energy taxes and endogenous technological change ...staff/wrkshop_papers/2006... · tions of the us...

Energy taxes and endogenous technological change(Preliminary: Please do not cite)

Pietro F. Peretto∗

Department of EconomicsDuke University

November 27, 2006

Abstract

This paper studies the effects of a Pigouvian tax on energy useon the growth path of a model economy where market structure is en-dogenous and jointly determined with the rate of technological change.Because this economy does not exhibit the scale effect (a positive rela-tion between growth and aggregate effort in R&D), the tax has no effecton the economy’s steady-state growth rate. It has, however, importanttransitional effects that give rise to surprising results. Specifically, un-der plausible conditions a small tax can raise long-run consumptionand welfare despite the fact that — in line with standard intuition — itlowers consumption in the short run.

Keywords: Endogenous Growth, Market Structure, Energy Taxes,Environment.JEL Classification Numbers: E10, L16, O31, O40

∗Address: Department of Economics, Duke University, Durham, NC 27708. Phone:(919) 6601807. Fax: (919) 6848974. E-mail: [email protected].

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1 Introduction

This paper studies the effects of an energy tax on the growth path of amodel economy where market structure is endogenous. Of particular in-terest is the interaction between changes in the inter-industry allocationof resources between manufacturing and energy, and the intra-industry ef-fects that characterize the manufacturing sector. The latter are importantbecause the manufacturing sector is the engine of growth of the economy.

There are several reasons why such an analysis is worthwhile.

• The recent spike in the price of oil prompted a debate about the short-and long-run prospects of the economy. Despite, the tone of impend-ing doom that some of the commentary took, the economy seems tohave taken the hit in stride. The price of oil seems now to be headingdown to its level of 2-3 years ago. Accordingly, the debate has shiftedits focus away from rising gas prices to some other feature of the econ-omy. The question, however, remains. What does the price of oil doto the economy in the short and long run? The evidence is mixed.Hamilton (1988, 2003) argues that it explains most of the fluctua-tions of the US economy; Barsky and Killian (2002, 2004) and Killian(2006a, 2006b), in contrast, argue that it matters very little. It is fairto say, however, that the conventional wisdom outside of economicsis in line with Hamilton’s view — that is, energy prices, in general,and the price of oil, in particular, drive business cycle fluctuations andlong-run growth.

• One of the arguments usually triggered by a spike in the price ofoil is the need to reduce dependence on foreign supply. In light ofthe war in Iraq and the escalating tension with some large oil pro-ducers in the Arab world (e.g., Iran) and in Latin America (e.g.,Venezuela), this argument acquires more and more traction for po-litical and international-affairs reasons. The question, of course, ishow to accomplish a reduction of the dependence on foreign oil over areasonable time-frame without harming the economy.

• Last but not least, there is the issue of pollution and global warming.Energy production and use is one of the main sources of harmful emis-sions, in general, and the main source of CO2 emissions, in particular.CO2, in turn, is the main factor in the building up of greenhouse gasesthat according to the growing consensus among natural scientists drivethe rise in global temperature. The issue is far from settled, of course,

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and there is sharp disagreement between those who argue that globalwarming is man-made and those who argue that it is not, or that theprojected costs of reducing it far outweigh the benefits; see, e.g., theheated debate following the recent release of the Stern Report by theUK government. Nevertheless, the notion that reducing pollution isdesirable, and therefore that appropriate interventions in the energysector are called for, is widely accepted.

In summary, there is ample motivation for studying the role of energyprices in the economy. In light of the stated goal of reducing the economy’senergy intensity (the ratio of energy use to GDP) without inflicting undueharm, moreover, understanding the role of specific instruments like energytaxes becomes very important. Perhaps surprisingly, the literature has notpushed hard in trying to bring to bear on these issues the power of modernendogenous growth theory. [Need a footnote here with references anda short statement about what is done and what is left out.] Thispaper tries to fill this gap.

[Need here transition statement from literature review to whatI do. Also elaborate on the two main points below to provide ashort description of mechanism and discuss the intuition.]

The main points of the paper are:

• The reallocation of resources from energy to manufacturing does notraise long-run (steady-state) growth because there is no scale effect.There are, however, transitional effects that generate important leveleffects.

• There exist technological offsets that reduce the cost of the policyin terms of real expenditure. Under plausible conditions a small taxraises long-run real expenditure despite the fact that it lowers it in theshort run.

The paper is organized as follows. In Section 2, I discuss the setup ofthe model. In Section 3, I construct the equilibrium of the market economy.In Section 4, I discuss the dynamic effects of the energy tax in a small openeconomy fully dependent on foreign supply of oil. In Section 5, I discuss thecase of a closed economy that must rely on domestic supply and thus facesscarcity. I conclude in Section 6.

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2 The model

2.1 Overview

I consider an economy populated by a representative household that supplieslabor services inelastically in a competitive market. The household can alsofreely borrow and lend in a competitive market for financial assets.

Manufacturing firms hire labor to produce differentiated consumptiongoods, undertake R&D, or, in the case of entrants, set up operations. Inaddition to labor, production of consumption goods requires energy, whichis supplied by a separate, competitive sector. Energy use and productiongenerate pollution. The government taxes energy purchases and returns theproceeds in a lump-sum fashion to the household.1

The economy starts out with a given range of goods, each supplied byone firm. The household values variety and is willing to buy as many dif-ferentiated goods as possible. Entrepreneurs compare the present value ofprofits from introducing a new good to the entry cost. They only targetnew product lines because entering an existing product line in Bertrandcompetition with the existing supplier leads to losses.

Once in the market, firms establish in-house R&D facilities to producea stable flow of cost-reducing innovations. As each firm invests in R&D, itcontributes to the pool of public knowledge and reduces the cost of futureR&D. This allows growth at a constant rate in steady state, which is reachedwhen the economy settles into a stable industrial structure.

2.2 Households

The representative household maximizes lifetime utility

U(t) =

Z ∞

te−(ρ−λ)(s−t) log u(s)ds, ρ > λ > 0 (1)

subject to the flow budget constraint

A = rA+WL− POO + τE − Y, τ ≥ 0 (2)

where ρ is the discount rate, λ is population growth, A is assets holding,r is the rate of return on financial assets, W is the wage rate, L = L0e

λt,L0 ≡ 1, is population size, which equals labor supply since there is nopreference for leisure, and Y is consumption expenditure. In addition to

1This ensures that the government balances the budget without introducing feedbackeffects that are not the focus of this paper.

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asset and labor income, the household receives the lump-sum rebate of theenergy tax revenues, τE, where τ is a per-unit tax and E is aggregate energyuse. It also pays POO to foreigners for oil imports.

The household has instantaneous preferences over a continuum of differ-entiated goods,2

log u = log

"Z N

0

µXi

L

¶ −1

di

# −1

, > 1 (3)

where is the elasticity of product substitution, Xi is the household’s pur-chase of each differentiated good, and N is the mass of goods (the mass offirms) existing at time t.

The solution for the optimal expenditure plan is well known. The house-hold saves if assets earn the reservation rate of return

r = rA ≡ ρ+Y

Y− λ (4)

and taking as given this time-path of expenditure maximizes (3) subject toY =

R N0 PiCidi. This yields the demand schedule for product i,

Xi = YP−iR N

0 P 1−i di. (5)

With a continuum of goods, firms are atomistic and take the denominatorof (5) as given; therefore, monopolistic competition prevails and firms faceisoelastic demand curves.

2.3 Manufacturing: Production and Innovation

The typical firm produces one differentiated consumption good with thetechnology

Xi = Zθi F (LXi − φ,Ei) , 0 < θ < 1, φ > 0 (6)

where Xi is output,3 LXi is production employment, φ is a fixed laborcost, Ei is energy use, and Zθ

i is the firm’s TFP, a function of the stock

2For simplicity I omit a term representing preference for environmental quality. Thereason is that it is not necessary to develop my argument about energy taxes since I focuson the response of the decentralized market equilibrium and not on the socially optimalpolicy. I discuss the role of pollution externalities in Section 5.

3To simplify notation, I use Xi to denote both the demand for and the production ofgood i.

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of firm-specific knowledge Zi. The function F (·) is a standard neoclassicalproduction function homogeneous of degree one in its arguments. Hence,(6) exhibits constant returns to rival inputs, labor and energy, and overallincreasing returns. The associated total cost is

Wφ+ CX(W,PE + τ)Z−θi Xi, (7)

where the function CX (·) is a standard unit-cost function homogeneousof degree one in its arguments. The elasticity of unit cost reduction withrespect to knowledge is the constant θ.

The firm accumulates knowledge according to the R&D technology

Zi = αKLZi , α > 0 (8)

where Zi measures the flow of firm-specific knowledge generated by an R&Dproject employing LZi units of labor for an interval of time dt and αK is theproductivity of labor in R&D as determined by the exogenous parameter αand by the stock of public knowledge, K.

Public knowledge accumulates as a result of spillovers. When one firmgenerates a new idea to improve the production process, it also generatesgeneral-purpose knowledge which is not excludable and that other firms canexploit in their own research efforts. Firms appropriate the economic re-turns from firm-specific knowledge but cannot prevent others from usingthe general-purpose knowledge that spills over into the public domain. For-mally, an R&D project that produces Zi units of proprietary knowledge alsogenerates Zi units of public knowledge. The productivity of research is de-termined by some combination of all the different sources of knowledge. Asimple way of capturing this notion is to write

K =

Z N

0

1

NZidi,

which says that the technological frontier is determined by the averageknowledge of all firms.4

The R&D technology (8), combined with public knowledge K, exhibitsincreasing returns to scale to knowledge and labor, and constant returns toscale to knowledge. This property makes constant, endogenous steady-stategrowth feasible.

4For a detailed discussion of the microfoundations of a spillovers function of this class,see Peretto and Smulders (2002).

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2.4 The Energy Sector — and the rest of the world

Energy firms hire labor, LE, to extract energy from natural resources (e.g.carbon, oil, gas), O. The energy-generation technology is E = G (LE, O),where G (·) is a standard neoclassical production function homogeneous ofdegree one in its arguments. The associated total cost is

CE (W,PO)E, (9)

where CE is a standard unit-cost function homogeneous of degree one in thewage W and the price of resources PO.

To fix ideas, I refer to resources as “oil” and assume that the domesticsupply of oil is zero. Therefore, energy producers tap the world marketand face an infinitely elastic supply at price PO. In other words, I thinkof this as a small open economy fully dependent on the world oil market.Corresponding to oil purchases, then, there is a flow of payments to the restof the world.

For simplicity, I ignore international assets flows and any other sort ofinteraction with the rest of the world, e.g., technological spillovers, import-export of goods other than the exchange of labor services (my numeraire,see below) for oil, and so on.5

This is the simplest way to model the energy sector for the purposes ofthis paper. Energy is produced with labor and natural resources purchasedat a given price in the world market. The energy sector competes for laborwith the manufacturing sector. This captures the fundamental inter-sectoralallocation problem faced by this economy. Energy purchases are taxed on aper-unit basis. This affects manufacturers’ production costs, their demandfor energy and labor, and the general equilibrium path of the economy.

3 Equilibrium of the Market Economy

This section constructs the symmetric Nash equilibrium of the manufactur-ing sector. It then characterizes the equilibrium of the energy sector. Fi-nally, it imposes general equilibrium conditions to determine the aggregatedynamics of the economy. The wage rate is the numeraire, i.e., W ≡ 1.

5 It is also possible to think of this as a small open economy that takes as given theworld interest rate. Since the model has the property that the domestic interest ratejumps to its steady state level, given by the domestic discount rate, as long the small openeconomy has the same discount rate as the rest of the world the equilibrium discussed inthe paper displays the same properties as an equilibrium with free financial flows.

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3.1 Equilibrium of the Manufacturing Sector

3.1.1 Incumbents

The typical manufacturing firm maximizes the present discounted value ofnet cash flow,

Vi (t) =

Z ∞

te−

R st r(v)vΠi(s)ds.

Using the cost function (7), instantaneous profits are

Πi = [Pi − CX(1, PE + τ)Z−θi ]Xi − φ− LZi ,

where LZi is R&D expenditure. Vi is the value of the firm, the price ofthe ownership share of an equity holder. The firm maximizes Vi subject tothe R&D technology (8), the demand schedule (5), Zi(t) > 0 (the initialknowledge stock is given), Zj(t

0) for t0 ≥ t and j 6= i (the firm takes asgiven the rivals’ innovation paths), and Zj(t

0) ≥ 0 for t0 ≥ t (innovation isirreversible). The solution of this problem yields the (maximized) value ofthe firm given the time path of the number of firms.

To characterize the typical firm’s behavior, consider the Current ValueHamiltonian

CVHi = [Pi − CX(1, PE + τ)Z−θi ]Xi − φ− LZi + ziαKLZi ,

where the costate variable, zi , is the value of the marginal unit of knowledge.The firm’s knowledge stock, Zi, is the state variable; R&D investment, LZi ,and the product’s price, Pi, are the control variables. Firms take the publicknowledge stock, K, as given.

Since the Hamiltonian is linear, one has three cases. The case 1 > ziαKimplies that the value of the marginal unit of knowledge is lower than itscost. The firm, then, does not invest. The case 1 < ziαK implies thatthe value of the marginal unit of knowledge is higher than its cost. Sincethe firm demands an infinite amount of labor to employ in R&D, this caseviolates the general equilibrium conditions and is ruled out. The first orderconditions for the interior solution are given by equality between marginalrevenue and marginal cost of knowledge, 1 = ziαK , the constraint on thestate variable, (8), the terminal condition,

lims→∞ e−

R st r(v)vzi(s)Zi(s) = 0,

and a differential equation in the costate variable,

r =zizi+ θCX(1, PE + τ)Z−θ−1i

Xi

zi,

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that defines the rate of return to R&D as the ratio between revenues fromthe knowledge stock and its shadow price plus (minus) the appreciation(depreciation) in the value of knowledge. The revenue from the marginalunit of knowledge is given by the cost reduction it yields times the scale ofproduction to which it applies. The price strategy is

Pi = CX(1, PE + τ)Z−θi − 1 . (10)

Peretto (1998, Proposition 1) shows that under the restriction 1 > θ ( − 1)the firm is always at the interior solution, where 1 = ziαK holds, andequilibrium is symmetric.

The cost function (7) gives rise to the conditional factor demands:

LXi =∂CX(W,PE + τ)

∂WZ−θi Xi + φ;

Ei =∂CX(W,PE + τ)

∂ (PE + τ)Z−θi Xi.

It is useful to define the shares of the firm’s variable costs due to labor andenergy:

SLX ≡

WL

CX(1, PE + τ)X=

∂ logCX(W,PE + τ)

∂ logW;

SEX ≡

(PE + τ)E

CX(1, PE + τ)X=

∂ logCX(W,PE + τ)

∂ log (PE + τ).

Then, the price strategy (10), symmetry and aggregation across firms yield:

LX = Y− 1

SLX + φN ; (11)

E = Y− 1 SE

X

PE + τ. (12)

Note that SLX +SE

X = 1. Since the wage is normalized to 1, these shares arefunctions of the after-tax price of energy only.

Also, in symmetric equilibrium K = Z = Zi yields K/K = αLZ/N ,where LZ is aggregate R&D. Taking logs and time derivatives of 1 = ziαKand using the demand curve (5), the R&D technology (8) and the pricestrategy (10), one reduces the first-order conditions to

r = rA ≡ α

·Y θ( − 1)

N− LZ

N

¸. (13)

This defines the rate of return to cost reduction.

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3.1.2 Entrants

To characterize entry, I assume that upon payment of a sunk cost βPiXi inunits of labor, an entrepreneur can create a new firm that starts out its ac-tivity with productivity equal to the industry average.6 Once in the market,the new firm implements price and R&D strategies that solve a problemidentical to the one outlined above. Hence, entry yields value Vi. Takinglogs and time-derivatives of Vi yields

r =ΠiVi+

ViVi,

which is a perfect-foresight, no-arbitrage condition for the equilibrium of thecapital market. It requires that the rate of return to firm ownership equalthe rate of return to a loan of size Vi. The rate of return to firm ownership isthe ratio between profits and the firm’s stock market value plus the capitalgain (loss) from the stock appreciation (depreciation).

In symmetric equilibrium the demand curve (5) yields that the cost ofentry is β Y

N . The corresponding dmand for labor in entry is LN = Nβ YN .

The case V > β YN yields an unbounded demand for labor in entry, LN =

+∞, and is ruled out since it violates the general equilibrium conditions.The case V < β Y

N yields LN = −∞, which means that the non-negativityconstraint on LN binds and N = 0. A free-entry equilibrium requires V =β YN . Using the price strategy (10), the rate of return to entry becomes

r = rN ≡ 1

β

·1 − N

Y

µφ+

LZ

N

¶¸+ Y − N . (14)

The dividend price ratio in this rate of return depends on the gross profitmargin 1 . Anticipating one of the properties of the equilibria that I studybelow, note that in steady state the capital gain component of this rate ofreturn, Y − N , is zero. Hence, the feasibility condition 1 > rβ must hold.This simply says that the firm expects to be able to repay the entry costbecause it more than covers fixed operating and R&D costs.

Interestingly, neither the return to cost reduction in (13) nor the returnto entry in (14) depend on factors related to the energy market. It is useful toreview why. The production technology (6) yields a unit-cost function thatdepends only on input prices and is independent of the quantity producedand thus of inputs use. Since the pricing rule (10) features a constant markupover unit cost, the firm’s gross-profit ratio (revenues minus variable costs

6See Peretto and Connolly (2004) for an interpretation of this assumption.

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divided by revenues) is independent of input prices. Equations (13) and(14), then, capture the idea that investment decisions by incumbents andentrants do not respond directly to conditions in the energy market becausethey are guided by the gross-profit ratio. The state of the energy markethas at most an indirect effect through aggregate spending Y .

3.2 Equilibrium of the Energy Sector

Given the cost function (9), competitive energy producers operate along theinfinitely elastic supply curve

PE = CE (1, PO) . (15)

In equilibrium, then, energy production is given by (12) evaluated at thispre-tax price. Defining the share of oil in energy costs as

SOE ≡

POO

CE (1, PO)E=

∂ logCE(W,PO)

∂ logPO,

I can write the associated demands for labor and oil as:

LE = E∂CE(W,PO)

∂W= Y

− 1 PESEX

PE + τ

¡1− SO

E

¢; (16)

POO = E∂CE(W,PO)

∂PO= Y

− 1 PESEX

PE + τSOE . (17)

The share SOE depends only on the exogenous price of oil. (Recall that OPO

is a flow of payments to foreign suppliers.)

3.3 General equilibrium

The model consists of the returns to saving, cost reduction and entry in (4),(13) and (14), the labor demands (11), (16), and the household’s budgetconstraint (2).7 Since the latter implies labor market clearing, equilibrium

7It is immediate to see that the household’s budget constraint in (2) is in fact theeconomy’s resource constraint since it includes the transfer of energy tax revenues from thegovernment and the flow of payments to foreigners for oil imports. Using the expressionsfor factor prices derived so far and the normalization W ≡ 1, this constraint becomes thelabor market clearing condition

L = LX + LZ + LE + LN + LO,

where LN = βY · N and LO = POO. The last term is the amount of labor income (i.e.,labor services) that the economy gives to foreigners to pay for oil imports.

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of the goods market follows from the fact that firms choose a point on theirdemand curves, and I have already imposed energy market equilibrium, toconstruct the general equilibrium of the economy, I now only need to lookat the financial market.

Assets market equilibrium requires equalization of all rates of return (no-arbitrage), r = rA = rZ = rN , and that the value of the household’s portfolioequal the value of the securities issued by firms, A = NV = βY . Thus, theeconomy features a constant wealth to expenditure ratio. This propertyand log utility deliver a result that simplifies dramatically the analysis ofdynamics. Substituting A = βY into (2) and using the rate of return tosaving in (4), I obtain

0 = β (ρ− λ) +L− POO + τE − Y

Y.

The equilibrium conditions for the energy market — in particular, the oildemand schedule (17) — yield

Y

L=

1

1− β (ρ− λ) +¡PESO

E − τ¢EY

≡ y∗, (18)

whereE

Y=− 1 SE

X

PE + τ.

This term depends only on parameters and the exogenous price of oil PO.Since y∗ is constant, then, the interest rate is r = ρ at all times.

The denominator of (18) identifies two important channels through whichthe energy tax affects expenditure. The first is the tax-revenue effect, cap-tured by the term −τ EY . This says that because the household receivesthe lump-sum rebate of energy tax revenues, expenditure on comsumptiongoods rises with τ as long as the economy operates on the upward slopingportion of the tax revenue curve. The second channel is the import-savingeffect, captured by the term PES

OEEY . This says that because the tax re-

duces energy intensity EY , it reduces demand for oil and therefore the flow of

payments to foreigners. This frees up resources for domestic consumption.

3.4 Dynamics

Because population grows, it is useful to work with the variable n ≡ NL . The

results just derived allow me to solve (8) and (13) for

Z = αLZ

N=

y∗

n

αθ ( − 1) − ρ. (19)

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An important feature of this equation is that LZN = 0 for

n ≥ n ≡ y∗αθ ( − 1)

ρ.

This is the familiar effect of the number of firms on the individual firm’smarket share and thus on the incentives to do R&D.8

I now substitute these results into (14) to obtain

n =

1β

h1−θ( −1) − ¡φ− ρ

α

¢ny∗

i− ρ n < n

1β

h1 − φ n

y∗

i− ρ n ≥ n

.

The general equilibrium of the model thus reduces to a single differentialequation in the mass of firms per capita. Figure 1 illustrates dynamics.9 Ifαφ > ρ, the entry rate is always falling. In contrast, if αφ ≤ ρ the entryrate is initially rising or constant, until the economy crosses the threshold nwhen the entry rate starts falling. In all cases, the economy converges to

n∗ =

1−θ( −1)−ρβ

φ− ρα

y∗ for1−θ( −1)−ρβ

φα−ρ < θ( −1)ρ

1−ρβφ y∗ otherwise

. (20)

These dynamics make clear that φ > 0 kills the possibility of endogenousgrowth through product proliferation because the term φN on the righthand side of the resources constraint implies that the equation cannot holdfor a given labor endowment if N grows too large.10

The solutions in (20) exist only if the feasibility constraint 1 > ρβ holds.The interior steady state with both vertical and horizontal R&D requiresthe more stringent conditions αφ > ρ and

ρβ +θ ( − 1)

<1< ρβ +

αφ

ρ

θ ( − 1).

It then yieldsy∗

n∗=

φ− ρα

1−θ( −1) − ρβ(21)

8See Peretto (1998, 1999) for a detailed discussion of this property in this class ofmodels.

9For simplicity I ignore the non-negativity constraint on N . I can do so without loss ofgeneraility because population growth implies that the mass of firms grows all the time.See Peretto (1998) for a discussion of this property.10See Peretto and Connolly (2004) for a detailed discussion of this property in Schum-

peterian models of endogenous growth.

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so that

Z∗ =φα− ρ

1−θ( −1) − ρβ

θ ( − 1) − ρ. (22)

Notice how the steady-state growth rate of productivity in manufacturingis independent of conditions in the energy market because the sterilizationof the scale effect implies that it does not depend on the size of the manu-facturing sector and therefore on the inter-sectoral allocation of labor.

To perform experiments, I shall focus on this region of parameter spaceand work with the equation

n = ν −³φ− ρ

α

´ n

βy∗, ν ≡ 1− θ ( − 1)

β− ρ.

This is a logistic equation (see, e.g., Banks 1994) with growth coefficientν and crowding coefficient

¡φ− ρ

α

¢1

βy∗ . Using the value n∗ in (20) — also

called carrying capacity — I can rewrite it as

n = ν³1− n

n∗´, (23)

which has solution

n (t) =n∗

1 + e−νt³n∗n0− 1´ , (24)

where n0 is the initial condition.

4 The effects of the energy tax

In this section I analyze the effects of the energy tax. I begin with a dis-cussion of the conditions under which it raises consumption expenditure sothat the market for manufacturing goods expands. Next, I show how the re-allocation of resources associated to that expansion affects the path of TFP,the CPI and therefore of (real) consumption per capita. Finally, I show thatwelfare is a hump-shaped function of the tax.

4.1 Expenditure

The first step in the evaluation of the effects of the energy tax on growthand welfare is to assess its effect on expenditure. The following property ofthe energy demand function (12) is quite useful.

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Lemma 1 LetEX ≡ −

∂ logE

∂ log (PE + τ).

Then, EX ≤ 1 if

∂SEX

∂ (PE + τ)≥ 0,

which is true if∂LE

∂ (PE + τ)≤ 0,

that is, if labor and energy are gross complements in (6).

Proof. See the Appendix.

In words, this says that energy demand is inelastic, i.e., EX ≤ 1, when the

energy share of manufacturing cost, SEX , is non-decreasing in the after-tax

price of energy. Energy demand, conversely, is elastic when the the energycost share in manufacturing is decreasing in PE + τ . The effect of the after-tax price of energy on the energy cost share, in turn, depends on whetherlabor and energy are gross complements or gross substitutes. I now use thisresult to derive one of the key ingredients for the analysis of the growth andwelfare effects of the tax.

Proposition 2 Assume that the production technology (6) exhibits grosscomplementarity between labor and energy so that energy demand is inelas-tic, E

X ≤ 1. Then, y∗ (τ) is a monotonically increasing function with domainτ ∈ [0,∞) and codomain [y∗ (0) , y∗ (∞)), where:

y∗ (0) =1

1− β (ρ− λ) + −1SOES

EX (0)

;

y∗ (∞) = 1

1− β (ρ− λ)− −1SEX (∞)

, SEX (∞) = 1.

Assume, in contrast, that labor and energy are gross substitutes so that EX >

1. Then, y∗ (τ) is a hump-shaped function of τ with the same domain andcodomain as before except that SE

X (∞) = 0.


The mechanism driving this result highlights the importance of the sub-stitution possibilities between labor and energy in manufacturing. If labor

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and energy are gross complements, higher manufacturing employment re-quires higher energy use, and this dampens the negative effect on energydemand of the increase in the after-tax price of energy. If, in contrast, laborand energy are gross substitutes, higher manufacturing employment requireslower energy use and thereby amplifies the negative effect of the after-taxprice increase on energy demand.

To see this property in sharper detail, it is useful to consider the followingexample.

Example 3 Consider a CES production function of the form

Xi = Zθi [(LXi − φ)σ +Eσ

i ]1σ , σ ≤ 1

As is well known, this contains as special cases the linear production function(σ = 1) wherein inputs are perfect substitutes, the Cobb-Douglas (σ = 0)wherein the elasticity of substitution between inputs is equal to 1, and theLeontief (σ = −∞) wherein inputs are perfect complements. The associatedunit-cost function is

CXi = Z−θi

hW

σσ−1 + (PE + τ)

σσ−1iσ−1

σ.

From this one derives (recall that W ≡ 1):

SEX =

1

1 + (PE + τ)σ

1−σ;

EX = 1 +

σ

1− σ

1

1 + (PE + τ)−σ

1−σ= 1 +

σ

1− σ

¡1− SE

X

¢.

Hence, σ ≤ 0 yields EX ≤ 1 and dy∗

dτ > 0 for all τ . In contrast, 0 < σ ≤ 1yields E

X > 1 and dy∗dτ > 0 for

τ < τ ≡ PE1 + SO

EEX

EX − 1

= PE

·1− σ

σ

1 + SOE

1− SEX

+ SOE

¸.

The main message of Proposition 1 is that expenditure rises with thetax when energy demand is inelastic because the induced fall in energy useis not so dramatic that total tax revenues fall. In other words, the economyoperates on the upward sloping part of the energy tax revenue curve. Asdiscussed previously, this tax-revenue effect adds to the import-saving effect

16

due to the fact that lower energy use implies lower demand for oil. Anotherway to think about this property is to observe that to preserve equilibrium ofthe labor market the manufacturing sector must absorb the labor expelled bythe energy sector. This requires aggregate demand for consumption goodsLy∗ to rise. The increase in demand is fueled by the increase in disposableincome due to the lump-sum rebate of the energy tax revenues and to thelower flow of payments to foreigners for oil imports.

4.2 The CPI: how the cost of energy and TFP affect con-sumption

Consider now the effects on consumption. The price index of a basket ofconsumption goods — the CPI of this economy — is

PY =

·Z N

0P 1−j dj

¸ 11−

.

Accordingly, the price strategy (10) yields real expenditure per capita as

y∗

PY=− 1 y∗

c∗N

1−1Zθ,

where to simplify the notation I define c∗ ≡ CX (1, CE (1, PO) + τ). Thisunit cost is pinned down by exogenous parameters.

Why look at real expenditure? Because it measures the flow of consump-tion in the utility function (3) and thus is relevant for welfare. Moreover,one can reinterpret (3) as a production function for a final homogenous goodassembled from intermediate goods and define aggregate TFP as

T = N1−1Zθ. (25)

Accordingly,

T (t) =1

− 1N (t) + θZ (t) .

In steady state this gives

T ∗ =λ

− 1 + θZ∗ ≡ g∗, (26)

where Z∗ is given by (22). Observe how g∗ is independent of conditions inthe energy market and of population size and growth.

17

A nice feature of this model is that I can compute TFP in closed formalong the transition path. To bring this feature to the forefront, notice thataccording to (21) in steady state

y∗

n∗=

y0n0⇒ y∗

y0=

n∗

n0.

Now define

∆ ≡ n∗

n0− 1 = y∗

y0− 1.

This is the percentage increase in expenditure that the economy experi-ences in response to changes in fundamentals and/or policy parameters. Itfully summarizes the effects of such changes on the scale of economic activ-ity. The following proposition characterizes how changes in scale affect themanufacturing sector.

Proposition 4 At any time t > 0 the log of TFP is

log T (t) = log

µZθ0n

1−10

¶+ g∗t (27)

+γ∆

ν

¡1− e−νt

¢+

1

− 1 log1 +∆

1 +∆e−νt,

where

γ ≡ θαθ ( − 1) y∗

n∗= θ

θ ( − 1) φα− ρ1−θ( −1) − ρβ

.

Moreover,

d log T (t)

dτ=

·γ

ν

¡1− e−νt

¢+

1

− 11

1 +∆

1− e−νt

1 +∆e−νt

¸d∆

dτ> 0.


The TFP operator in (27) has two transitional components. The first isthe cumulated gain from cost reduction, Z (t), the second is the cumulatedgain from product variety per capita, n (t). The mechanism driving thesecomponents is quite intuitive: a tax on energy use changes relative inputprices and induces manufacturing firms to substitute labor for energy in theirproduction operations. This standard effect is associated to an increase ofaggregate R&D employment, the sum of cost-reducing R&D internal to thefirm and entrepreneurial R&D aimed at product variety expansion. Despitethis reallocation, however, steady-state growth does not change because the

18

dispersion effect due to entry offsets the increase in aggregate R&D. AverageR&D, in other words, does not increase. This follows from the fact that theincrease in the size of the manufacturing sector, measured by the rise inaggregate expenditure on consumption goods, raises the returns to entry.Over time the larger number of firms generates dispersion of R&D resourcesand sterilizes the scale effect. It follows that the growth rate is independentof the size of the manufacturing sector.

Summarizing, the energy tax reallocates labor from the energy sectorto the manufacturing sector. The increase in the size of the manufacturingsector generates a temporary growth acceleration.

If one ignores these effects of endogenous technological change — if oneholds T constant — consumption depends only on y∗/c∗. The followingproposition characterizes this case.

Proposition 5 Assume that technology does not adjust in response to theenergy tax. Then two cases are possible:

1. SOE

EX (0) ≥ −1

£1 − β (ρ− λ)

¤, which implies that

d

dτ

µlog

y∗

c∗

¶≥ 0 for τ ≤ τ ,

where τ solves

PESOE − τ

PE + τEX = − 1

·1 − β (ρ− λ)

¸;

2. SOE

EX (0) < −1

£1 − β (ρ− λ)


d

dτ

µlog

y∗

c∗

¶< 0 for all τ .


This is an important result. It says that endogenous technological changeis not necessary to obtain welfare gains from the energy tax if the tax issufficiently small. The reason is that the import-saving effect alone canpotentially offset the rise of the manufacturing unit cost. To see the intuitionmore clearly, consider the CES technology in Example 2 and observe that

d

dτ

µlog

y∗

c∗

¶≥ 0 for PES

OE − τ

PE + τ

1− σSEX

1− σ≥ − 1

·1 − β (ρ− λ)

¸.

19

The left-hand side of the second expression is a decreasing function of τ thatcuts the horizontal axis at PESO

E = τ . It follows that if one assumes

SOE

1− σSEX (0)

1− σ> − 1

·1 − β (ρ− λ)

¸, σ ≤ 1,

there exists a τ ∈ £0, PESOE

¤such that y∗/c∗ is increasing in τ for τ ≤ τ .

This condition holds if with no government intervention, i.e., τ = 0, theshare of oil in energy production costs, SO

E , is large, the share of energy inmanufacturing costs,

SEX (0) =

1

1 + Pσ

1−σE

,

is small and the elasticity of substitution between energy and labor, 11−σ ,

is high. While the first two requirements are plausible, the third is not.In fact, labor-energy complementarity means 1

1−σ ≤ 1 in which case 1 −β (ρ− λ) ≥ −1 is sufficient to make the condition fail because the left-handside is a number less than 1. Then only case 2 in the proposition applies.It seems, therefore, that although not necessary, endogenous technologicalchange plays a crucial role in the determination of the welfare effects of theenergy tax.

4.3 Welfare

I now investigate how the long-run effects of endogenous technological changeoffset the short-run cost (if any) of the energy tax. As mentioned in Sec-tion 2, to emphasize the importance of endogenous technology, I ignore fornow environmental quality in the household’s preferences so that the welfaregains of the tax stem solely from the fact that the reallocation of resourcesfrom energy to manufacturing accelerates temporarily the pace of endoge-nous technological change and yields a long-run TFP gain.

Proposition 6 Let log u∗ (t) and U∗ be, respectively, the instantaneous util-ity index (3) and welfare function (1) evaluated at y∗. Then, a path startingat time t = 0 is characterized by:

log u∗ (t) = logy∗

c∗+ g∗t (28)

+γ∆

ν

¡1− e−νt

¢+

1

− 1 log1 +∆

1 +∆e−νt;

20

U∗ =1

ρ− λ

·log

y∗

c∗+

g∗

ρ− λ+

γ∆

ρ− λ+ ν

¸(29)

+1

− 1Z ∞

0e−(ρ−λ)t log

1 +∆

1 +∆e−νtdt.


The first term in (29) captures the role of steady-state real expenditurecalculated holding technology, T , constant; the second captures the role ofsteady state growth of income per capita, g∗; the third is the contributionfrom the gain in firm-level productivity, Z, due to the transitional accelera-tion of firm-level cost-reduction; the fourth term captures the contributionof the gain in n due to the transitional acceleration of product variety expan-sion. Figure 2 illustrates the underlying path of the log of consumption inthe case in which the CPI jumps up on impact. It thus identifies very sharplythe fundamental trade-off driving the role of the energy tax. There is aninstantaneous drop in consumption, followed by faster than trend growth,with eventual convergence to a higher steady-state growth path, parallel tothe initial one. Thus, all the gains from the energy tax stem from leveleffects spread over time. This is where the model’s property that the tran-sition path has an analytical solution comes very handy since it allows oneto see exactly how the intertemporal trade-off plays out.

I now show that under plausible conditions endogenous technologicalchange yields that welfare is a hump-shaped function of τ , as in Figure 3.

Proposition 7 Assume that the production technology (6) exhibits grosscomplementarity between labor and energy so that energy demand is inelas-tic, E

X ≤ 1. Consider an economy on the equilibrium path induced by theintroduction of an energy tax τ > 0, starting from the equilibrium with noenergy tax, i.e., τ = 0. Assume that the conditions for Case 1 from Proposi-tion 5 apply. Then, U∗ is a hump-shaped function of τ . Assume, in contrastthat the conditions for Case 2 in Proposition 5 apply. Then, U∗ is a hump-shaped function of τ iff

γ + ν−1

ρ− λ+ ν> − 1

·1 − β (ρ− λ)

¸. (30)

Otherwise, U∗ is decreasing in τ for all τ ≥ 0. A sufficient condition forinequality (30) to hold is

ρ− λ <θ ( − 1)

β.

21


The second inequality in this proposition provides a sufficient condi-tion for existence of an “optimal” energy tax τ∗ > 0. The interpretationof this condition is that the economy has growth-favoring fundamentals inthat technological opportunity is high and households are patient. The firstaspect implies that R&D is productive, the second that households demanda low reservation interest rate to fund it. Consequently, the economy enjoysfast steady-state growth. Moreover, the fact that households are patientimplies that the long-run gains due to the temporary growth accelerationdue to the energy tax is more likely to offset the short-run losses due totemporarily higher production costs. Specifically, the right-hand side of theinequality is large when:

• the elasticity of cost reduction θ is high, which means that knowledgeaccumulation translates into large reductions in production costs;

• the entry cost relative to initial revenue β is low so that product va-riety expansion does not require a large diversion of resources fromproduction and cost reduction;

• the market is competitive, because the elasticity of product substitu-tion is high, so that a given cost reduction yields large gains in salessince customers are price sensitive.

Overall, these conditions lead to fast steady-state (endogenous) productivitygrowth. More importantly, they lead to a stronger response of TFP growthto instantaneous market size.

5 Pollution

The previous section has shown that there exists an optimal tax abstractingfrom the benefits of pollution reduction. This section investigates the roleof pollution, a by-product of economic activity. Denote emissions per firmwith qi. Labor use does not pollute while energy use does.11 Assuming a

11 If the damage that the firm inflicts on the environment is a function of the level ofoutput, qi = Xi, environmental quality falls without bound since output grows forever.This is not a useful assumption if one does not specify an abatement technology thatallows a growing economy to invest resources and reduce pollution. (For an example ofwork that follows this approach, see Peretto 2006a.) This is particularly important if thereis a minimum level of environmental quality below which economic activity itself is not

22

linear relation, qi = Ei, to simplify the algebra, I can write the pollutiondamage function as

D(q1, ..., qN) =

Z N

0qidi =

Z N

0Eidi = E.

In this formulation pollution depends only on aggregate energy use. Allow-ing for dependence on production of energy as well (which in equilibriumequals use) does not add insight to the analysis.

To see how pollution affects utility, I augment (3) as follows

log u = log

"Z N

0

µXi

L

¶ −1

di

# −1

− ξ log (1 +D (E)) , > 1, ξ > 0 (31)

where as before is the elasticity of product substitution, Xi is the house-hold’s purchase of each differentiated good, and N is the mass of goodsexisting at time t. The new parameter ξ is the elasticity of utility withrespect to pollution. D = 0 corresponds to a pristine environment; anypositive D produces a “bad” that reduces the flow of utility. Pollution ex-ternalities depend on aggregate energy use according to the linear functionconstructed above.

Recall that in equilibrium (12) yields

E = Ly∗( − 1)(PE + τ)

SEX .

This expression says that pollution damages, D = E, grow over time becauseof population growth. Therefore, this economy exhibits the property thatthe extensive component of growth of income per capita — product varietyexpansion, which is tied to population growth — has a dirty side, while theintensive one — cost reduction, which is not related to population growth— does not. The reason is straightforward. Intensive growth is driven byHicks neutral factor augmenting technological change that does not requirean expanding resource base. In contrast, extensive growth is tied to the re-source base — the labor endowment — because the expansion of the variety ofproducts implies replication of fixed labor costs. In fact, population growth

sustainable. In the context of this model a more reasonable proxy for emissions per firmis energy use since pollution does not rise with output if output growth comes from TFPas opposed to more input use. It is the fact that energy use ultimately involves burningof fossil fuels that makes it so important for pollution. Higher ouptut at unchanged useof energy inputs should not per se contribute to emissions.

23

drives product variety expansion. This semi-endogenous component of therate of growth of income per capita has the implication that in the presenceof infinitely elastic supply of oil it also drives energy use and thus pollution.(If oil supply is finite, then it drives explosive growth of the price of oil; seePeretto (2006c) for an analysis of this case.)

It is desirable to take action to offset the dirty side of economic growth.Suppose, for example, that the government allocates a fraction 1− f of theenergy tax revenues to abatement activities that clean up emissions. Thenit is possible to construct an equilibrium where D = E − (1− f) τE = 0 sothat the loss of utility from pollution is eliminated, i.e., −ξ logD = 0. Thisrequires setting the lump-sum rebate of tax revenues to the household at

fτE = (τ − 1)E.Accordingly, (18) becomes

y∗ =1

1− β (ρ− λ)− £PESOE − (τ − 1)¤ −1 SEXPE+τ

.

Now observe that this policy eliminates pollution damages from the welfarefunction and leaves the remainder of the analysis virtually identical to that ofthe previous section. The only difference is that now there is the additionalconstraint that the tax must generate sufficient revenues to pay for fullycleaning up the environment. This is why in the expression above the termτ−1 appears in place of τ . It is then straightforward to replicate the previousanalysis and show that welfare is again hump-shaped in τ .

This type of intervention corresponds to the “Kindergarten Rule” dis-cussed by Brock and Taylor (2003, 2005). Notice that this is not a first-bestpolicy since I am not solving a social planning problem taking into accountall distortions. The point of this example is simply to show that introducingpollution externalities does not change the substance of the analysis of theprevious section.

6 Extensions and other questions

As a robustness check one can investigate the following extensions of themodel discussed in this paper.

• The most obvious, and interesting, extension is the case of endogenouslabor supply. In particular, I can show that with log utility definedover consumption and leisure the qualitative results are exactly thesame.

24

• As shown in Section 5, population growth generates escalating pollu-tion. This requires targeted interventions. I discussed one that fullyeliminates pollution under the assumption that pollution is a flow. Theanalyisis can be easily extended to the case of stock pollution with nosubstantial change in the main insight. It can also be extended to thecase of costly abatement undertaken by firms (see, e.g., Peretto 2006a).This entails a trade-off between allocating resources to cost-reductionand product-creation as opposed to emissions abatement.

• Population growth also puts pressure on the resource endowment andthus generates escalating rents. I bypassed this complication by usingthe small open economy assumption, which in practice turns a finitesupply of oil into an infinite elastic one. This is convenient but mightraise doubts. It turns out, however, that the results are robust andcarry over to the case of scarcity (see Peretto 2006c).

• It is worth emphasizing that at issue here is the behavior of the pricesof resources, not whether the economy is capable of long-run growth.Escalating resource prices are consistent with long-run growth in thepresence of scarcity. In fact, escalating prices is precisely how the econ-omy copes with resource scarcity. My assumption that TFP in man-ufacturing is Hicks neutral implies that growth of income per capitais feasible at unchanged physical inputs use, and thereby makes quitestarkly the point that the scarcity signaled by escalating resource pricesis due to population growth, not to growth of income per capita.

• An important aspect of scarcity is that it requires resource-augmentingtechnical change in the energy sector in order to offset the upward pres-sure on prices due to population growth. There is then an additionaltrade-off because resource-augmenting R&D in the energy sector com-petes for resources with R&D in manufacturing. On the other hand,this gives cumulative effects along the vertical production structure sothat cost reductions in energy production ultimately show up in theprices of consumption goods. I expect the basic result not to change,but need to work this case out carefully to check.

• It is possible to extend the model to the case of an economy with anon-zero endowment of oil, and look at how energy taxes bring abouta reduction of dependence on foreign sources (see Peretto 2006b). Thisreduction could be engineered as a smooth transition were eventuallyimports go to zero and the economy relies only on domestic sources.

25

7 Conclusion

To be written.

8 Appendix

8.1 Proof of Lemma 1

Observe that

EX ≡ −

∂ logE

∂ log (PE + τ)= 1− ∂ logSE

X

∂ log (PE + τ)= 1− ∂SE

X

∂ (PE + τ)

PE + τ

SEX

so that EX ≤ 1 if

∂SEX∂ (PE + τ)

=∂

∂ (PE + τ)

µ(PE + τ)E

(PE + τ)E + LE

¶≥ 0.

This in turn is true if¡1− SE

X

¢ ∂ ((PE + τ)E)

∂ (PE + τ)− SE

X

∂LE

∂ (PE + τ)≥ 0.

Recall now that total cost is increasing in PE + τ so that

∂ ((PE + τ)E)

∂ (PE + τ)+

∂LE

∂ (PE + τ)> 0⇒ ∂ ((PE + τ)E)

∂ (PE + τ)> − ∂LE

∂ (PE + τ).

It follows that∂LE

∂ (PE + τ)≤ 0

is a sufficient condition for EX ≤ 1 since it implies that both terms in the

inequality above are positive.

8.2 Proof of Proposition 1

Use (12) and the fact that Y = Ly∗ to rewrite (18) as

y∗ =1− ¡PESO

E − τ¢EL

1− β (ρ− λ). (32)

26

Then,

dy∗

dτ= − 1

1− β (ρ− λ)

d¡¡PES

OE − τ

¢EL

¢dτ

=1L

1− β (ρ− λ)

·E − ¡PESO

E − τ¢ ∂E∂τ

¸=

EL

1− β (ρ− λ)

·1− ¡PESO

E − τ¢ ∂E∂τ

1

E

¸=

EL

1− β (ρ− λ)

·1− PES

OE − τ

PE + τ

∂ logE

∂ log (PE + τ)

¸=

EL

1− β (ρ− λ)

·1 +

PESOE − τ

PE + τEX

¸=

EL

1− β (ρ− λ)

PE¡1 + SO

EEX

¢+ τ

¡1− E

X

¢PE + τ

.

This expression is positive if EX ≥ 1. If, in contrast, E

X > 1 this expressionis positive if PESOE ≥ τ while it changes sign at

1 +PES

OE − τ

PE + τEX = 0

if PESOE < τ . It follows that y∗ is a hump-shaped function of τ .


Taking logs of (25) yields

logT (t) = θ logZ0 + θ

Z t

0Z (s) ds+

1

− 1 logN (t) .

Using the definition n ≡ Ne−λt, the expression for g∗ in (26) and addingand subtracting Z∗ from Z (t), I obtain

log T (t) = θ logZ0 + g∗t+ θ

Z t

0

hZ (s)− Z∗

ids+

1

− 1 logn (t) .

27

Using (19), (24) and the definition of ∆ I rewrite the third term as

θ

Z t

0

³Z (s)− Z∗

´ds = θ

αθ ( − 1) Z t

0

µy∗

n (s)− y∗

n∗

¶ds

= γ

Z t

0

µn∗

n (s)− 1¶ds

= γ∆

Z t

0e−νsds

=γ∆

ν

¡1− e−νt

¢,

where

γ ≡ θαθ ( − 1) y∗

n∗= θ

θ ( − 1) φα− ρ1−θ( −1) − ρβ

.

Using (24) and the definition of ∆ I rewrite the last term as

1

− 1 logn (t) =1

− 1 logn∗

1 +∆e−νt

=1

− 1 logn0 +1

− 1 logn∗n0

1 +∆e−νt

=1

− 1 logn0 +1

− 1 log1 +∆

1 +∆e−νt.

These results yield (27). Taking derivatives with respect to τ and observingthat

d∆

dτ=

dy∗

dτ

1

y0> 0

yields d log T (t)dτ > 0.


Observe thatd

dτ

µlog

y∗

c∗

¶< 0⇔ dy∗

dτ≥ y∗

SEX

PE + τ.

28

Now use (32), (12) and the fact that Y = Ly∗ to rewrite the second inequalityas

E

L

·1 +

PESOE − τ

PE + τEX

¸≥ E

L

1− β (ρ− λ)−1

1− 1 + − 1 PESOE − τ

PE + τEX ≥ 1− β (ρ− λ)

− 1 PESOE − τ

PE + τEX ≥ 1 − β (ρ− λ) .

The right-hand side of this expression is positive because 1 − β (ρ− λ) > 0since the feasibility constraint 1 − βρ > 0 holds. The left-hand side is adecreasing function of τ . Assuming

SOE

EX (0) ≥ − 1

·1 − β (ρ− λ)

¸,

then, there exists a τ such that y∗/c∗ is increasing in τ for τ ≤ τ . Otherwise,y∗/c∗ is decreasing in τ for all τ ≥ 0.


I first use (27) and the definition of ∆ to write (3) as

log u∗ (t) = logy∗

PY (t)

= log− 1

+ logy∗

c∗+ log T (t)

= log

µ − 1Zθ0n

1−10

¶+ log

y∗

c∗+ g∗t

+1

− 1 log1 +∆

1 +∆e−νt+

γ∆

ν

¡1− e−νt

¢.

Without loss of generality, I set

− 1Zθ0n

1−10 = 1

and obtain (28). I then substitute this expression into (1) and write

U∗ =

Z ∞

0e−(ρ−λ)t

·log

y∗

c∗+ g∗t

¸dt

+γ∆

ν

Z ∞

0e−(ρ−λ)t

¡1− e−νt

¢dt

+1

− 1Z ∞

0e−(ρ−λ)t log

1 +∆

1 +∆e−νtdt.

29

The first and second integrals have straightforward closed form solutions.(The third is solvable as well, but it entails a very complicated expressioncontaining the hypergeometric function and is not worth using since it addsno insight and does not simplify the algebra in the analysis below.) Hence,I obtain (29).


Let y0 = y∗ (0) denote y∗ evaluated at τ = 0. Observe then that:

∆ (0) =y∗ (0)y0− 1 = 0;

∆ (∞) = y∗ (∞)y0

− 1 =−1SE

X (∞)1− β (ρ− λ)− −1SE

X (∞)> 0;

U∗ (0) =1

ρ− λ

·log

y∗ (0)c∗ (0)

+g∗

ρ− λ

¸> 0;

U∗ (∞) =1

ρ− λ

·log

y∗ (∞)c∗ (∞) +

g∗

ρ− λ+

γ∆ (∞)ρ− λ+ ν

¸+

1

− 1Z ∞

0e−(ρ−λ)t log

1 +∆ (∞)1 +∆ (∞) e−νtdt

= −∞.

Note that the restriction U∗ (0) > 0 makes sense (and does not requirespecial assumptions beyond log y∗(0)

c∗(0) > 0), while U∗ (∞) = −∞ follows from

that fact that y∗ (∞) is finite and c∗ (∞) = ∞. Next, use the propertiesderived in Proposition 4 and note that dU∗

dτ > 0 iff

d

dτ

µlog

y∗

c∗

¶+

γ

ρ− λ+ ν

d∆

dτ+

ρ− λ

− 1Z ∞

0

e−(ρ−λ)t

1 +∆

1− e−νt

1 +∆e−νtd∆

dτdt > 0.

Now observe that U∗ (∞) = −∞ means that the function U∗ eventuallymust be decreasing. Moreover, the second and third terms in the inequalityabove are always positive. The first term is either always negative or startsout positive and then becomes negative. It follows that dU∗

dτ changes signexactly once. Therefore, U∗ is hump-shaped in τ iff the inequality holds ina neighborhood of τ = 0, otherwise it is always decreasing in τ . This yieldstwo cases:

30

1. SOE

EX (0) ≥ −1

£1 − β (ρ− λ)


d

dτ

µlog

y∗

c∗

¶|τ=0> 0.

Then dU∗dτ |τ=0> 0 and U∗ is hump-shaped in τ .

2. SOE

EX (0) < −1

£1 − β (ρ− λ)


d

dτ

µlog

y∗

c∗

¶|τ=0< 0.

Then, dU∗dτ |τ=0> 0 requiresd log c∗

dτ<

µ1

1 +∆+

γ

ρ− λ+ ν

¶d∆

dτ

+ρ− λ

− 11

1 +∆

d∆

dτ

Z ∞

0e−(ρ−λ)t

1− e−νt

1 +∆e−νtdt,

which one can rewrite asd log c∗dτ

d log y∗dτ

< 1 +γ

ρ− λ+ ν+

ρ− λ

− 1Z ∞

0e−(ρ−λ)t

1− e−νt

1 +∆e−νtdt

sinced∆

dτ

1

1 +∆=

d log y∗

dτ.

Letting τ → 0,

−1y∗ (0)

< 1 +γ

ρ− λ+ ν+

ρ− λ

− 1Z ∞

0e−(ρ−λ)t

¡1− e−νt

¢dt.

This yields

− 1·1 − β (ρ− λ)

¸<

γ + ν−1

ρ− λ+ ν.

Using the definitions of ν, γ and g∗ and a little bit of algebra, thisinequality reduces to

θZ∗ >λ

− 1·β (ρ− λ)− θ ( − 1)¸

.

Under the restrictions imposed in Section 3 that guarantee Z∗ > 0, asufficient condition for this inequality to hold is

β (ρ− λ) <θ ( − 1)

.

31

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n

n

n n*

Figure 1: General Equilibrium Dynamics

log )(* tu

Path with τ>0, T endogenous

Path with τ=0

Path with τ>0, T constant

t t=0

Figure 2: The path of consumption

)(τ*U

τ τ*

Figure 3: The energy tax and welfare

energy taxes and endogenous technological change ...staff/wrkshop_papers/2006... · tions of the us...

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