resources economics. jon m.conrad. chapter 1

BASIC C

ONCEPTS A

BOUT

RESOURCES ECONOMIC

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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?


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.


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


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.


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.


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.


• 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.


• 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.


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.


• 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.


• 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.


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.)


• 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 .


• 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.


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.


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.


•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.


•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


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


t Period (years)

Yt Harvesting

Xt Fish stock in period t

Xt+1 Fish stock in period t+1


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.

resources economics. jon m.conrad. chapter 1

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