Non-Renewable Resources - Environmental Economics - Lecture Slides, Slides of Environmental Economics

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Environmental Economics 2

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

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)

Hamiltonian

H [ q , x ,^ λ^ , t ]=^ π[ q , x , t ]+^ λ g ( x , q , t )  profit plus change in stock valued by shadow price. To maximise => = 0 ∂ ∂ q H and λ

= − ∂ ∂ 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.

• 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=

(0.92 in t=0 does not equate to £1 in t=1 => try

it!)

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!

Time

Price

r

P t

dP dt

Quantity

The Basics (3): Backstop exists

Time Price r P t dP dt = ( ) / Quantity Price of Backstop (^) • 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 the Model (3)

Price

Time

New Discoveries

Effect 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.

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 - - μ t

Non-Renewables

Assumptions:

• good can be extracted costlessly and no direct benefit from holding stock (ie vt =
  1. => Abritrage equation becomes: r(t)=

μ t^ /^ μt The equilibrium, where the firm is indifferent between which asset held, can only occur if price of the asset appreciates, ie

μ 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.

Growth function

Equilibrium

Assume that equilibrium is reached where the growth in each period equals the harvest rate, ie ( )= ( *, *)= 0

xt g x q 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 pg x q W pgx q dt e rt Where e

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

− r pg e dx dW (^) rt x