Malthus and Climate Change:Betting on a Stable Population

8/9/2019 Malthus and Climate Change:Betting on a Stable Population

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.Journal of Environmental Economics and Management 41, 135161 2001

doi:10.1006rjeem.2000.1130, available online at http:rrwww.idealibrary.com on

Malthus and Climate Change:

Betting on a Stable Population1

David L. Kelly

Department of Economics, Uni ersity of Miami, Box 284126, Coral Gables, Florida 33124

and

Charles D. Kolstad

Department of Economics and Bren School of En ironmental Science and Management,

Uni ersity of California, Santa Barbara, California 93106

Received August 10, 1999; revised October 8, 1999; published online December 7, 2000

A standard assumption in integrated assessment models of climate change is that popula-tion and productivity are growing, but at a decreasing rate. We explore the significance of the

assumption of population and productivity growth for greenhouse gas abatement. After all,there has been no long run slowdown in the growth of productivity over the past fewcenturies, and the rate of population growth has actually been increasing for the past 19centuries. Even if either of these growth rates were expected to slow, by how much is subjectto great uncertainty. We show computationally that such continued growth greatly increasesthe severity of climate change. Indeed we find that climate change is a problem in large partcaused by exogenous population and productivity growth. Rapid reductions in growth makeclimate change a small problem; smaller reductions in growth imply climate change is a veryserious problem indeed. Analogously, reductions in the growth rate of population can beeffective in controlling climate change. 2000 Academic Press

Key Words: climate change; population growth; productivity; integrated assessment.

1. INTRODUCTION

A standard assumption in computational models of climate change integrated.assessment models is that population and productivity are growing, but at a

w xdecreasing rate. For example, a survey in 15 finds that most researchers assume alarge slowdown in productivity growth over the next century. The mean estimate of

the survey is that productivity growth will slow by more than half by 2025 w x .Nordhaus 16 assumes productivity growth slows by half every six decades .Similarly, the Intergovernmental Panel on Climate Change report hereafter IPCC,

w x w x.1990 6 , and IPCC, 1992 7 estimates population growth will slow from 22% to18% this decade, with zero growth by the year 2200, largely based on United

w x w xNations 20 or World Bank 22 predictions.

1Research supported by US Department of Energy Grants DE-FG03-94ER61944 and DE-FC03-

90ER61010, the latter through the Midwestern Regional Center of NIGEC. We thank Robert Deacon,Alan Manne, and seminar participants at the 1996 California Workshop on Environmental and NaturalResource Economics. Research assistance from Aran Ratcliffe is gratefully acknowledged.

135

0095-0696r00 $35.00Copyright 2000 by Academic Press

All rights of reproduction in any form reserved.


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KELLY AND KOLSTAD136

In this paper we explore the significance of the assumption of population andproductivity growth for greenhouse gas abatement. After all, there has been nolong run slowdown in the growth of productivity over the past few centuries, andthe rate of population growth has actually been increasing for the past 19

centuries. While most analysts expect a turnaround in the growth of populationand productivity, exactly by how much and when is highly uncertain. We showcomputationally and theoretically that such continued growth greatly increases theseverity of climate change. The conclusions are startling: an almost a one-to-onecorrespondence exists between population and productivity growth assumptions,the degree of climate change, and hence the optimal response to climate change.Indeed we find that climate change is a problem in large part caused byexogenous population and productivity growth. By examining this question we alsogain important insights about how assumptions in the growth of population andproductivity drive the results that are common to almost all integrated assessmentmodels.

This paper also extends the recent literature on the effect of population on w x.externalities see for example 5 to the case of a stock pollutant in a production

externality. Larger future populations increase the damage from current emissions,at least under the standard utilitarian paradigm. One can simultaneously observethe marginal damage from an additional unit of pollution, as well as the marginaldamage from an additional person, with the mechanism being the increased

damage from a larger population which suffers the pollution in future years. .In most integrated assessment models, emissions of greenhouse gases GHGs

.are proportional to the gross world product GWP . Most models also assume theproportion declines over time, to reflect compositional changes in output, exoge-nous technical progress in abatement, or switching to fuels which release less

w xGHGs. Models which make this assumption include 10, 16 . Other models such asw x12, 19 have an energy sector and model emissions as proportional to energyconsumption. In the long run, energy usage rises with GWP, hence emissions and

.GWP rise together. Other models primarily from the natural science literaturew xsuch as 11 let emissions be exogenous. However, the exogenously specified

emissions path is calibrated from projections which assume emissions are propor-tional to GWP. Hence nearly all integrated assessment models directly or indirectlymake GHG emissions proportional to GWP.

The GWP is in turn a function of the different production inputs: capital, labor,and possibly an energy input. The world productivity variable and value share ofeach input are calculated using the Solow procedure, or growth accounting.

According to growth accounting, productivity is simply the difference between thechange in output and the change in production inputs at their average shares. Useof average shares in growth accounting in turn implicitly assumes that the economyis on the long run balanced growth path. By balanced growth, we mean that capital

.and labor in productivity unit terms, hereafter adjusted labor are growing at thesame rate. This is equivalent to a constant capital to labor ratio. Thus output per

. 2adjusted labor unit also grows at the same rate.

2 w x A large literature exists which tries to explain long run growth in output 1 provides an excellent.summary .


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CLIMATE CHANGE AND POPULATION GROWTH 137

To see why the balanced growth result holds in an optimal growth model,suppose capital were growing at a slower rate than adjusted labor. Then over timethe economy accumulates relatively more labor resources than capital resources.But then since labor has diminishing marginal returns, an extra unit of capital

produces increasingly more output relative to another unit of adjusted labor.Eventually, the returns to investing in this capital relative to consumption becomelarger so that the economy invests at a greater rate. But then capital grows at agreater rate, moving back to the growth rate for adjusted labor. Hence balancedgrowth holds at the long run steady state.

Now add climate change to a model in which balanced growth holds in the longrun and suppose for now abatement is zero. Since emissions are proportional toGWP, emissions must also grow at the same rate as output, capital, and adjustedlabor. However, since emissions cause damage from climate change, the return tocapital is lower. Damage from climate change reduces future output for a givenamount of capital. Climate change does not affect the return to consumption. Ifthe growth rate in adjusted labor is zero at the steady state, the growth rate incapital and emissions is also zero at the steady state and balanced growth holds,but with a lower capital to labor ratio. The capital to labor ratio will offset thedamage from climate change with the loss in consumption from having less capital.

w xWeitzman 21 refers to the difference in capital between a model without anenvironmental externality and a model with an environmental externality as the

environmental drag. If investment in abatement is possible, the capital to outputratio will be closer to the capital to output ratio in the optimal growth model withsome steady state investment in abatement.

In this paper, we argue that the reduction in the return to capital caused bydamage from climate change is very small under standard assumptions. Hence theoptimal solution found by climate change modelers is generally to grow at nearlythe no-climate change rate. Since marginal abatement costs go to zero as abate-ment goes to zero, some amount of abatement is optimal.

Why is the damage from climate change small? This is basically becauseresearchers assume that growth in adjusted labor stops long before climate changebecomes a real problem. If adjusted labor stops growing, then capital stopsgrowing, which implies output stops growing, but then so too do emissions, andtherefore so do temperature change and damage from temperature change, all

without cost.Conversely, imagine a world in which adjusted labor units grew at a constant rate

forever. Then capital grows at a constant rate, and therefore so does output and

uncontrolled emissions. Because damages are convex, abatement approaches 100%to keep net emissions finite. That is, emissions control approaches one anduncontrolled emissions goes to infinity, but net emissions remains finite. A cornersolution of 100% emissions control differs greatly from emissions control whenadjusted labor stops growing.

From these two thought experiments, one can see that assumptions about thegrowth in adjusted labor drive climate change results, as well as optimal policies.

Indeed adjusted labor growth determines investment levels, emissions growth or.emission control , GHG levels, and temperatures. We find that if the growth rate

in adjusted labor declines by 16% per decade instead of 31% per decade assumed .in IPCC 1990 , then the total steady state temperature increase rises from 5.83C


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to an unprecedented 16C.3 Furthermore, control measures are severe: abatementnearly doubles, eventually rising to 26% of emissions, and the optimal capital levelfalls 36% below the balanced growth level. If the growth rate of adjusted labordeclines by 31% per decade, then abatement rises to at most 14% of emissions and

capital spending is at most 7% less than balanced growth.Implications for near-term climate policy are significant, though less spectacular.

Higher rates of population growth imply as much as 50% higher current marginaldamages from carbon. If a carbon tax were used to implement the optimal level ofemissions, the carbon tax would be set to the marginal damage. Though marginaldamage from carbon is in the neighborhood of $5 per ton in 1995, the marginaldamage of one more person is roughly two orders of magnitude higher, suggestingthat attention to factors leading to population stabilization are in order for

addressing climate change. Indeed, efficiency requires both a carbon tax and a tax .on children or equivalent policies .While we use a model which is similar to standard and well-known models of

climate change to facilitate comparisons, we believe the results apply generally tonearly all climate change models, because nearly all climate change models makesimilar assumptions about growth in population and productivity. One way tointerpret our results is as a critique of climate change models which assumeslowing growth in population and productivity. Those assumptions drive the results

and are at the same time difficult to justify. Another interpretation of our results is that efforts at GHG control might beappropriately supplemented by efforts to reduce population growth. While popula-tion control is sometimes mentioned as an important part of policies for otherenvironmental problems, population control is rarely considered for climate changepolicy. This is especially surprising given the long run nature of climate change.

w xOur analysis parallels 5 , which addresses child bearing choices and externalities inw xan economy with a consumption externality. Harford 5 shows that both pollution

taxes and child bearing taxes are needed to achieve the Pareto optimum, with theoptimal tax per child equal to the present value of the pollution taxes paid by thechild.

In the next section we develop a simple theoretical framework from which weanalyze how productivity and population growth affect decisions about investmentand emissions. In the subsequent section, we give computational results foremissions and investment decisions under alternative specifications about thegrowth of adjusted labor. In Section 4, we derive computationally the 1995 control

rates under alternative specifications about the growth of adjusted labor. InSection 5 we derive the marginal damage caused by an additional person. The finalsection is concluding remarks.

3 .For reference, IPCC 1992 reports the current warming trend is about one half of one degree overthe last 100 years. An increase of 34 degrees above modern temperatures last occurred an estimated5000 years ago. We assume the structural model does not change if GHG levels rise above two times

.pre-industrial levels in some cases considered here, GHG levels increase six-fold . In reality, a six-fold

increase in carbon would likely exceed the total carbon content of the planet and a 16 degreetemperature change would likely cause damage far in excess of that given in the model. Our point is notto predict a temperature change but simply to demonstrate the sensitivity of the model to assumptionson population and productivity growth.


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2. A REPRESENTATIVE MODEL

Consider a simple model of climate change, the notation and structure of whichw xis similar to 16 . Let time in years be discrete with t s 0, 1 . . . . For computational

reasons our simulations are in decades, which changes the model only through afew of the parameter values noted in Table I. The population consists of Lt .identical representative consumers consumers are not differentiated over regions .

.Consumers value consumption of a composite commodity C .t .The consumer faces several resource constraints. Consumers use capital Kt

and labor to produce the consumption good. Productivity is proportional to A t .technology . Production is CobbDouglas:

Q sA KL1y . 2.1 .t t t t

.Here g 0, 1 is the capital share. We let l denote the productivity weightedt .labor units adjusted labor :

.1r 1yl s A L . 2.2 . .t t t

Therefore we can rewrite the production function as

Q sK

l

1y

2.3 .t t t

.Emissions of GHGs E cause increases in atmospheric temperature above at .base temperature T , which causes environmental damage through reducedt

TABLE I

Parameter Values in Decades

Parameter Value

. 1 logarithmic 0.25

b 0.06861b 2.8872 0.014871 22

. 6.4 0.64 per yearm . 0.11 0.011 per year

. 0.083 0.0083 per yearM . 0.6513 0.1 per yearK

0.4181Tr 1.33681r 0.09442r 0.023M 590b

. 0.7441 0.97 per year .g 0.3653 0.032 per yearl, 1985 .g 0.0823 0.008 per year

, 1985

l 5.86261985 0.3731985

Note. The decline rate is varied asldiscussed in the text.


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.production. A fraction of the production can be spent on abatement which hasta convex cost function and reduces emissions linearly. The abatement level is the

.primary concern of policymakers, especially in the current period. Net income Ytis the fraction of production which is not spent on emission control.

1 yb b 2 .t 1 1yY ' Q ' K l . 2.4 .t t t t2D T 1 q T .t 1 t

.In Eq. 2.4 , b , - 1 and b , ) 1 are positive constants.1 1 2 2The consumer faces the overall resource constraint that total income equals

consumption plus net investment. Let be the depreciation rate; thenK

C s Y yK q 1 y K . 2.5 . .t t tq1 K t

.Uncontrolled emissions emissions before abatement in a given period dependlargely on the specific mixture of fuels consumed. Some analysts suggest that in thefuture we will rely increasingly on energy sources that emit less GHGs per unit ofreleased energy and that output will continue to become less energy intensive. Inorder to make comparisons between our results and previous results and also forsimplicity, we let this process be exogenous. In reality, the mixture of fuels

consumed is likely to be sensitive to the tax or other GHG control policy indeed

w xManne and Richels 13 estimate that if consumers believe with 50% probabilitythat emissions will be reduced to 1990 levels in 2010, consumers will reduceemissions in the year 2000 as if a tax of $17.50 per ton of carbon were already in

.place . The emissions intensity of output corresponds to the variable . Wetassume

s 1 yg 2.6 . .t , t ty1

g s 1 y g . 2.7 . . , t , ty1

Thus the emissions intensity of output improves at a declining rate. Note thatimprovements in the emissions intensity of output also implicitly represent sectorshifts in output from goods which are GHG-intensive to produce to goods whichare less GHG-intensive to produce.

The atmosphere absorbs fraction of emissions. Let M denote the stock ofm tGHGs above preindustrial levels and E the emissions; thent

E s 1 y Q 2.8 . .t m t t t

M sE q 1 y M . 2.9 . .tq1 t M t

.Here is the depreciation of the stock of GHGs primarily into the ocean . WeMlet emissions be proportional to output and not net income Y. Hence, the purchasetof abatement is not emission free; a factory must make the CO scrubber. This2simplifies theoretical analysis considerably because climate change does not causea reduction in emissions and therefore future climate change through decreasednet income.

We assume a two layer model for temperature change. Let surface temperature . .be the top layer T and deep ocean temperature O be the second layer, botht t

measured in degrees Celsius above a base year. Let M denote the preindustrialb


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level of GHGs; then the temperature model is

MtT s 1 y T qr log 1 q qr O 2.10 . .tq1 T t 1 2 t /Mb

O s O qr T y O . 2.11 . .tq1 t 3 t t

.Here r and r , r , g 0, 1 are positive constants.1 2 3 TPreferences for the consumption good are specified by a twice differentiable and

concave utility function. We assume utility has a constant relative risk aversion .CRR specification:

C1y y 1tw xU C s G 0. 2.12 .t

1 y

Here the limiting case ofs 1 is logarithmic utility. The welfare function W is thediscounted sum of per-capita utilities over the infinite horizon. Let denote thediscount rate; then

CttWs U . 2.13 .Ltts0

Many integrated assessment models use a Ramsey aggregate welfare function. TheRamsey welfare function weights utility at period t by the size of the population:

CttWs L U . 2.14 . tLtts0

The former welfare function maximizes the utility of a representative consumer ofthe current generation while the latter maximizes the aggregate utility of all

consumers including those yet to be born. One can show that both the Ramsey andper-capita specifications yield the same stationary values, since the steady state isindependent of the functional form of the utility function. Hence most of theresults apply to either specification. However, the 1995 abatement decisions underRamsey utility are higher than under per-capita utility, because the social plannerdiscounts the future less. When calculating 1995 abatement decisions, we considerboth the Ramsey and per-capita welfare functions.

.We can rewrite the utility function 2.12 in terms of adjusted labor units.

1y 1r1y. 1yC A Ct t ty 1 y 1

lC ltt t1y.r1y.U s sAtL 1 y 1 y t 0

qa s U c qa . 2.15 . .0 t 0

Here

A1y.r1y. y 1 Ct ta s , c s . 2.16 .0 t1 y l t


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Hence

t 1y.r1y. w xWs A U c qa 2.17 . t t 0ts0

t yr1y. w xWs A l U c qa . 2.18 . t t t 0ts0

. .Equations 2.17 and 2.18 show that productivity increases and population in-creases have similar effects on utility, with two possible exceptions. First, aproductivity advance provides extra per person consumption which under the CRR

specification is captured additively by the term a . Thus although a productivity0advance increases per-capita utility, the optimal decisions are unchanged. Second,if the elasticity of substitution is not equal to one, then higher future returns tocapital caused by future productivity advances bias decisions toward future con-

. .sumption essentially raising the discount rate . In Eq. 2.18 population growthalso biases decisions toward future consumption as the planner maximizes utility ofall consumers including those not yet born.

Because a is a constant, we may drop the a terms from the welfare function.0 0

Note that the only difference between the welfare functions W and W is the termAyr1y.l , which serves to alter the discount rate, consistent with our previoust tdiscussion. Hence the problem is to choose an investment level K , a consump-tq1tion path C , and a level of GHG abatement which maximizes the welfaret tfunction, subject to the environmental and economic constraints.

After substituting for consumption using the budget constraint we rewrite theoptimization problem recursively using Bellmans equation. First we rewrite theexogenous variables and l as functions of the integer-valued state variable ,t

which indexes the position of the exogenous variables. Let k sK rl denote thet t tcapital to labor ratio and let g denote the growth rate in adjusted labor, whichl, t

we assume is a decreasing function of time. Let : 5 denote the presentdiscounted sum of utility given that the planner makes optimal decisions at each

. point in time i.e., the value function . Then for problem W problem W is.analogous ,

. .1y r 1y S s max A . .k , .

=U k q 1 y k y 1 qg k q S . 2.19 . . . . .k l 5D T .

Here primes denote next periods value and S is the vector of state variables:

w xS s k M T O .t t t t t t


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The maximization is subject to

k 1 y k l q 1 y M . . . .k m m

M

M1 y Tqr log 1 q qr O' S s G S, , k ' . 2.20 . . .T T 1 2 /MbOO s O qr Ty O . 3

q 1

Two first order conditions, the three environmental constraints, and the exoge-nous behavior of the growth in adjusted labor and energy efficiency characterizethe time series behavior of the model. In the Appendix, we establish that the valuefunction is increasing in the economic states k and , and decreasing in theenvironmental states M, T, and O. Differentiating the right hand side of the

. .Bellmans equation 2.19 with respect to and k assuming an interior solutiongives the first order necessary conditions for the optimal abatement and invest-ment:

c G S, , k M . . . .1y r 1yA U c s 2.21 . . .

M

G S, , k . . . .1y r 1y1 qg A U c s . 2.22 . . . . .l

k

.The left hand side of 2.21 is the marginal cost of abatement. Abatementreduces income which could be used for consumption. The right hand side is themarginal benefit of abatement: increased abatement reduces GHG concentrations

in the following period, which in turn reduces the damage from climate change thepartial of with respect to M is the marginal loss of value from an incremental

. .increase in M . Similarly, the left hand side of Eq. 2.22 represents the marginalutility of consumption, while the right hand side represents the marginal utility ofinvestment. These must be equal for an optimal investment decision.

We can gain some insight as to how abatement changes with the growth rate in .adjusted labor by examining the first order condition for optimal abatement 2.21

.at the steady state. At the steady state, the right hand side of 2.21 , the marginal .benefit, is the details of the derivation are in the Appendix

T M D T . . . 1y.r1y. 2MB s A U c Q . 2.23 . .3 m 2M T D T .

Here is a positive constant and c is the stationary consumption level per unit of3adjusted labor, and is the stationary value of abatement. Since the stationarymarginal benefit of abatement is convex in the stationary gross output Q, thestationary marginal benefit of abatement is convex in adjusted labor. Hence theincentives to raise stationary control rates become stronger when the stationary

adjusted labor is higher. The stationary marginal cost of abatement the left hand ..side of 2.23 is linear in Q. Hence the marginal cost of abatement also rises with


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adjusted labor. The stationary first order condition reduces to

T D .s Y. 2.24 .3 m

M M T T . .

Hence since marginal costs are increasing in , as the stationary level of adjustedlabor rises, the stationary abatement level rises.

. .Equation 2.24 suggests a direct relationship between output Y and abate-ment. In the optimal growth model, growth in output is determined by growth in

.adjusted labor balanced growth . To see how growth in output in the optimalgrowth model relates to growth in output in our model, we examine the marginal

. . . value of investment in 2.22 , which is S rk. After evaluating 2.19 at the

optimal abatement and investment decisions, we differentiate both sides withrespect to k and get

S C G S, , k M . . . . .1y r 1ysA U c q . 2.25 . . .

k k M k

The marginal utility of investment consists of two terms. The first term is themarginal benefit of investment: investment increases future capital stocks which

.provides income and therefore consumption. But Eq. 2.25 has a second termwhich is not present in the standard optimal growth model. The second term is themarginal damage from an increase in the capital stock. A marginal increase in thecapital stock causes higher emissions next period, which increases the stock ofGHGs. An increase in the stock of GHGs in turn increases the temperature andcauses a loss of utility through reduced output.

Consumption has no effect on emissions. Climate change reduces the marginalutility of investment; hence the optimal decision is to consume more and invest lessrelative to the decision without climate change. We will argue, however, that given

current assumptions about the growth in population and productivity this term issmall and so investment is largely unchanged from the optimal growth model.

3. ALTERNATIVE ASSUMPTIONS ABOUT ADJUSTEDLABOR GROWTH

3.1. Decreasing Growth in Adjusted Labor

Next we analyze the deviation from balanced growth levels and the marginal costof abatement under different assumptions about the growth of labor and productiv-ity. We assume adjusted labor grows at a continuously decreasing rate.4

l s 1 qg l 3.1 . .t l , t ty1

g s 1 y g . 3.2 . .l , t l l , ty1

For 0-

-

1 we have a decreasing growth rate of adjusted labor. In the nextlsubsection, we consider the limiting case of s 0, which is constant growth ofl

4 w xThis specification fits the assumptions of most climate change modelers reasonably well. See 2 .


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population and productivity. We will also look at the other limiting case of nogrowth, or s 1.5l

Given that 0 - - 1, the model has a globally stable stationary state. In thelstationary world, all variables are constant. The economy converges to the station-

ary state from below; i.e., capital, labor, productivity, emissions, temperature, andGHG concentrations all increase until the stationary values are attained. As capitaland the exogenous variables increase, the stock of GHGs increases. Increases inthe stock of GHGs cause increases in the atmospheric temperature. This causes

.damage, which decreases the benefit of investment in Eq. 2.25 . Hence thestationary state is an upper bound for the effect of climate change on capitalinvestment.

To get an idea about the maximum values for investment and abatement, wenumerically calculate the stationary investment levels with and without climatechange. Table I gives the parameter values, which are close to those commonly

w x.used in integrated assessment models most are from 16 . The baseline value of s 0.31 is calibrated to match average values used by climate change researchers.lIn the Appendix we give the solution procedure for the stationary values of thecontrols and states, using the first order conditions, constraints, and stationaryconditions. After solving for the stationary state, we find that climate changedecreases the stationary state capital level by only 7%.

However, this is not so if the growth rate of adjusted labor fails to slow down as

.significantly that is, if falls below 0.31 . Table II highlights values of stationarylvariables as the growth rate in adjusted labor is varied. The first column of Table IIgives the decline in the growth rate of adjusted labor. The second columnrepresents the percentage change in stationary capital levels from the model

without climate change. The third column is the stationary abatement level. Thefourth column is the stationary atmospheric temperature. The final column is thestationary value of adjusted labor. If adjusted labor slows down by 16% per decadeinstead of 31%, we see increased environmental drag: the stationary state capital

level falls by 36%.In other words, if population and productivity growth more closely follow

historical growth rates and fail to slow down significantly, then climate change has

5 We parameterize the elasticity of substitution equal to one, so productivity and population growthaffect decisions in the model identically, except in the case of the Ramsey welfare function wherepopulation growth adds to the weight of future consumption by essentially increasing the discount rate.

.Thus when varying we also vary the welfare function as in Eq. 2.18 .l

TABLE II

Stationary Values and l

% in k T ll

1.00 y0.17 0.03 0.8 28.70.31 y6.88 0.14 5.83 100.420.28 y9.20 0.16 6.84 140.67

0.25 y12.69 0.18 8.20 215.700.22 y17.62 0.21 9.95 361.790.19 y25.35 0.23 12.52 742.340.16 y36.00 0.26 16.02 1895.37


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very large effects. An interesting case is the special case of no growth in adjustedlabor. Here temperature rises less than 1 degree most of this due to already

.present inertia in the climate . Hence we can conclude that climate change is aproblem in a sense caused by adjusted labor growth.

Just as the stationary state is the upper bound for capital and the deviation fromthe no-climate change growth path, the stationary state also represents the maxi-mum climate change and the maximum abatement levels. Without a significantslowdown in adjusted labor, we see in Table II that emissions, climate change, andabatement all rise significantly. If adjusted labor growth slows by 16% per decadeinstead of the baseline 31%, the stationary abatement nearly doubles from 14 to26%. The total climate change nearly triples from 5.83 to 16 degrees. Of course,the model is not specified to handle such a large increase in GHG concentrationsand temperature. GHG concentrations in this case would likely exceed the totalcarbon stocks in the planet and the temperature change would likely be catas-trophic. Our point is not to predict a 16 degree temperature change if populationgrowth and productivity growth continue for longer than expected, but instead topoint out the sensitivity of the model results to assumptions on population andproductivity growth.

In other words, climate change is a much more serious problem. In effectintegrated assessment models rely on a significant slowdown in population andproductivity to solve the climate change problem. Since population slows before

climate change becomes a serious problem, only modest abatement is required.Figures 1 and 2 illustrate the stationary results under various assumptions about

the decline in the growth rate of adjusted labor. Here we consider a range of valuesfor the growth rate of adjusted labor between a decline of 16% per decade and

.31% per decade well above the historically observed negati e values for .l

FIG. 1. Optimal stationary capital levels and GHG concentrations as a function of .l


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FIG. 2. Optimal stationary state abatement and temperature as a function of .l

Included in the simulations is the baseline value of 31%. Figure 1 shows thestationary capital levels of the model with and without climate change as a functionof the decline in the growth rate of . Under standard assumptions the environ-lmental drag is small: only modest differences exist between the stationary capitallevels with and without climate change. The difference magnifies exponentially asthe decline rate shrinks. Figure 1 also shows GHG concentrations are convex in .l

Figure 2 gives the optimal stationary abatement levels as a function of .lTemperature is also convex in .l

3.2. Constant Population and Producti ity Growth

A common assumption in the economics literature is constant population growth.Constant population growth can be motivated by the difficulty in predicting futurepopulation growth. If future population growth rates are unpredictable, then

todays growth rate is the best estimate of future growth rates. Furthermore, acommon specification for the behavior of productivity is to assume that the growth w x.rate of productivity is stochastic around a constant rate for example see 8 . If we

drop the stochastic part of productivity, we can model adjusted labor as growing at .a constant rate. See Fig. 3.

Suppose, then, adjusted labor grows at a constant rate; that is, s 0. Keep inlmind that population growth may cease, but if technological change continues,adjusted labor keeps growing. Recall that the budget constraint in per-capita termsis

1 yb b 21 t c s k q 1 y k y 1 qg k . 3.3 . . .t t K t l tq121 q T1 t


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FIG. 3. Historical and predicted population growth. Shown is the data set from the SmithsonianInstitute; the United Nations data are similar.

Further,

E s 1 y kl 3.4 . .t t t t t

M s E q 1 y M . 3.5 . .tq1 m t M t

With constant growth of population and productivity the emissions level has aninterior stationary value whereas the control variable does not. To see this, note

that if emissions grow unbounded, then so to does the atmospheric temperature.This leads to zero consumption, which cannot be a maximum since the planner canalways choose s 1 at cost less than gross income. Hence an upper bound existsfor emissions and temperature. Emissions and temperature are bounded below byzero. Hence we substitute out for the abatement level and use emissions as thecontrol variable. The resource constraint is then

b 2Et1 yb 1 y1 / k lt t t c s k q 1 y k y 1 qg k . 3.6 . . .t t K t l tq121 q T1 t


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Emissions, the emissions intensity of output, and the capital to labor ratio are allstationary; hence as adjusted labor approaches infinity, the cost function mustapproach b or s 1. One hundred percent abatement in the long run is quite1different than the standard policy result of modest abatement in the long run.

The reader may find it unlikely that improvements in productivity generate .strong effects on uncontrolled emissions, because anecdotally improvements in .productivity come largely in non-GHG-intensive sectors e.g., services . Harberger

w x3 finds a few industries which vary randomly from decade to decade generatemost of the observed change in productivity for the economy. Thus in the future

w xproductivity may again be in GHG-intensive industries. Harberger 3 also findssignificant productivity growth in manufacturing and other GHG-intensive indus-tries throughout the post-war period. Our model assumes emissions intensity

.improves over time, so in the future most 83% productivity advances are emissionfree. Specifically, we assume that g s 0.8% per year, with a decline rate of

, 1985

s 1.1% per year. This implies the emissions intensity of output declines from t

0.373 to a steady state of 0.17. The decline rate of the emissions intensity of outputis calibrated from historical data as about halfway between the historical decline

. . w xrate of developed 1.5% and developing 1% countries in 1965 see 16 for a.detailed discussion . Thus growth in population and productivity drives results

despite exogenous changes in fuel use and changes in sectoral composition.On the other hand, the decline in emissions intensity of output eliminates much

of the climate change problem without cost, similar to a slowdown in population .and productivity growth. Equation 2.8 indicates that emissions are proportional to

both the emissions intensity of output and adjusted labor. However, changes inadjusted labor invariably generate more emissions in the future as the capital stockrises to maintain balanced growth. Thus the results do not depend as critically onthe decline in emissions intensity as the growth in adjusted labor.

4. IMPLICATIONS FOR CURRENT PERIOD ABATEMENT

The economy and climate converge to the stationary investment and abatementlevels far into the future. Due to long lags, especially in the response of the ocean,the climate does not converge to a stationary equilibrium for hundreds of years.Similarly, if adjusted labor continues to grow, the economy takes longer to reachstationary levels. One important policy issue is the optimal abatement level right

now.Tables III and IV show the current period and future control rates obtainedafter solving the Bellman equation numerically for several values of . Kelly andl

w xKolstad 9 give the details of our numerical solution method. Current abatement islower under the per-capita welfare function versus the Ramsey welfare function.Hence we consider both welfare functions here.

Table III indicates that if we assume adjusted labor growth slows by 16% perdecade as opposed to 31%, 1995 abatement increases slightly. By the year 2055,abatement levels given adjusted labor grows at 16% are already 18% higher thanabatement levels given adjusted labor grows at 31%. Hence assumptions aboutfuture adjusted labor growth significantly affect even current abatement decisions.Investment levels also quickly rise as the value of decreases.l


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TABLE III

.19952095 GHG Abatement

. .W per capita W Ramsey

1995 2025 2055 2095 1995 2025 2055 2095l

1.000 0.0250 0.0264 0.0288 0.0295 0.0250 0.0264 0.0288 0.02950.310 0.0596 0.0801 0.0984 0.1138 0.0880 0.0974 0.1065 0.11540.250 0.0605 0.0820 0.1036 0.1256 0.0928 0.1074 0.1212 0.13610.225 0.0605 0.0809 0.1021 0.1242 0.0961 0.1104 0.1253 0.14480.190 0.0630 0.0839 0.1063 0.1331 0.1082 0.1238 0.1392 0.16090.160 0.0633 0.0874 0.1160 0.1548 0.1123 0.1281 0.1449 0.1716

The result that changes in the slowdown of adjusted labor affects currentwabatement decisions is perhaps surprising. For example, Manne and Richels 14,

xFig. 16 find near term abatement decisions are not sensitive to a 1% higherpotential GDP growth rate from the year 2030 onward. On the other hand, the

w xNordhaus 16, Table 6.2 model overall is most sensitive to the growth ratevariables in population and productivity. However, the 1995 control rate is more

.sensitive to a few other variables such as the discount rate than to the declinerate in population and productivity. Such results are in fact consistent with our

results, after some careful analysis. Changes in the decline rate affect near term .growth rates in emissions as observed in the booming 1990s . For example, Manne

w xand Richels 14 consider changes in the growth rates after 2020.

5. CARBON AND BABY TAXES

Here we examine the relationship between the carbon tax and the damagecaused by population growth. The idea is to examine the relative effectiveness of

carbon taxes versus policies which restrict population growth. Our model haspopulation growth being exogenous; yet we can still compute the marginal damagecaused by an additional person.

Consider the competitive version of the above model, in which there exist Ltfirms, each of which rents capital and labor from households at rates r and w ,t trespectively, in order to produce the consumption good. Because we assume

constant returns to scale, profits are exactly zero. A carbon tax x in trillions oft

TABLE IV

19952095 Investment Levels in Trillions of Dollars

. .W per capita W Ramsey

1995 2025 2055 2095 1995 2025 2055 2095l

1.000 19.43 19.90 19.92 19.92 19.43 19.90 19.92 19.920.310 32.31 74.20 128.30 200.13 41.86 83.39 132.55 193.36

0.250 32.66 75.11 140.55 243.92 48.93 107.96 189.48 132.550.225 32.83 75.54 141.22 244.18 50.98 115.72 211.63 375.310.190 34.70 81.30 158.80 299.07 57.95 127.95 233.40 437.380.160 35.01 85.53 189.78 429.72 58.10 129.51 249.08 522.24


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.dollars per gigaton CO motivates firms to use the abatement technology . The2 tperiod-by-period optimization problem of firm i is

Q Qi t i tmax s yr K yw L yx E y y 1 . 5.1 . . .t t i t t i t t i t i t

5 D T D T , K , L . .it it it t t

The first order conditions reduce to

.i t y1 1y y1 1yr s A K L yx 1 y A K L 5.2 . .t t i t i t t t t t i t i tD T .t

.i t y yw s 1 y A K L yx 1 y A K L 5.3 . . .t t i t i t t t t t i t i t

D T .t . i t

y s x . 5.4 . m t tD T .t

General equilibrium requires that supply equals demand in the capital and labormarkets, and that firms are identical. Hence K sK rL , L s 1, and s .i t t t i t i t tSubstituting these conditions into the first order conditions gives

.t y1 y1r s k yx 1 y k 5.5 . .t t t t t tD T .t

.t 1y2.r1y. 1y2.r1y. w s 1 y A k yx 1 y A k 5.6 . . .t t t t t t t tD T .t

. ty s x . 5.7 . m t t

D T .t

So the carbon tax creates incentives to use abatement technology and reducesincentives for labor and capital usage. Finally, suppose the government creates abalanced budget by returning tax revenues as lump-sum transfers, TR , to con-tsumers. The government budget constraint sets x E s TR . Consumers thus maket t tconsumption and investment decisions as before, but receive income from capitaland labor rentals and transfers.

By comparing the first order conditions of the consumer and firm with the first

. .order conditions associated with the social planners problem 2.21 and 2.22 , wesee that the carbon tax that equates the first order conditions associated with thesocial planners problem and the first order conditions associated with the competi-tive problem is

Mtq1x s l . 5.8 .t t

U c .t

Thus the above tax induces the optimal allocation. The optimal carbon tax isthe marginal loss of utility per ton of carbon converted to dollars. Note that theoptimal carbon tax increases directly with adjusted labor, again indicating the


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.sensitivity of the model to adjusted labor. Additionally, Eq. 5.7 implies thatthe optimal tax equals the marginal cost per unit of emissions reduction:

. tx s . 5.9 .t

D T . m t t

Given the optimal carbon tax, we can calculate the marginal damage peradditional person, or the optimal baby tax which is a proxy for population control

.policies . Consider the Bellmans equation from the social planning problem,modified to include an additional parameter which is the number of people in thebase year L :0

. .1y r 1y w x S ; L s max A U c q S; L . 5.10 . . . . 50 0k ,

Consider a small increase in population, but suppose the marginal person entersthe world with enough capital to leave the capital to labor ratio unchanged which

.allows us to focus directly on the externality . The marginal damage is thederivative of the value function with respect to an increase in the initial population.

M S ; L s S; L q S; L . 5.11 . . . .0 0 0

L M

L L0 0 0

.Here we evaluate the controls at the optimum. Equation 5.11 implicitly definesthe marginal damage from an extra person, which results from increased emissionsnow and into the infinite future, as well as more descendents who also increaseemissions in the future. Starting from an initial year of t s 1995 we recursivelysolve for the marginal damage:

Miq1iytq1S ; L s S ; L . 5.12 . . .t 0 iq1 0L M L0 iq1 0ist

.From Eq. 2.9 ,

M E Miq1 i is q 1 y . 5.13 . .M

L L L0 0 0

We also obtain M rL recursively:iq1 0

iM Eiq1 jiyjs 1 y . 5.14 . . M

L L0 0js0

We convert the damage from utility units to dollars per adjusted labor units bydividing by marginal utility and then to dollar units by multiplying by adjustedlabor:

Miq1S ; L .iq1 0 M Liq1 0iytq1MD s l . 5.15 . t

U c .tist


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.Substituting in Eq. 5.9 gives

M . i iq1iytq1MD s l 5.16 . t D T L .

m i i 0ist Miq1iytq1s x 5.17 . i

L0ist

i Ejiyjiytq1s 1 y x . 5.18 . . M iL0ist js0

Apparently, the marginal damage per additional person is the sum of themarginal additions to the stock of GHGs created by the additional person and alldescendents in each year, multiplied by the cost of the emissions, the discountedtax in each year. An extra person creates emissions into the infinite future, whileproduction creates emissions only in the current period. Thus the population taxmust consider emissions in the future, while the carbon tax considers only current

w xemissions. This is similar to the recent results of 4, 5 on population and stockexternalities. The population tax is thus quite large relative to the carbon tax,about $270 per person versus $6.87 per ton in the base case maximizing theRamsey welfare function. Of course, firms pay the optimal carbon tax each decade,

while consumers pay the population tax only once per lifetime. The present .discounted value of eight carbon tax payments starting in 1995 corresponds to

$35.10, still much less than the population tax. Table V gives the optimal taxes forboth CO and for population growth.2

The population tax reflects the fact that there are two ways to damage the globalenvironment: directly emitting GHGs and indirectly emitting GHGs by generatingmore people. Giving birth to one more person creates emissions from that

additional person as well as emissions from the progeny of that additional person.This suggests that an additional tool to use in managing the greenhouse problem ismanaging population growth. Indeed, a carbon tax alone is not sufficient to induceoptimal emissions, given the second implicit externality of population growth.Population growth can be managed in a variety of ways, including birth controlprograms, promotion of social security programs to reduce the reliance on childrenin old age, and development assistance to increase incomes and thus, presumably,birth rates. Table V suggests that if an additional birth can be avoided at a cost of

$200 to $800, then it is desirable to do so.

TABLE V

Efficient Tax Rates on Population and CO and Net Emissions per Person, 19952

Baby tax Carbon tax 1995 emissions . . .Welfare fn. $ per person $ per ton per person tonsl

Per-capita 0.31 $226.00 $3.30 0.7056Per-capita 0.16 $541.23 $3.70 0.6909Ramsey 0.31 $270.14 $6.87 0.7115Ramsey 0.16 $770.71 $10.90 0.7303


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Several caveats are appropriate here. Our model assumes a homogeneouspopulation and the baby tax in Table V reflects average world incomes. High birthrates occur in lower income countries where per capita GHG emissions are lower.Thus the efficient baby tax in lower income countries is undoubtedly lower than

the averages in Table V. Our model ignores all other externalities of populationgrowth. Population growth certainly generates positive externalities such as morerapid productivity growth by increasing the knowledge base of the economy. Inindustries with increasing returns, population growth reduces costs per capita. Wehave also neglected all other pollution externalities caused by population growth.

6. CONCLUSIONS

We have shown that the predictions of climate change models are very sensitiveto assumptions about the growth in population and productivity. Furthermore,integrated assessment models generally assume that population and productivity

will grow at an increasingly slower rate and consistently below historical rates.There are two possible justifications for slow growth in population.

w xForecasters of population growth such as 22 often predict population growth will slow due to an income effect. As developing countries grow and increaseincome, incentives for large families decline, as has happened in the developed

world. This implies that a possible policy response to climate change is to takesteps to help developing countries reduce population growth through developmentaid, rather than specifically limiting or taxing greenhouse gas emissions. This wouldhave numerous other benefits, such as helping to control other environmentalproblems brought about in part by a large population. This is supported by the highmarginal damage brought about by introducing one more person into the popula-tion. However, several problems might arise. First, population growth in someregions might be the result of other factors unrelated to income. Second, the cause

of heterogeneity in income growth around the world is unclear. As noted in thew xIntroduction, 18 and others have found the post-war record of progress in

economic development to be mixed at best. In any case, population might continueto increase in spite of a policy to help developing countries increase income.

But the most problematic part of the idea that higher per-capita income will slowpopulation growth from the perspective of the integrated assessment modeler is

that integrated assessment models assume productivity and therefore per capita.income also stops growing. In other words the world economy must experience

enough technological change to slow the growth of population, but not so muchgrowth that emissions rise substantially. .Perhaps some modelers have implicitly an alternative idea: that population

growth will slow or stop because, in a Malthusian way, the strain on environmentalresources will become too great to support population growth beyond a certainpoint. One of these environmental resources is the condition of the climate. Thuscausality plausibly can run in two directions for the relationship between popula-tion growth and climate change. Population growth affects the climate, as noted inthis paper, but increased climate change can also affect the size of the population.

Regardless of the reasoning for the assumption, there are two possible solutions.The first is to reduce the drain on resources per-capita with climate changepolicies. The second is to slow the total population growth so that the total drain


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on resources is not so great. Climate change models implicitly pursue both thestrategy of abatement and the strategy of limiting population growth. The socialplanner endogenously mitigates long term climate change by traditional abatement,but climate change is also reduced because population growth ends exogenously.

The latter effect is very strong: a population of 10 billion puts a strain on theclimate resource, but nothing like a population of 20 billion. A fruitful and morerealistic direction for further research would be to take this into account. Ifabatement reduces per-capita emissions, then perhaps conditions will result in alarger population; the net effect would be a larger asymptotic population and littlelong run effect on climate change.

Perhaps the only justification for the assumption that technological change willstop is the idea that in the long run all productivity advances are in non

.GHG-intensive industries e.g., services . Anecdotally improvements in productivitycome largely in areas of the economy which are not intensive in GHGs i.e., service

. w xindustries . As shown in 3 , productivity growth is often concentrated in a fewindustries which vary randomly from decade to decade. Thus, productivity growthin the next decade may again be in GHG-intensive industries. Although servicescompose much of recent productivity growth, manufacturing productivity has alsobeen strong. The strong productivity-driven growth in the 1990s has produced alarge increase in GHG emissions, due to higher consumption of gas and electricityby wealthier consumers.

Because climate change is a global problem and because electricity and gasconsumption generate much of the GHG emissions, developed countries cannoteasily export all GHG emissions. Finally, integrated assessment models are cali-brated such that a high percentage of growth of productivity results in no GHGemissions, which is consistent with current and historical data. Still, in the very longrun scarcity of fossil fuels and extensive regulation may result in greater improve-ments in emissions intensity than is observed in the past. A fruitful direction forfurther research is to include an endogenous model of technical change.

APPENDIX: PROOFS OF MAIN RESULTS

A.1. Proofs of Theorems which Sign the Deri ati es of the Value Function

We use a proof by induction technique to sign the derivatives. Let be aniy1initial guess of the value function. Let the next guess bei

w x S s max U S, , k q G S, , k . 7.19 4 . . . .i iy1k ,

Given the assumptions, for any continuously differentiable , iterations on the0above equation converge to the unique value function.

.PROPOSITION 1. Let r ) 0 and , , r g 0, 1 . Then the partial deriati es of2 M T 3 .the alue function S with respect to M, T, and O are negati e.

We need only show that

S .iy1- 0 i s 2, 3, 4 7.20 .

Si


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implies that

S .i- 0 i s 2, 3, 4. 7.21 .

Si

To find the derivative of the value function with respect to the stock of GHGs, we .evaluate Eq. 7.19 at the optimum s * and k sk*. Taking the derivative of

the resulting equation with respect to M gives

S G S, *, k* T G S, *, k* . . . . .i iy1 iy1s 1 y q . .M

M M M T

7.22 .

Since the derivative of temperature change with respect to the stock of GHGs ispositive, the above equation implies that if the derivatives of with respect toiy1

M and T are less than zero, then the derivative of with respect to M is negative.iBecause is the unique, globally stable fixed point of the Bellmans equation, wehave by induction

S .- 0. 7.23 .

M

For the derivative of the value function with respect to temperature, we again .evaluate Eq. 7.19 at the optimal controls and take the derivative with respect to

T. We have

S U C* C . .i 1y.r1y.sAT C T

G S, *, k* G S, *, k* . . . .iy1 iy1

q 1 y q r . .T 3T O7.24 .

Here C* is the consumption when the controls are evaluated at the optimum. Notethat derivative of consumption with respect to T is negative, r is positive, and30 - - 1. Hence the above equation implies that if the derivative of withT iy1respect to T and O is negative, then the derivative of with respect to T isinegative. Hence using the assumption of the proposition we have by induction

S .- 0. 7.25 .

T

The derivative of the value function with respect to O is

S G S, *, k* G S, *, k* . . . . .i iy1 iy1s r q 1 yr . 7.26 . .2 3

O T O

Here r is positive and 0 -r - 1. Hence the above equation implies that if the2 3derivative of with respect to T and O is negative, then the derivative of iy1 i

with respect to O is negative. Hence using the assumption of the proposition we


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have by induction

S .- 0, 7.27 .

O

which completes the proof.Now that we have established the sign of the derivatives with respect to the

environmental states, we can establish the sign of the derivatives with respect tothe economic states. The proof requires a slightly more subtle argument, sincethere are terms with opposite signs in the derivative. Essentially, an increase in kor causes an increase in production, but also an increase in pollution. Therefore,the strategy is to increase the control enough so that pollution is unchanged, andthen show that after paying the cost extra production exists which can be con-

sumed. We omit the proof of the sign of the derivative with respect to , since it isanalogous to the proof with respect to k.

PROPOSITION 2. Suppose b ) 1 and 0 -b - 1. Then the partial deri ati e of2 1the alue function with respect to k is positi e.

Proof. Suppose the contrary; then for some k there exists an ) 0 such that

S k q - S k . 7.28 . . . . .

Since all states are constant except for k, for ease of notation we drop the S and . just use k . Now at the optimum, we have

k sA1y.r1y.U k , kXU , U q G k , kXU , U 7.29 . . . . .1 1 1 1

k q sA1y.r1y.U k q , kXU , U q G k q , kXU , U . 7.30 . . . . .2 2 2 2

Here U is the optimal abatement when the capital level is k and U is the1 2optimal abatement when the capital level is k q and the same for the investment

decision. By the definition of the maximum, we have

k q GA1y.r1y.U k q , kXU , U q G k q , kXU , U . 7.31 . . . . .1 2 1 2

XU . XU U .We now choose a level of abatement so that G k q , k , s G k, k , . 1 1 1We equate the next period stock of GHGs to achieve the equality:

1y X XU M s 1 y k l q 1 y MsM . . . .1 M 1 M 2 1ys 1 y k q l q 1 y M 7.32 . . . . . .M M

kU

s 1 y 1 y . 7.33 . . 1 /k q Now by definition of the maximum we have

k q GA1y.r1y.U k q , kXU , q G k q , kXU , 7.34 . . . . .1 1

sA1y.r1y.U k q , kXU , q G k , kXU , U 7.35 . . . .1 1 1XU XU U1y.r1y.s A U kq , k , y U k , k , q k . 7.36 . . . .1 1 1


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Hence, if we can show that the difference in utilities is non-negative, we have a .contradiction of 7.28 . Since utility is strictly increasing, we have

U k q , kXU , G U k , kXU , U . .1 1 1

if and only if

. 1y XUk q l q 1 y k q yk . . . .K 1

D T .

U .1 1y XUG k l q 1 y k yk . 7.37 . . .K 1

D T .

Hence it is sufficient to show

U k q G k . 7.38 . . . . 1

Plugging in for the cost function and and rearranging give

b 2k k

U Ub 21 Gb 1 y 1 y q 1 yb . 7.39 . . .1 1 1 1 / / /k q k q Now it is easy to show that given the assumptions of the proposition, the derivativeof the right hand side of the above equation with respect to is strictly positive.Hence the maximum value of the right hand side occurs at s 1. Substituting infor s 1 gives

k1 Gb q 1 yb . 7.40 . .1 1 /k q

According to the assumptions of the proposition, the above equation holds sincethe ratio of capital stocks is less than one. Hence the difference in utilities ispositive, which gives the contradiction.

A.2. Deri ation of Stationary Values

We next derive the derivatives of the value function at the stationary point

S s S s S. Clearly s , with l and taking on their limiting values. Let idenote the partial derivative of the value function with respect to state i, evaluatedat the stationary point S. The first order conditions at the stationary value of thestates and controls are

C M1y.r1y.A U C q s 0 7.41 . .M

1y.r1y.yA U C q s 0. 7.42 . .K

Here the last equation follows because the growth rate of adjusted labor is zero atthe steady state.


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.The derivatives of the value function at the optimal controls with respect to thestates evaluated at the stationary state are

C M1y.r1y. sA U C q 7.43 . .K M

k kT

s 1 y q 7.44 . . M M M T M

C1y.r1y. sA U C q 1 y q r 7.45 . . .T T T 3 O

T

s r q 1 yr . 7.46 . .O 2 T 3 O

. .Equations 7.43 7.46 form a linear system of four equations and four un-knowns. The solution is then

C C M T1y.r1y. 1y.r1y. sA U C q A U C 7.47 . . .k 3

k T k M

T C1y.r1y. s A U C 7.48 . .M 3

M T

C1y.r1y. s A U C 7.49 . .T 2

T

C1y.r1y. s A U C , 7.50 . .O 1 2

T

where

r2 s 7.51 .1 1 y 1 yr .3

1 s 7.52 .2 1 y 1 y y r .T 3 1

2 s . 7.53 .3 1 y 1 y .M

Substituting the derivatives of the value function with respect to the states intothe first order conditions gives

C M T Cq s 0 7.54 .3

M T

C C M T2y1 q q s 0. 7.55 .3

k T k M

To complete the model, we use the equations that govern the change in GHGlevels, atmospheric temperature, and ocean temperature, evaluated at the station-


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ary points:

M 1yMs 1 y k l 7.56 . .M

r M1Ts log 1 q 7.57 . / yr MT 2 b

O s T. 7.58 .

.After substituting out for the ocean temperature, the first order conditions 7.54 . . .and 7.55 and the constraints 7.56 and 7.57 form a non-linear system of four

equations with unknowns k, M, T, and . The numerical solution to the abovesystem is analyzed in Section 3.1.

APPENDIX B: LIST OF SYMBOLS

coefficient of relative risk aversion capital shareb cost function parameter

1b convexity of cost function2 damage from a doubling of greenhouse gasses1 convexity of damage2 atmospheric retention ratiom decline rate in exogenous emissions intensity decline

depreciation of greenhouse gassesM depreciation rate of capitalK depreciation of atmospheric temperatureTr climate sensitivity parameter1r warming parameter2r warming parameter3

.M natural non-anthroprogenic greenhouse gas concentrationb discount rateg growth rate in labor-efficiency unitslg growth rate in emissions intensity of output

l labor-efficiency units

K capital stockM greenhouse gas concentrationT atmospheric temperatureO ocean temperature position of exogenous variablesQ gross outputY net outputC consumption

E emissionsL population growth parameterA productivity parameter value function


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W indirect utilityS vector of state variablesG constraintsMD marginal damage

x carbon tax

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.Climate Change, pp. 37, Stanford University 1995 . .3. A. Harberger, A vision of the growth process, Amer. Econom. Re . 88, No. 1, 132 1998 .

4. J. D. Harford, Stock pollution, child-bearing externalities, and the social discount rate, J. Eniron. .Econom. Management 33, 94105 1997 . .5. J. D. Harford, The ultimate externality, Amer. Econom. Re . 88, 260265 1998 .

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.7. J. Houghton, G. Jenkins, and J. Ephraums Eds. , Climate Change: The IPCC Scientific Assess- .ment, Cambridge Univ. Press, Cambridge, UK 1992 .

.8. C. Jones, Time series tests of endogenous growth models, Quart. J. Econom. 110, 495525 1995 .9. D. L. Kelly and C. D. Kolstad, Bayesian learning, pollution, and growth, J. Econom. Dynam. Control

.23, No. 4, 491518 1999 .

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12. A. Manne, R. Mendelsohn, and R. Richels, Merge: A model for evaluation regional and global .effects of ghg reduction policies, Energy Policy 23, No. 1, 1734 1993 .

13. A. Manne and R. Richels, CO hedging strategiesthe impact of uncertainty on emissions,2 presented at the OECDrIEA Conference on the Economics of Climate Change, Paris 1416June 1993.

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.hedging strategies, Energy J. 16, No. 4, 137 1995 .15. W. Nordhaus and G. Yohe, Future carbon dioxide emissions from fossil fuels, in Changing

.Climate, National Academy Press, Washington, DC 1983 .16. W. D. Nordhaus, Managing the Global Commons: The Economics of Climate Change, MIT

.Press, Cambridge, MA 1994 .17. D. North, Institutions, Institutional Change, and Economic Performance, Cambridge Univ. Press,

.Cambridge, MA 1990 .18. D. North, Some fundamental puzzles in economic historyrdevelopment, mimeo, Washington

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Malthus and Climate Change:Betting on a Stable Population

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