PART ONE
ENVIRONMENTAL ECONOMICS
CHAPTER ONE
1.0
Scope of Environmental Economics
What is environmental economics?
How environmental economics is different from natural resources economics? The
definition of environmental economics is narrow and broad.
1.1
Narrow Definition
Environmental Economics is distinct from the sister discipline of natural resource
economics. It is in the narrow sense concerned with:

Welfare economics

Economics of pollution

Valuation theory, and

Environmental policy
Narrowly defined, environmental economics is concerned with what we sometimes
call brown issues, the topics given above.
1.2
Natural Resources Economic (NRE) in Narrow Sense
NRE is concerned with optimal utilisation of natural resources economics and is
concerned with what are sometimes called green issues.
1.3
Broad definition of environmental economics
Not distinct from natural resource economics in broad sense, environmental
economics in concerned with:

Welfare economics

The economics of pollution

Valuation theory

Environmental policy and

Optimal utilisation of natural resources
1
Economic objective
Social objective
1.4
Ecological objective
The Nature of Welfare Economics
Concerned with what ought to be; that is welfare economics is based on normative
economics. Focuses on using resources optimally so as to achieve the maximum well
being for the individuals in the society. source: Just Hueth and Schmitz (1982).
1.5
Basic issues in Welfare Economics
Deals with 3 questions
1. How should society’s resources be used and what social organization should
society adopt. Social organization refers to market system, mixed economy or
socialization.
2. How can we tell whether any change we make is for the better?
3. What would be the properties of an acceptable welfare functions (Arrows
Impossibility Theorem?
1.6
The Meaning of Efficiency and Social Justice in Welfare Economics
The definition of efficiency in welfare economics owes much to the economist
Vilfredo Pareto. Therefore an efficiency social state is often called Pareto optimal.
1.6.1 Pareto efficiency (optimality)
A situation is efficient or Pareto optimal if it is impossible to make one persons better
off except by making someone else worse off.
Source: Layard and Walter (1978) P.7
Alternatively, Pareto Efficiency can be defined as a situation in which it is impossible
for one individual to gain without another individual incurring a loss.
2
1.6.2 Pareto inefficiency
This is a situation in which it is possible for someone to gain without another
individual incurring a loss.
1.6.3 Pareto improvement/pareto superior move
If an economy is inefficient it is possible to make at least one person better off at no
cost to anyone else.
1.6.4 Three conditions for efficiency
Three conditions are necessary for efficiency:
First condition is, efficient consumption/exchange efficiency
Efficient consumption requires that all individuals place the same relative value on all
products (value being assessed at the marginal i.e. all individuals have some marginal
utility).
source Layard P.10
Graphically
Exchange efficiency is described by the tangency between two indifference curves.
Qy O2
D
C
B
A
1
2
IV
3
4
QX
III
I
II
From Z to A is pareto improvement because individual number 2 will gain without
individual no 1 incurring a loss. Similarly from Z to B.
NB exchange efficiency occurs on the contract curve: points A, B, C, and D.
Formally
3
2
n
MRS 1XY  MRS XY
 ....  MRS XY
Where i = 1, 2, …., n are individuals consuming X and Y commodities.
Alternatively;
i
j
for all consumers i and j in the economy, on commodities X and Y
MRS XY
 MRS XY
Second condition: Efficiency production
Efficient production requires that marginal rate of substitution between functions of
production be the same in all industries.
Source Laynard and Walter P. 12
Graphically
Production efficiency is described by the tangency between two isoquants.
Edgeworth - Bawley Box
Figure 2.
Production of X and Y using inputs labour and capital
QY
D
C
B
A
1
IV
2
K
3
4
II
III
I
QX
Efficiency locus
NB: Production efficiency occurs on the efficiency loci: A, B, C, and D. These
indicate the rate at which marginal rate of transformation of X and Y is equal by
varying labour and capital.
4
Third Condition, Efficient product mix
Product - mix efficiency requires that the subjective value of commodity X in terms of
commodity Y should equal the two marginal costs.
Graphically
E
D
B
C
NB: Exchange and production efficiency occur at points D and E, where:
MRS XY  MRTXY
Formally
MRS YX  MRTYX
Where MRS YX 
Ux
UY
MRTYX 
Yk
Xk
Additional Condition: Forth condition
Social justice and the social optimum
There are numerable efficient configurations of the economy, depending, among
others, on the initial distribution of resources.
However, efficiency does not constitute a social optimum.
To obtain the social optimum we need the utility frontier of the economy, known as
the Grand utility frontier. We also need the social welfare function.
To obtain the grand utility frontier we take each feasible output mix from the
transformation curve, and construct its utility frontier. This gives a family of utility
frontiers. The overall utility frontier is then the outer envelope of all these frontiers:
The slope of the grand utility frontier is
5
U 2x
U 1x
U2
Utility frontier for (X0,Y0)
Grand utility frontier of Pareto optimal points
Utility frontier for (X1,Y1)
Utility frontier for (X2,Y2)
U1
The social welfare function is given as

W  W U 1 ,U 2

This gives rise to what is sometimes called the Iso-welfare curve.

W0  W U 1 ,U 2

With a slope
dw / u 1 Wu 1

dw / u 2 Wu 2
To obtain the social optimum (or constrained “BLISS POINT”) we use the social
welfare function to pick the preferred utility mix. At the optimum, the slope of the
isowelfare curve equals the slope of the grand utility frontier, i.e.
U 2 x Wu 1

 social optimum
U 1 x Wu 2
This is the fourth optimality condition. It is the condition for social justice. This
therefore implies that for social justice we need fourth conditions.
6
CHAPTER 2
A MODEL OF EXTERNALITY IN CONSUMPTION
2.0
CONDITIONS REQUIRED FOR A MARKET SYSTEM TO
EFFICIENTLY ALLOCATE RESOURCES
1. Markets exists for all goods and services exchanged.
2. All markets are perfectly competitive.
3. All transactors have perfect information.
4. Property rights are fully assigned.
5. No externalities exist.
6. All goods and service are private goods i.e. there are no public goods and common
properly resources.
7. Long run average cost are non-decreasing that is there are no natural monopolies.
8. All firms are profit maximisers and all individuals are utility maximisers (profit
maximization is a subset of cost minimization)
9. Transactions costs are zero.
10. All relevant functions satisfy convexity conditions (utility functions, production
functions).
In reality however we find:
1) Missing futures markets
2) Imperfect information
3) Information asymmetry among transactors i.e. imperfect information
4) Property rights are not fully assigned absence of property rights
5) Externalities are present, especially in the use of environment goods
6) Public goods and common properly resources
7) LRAC are decreasing i.e. we have natural monopolies
8) Not all firms or individuals are profit maximisers or utility maximisers
9) There are positive transactions costs
10) Not all relevant functions satisfy convexity conditions (some isoquant are
7
Conclusion
Therefore one of the fundamental messages of environmental economics is that we
must be pessimistic about the outcomes of unregulated markets.
2.1
Distinction between Private and Social Values
Private costs: these are costs that are born to undertake an activity b the economic
agent undertaking the activity.
Social costs: there are costs that are born to undertake an activity by the economic
agent undertaking the activity plus costs imposed on all other economic agents.
Social costs = private costs + external costs therefore SC > PC
Private Benefits: Benefits obtained from the realization of an activity by the
economic agent undertaking the activity plus benefits obtained by all other economic
agents.
Social benefits = private benefits + external benefits
therefore SB > PB
PARETO EFFICIENCY AND EXTERNALITIES
In the presence of externalities, a competitive equilibrium is not pareto efficient. In
other words in the presence of externalities, the pareto efficiency conditions are not
the same as the competitive profit maximization or utility maximization conditions.
2.2
TECHNICAL DEFINITION OF EXTERNALITY
Externality is the case where the actions of one economic agent affect the utility or
production functions of another agent.
Non-technical definition of externality
An externality is the impact of one person’s actions on the well being of a bystander.
If the effect of a bystander is adverse, it is called a negative externality. If the effect is
beneficial, it is a positive externality.
8
2.3
Types of Externalities
Pecuniary externalities: prevail when changes in industrial output induce changes in
the price of one or more inputs employed by firms in the industry. This is not a cause
of market failure. They are captured by market system so there no need to worry.
Technological externalities: prevail when at least one of the variables in production
or utility functions of an agent falls under the control of external economic agents.
This is a cause of market failure.
Positive externalities occur when there are external benefits. In this case social
benefits are greater than private benefits. Negative externality occurs when there are
external costs. In this case social costs are greater than private costs.
Unidirectional externalities occur when externalities occur in one directional. They
are unilateral. Reciprocal externalities occur when externalities are reciprocal e.g. in
the case of negative externalities each of the agents damages and gets damaged by all
the other agents through its actions e.g. acid rain in the Europe and industrial
pollution.
2.4
Impact Of Externalities On The Markets System
When there are external benefits, a market system that is unregulated would underproduce the (desired) quality of a given goods. In other words
Figure : External Benefits
prices
MSC (MPC) = SS curve
MEB
MSB = D1 MSB = D
Qm
Qs
9
MEB = marginal external benefits, MPB = Marginal private benefits,
MSB = marginal social benefits
Figure : Alternative Method of Depict a Positive Externality
P
MC
MSB
MR
Qm
Qs
Observation
Area A is social loss, MR is lower than MSB because the firm does not tae into
account the value of the benefits its activity creates for others by the externality. The
firms produces at the output where MR = MC instead of choosing the social optimum
where MSB = MC.
Conclusion
When there are positive extenality not enough of the activity is undertaken by the
market system.
When There Are External Costs
Where there are external costs an unregulated market system would over produce the
undesired quality of a given goods.
Another way (alternative way) of depicting negative externality
Observation
10
Area B is social inefficiency, MC is lower than MSC because the firm does not take
into account the value of the damage imposed on others by the externality. The firms
produces where MR = MC instead of the social optimum where MR =MSC.
Conclusion
When there are negative externalities too much of the activity is undertaken by the
market system.
11
CHAPTER 3
3.0
POLICIES AIMED AT INTERNALISING EXTERNALITIES
THE BASIC ISSUES
1) Taxes versus subsidies which ones are better
2) What form should a Pigouvian Tax take?
a) Tax on output
b) Tax on inputs
c) Tax on emissions
3) What other policies generate efficiency
3.1
Taxes Versus Subsidies Which Ones are Better
The basic issue in this case is “should an economic agent causing an externality be
taxed or subsidized in order to induce the economic agent to internalize the
externality?”
The following model by Baumol and Oates shades light on this issues. Let a firm’s
cost curve be defined as
Total cost, TC = TC(Q)
Average cost = AC
(1)
TC Q 
Q
Marginal cost, MC =
(2)
TC Q 
(3)
Q
Case # 1: A tax scheme, a pigouvian tax, t, is levied per unit of output. Then that
changes equation (1) to (3)
TCt  TC Q   tQ
M Ct 
TC Q 
t
Q
(4)
(5)
CASE # 2: A subsidy scheme such that
S  S Q * Q where Q * is the social
optimum and Q is the private optimum. The, equation (1) to (3) become.
12
TC s  TC Q   S Q * Q 
M Cs 
TC Q 
S
Q
(6)
(7)
M Cs  MC with subsidy
In order to compare the two schemes, we set t = s from (6) and (7).
Therefore if t = s, MC is the same in both equation (6) and (7)
CONCLUSION
a) Since AC is lower under subsidy scheme it will encourage an increase in
output (Q).
b) For a positive externality, social optimum requires an increase in Q; therefore
a subsidy is recommended
c) For a negative externality social optimum requires a decrease in Q, therefore
A Pigouvian Tax is recommended.
What form should a pigovian tax take?
Consider a model of externality in production where;
Firm 1’s output are:
(X, Z)
X = F(Lx , Kx)
Z = Z(Lx , Kx, KA)
Firm 2’s output are:
(Y)
Y = G(LY , KY, Z)
where Z = Z(Lx , Kx, KA)
Assume that
G
< 0: negative externality
Z
13
Firm 1
(a) Profit maximisation conditions
The objective function is:
Max  1  Px F Lx , K x   WLx  rK x  K A
F .ON.Cs
 1
F
 Px
W  0
Lx
Lx
(1)
 1
F
 Px
r  0
K x
K x
(2)
 1
 r  0
K A
(3)
(b) Pareto Efficiency Conditions Are As Follows
 1
F
G Z
 Px
 Py
W  0
L x
L x
Z L x
(1)
 1
F
G Z
 Px
 Py
r 0
K x
K x
Z K x
(2)
 1
G Z
 Py
-r 0
K x
Z L x
(3)
Firm 2
Objective function
Max  2  PY GLY , K Y , Z Lx , K x , K A   WLY  rKY
F .ON.Cs
 1
F
 Py
W  0
L y
L y
(1)
 2
G
 Py
r 0
K y
K Y
(2)
14
CASE # 1: A Tax On Output X
 1  P F F L X , K X   WLx  rK x  K A
where P F  Px t   PX  PY
t  PY
G
X
G
X
G 

 1   Px  Py
 F Lx , K x   WLx  rK x  K A
X 

F .O.N .Cs
 1 
G  F
  Px  PY
W  0

L x 
X  L x
(1)
 1 
G  F
  Px  Py
r 0

K x 
X  K x
(2)
 1
r 0
K A
(3)
NB: This is a suboptiomal solution because it does not generate the pareto efficiency
conditional for firm 1.
CASE # 2: Tax on inputs Lx and Kx
Tax  t  Py
G
G
, andt  Py
Lx
K x

G
 1  PX F Lx , K x    w  Py
Lx



G
 Lx   r  Py
K x


F .O.N .Cs
15

 K x  KA

 1
F
G
 PX
 W  PY
0
L x
L x
L X
 PX
F
G
 PY
 w` 0
L x
L X
(1)
 1
r 0
K A
 1
F
G
 PX
 r  PY
0
K x
K Y
K X
 PX
F
G
 PY
r 0
K x
K X
(2)
 1
 r  0
K A
(3)
NB: This is a sub-optimal solution.
16
CHAPTER 4
4.0
THE ECONOMICS OF POLLUTION
Types of pollution
There are essentially three of pollution
i.
Air pollution
ii.
Water pollution and
iii.
Solid waste
Most of the substances, which cause pollution, are naturally present in the
environment in low (background) concentration are usually considered to be
harmless. Thus, a particular substance is considered to be a pollutant only when its
concentration is relatively high and causes adverse effects.
Pollution is defined as the presence of certain substances beyond the absorptive
capacity of the earth or environment.
4.1
The rotation between emission and pollution damage
Absorptive capacity of the environment (3)
Pollution Accumulation (2)
Pollution damage (4)
(1) Emission
(1) Can either be 3 or 4
(4) Depends on 3
If: 1 > 3 then 2 and 4 occur (i.e. pollution accumulate and cause damage)
4.2
Efficient Pollution Abatement
Occurs where the marginal abatement cost is equal to marginal damage cost due to
pollution.
17
Definition
Marginal Abatement Cost (MAC)
Is the cost of abating (controlling) an extra unit of pollution. It is also known as
marginal control cost (MCC)
Marginal Pollution Cost (MPC)
Is the health or environmental damage caused by an extra unit of pollution. It is also
known as marginal damage cost (MDC).
The relationship between Marginal Damage Cost and quantity of pollution is as
follows.
DISCUSSION: “There can never be zero pollution if abatement cost are considered”
Amount ($) MC
MDC
Quantity of pollution
NB: The marginal damage does not start at zero level of pollution because of the
ability of the environment to assimilate a certain amount of pollution without any
damage.
18
The relationship between the marginal cost of abatement and the amount of pollution
is as follows:
Amount ($) MC
MAC
B
O
A
Quality of pollution
NB: The higher the MAC, the lower the quality of pollution, and vice versa. The more
we spend to control pollution, quality of pollution is less.
OPTIMAL CONTROL OF POLLUTION
MC
MAC
MDC = MPC
cost
Total damage cost
Total control cost
Quality of pollution emitted
A
Q*
B
At A MAC > MDC
At B MAC < MDC
At Q* MAC = MDC
19
Point A means we are spending for too much on controlling pollution compared to the
damage caused by pollution.
Point B means that we are spending far less on controlling pollution than the damage
caused by pollution
As long as MAC ≠ MDC, there will be an incentive to change i.e. Market forces will
not be balanced.
Equilibrium occurs at Q* where MAC = MDC equilibrium is a solution in which
market forces are balanced such that there is no incentive to change.
Optimal Level Of Pollution Control
A contrast of perspectives
MDC
MC
MAC
Quality of
pollution
Q*
ecologists
economic
pollution
capitalist
For ecologists, ecological optimum occurs where there is zero tolerance.
For capitalist, optimum occurs where there will be zero amount of money spent on the
control of pollution to maximize profits.
20
Summary Of The Basic Results Concerning Externalities And Pollution
Abatement
i.
Private optimum occurs where
MPB = MPC
ii.
Social optimum
MSB = MSC
iii.
Optimal pollution control
MAC = MPC or MAC = MDC
Formal derivation of the principles of the economic of pollution
The purpose in this case is two fold:
1) To show that, for the individual economic agents, optimal pollution control
occurs where MAC = MPC (MDC).
2) To show that, for society, optimal pollution control entails equating the
marginal cost of reducing pollution across the economic agents causing
pollution.
Formal Derivation Of The Principle Of Optimal Pollution Control
1) The optimal level of pollution control occurs where the marginal abatement
cost is equal to the marginal damage cost due to pollution
2) Suppose the objective of social policy is to reduce pollution to a certain level,
say, by 50%.
The optimal policy to achieve this goal is to minimize social costs given the target
reducing in pollution. This entails equating the MC of reducing pollution across firms
FORMALY
Suppose we have n different firms in a large city
Let Zi = pollution by firm i
n
  Z i Total pollution
i 1

The target as to reduce the total pollution to Z say 50%.
21

n
_
Constraint is:   Z i  Z

i 1
Cost for firm i is as follows
C i Qi , Z i 
C i
where C 
0
Q
i
Q
C Zi 
C i
 0 : Marginal cost of reducing pollution MAC
Z
Cost of MC
MC
Output, Q
C Zi 
C i
0
Z
Cases of an individual firm
Costs of MAC
MAC
Pollution, Z
22
Furthermore
n
 C Q , Z : total cost
i
i
i 1
i
The objective of society is to:
min
zi

n
n
i 1
i 1
 C i Qi , Z i  s.t.  Z i  Z

NB : Min F X  MAX  F  X 
Therefore the objective may be restated as
max
zi
n
n
i 1
i 1

 C i Qi , Z i  s.t.  Z i  Z

Lagrangean function is:
n
 n

L   C i Qi , Z i     Z   Z i 
i 1
i 1


F .O.N .C
L C i

  0
Z i Z i
-
(1)
C i

Z i
(2)
shadow price of pollution MDC
MAC
Since is common to all firms
 C i
  
 Z i
MAC I
  C j
  

  Z j




(3)
MACi
Conclusions
1. For the individual firms, the optimal pollution control occurs where MAC =
MDC, equation (2) above.
23
2. For society as a whole, optimal pollution control entails equating the marginal
cost of reducing pollution across firms equation (3) above.
These principles can also be derived from profit maximization condition as follows:
Let Pi = price of output of firm i
 i  Pi Qi  C i Qi , Z i 
the objective function is
 P Q
n
max
Qi , Z i
i
i n
n
i

 C i Qi , Z i 
_
s.t. Z i  Z

i 1
n


L   Pi Qi  C Qi , Z i 
i
i n
 n
 L C i
  Z   Z i 

  0
Z
i n

 Z i
F.O. N.Cs
L
C i
 Pi 
0
Qi
Qi
L C i

  0
Z i
Z
(1)
Vi
(2)
From equation (2), we get
C i


Z i
(3)
MDC
MAC
Since (3) holds for all firms.
C i
C j

(4)
Z i
Z j
Equality of MAC across firms

24
Vi
(2)
Reference On Policies On Externalities And Pollution Abatement
Hanley, Nick, Jason F. Shogren and Ben White (1997)
Environmental economics in theory and practice London
Macmillan Press Ltd chapter 3 to 5
4.3
Tradeable Pollution Rights
The purpose in this case is to show that when there are tradable pollution rights, a
profit maximization firm will equate the MC of reducing pollution to the marginal
benefit of reducing pollution MAC = MBA.
Where MAC = Marginal abatement costs
MBA = Marginal benefit of abatement
A Model Of Tradeable Pollution Rights
Let Zi = pollution by firm i
Ri = the number of pollution rights given to firm i.
The pollution by firm i cannot exceed the rights given to firm i.
If firms can trade the rights to pollute, let the price of pollution rights be denoted by P.
furthermore, let
R0 = The amount of rights given to a firm, e.g. 40
Ri = The amount of rights each firm retains (keeps to itself) e.g. 25
 R0  Ri  Rights the firm sells: (40 –25) = 15
 PR R0  Ri  Revenue from selling pollution rights PR (15)
The π function of firm i is
 i  Pi Qi  C i Qi , Z i   PR R0  Ri 
The Objective Function Is
Max i  Pi Qi  C i Qi , Z i   PR R0  Ri 
s.t.Z i  Ri

The Langrangean function
L  Pi Qi  C i Qi , Z i   PR R0  Ri   Ri  Z i 
25
F. O. N. Cs
L
Ci
 Pi 
0
Qi
Qi
(1)
L C i

λ0
Z i Z i
(2)
L
  PR  λ  0
Ri
(3)
from (2), we obtain

C i
λ
Z i
( 2' )
From (3), we get
PR  λ
( 3' )
from ( 2' ) and ( 3' ) we obtain

C i
 PR
Z i
(4)
Marginal benefit of reducing pollution MBA
Marginal cost of reducing pollution MAC
Conclusions
When there are tradable pollution rights a profit-maximizing firm will equate the MC
of reducing pollution to the MBA (equation 4 above). This is at the desired level of
pollution.
Tradeable pollution rights are considered to be an easier option then taxing pollution
emission, because the role of the planner in this case is to decide on the level of
pollution, Zi and then issue Ri.
26
CHAPTER 5
5.0
OPTIMAL UTILISATION OF NATURAL RESOURCES
Optimal simply means efficient. Therefore, optimal utilisation of natural resources is
about efficient use of natural resources.
5.1
The Methodology Of Environmental Economics (EE)
EE is all about the use of limited resources over long periods of time.
Soruce Rogre Perman, Yue Ma. And James
Mc Gilvary (1996) Natural Resources and Environmental Economics, Londru:
Longman PXVIII
“….. the subject is eclectic based predominantly on conventional neoclassical micro –
economics foundations, but drawing significantly from developments in het natural
and physical sciences”.
Introduction To Dynamic Optimization
Definition: static optimization is about optimal use of resources at a given point in
time.
Definition: dynamic optimization is about optimal use of resources overtime.
Static Optimization
1. Concerned with optimal resource allocation at a given point in time.
2. The optimization problem in this case is to choose instruments from an opportunity
set to maximise a given function.
3. The problem is this case is called a mathematical programming problem.
Dynamic Optimisation
1. Concerned with optimal resource allocation over time.
2. The optimization problem is to choose time path of control variable from a control
set.
3. The problem is this case is called an optimal control problem.
Formal Statement Of The Optimal Control Problem
A formal statement of the optimal control problem is comprised of the following:
27
TIME, the STATE VARIABLES, the CONTROL VARIABLES the EQUATIONS OF
MOTION, the determination of TERMINAL TIME, and the OBJECTIVE FUNCTION.
5.2
Time
Time, t, is measured in continuous units and is defined over the internal, from the
initial time, t0 which is typically given, to terminal time, T which often must be
determined. That is:
t = 0, 1, …..T
where
t = o : is the present time
t = T: is the terminal time
5.3
State Variables
How much is remaining after extraction or harvested
A state variable is a variable describing the system in a given period in time. Each
state variable is a function of time. Thus;
Xt = Discrete-Time value of X in period t
X(t) = Continuous- time value of X in period t.
5.4
Control Variables
Is also known as an instrumental variable. It represents a variable for which we
have to choose a time path. A control variable is also a function of time
Yt = Discrete-Time control variable in period t
Y(t) = Continuous-time control variable in period t.
5.5 Equations Of Motion
An equation of motion is a difference equation (discrete – time) or a differntial
equation (continous – time) defining the change in the state variable from period t
to (t + 1), t = 0, 1 ……T – 1.
Thus
Xt+1 – Xt = F ( Xt, Yt): Discrete time
28
X  F  X t , Y t 
.
5. 1
Objective Function
An objective function is a function to be maximized. It represents the economic
returns in a given period. It consists of two parts:
(i) The intermediate function
(ii) The final function
(i) The intermediate function
Is a function of the state variable, the control variable, and time.
Formally:
V  X t , Yt , t  : Discrete time
V  X t , Y t , t  : Continuous time
(ii) The final function
Is a function of terminal state and terminal time.
Formally:
F  X T  : Discrete time
F  X T  : Continuous time
Therefore, we have
J  V  X t , Yt , t   F  X T 
or
J  V  X t , Y t , t   F  X t 
The problem is essentially that of determining the optimal values of yt, from t = 0,
1,….., T - 1. The values of yt will, via the difference equation, imply values for xt
from t – 1, …., T - 1
Summary
Choose, values of yt during t = 0,…. T - 1
29
xt during t = 1,…. T - 1
xT at t = T
Lagrangean Function is
T 1
L   V (.)  t 1  X  F (.)  X t 1   F (.)
(1.32)
t 0
Note that t 1 is the multiplier associated with X t 1 . The Langrangean multiplier is
put on X t 1 because the difference equation is a constraint equation which serves to
define X t 1 .
y t for t  0,....., T  1
xt for t  1,....., T
t for t  1,....., T
It is therefore possible to solve the equations simultaneously.
For conditions – time the above problem because
t = 0, ….., T: Set of time period
t = 0: initial period
t = T: Terminal time
X(t) = State variable
Y(t) : critical variable
V(.) = V(x(t), y(t), t) : Net economic return
F (x(T) : Final function
X  f x(t ), y(t )  : Equation of motion
.
x (0) = a : initial condition
30
Objective Function
Max
T

 V  x(t ), y (t ), t dt F  x(t ) 
 y t , xt  0
Subject to X  f x(t ), y (t ) 
.
x(0)  a
The integration operator has replaced the summative operation
Taxonomy of dynamic optimization
Problems
(Distribute relevant photocopies)
The Hamiltonian
In dynamic analysis, we use Hamiltonian rather than the Lagrangean because the
Hamiltonian is concise, yet yields the necessary first order conditions.
However the two functions are related as follows. The Hamiltonian is a subset of the
Lagrangean
Discrete - time Hamiltonian
Suppose we have the following objective function
T 1
MaxV xt , y t , t   F  X T  s.t. X t -1  X t  f xt , yt 
t 0
X 0  a........Given
The Hamiltonian is defined as follows
H xt , yt , t 1 , t   V  X t , yt , t   t -1 f xt , yt 
It is possible to write the F.O.N.C.s directly as partial derivations of the Hamiltonian
From the current value Hamiltonian, we obtain F.O.N.Cs for a maximum which we
solve for the optimal values of xt , yt and t.
The F.O.N.Cs are
31
~
H
0
y t
(1)
~
H
t 1  t  xt
(2)
~
.
X t 1  X t 
H
  t 1 
(3)
T  F 1  X T 
(4)
X0  a
(5)
32
CHAPTER 6
6.0
Discounting and Compounding in Environmental Economics
Discounting is a technique for calculation the present value of a future stream of net
income. It is opposite of compounding.
Compounding
A technique for calculating the future value of a present stream of net income.
Relationship between discounting and compounding
Discounting
PV
FV
Compounding
PV = present value
FV = future value
NB. Most optimal controls problems deal with the issue of discounting.
6.1
Discrete time
Let Nt = Future net incomes, t = 0,1,…….,T
The present value (PV) of Nt can be calculated as follows, PV  N
T
N   N t / 1   
t
t 0
T
  P t N t where P 
t 0
1
 discount factor
1  δ 
δ  discount rate
33
6.2
Continuous time
Let N(t) future net income
t = 0,1,……., T
The present value (PV) of N(t) can be calculated as follows; PV  N :
T
N   N (t )e rt dt
O
where
e  rt continuous time discount factor
r = continuous time discount rate
NB: sometimes r is denote by 
Negative Growth
yx
xt
xt
yt
10
t
V .e  rt
r = Rate of decay
34
Positive Growth
xt
xt
yt
yt
V .e  rt
r = rate of appreciation
6.3
Discrete – Time Optimal Control Problem With PV Objective
max
T 1
 yt , xt  P tV xt , yt   P T F ( XT )
T O
s.t. X t 1  X t  f  xt , y t 
XO  a
Things to note are
Time, t, is not an argument any more in V (.). This is because we are now dealing
with present and not future values.Both V(.) and F (.) are discounted, by and
respectively
Since we discount V(.) and F (.) we also need to discount the following expression in
the Lagrangrean
t 1 xt  f xt , yt   xt 1 
Therefore, we have
t 1 xt  f xt , yt   xt 1 
This discounting is up to period t + 1, because t 1 is on xt 1
Since V(.) is discounted to period t, by P , we discount the multiplier term further by
P. therefore the discrete – time lagrangean is
35
 P V x , y    x
T 1
L
t
T O
6.4
t
t
t 1
t

 f xt , y t   xt 1    T F ( XT )
Continuous Time Optimal Control Problem With Present Value
Objective
T
MAX  V  xt , y t e t dt  F ( X T )e T
0
s.t. X  F  xt  y t 
.
X o   a
NB:   r discount rate
The continuous - time current value Hamiltonian is
H xt , y t ,  t   V xt , y t    t F xt , y t 
~
Where  t   et  t 
Current value shadow price of xt 
As the resources get scarce, the shadow price is surplus to appreciate (i.e. why is
positive)
36
CHAPTER 7
7.0
PRINCIPLES OF THE OPTIMAL UTILISATIION OF NATURAL
RESOURCES
Taxonomy of resources (1)
Resources
Non-renewable resources
(NRR)
Renewable resources (NR)

Metal ore
- forests

Coal
- fish

Oil
- wildlife

Stone
- rainfall

Clay
- solar energy

Groundwater
NB : Focus is on economically significant rate of generation or regeneration.
7.1
Taxonomy Of Natural Resources (2)
Stocks (NRR)
flows (RR)
No deterioration
with deterioration
use dependent
use dependent
Metal core
oil
timber
rainfall
Coal
gas
groundwater
rive – flow
Stone
plant nutrients
fish
solar
wildlife
energy
Clay
37
Sources function. This refers to the role of the environmental as a source of raw
materials, energy, and environmental services.
Sink functions: This refers t the role of the environmental as a dumping ground for
waste and emissions.
Stocks: Non-renewable resources, which cannot be increased in supply within a
meaningful, time horizon.
Flows: Renewable resources, which can be, increase in supply within a meaningful
time horizon.
Optimal utilization of renewable and non-renewable resources
Renewable and non-renewable resources have different characteristics. Therefore the
principles of the optimal utilization of these resources are also different.
7.2
Characteristics On Non –Renewable Resources
i.
They exhibit no growth or regenerative process therefore they are depletable
ii.
Depletable resources are those whose stock cannot be augmented in a
reasonable time frame.
iii.
Current use of depletable resources precludes future use. There is an
opportunity cost i.e. foregone future benefits
iv.
For a resource, which is finite (fixed) in stock, the cost of extraction rises
overtime. This reflects increasing scarcity and the rise in the opportunity cost
of current consumption.
v.
In response to the rising cost of extraction, the quantity extracted falls
overtime, until it finally goes to zero.
38
Graphically Illustration Of (Iv) And (V)
Extraction
quantity
costs
extracted
Time
Time
Critical Issues In The Optimal Utilization Of Non-Renewable Resources
i.
How to allocate dwindling stocks among generations
ii.
How to re-cycle those which can be re-cycled
iii.
The transition to renewable substitutes, if any are available
Critical Issue # 1
How to allocate dwindling stocks among generations
This issue is about determining the conditions for optimal depletion of non-renewable
resources
Environmental economics has, through the years, developed the following conditions
for optimal depletion of non-renewable resources
Conditions # 1
Price = Marginal (extraction) cost + opportunity cost. This implies that less of the
non-renewable resource will be extracted today than if it were renewable resources.
For non-renewable. resources, the critical issue is how to determine the optimal rate
of depletion why? As long as resource use continue a non-renewable resource will be
depleted overtime. The fundamental issue in this case is therefore how to determine
the optimal rate of depletion.
The opportunity cost of utilization of non-renewable resources
The opportunity cost of utilization of non-renewable resources is the foregone future
benefits. Since the resource stock is fixed, the use of non-renewable resources sets in
39
motion a process of depletion of the stock. Therefore, current use of a depletable
resource precludes future use.
Opportunity cost and optimal depletion
When the opportunity cost of the use of a non-renewable resources is considered the
quality extracted will be such that.
Price = marginal cost + opportunity cost
General rule for profit maximization
Marginal revenue = marginal cost.
By definition
  TR  TC  0
(1)
MR 
TR
Q
(2)
M C
TC
Q
(3)

Max  = Max (TR – TC) > 0  MR – MC = 0 or MR = MC
Examples different market structures
1. Competitive firm
MR = d = P
; d = demand
 Max  rule is : MR = P = d = MC
or simply :
P = MC
Price
MC
d = MR
Pc
Qty
Qc
40
2. Monopoly
MR  P
 Max  rule is : MR = MC
MC
Pm
D
MR
3. Monopoly and a competitive firm
MC
Pm
PC
D
MR
Qm
QC
Qty
Based on the foregoing analysis, we add opportunity cost of marginal cost in nonrenewable resource use to obtain the following results.
The case of constant MC
price
Marginal cost + opportunity cost
Marginal cost
Qty
QN
QR
41
QN = quantity extracted if this were a non-renewable resource
QR = quantity extracted if this were a renewable resources
The case of rise MC
MC + OPP. C
price
MC
PN
PR
MR
QN
QR
Qty
Conclusion
When the opportunity cost is considered the optimal depletion of a non renewable
resource implies that less of the resource will be extracted today than if the resource
were renewable i.e. QN < QR.
Conditions # 2
The second conditions for the optimal depletion of a non-renewable resource is that
the present value of the royalty must be the same in all periods. Alternatively, the
second conditions states that:
The royalty must rise at the same rate as the rate of the interest.
The second condition of optimal depletion describes the behaviour of the opportunity
cost overtime. This is the same as saying the behaviour of the royalty over time.
Note the following:
The net social benefit from extracting a depletable resource is called ROYALTY.
Royalty is defined as the difference between price (or what consumes are willing to
pay) and the cost of extraction.
42
More generally, for any commodity, the net (social) benefit in a single period is
conventionally measured as the difference between what consumers are willing to pay
for a good and what it costs to produce.
Simply stated
The net benefit is the difference between the willingness to pay and the cost of
production.
Geometrically
The net benefit is the integral of the demand curve (the total willingness to pay)
minus the total cost.
Suppose that the relevant details of a non – renewable resource, e.g. a mineral are as
follows.
Xo = 10 tons : extractive quantity; initial reserves
Qt = q0 , q1: quality extracted in periods 0 and 1 respectively.
Pt = 20 – qt : demand for the mineral output
MC = US$5 per ton:
Marginal cost of extraction: assumed to be constant
r = 0.10: the rate of discount (interest rate)
t = 0, 1 : set of time periods, assumed to be only two periods.
XT = X2 = 0 : Final function
The net benefit, or the difference between the willingness to pay and cost, can then be
written as follows:
In period 0
q0
q0
0
0
 20  q dq   5dq integral demand minus integral of the cost.
Or simply
q0
 20  q   5dq
0
In period 1
q1
 20  q   5dq demand cost 
0
The objective functions is
43
Max
q0 , q1
q0
q0
0
0
 20  q   5dq  
20  q   5dq
1  0.10
q0  q1  10
s.t.
Max net between in period zero and period 1.
NB: 1
 1 

1  r   1  0.10  : Discount factor.
The Lagrangean function is
L
q0
q1
0
0
 20  q   5dq  
20  q   5dq   10  q
1  0.10
F.O.N.Cs
L
 20  q 0   5    0
q 0
(1)
L 20  q1   5

  0
q1
1.1
(2)
L
 10  q 0  q1  0

(3)
we then solve for q0* , q1* , * , P0 and P1 as follows.
From (1)
15  q0  
1'
from (2)
15  q1

1.1
From
2'
1' and 2'
15  q0 
15  q1
1.1
Solving for q 0 we get
44
0
 q1 
 15  q1 
q 0  15  

 1.1 
16.5  15  q1
1.1
1.5  q1
q0 
1.1

1"
1" into (3) and solving for
Substituting
q1* , we get
10  1.5  q1 
 q1  0
1 .1
10 - 1.5  q1   1.1.q  0
9.5  q1  1.1.q1  0
9.5  2.1q1
q1 
9. 5
2 .1
2"
 q1*  4.5
Substitute
q0 
2' into 1' yields
1.5  4.5
1 .1
 q 0*  5.5
From 1' solving for  , obtain
  15  5.5
 *  9.5
Summary
45
q 0*  5.5
q1*  4.5
*  9.5
Substituting the values of q 0* and q1* into the demand function
Pt  20  qt
For period 0
For period 1
P0  20  q0
P1  20  q1
= 20 – 5.5
= 20 – 4.5
 P  14.5
*
0
 P1*  15.5
Overall summary
In addition we have P0*  14.5
P1*  15.5
CASE # 1
Discounting the royalties
Royalty in period 0:
P0  MC  14.5  5
 Royalty  9.5
Royalty in period 1 :
P1  MC  15.5  5
 Royalty  10.5
Finding the net present value of the royalty in period 1:
46
P1  MC 15.5  5

1.1
1.1

10.5
 95
1.1
 Discounted royalty of period 1  9.5
 P0  MC 
P1  MC
 9.5
1.1
The present value of the royalty is the same in both periods.
More Generally
P0  MC 
P1  MC Pi  MC

1.1
1  r t
if you deplete your resources optimally the royalty is the same for all periods.
Thus, the second condition for optimal depletion is that the present value of the
royalty must be the same in all periods.
CASE # 2
Compounding the royalties
Royalty in period 0:
P0  MC  14.5  5
 Royalty  9.5
Royalty in period 1 :
P1  MC  15.5  5
 Royalty  10.5
Finding the future value to the royalty in period 0:
9.5 x
10
 0.95
100
 future value of
47
9.5  9.5  0.95
 10.45  10.5
 1  r  P0  MC   P1  MC
n
By growing at the rate of interest, the royalty is the same in both periods.
Conclusion
Another way of stating the second condition for optimal depletion is that the royalty
must rise at the rate of interest. Otherwise if beyond that rate you deplete the
resources.
Condition # 3
The third condition for the optimal depletion of a non-renewable resource is that,
along at optimal depletion path, the price of the resource is equal to the marginal cost
of extraction plus royalty.
From condition # 2 and for the period 0
Case:
P0  MC 
P1  MC
1  r 1
or
P0  MC  P0  MC 1  r 
1
Solving for P1
P1  MC  P0  MC 1  r 
1
more generally
Pt  MC  P0  MC 1  r 
t
where
Pt  price in period t
M C  M arginal cost of extraction
P0  MC 1  r t
 royalty from the resource componded at the rate of interest, r.
48
Therefore the third condition for the optimal depletion of a non renewable resource is
that, along an optimal depletion path, the price of the resource is equal to the marginal
cost of extraction plus royalty.
Reference
Fisher, Anthony C. (1981). Restorer and environmental economics, Cambridge,
England: Cambridge University Press.
49
CHAPTER 8
8.0
Renewable Resources (RR)
8.1
Characteristics of RR
They have the capacity for reproduction and growth, e.g. plant or animal population
they are also inanimate mass or energy source subject to constant or periodic inflow
e.g. water, wind or solar radiation.
Things to note:

For a renewable resource, growth or FLUX is assumed to take place at a
“significant” rate when viewed from man’s economic time scale.

For renewable resources state variables are often called a standing stock or the
biomass of the population.

If the age structure, sex ratio, or other population characteristics are constant,
the optimal control problem, in this case, will require more than one state
variable.

For simplicity, however, it is assumed that the resource under consideration
can be described by one state variable.
8.2
Growth functions
Suppose that we have s biological resource stock whose size at time, t, is denoted but
Xt, in discrete time or X(t), in continuous time,
In the absence of harvesting, the dynamics of the resource stock can be described by
the difference equation.
X t 1  X t  F  X t 
It can also be described by the differential equation
X t  .
 X  F  X t 
t
These equation simply say that change in the resource stock depends on the current
stock size X t or Xt  .
8.3
Density Dependent
The growth function F  X t  is defined over the interval and it is usually assumed that
there exits two values.
50
XL < XU for which the following is true
 0


 0
F  X t 

 0


0  Xt  XL
X L  X L  XU
X t  XU
Where
X U = Upper limit of X
X L = Lower limit of X
X t = Value of X in t
Alternatively, these properties of F(Xt) may be stated as:
F ( X )  0 if 0  X  X


F ( X )  0 if X  X
-
-
F ( X )  0 if X  X
Where

X  X u : upper limit
X  X L : lower limit

51
F(X)
F(Xt)
Growth of X
(2)
XL
XU
(3)
Depending on the values of XL and the characteristics of F(Xt), we have three cases as
follows.
Case # 1
XL = 0 and F (XL) is strictly concave from below. For such a growth function, the
proportional growth rate:
rX  
f X 
is a decreasing function of X, that is, F " 0
X
This growth function is said to be purely compensatory.
Graphically
F " 0
growth of X
XL
XU
52
Population of X
CASE # 2
XL = 0 and F(Xt) is initially convex then concave, in other words, it has an inflection
point. This growth function is said to be DEPENSATORY.
Growth of X
Inflection point
XL
Population of X
Xk
CASE # 3
XL = 0 and F(Xt) is initially convex and than concave. This growth function is called
CRITICAL DEPENSATION and XL is called the MINIMUM VIABLE
POPULATION.
F(x)
F(Xt)
Growth of X
XL
XU
Population of Y
53
8.4 Specific Forms Of The Growth Functions
There are many possible functional specifications for F(X(t). The best known are:
i.
The logistic growth model
ii.
The Gompertz growth model
(i) The Logistic Growth Model
This model was first proposed in 1830 by P.F Verhust in relation to human
population. When written as a differential equation, the Logistic Growth model takes
the following form.
X  F  X t   rX t 1  X t  / K 
.
Where
r = the growth rate of the resource, X.
k = the environmental carrying capacity or saturation point.
Specified in this manner, the logistic equation is purely compensatory, i.e. XL = 0,
F(X(t) is strictly concave and Xk = K.
X t   K from any X (0) >0 as t   ,
that is him X t   K provided that X (0) >0 t  
Graphically
Growth rate of X

F  X   rX 1  X
Xt = 0
k

Xk = k
54
Oty
Population of X
Logistic growth model
time
(ii) The Gompertz Growth function
when with as a differential equation the Gompertz function takes the form:
X  F  X t 
.
 r Xt  h k/X t 
Where
r = growth rate of the resource, X.
k = the environmental carrying capacity
8.5
Limitations Of These Growth Models
These models do not indicate the stochastic nature of the problem (EL NINO
EFFECT) or the age and sex distribution of the population.
The models do not show competing or complementary species.
For example
X  F  X t 
.
 r Xt  1 - Xt  / K   sY t 
where
y(t) = compounding or complementary species
s = growth rate of competing or complementary species
55
Case # 1
.
X
If
 s  0  s  0
Y t 
 Y t   X 
.
In this case, X t  and Y t  are complementary species.
Case # 2
.
X
If
 s  0  s  0
Y t 
 Y t   X 
.
in this case, X t  and Y t  are competing species.
Reference
CLARK, COLIN W. (1990) MATHEMATICAL
BIOECONOMICS, OPTIMAL MANAGEMENT OF RENEWABLE RESOURCES
New York: John Wiley and Sons Chapter 1
56
CHAPTER 9
9.0
i.
PRODUCTION AND YIELD FUNCTIONS
Harvest or yield
When a renewable resource is harvested it is assumed that the harvest rate is a
function of economic inputs devoted to harvesting and of the available stock. That is
Y t   H Et , X t 
where
(1)
Y t  = the production function or rate of harvest
E t  = effort, that is, aggregate measure of various inputs
X t  = resource stock, such as forests, fish or wildlife.
(ii) Rate of growth of resource stock
With harvesting, the rate of growth of the resource stock must now reflect:
The resource stock, F  X t 
The harvest Y t 
Thus, we have
X t 1  X t  F  X t   Yt
or
X  F  X t   Yt
.
(2)
.
This simply states that the growth in resource, X , is a function of the resource stock
F  X t  minus the rate of harvest, Y t  .
Note that, without harvesting, equation (2) becomes simply
X  F  X t 
.
The way the Logistic and Gompertz models are stated above.
(iii) Sustained Yield Function
By sustained yiled, we mean that X, Y and E all remain constant overtime.
Therefore, the sustained yield function is an equilibrium concept expressing
sustainable harvest (YIELD) as a function of effort.
From (1) and (2), we obtain
X  F X   Y  0
.
(3)
57
.
Since X, Y, and E are constant X  0
Y  H ( E, X )
(4)
Eliminating X from (3) and (4) gives the sustained – yield function
Y  Y E 
(5)
Example
Suppose we have the logit growth functions
X  F  X t 
.
 r Xt  1 - Xt  / K 
(6)
And the following production function commonly used in Fisheries models.
Y t   qEt X t 
(7)
where q = constant
the assumption behind (7) are:
(i)
the catch per unit of effort (Y/E) is directly related to the density of fish in
the fishery, and Y t  / Et   qX t 
(ii)
the density of fish is directly proportional to the abundance of fish X(t).
From (6) and (7), we obtain the rate of growth function as:
X  F  X t   Y t 
.
 r Xt  1 - Xt  / K  - Y t 
(8)
where
Y t   qEt X t 
The sustained yield function is
X  F X   0
.
 r X 1 - X / K   Y  0
 r X 1 - X / K   Y
(9)
But, from (7)
58
Y  qEX
Substituting this value of Y into (9), we get
 r X 1 - X / K   qEX
Solving for X, we obtain:
rx 
rx 2
 qEX
k
rx 
rx 2
 qEX  0
k
rx


x r   qE   0
k


r
rx
 qE  0
k
Simplifying yields
 qE 
X  K 1 

r 

(10)
Substituting (10) into (11), the sustained yield function is:
Y  qEX
 qE 
Y  qEK 1 

r 

(11)
This is the Schaefer Fisheries model
In equation (11)
qE = relative rate of harvest
r = intrinsic rate of growth of fish stock (exogenous).
Note that if:
qE = r, then Y = 0
Interpretation
In equation (11), if the relative rate of harvest (qE), exceeds the rate of growth of the
fish stock, r, then the population will be driven to extinction and the yield will
become zero.
59
Graphically
yield of y(t)
yield – effort curve
Effort (e(t))
The Yield Effort Curve
It is concave or bell shaped. It describes the amount of a given resource harvested in
relation to effort. The yield rises, reaches the maximum, and then begins to decline as
effort increases. Eventually the yield becomes zero and the renewable resources may
be driven to extinction because the rate of harvest exceeds the rate of regeneration of
the resource stock.
Conclusion
One of the critical issues in the optimal utilization of renewable resources is that,
although the resources are renewable, they can be driven to extinction (depleted) if the
rate of harvest exceeds the rate of regeneration of the resource stock.
Note that the yield – effort curve is different from the population growth rate curve:
X  F  X t   r Xt  1 - Xt  / K 
.
F(X)
Growth rate of X
k
Population of X
60
From equation (11):
 qE 
Y  qEK 1 

r 

Y qEK  qE 

1 

E
E 
r 
 qE 
 qK 1 

r 

or
Y
K
 qK  q 2 E
E
r
In econometric form:
Y
 a  bE
E
61
CHAPTER 10
10.0
MANAGEMENT OF RENEWABLE RESOURCES:
A COMPARISON BETWEEN THREE EQUILIBRIA
The principles of the optimal use of renewable resources are often derived from
comparing three types of equilibria.
a. The competitive (economics) optimum.
b. The maximum sustainable yield (biological optimum)
c. The common – property resource equilibrium (free access)
MR 
TR
E
Graphically :
TC = MC
A
B
TR
E*
EMSY
E
EFFORT
Note that the yield – effort curve is, in essence, the total revenue curve. This is
because, it is from fish harvest that revenue is obtained.
Therefore, MR 
TR
, the slope of the total revenue curve.
E
The TC curve deficits the total cost curve representing the cost of fishing, such as
wages and salaries capital costs, and so on.
The TC curve is directly proportional to effort (economic inputs) because the mkore
the effort, the more the cost of fishing.
Since the TC curve is a straight line, the marginal cost is everywhere coincidental
with the TC line.
By definition:
MR 
TC
: Slope of the TC curve
E
62
(i) The competitive (economic) optimum
Based purely on economic considerations, profit maximization occurs at E* where
MR = MC.
It is important to note that, the level of effort is the lowest among three equilibria
considered here.
One model for probit maximum is as follows.
Let: PY (t )  cE (t )  Net revenues
Where
P = price per unit of the resource after harvest,
c= per unit cost of effort
 u  X t , E t   PY t   cE t 
 PH  X t   E t   CE t 
 
TR
TC
In other words, society derives utility from net revenues
The objective is to
T
E t , X t    PH  X t , E t   cE t e t dt
Max
0
s.t X  F  X t   H xt E t 
.
X 0  X
Assignment
State the current value Hamiltonian and obtain F.O.N.Cs
(ii) The maximum sustainable yield (MSY)
(Biological optimum)
There are two definitions of MSY, and these give rise to two different ways of
formally deriving MSY.
Definition #1
MSY occurs when the growth rate of a resource reaches a maximum
It is represented by the highest point on the F  X t  curve
Definition 2
63
MSY is the highest possible yield without depleting the resource.
It is represented by the highest point on the yield – effort curve (or the TR curve)
Mathematically, there are two ways to obtain the MSY
By maximizing the sustained yield function Y  F x  which
Requires that F '  X   0
ii) by maximizing the sustained yield function Y  Y x  which require that
Y ' t   0 .
CASE # 1
MSY where Y  F x  and F '  X   0
Suppose we have the logistic function
X  F  X t 
.
 r Xt  1 - Xt  / K 
(1)
Simplifying, we get
F  X t   r Xt  -
r 2
X t 
k
(1' )
From (1' ) we get
F
 F '  X t 
X t 
r
 r - 2 Xt 
k
MSY requires that F '  X   0
r
 r - 2 Xt   0
k
Since we are dealing with sustained yield, we drop the t notation:
 F ' X   r - 2
rX
0
k
Solving for X, we obtain
rk - 2rX
0
k
Simplifying, we get:
X MSY 
k
2
64
Substituting X MSY into the sustained yield function
X  F X   Y  O
.
or X MSY 

 rX 1  X
k
2
K

 k
K 2
 r 1 
K
2


 YMSY 





rk
4
F  X   r X 1 - X / K 
Growth rate of X
X MSY 
k
2
Population of X
CASE # 2:
MSY where Y  Y E  , and Y ' E   0
Consider the sustained yield function
65
k
X  F X   Y  0
.
F  X   r X 1 - X / K  - Y  0
Where
Y  qEX

 rx 1  x
k
  qEX
Solving for x, we get
 q 
X  K 1  E 
 r 
Substituting into the equation
Y  qEX , and
Solving for Y we obtain
E

Y  qKE 1  q 
r

(11)
Simplifying, we get
Y  qKE 
q 2 KE
r
To obtain MSY, we get
Y
0
E
Y
E

 qk 1  2q   0
E
r

Solving for E, we obtain
E MSY 
r
2q
66
Graphically
YMSY
YIELD Y(t)
E MSY 
r
2q
effort
Note that at YMSY, the level of effort is greater than at the economics optimum, E*.
Clearly then, the MSY is a sub-optimal management strategy.
(iii) The common property Resource Equilibrium (Free Access) (CPRE)
The CPRE occurs where net economic returns are zero. This is at E , where
TR = TC. Why
For E  E , such as point A, TR > TC; there are positive economic profits
Given free access, there will be an expansion of effort, E.
For E  E , such as point B, TR > TC, there are positive economic profits
(economic losses) and therefore a decrease in effort.
This, the CPRE will occur at E , where
TR > TC
Note that
E  EMSY  E *
This is often referred to as the tragedy of the common.
The Tragedy of the common
Simply stated, the concept of the tragedy of the common says that a common –
property resource gets used by everybody until is it of no use to nobody.
67
CHAPTER 11
11.0 THE ECONOMICS OF FOREST MANAGEMENT: THE FACISTMANN
MODEL
The basic problem is how to determine the optimal rotation period. Rotation simply
means the period from planting to cutting of a forest stand (a group of trees)
Case 1: Single Rotation
Consider the following description of a first stand
V(t) = stumpage value, i.e. commercial value of stand
C = cost of felling the trees
V(t) – C = Net value of the stand
e.g.  = v(t) – C
Objective Function is
max
t
 t   V t   C e t
F .O.N .C

 V ' t e t   V t   C e t  0
t
V ' t e t   V t   C e t
Dividing both sides of the equation by e t
V ' t    V t   C 
(2)
Equation (2) is the Faustmann formula for single rotation
The equation states that we should harvest the stand at time t*.
When the rate of growth of the stand V ' t  , is equal to the interest that could be
earned if the net value from cutting the stand is invested at the rate of  . The
equation depicts the opportunity cost of capital now tried up in the trees, that is
 V t   C  . However this equation does not show the opportunity costs of the site
tied up in the production.
Case 2: Multiple Rotation
Given the present value of the stand
68

PV   e  kT V t   C 
K 1

V T   C
e T  1
The objective is to
max
T
V T   C
e T  1
 T  
.F .O.N .C



e t  1 V ' T    V T   C e t

0
2
T
e t  1
e
t
e

 1 V ' T 
t

1
2



V T   C e 
e  1
T
T
2
Simplifying, we obtain
V ' T  e T V T   C 

e t  1
eT  12
Cross multiplying
V ' T 
e T e T  1

V T   C
eT  12

 e T
e
T
 1
Then we add a complex zero    to the numerator on the right hand side:
69
V ' T 
 e T    

V T   C
e T  1

 e T  1  
e T  1
V ' T 

   T
V T   C
e 1
M ultiplying by V T   C , we obtain
V T   C 
V ' T    V T   C     T

 e 1 
(4)
Equation (4) is the Faustmann formula for multiple rotation. The equation states that
we should harvest the stand at T* when the marginal increment to the value of the
trees, V ' T  , is equal to the sum of the opportunity cost of the investment tied up in
the standing trees,  V T   C , and the opportunity cost of the investment tied up in
V T   C 
the site,   T
.
 e  1 
In the equation
V ' T  = The increase in the net value of the standing forest
 V T   C  = The interest that can be earned if the net revenue from cultures the
stand i.e. V T   C , is invested at an interest rate .
V T   C 
= the interest that can be earned if the present value of the stream of
T
 e  1 

future revenues
V T   C
, is invested at an interest rate .
e T  1
Assignment
(1) Single rotation
Optimize and interpret
max
t
 t   P  h V t   C e T
(2)
Multiple rotation
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Optimize and interpret

NPV   e  KT  P  h V t   C
K 1
P  h V t   C
e T  1
(3) Comparative statics
Optimize and interpret
When C rises from C to C ' : Such that


NPV   e  KT  P  h V t   C '

K 1

P  h V t   C '
e T  1
where
C = Old planting costs
C ' = new planting costs
C' > C
What is the effect of the rise in planting costs on the optimal rotation age? Is it to
lengthier or shorten T*? Justify your Answer.
Hints
Compare the increase in the net value of the standing forest, P  hV t   C ' in (2 and
(3). The one with greater P  hV t   C ' has shorter T*.
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