an industrial economy's response to declining energy resources

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An Industrial Economys Response To Declining EnergyResources

David Comerford

5th April, 2012 - preliminary, comments appreciated

Abstract

This paper explores how a decline in the availability of easy to exploit energy resources

feeds into economy-wide prices that affect the profitability of energy production. A modeleconomy is constructed in which energy is produced with capital services, which are them-

selves produced ultimately with energy inputs. In such an economy, removing the easy

resources causes energy prices to rise, but it also causes the price of capital services to rise.

After an energy quality shock, more marginal energy resources will only be exploited as a

substitute for the lost easy resources if the energy price to capital services price ratio rises.

It is found that there are conditions related to the returns to scale in the capital services

sector which cause the energy price to capital services price ratio to fall as easy energy re-

sources become unavailable. As the price ratio falls, the profitability of exploiting marginal

resources falls, these resources are abandoned, and the economy can collapse.

JEL Classification: Q30, Q43;

Keywords: Non-Renewable Resources, Energy, General Equilibrium, Returns to Scale;

Corresponding author: David Comerford, School of Economics, University of Edinburgh, 31 Buccleuch Place,Edinburgh, EH8 9LN, Scotland, UK; Email: [email protected]

I would like to thank the Scottish Institute for Research in Economics (SIRE) for financial support via aconference participation grant to present this work.

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1 Introduction

The purpose of this paper is to explore the economys response to being faced with only lowerquality energy resources, with a view to characterising the situations under which this is prob-lematic. Problematic in this context is interpreted as an inability to use all available resoures,or to a price signal that makes the use of the resources less efficient, and does not incentivise theuse of alternatives, as the resources becomes scarcer. Capital assets are used to supply energyto the economy, but the manufacture of capital assets can be an energy intensive process. Ifhigher quality energy resources are no longer available, and the use of lower quality resources isto be expanded by applying more capital inputs to their exploitation, then the relative energy(output) to capital (input) price movement will have to be consistent with this expansion. Athree good economy is therefore of the minimum complexity required to investigate this issue,and given the feedbacks between available energy resources and economy wide prices, a generalequilibrium approach is appropriate1.

The theoretical natural resource economics literature usually has fairly sanguine conclusionswith regards to the exhaustion of non-renewable resources. Since Hotellings (1931) [10] article,

the first principle is that resource prices rise at a compound rate related to the rate that canbe earned by extraction and investing the financial proceeds. This rising price ensures thatresources that are initially unprofitable to exploit eventually become profitable, and that thereare incentives both to economise on the use of the resources and to develop alternatives. Thisfoundation to the literature has subsequently been built upon, e.g. Holland (2008) [9] describesmodels of resource extraction that generate a peak in the extraction rate during the extractionperiod using a partial equilibrium approach (since interest rates and backstop prices do notdepend upon energy used in the aggregate economy); Dasgupta & Heal (1974) [8] extend TheHotellings framework to a general equilibrium setting without changing the conclusion thatnon-renewable resource prices rise without bound as they become more scarce; and Aghionand Howitt (1998) [2] describe a two sector general equilibrium model in which growth can besustained despite declining availability of non-renewable natural resources, that are essential

for production, through investment in intellectual capital. In all these cases, and in generalthroughout the literature, natural resource scarcity is accompanied by a rise in their price2.Holland claims that price movements will be smoothly increasing because oil is virtually costlessto store in its natural reservoir ... even completely myopic firms without secure property rightswould wait to produce from these [higher cost] deposits until the price were high enough to coverthe extraction costs. This statement reveals, I believe, a possible shortcoming in this approach:yes, the resources can be left in the ground at zero cost, but there is no guarantee that the capitalassets that are used to extract these resources will be reasonably priced in future. It may be thecase that as resources become scarce, capital prices rise faster than energy prices, and so lowerquality energy resources can never be profitably exploited.

The analysis in this paper reveals a crucial role for economies of scale in determining howthe economy responds to declining availability of energy resources: Constant returns, or a lowlevel of increasing returns, in the capital services sector are consistent with energy scarcity

causing energy prices to rise faster than the prices of capital services, and so for the economy toprofitably expand into lower quality energy resources; However, if the degree of returns to scalein the capital services sector is strong enough then the reduction in this sectors productivity,

1As Ayres (2001) [1] shows however, prices in a three sector economy do not necessary reflect marginal con-tribution and can respond in complicated ways to changing factor abundances - so it need not be the commoditythat is becoming scarce whose price responds most strongly to its own rising scarcity.

2A possible exception to this is Gaitan, Tol & Yetkiner (2004) [19] which finds that the resource to capitalprice ratio tends to a constant, but this result may be driven by their use of a constant savings ratio rather thanbeing the result of some optimal plan.

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caused by the restriction in its factor inputs, boosts the price of capital services by more thanthe price of energy. The supply of energy therefore contracts rather than expands at the margin,and eventually the economy can collapse.

The contribution that this paper makes is to draw attention to the possibility that pricemovements in response to scarcity may not be favourable to bringing on substitutes or for usingcapital intensive but energy efficient alternatives, a possibility which is not considered in theexisting literature. The paper introduces the three sectors of the economy in sections 2 - 4and makes some basic propositions about the economy in section 5. Section 6 presents someillustrative results from this basic model and section 7 does likewise with a generalised model.Section 8 discusses the interaction between conclusions from these models and the incentivesto innovate, and section 9 concludes. Full proofs of section 5s propositions are given in theappendix.

2 The Energy Sector

The Hotelling model takes the incentives of an energy market participant as given, and generatesa rising price as the resources are depleted. This rising price is at odds with the observed pricehistory of non-renewable resources, and some economists e.g. Barnett & Morse (1963) [12], andSimon (1996) [15], have concluded from this price history that it is evidence of declining ratherthan increasing scarcity of energy resources. The explanation for this is usually technologicaladvances. However, Hamilton (2011) [7] details the history of global crude oil production overthe last century and a half and finds that the production increases have been achieved mainlythrough the exploitation of new geographic areas, rather than predominantly through techno-logical advances as applied to existing sources. As the scope for adding to production from newgeographical areas declines, the suggestion is that the era of rising production could soon end.There are two effects going on: depletion and technological progress; and there is some disputeabout which of these effects is winning.

A set of data that is broadly consistent with Hamiltons interpretation is the energy return

on energy invested (EROI) for fossil fuels over the past century (see figure 1). EROI can beconsidered as a technologically adjusted index of the cost of obtaining energy resources. So forexample oil and gas from 1930 had an EROI of (greater than) 100 : 1 and so obtaining 100 boe(barrels of oil equivalent) required spending energy (including the energy embodied in the capitalused to extract the energy) that contained 1boe so that gross energy production would havehad to be 101boe to supply the final economy with this 100boe. By 2005 oil and gas EROIwas 15 : 1 and supplying the final economy with 100boe would have required gross energyproduction of 107boe. This increase in the cost of supplying the same amount of energy comesdespite improvements in technology over the period. Extracting deep water oil in 1930 wouldnot have cost an extra 6% over the oil that was being extracted at that time, rather it wouldnot have been possible at all with the technology available. It is in this sense that EROI can besaid to be a technologically adjusted index of the cost of obtaining these resources, and this data

suggests that, even allowing for technological advances, the resources that we are extracting arebecoming more costly.

However, given that there is some expectation (by at least some people) that technology willwin out over depletion, and some dispute about whether the evidence of the 20th century isconsistent with this belief, in the model presented in this paper I look at economies in whichtechnological progress and depletion are exactly offset. The intellectual experiment that is thenexplored is how these economies compare with each other, given that they only differ in theavailability of high quality (low cost) energy resources.

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Figure 1: Figure from Murphy and Hall (2010) [5]

The availability of low quality (high cost) energy resources is taken to be infinite. This isbroadly consistent with the evidence in that the remaining fossil fuel resource is massive, butis increasingly in lower quality deposits. Figure 2 also shows that we are a long way from anylimits in the availability of fossil fuel resources, notwithstanding any efforts on our part to leavesome resources unused because of climate change concerns.

Energy resources with higher costs of production and/or low EROI tend to be more capital

intensive. As an illustrative example we can consider the wooden derricks used for Pennsylvanianoil production in the 19th century against the deep water drilling rigs used today in places likethe Gulf of Mexico; or we can compare the pick and shovels used for easily accessible coal seams,to the machinery required for mountain top removal in the Appalachian Mountains. The modelpresented here effectively takes this as its definition of energy quality.

Energy resources are owned by the agents in the economy who auction the right to exploitthese resources for one period, at the start of each period, to a continuum (0 ,) of potentialenergy firms. The agents therefore extract all the surplus and the license fee for resource j,Fj, will be equal to the one period profits that can be made from exploiting this resource. Theresulting energy market is competitive (i.e. price taking) with a continuum of differentiated firms,j, each producing homogenous output, Ej using capital services Qj in a decreasing returns toscale production function that also exhibits costs indexed by j i.e. high j firms are exploitingpoorer quality energy resources than low j firms and so, for a given quantity of inputs, Qj ,

they produce a lower quantity, Ej of outputs. The production function is:

Ej = Qj j , (0, 1) (1)

Once they have paid the license fee, Fj , firms maximise profits, j = pEEj P Qj , where pE isthe unit energy price and P is the unit capital services price, by optimising over the quantity ofinputs used, taking the prices as given. This gives:

Qj =pE

P

11

(2)

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Figure 2: Adapted from Brandt & Farrell (2007) [3] by Murphy (2011) [13]. Shows resourceswith their production cost. Proven reserves are dark bands on left, uncertain resources are lighterbands on right.

This is independent of j i.e. all energy firms use the same quantity of inputs. Therefore profitsand energy output are both decreasing in j. The value of a firm is zero however since Fj = j . Wedefine j on [e, r] where e 0 is the parameter representing the highest quality energy resourcesavailable, whilst r is an endogenous variable that is defined by r = 0 i.e. there is free entry inthe energy sector and firms continue to enter, and pay a positive license fee, until the marginal

firm makes zero profits and the agents can no longer extract any surplus. This gives:

r = Qj (1 ) =pE

P

1

(1 ) (3)

The total inputs and outputs from the energy sector are calculated by summing over the firmsfrom e to r i.e.

QE =

re

Qjdj =pE

P

11

(r e) (4)

E =

re

Ejdj =pE

P

1

(r e)1

2(r2 e2) (5)

=re

jdj = pEE P QE =re

Fjdj = F (6)

The energy sector uses capital inputs, and its output responds endogenously to the relation-ship between output energy prices and input capital prices. High quality resources are thosewhich require low capital inputs per unit of energy produced whereas low quality resources re-quire higher capital inputs per unit of energy produced. There is no limit imposed upon energyavailability, however these unlimited resources will be of increasingly poor quality. If it is optimalto exploit a particular resource, then it is optimal to exploit every resource of higher quality,

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and so the available high quality resources are always exploited. Exploitation of lower qualityresources is an increasing function of the energy price to capital price ratio.

The agents assume that energy quality, e, is a constant. This is equivalent to the agents

believing that the forces of depletion and technological progress are equal and opposite. Thecomparative static analysed is the steady state of an economy with a particular level of highestquality available energy resource, e. This can be compared to the equivalent steady state of aneconomy with a lower highest quality available energy resource. We can imagine that this isthe same economy after an energy quality shock and consequent resetting of beliefs (i.e. thattechnological progress and depletion are equal and offsetting, at the new lower highest qualityavailable energy resource level). If this energy quality shock leads to a rise in the ratio of energyprices to capital prices then the usage of lower quality resources will expand to (partially) offsetthe loss of the high quality resources.

3 Returns To Scale In The Capital Services Sector

There is some evidence to suggest that manufacturing industries behave as if they are subjectto increasing returns to scale. Hall (1989) [6] explains the correlation of factor productivitywith exogenous demand shocks using increasing returns and finds that increasing returns areparticularly evident in the aggregate economy and in manufacturing sectors. Caballero & Lyons(1989) [11] split the returns to scale evident in the aggregate economy into internal, firm level,constant or decreasing returns to scale, and positive external returns to scale. Their best estimateof the degree of scale economies in the US is that a sector which increases its inputs by 10% willsee an increase in output of 8%, but if the whole economy increases its inputs by 10% then outputwill rise by 13%3. Basu & Fernald (1997) [4] explain similar data as [11] as a reallocation effecttowards more efficient firms rather than any real increasing returns, but agree that if we modelthe aggregate economy as a representative firm then increasing returns to scale are appropriate.

Models with increasing returns are widespread e.g. economic geography models with agglom-eration effects; business cycle models may use increasing returns as a partial explanations for size

of fluctuations; some endogenous growth models rely on increasing returns to scale; and increas-ing returns to scale are ubiquitous in new-trade models of intra-industry trade4. The interactionbetween possible increasing returns to scale and resource constraints has not been considered.However, it is easy to imagine that they may be important: for example perhaps the ability andprofitability of deep oil drilling is only possible because there is a full manufacturing supply chainthat is predicated on the existence of automobile and aerospace industries. Perhaps if there wasno cheap oil available, the contraction of the automobile and aerospace sectors could affect themanufacturing supply chain in such a way as to drastically increase costs/decrease productivityin the sector that manufactures equipment for deep oil drilling. This in turn may mean that,without cheap oil sustaining the automobile and aerospace sectors, the deep water drilling sectoris unprofitable, and so it would not exist in cheap oils absence.

In this section we present a general returns to scale capital services sector which converts the

economys capital stock into capital services using the production function:

Q = K, 1 (7)

3i.e. in the notation of equation 7, this translates as 1.3 (ignoring the fact that capital services are only asubset of the whole economy, which is also subject to diseconomies of scale caused by declining energy resourcequality).

4In the empirical work of Mohler & Seitz [14], the elasticity of substitution in the Dixit-Stiglitz productionof European economies is found to lie in the range 3 - 5, which corresponds to a returns to scale parameter, (1.25, 1.5)

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where K is the capital stock, Q are the capital services supplied, and determines the degree ofreturns to scale. is a parameter that will be used to normalise the output from this sector suchthat at a particular capital stock and energy resource availability, (K, e), the capital services

supplied to the economy will be independent of degree of returns to scale, . Zero profits aremade in this sector, and so if a rental rate for capital, , is paid to the agents in this economy,while the price that this industry supplies capital services for is P, then the following relationshipholds:

P Q = K (8)

i.e. = P K1 (9)

Section 7 is used to present a specific microfounded form for the returns to scale in thecapital services sector using Dixit-Stiglitz aggregation of a monopolistically competitive sector[17]. Dixit-Stiglitz production is a conflation of internal economies due to fixed costs, and externaleconomies due to the benefits of product variety. The more simple model presented here, andused in sections 4 - 6, abstracts from the source of scale economies.

4 The final goods market and the whole economy

The final goods market in this economy is a perfectly competitive, constant returns to scale,Cobb-Douglas economy which uses labour, L, (with effectiveness/technology parameter A), en-ergy, E, and capital services QF to produce output:

Y = EQF(AL)1 (10)

Being perfectly competitive and exhibiting constant returns to scale means that, in equilibrium,factors must be paid their marginal product i.e.

w = (1 )A E

ALQ

FAL

= (1 )

Y

L (11)

pE = E

AL

1QFAL

=

Y

E(12)

P = E

AL

QFAL

1=

Y

QF(13)

such that Y = wL +pEE+ P QF (14)

Capital accumulation dynamics are introduced in a very simple way: a constant savings rate,s, and depreciation rate, , as in the Solow model [16]. The assumption of a constant savingsrate will be generalised in Section 7.

Kt+1 = Kt(1 ) + sYt (15)

To complete the model economy we now only have to consider the accounting identities linkingthe sectors. Capital services are fully utilised and the agents receive labour income, capital rentsand energy sector license fees, which buys the output of the economy i.e.

Q = QE + QF (16)

and Y = wL +pEE+ P QF = wL + K + F (17)

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5 Some propositions on this model economy

Full proofs of all the propositions in this section are given in Appendix 1.

Proposition 1. For any given capital stock, K 0, the market equilibrium exists and is unique.There are therefore no problems of interpretation with a multiplicity of equilibria. As we

shall see though, there are multiple steady states. The equilibrium price ratio as a function ofcapital stock is given by:

K = Q =1

2

(1 )(1 + + 2

)z

z +2

1 +

1 +

1 + 21+

e

1

z +

e

1

11

so K =c0

1

z1 (z + g(e))

1 (z + h(e))

1( 11) (18)

where c0, g , h 0, g, h > 0, g h, e 0

The variable z is a positive transformation of the energy to capital services price ratio:

z =pE

P

1

e

1

z (0,) therefore represents all possible price ratios that are associated with positive outputfrom the energy sector. Clearly K(z) is a bijection on z > 0.

Proposition 2. K = 0 is a steady state.

Phase space is characterised by the following expression:

K(zt) = sY(zt) K(zt)

= s(AL)11

2

+

(1 2)+z+t

zt +

2

1 +

e

1

+zt +

e

1

( 11)

c0

1 z 1 (z + g(e)) 1 (z + h(e)) 1 ( 11)

= s(AL)1c1z+t (zt + f(e))

+(zt + h(e))( 11)

c0

1

z1 (z + g(e))

1 (z + h(e))

1 (

11)

= s(AL)1c1z+t (zt + f(e))

+(zt + h(e))( 11)

1

sc1(AL)+1

c0

1

z1

t

zt + g(e)zt + f(e)

+(zt + g(e))

1(zt + h(e))

( 1)(11)

i.e. K(zt) = H(zt)[1G(zt)] (19)

where c1, f 0, f > 0, f g h, e 0

and H(0) = 0 , H > 0 H > 0 zt > 0

Proposition 3.

=1

21+

+ such that >

, limz0 K(z) is negative andlimzK(z)is positive. For > and e = 0, G(zt) is monotonically decreasing and so there is only one(unstable) steady state K > 0.

The phase diagram for this extreme case of Super Strong Increasing Returns To Scale is shownin figure 3. Since no stable productive economy (i.e. Y > 0) exists even when all resources areavailable under these parameters, and we are interested in how economies that can exist copewith resource restrictions, we do not consider this case further.

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Figure 3: At e = 0, economies with > , i.e. Super Strong IRS have two steady states: astable state at K = 0 and a higher unstable state.

Proposition 4. For 1 < , limz K(z) negative. For 1 < and e = 0,

limz0 K(z) is positive (and so K = 0 is unstable), G(zt) is monotonically increasing and sothere is a unique stable steady state K > 0.

The phase diagram for e = 0 and Returns To Scale that are not Super Strong, is shown infigure 4. We can find an analytic expression for the relative price at the stable steady state,

Figure 4: At e = 0, economies with 1 < have two steady states: an unstable state atK = 0 and a stable state at K > 0.

z > 0 with e = 0. The steady state capital stock is then K(z).

z =sc1

(AL)1

c0

1 1( 1)( 1

+1)2

(20)

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Proposition 5. = 1+

< such that (, ) and with e > 0, limz0 K(z) is

negative (and so K = 0 is stable). For 1 < and e > 0, limz0 K(z) is positive (and soK = 0 is unstable).

With 1 < the phase diagram in figure 4 holds for e > 0 (though we cannot strictlyprove monotonicity for G(z), the signs at very low and very high z have been established). How-ever, for < < , Strong Increasing Returns To Scale, the phase diagram changes to thatshown in figure 5.

Figure 5: At e > 0, economies with < < , i.e. Strong IRS, may have three steadystates: stable states at K = 0 and K > 0 with an unstable state between.

Proposition 6. For 1 < , a stable steady state K > 0, always exists.

Proof. Previous analysis has shown that, for 1 < , and general e 0, limz0 K(z)is positive and limzK(z) negative. Further K(z) is a continuous function of z with nosingularities over z (0,). Therefore there must be some z (0,) for which K(z) = 0and K(z ) > 0, K(z + ) < 0 for very small . Further, z > 0 K(z) > 0.Therefore there is a stable steady state for the economy with 1 < and general e 0.

Significantly, we cannot prove the existence of a non-zero steady state for Strong IncreasingReturns to Scale. All we know here is that for e > 0, limzt0 K(zt, e > 0) is negative, and thatlimztK(zt, e > 0) is negative. Therefore it is possible that the phase diagram for Strong

IRS looks like that shown in figure 6, i.e. with a globally stable K = 0 steady state. Indeed, wecan experimentally construct a particular economy with Strong Increasing Returns to Scale, thenwith e = 0 we can evaluate the K(z) function and see that the result is akin to figure 4. Thenas we raise e, we observe that the experimentally constructed phase diagram looks like figure 5.And as we raise e further, phase space looks like figure 6, i.e. there is a point in the e parameterspace at which the economy collapses as the K > 0 steady state ceases to exist. We can describethe transition from figure 5 to figure 6 as a collapse because there is a discontinuity in the steadystate that the economy can reach. Once we raise e past a critical value, the steady state changesdiscontinuously from K > 0 to K = 0. This is unlike Constant, or Weak Increasing, Returns

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to Scale in which the economy exists at some positive level of production, irrespective of theseverity of the resource restrictions, e, that are imposed.

Figure 6: At e > 0, economies with < < , i.e. Strong IRS, may only have one stablesteady state at K = 0.

6 Basic Model Results

In this section we present illustrative results from the basic model developed in the preceding

sections. We want the economies considered to differ only in their returns to scale, , but fortheir results to be comparable. Therefore we normalise the economies so that they coincide fore = 0. Let = 1 = 1 and define = (K

)1 where K is given by equation (20) for = 1, = 1, and e = 0.

Figure 7 shows how the index of the marginal, zero profit energy firm, r, varies with the indexof the highest quality available energy firm, e. These comparative statics show how the steadystate of the economy varies with and e - they do not show timepaths. The parameters usedto generate these results, and the results from the following section, are listed in Appendix 2.If we imagine that the economies along a specific line are connected by a series of depletionshocks (as discussed in Section 2), then a rising r(e) curve indicates that the use of lower qualityresources substitutes for the high quality resources that are no longer available to the economy.A falling r(e) curve indicates that marginal resources are abandoned as high quality resources

cease to be available.These results are somewhat unsatisfactory in that there is such a clear distinction betweeneconomies that will collapse under resource restrictions and economies that will not: from theoutset at low e, the strong IRS economy starts abandoning marginal resources as high qualityresources become unavailable. This is not suggestive of the real world, but the results generateddo however illustrate the salient features of the model: it is possible that the use of low qualityresources does not expand as high quality resources become unavailable, and this occurs becauseof price incentives.

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Figure 7: r vs e for three economies: CRS with = 1, Weak IRS with 1 < < , and StrongIRS with > .

We now examine the assumption that the agents believe depletion and technological ad-vances in reducing costs in the energy sector exactly offset. If the true state of the world isthat depletion will outpace technological progress, then is this belief sub-optimal? In the aboveresults, the profits at the energy firm exploiting the resources at j = 0 (and hence the licensefees that the agents can extract) are rising with e only for the CRS or Weak IRS economies. ForStrong IRS economies, these profits are falling with e, so why would any (risk neutral) agentseek to save these resources against future scarcity? In other words, this is a poor model in the

sense that agents myopia is sub-optimal precisely in the cases that it doesnt tell us anythinginteresting. In the interesting cases, where economies can cease to function because of lack ofresource availability, the agents myopia is optimal - they cannot gain by hoarding because theprice of extracting the resources is going to rise by more than the price that they can sell themfor. This is a general result in that the defining feature of the economies that will collapse isthat, at least for e greater than some e, r(e) < 0. Since r is defined as a zero profit condition,then for this to hold there must be some range in which firms/licensing agents with resourcesj > j have no incentive to hoard because their profits are falling in e. Figure 8 illustrates thisgeneral case schematically by showing an economy where 0 is rising with e but which r

(e) < 0.

7 A generalised model and the impact of policy

In this section we generalise the model to produce more interesting results and then show that thecollapse of a Strong IRS is due to prices rather than to any fundamental limits. In generalisingthe model we lose the ability to interrogate it analytically and since were just producing anumerical result we may as well build in microfoundations for the capital services sector. We canexperimentally verify that the propositions of Section 5 appear to hold, but we have not provedthat this is the case in general. The generalisations that we are adding are to endogenise savingsand to have two input factors for the capital services sector: capital stock and energy.

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Figure 8: Schematic showing that if r(e) < 0 then there must be some resources for which thereis no incentive to hoard.

Savings are endogenised by giving the risk neutral agents some rate of time preference andassuming full depreciation. At steady state, the rental rate of capital, , will be a constant equal

to the inverse of the agents discount factor. Rather than the price of capital varying and aconstant proportion of output being saved as in the model of the previous sections, now we havethe price of capital constant and the steady state savings rate, s = K

Y, varying (increasing) with

the parameter e5.The capital services sector is generalised by splitting it into two. A perfectly competitive

aggregation sector buys the output of an imperfectly competitive, disaggregated, capital servicessector. The aggregation sector has production function and profits as follows:

Q =n

0

q1

i di 1

, > 1 (21)

A = P Q

n0

piqidi = 0 (22)

5

The assumption of risk-neutrality is uncontroversial here since the agents expect no change in the value ofthe parameter e. It may be a more controversial assumption in a model in which agents rationally anticipatedepletion outpacing technological progress. We could postulate an equation of motion et+1 = f(et, Et), fEt > 0,and have the agents chose the amount of energy used today to balance present consumption against the valueof future consumption. However in order to generate catastrophic/collapse outcomes, it would be necessary thatthe agents not make these choices based on CRRA utility optimisation - and we should not want to exclude thispossibility without good reason. Weitzman (2009) [18] shows that infinite marginal utilities at zero consumptionand the possibility of catastrophe do not mix. We do not have good models of how people behave under suchextreme conditions and it is not the purpose of this paper to propose such a model. All this paper seeks to achieveis to signpost that extreme conditions may lie in this direction. The analysis of Section 6 on the incentives tohoard suggests that even a forward looking model with risk neutral agents and Strong IRS may still collapse.

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The disaggregated capital services sector consists of measure n (endogenous) firms each produc-ing a differentiated good with some monopoly pricing power. The demand schedule that eachmonopolist faces, and their production and profit functions are:

qi = Qpi P

(23)

qi = (Ei K

1i f) (24)

i = piqi pEEi Ki (25)

Where f is fixed costs. Solving this model, assuming free entry drives the profits of each monop-olist to zero, gives:

EQ =

1

pE

K (26)

P =

1

(1 )

1

pE

1

(f)1

1 K1

1 (27)

Q = ( 1)f

1

pE 1

(f)

1 K

1 (28)

The only other change from the model presented previously is that the total output of the energysector now has to be split across the final goods and capital services sectors, EF + EQ. Definingf = f() allows us to normalise the economies with different s so that they all coincide fore = 0. Simulating this model (with the parameters detailed in Appendix 2) gives:

Figure 9: r vs e for three economies: CRS, Weak IRS, and Strong IRS.

We now initially have the (infinitely) abundant low quality resources providing a substitute forunavailable high quality resources under strong IRS. Eventually as e continues to rise, the r(e)curve has a turning point and as e rises further the economy starts to abandon the marginalresources despite their abundance, and ultimately the economy collapses. This can be seen moreclearly by zooming into figure 9:

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Figure 10: As Figure 9 but zoomed in.

This is a much more plausible result than that presented in Section 6, in that our experienceof the real world does not lead us to automatically allocate either of the CRS/Weak IRS orStrong IRS lines to an implausible category. As described in Appendix 2, the degree of returnsto scale needed for Strong IRS and collapse is much lower in the generalised model of this sectionthan in the simple model of the previous sections6.

Is the collapse of the economy due to fundamental limits in how the technologies of the sectorscan exploit the resources available? No, and as an example of this we reproduce figure 9 withthe same Strong IRS economy operating at a point e > e, where e is the point at which this

IRS economy collapses. This is done by allocating factors to industries rather than by using theprice mechanism for allocation. All technological conditions are satisfied and it is a steady statein the sense that consumption is the same as that achieved by the CRS economy and savings aresufficient to replace the depreciating capital stock.This illustration of how allocating the factors under a command economy system can outperformallocation using prices is an example a more general phenomenon: this collapse is not due tofundamental resource limits but is rather due to a failure of the price mechanism to allocate theseresources efficiently when there are returns to scale in the economy and factors feed-back uponthemselves by being both inputs and outputs. The economy does not collapse under CRS, andthe market allocation cannot be improved upon - this follows from The First Welfare Theorem(all sectors are price taking under CRS). However, the more at risk of collapse the economy isfrom the interaction of increasing returns and scarce factors, the more amenable this situation is

to policy intervention. If we were to impose lump sum taxes on the agents and use the proceedsto subsidise energy production, then the pareto-optimal tax rate is zero for CRS and is increasingin the degree of scale economies.

6Indeed the Strong IRS line in figures 9 and 10 was generated with a returns to scale of 1.3 - which is thenumber estimated by Caballero & Lyons [11] - although no calibration has been performed.

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Figure 11: As Figure 9 with a command economy.

8 Innovation

When we imagine technological solutions to energy scarcity, we often think of high technologygoods that use energy more efficiently. This is the situation that Aghion & Howitt were abstract-ing from. A real world example of this could be modern cars with computer optimised engines,as opposed to simpler vehicles that use petrol inputs much less efficiently. In Aghion & Howittsmodel, as energy became scarce and expensive there was an increasing incentive to develop thistechnology. However, in the model presented in this paper I suggest that, if we live in a world

of strong IRS, then energy scarcity may not motivate us to use this advanced energy efficienttechnology, because the price of the high-tech computer optimised capital goods could rise bymore than the price of energy.

Specifically, we could formulate two alternative production technologies for final goods: atechnology that used energy very efficiently by applying lots of capital services, and a technologythat used capital services efficiently by applying lots of energy. There would be some price ratioat which these technologies used energy and capital services in the same ratio to produce thesame output level, and this price ratio would be the price ratio that the economy switched fromone technology to the other. Considering only the model of Section 7, rising e always causesrising r in the CRS and Weak IRS economies, and causes rising r in the Strong IRS economyinitially. Rising r occurs because of a rising energy to capital services price. The switch pricewill eventually be reached for the CRS and Weak IRS economies and they will ultimately usethe energy efficient technology. The switch price may or may not be reached under Strong IRS,but even supposing that it is, further declines in high quality energy availability could see theswitch price being reached again on the way down i.e. the Strong IRS economy may choose neverto take up the energy efficient technology, and even if the economy does adopt it, it may thenabandon it. This is intuitive - we can well imagine that productivity in advanced sectors dependson sufficient scale, and if scale is hit hard enough by a shortage of energy, then advanced energyefficient products may not be available.

The same issue arises for a putative backstop technology. We could suppose that some non-depletable backstop was available at some energy to capital services price ratio. For expositional

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purposes let us suppose it is large scale deployment of solar panels in deserts. Again the pro-ductivity of the sectors that can produce the solar technology depends upon the scale at whichit operates. At low levels of scarcity there is large demand for semiconductor technology so this

sector is large and productive. It appears that the solar backstop is feasible but the energy priceis too low to justify its deployment. Energy scarcity rises and energy prices rise. However capitalsectors across the economy contract and productivity falls. The pricing is such that despitethe rise in energy prices, we are no closer to profitably deploying the backstop technology. Theeconomy eventually collapses for lack of energy, and at no point was it profitable to deploy thebackstop technology. The description here only applies to the Strong IRS economy, under CRSor Weak IRS, the backstop technology will eventually be deployed.

In general, this story applies to any innovation effort that may allow an economy to grow orcontinue at the same level under resource restrictions. If the benefits to innovating are positivelyrelated to the energy price, but the costs are positively related to the capital services price, thenit will eventually be optimal to undertake the innovation effort under CRS or Weak IRS. It maybe the case that it is never be optimal to undertake the innovation effort under Strong IRS. Asin Section 7 this problem is amenable to policy intervention. Subsidies can support the scale

of industry so that innovations or technologies are within the reach of a Strong IRS economy,whereas they may be out of reach without policy interventions. This is therefore (theoretical)support (though not necessarily support in any specific case, or for our real world economy) forsubsidies, e.g. renewables feed-in-tariffs, which may create an industry of sufficient scale to beprofitable.

9 Conclusion

We have shown that it is possible that the price movements caused by declining energy resourcesmay not be conducive to the exploitation of more marginal resources, in contrast with the basicHotellings model and almost all of the non-renewable natural resources and energy literature.And we have shown that increasing returns to scale, as estimated as occurring in, and often

assumed for, industrial economies, is a sufficient condition for this phenomena to be manifest.Further, we have discussed how this phenomena may impinge upon innovative or technologicalsolutions to future energy shortages - and the interaction does not mitigate the problem. However,the more that this phenomena is a real problem, the more it is amenable to policy intervention- which does allow society to mitigate the problem through activist policy.

Future research must focus on testing these conclusions in a model in which the energy shocksare rationally anticipated by the agents, and on calibrating such models to uncover whether theysuggest we live in a world of Strong Increasing Returns to Scale or not.

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References

[1] R Ayres. The minimum complexity of endogenous growth models: the role of physicalresource flows. Energy, 2001.

[2] Aghion P. Howitt P. Brant-Collett M. Garcaia-Peanalosa C. Endogenous Growth Theory.1998.

[3] A Brandt A Farrell. Scraping the bottom of the barrel: greenhouse gas emission conse-quences of a transition to low-quality and synthetic petroleum resources. Climatic Change,2007.

[4] S Basu J Fernald. Returns to scale in u.s. production: Estimates and implications. Journalof Political Economy, 1997.

[5] D Murphy C Hall. Year in review - eroi or energy return on (energy) invested. Annals ofThe New York Academy of Sciences, 2010.

[6] R Hall. Invariance properties of solows productivity residual. NBER Working Paper, 1989.

[7] J Hamilton. Oil prices, exhaustible resources, and economic growth. Prepared for Handbookof Energy and Climate Change, 2011.

[8] P Dasgupta G Heal. The optimal depletion of exhaustible resources. Review of EconomicStudies, 1974.

[9] S Holland. Modeling peak oil. The Energy Journal, 2008.

[10] H Hotelling. The economics of exhaustible resources. The Journal of Political Economy,1931.

[11] R Caballero R Lyons. The role of external economies in u.s. manufacturing. NBER Working

Paper, 1989.

[12] H Barnett C Morse. Scarcity and Growth. Johns Hopkins Univ Pr, 1963.

[13] T Murphy. Peak oil perspective. physics.ucsd.edu/do-the-math/2011/11/peak-oil-perspective,2011.

[14] L Mohler M Seitz. The gains from variety in the european union. Munich Discussion Paperhttp://epub.ub.uni-muenchen.de/11477/, 2010.

[15] J L Simon. The Ultimate Resource 2. Princeton University Press, 1996.

[16] R Solow. A contribution to the theory of economic growth. The Quarterly Journal ofEconomics, 1956.

[17] A Dixit J Stiglitz. Monopolistic competition and optimum product diversity. AmericanEconomic Review, 1977.

[18] M Weitzman. On modeling and interpreting the economics of catastrophic climate change.The Review of Economics and Statistics, 2009.

[19] B Gaitan R Tol I Yetkiner. The hotellings rule revisited in a dynamic general equilibriummodel. University of Hamburg Working Paper, 2004.

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Appendix 1: Proofs of propositions

Proposition 1. For any given capital stock, K 0, the market equilibrium exists and is unique.

Proof. Given K, the total capital services available to the economy, Q = K, is also given. Itis then useful to perform a positive transformation on the energy price to capital services priceratio:

z =pE

P

1

e

1

Now consider the demand for capital services from the energy industry as a function of z:

QDE = (1 )z(z +e

1 )1

Clearly QDE (0) = 0 ,dQDE

dz> 0, z > 0

and QDE as z

and so we have demand monotone in relative prices. Given a relative price level, z, and assumingthat the energy industrys demand for capital is satisfied, we have the energy output as a functionof z:

E =1

2(1 2)z

z +

2

1 +

e

1

Clearly E(0) = 0 ,dE

dz> 0, z > 0

and E as z

The supply of capital services to the energy industry can be determined given the total capitalservices available and the demand from the final goods sector:

QSE = QQF

P = Y

QF

pE = Y

E

i.e. z =

QF

E

1

e

1

so QSE = Q

E(z +

e

1 )1

Clearly QSE as z

and QSE(0) = Q

with dQSE

dz< 0

So, XS D = QDE QSS

with XSD(0) = Q

XS D as z

anddXSD

dz> 0 z > 0

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so, by continuity and monotonicity, there is a root for XSD and it is unique. Therefore thereexists a market equilibrium and it is unique.

Proposition 2. K = 0 is a steady state.

Proof. At K = 0 steady state, we have Q = 0. If we have z = 0 then we will have QDE = 0 and

E = 0 so QSE = Q

E

z + e1

1

= 0. This is a steady state since Kt+1 = (1)Kt+sYt =

(1 ) 0 + s 0 = 0.It remains to show that zeq(K = 0, e) = 0. Clearly K(0) = 0, i.e. z = 0 K = 0. Since K(z)is a bijection, we also have K = 0 z = 0. Therefore we do indeed have z = 0 if K = 0 and soK = 0 is a steady state.

Proposition 3. = 121++

such that > , limz0 K(z) is negative andlimzK(z)

is positive. For > and e = 0, G(zt) is monotonically decreasing and so there is only one(unstable) steady state K > 0.

Proof. Have K = Kt+1 Kt = sYt Kt = H(zt)[1 G(zt)] and since K(z) is a bijectionwith K > 0, Kt+1 > Kt zt+1 > zt i.e. sign(z) = sign(K). H(0) = 0 and H(z) >0, H(z) > 0, z (0,), so the sign of, and roots of, K(z) depend upon the sign of theterm in brackets. We define as the degree of increasing returns required such that G(z) ismonotonically decreasing on z [0,) for all parameter values when the parameter e = 0. ForG(z) < 0 we need > , where:

=1

21+ +

(29)

Given this parameter restriction, e = 0 limz0 G(z) = , limz G(z) = 0, and G(z) = 1

occurs only once for unique z because of monotonicity. Further, we also have limz0 G(z) = and limzG(z) = 0 even for e > 0 (where we cannot definitively claim monotonicity).

Therefore, the proposition holds, K = 0 is stable e [0,) and, at least for e = 0, there is aunique K > 0 steady state that is unstable.

Proposition 4. For 1 < , limz K(z) negative. For 1 < and e = 0,

limz0 K(z) is positive (and so K = 0 is unstable), G(zt) is monotonically increasing and sothere is a unique stable steady state K > 0.

Proof. We have already seen that > K = 0 stable. Here we consider only 1 < <

and e = 0.We have:

K(zt) = s(AL)1c1z

+t (zt + f(e))

+(zt + h(e))( 11)

1

sc1(AL)+1

c0

1

z1

t zt + g(e)

zt + f(e)+

(zt + g(e))1(zt + h(e))

( 1)( 1

1)

= H(zt)[1G(zt)]

Taking limits with 1 < we get limz0 K(z) is positive and limz K(z) is negative,e 0 since limz0 G(z) = 0 and limzG(z) = . Further:

K(zt, e = 0) = s(AL)1c1z

2+( 1+1)

t

1

sc1(AL)+1

c0

1

z( 1)( 1

+1)2

t

= H(z, e = 0)[1G(z, e = 0)]

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Given the parameter restriction 1 < holds, the function G(z, e = 0) is monotonicallyincreasing in z from G(0, e = 0) = 0, to limz G(z, e = 0) , so G(z

) = 1 occurs only oncefor unique z. Therefore, for e = 0 there is a unique, stable, K > 0 corresponding to K(z).

Proposition 5. = 1+ < such that (, ) and with e > 0, limz0 K(z) is

negative (and so K = 0 is stable). For 1 < and e > 0, limz0 K(z) is positive (and soK = 0 is unstable).

Proof. We have:

K(zt) = s(AL)1c1z

+t (zt + f(e))

+(zt + h(e))( 11)

1

sc1(AL)+1

c0

1

z1

t

zt + g(e)zt + f(e)

+(zt + g(e))

1(zt + h(e))

( 1)(11)

= H(z)[1G(z)]

Therefore, we can take the limit as z 0 with e > 0:

limzt0

K(zt, e > 0) = s(AL)1c1z

+t f(e)

+h(e)(11)

1

sc1(AL)+1

c0

1

z1

t

g(e)f(e)

+g(e)

1h(e)(

1)( 11)

The quantity in the square bracket is positive if 1 > 0 but negative if 1

< 0.

Therefore = 1+

and K = 0 is unstable for 1 < . For > , K = 0 is stable.

Since (0, 1), clearly

=1

+ 0, always exists.

Proof. Previous analysis has shown that, for 1 < , and general e 0, limz0 K(z)is positive and limzK(z) negative. Further K(z) is a continuous function of z with nosingularities over z (0,). Therefore there must be some z (0,) for which K(z) = 0and K(z ) > 0, K(z + ) < 0 for very small . Further, z > 0 K(z) > 0.Therefore there is a stable steady state for the economy with 1 < and general e 0.

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Appendix 2: Parameters of simulated economies

22

an industrial economy's response to declining energy resources

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