HartwickOlewiler1986 TOC Ch3

8/17/2019 HartwickOlewiler1986 TOC Ch3

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THE ECONOMICS

OF

NATURAL

RESOURCE

USE

John

M

Hartwick

Queen s

University

Nancy

D

Olewiler

Queen s U niversity

tfj

ARPER ROW, PUBLISHERS , New York

Cambr idge Philadelphia San FrancIsco

London Mexico City Sao Paulo Syd ney

8 7


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Nature s resources

are not

so

much

an inheritance/rom our parents, as a loan/rom

our

children.

We

dedicate

this

book

to

our parents

and children.

Sponsoring Editor: John Greenman

Project Editor: Mary G. Ward

Cover Design:

ohn

Hite

Text Art: Artset Ltd.

Production: Debra Forrest

Compositor: Progressive Typographers

Printer and Binder: R. R Donnelley Sons Compan y

THE ECONOMICS OF NATURAL RESOURCE USE

Copyright © 1986 by Harper Row, Publishers, Inc.

All rights reserved. Pri nted in the United States of America No

part of

his book

may be used

or

reproduced in any manner whatsoever without written permission,

except in th e case

of

brief quotations embodied in critical articles and reviews. For

informati on address Harper Row, Publishers, Inc.,

10

East 53d Street, New York,

NY

10022.

Library of Congress Cataloging in Publication Data

Hartwick, John M.

The economics of natural resource use.

Includes bibliographies.

1 Natural resources. 2 Environmental policy.

I. Olewiler, Nancy D. II. Title.

HC59.H3558 1986 333.7 13 85-14076

ISBN 0·06-042695-0

85

86 87 88

9 8 7 6 5 4 3 2 I

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T

ONTENTS

PREFACE xi

1

2

3

Approaching the Study

o

Natural Resource Economic

Introduction, 1

Decision-Making over Time, 3

Interest Rate, 4 Compounding, 4 Discounting or Getting Present

Value, 5 Techniques of Analysis, 7

,

Property Rights and Natural Resource

Use

8

Welfare Economics and the Role o Government, 11

Resources in the Economies of the United States, Canada, and Other Nations

Summary, 20

Land Use and Land Value

Introduction, 22

The Concept o Economic Rent, 23

Rent on Homogeneous Land, 24 Plots

of

Differing

Quality 27 Institution al Arrangements: The Role

of

Market Structure and

Property Rights in the Determination of Rent,

31

Location and Land Value 35

The Price

of

an Acre

of

Land,

35

Efficient Land Use with Two Competing

U ~ e s

36

Intermediate Goods and Mixing Land Uses,

38

Neoclassica

Land Use Patterns,

39

Institutional Arrangements

and

Land Use

Patterns, 42 Transportation Costs and Aggregate Land Rents, 45

Summary, 46

Nonrenewable Resource Use: The Theory o Depletion

Introduction, 49

The Theory of the Mine,

Cl

Extraction from a Mine Facing a Constant Price,

51

Profit Maximization

for the Mine, 54 Quality Variation Within the Mine, 57

Extraction by a Mineral Industry, 59

The Hotelling Model,

59

Exhaustibility

and

Welfare: Demand Curves and

Backstop Technology, 63 A Model

of

a Competitive Nonrenewable

Resource Industry,

65


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vi

4

5

6

CONTENTS

Extensions of the Industry Model, 67

Changes in Extraction Paths Under Altered Conditions 67 AnJncrease in

Extraction Costs

67

A Rise in Interest Rates 68 The Introduction of

Taxes 68 Declining Quality in the Stock

70

Deposits of Distinct and

Differing Quality 72 Setup Costs for the Mine and Industry 74

Summary, 77

Market Structure and Strategy in Nonrenewable

Resource Use

Introduction,

88

Market Structure in a Static Environment, 89

Monopoly versus Perfect Competition 89 Cournot versus Stackelberg

Models o f Oligopoly 91

Market Structure

in

a Dynamic Environment, 95

The Monopoly Resource Supplier 95 Dominant Firms and Fringe Sellers:

A Case of Oligopoly 100

Backstop Technologies

and

Market Structure, 106

A Backstop Owned by the Monopolist 106 A Resource Monopolist Facing

a Competitive Backstop Technology 107 Strategic Behavior in Developing a

Substitute

109

Recycling as an Alternate Source of Supply 112

Market Structure and Common Pool Problems with Oil and Gas Resources, 113

Summary, 117

Uncertainty and Nonrenewable Resources

Introduction, 119

. Economics and Uncertainty, 120

Uncertainty Concerning Stock Size 122

Two Deposits of Unknown Size 125

Uncertainty

and

Backstop Technologies, 127

Uncertain Date of Arrival of he Backstop Technology

131

Option Values

of Minerals and Coping with Price Uncertainty 132

Exploration for New Reserves, 133

General Issues and Externalities 133 Exploration and Risk

Pooling 137 Exploration

ofa

Known Deposit

of

Unknown Size

138

Summary,

141

Economic Growth and Nonrenewable Natural Resources

Introduction, 144

Natural Resource Scarcity, 145

Measures of Natural Resource Scarcity 147 Real Unit Costs 147 The

Real Price of the Resource

148

Resource Rents

151

Concluding

Comments 154

Economic Growth and Depletable Resource Use 157

The Derived Demand for Nonrenewable Resource Flows 158

Models of Economic Growth with Depletable Resources

161

Normative

Investing

88

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CONTENTS

7

8

9

Resource Rents and Intergenerational Equity 165 Population Growlh

Pollution and Technical Progress 167 Positive Approaches to Economic

Grqwth with Nonrenewable Resources 168

Limits to Growth, 169

Projections and Predictions 169 A Trend Displaying Arithmetic

Growth

170

Geometric Growlh

172

The Limits to Growth

174

Summary 179

Issues in the Economics

of

Energy

Introduction, 183

Short-Run Effects of Energy Price Increases, 189

Long-Term Effects of High Oil Prices: Supply Effects,

191

Possible OPEC Actions: The Wealth-Maximizing Model 192 Possible

OPEC Actions: The Target Revenue Model

197

Effects of Rising Oil Pric

on Energy Supply 203 Estimation of Supply Curves of Nonrenewable

Resources 205 Specific Energy Resources 2 0

The Demand for Energy, 221

The Importance pfEnergy in the Economy 222 Price Elasticities of

Demand for EnergY

224

Energy Consumption and Conservation over

Time

228

.

Energy Policy

and

the Energy Crisis, 231

Price Ceilings 233 Policies to Reduce Dependence on Oil Supplies 236

Summary, 238

The Economics of the Fishery: An Introduction

Introduction, 243

A Model of the Fishery,

24El

Fishery Populations: Biological Mechanics 247 The Bionomic Equilibrium

in a Simple Model 252 Harvesting Under Open Access 255 The

Industry Supply Curve for a Common Property Fishery 261 Socially

Optimal Harvests under Private Property Rights 263

Fishery Dynamics, 268

The Discounting

of

Future Harvests 268 Dynamic Paths in the Fishery:

The North Pacific Fur Seal 276

Extinction of Fish Species, 2 4

The Bioeconomics of he Blue Whale: A Case of Near

Extinction 284 Socially Optimal Extinction

287

Summary, 287

Regulation of the Fishery

Introduction 292

The Economics of Fishery Regulation, 293

The OptirnalTaxes o n the Fishery 293 A Tax on the Catch 296 A

Tax on Fishing Effort 298 A Quota on the Catch and Effort 300 So

Ownership of he Fishery: The Cooperative 304


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viii

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CONTENTS

Fishery Regulations in Practice, 304

Fisheries Management Councils in the United States, 306 The Oyster

Fishery, 310 Spanish Sardines, 314 The Pacific Halibut, 318

Summary, 323

Special Topics in the Economics of the Fisheries 326

Introduction, 326

Varieties of Biological Basics, 327

Salmon, 327 Environmental Factors in Fishery Biological

Mechanics 33 Interaction among Species 33

International Fisheries, 340

A Two-Country Model, 340

Uncertainty

in

the Fishery, 343

Summary, 344

Forestry

Use 348

Introduction, 348

Some Biological Basics of Trees

and

Forests,

351

The

Optimal Rotation Period for the Firm, 355

Another Approach to the Choice

of

Rotation Period, 360 Rotation Periods

in Different Environments, 360 Optimal Harvests in a Mature Forest, 364

The Forest Industry: Price Determination with Many Different Forest Types, 365

The Relative Profitability of Forestry Use in Different Locales, 365 Effects

of

Taxes on Forestry Use, 367 Property Rights and the Incentive to

Replant,

368

Forestry

~ r c t i c e s

and Policies, 368

Mean

Annual Increments and Allowed

Cuts

in National

Forests, 372 Policies and Practices

of

the

U S

National Forest

Service, 374 The Social Losses from Public Forest

Management, 376 Do Private Firms Harvest Too Quickly?, 378

Summary, 379.

An

Introduction to Environmental Resources:

Externalities

and Pollution

382

Introduction, 382

A Taxonomy of Externalities, 386

Public versus Private Externalities 386 Externalities in

Consumption, 394 Externalities in Production, 402

Distribution Di mensions

o

Intermllizing Externalities 406

Pollution Control Mechanisms, 4 6

Topics in the Theory of Environmental Controls, 410

Income

Alternative

Taxes versus Subsidies, 411 Taxes versus Standards, 414

Summary, 421

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CONTENTS

3

Pollution Policy

in Practice

Introduction, 425

Environmental Quality

and

the Valuation

of

the Marginal benefits from Pollu

Abatement, 426

General Issues, 426 Water Quality and Recreation Benefits, 427

Bees and Blossoms, 432

Standards versus

Fees

in Practice, 435

France and the Netherlands: Fees for Emissions, 436 The,United States:

Water Pollution Standards, 438

Marketable Permits: The New Wave in Pollution Control?, . 442

Institutional Background, 443 Characteristics of Marketable

Permits,

446

The Ambient'Based ystem,

448

The Emissions-Base

System, 449 The Offset System, 451 Costs of Alternative Permit

Systems, 453 Acid Rain: Can Marketable Permits Help?, 455

Summary, 457

4 Governmental

Regulations

and

Policy in Natural

Resource

Use

Introduction, 461

Promoting Efficiency

and

Equity

in

Natural Resource

Use

464

Efficiency and Market Failure in a Static Economy, 465 Efficiency over

Time, 469 Mitigating the Effects

of

Market Failures,

471

Second B

Policies, 474 Equity Concerns, 475

Regulatory Agencies and

the

Efficacy

of

Regulation,

476

Regulatory Agencies in the United States and Canada, 476 The Efficienc

Regulation, 478 The Regulation and Deregulation

of

Natural Gas Prices

the United States: A Case Study,

482

Concluding Remarks: Policy Formation and the Political Process, 491

Summary; 493

Appendixes

References

Author Index

Subject Index


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8

L ND USE ND LAND VALUE

3. The derivation ofR involves summin g rent per acre over all acresor integrating over a series

of infinitesimal circular rings or annuluses. Before integrating,

R =

XI 21lXYl

Pl tx

-

ZIW) dx X2 21lXY2

P2 -

t2X

-

Z2W) x

o Yl

JX

Y2

and

R in the text results after carrying

out

he integration operations. Below we maximizeR

by differentiating with respect to Xl and X2

and

set the expressions equal to zero.

4. An early statement of his was Mohri ng 1961). See the extensive discussion

in

Arnott and

Stiglitz 1979).

chapter

Nonrenewable

Resource Use

The Theory

of

Depletion

INTRODUCTION

Nonrenewable resources include energy

supplies-oil

natural gas

coal-and nonenergy minerals-copper nickel, bauxite, and zinc, t

These resources are formed by geological processes

that

typically. ta

years, so we can view these resources for practical purposes

as

having a

reserves.

That

is, there is a finite amount

of

he mineral in the groun

removed cannot be replaced.

1

Nonrenewability introduces some new

issues into the analysis of production from the mine or well that do n

production of reproducible goods such as agricultural crops.

A mine manager must determine not only how to combine v

inputs such as labor and materials with fixed capital as does the farm

quickly to run down the fixed stock

of

ore reserves through extraction

o

A unit

of

ore extracted today means that less in total is available for to

plays an essential role in the analysis. Each period is different, because t

resource remaining

is

a different size. What we are concerned with in

analysis of nonrenewable resources is how quickly the mineral is ext

the

flow

of production is over time, and when the stock will be exhau

In this chapter, we determine the efficient extraction path of he r

amount extracted in each time period. First, we examine the behavior

ual mine operator; We then examine how a social planner would exp

deposit. Finally, we develop the extraction profile

of

a mining

industr

we assume that perfect competition prevails in every market. We deriv

mineral output, prices, and rents over time under varying assumpti

nature of he mining process. The competitive equilibrium over time i

the socially optimal extraction path.


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5

NONRENEWABLE RESOURCE

USE

THE THEORY

OF

DEPLETION

Our initial moqel is very simple

and

abstracts c.onsiderably from reality so that

we can identify and examine basic concepts.

The

assumptions are gradually relaxed

so that we can deal with increasingly complex

but

more realistic models. Relaxing the

assumption of perfect competition is done in Chapter 4 and of certainty in Chapter 5.

In examining the mine's and industry's extraction decision, we also illustrate the

effects on

output

and prices over time of changes in particular variables affecting the

mining process. What will be the effect

on

extraction over time of, for example, a

change in extraction costs, the introduction

of

setup or capital costs, different quali

ties

of

ore, a change

in

the discount rate, the imposition

of

axes?

THE TH ORY

OF THE

MIN

We begin with a simple model

of

esource extraction from

an

individual mine which

operates in a perfectly competitive industry. The mine owner will seek to maximize

the present value of profits from mineral extraction in a

manner

similar to that of a

manager

of

a plant producing a reproducible good. An output level must be chosen

that

maximizes the difference between total revenues - the discounted value of

future extractions q q2 Q3 etc., inultiplied

by

price,

p and

total

cost- the

discounted value

of

dollars expended in getting each q out of the ground.

The

presence of the finite stock of

the

.mineral modifies the usual maximization condi

tion; marginal revenue

MR)

equals marginal cost

MC),

in three

fundamental

ways.

Suppose we compare farming to copper extraction.

The

owner of the copper mine

faces

an

opportunity cost not encountered by the farmer. This is the cost of using up

the fixed stock at any point in time, or being left with smaller rem aining reserves. To

maximize profits, the operator must cover this opportunity cost of depletion. For a

competitive firm manufacturing a reproducible good,

the

conditions for a profit

maximum are to choose output such thatp( =

MR)

= Me The nonrenewable re

source analogue requires p = MC he opportunity cost of depletion.

How

then

would the mine owner measure this opportunity cost?

t

is the value of the unex

tracted resource, a resource rent related

to

those discussed in Chapter 2.

The second feature

that

differentiates nonrenewable resources from reproduc

iblegoods concerns the value of he resource rent over

time.

Deciding how quickly to

extract a nonrene wable resource is a type

of

nvestment problem. Suppose one has a

fixed amount of money to invest in some asset, be it a savings account, an acre of

land, a government bond, or the stock

of

a nonrenewable resource in the ground.

Which asset is purchased (and held

on

to over time) depends on the investor's

expectation of the rate

of

return

on

that asset----the increase in its value over time.

The investor obviously wants

to

purchase

the

asset with

the

highest rate of return.

However, in a perfectly competitive environment with

no

uncertainty, all assets

must, in a market equilibrium, have the same rate of return.

To

see how this is so, consider

what

would

happen

if

the economy had

two

assets, one that increased in value 10 percent per year, the other at 20 percent per year.

Assume there is no risk associated with either

asset

No one would invest in the asset

earning only 10 percent; everyone would want the asset earning 20 percent The price

of

the high-return asset would then increase,

and

the price

of

the low-return asset

would decrease until their rates of return were equalized.

THE THEORY

OF

THE MINE

What

exactly is

the

rate of eturn

to

a nonrenewable resource?

Th

to the mine is the resource rent-the value of he ore in the ground. W

positive

discount rate, the rent is positive

and

rises

in nominal

valu

occurs. If the resource rent did

not

increase in value .over time,

purchase the mine, because the rate

of

return

on

alternative assets w

valuable.

In

addition, the owner of an existing deposit would

attempt to

. are as quickly as is technically feasible. Why should

one

hold on to ore

that is increasing in value at a rate less than

can

be earned on, say, a sav

Alternatively, if the value

of

he oreis growing at a rate in excess

of

w

earn in an alternative investment, there

is no

incentive to extract at all

ground is then more valuable to the mine owner

than

are extracted.

To

extraction then,

the

rental value of he mineral must be growing at th

that

of

alternative assets.

There is

one

final condition imposed ottthe mine owner that d

with reproducible goods. The total

amount

of the natural resource

time cannot

exceed its total stock of reserves. We call this the stock

c

Let us draw thes.e strands together for the first time. Suppose a mi

plan of quantities extracted roughly worked out by a rule of

thumb

make the extraction plan somewhat tighter. Should he extract one m

in this year's liftings or leave it for next year's liftings. Ifhe takes it ou

gets $10 profit, he can that profit (rent) in the

bank

at, say, 8 per

1 O(1.08) = $10.80 next year. If he leaves it

in the

ground and takes

it

he foresees t h ~ he will get a different price

and

can reap a profit (rent)

case he will

make

more money by deferring extracti on ofthe extra ton u

(If

he were

to

get only $10.75 upon extracting next year,

it

would

currently

and

sell

the are

this year.) By doing this calculation repeat

owner arrives at the best extraction plan year by year. Let us now

important

features of nonrenewable resource extraction in

more

deta

Extraction

from a

Mine

Facing a

onstant Price

One of the earliest economic analyses of mineral extraction appeared

article by L. C. Gray. In Gray's model, the owner

ofa

small mine has

much are

to

extract and for how long a period of ime. To solve this p

made a number

of

simplifying assumptions. First, he assumed

that

th

ofa

unit

of the mineral remained' constant (in real terms) over

the

lif

The producer knew the exact

amount of

reserves

in

the mine (the s

extraction. All the are was

of

uniform quality. Extraction costs

then

dep

the quantity removed.

We

could

view Gray's

mine

as a gigantic

blQ.Ck of

pure

copper. P

constant forever, while the marginal cost

of

cutting off a piece

of

coppe

size of

he

piece cut off. If 1 ton ofcopper is cut off, it will cost $500

to

tons are cut offat once, the extraction costs could be $10,000. The econ

is to cut. off appropriate quantities in each period in order to maximi

value

of

profits available from the stock

of

the mineral. The mode


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5

NONRENEWABLE RESOURCE USE: THE THEORY OF DEPLETION

appeal, because in many mineral markets we do observe relatively constant prices

over long periods of ime.

To determine th e efficient extraction path for the mine, we start with a simple

illustration. Suppose the mine will operate for two periods only.

The mine

owner

must

determine how

much

copper

to

chop off he block today

and

tomorrow, using

the three conditions identified earlier.

2

For the two-period case, these conditions can

be

stated

as:

1. Price

= M

rent

in

each

period(in

present value).

2. Rent today = the present value of rent tomorrow.

3. Extraction today

+

extraction tomor row

=

total stock of reserves.

The solution is shownin Figure

3.1.

Assume that the mine has aU-shaped

average cost curve AC)

and an

upward':'sloping marginal cost curve

MC)

over some

output range.

3

The constant price is shown as p.

Output

today is designated q O),

output tomorrow is q

T),

where

T

signifies the end of the mining operation - the

length of ime the mine operates (in this case, two periods). Given these curves

and

the total stock of ore; there will be a unique solution to the extraction problem that

satisfies all three conditions.

Me

p ~ ~ ~ ~ ~ ~ ~

o

q(t)

Figure 3.1·

Mineral

extraction

in

two periods

when

there

is

a constant

price.

The mine

operator extracts

the amount

q O) today and q T) tomorrow. The sum of q O) plus q T)

completely exha usts the mine s reserves. Rents are R O) today and

R T)

tomorrow,

where

it

must

be

the

case that [R O)] 1

)

= R T),

where

r is

the

interest

rate

on

alternative assets. Output levels

q 1)

and

q 2)

illustrate a plan that is not feasible

because

q 1) q 2)

exceeds the total stock of reserves. .

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THE THEORY OF THE MINE

The

mine ·owner must pick

an

initial

output

level where

p =

M

resource rent obtained at the output level q O)

isR O).

This is conditio n

condition i defines rent as the difference between price and marginal c

period, extraction must equal q(T)

and

the rent will be R(T).

It

must b

R O)

= R(T)/( r), where r is the market interest rate or discount

return

on

any alternative asset.

4

This is condition

2.

Ifrents did

not

ri

. interest extraction would not occur in both periods. If ent rose mor e s

interest'rate the entire stock of ore would be extracted

in

the initial

proceeds of he sale invested in some other assets whose value would ri

interest (e.g., a savings account). If rent rose faster than the rate of nte

stock

of

ore would be held

in

the ground until the last

moment in

extracted. In this case, the min e is worth more unextr acted because th

on

holding ore in the ground exceeds the return on alternative inves

the rental value of he min e is growing

at

exactly the same

rate

as the

assets extra ction will either be as fast as possible or deferred as lon

Finaliy

,

output

today

and

tomorrow

must

be chosen such

that q O

where S is the stock of mineral reserves. This is condition 3. For a giv

there will be only one level of initial output

and

hence final output t

these conditions.

To see that q O) is unique,. consider a case where the mine m

initial

output

level greater.

than q O),

say

q 1). The rent

will

then

be

R

than R O). Output in the second period must then be such that

R

1) =

This occurs at output q 2). The mine o wner will then have satisfied tw

conditions

1 and

2),

but

notice that condition 3 is violated.

The sum

must exceed S because they are both larger than previous outputs

extraction plan simply is

not

possible.

The

owner

cannot

extract

more

in the mine. Suppose the sum of q O) q(T) is less

than

S. Then the m

remaining in the ground after extraction ceases, and revenue will

unextracted ore. A slightly higher extraction r ate would yield additio

This example can easily be extended to many periods of operation

three conditions must be met. In addition, we can also tell when the m

operation - how long Tis. Refer again to

FigureJ.l.

It s not a coincid

is at the

point

where

M =AC.

This

point

is calleda

terminal

con

nonrenewable resource extraction problem. It has a clear economic

Consider any

output

level to the right

or

left of his point.

foutput in

t

is to the right

of q T),

the last unit of he mineral extracted will yield a m

p - C [q(T)] where C [q(T)] is marginal cost at q T). Each ton of he

in the last period will contribute an average rent of

pq(T)

- C q T»

q(T)

or p - AC. By inspection, we can see

thatthe

average rent exceeds the

It would therefore increase the present value of profits (rents)

if

the

moved more tons of ore from the last period into the first period

marginal rent isgreater

than

average rent

at

the last period

MC


8/24

5


the optimal extraction plan must have the numb er ofton s

in

the last period such that

average rent

is

equal to marginal rent; that

is,

pq(T)

-

C q T»

p _ C (q(T»

q(T)

In terms ofFigure 3.1, average and marginal costs are equal at

q(T),

and

q(T)

is

also that outp ut which combined with q O) exhausts the. mine. In the many-period

case, the time to depletion will. be such that all three extraction conditions are

satisfied, plus the terminal condition. This will determine a unique

T.

Table

3.1

provides a numerical example

of

the efficient extraction path in a many-period

situation. Price

is

constant

at

$10, and marginal cost rises at a 45-degree line from

M = 1 when quantity extracted in zero. In this example, 14.8 tons are extracted.

over seven periods, with quantity declining toward the final period

of

extraction:

q O) =

3.9197,

q 1)

=

3.4117,

q 2)

=

2.853,

....

Between consecutive periods

(p

-

me)

1 + r)

= p

- me where

r =

0.1, a

10

percent rate

of

interest. This is our

condition that rent rises at the rate

of

interest. I n the last period

p

- me =

p

- ae,

which is the condition for extracting the optimal

amount in

the last period (the

terminal condition). Total profits evaluated

in

present value in period 0 are

$94.4604. .

Profit

Maximization

for the

MineS

rofit maximization

involves making revenues large in relation to costs of produc

tion. There

is

a series

of

evenues minus costs each year or period into the future. Each

instant in time is slightly different, since depletion of he stock is occurring year by

year. Discounting with the current interest rate makes each annual profit value

comparable to others at the date at the beginning of extraction. In the absence of

discounting, recall

that

profit in year 8

in

the future would be no t comparable with

profit in year 11. Each nominal value is different at anyone point

in

time

in

the

absence

of

discounting.

The conditions discussed for the mine facing a constant price will also hold

in

this more general model. For the mine owner, total discounted profits (the present

value

of

profits) are

.

,=P

•

q O)

- C q O»

+

p •

q J)

-

C q J»] J

: r

+

[p

•

Q 2)

-

C q 2»] J

: r)

+

...

+

[p

. q(T) - C(q(T»] _l_)T 3.1)

1 +

r

Equation

3.1

is what the mine owner wants to maximize subject to the stock con

straint which requires that

q O)

+ q(l) +

...

+ q(T)

S

(3.2)

I-

:;

0-

U

-

Q

or:

I

C

oS

.2:

;<

U

CQ

I

oS

u

CQ

0-

Q,

0-

*

Z

:i

-0 '100 '11 .0 V) 01

e

oo-NOV)

..... ~ r - O ' I

i38

; ;NNNN

~ I I

' - ' I : l ,

.

'

... 0

ell

V)

0 1 - 1.0

o ...

N M r - o o o o ~

15

o o o o \O r - O M

' ( ; lo

- 0 ' 1

..... oor-r-

..

'0

0 0 0 '

o o N

..... 01

i:l S

1.0 M V) ..........

OO.nNOO..,fo.;

..... NNMM

~ p . .

Po

]0

+-' 0

cu 0

, :::I'':

S

0

V)

01

.....

\0

$

N M r - o o o o ~

S\O

o o o l . O r -O M

V)

.....

01 .....

oor-r-

\0

~ . 8

ooo .....

o o N

..... OI N

tl'5

.... 1.0 M

V)

..........

0

o o V ) N o o ~ O ' I 00

'EPo

·....;NNMM

:

S

' 0

~ < l : l

' ' :

c u ~

PoOl)

s s

V)

01 ..... 1.0

~ : 1 2

N M r - o o o o ~

o ...

' 0

o o o l . O r -O M

t l ~

.... 01 ..... oor-r-

.. ;>.

' 00

.....

o o N

..... 01

..... 1.0 M V) ..........

Ocu

00 V) N 00 01

>

;Nc...)c...)..,f..,f

'

s

§ , ~

V)

- 0 1

.... 0

oo\OM--V)

00

...

0

--MNO I\O

:: :.s

0 ' I ~ ~ ~ ; : : : l 8 1

' 0 ]

o o M \ o ~ o o o o

~ o : j

- ~ r -

..... V ) o

g

lJ

00r- :00.n .n

~ c u

< ]

*

1 . O v ) ~ M N ' O

0


9/24

6


Equation (3.2) says that the sum ofthe quantities ofore extracted must not exceed the

total stock of reserves available. The symbols used in Equati ons (3.1)

and

(3.2(are

defined as follows: .

p

is

the constant price per ton for the mineral

q(t) is the quantity extracted in time t

qq(t)]

is the total cost

of

extracting

q(t)

tons

of

the mineral

t indicates the time period. Today is time 0, the next period is time

1,

and so on.

One can think of these periods as years

r is the discount or nterest rate, which is assumed to remain constant over time

T

is the

number of

periods over which the mine will be operated

S

is the total stock

of

mineral reserves

All variables are interpreted in real (constan t dollar) terms.

Maximizing this profit stream subject to the stock constraint on total output

yields .

p - c (q(O» = k

Cr

[p-c (q(1))] k

C

r

p

- c (q(T))]

=

k

(3.3)

where c' means

dC/dq

and kis a constant dependent on the stock size S

(k s

called the

shadow price of a unit of stock). The principle in action here is that the discounted

value

of

he marginal

ton

taken

out in

any period

mustbe

the same for

an

extraction

program to be profit-maximizing. If his were not the case, the mine operator could

increase the return to the mi ne by shifting production to where the marginal ton earns

a higher discounted value. p - c (q(t» is the value of he marginal or last ton taken

out in period t.

(1/1

+ r)t[p - c (q(t»] is its discounted value.

. For adjacent periods in time we have

(

1 t

(

1

t+l

1 +

r

. [p -

c (q(t»]

= 1 +

r

[p - c (q(t + 1)]

or

[p -c (q(t + 1»] - [p - c (q(t»]

=r

[p

-

c (q(t»]

(3.4)

wh ch says

that

the percentage change

in

p -

c'

between periods

must

equal the rate

·of mterest.

p

- c' is the

rent

on the marginal ton extracted. So

we

have the basic

~ f f i c i e n c y condition: The percentage change in rent across periods equals the rate of

lnterest. .

The terminal condition requires. that the quantities chosen are those which

maximizt; discounted total profits so that the average profit in the last period equals

the margmal profit on the last ton extracted. This tells us how to terminate the

r

I

THE THEORY OF THE MINE

sequence q(1), q(2), q(3). This condition combined with the stock c

sure that the sum of the qs equals the original stock. These two con

pq(T)

- C(q(T» = p _ c [q(T)]

q(T)

q(l) + q(2) + + q(T) = S

These two conditions, in addition to th e percentage change in rent

the profit-maximizing number of periods over which to exhaust th

Now

we turn to mines with

an

ore quality which declines as e

deeper into the mineral material

.

We are going to associate costs o

processing with each unrefined

on,

not with a batch

of

homogen

did above. We no longer have a large homogeneous block of coppe

,

.I

Quality Variation Within the ine .

In the previous section, it was assumed that the cost of extraction r

units of ore were extracted at anyone time. Suppose now that the ore

pure cop per as before, but consists

of

metal

and

waste rock. The me

throughout the waste rock in

e a m s

of varying thickness. The min e o

to extract from t he thickest seams first, where the ratio of metal to w

highest. Suppose the deposit is laid down with richest seams on top. A

mined, the thic ker seams are depletedand more waste rock must be r

increasingly thinner seams. Mining costs rise per unit of metal p

because the metal content ofthe ore diminishes while the rock conten

means

that the

marginal cost

of

extracting

and

processing

each ton

o

Extraction costs per ton shift up (increase) as subsequent am

extracted. The flow condi tion for efficient extraction of he mineral (

2, or Equati on 3.4) is unchanged, but now holds fo r a single ton of

quality (seam thickness). The mine owner can no longer slide down

curve by extracting smaller amounts of ore over time to satisfy th

efficient extraction because the marginal cost

of

extraction increas

each incremental ton of ore processed. To see what happens to the e

this case, we turn to Figure 3.2.

The mine represented in Figure 3.2 illustrates a case where o

tinuously decreasing, and we are examining extraction over two

There are thus two curves of extraction cost per ton, one for period

period

(t +

1), where the curve for (t

+ 1)

lies everywhere above

indicating that it will· cost more to extract and process additional

time.

The

mine owner must determine how

much

ore to extract

(t + 1) by following the flow condition.

The

quantity extracted in

such that the rent on the last ton n the period will be exactly equal to

could obtain

if

extracted hi the next period, discounted by

1

+ r

Figure 3.2, the rent on the marginal ton in he first panel ofthe figure

if

output

is

chosen at q(t) and the price is

p(t). If

he mine o wner is

between extracting the marginal ton in period

t

or in period (t + 1), i

~ ' - - -


10/24

58


USE:

THE THEORY OF DEPLETION

p t+

I) I- c____ -_____

a

p t)1---------- ------

d

o q t)

o

q t + I

Period

t

Period t

+

I

Figure 3.2 . Quantity

q t)

in period

t

is set so that distance

ab

equals distance cd divided by 1

+

. The

marginal ton in period

t

gets rent ab and would get rent cd if it were extracted

in

period

t +

1

that the rent in

t + 1

is equal to

cd,

where

cd

= ab 1

+ r).

The marginal ton in

period

t

is, we emphasize, the least marginal ton in period

t+

1 because now

every

ton

is of a different quality or has a different extraction and processing cost.

This is what is illustrated by the higher marginal cost curve in the second panel of

Figure 3.2.

The important implication of this analysis is that the market price must rise

over time if extraction

of

lower-quality, higher-cost ore

is

to occur. If the mineral

price does

not

rise to

p t

+

1

in

the

period

t

+

1 ,

no extraction will occur

in

that

period. Extr action will end with

p

) equal to the extraction cost on the last ton taken

out in period

t.

This possibility gives rise to another important distinction about the

.

end of

the mining operation. In the case

of

uniform ore quality,

we

argued that

mining would cease when all the ore was removed. We

can

call thisphysicaldepletion

or exhaustion.

Suppose, however, that the ore is_not of uniform quality and the costs of

extracting additional units rise, as Figure 3.2 shows. If he market price does not rise

sufficiently to ensure that extraction proceeds from one period,to another, the mine

will shut down. Ifin period t + 1 the price is constant atp t), ore will be extracted to

the point that

p t)

equals extraction cost. The mine is then said to have

economic

depletion in period t.

t

simply does not pay

he

mine owner to extract

any

ore beyond

, the quality indicated at

q t),

given the extraction cost curves

and

the price

p t).

A

higher rate of return can be' earned by taking the rent,

ab, and

investing it in an

alternative asset which earns the market interest rate of r percent per year.

In

a many-period model, the length

ofthe

extraction period will be determined

by the time path ofprices.

For

two prices

inany

consecutive periods, there will be only

one· value

of

cost per ton and hence output that satisfies the flow condition. The

optimal life

of

the mine, the length of time to depletion (whether economic

or

physical), will then be determined by linking together quantities over subsequent

periods until the flow condition no longer is satisfied, or the mine runs out

of

ore.

.

I

..;.,

I

EXTRACTION BY A MINERAL INDUSTRY

Mines frequently h ~ v valuable by-products. The mining of n

yields profitable amounts of gold and platinum. It is straightforwa

model of the mi ne to incorporate this s ituation. Valuable rock, So

that for each scoop of size Q, there is K Q of nickel and G Q ofgold

suppose that costs rise with the amount of rock, Q, processed. ForPK

. or nickel, and PG gold, we get a revised pricing rule for optima

(

1

t (

,dK dG·

d e

1 + r PK dQ t) +PG dQ t) - dQ t) = constant fo

Now a weighted sum

of

prices net ofcosts increases at the rate

of

nte

owner to be maximizing the discounted present value

of

profit:

marginal cost

of

hoisting quantity Q t).

We haveseen that when quality variation is introduced in eit

above thata simple rule such as rent rising at the rate of nterest m

in an essential way. In other words, simple formulas are inadequate f

real world extraction programs. Fimi.lly, note that because Gray's mo

firm, there is no discussion of he resource market and how (or if) m

rise to satisfy the flow condit ion and allow produc tion to occur ove

ine this important issue, we now turn to a model of a mineral indu


The Hotelling Model

In 1931, Harold Hotelling wrote a classic paper which examined th

tion of a nonrenewable resource from the viewpoint

of

a social plan

had

as its goal

the

maximization

of

social welfare from the produc

The model was at the industry level rather than that ofthe single min

Hotelling arrived at the same condition for the efficient extractio

namely, that the present value of a unit of a homogeneous but f

mineral must be identical. regardless of when it is extracted. This

conditions 1 and 2, which we call thejlow condition. Together wi

straint and terminal condition, the optimal extraction plan for th

resource can be determined at the indust ry level as well as for the

Hotelling viewed the problem of

how to extract a fixed st

resource from the vantage point of a government social planning

showed

that

a competitive industry facing the same extraction co

curve as the government, and having perfect information a bout res

arrive

at

exactly the same extraction

path

for the minera1.

6

The

ef

path determi ned by each firm acting independently in the competi

yield the socially optimal extraction path. We first examine the pl

and then show why it is achieved in a competitive industry. As bef

simplifying assumptions are made and then gradually modified t

with practical relevance. .

When we deal with an industry rather than a single mine, the m

no longer be· treated as a constant. Rather, it is assumed that the

negati vely-sloped demand curve. The greater the industry output, th


11/24

60

NONRENEWABLE RESOURCE USE: THE THEORY

OF

DEPLETION

will have to b e if we are to have an equilibrium

in

any mineral market at any given

point

in

time. Hotelling assumed that prices would adjust so that a mineral market

would be in equilibrium at every point in time; supply must always equal dem and.

7

We can think of the Hotelling model as examining world production

of

oil, nickel,

copper, or some other mineral. As before, the stock of mineral reserves is of known

size, and all units of the mineral are homogeneous. We assume a unit of the stock

costs c dollars to extract and refine and that this cost is constant for all units of the

~ t o c k in the reserve endowment, S. Once again, it is as

if

we had a huge block

of

copper that cost c per

ton

to chip off. We want to find the rate of extraction

that

maximizes social welfare

and

completely exhausts the stock.

(If

c

=

0, the analysis is

qualitatively unchanged.

If

c

ncreases with the quantity extracted, we are back with

Gray's cost assumption, which is an unnecessary minor complication at this point.

See question 4 at the end of the chapter.)

The

problem is to maximize society's wealth

W) or net

return from mineral

extracti on, which is defined' as ' '

W = B q O»

+

B q 1» C

r

+

B q 2)) ( 1 ,)

+ ... + B q T)) CJ (3.7)

Again, there is the,resource stock constraint which requires that

(3.8)

q O) +

q(l)

+ q 2) +

. . .

+

( T ) ~ S

B

q t»

is the consumer plus producer surplus obtained n period

t

from the extraction

,

of

output,

q(t).

This social surplus is simply the area und er

the demand

curve up to

quantityq(t) and

above the constan t costs of extracting

q t).

This problem canbe solved in a manneranalogous to that presented previously.

As before, the solution requires

that

he flow condition, terminal condition, and stock

constraint be met. The crucial distinction between Hotelling's model and Gray'sis

that

we must examine the

demand

curve explicitly

and

derive a unique price for the

resource

in each

period the mineral ind ustry operates.

In

addition, we

can

make the

stronger statementabout the socially optimal as opposed to just the mine's efficient

extraction path. 'We now derive the solution to the maxim ization of Equati on (3.7)

subject to the stock constraint, Equati on (3.8), for a linear demand curve.

In maximizing social welfare, the pl anner must decide what the net benefits are

of

extracting some

of

the mineral today as opposed

to

tomorrow. Therefore, the

planne r will want to measure the change

in

the social surplus as one more unit of he

mineral is produce d today. Consider Figure 3.3.

The

social surplus for the last unit

extracted is simply the difference between the market price and the marginal cost of

extraction, c. If q t) is extracted

in

period

t,

society gains

the amount

for

the

q(t)th

unit extracted and the amount under the demand curve

and

above c for all previous

or

inframarginal units extracted. How muchwill the planner choose to extract

in

each

period?

As before, the flow condition provides an answer. To maximize social welfare, it

must

be the case

that

the ne t benefit to'society fr'om the last

unit

extracted

in

each


p t) I ~ - - - - - - - -p t

+

1) r ~ ~

c r ~ ~ b L _

q t)

Period

t

(a)

D

.i

q t

Period

(b)

Figure

3.3 Rent per ton

in

period

t is

distarlce

8?(1

+

) = de.

Price

is

higher in period

t

+

1

m an[d

(ent per ton

in

period

t

+

1

Yields the optimal path of q s to extract.

Ing p t)

- c] 1

+

)

= [p t +

1) -

period

.is

exactly equal in present value ter .

otherwIse would entail foregoing

th

. ms In each penod

of

e

net benefit of he marginalunit extr: ~ ~ ~ I m . u m benefits possible. A

the case that the present value ofth c e IS SImply the resource ren

F th e rent on the

marginin

e h .

or , e two-pe nod case,

q(t)

and

q(t

+ 1)

,

ac

peno

,

must

be chosen

such tha

p t) - c

=

[pet

+

1) '- c] _1_)

. : 1

+

r

F I g u ~ e .

3.3 i,llustrates one pair

of

out

uts ,

. ,

.

condItIOn gIven

D,

the

demand

c p d

or

the two

?enods WhICh

as.

can be s.een in Figure 3.3, that ~ ~ ~ :

c t n ~

r. N o t I c ~ the fl?w.co

WIth

a statIOnary demandcurve

th ,

I era pncemustnse over tIm

if

he

quantity extracted

d e l i n ~ s

~ : . ~ ~ ; ; ; : y price, and hen,: th

must be less than that in period

t

t .

h e r e f o r ~

extractIOn

Equation (3.9). ' 0 ensure that thepnce rises just e

We can rewrite Equat ion (3.9) to obtain

[pet

+ 1)

- c]

+ [pet) -

c]

. . [p t) -

c]

= r

Wntten In this, form we see that as p . .

eq It h , ce nses rent per ton g

.

ua

0

t e rate o

nterest. This is often re fe r i r ~ w ~ ov

SImply

Hotelling s

rule. We sketch the • i c ~ ed. to Hotellmg s

r

path shown is the one that

rna

p. path In FIgure 3.4. How

ren.ts

must

grow

at

the rate

of

~ ~ : ~ ~ ~ s

M ~ I h l

welfare?

All

HoteUing's

whIch satisfy Hotelling's rule? Yes

b'

t Ig

t.

there be dozens of diffe

the help

of

he stock

o n s t r a i ~ t

an'd ~ r ~ m q u e

p ~ t ~ of

output ca.n

stant, the planner,will want to ensure that ~ ~ o n ~ I t I O n .. f x t r ~ p t I O

e mIneral IS removed


12/24

62

NONRENEWABLE RESOURCE USE: THE THEORY O DEPLETION

p L ~ ~ ~

$

per ton

2

3

4

T

o

Time

.

..

b t n periods

t

a rate equal to r the

Figure 3.4 P r i c ~ ~ i ~ U I S d c o s t c ~ s p r : ~ ~ S i ~ 9 ~ ~ ~ o n e n t i a I l Y t rate r n the optimal

rate of nterest. This Yle s a ren

program of extraction. . . .

. fi in rents. The constant cost s s u m p ~ l n

in the gro ;lnd, ~ h e mme w n ~

w ~ l

t ; ~ ~ e ;osts the same to extract (in nommal

is crucialm thIS a : g u m ~ n t ac um net to leave ore behind. We know tha t the

terms); therefore,

It

cannot pay the

p l ~ n

. d must exactly equal the total stock

. sum of he amounts extracted m each tIme peno "

of the mineral reserves S) . . 3 h t 'th the linear demand curve there

We can a l ~ o

ee, f r ~ m

FIgure ? i ~ i ~

:

buy more of he mineral, The price?

some price, call It

p,

at

h l ~ h

no one

t

that

e m n d

for the good is choked off at

thIS

is often called the

choke

przce,

mean

have the stock

of

the mineral

go

to zero at

point. Ideally, the planner would see to Therefore the plann er would seek to have

exactly the point that demand goes ~ r o d t h e r w i s ~ deprives society of maximum

the last unit of output extracted at p. Of 0 0

benefits. - .ven the fixed stock

S,

to find just that

We can then

w o ~ k

b a ~ k w a r d fr.om Pci line so that rent increases at rate rand

initial output q O), whIch WIll, over 1 ~ ~ i one such extraction and hence rent path

outputs sum to the stock of reserves. y . 1 rplus available to society and hence

exists.

It

will yield the r g e ~ t ~ m o u n t ~ ~ s ~ ~ ~ J:termine the length dftime the i ~ e

be the optimal plan. In

d d l t 1 o n ~

we c h' h the price path intersects price p

wIE

operates. In Figure 3.4, the pomt at w lC . file' T Once the price reaches p,

determine the unique duration f t h e h e x t ~ a c t l O ~ ~ ~ ~ x t ; a c ~ i o n will cease. Extraction

there will be no more demand for t e mmera , .

ends at time T.

.

1 dition discussed earlier is also met

at T.

T h ~ r ~ n t

Note also that the termma con

n the average ton at

T.

This condItIon

on the m a r g i ~ a l

to.n

must be equal to the

r e

0

T

must equal zero. Table 3.2 gives

.a

also implies m thIS case that .output at tl. . th for the industry. Rent per ton IS

numerical example of an

o p t 1 ~ a l e x t r a c t 1 o ~ p: +

r

= pet + 1)

_

c.

This is the con

pet) - cand for consecutive e n o ~ s

p t)

- T ~ ~ deJand curve intersects the vertical

dition that rent rises at the rate of mterest. .


;<

Table 3.2

AN INDUSTRY EXTRACTION EXAMPLE*

pet) - c

pet)

q t)

(p

-

c)xq

6

9

10

0

0

5

8.181818

9.181818

.818182

6.6942162

4

7.4380165

8.4380165

1.5619835

11.618059

·3

6.761833

7.761833

2.2388167

15.134111

2

6.1471211

.7.1471211

2.8528789

17.536992

1

5.5882919

6.5882919

3.4117081

19.065621

0

5.0802654

6.0802654

3.9197346

19.913292

14.802654

I

A

stock of

14.8

tons

is

extracted· over seven periods resulting in the maximization of t

consumer surplus. Extraction cost per ton is constant at $1 and the industry demand curve

is

linea

back from period 6 to period

O

(Parameters: r 1, c

=

1, q

= 10

- p, S 14.802654.)

.i

axis at

p

=

10, when

q

= O

This a l u ~

p

= 10

is the

choke

price.

Ove

14.803 tons are extracted.

Totalprojii

evaluated at period 0 in prese

$75.20. (We have no t illustrated that in period 6;

p 6)

- c

= [B q 6»

or marginal welfare from extracting

q 6)

equals average. welfare.

tricky, since

q 6) =

0 in order to satisfy this basic end point conditi

Exhaustibility and Welfare: Demand Curves and Backstop Technology

What would happen if he world were to run out ofoil or any nonrene

one day? What does the Hotelling model tell us about this occurrence

complete exhaustion on society depends on the technology of produ

resources

in

production and can be reflected in the demand curve for

crucial question is whether substitutes for the resource exist or whethe

so necessary to the production processofother goods that once it is de

goods will also cease to be produced. Our model with the linear dem

choke price says that a substitute exists. The choke price is that pric

users of he good will switch entirely to the use of he substitute good.

may be another nonrenewable resource such as oil shale as a substitu

tional crude oil, or it may be a reproducible good such as solar e

substitute exists, society and economic systems will not collapse wh

out; they will shift to the substitute commodity.

What ifthere isno substitute for the depletable resource? In this s

available quantities of he resource dwindle, prices would begin to ris

We can characterize this with a nonlinear demand curve, sayan isoela

does not have a positive intercept

9

(see Figure 3.5). We will not have

running out of the resource in this case, because we never will in fin

society's viewpoint, however, this

is

not a very desirable· situation b

suggests is that

as

the resource quantity extracted gets smaller and sm

will rise to higher levels. We can thin k of extracting oil by the bucket

and finally by teaspoons and eyedroppers while the price climbs contin

infinity. This

is

asymptotic depletion.

Asymptotic

means that two l

each other more and o ~ e closely

as

time passesbut never touch.) This


13/24

6


Price

p t)

Quantity q t)

. . elastic demand curve. As the quantity

FlgtUrcet ed5 i ~ ~ i l : ~ e s , the price rises toward infinity. Ex

ex ra . . f

t

t'me

haustion does not occur In

1m

e I '

. l' t The co·nventional view is that

.

th

r un rea is

IC.

a mathematical m o d e ~

but

is

ra

edb. the production

of

a substitute r o d u c ~ .

exhaustion of most mmerals is f o l l o ~ e

Y d

t . nto the extraction model. It is

We can incorporate t h ~ substitute p r ~ a

~ c ~ s t o p

technology a technique for

common to think of he substitute product a fl power that becomes feasible to

producing energy at a

~ o n s t a n t

cost ~ ~ c ~ a ~ ~ i ~ : ~ ~ ~ h e s a certain level. Feasible here .

implement once the pnce o r o n v ~ n 10 . cover their costs.

10

Suppose the back

means that the producers the

a c k s : ~ f t ~ ~ n

commodity at a price of 2, per t o ~

stop technology· can p r o v ~ d e the s u ~ d ced with cons tant costs. ThiS case is

forever,because the substitute can e pro u .

illustrated

in

Figure 3.6.

Price

Z

________

Backstop supply

D

Quantity

q

I provides a resource substi-

Figure 3.6 The

a c k s t o ~ I ~ ~ ~ : t ~ ~ ~ ~

point at which price per ton

tute at $Z per ton. Rebntt,w

t

It take over as exhaustion occurs at

equals$Z and the

s,u

s I u e

that joinin g of price and

Z.

.


All demands below Z are satisfied by

flows

from the nonren

stock and those

at Z are supplied by the backstop technology. One

demand curve in Figure 3.6

as

one for energy, derived from conven

The backstop is energy from fusion or solar power (two possible su

For a planner controlling both sources of supply - the exhaustib

backstop technology - the optimal program will be to consider Z a

for conventional oil and to arrange to exhaust this oil in the Hotellin

out in the previous section. At the moment of exhaustion, price will

per ton and the backstop technology brought on line. 11

Model of a ompetitive Nonrenewable Resource

Industry

We have argued that the socially optimal extraction path would

planner organized production in the industry. Would a decentraliz

industry replicate the socially optimal program

of

extraction? Supp

large number of mines or oil wells, each owned by a different perso

owners coordinated their actions, we would have what is called

competitive setting. Each owner would be faced with this decision: Sh

sell a ton

of

ore this period and earn p t)

-

cdollars

of

profit, or shoul

next period and extract, sell, and receive p t

+

1) - cdollars of prof

the prices

p t)

and p t +

1)

were such that Equation (3.9) was

sa

p t)

- c

= p t + 1)

- c)/ 1

+

r), each owner would be indifferent

this period

or

next. Since all owners

and

deposits are identical (ther

differences among mines), all are indifferent.

If there was a general tendency to wait until next period by

out put would fall and the curre nt price would rise. Sellers would th

able to sell now and

put

their rents (

or

profits) into

an

asset earning

r

was a tendency to sell a lot ofore in the period, however, the current

and

mine operator s would be reluctant to sell until future periods. Th

conditi on will be met by each firm seeking to maximize profits to ens

each period, and the market forces of supply and demand will

conditiop is met.

Will a competitive industry which is on an equilibrium path

optimal terminal condition? To show that it does, we assume that th

price paths were not optimal then argue that this cannot happen in th

model. If he price path were not the optimal path, there would be aj

up or down at the transition to the backstop technology. Consider Fi

tion commences at t = 0 with an initial price of p(O). The rent in

R(O)

F

p(O) - C, grows at rate r utitil the resource is exhausted at tim

that at T, the market price ofthe resource appears to have risen abov

occur, because with the backstop technology, no consumer will pay

the resource. It therefore means that the resource will be economic

time

T

which

is

less than

T.

But this cannot be a plan that maximizes profits, because ore w

ground

that

could have been extracted.

If

all firms know that th

emerging, theywill alter their extraction plans to shift production to


14/24

NONRENEWABLE RESOURCE USE: THE THEORY

OF

DEPLETION

P

I

I

I

I

I; per

ton

I

I

I

pCO)

I

I

p (O)

I

I

R (O)

I

c

teO)

T T

Time

. . um down

in

the price at the time of

Figure

3.7 A pnce path ~ t h

~ i ~ n

:OUld be shifted away from later pe-

e x h a u s t i o ~ c a n n o ~ occur: dX p revent economic depletion at T .

riods and Into earlier peno

. .. '(0) which yields the

. 111 wer the 1mtlal pnc e

o

say

p

, _ .

increase in curre nt productIOnWI ; h exhaustion occur at precisely p. A SImilar

rent R (O). t will grow at rate ran . ave h - that is firms run out

of

ore before

ment applies if the price path fmls to reac p,

argu b lemented

the backstop technology can e

1 ~ p th

t all' mine operators have perfect foresight

What these

a r g u m ~ n t s r e ~ U l r e

1S .

To

be able to see

that it

is profitable. o

about the price

of

the

m e r a ~

m all

pen

.

d' viduals must know what the pnce

shift production from one penod to the other, ~ a ~ decades it is difficult

to

imagine

will be Given that mines can produce for seve ' d o r de'centralized opera tion of

. . h

t

es the centra

1ze

that perfect foresIght c arac e ~ z d that marke t forces will ensure that

mineral markets. Some c o n o m ~ s t s haveha:gued 1 n decentralized situations, even 1f

'l'b . w1ll be ac

1eve

. ' .

perfect foresight eqUlI

num

d fi .

nt

information. Ifperson i s foreSIght 1S

some participants in the market have e C1e, , ,

l

I

EXTENSIONS

OF THE

INDUSTRY MODEL

deficient relative to person} s,} could make a profit by signing a cont

, from

i

in the future at specific conditions. The informational defi

competed away by the forces

of

profit maximization and free entry i

We now consider some modifications

of

the basic model, but' still

participants act with perfect foresight. (Imperfect information i

Chapter

5.)

EXTENSIONS OF THE INDUSTRY MODEL

Changes in Extraction Paths Under Altered Conditions

We will show how the price and extraction paths of a competit

affected by: (1) a rise in the cons tant costs of extraction; (2) an increa

(discount) rate; and (3) the introduction of axes. In each case, we c

rium paths under two different assumptions. Biagrammatic techniq

to

derive the results.

n Increase in Extraction Costs

In

Figure 3.8, we examine the case where the costs

of

extraction, whi

are higher than initially assumed. How will these higher costs affec

and price paths? In , this and all subsequent cases,

we

look

at,

the

extraction had yet occurred. We could modify the results

if he cost (o

ter) change occurred at some time after the mine had begun to o

situations, it will matter whether the mine owner antic ipated the cha

f

a d

p ~ ~ ~

per ton

p (O)

p(O)

c f----_+_------------_+_---+-----

c f 4 ~ _ _ _ T

teO)

I·

I

I

T

Time

Figure 3.8 The comparative statics

of an

increase

n

extraction costs

effect on the price path. If costs of extraction increase from c toe , the m

industry will respond by reducing output

in

the initial period so that th

source price rises fromp(O)

top (O).

Production s then increased again

in

periods. A higher extraction cost will lengthen the time

to

depletion.


15/24

8


USE:

THE THEORY

OF

DEPLETION

discussion question 7). Suppose the initial situation is

an

industry facing constant

costs of c and a choke price ofp see Figure 3.8). The price path t hat satisfies the flow

and

terminal conditions along with the stock constraint is

aa .

Now, what would

happen if costs were

c

rather than

c,

where

c

>

c?

If he industry tried to follow the

path aa it would not be maximizing profits. Along the path aa

'

ren t will grow faster

than rate r when costs equal c . Each mine owner then decreases curren t output

(because rents in the ground exceed returns from extraction), the industry supply

declines, and the initial price,

Po,

rises to Po in Figure 3.8.

What happens to extraction over time? Ifmine owners produced less outpu t in

every period when costs are c rather than c, the price pat h would look like df. In this

situation, the choke price

pis

reached when there is still ore remaining

in

the ground.

This cannot be an optimal plan because the terminal condition is not met. What it

means is that at some point in the extraction path, m ines must begin to increase their

rate of production, which will cause the mark et price to rise less rapidly than when

costs were

c.

This will ensure that physical depletion occurs atp. Thus, the path dd

results.

t

s important to notice tha t the increase in costs results

in

a

lengthening

ofthe

time to depletion. The backstop is reached at T , which exceeds

T.

There is an economically intuitive explanation for the effect on the extraction

and hence the price path when costs c h a n g ~ . If the cost of extraction is higher, in

present value terms it will benefit each firm to postpone extraction. Iffirms postpone

incurring the costs, the rents will be larger than if the same path as aa is followed.

Production is reduced in the early periods and increased in later periods. The present

value of the mine has fallen due to the increase in costs, but the stock of ore is still

physically depleted and the terminal condition is met. .

Rise in Interest Rates

Suppose that the rate of return on investing

in

assets alternative to mineral extraction

rises. Wh at will be the response

of

he mining firms?

In

Figure 3.9, suppose the price

path prior to the increase in the nterest rate

is

aa . Ifthe mine operators contInued to

follow this path, the mines would be earning a lower rate

of

return over time

than

available elsewhere. The way to avoid this loss is to shift production to the present.

The mi ne owners will extract more ore in the initial period, thus driving down the

market price to, say, p (O).

Thereafter, less ore will be extracted so that the rate

of

return

on

the remaining

ore, rises at the now higher interest rate. This means, however, that the time to

depletion must fall. The extraction path dd will start from a lower initial price than

did aa and will rise more steeply so that it hits th e choke price at T which is less than

T. Any other path would not maximize the profits

of

he mines, given the new interest

rate.

The ntroduction of Taxes

Suppose the government decided to impose various taxeson the minin g industry.

What would

be their effect on the extraction

and

price paths,

and

the time to deple

tion? We conside r two types

of

axes: a tax on the mineral rent, the difference between


;,

l i r - - - - - - - - 4 ~ - f - - -

$

per ton

p(O)

p (O)

c r - ~ - - - - - - - - - -

t(O)

r T

Time

~ i g ~ r e 3 . ~

The

c o m p a r a t i v ~

statics

o f

. .

,rise n the Interest rate will increase the 0

an

I n c r ~ a s e In the Interest ra

the future (the present value of fut PP?rtumty cost of extracting

~ h e present, and the initial price a ~ e

~ n t s

S lower). Extraction s shift

1

n

the ground increase

in

value at a hi e smaller amounts of ore rema

IS s t e e p ~ r than aa . Depletion occu gher ra e th.an ~ e f o r e , so the pat

exhaustion point ~ n d e r a lower i n t e r ~ ~ t a : a ~ e which S sooner than T

, price and' marginal cost and a ro alt

production.

12

A rent t a ~ will ha y y tt tax at a constant rate,

already in existence. There is no no e. ect on the extraction dec

offset ~ e d ~ c l i n e in the present a l u : ~ ~ ~ ~ ~ th.e rate of x t r a c t i o n ov

ment

wIll

SImply collect som e

of

he . e resultmg from the t

same manner as before the tax ' mmeral rent, and production wi

T

. 0 see that this is so, we present the flow . . .

Imposed. Let a be the tax rate levied 0 . IcondltlOn reqUIred a

becomes n mmera rents or profits. The n

-t

(p(t)

- c)

1

a) =

1

- a ) ( p ( ~ - c) (_1_)

. ' 1 + r

e c a u ~ e rent m each period is taxedexactl th

both SIdes of Equation (3 11) Th Y e same, the term (1 -

h

. . ere IS no way the mi

s i mg p oduction. ne operator can

W ~ l l e a rent tax is said to be

neutral

"

because It does not alter the path th

or

nondlstortmg to the

?n discovery of new mines. The

~ a : ~

c:nnot be said b o u ~ t h e

m . d l v l ~ u a l s

and firms to explore for

;ew

t

e tax rate,

t ~ e

less mcen

WIth dIfferent taxes levied on their out mmeral

depOSIts.

Two id

m ~ r k e t

~ a l u e . The rent tax reduces

t h e ~ e ~ ~ : : P £ e s e n t

two a s ~ e t s

of

mmeral m the ground _ because th ' : ' rom exploratlOn - t

dec} , . h " e expected payoff fro d .

, mes

WIt

the tax. We will come back' .

..

m Iscovenn

and uncertainty in Chapter 5. ' to

thIS tOPIC

m our discussion


16/24

7


N ow let us examine the effect

of

he imposition of a royalty on the total value of

mineral extraction. Th e government now taxes the total revenues

of

each mining

firm at, say, rate

y.

The introduction of he royalty has an effect analogous to arise in

the cost of extraction.

If

he firm postpones the extraction

of

some ore to the future,it

can then reduce the effect

of

he royalty on the present valueof ts rents. To see this,

we rewrite Equation (3.11) to obtain the flow condition after the royalty.

1

1

- y) p t»

- c = [ 1

- y) p t»

- c] 1 +

r)

(3.12)

Notice that there is no way we can cancel the term 1 -

y

from bot h sides of

Equation (3.12). The royalty reduces the price received by the firm for each unit

of

mineral sold and thus, the present value of he mine. If sales are pos tponed, the effect

of his reduction will be minimized because of he discount factor. The result is a time

path

of

extraction similar to tha t shown

in

Figure 3.8; the initial output will fall,

and

price will rise. Later

in

time, ext raction will rise again,

and

price will rise less quickly

than in the case without a mineral royalty. Again, the time to depletion

of

the fixed

stock

is

lengthened. The size

of

this effect depends on the magnitude

of

costs.

f

extraction cost is very small, the distorting effects

of

he royalty are relatively small.

Numer ous other comparative static exercises can be done (see discussion prob

lem 8). These exercises will help

in

understanding the model

of

he industry and also

enable the reader to try applications of he model to some real-world events, such as

the introduction

of

new energy taxes, a fall

in

the cost of producing oil from oil shale

deposits, and so on. We now turn to some further extensions of he industry model

that make it more compatible with real-world

o ~ s e r v a t i o n s

eclining Quality of the Stock

Suppose the mineral industry finds its stock of ore declining in quality. Deposits of

poorer quality must be brought on stream as the high-quality reserves are exhausted.

As

in the case

of

the single mine, quality decline can be viewed as requiring the

removal of more waste rock per unit of ore extracted to get at the thinner seams of

metal. Thus the average cost of producing metal increases as more

of

he mineral is

extracted. If each ton has a specific extraction and processing cost associated with it,

and these costs rise

as

more mineral is extracted, the

flow

condition

of

Equation (3.9)

remains the same,

but

its interpretation is modified.

For a specific

ton

of he mineral, the flow condition requires the rent on tha t ton

in

period t to be equal to the discounted rent on that same ton if t were extracted

in

period (t

+ 1).

In period

t

this

ton

will be the marginal ton extracted, whereas

in

.period (t

+ 1) it

will be the most inframarginal ton extracted. We illustrate the effects

on

the industry

in

Figure 3.10.

In Figure 3.10, the rent on the marginal

ton

in period t is

abo

This same ton

would obtain the rent

ofge

in period t

+

1. Distancege must be equal to the amount

abe1

+

r),

if

the owner

of

the marginal

ton

is

to be

ndifferent between extracting in

period

t

or period t

+

1). All owners of ore inframarginal to the marginal ton earning

ab at

t will extract their ore

at

that po int simply because their rent

in

period t exceeds


$

per

ton

o

q (t) q(t)

Period t

D

$

per

ton

p(t

+ 1)

e

o

q(t

+

I)

Period t + 1

Figure3 10 The marginal ton extracted

in

period t h

earn rent p(t

+

1) _ c

in

period t

+

1 or dis as rent p e ~

~ o n

of ab or p t) - c

to extract

is [p t)

_ c 1 + )

= [p(t

+

1) ~ a ~ ] c ~ g ~ ~ ~ ~ e r ~ ~ e c o n d l t l o ~

causing

q t)

to

quality. , . e ~ I s l n g extraction cost per ton

t ~ e rent they could obtain by waiting until period t

+

1). The inf

q ~ ) h f ? \ e x a ~ p l e earns a rent of a b

in

period t. It would earn gei

w /cfi ;ss h ~ n a b 1

+

r). The flow condition for extraction in bo

sa IS e or t IS ton, and hence the ton is extracted in eri d A

applies all tons to the left

of q(t)in

Figure 3.10.

pot

SI

. . D ~ s t a n c ~ ab is.

r e ~ t

per ton. Area cbl s the Ricardian or

ansmg rom t e vanatIOn m ore quality. The marginal ton of ore in

e a r ~ s

a rent equal to

pet

+

1)

- c .

Ifthe

flow condition

is

then linked

~ e n o d s h the

m o u n t

ofo:e be extracted

at

each different quality wil

or

t

eac

t

d ~ e h n o d The ?nc.e m each period is determined by the agg

ex rac e

, t

US, the pnce

IS

endogenous.

t die ave yet to x a ? I ~ ~ e the terminal condition for the problem

.0d ep etlon, and e ~ c ~ mlt al q t). Because there are now many grad

m hustrr we

A

must d ~ s t l l ~ g U 1 S h between the possibility of economic

aus I O ~ ssu?Ie m eIther case that there is a linear demand curve a

Ph

If ~ e r e P E y s l ~ a l exhaustion, the output will go to zero in the las

t epncehltsp WIth

(T)

- 0 .

. q - marginal rent equals average rent on t

or:. : m ~ ~ e d t i We can then work backward to find the un ique first per

sa

IS

es ow and

e r m l ~ a l c o n d ~ t i o n

with complete physical exh

the fi ~ y : I C a l e ~ h a u s ~ o n wIll occur m any industry ifthe marginal co

. IS

ess

t .an (or equal to) the choke price.

If,

however

quahtles

WIth

extractIOn costs m excess

of

the choke . I

h·· ·

. pnce, econom

o c c u ~ s n t I s l t u a t I O n ~ the mming sequence ends when the choke p

marginal cost of extractIOn. There will be an ore grade at whl'ch m

fin 1 t' . mm

a quan Ity extracted from all mmes is that which yieldsp- - c

=

0 A

backward fi th . .

d

..

. rom

IS

quantIty to find the initial output level that sat

con

ItIOn

m each penod.


17/24

72


Deposits

of

Distinct and Differing Quality .

N owwe examine the case where each deposit has ore of a unif0rD?-

q ~ a l i t y

but i f f e ~ s

in quality from other deposits. What is an optimal plan for e x p l O 1 t a t ~ o n of ore m thIS

case? We can tr eat ore qual ity as signifying different costs

of

extractIOn process

ing. These costs

of

extraction could be due to ore grade

and s e ~ m

t h I c k ~ e s s

as

discussed earlier, or simply to the fact that deposits are o ~ a t e d ~ t ~ I f f e ~ ; n t d ~ s t ~ n c e s

from a central market. Trans porta tion costs give the deposIts a dIstmct quah ty ; the

metal is still of uniform quality. .

Consider an example

of

two deposits of different quality within a competlt.Ive

industry. Each deposit is within itself of uniform quality.

e ~ o s i t

1 has. extractIOn

costs

of

c and

an

initial stockof eserves equal to 8

1

tons. DepOSIt 2 has umt costs of C

2

and

rese;ves equal to 8

2

,

As

long as the demand curve remains s ~ a t i o . n a r y only the

low-cost deposit will be exploited initially. Why? Suppose deposl.t

lIS

the low-c?st

deposit. Its extraction costs are shown in Figure 3: 11 as c

i

. ExtractIOn

fr0D?-

deposIt 1

commences at T where the initial rent earned IS the amount abo DepOSIt 2 clearly

cannot come on t r e a m at

Tl because

at

price

PI

it will incur a substantial loss. Its

extraction costs of C2 greatly exceed the initial price.

How is this initial price set? Why is n't the initial price high enough to allow

b?th

deposits to o p ~ r a t e ? One way to see why the initial price is ~ e s s than the extractI?n

costs of he high-cost deposit is to work backward from the tIme when b oth ~ p o ~ ~ s

are exhausted - that is,

we

use the terminal condition. As long as the choke pnceP

S

greater than C2, we know that physical exhaustion must

c c u r .

At T, q(T) = 0. The

rent at

T

would be equal to

p

-

C

I

for deposit 1 and

P - C

2

for deposIt 2 If both

extracted their last tpn of ore

at

this point. Deposit 1 s rent greatly exceeds that of

a

p

I

I

g per ton

I

I

I

I

I

I

I

P2

~ - - - - - - - - j l d

I

2

I

I

I

I

I

I

----1

b

Pi

b

I

T

Time

Figure

3 11 The

low-cost deposit is extracted

b e t w e ~ ~ T

and T ·

Upon

exhaustion

the

second

deposit is

worked

until

it

is exhausted

at time

T

P S the

cost of

the

backstop.

Rents

rise over each phase at the rate of interest.

EXTENSIONS

OF

THE INDUSTRY MODEL

deposit 2. Can we then arrange extraction in each period prior to T

condition

is

met for each deposit? The answer

is

no

if

each

simultaneouslyand yes if they operate sequentially.

Consider simultaneous extraction. T he price

in

each period m

for each deposit if both are to sell any metal. There is then no wa

condition can be met for both firms. Working backward from

T,.

the

. cannot simultaneously be the same for each deposit with different

and the same market price. In particular, if a path such as a C is fo

deposit 1 will not be falling fast enough from T. Only deposit 2 can

satisfy the flow condition.

If here is sequential extraction, then for each deposit both th

and

terminal condition will be met, along with physical exhaustion

ward from T where the choke price is reached, deposit 2 will be ex

interval

T2

to T. Tis defined by the choke r i c e ~

T2

is defined by the in

2 can earn to satisfy the flow condition for each period and exhaust

rent for deposit 2 the amount

cd,

will be that which comp ounded a

terminal rent a d as the ore body is depleted.

T2

is also the time

physically exhausts its reserves.

It will not pay the owners

of

deposit 1 to extract once the price

because they know that th en deposit 2 can begin extraction. Ifboth

extract simultaneously, the market price would not rise fast enou

satisfy the flow condition,

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HartwickOlewiler1986 TOC Ch3