Resources Economics. Jon M.Conrad. Chapter 1
description
Transcript of Resources Economics. Jon M.Conrad. Chapter 1
BASIC C
ONCEPTS A
BOUT
RESOURCES ECONOMIC
S
MA
RT A
CE
L I A S
UÁ
RE
Z- V
AR
EL A
MA
CI Á
J OS
É G
ÁM
EZ
BE
L MO
NT
E
CA
RL O
S F
ER
NA
ND
O R
OJ A
S T
RA
VE
RS
O
INDEX
1. Renewable, Nonrenewable and Enviromental Resources
2. Discounting
3. A Discrete-Time Extension of the Method of Lagrange Multipliers
4. Exercises
RENEWABLE, NONRENEWABLE, AND ENVIROMENTAL
RESOURCESEconomics might be defined as the study of how society allocates scarce resources.
The field of resource economics would then be the study of how society allocates scarce natural resources.
A distinction between resources and environmental economics is necessary to
continuous our analysis.
QUESTION 1: What is the central subject in the field of resource economics?
RENEWABLE, NONRENEWABLE, AND ENVIROMENTAL
RESOURCESEnvironmental Economics is concerned with the conservation of
natural environments and biodiversity.
Natural Resources
But our study is about Renewable
resource
Nonrenewable resource
Must display a significant rate of growth or renewal on a
relevant economic time scale.
An economic time scale is a time interval for which
planning and management are meaningful.
A critical question in the allocation of natural resources is “How much of the
resource should be harvest today, and in each period?”
Dynamic optimization
problem
QUESTION 2: What is the economic distinction between renewable and non renewable resources?
QUESTION
A renewable resource must display a significant rate of grown or renewal on a relevant economic time scale. An economic time scale is a time interval for which planning and management are
meaningfuly.
RENEWABLE, NONRENEWABLE, AND ENVIROMENTAL
RESOURCESDynamic
Optimization Problem
Maximize some measure of net economic value
Solution: schedule or “time path” indicating
optimal amount to be harvested in each period.
The optimal rate of harvest in a particular period may be
zero
If a fish stock has been historically mismanaged, and the current stock is below what is deemed optimal, then
zero harvest may be best until the stock recovers.
Example
We are going to analyse the next figure, where we
suppose that the economy is composed by two resources: one Nonrenewable and the
other renewable
RENEWABLE, NONRENEWABLE, AND ENVIROMENTAL
RESOURCES
1. Assume that fish stock is bounded by some “environmental carrying capacity”, denoted by K.
2. K ≥ Xt ≥ , then F(Xt) might be increasing as Xt goes from a low level to where F(Xt) reaches a maximum sustainable yield,
at XMSY; and F(Xt) declines as Xt goes from XMSY to K.
Analysis process: renewable reource
RENEWABLE, NONRENEWABLE, AND ENVIROMENTAL RESOURCE
1. Assume that fish stock is bounded by some “environmental carrying capacity”, denoted by K.
2. K ≥ Xt ≥ , then F(Xt) might be increasing as Xt goes from a low level to where F(Xt) reaches a maximum sustainable yield,
at XMSY; and F(Xt) declines as Xt goes from XMSY to K.
Analysis process: renewable resource
3. The change in fish stock in two periods is the difference between Xt+1 - Xt = F(Xt) - Yt
4. If harvest exceeds net growth [Yt > F(Xt)], fish stock declines (Xt+1 - Xt < 0), and if it is less than net growth [Yt < F(Xt)] the
fish stock increases, (Xt+1 - Xt > 0). Yt, flows to the economy, where it yields a net benefit. Xt+1 forms the inventory (stock
for the next period), and it also will produces a benefit. Harvest decision is a
balancing of current net benefit.
RENEWABLE, NONRENEWABLE, AND ENVIROMENTAL RESOURCE
RENEWABLE, NONRENEWABLE, AND ENVIROMENTAL RESOURCE
Analysis process: nonrenewable resource1. The stock in period t+1 is= Rt+1 = Rt – q. The amount
extracted flows into the economy, where it generates net benefits. 2. Consumption of the nonrenewable resource generates a
residual waste, proportional to the rate of extraction (1 > α > 0). For example extraction of a deposit of coal, when we
consume it, we produce CO2. 3. This residual waste can accumulate as a stock pollutant, Zt.
RENEWABLE, NONRENEWABLE, AND ENVIROMENTAL RESOURCE
Analysis process: nonrenewable resource1. The stock in period t+1 is= Rt+1 = Rt – q. The amount
extracted flows into the economy, where it generates net benefits. 2. Consumption of the nonrenewable resource generates a
residual waste, proportional to the rate of extraction (1 > α > 0). For example extraction of a deposit of coal, when we
consume it, we produce CO2. 3. This residual waste can accumulate as a stock pollutant, Zt.
4. If the rate, αqt , exceeds the rate at which it is assimilated, -γZt (γ, is called the assimilation coefficient, 1 > γ > 0), the stock of pollutant will increase, whereas if the rate of generation is less than assimilation, then the stock
will decrease.
This Damage, the Pollution, generates a cost imposed on the economy. (Coase’s Theorem)
The amenity value, it’s an additional service.
RENEWABLE, NONRENEWABLE, AND ENVIROMENTAL RESOURCES
DISCOUNTING• When we attempt to determine the optimal allocation of natural resources over time, most individuals
prefer receiving benefits now than the same benefits later.
• In order to induce these individuals to save (providing funds for investment) an interest payment must be offered.
• This societies will create “markets for loanable funds” where the interest rates are like prices and reflect, in part, society´s underlying time preference.
RENEWABLE, NONRENEWABLE, AND ENVIROMENTAL RESOURCES
• An individual with positive time preference will discount the value of a note or contract which promises to pay a fixed amount of money a some future date.
Example:A bond which promises to pay 10.000 $ , 10 years from now in this kind of society is not worth 10.000$ today.
The current value will depend on the credit rating of the government or corporation promising to make the payment, the expectation of inflation, and the taxes would be paid on the interest income.
RENEWABLE, NONRENEWABLE, AND ENVIROMENTAL RESOURCES
• If the payment will be made with certainty, there is no expectation of inflation, and there is no tax on earned interest, then, the bond payment would be discounted by a rate that would approximate society’s “pure” rate of time preference. (δ)
• The risk of default (nonpayment), the expectation of inflation, or the presence of taxes on earned interest would raise private market rates of interest above the discount rate.
RENEWABLE, NONRENEWABLE, AND ENVIROMENTAL RESOURCES
Example:
Discount rate =3%, (δ) = 0.03,
“discount factor” 𝜌 = 1/(1 + δ) = 1/(1 + 0.03) =0.97.
The present value of a $10,000 payment made 10 years from now would be:
This should be the amount of money you would get for your bond if you wished to sell it today. That is the same the amount you would need to invest at a rate of 3%, compounded annually, to have $10,000, 10 years from now.
RENEWABLE, NONRENEWABLE, AND ENVIROMENTAL RESOURCES
• The previous example can be generalized to a future stream of payments in a straightforward fashion:
Where Nt are the payments made in year t and t=0 is the current year.
N=The present value of this stream of payments.
RENEWABLE, NONRENEWABLE, AND ENVIROMENTAL RESOURCES
• A particular case could be when N0 = 0 and Nt = A for t = 1, 2, . . . , T.
In this case we have a bond which promises to pay A dollars every year, from next year until the end of time. Such a bond is called a perpetuity, and with 1 > 𝜌 > 0,
when δ > 0, equation
becomes an infinite geometric progression which converges to N = A/ δ. This special result might be used to approximate the value of certain long-lived projects or the decision to preserve a natural environment for all future generations.
RENEWABLE, NONRENEWABLE, AND ENVIROMENTAL RESOURCES
Example:
If a proposed park were estimated to provide A = $10 million in annual net benefits
into the indefinite future, it would have a present value of $500 million at δ = 0.02.
(N0 = 0 and Nt = A for t = 1, 2, . . . , T.)
RENEWABLE, NONRENEWABLE, AND ENVIROMENTAL RESOURCES
• In some resource allocation problems, it is useful to treat time as a continuous variable, where the future horizon becomes the interval T ≥ t ≥ 0. If A dollars is put in the bank at interest rate δ, and compounded m times over a horizon of length T, the value at the end of the horizon will be given by
where n = m/ δ .
RENEWABLE, NONRENEWABLE, AND ENVIROMENTAL RESOURCES
• If interest is compounded continuously, both m and n tend to infinity and [1 + 1/n]n tends to e. Then, the present value of a continuous stream of payments will be:
• If N(t) = A (a constant) and if T →∞ this equation can be integrated directly to yield
which is interpreted as the present value of an asset which pays A dollars in each and every instant into the indefinite future.
RENEWABLE, NONRENEWABLE, AND ENVIROMENTAL RESOURCES
Discounting has an important ethical dimension:
The way resources are harvested over time
The evaluation of investments or policies to protect the
environment
Welfare and options of future generations.
RENEWABLE, NONRENEWABLE, AND ENVIROMENTAL RESOURCES
To ignore time
preferences
Inefficiencies
Reduction in outputs and wealth
•A society’s discount rate would reflect:
Its collective “sense of immediacy”
Its general level of development.
RENEWABLE, NONRENEWABLE, AND ENVIROMENTAL RESOURCES
•A society where time is the essence or where a large fractionof the populace is on the brink of starvation would have a higher rate discount.
•Higher discount rates more rapid depletion of nonrenewable resources and lower stock levels for renewable resources.
•High discount rates can make investments to improve or protect environmental quality unattractive.
High rates of discount will greatly reduce the value of harvesting decisions or investments.
RENEWABLE, NONRENEWABLE, AND ENVIROMENTAL RESOURCES
•Benefits in a near term future are more valuable than in a long term future which can motivate the individuals to consume the scarce resources, leaving an impoverished inventory of natural resources, a polluted environment, and very few options to change their economic destiny.
•On the other hand, this resources will have been invested in generating both physical and human capital. These alsowill benefit future generations.
•Determining the “best” endowment of human and natural capital to leave future generations is made difficult because we do not know what they will need or want.
3. A DISCRETE TIME EXTENSION OF THE METHOD OF LAGRANGE MULTIPLIERS
Recall that:
Xt is the stock of the renewable resource in year t
Yt is the harvest of the renewable resource in year t
F(Xt) is the net growth function of the renewable resource in year t
With all this, the resource dynamics can be explained by the First –Order difference equation:
The net benefits for period t are a function of the stock of resource and the harvest level:
• Higher levels of harvest yields higher net benefits:
• Higher resource stock makes it easier to search and harvest, reducing costs.
),( ttt YX
We want to choose the “best” harvest strategy, which is the one that maximizes the present value of net benefits.
• To obtain the present value , we must use a discount factor:
Where δ is the periodic rate of discount. Time invariant
• The inicial stock of resource X0 is known and, therefore, given.
• We must also take into account that candidate harvest strategies must satisfy the First-Order Difference equation:
)1/(1
With all this information, we can find the Optimal Harvest Path (Yt, t=0,…,T),
we must solve the following optimization problem:
This is a constraint optimization problem, so in order to solve it, we must
first build the Lagrangian:
Lagrangian Multipliers (λt) are interpreted as “shadow prices”, becausetheir value indicates the marginal value of an incremental increase in Xt inperiod t.Therefore, λt+1 is the value of an additional unit of Xt+1 period t+1
This value is discounted one period, to obtain its value at period t.
The difference equation, included implicitly in the Lagrangian,
is defining the level of Xt+1 that will be available in period t+1.
This is a constraint optimization problem, so in order to solve it, we must
first build the Lagrangian:
Both net benefits in period t and the discounted value of the resource stock in period t+1 are discounted back to the present. t
Now, we obtain the F.O.C (First Order Conditions):
Economic interpretation of the F.O.C
∀t =0,1,…,T
First, we must simplify:
As we can see, the third equation is simply the First- Order Difference
Equation
WHAT ABOUT THE OTHER TWO EQUATIONS?
∀t =0,1,…,T
First equation:
Marginal net benefit of an additional unit of the resource
harvested in period t USER COST
QUESTION 3: WHAT IS MEANT BY THE TERM USER COST?IF USER COST INCREASES, WHAT HAPPENS TO THE LEVEL OF HARVEST OR EXTRACTION TODAY?
Second equation:
Value of an additional unit of the resource in
period t
Marginal net benefit of an additional unit
of the resource in the current period
Marginal benefit that
an unhavested unit will yield in the next period,
discounted.
Finally, to solve the system, we must find the path of harvest Y = (Y0, Y1,…,YT);
stock X = (X0,X1,…,XT ,XT+1) and Lagrangian multipliers λ= (0,1,…, T ,T+1).
For this, we should take into account that:
X0 is the initial stock of the resource, which is known. λT+1 is the shadow price of the natural resource in period T + 1: But since
the managers are not going to exploit the resource in period T +1; an additional unit of resource in period T + 1 does not have any value for them. Therefore λT+1 = 0
With this information we can solve the system with 3(T+1) equations
We are going to focus on the case in which T ∞
We have a optimization problem with an infinite
large system of equations and an infinite number of
unknowns
This problems usually have a transitional period
(τ), after which parameters are
unchanging.
Steady State Optimum:
Xt+1= Xt = X*Yt+1= Yt = Y*λt+1= λt = λ*
Taking this into account, our F.O.C become a system of three equations with
three unknowns:
Substituting the first equation in the second and after some manipulation we
obtain the “FUNDAMENTAL EQUATION OF RENEWABLE RESOURCES”:
Marginal net growth rate. Marginal stock effect
Resources internal rate of discount
Marginal net growth rate. Marginal stock effect
Resources internal rate of discount
Resource internal rate of return Resources internal rate of discount
The rate of return on investments elsewhere in the economy
Therefore, we have two equations in the steady-state optimum:
Low growth rate of the resource (F(X)) combined with high discount rate(δ)
Harvesting costs for the last members of the population are less than their price
Marginal stock effect Resources internal rate of discount
Marginal stock effect Resources internal rate of discount
RENEWABLE, NONRENEWABLE, AND ENVIROMENTAL RESOURCES
Exercise 1: Suppose the dynamics of a fish stock are given by the difference equation (written in “iterative” form) Xt+1=Xt + rXt(1-Xt/K)-Yt , where X0=0.1, r=0.5, and K=1. Management authorities regard the stock as being dangerously depleted and have imposed a 10-year moratorium on harvesting (Yt=0, for t = 0, 1, 2,…, 9). What happens to Xt
during the moratorium? Plot the time path for Xt (t=0,1,2,…,9) in t – X space.
t Y X(t) X(t+1)0 0 0.1 0.145
1 0 0.145 0.2069875
2 0 0.2069875 0.289059337
3 0 0.28905934 0.391811356
4 0 0.39181136 0.510958964
5 0 0.51095896 0.635898915
6 0 0.63589892 0.751664657
7 0 0.75166466 0.844997108
8 0 0.84499711 0.910485605
9 0 0.91048561 0.951236389
RENEWABLE, NONRENEWABLE, AND ENVIROMENTAL RESOURCES
t Period (years)
Yt Harvesting
Xt Fish stock in period t
Xt+1 Fish stock in period t+1
RENEWABLE, NONRENEWABLE, AND ENVIROMENTAL RESOURCES
0 1 2 3 4 5 6 7 8 90
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.91
Fish stock
Fish stock
Tiempo (t)
Fish stock increases to values close
to 1
EXERCISES
Exercise 2: After de moratorium the management authorities are planning to allow fishing for 10 years at a harvest rate of Yt= 0.125 (for t = 10, 11,…, 19). Suppose the net benefit from harvest is given by πt = pYt-cYt/Xt, where p =2, c = 0.5, and δ= 0.05. What is the present value of net benefits of the 10-year moratorium followed by 10 years of fishing at Yt = 0.125?
t Y X X(t+1)
0 0 0,1 0,145
1 0 0,145 0,2069875
2 0 0,2069875 0,289059337
3 0 0,289059337 0,391811356
4 0 0,391811356 0,510958964
5 0 0,510958964 0,635898915
6 0 0,635898915 0,751664657
7 0 0,751664657 0,844997108
8 0 0,844997108 0,910485605
9 0 0,910485605 0,951236389
10 0,125 0,951236389 0,84942925
11 0,125 0,84942925 0,788378849
12 0,125 0,788378849 0,746797669
13 0,125 0,746797669 0,716343124
14 0,125 0,716343124 0,692940951
15 0,125 0,692940951 0,674327845
16 0,125 0,674327845 0,659132747
17 0,125 0,659132747 0,646471131
18 0,125 0,646471131 0,635744235
19 0,125 0,635744235 0,626530986
πt Net actual P= 2
πt= 0 0 C= 0,5
πt= 0 0 δ= 0,05
πt= 0 0 ρ= 0,952380952
πt= 0 0
πt= 0 0
πt= 0 0
πt= 0 0
πt= 0 0
πt= 0 0
πt= 0 0
πt= 0,184296038 0,11314178
πt= 0,176421182 0,103149811
πt= 0,170723393 0,095065174
πt= 0,166309326 0,088197387
πt= 0,162751309 0,08220047
πt= 0,159804724 0,076868805
πt= 0,157315113 0,072067866
πt= 0,155178433 0,067703836
πt= 0,153321282 0,06370816
πt= 0,151690025 0,060028894
0,822132182
N = A/ δ
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 190
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fish stock
Fish stock
Tiempo (t)
Fish Stock increase until t=10; then,
when the moratory finish, fell down rapidly, and in
period 17, 18, the function is going to the steady state.