The Economics of

Nonrenewable ResourcesJeffrey Krautkraemer (JEL 1998)

Nonrenewable resource scarcity

Introduction

Natural resources: Renewable: capacity for regeneration (fish, clean water,

forests) Non–renewable: regeneration only at geological time

scale, not human (oil, gas, copper)

Leading to Issue of Sustainability: Non–declining (per capita) consumption or utility

are non–renewable resource use and sustainable growth incompatible?

Introduction (2)

Set–up:

1. Efficient use of non–renewable resources: perfect competition and non–renewable resource exploitation

1a. A simple model (the mine)

1b. A general model (optimal control)

2. Complicating factors

1a. The mine – Lagrangean method

Model of non–renewable resource extraction:

Firm i owns a mine with initial stock size: Si,0, and aims to maximize net benefits (profits) over infinite time horizon

by choosing the optimal amount extracted (qi,t 0) in every period

Net benefits: V(Si,t, qi,t) = B(Si,t, qi,t) – C(Si,t, qi,t)

Discount factor (0 1)

Objective function:

Note: constraint implies Si,t – Si,t+1 = qi,

0,0 ,0 ,,,, s.t. ),(),(max it tit

t titititi SqSqCSqB

The mine (2)

Simplest case – perfectly competitive mining industry: All firms (mine owners) identical in all respects; Perfect competition; sales price Pt

Same to each individual mine owner Assume industry D: Implies max price of

Just sales, no other benefits (or costs) of extraction, no ‘stock effects’

B(Si,t, qi,t) = Pt qi,t; C(Si,t, qi,t) = 0 V(Si,t, qi,t) = Pt qi,t

Objective function for every firm:

)())(( tQePtQP

P

0,0 ,0 , s.t. max it tit

t tit SqqP

i

iSS 0,0

The mine (3) – first order conditions

Lagrangean:

where is the Lagrangean multiplier (or the shadow price of the resource)

First order conditions: L / qi,0 = P0 – = 0

L / qi,1 = P1 – = 0

L / qi,2 = 2 P2 – = 0

… L / qi,T = T PT – = 0

So: t Pt – = 0 for all t = 1, …, T, and hence

P0 = P1 = 2P2 = … = t Pt = … = T PT

0

,0,0 ,t

tiit

t tìt qSqPL

The mine (4) – arbitrage across time

What if t Pt > for any t = 1..T, and t Pt in all other t’s? Present value of marginal benefits in period t larger than in

all other periods Extract everything in period t, zero in all other periods…

But then that holds for all firms in the industry… Industry demand: Price very low in period t, very high in all other periods… Arbitrage until P0 = P1 = 2 P2 = … = T PT

Moving resource extraction between periods until, in equilibrium, each individual firm is indifferent when to extract arbitrage

)())(( tQePtQP

The mine (5) – Hotelling Rule

Interpretation Pt = Pt+1 ?

Note that 1 / (1 + r) Pt+1 = (1 + r) Pt

Hotelling Price path: First, Pt+1 = (1 + r) Pt Pt+1 – Pt = rPt

Continuous time equivalent:

Straightforward integration:

rdtP

dPr

P

dtdPrPP

/

rtCetPrtKPrdtdPP

)( ln1

The mine - Hotelling’s Rule for extraction

Resource stocks are “natural assets” I.e. the form which assets take in the initial state of the economy (Solow)Assets must be competitive against all other assets – marginal rate of return on holding asset must be equal to market rate of return (r)Extraction paths for resource stocks will generate the price path – and so we would expect extraction path to generate the return r on the initial price P

First point of natural resource economics is that natural resources operate as assets in economy

The mine (6) – terminal condition and path

Assume that the price that can be charged for the resource is bounded at

the maximum (or backstop) price for this resource (?)

Unknown variables: C and T…

We know: Q(T) = 0, and also

Insert in

Price path:

Given the price path, we can now find the industry extraction path

PTP )(rTrTrt ePCPCeTPCetP )()(

)()( TtrePtP

)())(( tQePtQP

P

The Mine – Backstop Price

Resource stock prices rise because they are assetsResource prices are bounded because there are substitutesThe point at which the initial price of a substitute outcompetes a resource renders it economically obsolete (choke off price)Extraction path targets this backstop price and works backwards to the present (in recognition the resource’s role as an asset)

Extraction paths and end of use (of a particular resource) are determined by the identification of substitutes to meet the needs identified by the D curve

Role of Exploration Role of Technological Change

The mine (7) – solving paths of extraction (Q)

Combine and

Constraint: ,

To find T, we just need to solve the integral of Q(t) for T

tTr

Ttr

tQePeP TtrtQ

)()()(

)()( tQePtP )()( TtrePtP

00)( SdttQ

T

t

i

iSS 0,0

The mine (9) – Terminal (depletion) Time

So,

Integration:

And hence

00)( SdttQ

T

t

T

t

T

t

T

tT

rtTt

rdttT

rdttQS

0

2

0

2

00 22

1)(

r

ST 02

The Mine – Terminal Conditions

Cake eating models are about the allocation of finite resources across time (who gets to eat the cake?)Assume that the definition of the relevant time horizon is that the resource should be economically extinct at the end of it (no sense in leaving leftovers for non-existent future)Work backwards from absolute depletion toward present recognising that the resource must meet marginal asset returns

The optimal extraction of resources is equivalent to the optimal allocation of their usefulness across the relevant time horizon

Hotelling showed that there were forces within the economy to prevent the first owners/users from eating the entire cake in the first period

The Mine - Complete solution:

Depletion time:

Price path:

Industry extraction path:

Consider the example of two different initial stock levels

(same extraction paths but aggregating to different S)

rtSr

tr

SrtQ

00 22

)(

00 2/2)( SrrtrStrTtr ePePePtP

rST /2 0

Q t

t

P

T

T

tTr

tQ

)(

P

QePQP )(

450

t = t

TtrePtP )(

The mine (11)• Sensitivity analysis:

• How does optimal extraction path change if (some of the) parameters change?

• Parameters:

• First: r increases:

higher rate of price increase

more extraction in early periods, faster depletion.

• Graphically…the curve first shifts in, but then also shifts down (to indicate faster early depletion)

Greater pressure on resources to perform as assets

PSr ,,, 0

Q t

t

PP

450 T

T

QePQP )(

t = t

tTr

tQ

)(

TtrePtP )(

The mine (13)• Discovery of additional reserves S:

– r unchanged, Q(t) should be increased in all t to prevent being hit while there are still reserves left

– Price jumps down, price path becomes flatter (because 100r% of a smaller sum of money is smaller in absolute terms than 100r% of a larger sum)

– Depletion postponed (higher T);

– As we saw above

rST /2 0

P

Q t

t

P

T

T

tTr

tQ

)(

P

QePQP )(

450

t = t

TtrePtP )(

The Mine - Hotelling Diagram

A Hotelling diagram charts out the extraction paths and price paths for a given D, given S, and given P barThe diagram illustrates how the market will itself allocate resources across time, given their function as assetsThe primary force that causes the path to deviate from this:

“technological change”

- meeting needs represented by this D differently- exploration for additional stocks- identification of new substitutes

Technological change is the means by which an economy alters the role and extraction paths of different resources (“peak oil”, “peak whale oil”)

The mine (15)• Demand Shift (Lower ): D function shifts out (flatter,

intercept unchanged):

– along original price path, for any price the quantity demanded of resource is higher reserves depleted before is hit

– Current price should be increased to reduce current extraction, but depletion still occurs sooner

rST /2 0

P

The mine (16)• Lower backstop price ( ):

- in this specification: no effect here because of particular specification of the industry demand function (creates linear extraction path)

– In general (with nonlinear extraction path): initial price would be reduced to stimulate current consumption, resource depletion occurs sooner.

P

Moving toward Continuous Time NR Econ

The same points are made as beforeThese points are a bit more elegant when using dynamic optimisation techniques (used for growth theory in general)

The mine (17)More general case – away from simple Hotelling model

• What if there are positive extraction costs (C(q,S))?– Costs vary with amount extracted (Cq > 0, Cqq > 0)

– Extraction costs are stock–dependent: smaller stock, need to drill deeper (CS < 0, CSS > 0)

• And what if there are stock–dependent benefits too (B(q,S), BS > 0, BSS < 0)?

– Scenic value: larger stock, smaller cumulative extraction, landscape less devastated. Or greenhouse effect; larger stock, less cumulative extraction and emission of carbon…

• So, define V(q,S) = B(q,S) – C(q,S)

• We need to do optimal control…

1b. Optimal control• Optimal control: solving current value Hamiltonian…

• If confronted with

• Note that constraint implies

• Solution: write down current–value Hamiltonian

• Necessary conditions are:

qSqVSSqVH CV ),(),(

SCVS

qCVq

VrHr

VH

(2)

0 )1(

0

0

0

)( subject to ),(maxtt

rt dttqSdteSqV

)(tqS

Optimal control (2) – simple hotelling rule again• Let’s check: correct for simple case?

• Note V(q,S) = Pq, and

• Claim: solution is

•

•

•

• Combine:

))(),(()( tStqVtr(t) S 0))(),(( tStqH q

rP(t)(t)P(t)P(t)λ

0

0,

0

)( subject to )()(maxt

i

t

rt dttqSdtetqtP

)()()()()()())(),(( tqttqtPtSttStqVH CV )()( ttP

)(tr(t)

)(tqS

Optimal control (3) – basics again• Road map in Conrad and Clark:• Problem we need to solve:

• We end up with current value Hamiltonian

• And we show analogy with just straightforward Lagrangean approach (discrete time)

• For that, we need the present value Hamiltonian

)()())(),(( tSttStqVH CV

0

0

0

)( s.t. ),(maxtt

rt dttqSdteSqV

0

10

),(t

ttttt

ttt SqSSqVL

)()())(),(( tStetStqVH rtPV

Optimal control (4)

So, if confronted with

Write up current value Hamiltonian:

Necessary conditions for optimum:

So what is interpretation?

)()())(),(( tqttqtSVH CV

)()( s.t. ))(),((max0

)( tqtSdtetqtSVT

t

rttq

)( ;0 SCVSq

CVq VHrVH

Optimal control (5)

Suppose V(q,S) = Pq – c(S)

Foc’s …

Interpretation: Net marginal benefits (Vq) of extracting today need to be

equal to the shadow value (= user cost of current extraction) Net benefits of extracting an extra unit today (Vq) should be

equal to the net benefits of leaving it in the ground (Hotelling)

SCVS

qCVq

crHr

PVH

0

SS cPPPrcPrP )1(

qScPqqqSVH CV )(),(

Optimal control (6)So much for optimal control, back to the questions…

• What happens to price path if extraction costs vary with amount extracted (Cq > 0, Cqq 0)?

• Suppose C = (t) q(t), with d(t)/dt 0

• Hamiltonian:

)()( and )()()( ttrtttP

PrPPrP / whereˆ1ˆ)(

0/lim Note t P

)()( )()()( tqtq(t)ttqtPH CV

Optimal control (7)

So,

• If

– In the limit, rate of price increase is r

• If – Price may fall in early periods, but in the limit increases

at rate r

Price path may be U–shaped (if technical progress in extraction technology is sufficiently large)

rrP 1ˆ0

ˆ1ˆ0 rP

ˆ1ˆ rP

Optimal control (8)• And what if benefits or costs stock–dependent?

– BS > 0, BSS 0, or

– CS < 0, CSS 0

VS = BS – CS > 0

• Suppose BS = 0. Hamiltonian:

• Slower extraction, and maybe even depletion not economically feasible…

Sq CttrttCtP )()( and )()()(

rCr S /ˆ

λ(t)q(t))) C(q(t),S(tP(t)q(t)H CV

2. Complicating factors• World more complex than captured here…

– Endogenous exploration

– Extraction capital intensive

– Heterogeneity in resource quality

– Market imperfections

Complicating factors (2)• Endogenous exploration

– Finding new reserves not accidental... Is result of deliberate action

– More effort if current reserves are closer to depletion

• Consequence:

– Small reserves, high extraction costs

– more exploration effort new reserves discovered

– extraction costs fall price jumps down

– more extraction price increases again…

• Price path U–shaped

Complicating factors (3)• Extraction is capital–intensive

• C = C (q,S,K)

• Higher interest rate, always faster depletion?

• Higher interest rate -> less investments in extractive capital

• And backstop technology maybe also more expensive if interest rate is higher… even slower extraction

Capital costs complicate implications of higher r

Complicating factors (4)

• Heterogeneity across reserves:

– Differences in quality (higher and lower grade ores)

– Differences in extraction costs (higher and lower).

• Optimal extraction: most profitable option exploited first (discounting of future costs –extraction costs, or lower quality ores)

• Captured in making costs stock dependent

Meaning of Hamiltonian (2)• Hotelling valuation approach:

• Can we deduce anything about the value (W) of mining?

• Suppose B = P(t)q(t), C = (t) q(t) • Current value Hamiltonian:

T

t

rtdtetqtW0

)()()0(

(t)q(t) (t)q(t))P(t)q(t)H CV

Hamiltonian (3)• Current value Hamiltonian:

• Foc’s

• That makes life very easy…:

(t)q(t) (t)q(t))P(t)q(t)H CV

rtrt

PVS

PVq

eCet

Krtdrdtttdrdt

ttrHttr

tttPH

)0()(

ln1

)(/)(

)()()()(

0)()()(

rtrt ePet

P

)0()0()0()(

)0()0()0(

Hamiltonian (4)• Hotelling valuation approach:

• W(0) is firm’s current value (say at stock exchange) and is product of current price, current technical change and reserves

0

0

0

00

)0()0(

)()0()0(

)()0()0(

)()0()()()0(

SP

dttqP

dttqP

dttqdttqetW

t

t

tt

rt