Environmental Economics 2 Lecture 6 Non-Renewable Resources.

Environmental Economics 2

Lecture 6

Non-Renewable Resources

Overview of this part…

• Natural Resources theory:– Non-renewables– Renewables– Applied studies of theory – is it true?– Fisheries– Forestry

• Energy security – valuation issues

• Climate Change

Health warning

• This lecture contains a lot of maths.

• Later lectures won’t be quite so bad (promise!!!)

• Key: understanding the concepts, not necessarily the maths

• Though if you can handle the maths this would be great.

Readings

• This lecture largely based on chapters 7,8,9 of HSW

• More advanced – see Conrad and Clark chapter 3

• Other sources (go to if don’t follow above): Perman et al

Definitions

• Non-renewables – eg coal

• Renewables – eg fish stocks or flows (eg wind)

Concepts you will need to have an idea of…

• Hamiltonian – specialised form of Lagrangian – see Perman or Pemberton and Rau if you don’t understand this in this lecture.

• Market structures – monopoly, oligopoly, perfect competition – see basic micro text book to refresh if you’ve forgotten!

• Discounting – intertemporal issues. See Perman.

Hamiltonian

),,(],,[],,,[ tqxgtxqtxqH

profit plus change in stock valued by shadow price. To maximise =>

0q

H

and

x

H

These conditions and equation of motion (change in x) give a set of differential

equations which define an optimal solution. (it also has to satisfy travers ality conditions but we won’t go into this – see HSW 186-188)

Hamiltonian helps to solve the control problem. Similar to Lagrangian.

Discounting

• Social rate of time preference => reduce future values to reflect this.

• Usual notation: discount rate is r (occasionally i).• 1/((1+r)^t) gives you the multiplier to reduce any

value in time t to the current time.• Eg 1/1.08 may give you the multiplier for t=1 and

r=8% => £1 = £0.925926• Note r=8% does not lead to 0.92 in period t=1,

because 1.08 in t=1 would be equal to 1 in t=0 (0.92 in t=0 does not equate to £1 in t=1 => try it!)

The basics

rtP

dtdP

)(

/ • If LHS is greater than RHS it pays to reduce extraction

• If LHS is less than RHS it pays to increase extraction

• But note, adjustment is inherently unstable!

The basics (2): No Substitute for the ER

• Resource is progressively exhausted on the price path.

• Eventually resource may be exhausted but this can take infinitely long!

T im e

P r ic e

rtP

dtdP

)(

/

Q u a n t it y

The Basics (3): Backstop exists

T im e

P r ic e

rtP

dtdP

)(

/

Q u a n t i t y

P r ic e o f B a c k s t o p • Resource is progressively exhausted on the price path.

• But now when price reaches Backstop Price the producer must have nothing left.

• For this to work initial price must be ‘correct’

• Lower is r, higher is initial price and lower is extraction initially

Extensions to model:

• Effect of Extraction Costs– Now net income (price minus extraction cost)

must rise at rate of interest.– But if extraction costs fall, then initially

extraction increases.Backstop

Initial Price Path

Price Path After Fall in Cost

Faster Extraction

Slower Extraction

Price

time

Extensions to the Model (3)

Price

Time

New DiscoveriesEffect is similar to an decrease in the price of the substitute – you extract faster. With unanticipated discoveries we see the following pattern:

Extensions to Basic Model (2)

• Capital Costs– These are part of extraction costs and are sensitive to

interest rates. If ‘r’ rises then extraction costs rise, resulting in slower extraction. But higher is ‘r’ faster is extraction on Hotelling grounds.

• Technology Changes– If backstop price falls, extraction must increase.– If technology lowers extraction costs, extraction also

increases initially.

Economic Approach to Resource Use: Theoretical Background

• Capital Theory Approach• In equilibrium the returns from buying machines = the total return from

holding the numeraire asset =>

(vt+1 + μt+1)/ μt=1+ rt+1

• If out of equilibrium then there is the chance of pure profits from arbitrage.

• The own rate of return = rental income/price• If rate of return of using machines and numeraire good is different this

can only be accounted for by a change in the price =>

vt+1= rt+1 μt-(μt+1 - μt )

• So the difference in interest rates must be accounted for by a change in the price of the asset.

Capital Theory

• In the continuous form:

• where is the time derivative (increase or fall in the price of capital).

• This is the short-run equation of yield or the arbitrage equation.

v(t) = r(t) μt -

t

Non-Renewables

Assumptions: - good can be extracted costlessly and no direct benefit from holding stock (ie vt =

0) => Abritrage equation becomes:

r(t)=

t/ μt

The equilibrium, where the firm is indifferent between which asset held, can only occur if

price of the asset appreciates, ie

t>0 at the own rate of return of the numeraire asset.

This is called Hotelling’s Rule. For example, if a firm precommiting to supply a resource over a number of time periods then the forward price would have to satisfy Hotelling’s rule – i.e. a rise at least at the rate of return of the numeraire, otherwise firm better off extracting all in t=1 and investing the proceeds in the numeraire.

Renewables

Renewables have an additional complication in that stock grows through reproduction. This means we need a growth function.

)),(),(()( ttqtxgtx

x(t) is the stock, q(t) is the harvest rate. Commonly used form of growth function = logistic: At low stock rates, rate of growth is low, it peaks and then falls off. i.e. 0<x<xmax gx>0, gxx<0 (subscripts show partial derivatives)

Growth function

EquilibriumAssume that equilibrium is reached where the growth in each period equals the harvest rate, ie

0*)*,()(

qxgtx

This is a steady-state equilibrium. Hence the total value of the stock over an infinite time horizon, with constant prices and no harvest costs is :

0

*)*,(*)*,(

r

qxpgdtqxpgW e

rt

Where e-rt is the continuous time discount factor. Differentiating this with respect to stock gives the shadow price of stock:

r

pge

dx

dW xrt

*

EquilibriumSo the current value equilibrium is where

rte

The condition for a steady-state is as follows:

repg rtx

This is comparable to

v(t) = r(t) μt -

t

in that rt

xepg is the rent,

tis zero as stock is constant. The growth term is very important – it shows how the rate

of stock growth changes with the stock and therefore represents the return on retaining the marginal unit of stock. In market equilibrium, the market price of fish should equal

the shadow price, i.e. rtep and gx=r. gx is the own rate of return.

Market Structure

• The market structure of an industry may be important in determining the rate of extraction of a resource. Here we will take the earlier analysis further, building on mathematical techniques of comparative dynamics.

Competitive Industry

Assume extraction costs are zero. Large number of small firms, all price takers. For a firm to be indifferent between when to extract the marginal resoures, the price must rise at the discount rate – ie by Hotelling’s rule. Total stock declines at the rate of aggregate extraction and there exists some time in which the industry’s stock is exhausted Tc By assumption individual mine owner is indifferent between pc at t=0 and pc(0)ert at time t. Price at Tc must equal the price at which demand is exhausted. So the price equals the backstop price. So, the initial resource price is related to the final resource price by:

rTcbrTcrTcc

cc epefeTqfp )0())(()0(

Competition

Determining the price and extraction paths depends on finding Tc. Present value of the total profit pc(0)x0 is increasing in x0 so it must be optimal to extract all the stock. The time when all is extracted Tc is found by equating the integral of extraction with the initial stock.

dtepddttqTrpS TcrTc

bcTc

cbc )1(

00[)(),,(

Ie sum extracted must be equal to the initial stock. The results derived can be compared to those for social optimality – as if costs are zero then it collapses to the same objective – that of maximizing consumer surplus.

Consumer SurplusConsumer surplus is defined as the area under the inverse demand curve:

u(q)= q

dwwf0

)(

and the social planner’s objective function is to maximize the present value of consumer’s surplus over the life of the resource:

maximiseq Tc

rtdtequ0

)(

subject to: x(0)=x0

and qx

HamiltonianThe current value Hamiltonian (p188 HSW) is H=u(q)-μq The first order conditions are: u’(q) –μq=0 and the costate condition:

0

X

Hr

Note that u’(q)=p, so substituting for μ and rearranging the costate equation leads to:

p

pr

which is Hotelling’s rule. So, the socially optimal planner chooses the same

extraction path as the competitive industry as long as r equals the social rate of time preference.

Monopoly

Monopolist’s optimal control problem is:

maximiseqm Tm

rtmm dteqqf0

)(

subject to: x(0)=x0

and qmx

Hamiltonian

We can specify a current value Hamiltonian:

mmmm qqqfqH )(),(

Where μ is the costate variable that gives the marginal current value of a unit of stock. The first order conditions are:

mmmm

qqfqfq

H)(')( (1)

Which can be simplified by defining a revenue function R(q)=f(q)q R’(qm)=μ The costate condition is:

0

X

Hr

Hotelling Rule for Monopoly

Differentiating 1 wrt time, equated with the costate condition and R’(qm) is substituted for μ then:

rq

q

R

Rm

m

)(

'

'

This is Hotelling’s rule for the firm. This states that the rate of change in the marginal revenue on the last unit conserved must equal the rate of return on the numeraire asset. Monopoly equates the present value of the marginal revenue across the life of the resource, i.e. monopoly operates a form of perfect price discrimination through time. Initially, it supplies at a relatively low price on the less elastic segment of the demand curve; just before depletion it supplies at a point where the elasticity is high. The “markets” are separated by time and also by the fact that the price rises at less than the discount rate, so no incentive exists for speculative storage and resale of resource by traders.

From the Hotelling’s rule for the firm, it follows that the value of stock over time must be related by μ(t)= μ(0)e-rt and that μ(Tm)= μ(0)e-rTm where Tm is the time when the monopolist exhausts his stocks. As q(Tm)=0 then the first order condition of the Hamiltonian becomes: f(0)- μ(Tm)=0 recall demand is zero at the backstop price, so μ(Tm)=pb. As μ has to increase by the discount rate, μ(0)= pbe-rTm so for time interval [0, T m]

)()()0()( TmtrbrtrTmbrt epeepet

Equating this with the first order condition of the Hamiltonian f(qm)+ f’(qm) qm=pber(t-Tm)

The solution lies in solving for (qm) and integrating to determine the time take to extract the initial stock. General solutions are not simple. The initial rate of extraction increases with the discount rate and decreases with the backstop price. How can we compare monopoly and perfect competition? Assume a linear demand function….

Perfect Competition

pp

pdqb

1

)( where 0

Pb=backstop price. qpqfp b )(

Substituting into

dtepddttqTrpS TcrTc

bcTc

cbc )1(

00[)(),,( leads to

0)1(

0

)1()1( x

r

epTpdte

p rTcb

cbTcr

Tc b

Which can be solved for Tc using Newton’s method. Once Tc is determined the price path is p(t)=pb er(t-Tc) so by Hotelling’s rule p(t)=p(0)ert and extraction path is derived from the demand curve.

Monopoly

f(qm)+ f’(qm) qm=pber(t-Tm)

for a linear demand function becomes )1(

2)( Tmtr

bm e

pq

introducing the resource exhaustion constraint and integrating leads to

0)1(

02

)1()1(

2x

r

epTmpdte

p rTmbbTmr

Tc b

Comparison

0)1(

02

)1()1(

2x

r

epTmpdte

p rTmbbTmr

Tc b

Compare with competitive case:

0)1(

0

)1()1( x

r

epTpdte

p rTcb

cbTcr

Tc b

Tm>Tc as T-(1-e-rT) is increasing in T => Comparative dynamic time path for price and extraction.

Comparative dynamics

Figure 9.1 in Hanley, Shogren and White

Conclusions

• Initially competitive industry extracts more rapidly than monopoly but then less rapidly as the price increases towards the backstop price. Initial price for monopoly is higher than that for competitive industry. Then increases more gradually towards the backstop price,

• Rate of price increase for monopolist is less than discount rate, otherwise resource can be purchased by speculators, stored and sold at later date, so reducing monopoly profits.

Heroic Assumptions

• Non-linear extraction costs => solving for extraction path for competitive industry formidable problem

• Exploration issues

• Resource scarcity

• Uncertainty

Does it work?

• Will examine this in later lecture on empirical validation.