Non-renewable resource exploitation - The Economics Network

Non-renewable resource exploitation:externalities, exploration, scarcity and rents

NRE - Lecture 3

Aaron Hatcher

Department of EconomicsUniversity of Portsmouth

Externalities

Externalities

I External (social) costs

I Example: pollutionI E¤ects not covered by markets and so di¢ cult to valueI Trade o¤ between costs and bene�tsI MB = MCI Optimal pollution does not imply a fair distribution of costsand bene�ts!

Externalities

I External (social) costsI Example: pollution

I E¤ects not covered by markets and so di¢ cult to valueI Trade o¤ between costs and bene�tsI MB = MCI Optimal pollution does not imply a fair distribution of costsand bene�ts!

Externalities

I External (social) costsI Example: pollutionI E¤ects not covered by markets and so di¢ cult to value

I Trade o¤ between costs and bene�tsI MB = MCI Optimal pollution does not imply a fair distribution of costsand bene�ts!

Externalities

I External (social) costsI Example: pollutionI E¤ects not covered by markets and so di¢ cult to valueI Trade o¤ between costs and bene�ts

I MB = MCI Optimal pollution does not imply a fair distribution of costsand bene�ts!

Externalities

I External (social) costsI Example: pollutionI E¤ects not covered by markets and so di¢ cult to valueI Trade o¤ between costs and bene�tsI MB = MC

I Optimal pollution does not imply a fair distribution of costsand bene�ts!

Externalities

I External (social) costsI Example: pollutionI E¤ects not covered by markets and so di¢ cult to valueI Trade o¤ between costs and bene�tsI MB = MCI Optimal pollution does not imply a fair distribution of costsand bene�ts!

Externalities contd.

I A social planner would want a �rm to maximiseZ T

0[pq � c (q, x)� d (z , a)� θ (a)] e�rtdt

I Here z � αq is a pollutant generated by extraction; d (�) is adamage function and θ (a) are abatement costs

I Normalise α to 1 so that z = qI Optimality now requires

p � cq = λ+ dz , θa = �da

I If d (�) � d (z � a) then dz = �da and we can write

p � cq � θa = λ

for a �socially responsible��rm














I Normalise α to 1 so that z = q

I Optimality now requires
























I Firms must have an incentive to incur abatement costs

I How?

I pollution taxesI standardsI tradeable permits

I Backed up by enforcement (inspections and penalties)I Firms forced to internalise the (social) costs of pollutionI Note: there may be stock as well as �ow e¤ects


I Firms must have an incentive to incur abatement costsI How?





I pollution taxes

I standardsI tradeable permits




I pollution taxesI standards

I tradeable permits









I Backed up by enforcement (inspections and penalties)

I Firms forced to internalise the (social) costs of pollutionI Note: there may be stock as well as �ow e¤ects




I Backed up by enforcement (inspections and penalties)I Firms forced to internalise the (social) costs of pollution

I Note: there may be stock as well as �ow e¤ects





Exploration

Exploration

I Few general conclusions

I Costly and subject to uncertaintyI New discoveries may not be developed immediately,depending on extraction costs

I Models of exploration are complex!

Exploration

I Few general conclusionsI Costly and subject to uncertainty

I New discoveries may not be developed immediately,depending on extraction costs


Exploration

I Few general conclusionsI Costly and subject to uncertaintyI New discoveries may not be developed immediately,depending on extraction costs


Exploration

I Few general conclusionsI Costly and subject to uncertaintyI New discoveries may not be developed immediately,depending on extraction costs


Scarcity

Scarcity

I An economic de�nition of scarcity

I Must depend on demand (price) and extraction costsI A resource becomes more scarce if demand (price) increasesand less scarce if demand decreases

I Economic scarcity is decreasing if extraction costs increaseand prices don�t

I The shadow price λ (the di¤erence between price andmarginal cost) is regarded as the correct measure of economicscarcity

I A trend in the shadow price is the best indicator of resourcescarcity

I Sincep = λ+ cq ,

clearly p and λ can have opposite signs

Scarcity

I An economic de�nition of scarcityI Must depend on demand (price) and extraction costs

I A resource becomes more scarce if demand (price) increasesand less scarce if demand decreases




I Sincep = λ+ cq ,


Scarcity

I An economic de�nition of scarcityI Must depend on demand (price) and extraction costsI A resource becomes more scarce if demand (price) increasesand less scarce if demand decreases




I Sincep = λ+ cq ,


Scarcity





I Sincep = λ+ cq ,


Scarcity





I Sincep = λ+ cq ,


Scarcity





I Sincep = λ+ cq ,


Scarcity





I Sincep = λ+ cq ,


Rent capture

Rent capture

I Economic pro�ts � resource rent

I Government may try to capture (some) rent for society�sbene�t

I Common charges are taxes on revenues/quantities (severancetax) or pro�ts (royalty)

I With a revenue tax τ the �rm maximisesZ T

0[[p � τ] q � c (q, x)] e�rtdt

I E¤ect similar to an increase in the costs of extraction:extraction ceases earlier and more resource is left in theground

I A revenue tax is distortionary

Rent capture

I Economic pro�ts � resource rentI Government may try to capture (some) rent for society�sbene�t



0[[p � τ] q � c (q, x)] e�rtdt



Rent capture




0[[p � τ] q � c (q, x)] e�rtdt



Rent capture




0[[p � τ] q � c (q, x)] e�rtdt



Rent capture




0[[p � τ] q � c (q, x)] e�rtdt



Rent capture




0[[p � τ] q � c (q, x)] e�rtdt



Rent capture contd.

I With a pro�ts tax τ the �rm�s objective function isZ T

0[1� τ] [pq � c (q, x)] e�rtdt

I The tax does not change pro�t-maximising behaviour andhence is not distortionary

I But, a pro�ts tax reduces the present value of the resourceI Government may (also) sell extraction or prospecting rightsI Or, in some cases, the resource itself

Rent capture contd.


0[1� τ] [pq � c (q, x)] e�rtdt



Rent capture contd.


0[1� τ] [pq � c (q, x)] e�rtdt


I But, a pro�ts tax reduces the present value of the resource

I Government may (also) sell extraction or prospecting rightsI Or, in some cases, the resource itself

Rent capture contd.


0[1� τ] [pq � c (q, x)] e�rtdt


I But, a pro�ts tax reduces the present value of the resourceI Government may (also) sell extraction or prospecting rights

I Or, in some cases, the resource itself

Rent capture contd.


0[1� τ] [pq � c (q, x)] e�rtdt



Dynamic optimisation: the Hamiltonian


I De�ne an instantaneous pro�t function

π (q (t) , x (t)) � pq (t)� c (q (t) , x (t))

I The (social planner�s) problem is to

maxq

Z T

0π (�) e�rtdt

s.t. x = f (x (t) , q (t)) , x (0) = x0

I De�ne a Lagrangian function

L �Z T

0

�π (�) e�rt + µ (t) [f (�)� x ]

dt

whereµ (t) � λ (t) e�rt



π (q (t) , x (t)) � pq (t)� c (q (t) , x (t))


maxq

Z T

0π (�) e�rtdt

s.t. x = f (x (t) , q (t)) , x (0) = x0


L �Z T

0

�π (�) e�rt + µ (t) [f (�)� x ]

dt




π (q (t) , x (t)) � pq (t)� c (q (t) , x (t))


maxq

Z T

0π (�) e�rtdt

s.t. x = f (x (t) , q (t)) , x (0) = x0


L �Z T

0

�π (�) e�rt + µ (t) [f (�)� x ]

dt


Dynamic optimisation contd.

I De�ne a (present value) Hamiltonian

H (�) � π (�) e�rt + µ (t) f (�)

so that

L =Z T

0H (q, x , µ, t) dt �

Z T

0µ (t) xdt

I Integrate the �nal term by parts to obtainZ T

0µ (t) xdt = µ (T ) x (T )� µ (0) x0 �

Z T

0µx (t) dt


I De�ne a (present value) Hamiltonian

H (�) � π (�) e�rt + µ (t) f (�)

so that

L =Z T

0H (q, x , µ, t) dt �

Z T

0µ (t) xdt

I Integrate the �nal term by parts to obtainZ T

0µ (t) xdt = µ (T ) x (T )� µ (0) x0 �

Z T

0µx (t) dt


I Rewrite the Lagrangian as

L �Z T

0fH (�) + µxg dt + µ (0) x0 � µ (T ) x (T )

I Di¤erentiate w.r.t. q and x to get the FOCs

Lq = Hq = πqe�rt + µfq = 0

Lx = Hx + µ = πx e�rt + µfx + µ = 0

I Hence we require

πqe�rt = �µfq ) πqe�rt = µ

given f (�) � g (x)� q and hence fq = �1I This is equivalent to the familiar current period condition

πq = λ



L �Z T

0fH (�) + µxg dt + µ (0) x0 � µ (T ) x (T )




I Hence we require



πq = λ



L �Z T

0fH (�) + µxg dt + µ (0) x0 � µ (T ) x (T )




I Hence we require


given f (�) � g (x)� q and hence fq = �1

I This is equivalent to the familiar current period condition

πq = λ



L �Z T

0fH (�) + µxg dt + µ (0) x0 � µ (T ) x (T )




I Hence we require



πq = λ


I We also requireµ = �πx e�rt � µfx

I From µ (t) � λ (t) e�rt we can �nd

µ =�λ� rλ

�e�rt

I Hence, in current value terms, we have

λ = �πx � λ [fx � r ]

I For a non-renewable resource, fx = g 0 (x) = 0 and so

λ = λr � πx

I Or, without resource-dependent costs,

λ = λr




µ =�λ� rλ

�e�rt


λ = �πx � λ [fx � r ]


λ = λr � πx


λ = λr




µ =�λ� rλ

�e�rt


λ = �πx � λ [fx � r ]


λ = λr � πx


λ = λr




µ =�λ� rλ

�e�rt


λ = �πx � λ [fx � r ]


λ = λr � πx


λ = λr




µ =�λ� rλ

�e�rt


λ = �πx � λ [fx � r ]


λ = λr � πx


λ = λr


I What does the Hamiltonian represent?

I The present value Hamiltonian is

H (�) � π (�) e�rt + µ (t) f (�)

I Or, in current value terms

Hc (�) � π (�) + λ (t) f (�)

I The Hamiltonian measures the total rate of increase in thevalue of the resource

I π (�) is the net �ow of returns from the resourceI λ (t) f (�) is the increase in the value of the stock


I What does the Hamiltonian represent?I The present value Hamiltonian is

H (�) � π (�) e�rt + µ (t) f (�)


Hc (�) � π (�) + λ (t) f (�)





H (�) � π (�) e�rt + µ (t) f (�)


Hc (�) � π (�) + λ (t) f (�)





H (�) � π (�) e�rt + µ (t) f (�)


Hc (�) � π (�) + λ (t) f (�)





H (�) � π (�) e�rt + µ (t) f (�)


Hc (�) � π (�) + λ (t) f (�)


I π (�) is the net �ow of returns from the resource

I λ (t) f (�) is the increase in the value of the stock



H (�) � π (�) e�rt + µ (t) f (�)


Hc (�) � π (�) + λ (t) f (�)



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