above becomes the following static

problem:

max B(X) - C(X, S)

X,S

subject to: g(A, S) = X

where we have dropped the time subscripts because we only need to examine one of the

time periods if we are in steady-state.

-6-

The Lagrangian expression for this problem is:

L = B(X) - C(X, S) + λ [g(A, S) - X]

and the FOC's are:

(1)

dB X

dC X S

= − − =

( ) ( , )

λ 0

(2)

dC X S

dg A S

= − + =

( , ) ( , )

λ 0

(3)

g A S X

= − =( , ) 0

FOC (1) says that price (price = MB = B

(X)) equals the marginal cost of changing the

harvest level plus the user cost of changing the harvest level.

FOC (2) says that the value of marginal product of the stock (λg

) equals the marginal cost

of changing the stock level. The marginal cost associated with changing the stock level is

the change in the cost of finding the fish in the ocean as the stock level declines.

How does the steady-state stock level, S

, compare with maximum

sustainable yield stock level, S

msy

From FOC (1), λ = price - marginal cost with respect to X. Hence, FOC (2) can be

rewritten as:

(P - Cx)g

= C

Now, because C(X, S) falls as S increases, (i.e., C

< 0), we know that (P -

Cx)g

must be negative for FOC (2) to hold.

• Given P - Cx ≥ 0 in steady-state (which must hold for π ≥ 0), then it follows

that g

< 0 in steady state. Look at Figure 9.1:

- if g

< 0, then S

> S

MSY

Thus, the steady-state (second-best) level of fish stock, S

, is greater than

the level of fish stock that supports the MSY, S

MSY

. This is because maintaining

a larger stock of fish has the additional benefit of reducing the cost of fishing.

Note that the steady-state (second-best) level of fish harvest, X

, is less

than X

MSY

Were the Biologists wrong?

The factor that Biologists did not take into consideration is the economic reality that

the cost of fishing decreases as the fish population becomes larger. We can

reconcile the Biologists notion in the case where C

= 0 (i.e., no stock effects in the

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cost of fishing).

-8-

• When C

= 0, FOC (2) is revised to read λg

= 0, which implies g

= 0 since

λ > 0. As we can see, the harvest associated with the maximum

growth rate of fish, g

= 0, is X

MSY

Open Access and Competitive Behavior

Without barriers to entry, competitive fishermen will not consider the future implications of

today’s fishing. Instead, they will maximize profit and, if there is a steady state, will

harvest the resource according to:

Price = B

= C

price = MC

g(A,S) = X steady-state

Competitive industry members will not recognize that their harvesting depletes the

resource, thus increasing production cost. When a resource has open access, each

fisherperson believes that any fish he/she does not catch will simply be caught by another

fisher, so that the gains to keeping higher stocks will not be realized anyway. Therefore,

under competition:

(1) Stock is smaller than optimal.

(2) Output may be greater or smaller than is optimal.

(3) Social cost associated with extra harvesting will outweigh benefits from extra

fish in cases where open access output is greater than optimal output.

Figure 9.2 -- There can be more than one steady state.

-9-

Referring to Figure 9.2, under competition, if C

is relatively small and demand is big,

there may be a period when X

> X

and the resource can be depleted, for example, when

= X

. The marginal cost may be increasing as S declines and the possible output stock

combinations under competition will follow the line AB and the steady stock with X = S =

0 is at B.

When the open access outcome does not involve a steady state, the fish stock can be sent to

zero through extinction. This has come very close to occurring in Ocean resources where

many countries have competed against each other to catch certain species and have driven

many of them near extinction. Under open access, each fisher seeks to extract every

profitable unit he/she can, but is immizerated by the entry of new fishers that always drive

the profits of fishing to zero. Hence, the fisherperson is always trying to reduce the costs

of extracting the resource in order to gain temporary profit (private individuals are price-

takers, so only care about costs):

• The best way to lower extraction costs is to simply catch everything, then trash

unwanted fish species, and this is largely what we have seen

- Drag Netting is often used to dredge up all fish in an area

- Clear-cutting of forests is another example

If the marginal harvesting cost

→∞

as the stock approaches zero, a steady state may be

reached with a very small stock level. The elimination of buffalo from the American West

between 1860-1870 is an example of when open access and low C

lead to extinction.

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If C

is substantial and under the optimal solution λ/C

is small, competitive outcomes

will not be far away from optimal outcome. Several intervention policies alter the open-

access outcome including:

• Standards limiting the catch. Can be performed by:

- Exhaustive capacity restrictions (i.e., limited # fish per boat)

- Time restrictions.

• An output tax of λ.

- Recall that the only difference between open access and the socially

optimal solution is that open access fishers do not recognize the shadow

value of maintaining the stock.

• Tax on fishing effort (can be placed on labor or on the number of boats)

• Moratorium on Fishing

• Change Length of Fishing Season (only encourages highly concentrated effort)

• Regulations on Technology (i.e., no gill-netting; no using “fish finders”, etc.)

- Regulating technology, of course, is incredibly inefficient.

The Need For International Agreement:

One of the things that is necessary to control open access problems is an International

Authority to Monitor and Enforce controls on fishing so that fish populations are not

unsustainably harvested by open access:

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Renewable Resource Management in Steady-State

A Simple Bioeconomic Model of a Steady-State Fishery

Consider a price-taking fishing industry with:

Fish Stock (in biomass units): S

Growth Function of Fish Stock: g(m, n, S) = mS - nS

/2.

Note that m and n are two parameters which we represented with parameter A in the last

lecture. In this case:

• m might represent availability of food in the ecosystem

• n might represent the number of predators, but is constant, i.e., n ≠ n(S)

Fish Harvest (in biomass units): X

Total Benefits of Harvest: B(X) = P*X

Total Costs of Harvest: C(X, S) =

Note that, the larger the fish stock, the lower the cost of fish harvest.

Assuming that the fishery will be in steady-state, the fishery's economic problem can be

stated as:

max B(X) - C(X, S)

X,S

subject to: g(m, n, S) = X

First, note that we can find several important quantities without solving the optimization

problem:

Maximum Sustainable Stock

Maximum sustainable stock S

mss

occurs where g(m,n,S) = 0, hence:

mss

= 2m/n

Of course, S

mss

will only be achieved if the harvest X = 0.

Stock Level that Supports Maximum Sustainable Yield

The stock level that supports maximum sustainable yield, S

msy

, occurs where dg/dS = 0:

msy

= m/n.

Maximum Sustainable Yield

In steady state, X = g(m,n,S). So , to find maximum sustainable yield, substitute S

msy

into

g(m,n,S):

msy

= g(m,n,S

msy

) = m

/2n

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Optimal Steady-State Stock and Harvest Levels

To find the optimal steady-state stock and harvest levels, we need to solve the optimization

problem outlined above. The optimal choice problem can be solved using the Lagrange

multiplier method. The Lagrangian expression for the problem is:

L = B(X) - C(X, S) + λ [g(m, n, S) - X]

The FOC's are:

(1)

dTB(X)

−

dTC(X,S)

−λ = 0

or, by substituting, P - b/S - λ = 0

FOC (1) says: price - marginal cost of harvest - user cost = 0

(2)

= −

dTC(X,S)

+λ

dg(m,n,S)

= 0

or, substituting, -(-bX)/S

+ λ [m - nS] = 0

FOC (2) says: marginal benefit of - marginal cost of = 0

decreased harvest decreased growth

costs due to overcrowding

(3)

dλ

= g(m,n,S)− X = 0

or, substituting, [mS - nS

/2] - X = 0

FOC (3) says: growth - harvest = 0, which is simply the condition for a steady-state.

Solving for Steady-State Stock Level

We can solve the system of FOC's for the optimal steady-state stock, S*, as follows:

Solve FOC (3) for X. Plug the resulting expression for X into FOC (2) to get:

(4) -[-b(m/S - n/2)] + λ [m - nS] = 0

Putting aside expression (4) for the moment, solve FOC (1) for λ:

(5) λ = P - b/S

-13-

Now plug λ into (4) and solve for S* to get:

(6) S* = m/n + b/2P

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Solving for Steady-State Harvest Level

To find the steady-state harvest level, X*, plug the expression for S* into FOC (3) and

simplify:

(7)

−

Notice that the optimal harvest level X* increases with price P, decreases with the harvest

cost parameter b, decreases with growth function parameter n and increases with growth

function parameter m. That is:

• As the price of fish increase, the optimal harvest level increases towards X

MSY

• As the cost of harvesting decreases through improved technology, X* increases

• As the number of predators increases, the fish stock decreases, which implies that

the optimal harvest, X* decreases

• As the available food in the ecosystem increases, X* increases

Say we want to increase our fish harvest. We can thus do it in several ways:

• Subsidize investment in new technology

• Remove predator species ( this may lead to other, unintended consequences)

• Add fish food into the ecosystem.

Solving for Steady-State User Cost

The user cost in steady-state, λ* , can be found by substituting S* back into FOC (1):

(8)

= P −













Numerical Example

Suppose P = 1, b = 10, g(S) = .2S -.002 S

(m = .2 and n = .004) .

Maximum sustainable stock is S

mss

= 100.

The stock level that supports MSY is S

msy

= 50.

Maximum sustainable yield is X

msy

= 5.

Optimal steady-state stock S* = 55.

Optimal steady-state harvest X* = 4.95.

-15-

Notice how changing the parameters would affect S* and X*. For example, if harvesting cost

parameter b were to increase to b= 50, then the optimal steady-state stock would increase to

S* = 75 and the optimal steady-state harvest would fall to X

= 3.75.

A common policy goal of fishery managers for many years was to attain steady-state stock

msy

and to set steady-state harvest equal to X

msy

. This policy maximizes biological

growth and steady-state harvest. However, we often wish to maximize the value of the

biological growth and harvest. S* and X* maximize the value of the biological growth and

harvest. Notice that S* > Smsy and X* < Xmsy, i.e., the value of the biological growth

and harvest is maximized by harvesting less and leaving a larger stock in the ocean. Note:

this conclusion may not hold in cases where the interest rate is non-zero.

Open Access Market Failure

Suppose we have a competitive fishery with open access to the fish stock. Under open

access, each fisherperson will ignore user cost, and the resulting steady-state will occur at

an inefficiently low stock level. Under open access competition, fishers ignore the user

cost component of FOC (1) and instead harvest until:

(9) P - dTC/dX = 0

or, substituting and rearranging,

(10) P = b/S

This implies that the steady-state stock under open access competition, Sc, is:

(11) Sc = b/P

Given growth function g(m,n,S) = mS - nS

/2 and the fact that we are in steady-state (i.e.,

X = g(m,n,S)), we know that:

(12) X = mS - nS

Substituting equation (11) into equation (12), we get an expression for the harvest level

under open access competition, Xc:

(13)

−

[ ]

In terms of our numerical example, Sc = 10 and Xc = 1.8. Hence, open access

competition results in too little stock (i.e., Sc < S*), and too little harvest (i.e., Xc < X*).

If the stock were allowed to increase, then larger harvests could be supported. However,

under open access competition, no fisher has the incentive to reduce current harvest in

order to allow the stock to increase and harvests to be larger in the future. This is because,

under open access competition, if a given fisherperson reduces her current harvest, then

some other fisherperson would harvest the fish in the current period. And the “tragedy of

the commons” result occurs.

-16-

A Harvest Tax to Correct Open Access Market Failure

From FOC (1) we know that "price = extraction cost + user cost" at the optimal steady-state

harvest level. However, if open access competition exists, market failure will occur

because private firms will have no incentive to consider user cost when setting harvest

levels.

A policy to correct this market failure is a tax per unit of harvest equal to the user cost at the

optimal harvest level. In our case:

(14) tax per unit =

= P −











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Shifting the Growth Function - Fish Feeding

Various management activities can be undertaken to influence the growth of resources. For

example, the growth function of a fishery may be shifted upwards by feeding the fish, or

the growth function of a forest may be shifted upwards by fertilizing the forest. Private

firms may undertake these activities, as in the case of salmon raised in pens off the coast of

Norway, or Christmas trees grown by farmers in the mountains of North Carolina.

Alternatively, when the resources are public goods, it may be efficient for these activities to

be undertaken by the government. Indeed, in many countries, government-financed fish

hatcheries feed and raise fish in order to maintain high commercial or recreational fishing

levels. In either case, the economic issue is to find the optimal level of feeding.

Let the growth equation for a renewable resource be denoted g(S, A), where ‘A’ is a

variable representing the level of feeding. We assume that feeding increases growth, i.e.,

dg/dA > 0.

Assuming that the price per unit of feed is denoted by v, total cost becomes:

(15) C(X,S,A) = C(X,S) + vA

And the optimization problem is:

(16)

max

X,S,A

B(X) − C(X, S)− vA

subject to: g(S, A) = X

-17-

The Lagrangian expression for this problem is:

(17)

max

X,S,A,λ

L = B(X) − C(X, S)− vA + λ g(S, A)− X[

]

The first-order conditions include the familiar conditions:

(18)

−

− λ = 0

(19)

= −

+ λ

= 0

(20)

dλ

= g(S,A)− X = 0

but we also have a new first-order condition:

(21)

= λ

− v = 0

This new condition says that we should increase feeding until the value of marginal product

of feeding ( λ dg/dA) is equal to the price of feed v. FOC equations (18)-(21) represent a

system of four equations in four unknowns that can be solved for the optimal levels of X,

S, A, and λ in a similar fashion as in the last example.

Looking ahead . . .

Although the preceding steady-state models are elegant, they ignore several important

issues. For example:

• Resource populations can be driven to extinction (not just to a lower steady-state

stock level) under open access competition.

• Interest rates play a very important role in the management of renewable

resources and should be included in dynamic models.

• Managers are often interested in what stock and harvest levels will be on the way

from current conditions to an eventual steady-state. The way the the stock or

harvest level changes from one state to another in a dynamic system is referred to

as transition dynamics.

In the next lecture we investigate a more general model that can address these issues.