R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
Economics of Forests
So far, we have studied economics related to exhaustible
resources. We shall study economics of renewable resources –
forests and fishery in the next few lectures.
We shall take up the case of a renewable energy resource,
namely fuel wood. Unlike exhaustible resources, renewable
resources can get replenished over time. However, if our rate of
consumption is faster than the rate at which these resources are
replenished, then the stock of these resources will reduce over
time, and may lead to their eventual exhaustion.
The economic concepts that we shall be studying in this lecture
are applicable to such renewable resources as fuel wood and
biomass. Other renewable resources such as solar or wind are
generally not studied by supply and demand branch of
economics as their supply is virtually unlimited and not scarce
in that sense. As indicated in the introduction, economics is a
study of scarce resources.
Forests are the source for fuel wood. However, forests have
multiple uses and fuel wood is only one of them. We shall study
the general aspects of forestry, which are equally applicable
when forests are narrowly viewed as sources for fuel wood.
1
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
Additional benefits from forests include the timber for houses,
pulp for paper making, preservation of bio-diversity (housing a
variety of plants and animals), purifying the air especially by
consuming CO2 and releasing oxygen during photosynthesis.
Deforestation is a serious problem because it intensifies global
warming,
decreases
bio-diversity,
reduces
agricultural
productivity, increases soil erosion and desertification, and has
precipitated the decline of traditional cultures of people
indigenous to forests. Instead of forests being used on a
sustainable basis to provide the needs of the current as well as
subsequent generations, forests are being "cashed-in."
Unlike stocks of depletable resources, stock of renewable
resources such as trees can be increased over time. However,
there will be some finite level beyond with any given stocks of
renewable resources cannot increase. The amount of biomass in
an acre of land depends on two factors: on the input side, the
amount of light and nutrients in the soil will determine the
growth of biomass; on the external side, influence of disease,
decaying agents and accidents will limit the growth of biomass.
2
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
Some biological basics of trees and forests
As a tree grows, the amount of wood usable for commercial
harvests changes over time. The typical schedule of the tree's
growth over time is sketched in the following figure.
v(te)
Wood
v(t*)
Volume
v(t)
t*
te
Age of tree t
The figure plots the volume available from the tree v(t) as a
function of the tree's age. The volume grows over the age of
tree, until a maximum is attained at age te. Beyond this age, the
3
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
tree begins to decay from disease, insect predation, wind etc/
and collapses eventually.
Wood volume develops slowly in the early stages, as the tree
takes root. The rate of growth of wood volume (dv/dt) increases
in the early stages, reaches a maximum at the age t* and then
begins to decrease. The growth rate is zero at te, and then there
is negative growth. Thus, the variation of wood volume with
respect to its age follows a logistic pattern. A logistic pattern can
be mathematically represented as follows.
dv
 v
 r 1  
dt
 k
where  represents the instantaneous growth rate of biomass and
k is the carrying capacity of the forest (i.e., maximum biomass
the forest can support).
The volume-age schedule can be altered by arranging for the
optimal density of the tress on a plot, by fertilizing the land, and
by other activities such as thinning of trees and pest repression.
The schedule can shift in a variety of ways in response to human
intervention.
This
is
part
of
forestry
cultivation
and
management, generally known as silviculture.
A forest stand with trees of identical age can also have a similar
shape to this schedule. On a plot of land of a given size, many
4
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
trees can be grown so that there will be an aggregate v(t)
schedule representing the sum of many separate v(t) schedules.
If there are N trees, V(t) = Nv(t) only when we assume no
interaction effects. This is a unrealistic assumption since the
density of trees and other factors affect each one's growth.
However, for simplicity, let V(t) = Nv(t). We are assuming here
that all trees on the plot of land are identical in type and age at
any point in time. this is certainly a common feature of
cultivated forests, (e.g.), a forest of trees yielding large amounts
of firewood. Forests grown artificially solely for the sake of
getting firewood from it are called Energy Plantations.
The volume-age schedule shown earlier can also be represented
in terms of the rate of wood growth, dv/dt. As seen earlier, dv/dt
increases at early years in the life of a tree, reaches peak at the
age t*, and then reduces to zero at te.
5
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
dv/dt
t*
v(t*)
V(t*)
Age of tree t or
Tree volume v(t) or
Forest volume V(t)
6
te
v(te)
V(te)
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
Note that the same relationship holds when the wood volume
v(t) is used in the X-axis. For a forest with identical trees, it is
also possible to get similar relationship when the total volume of
the forest V(t) is used in the X-axis.
Before proceeding further, let us briefly discuss certain concepts
related to the economics of renewable resources, not necessarily
the renewable energy resources. I am interested in highlighting
the important concepts related to sustainability when we use
renewable sources.
Typically, the economic theory of renewable sources is built
using the model of fisheries. We shall see later that basic
concepts of forests are applied approximately well for the case
of fisheries also. For example, the relationship between the
growth rate of fishes in a fishery (dX/dt) and stock of fish X can
be given by the relationship shown below.
7
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
F(X) =
dX / dt
F(X*)
X'
X* = XMSY
Stock of fish, X
8
X''
k
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
When the stock of fish is small but positive, the biomass will
grow rapidly because there will be surplus nutrients (food). New
births will outnumber deaths. Growth will eventually reach a
maximum at X = X*. By then, food becomes scarce as the
limited food will be shared by a large stock now. We also
account for natural predation of fish by their natural predators –
other than human – in the deaths here. Beyond X*, growth rate
declines as deaths outnumber births beyond this point.
Equilibrium is reached at the point X = k, when the births and
deaths are in equilibrium. The value k is called the carrying
capacity of the fishery, which can be thought of as the
maximum population or biomass the habitat can support. Note
that the word habitat is used in a general sense. It represents the
place where the resource is grown; it is a forest for trees or it is a
fishery for fishes. The resource, tree or fish, is generally referred
to as biomass.
The same model can characterize a natural forest. In the absence
of
human
intervention,
forests
have
grown
following
approximately this pattern. However, the rate of growth of
forests will be very slow compared to fishes. The point k is not
the carrying capacity of the forests, the maximum number of
trees that the forest can support.
9
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
dV/dt or
dX/dt
X*
V*
Volume of forests V or
Stock of fish, X
10
k
for forest
k
for fishery
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
Let us now see what will happen when we introduce human
intervention to the biological equilibrium. Let us study the
binomic equilibrium, the concepts of renewable source literature
connecting biological and economic factors – an equilibrium
that combines the biological mechanics with economic activity
followed by human beings. Specifically, we now assume that
humans have begun to harvest the stock. Let us now consider
three harvest rates and their effects on the biomass population in
the habitat.
11
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
dX/dt
H1
dX2/dt
dX1/dt
H2
H3
X'
X1
X2
XMSY
Biomass, X
12
X''
k
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
Assume that the tock is in biological equilibrium at k. Let the
harvest from the habitat be H1 in the above figure. This
represents a harvest level that lies everywhere above the
biological growth function. This means that the stock is
removed from the habitat at each point in time than being
renewed. Obviously, no population can survive for long if more
are harvested than are renewed. Obviously, the stock will be
depleted to zero levels if this level of harvest is repeated
continuously. This is the extreme case of mining the habitat, and
the stock will be eventually depleted.
Actually, this is what has been happening to the stock of so
called non-renewable resources. The rate at which they are
extracted (harvested) is much higher than their rate of renewal.
The principles can be applied to renewable energy sources –
forests. If the rate of consumption of wood resources is much
higher than the rate at which they are renewed, we will witness
deforestation (what we see in many parts of the world now). Of
course, unlike coal or oil, it is possible to affect the rate of
renewability of forests by artificial means such as forestations
programmes, plantation programmes, or through the use of
fertilizers. Study of the best way by which the plantations can be
carried out to maintain continuous renewable supply over time is
an important area of the subject of forestry. We shall study these
details later.
13
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
Suppose now that the instantaneous rate of harvest is given by
H2. H2 touches the growth function dX/dt at its maximum point.
The point X* at which growth rate is maximum is called the
point of Maximum Sustainable Yield (MSY) from the
population. Suppose that the population is at equilibrium k
(where dX/dt is zero). In the next instant, if H2 quantity of stock
is harvested from the habitat and H2 is maintained continuously,
the stock will gradually decline from k to XMSY. The remaining
stock of biomass will now grow at the maximum rate, because
food and space are more ample now than at k. Note that the rate
of growth at XMSY is equal to the harvest (i.e. the rate at which
the stock is removed) H2. This means, if H2 amount of stock is
removed at any instant, the growth rate, which is equal to H2,
will make the stock to be replenished instantly. Thus, at the
MSY stock, the largest sustainable harvest can occur. That the
process of catching H2 amount of stock per unit of time can
continue indefinitely (so long as no exogenous changes occur).
In a very simple economic model, i.e., the model in which no
harvesting costs or discounting of future revenue from the
harvest are considered, the MSY is the most desirable
equilibrium for the fishery (or forestry).
Note that if the initial population were to the left of XMSY (say
X1) to begin with, a harvest of H2 per unit of time will deplete
the stock, because the population X1 cannot be sustained at a
14
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
harvest rate of H2. The maximum growth possible is dX1/dt,
which is less than H2. Thus, in every unit of time, H2 – dX1/dt
amount of stock will get reduced from the total stock.
Eventually, the stock will be depleted.
A harvest H3 is an interesting one. At this harvest, there are two
possible equilibria, X' and X''. Which will occur?
Suppose that harvesting has just began when the habitat is at
equilibrium at the point k. Obviously, when H3 is captured
where the growth rate is zero (at point k), total population
declines. Equilibrium occurs at point X'' where the rate of
growth is exactly equal to the harvest H3. Once this point is
reached, the harvest H3 can be sustained forever. Note that this
is not the maximum sustainable yield (XMSY).
Suppose now that the original population was not at k but
between X' and X'' when the harvest H3 has begun. The rate of
harvest H3 will be lower than the growth rate, and hence
population will begin to increase. Population will ultimately
stabilize at X''. Thus, for harvest H3, X'' is called the stable
equilibrium. This means that if there is a slight movement in the
stock size to the right or to the left of X'', the system will
ultimately return to an equilibrium at X''. The arrows in the
above figure toward X'' show that it is a stable equilibrium.
15
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
Suppose once again that the population lied to the left of X'
when harvesting H3 begins. This can happen, for example, due
to the destruction of forests due to fire or oil or chemical spills
in fisheries. If harvesting H3 is continued, we can see that the
growth rate is below H3, and hence H3 cannot be sustained. The
population will be extinct.
What happens at the point X'? Obviously, there will be
equilibrium for a harvest of H3 because the rate of growth just
equals the harvest. But, this will be an unstable equilibrium
because a slight movement of stock (which can occur in the
presence of some external factors – fire, predators, climate
change, etc.) can make the population either extinct or be stable
at X''. This is indicated in the figure by the outward going
arrows at the point X'.
Let us now go deeper in defining harvest. Harvest H is a
function of stock X and the effort E. Here, effort can be
considered as the aggregate of the factors of production –
capital, labour, materials, energy, etc. Obviously, harvest will
rise for a higher stock, i.e., H is an increasing function of X.
What about the relation between H and E?
16
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
Suppose we take a given level of effort, E'. The harvest will then
be an increasing function of stock X. The function is shown in
the figure in the previous page. The steady state equilibrium will
now occur at the stock X' and the harvest will be H'.
Let us see what happens when effort is increased from E' to E'',
say by bringing more labour for cutting trees or more machinery
and energy for the purpose. The slope of the harvest function
will now be higher because now it is possible to get a greater
harvest for a given increase in X. However, the amount of
harvest cannot be increased because of the limitation of the
biological growth process. Note from the figure that the steady
state harvest is the same as the harvest with E'. However, now
the equilibrium stock of biomass, X'' is now much lower. This
means that if more firms operate on the forests (which increases
effort), then the equilibrium level of forest stock for a given
harvest will be lower compared to the same harvest when only a
few firms operate in the forest. Or, the equilibrium forest stock
will reduce as more firms begin operating in the forest.
Let c be the constant, unit cost of effort. It may be the wage in a
labour dominated forest industry. Then, the cost of effort is cE
where E is the quantum of effort. Assuming that no other costs
are involved, we have total cost, TC = cE. If p is the price per
unit of timber, then pH is the total revenue TR from the harvest.
17
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
If we assume that forest is a common property and that no firm
controls the products of forests, then price will be fixed
exogeneously depending on demand and supply, and hence p
will be constant. Given the total revenue and total cost
functions, the optimal harvest will occur when TR = TC.
Let us plot the variation of total revenue with respect to E. We
first draw the variation of dX/dt with respect to E, which is
similar to the figures we have seen earlier. Suppose we start
with a biomass X = k, with no timber is harvested. As effort is
introduced, the stock of timber falls. Harvests at first rise as the
industry moves up the biological production function. They
reach a maximum at XMSY and then fall again. The variation of
dX/dt with respect to E is shown in the figure in the next page.
Note that, as TR = pH, and as H is the same as dX/dt, the same
variation captures the behaviour of TR for a constant p = 1. At p
= 0.5, TR will be below the dX/dt curve, and at p = 2, TR will be
above the dX/dt curve.
18
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
dX/dt
TR(p = 2)
dX/dt or
TR(p = 1)
TR(p = 0.5)
E
19
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
TR in $
TC = cE
H0=TR0
A
E0
E
20
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
Let us assume a constant stock X. Then, H = H(E), is a function
of effort. Let p = 1 so that TR curve is identical to H(E) curve.
For a given c, total cost is cE, and the curve is linear. The
equilibrium point is E0 where TC = TR. For any effort less than
E0, TC < TR. This will cause profits and hence additional firms
will enter the industry. Equilibrium occurs only at E0. The
harvest is H0, and total revenue TR0 = pH0 = H0.
Now let us see what happens at different prices. When the prices
is at $0.50 per unit, equilibrium is at point A corresponding to
effort E0 and harvest H0. When p = 1, the equilibrium is at (E1,
H1). Note that the harvests are obtained by mapping to the TR
curve when p = 1, which is also the curve showing the variations
in optimal harvests.
Let us now plot the variation of harvest with prices. Note that at
higher price of 2, the harvest has reduced resulting in the
backward bending supply curve. The supply curve shifts at a
point above (p1, H1) because HMSY > H1. Note that the supply
curve describes the maximum sustainable harvest of the forest
for given prices. It is quite possible to increase the supply curve
at higher prices, but will not be sustainable. Deforestation will
result.
21
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
TC
C
TR(p = 2)
HMSY
TR(p = pMSY)
H1
Harvest
&$
B
TR(p = 1)
and harvest
TR(p = 0.5)
H2
H0
A
E0
E2
E1
E
p2 = 2
pMSY
p1=1
p0=0.5
H2
H0
22
H1 HMSY
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
p2
D2
p1
D1
p0
D0
H2
H0
23
H1
R. Ramanathan/ Energy and Environmental Economics / HUT/ January-April 2001
At the demand function D0, equilibrium wil be (p0, H0). When
demand rises from D1 to D2, harvest will reduce to H2 to be sold
at the higher price p2. If the supply does not follow this trend,
the forest will be over exploited leading to deforestation. The
income preference for fuelwood suggests that higher prices,
people will switch over to better cleaner fuels, but if the society
has large poor people, exploitation will occur.
24