environmental policy and the equilibrium rate of unemployment

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Journal of Environmental Economics and Management 49 (2005) 132–156

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Environmental policy and the equilibriumrate of unemployment

Thomas Wagner�

University of Applied Sciences Nuernberg, Bahnhofstrasse 87, 90402 Nuernberg, Germany

Received 17 April 2000; received in revised form 23 June 2003

Available online 23 August 2004

Abstract

This paper integrates environmental policy instruments with the theory of equilibrium unemployment.We investigate the question of whether a low equilibrium rate of unemployment and a high quality of theenvironment are complementary policy goals or must be traded off. It turns out that an interval exists for atax on emissions where the two goals are indeed complementary. The tax stimulates the emergence of anabatement sector which provides pollution control and vacancies for the job seekers. For constrainedefficiency, the policy maker operates five instruments to internalize the environmental and the searchexternalities. A tax on emissions, employment subsidies and recruiting allowances for the pollutingindustries are sufficient to implement the first-best. The optimal emission tax is an increasing function of theworkers’ bargaining strength. For labor markets where workers have a strong bargaining position, theoptimal pollution tax may easily exceed the Pigouvian tax.r 2004 Elsevier Inc. All rights reserved.

Keywords: Environmental policy instruments; Emission tax; Equilibrium unemployment; Constrained efficiency

1. Introduction

In recent years, public opinion—which, in the past, often regarded environmental regulations as‘‘job killers’’—has gradually shifted towards supporting the hypothesis that a more stringent

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ress: [email protected] (T. Wagner).

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T. Wagner / Journal of Environmental Economics and Management 49 (2005) 132–156 133

environmental policy not only promotes a country’s international competitiveness but may alsoimprove its employment situation [5,19,23,31,32,39]. Discussions in the political arena have beenpreoccupied with conjectures about the employment effects of environmental policy, whileenvironmental economics has rarely addressed the issue of the impact of environmental policyinstruments on equilibrium unemployment. To investigate the relationship between the policygoals of a clean environment and low unemployment, the present paper integrates standardenvironmental policy instruments [3,8,20] with the theory of equilibrium unemployment[13,33,34].The economy of the model consists of two sectors: a production sector with a dirty industry and

a clean industry that emits an assimilative pollutant, and an abatement sector which providescleaning services for the polluting industries and vacancies for the job seekers.Depending on the level of the tax rate, the emission tax causes either a trade-off or a

complementarity between the two policy goals. Within the tax brackets where the goals arecomplementary, the tax reduces both the emissions and the equilibrium rate of unemployment.For low tax rates, on the other hand, we find a trade-off between the two policy goals. There aretwo reasons for this result. Firms in the polluting industries would rather pay the tax than controltheir emissions and workers faced with the low wages offered by the abatement sector do not findit attractive to search for cleaning jobs. Hence, an abatement sector, which could outweigh the jobdestruction effects of the pollution tax in the production sector, does not develop. We next showthat, under conditions like those of the Pigouvian economy, the policy goals of a cleanenvironment and a low rate of unemployment are complementary even in a command-and-control economy regulated by an emission standard.Empirical research has not produced clear cut evidence for the disputed trade-off

[1–2,4,16,19,21,27]. However, the results of Berman and Bui’s [4] research into the employmenteffects of local ‘‘air quality regulations’’ in the Los Angeles area seem to confirm our analysis.They note that, despite considerable investment in ‘‘abatement capital, y we find no evidence that

local air quality regulation substantially reduced employment, even when allowing for induced plantexit and dissuaded plant entry’’. According to Berman and Bui, all new regulations and everytightening of existing regulations induce large investments in abatement technology, and themeasured employment effects of air quality regulations are generally positive although they arenot statistically significant. They explain the adjustment of the demand for labor they observe inall polluting industries by the complementarities between abatement capital and labor inproduction that dominate the negative output effects of the air quality regulations. They alsoestimate the job destruction and creation effects of air quality regulations and again find nosignificant results, although they argue that the impact of new regulations on entry and exit islikely to be negative for the regulated industry.Although the extensive literature on the ‘‘double dividend’’ of an environmental tax reform

focuses on the employment effects of environmental policy, there seems to be only one paper—byBovenberg and v.d. Ploeg [10]—that describes the effects of such a reform within the frameworkof the theory of equilibrium unemployment. In their model, the government has a balancedbudget and uses the revenue of an ad valorem tax on energy consumption to lower the firms’payroll tax. The labor market is segmented. The natural rate of unemployment is positive in theofficial labor market which is a search market with a matching technology similar to the one weuse. In the informal neoclassical labor market, however, any job seeker instantly finds a job and

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T. Wagner / Journal of Environmental Economics and Management 49 (2005) 132–156134

the market clears at all times. The fiscal policy reform shifts the tax burden away from the officialsector to the informal sector. Nevertheless, for a ‘‘double dividend’’ to develop, the pre-reformenergy tax must be low. With a high initial tax rate the increase in the overall tax burden wouldmake the aggregate employment effect of the tax reform negative.In contrast to [10], in our two-sector model real income during job search is an exogenous flat

rate not indexed to the wages. Thus, the ‘‘double dividend’’ is neither a result of the assumptionsabout a job seeker’s utility function and the sources of his income during job search as in [10], nordoes it stem from swapping environmental taxes for distortionary taxes as in most of the literatureabout the ‘‘double dividend’’. It emerges because—at a sufficiently high emission tax—anabatement sector develops which provides both abatement services to the polluting industries andvacancies for the job seekers.The paper is organized as follows. In Section 2, we present the overall equilibrium of the

unfettered market with its search and environmental externalities. In Section 3 the polluters areforced to control their emissions and a market for pollution control services evolves. In Sections 4and 5 emission taxes and emission standards are discussed. Section 6 analyzes the first-bestallocation, Section 7 concludes. The appendix provides proofs of all propositions.1

2. The unfettered market

The entrepreneurs in this model organize production, sell output, offer vacancies, and recruitworkers. The workers employed carry out the production plans, the unemployed search for avacancy. Firms and workers are risk neutral and have an infinite time horizon. The labor marketsare search markets [13,35,11,17,36]. Frictions, heterogeneities, and informational imperfections—not explicitly modeled—force the market participants to search for matching partners.The labor force and the measure of jobs in the polluting industries are given [6–7,12]. The

capacity of the abatement sector is endogenous, adapting to shocks through a perfectly elasticinflow of vacancies [30]. Job seekers are mobile between industries and move to the location wherethey earn the highest income. Vacancies and filled jobs are immobile.2

Jobs: There are two polluting industries i ¼ C;X, where i denotes the location or thecharacteristics of a differentiated product produced by the firms of industry i. The exogenousmeasure of jobs in industry i is ki. Each firm consists of one job which is either vacant or filled. Afilled job produces the output yi and, as a by-product, the quantity Pi of a pollutant. Jobs in theindustry C produce with a more advanced technology than jobs in the industry X, in particular weassume3

ðCXÞ PX4PC40 and yCXyX:

These assumptions seem to be plausible because technologically improved process designs oftenpollute less and are more productive than less advanced technologies. For convenience, we willrefer to the industries C and X as the clean and the dirty industry, respectively.

1The appendix is available online as a supplement to this article at http://www.aere.org/journal/index.html.2Hosios [18] is a two-sector model with completely mobile jobs; Mortensen and Pissarides [25] discuss a case in which

even filled jobs are mobile between the sectors.3To prove the statements indexed by a* assumption (CX) is used.

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Matching technology: Time is continuous. The labor market is characterized by two-sidedsearch [12,24,26,29,30,34]. Trade in the labor market and production are separate activities. Jobs(workers) either search, if vacant (unemployed), or produce, if filled (employed). Of all the ki jobsof industry i, ei are filled and vi are vacant, ki ¼ ei þ vi. The labor force l is exogenous. Of the l

workers lP move to the economy’s production sector and lA to the abatement sector so thatl ¼ lP þ lA. For the present we assume that lP ¼ l, see Fig. 1. Of the lP (exogenous) productionworkers li move to the industry i, where ei are employed and ui search for a job, so thatli ¼ ei þ ui, i ¼ C;X, and lP ¼ lC þ lX.Vacant jobs and unemployed workers search for matching partners. All traders in the labor

market act as atomistic competitors. The transaction technology is modeled by a matchingfunction with the flow of matches mi per time unit as the dependent variable, and the measures ofjob seekers ui and vacancies vi as the independent variables, mi ¼ mðui; viÞ. Both industries i usethe same matching technology [22]. The matching function is continuously differentiable andconcave, has constant returns to scale, and strictly positive partial derivatives. The vacancies andjob seekers that are matched are randomly selected from vi and ui. Hence, movement fromunemployment to employment is a Poisson process with the transition rate mðui; viÞ=ui: From thehomogeneity of the matching function it follows that mðui; viÞ=ui ¼ mð1; yiÞ, so that the transitionrate pðyiÞ � mð1; yiÞ is a function of the labor market tightness, yi, where yi ¼ vi=ui is the ratio ofthe measure of vacancies and the measure of unemployed. The job seekers’ rate of arrival at avacancy is qðyiÞ � mðui; viÞ=vi ¼ mð1=yi; 1Þ, so that pðyiÞ ¼ yiqðyiÞ. With decreasing labor markettightness, the job seekers’ rate of transition into employment approaches zero, and their rate ofarrival at a given vacancy approaches infinity: pð0Þ ¼ qð1Þ ¼ 0 and pð1Þ ¼ qð0Þ ¼ 1.A job is created when a vacancy and a job seeker meet. At each moment of time, pðyiÞui of the ui

unemployed apply for a job. As soon as the worker is taken on, firm and employee begin toproduce until an idiosyncratic shock destroys the match. The job becomes vacant, and the workermoves to the submarket i with the highest income for job seekers. Firm-specific demand ortechnology shocks occur at a frequency l, which is the result of an exogenous Poisson process and

Fig. 1. Equilibrium E of the unfettered market.

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is identical for all jobs and industries. Unemployment persists in the steady state, since before allunemployed job-worker pairs are matched lei of the ei filled jobs are destroyed. The process of jobdestruction produces a permanent inflow into the pools of unemployed and vacancies.In the steady state, the inflow into unemployment equals the outflow so that lei ¼ pðyiÞui since

the measures for filled jobs and employed workers are identical. From this equation we derive4 thesteady-state measures of vacancies, job seekers, filled jobs, and the size of the labor force inindustry i as functions of the industry’s tightness yi and exogenous capacity ki.

5

Asset equations: Denote by Ji the net present value of a filled job, by Vi the value of a vacancy,by W i the value of an employed worker, and by Ui the value of a job seeker. In the steady state ofindustry i, the following asset equations hold:

rJi ¼ yi � wi þ lðVi � JiÞ; ð1Þ

rVi ¼ �c þ qðyiÞðJi � ViÞ; ð2Þ

rW i ¼ wi þ lðUi � W iÞ; ð3Þ

rUi ¼ b þ pðyiÞðW i � UiÞ; ð4Þ

W i � Ui ¼ b½ðJi � ViÞ þ ðW i � UiÞ; ð5Þ

where r is the real interest rate, b the UI benefit or the utility of leisure, l the separation rate, and cthe hiring costs. The labor productivity yi, the bargained wage wi, and the labor market tightnessyi of industry i are industry-specific.To interpret Eq. (1) note that a filled job with present value Ji is an asset owned by the firm. If

the job is destroyed, it moves into the pool of vacancies and the firm bears a loss equal to Vi � Ji.Therefore, the total return of the filled job is given by the sum of the current profit, yi � wi, andthe expected capital loss lðVi � JiÞ. By investing the capital tied up in the job in the perfectlycompetitive capital market the firm could earn a permanent income equal to rJi. In the steady-state, all arbitrage possibilities are exhausted. Therefore, in the steady state equilibrium of thelabor market, the permanent income of a filled job in industry i must be equal to rJi and thearbitrage Eq. (1) holds. Correspondingly, qðyiÞðJi � ViÞ in Eq. (2) is the expected gain for the firmif a vacancy is filled, an event which occurs with the endogenous flow-probability qðyiÞ. Takinginto account the hiring costs c, in the steady state a vacancy at location i earns a permanentincome rVi ¼ �c þ qðyiÞðJi � ViÞ.The asset Eqs. (3) and (4) can be interpreted analogously. The process of job destruction

changes the value of an employed worker from W i to Ui with rate l. Hence the worker’s expectedloss from the destruction of his job is lðUi � W iÞ and his permanent income, rW i, is equal to thesum of the negotiated wage wi and the loss lðUi � W iÞ. A job seeker’s expected gain from thetransition to employment is pðyiÞðW i � UiÞ. The job seeker’s return therefore equals the sum ofthe UI benefit b and the expected gain from a transition, see Eq. (4).Costly job search is the reason why a filled job earns a quasi-rent in the steady state. In the case

of a match, firm and applicant form a bilateral monopoly and bargain over the distribution of the

4See Lemma A.1 in Appendix A.5For convenience we write liðyiÞ ¼ lðki; yiÞ etc.

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rent. Eq. (5) describes the rule which governs the distribution of the present-discounted value ofthe match rent, ðJi � ViÞ þ ðW i � UiÞ, between firm and worker. The worker’s share of the rent isa constant fraction b, 0obo1, where b reflects the exogenous bargaining strength of the worker.Eq. (5) can be interpreted as the first-order maximization condition derived from the generalizedNash solution for the wage rate wi ¼ arg maxðW i � UiÞ

bðJi � ViÞ

1�b.Reservation wage Y: Eqs. (1)–(5) contain six endogenous variables for each industry: the labor

market tightness yi, the bargained wage wi, and the asset values Ji, Vi, W i, and Ui. Solving for theendogenous variables we obtain the permanent income of a job seeker in i as a function of thelabor market tightness:6

rUiðyiÞ ¼ b þ gðyiÞpi; ð6Þ

where pi ¼ yi þ c � b is the current match rent of an occupied job. Note that the asset Eq. (4) andthe income Eq. (6) are alternative expressions for a job seeker’s permanent income. In Eq. (4) theoption value component of the permanent income is represented by the expected capital gain fromtransition into employment. By contrast, in the income Eq. (6) the option value is expressed as theshare gðyiÞ of the current match rent pi for which gðyiÞ ¼ bpðyiÞ=dðyiÞp1, where dðyiÞ ¼

r þ lþ bpðyiÞ þ ð1� bÞqðyiÞ is a location-specific discount factor. The share gðyiÞ can beinterpreted as the endogenous bargaining strength of a worker at location i. gðyiÞ is a strictlyincreasing function of b and the labor market tightness satisfying gð0Þ ¼ 0 and gð1Þ ¼ 1.7

Workers are free to move between the two industries. Accordingly, in overall equilibrium, theexpected permanent income of job seekers in the two industries must be the same

rUiðyiÞ ¼ Y ; i ¼ C;X ð7Þ

where Y is the production sector’s reservation wage. Y is the minimum compensation a workerrequires in order to give up search and accept a job. In view of the income Eq. (6) and the mobilitycondition (7), the tightness in the labor market of industry i, yiðY Þ, can be shown to be a strictlyincreasing function of the production sector’s reservation wage Y.8

Assumption (CX) implies pCXpX. From this inequality and the Eqs. (6) and (7) it follows thatworkers in industry X have a bargaining strength at least as high as in industry C, gðyXÞXgðyCÞ.Thus, due to the monotonicity of gðÞ, we find yXðY ÞXyCðY Þ for Y with bpYpb þ pX. If Yob,no unemployed will search for a job. If on the other hand Y4b þ pX industry X is notcompetitive and does not offer vacancies.

Competitiveness: Industry i is competitive as long as its firms are willing to offer vacancies.Firms post vacant jobs if a vacancy’s permanent income is non-negative, rViðyiÞX0. Thepermanent income of a vacancy is9

rViðyiÞ ¼ �c þ dðyiÞpi; ð8Þ

6See Lemma A.2 in Appendix A.7See Lemma A.2. The worker’s endogenous bargaining strength, gðyiÞ, depends not only on b but also on the

prevailing labor market conditions expressed through yi. The tighter the labor market, the longer the expected duration

of a vacancy and the more resources a firm will have to invest in its hiring activities. Therefore, the tighter the labor

market, the larger the share of the match rent the firm is willing to sacrifice during wage negotiations.8See Lemma A.3.9See Lemma A.2.

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where the component representing the option value of the vacancy is expressed as the sharedðyiÞ ¼ ð1� bÞqðyiÞ=dðyiÞp1 of the current match rent pi.

10

Under the above assumptions, there are industry-specific reservation wages Y i, withboY iob þ pi, where the value of a vacancy in i approaches zero, since dðyiðY iÞÞpi ¼ c.11 IfY4Y i, industry i is not competitive because for the unemployed it is does not pay to search forvacancies in i which can only offer an income not exceeding Y i, see Fig. 1. If, on the other hand,YpY i, the industry advertises vacancies and is frequented by a positive measure of job seekers.Given assumption (CX) clean firms are more competitive than dirty firms, YCXYX —see also

the vertical axis of Fig. 1. To prove this statement let Y be a sustainable reservation wage for bothindustries. Then we know that yXðY ÞXyCðY Þ, so that dðyXðY ÞÞpdðyCðY ÞÞ due to themonotonicity of dðÞ. Thus, taking into account pCXpX and Eq. (8), we getrVCðyCðY ÞÞXrVXðyXðY ÞÞ. Hence, by virtue of the monotonicity of yiðY Þ the industry-specificreservation wage is higher in C than in X.

Environmental damage and equilibrium: To close the model we introduce the excess supplyfunction of the aggregate labor market. Since all endogenous variables now depend on thereservation wage, Y, we solve the two-industry model by determining the value of Y that clears theaggregate labor market, see Fig. 1.The labor force of industry i consists of the unemployed searching for a vacancy in i, uiðyiÞ, and

the workers employed in i, eiðyiÞ, such that liðyiÞ ¼ eiðyiÞ þ uiðyiÞ. Since yi ¼ yiðY Þ, the size of thelabor force of industry i is a decreasing function of the reservation wage Y, liðyiðY ÞÞ, see Fig. 1.Hence, given the (exogenous) labor force lP which stays in the economy’s production sector theexcess supply of the sector’s aggregate labor market, El, is also a function of Y:

ElðY Þ ¼ lP � lCðyCðY ÞÞ � lXðyXðY ÞÞ: ð9Þ

Using the measure of filled jobs, we obtain the emissions of industry i, DiðY Þ ¼ eiðyiðY ÞÞPi. Theaggregate emissions, D, and the equilibrium rate of unemployment, u, are:

DðY Þ ¼ DCðY Þ þ DXðY Þ; uðY Þ ¼ uCðyCðY ÞÞ þ uXðyXðY ÞÞ: ð10Þ

The pollutant generated in the process of production is a public bad and reduces the welfare ofall job seekers and workers alike. Environmental damage Z is measured in units of output and is afunction of the aggregate flow of pollutants D. The workers’ utility functions are assumed to beadditively separable with respect to their income and the damage Z(D). Hence the net utility of theemployed and the unemployed is rWi � ZðDÞ and rUi � ZðDÞ; respectively. Since workers are riskneutral and the environmental damage is not location-specific, the externality does not influencethe equilibrium allocation of the unfettered market [9–10,14,15,28].The overall equilibrium of the unfettered market is characterized by a reservation wage Y � for

which ElðY �Þ ¼ 0. Y � determines, in turn, the natural rate of unemployment u� ¼ uðY �Þ, and theaggregate emissions D� ¼ DðY �Þ. In view of assumption (CX) an overall equilibrium with anactive dirty and clean industry exists, see Fig. 1 point E, if we assume that the excess supply

10The share dðyiÞ reflects the endogenous bargaining strength of the firm and is a strictly increasing function of ð1� bÞand a strictly decreasing function of the labor market tightness with the limits dð0Þ ¼ 1 and dð1Þ ¼ 0. We will assume

throughout that the current match rent is higher than the hiring costs, pi4c. Then there always exist labor market

conditions with yi40 where firms in industry i are willing to post vacancies (see Fig. A.2 in Appendix A).11See Lemma A.3 and Fig. A.2 in Appendix A.

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function (9) for the industry-specific reservation wage YX satisfies ElðYXÞX0. Since the excesssupply function is continuous and strictly monotone, the overall equilibrium is characterized byan unique reservation wage Y �, with boY �pYX.

3. The market for pollution control

As a waste sink, the environment provides inputs which are overused if nature’s assimilationcapacity is free or legally assigned to the polluting industries without obligation to processemissions. In an economy where a regulator has the right to exclude polluters from (over-)usingnature’s services, firms are forced to control their emissions and therefore demand abatementservices. We now extend the model by introducing an abatement sector which provides pollutioncontrol and vacancies for the job seekers.First, we focus again on the economy’s production sector and deal with the cleaning service as a

factor of production, which is traded on a competitive market. Next, we present the model of theabatement sector. Third, we discuss the overall equilibrium. We establish that there exists a lowerbound to the price for abatement services kl, see Fig. 2, below which it does not pay for the jobseekers to leave the polluting industries and move to the abatement sector. For cleaning pricesabove kl , the abatement sector is active and supplies emission control and posts vacancies in thelabor market. Next we show, that there exists a cleaning price kn, see Fig. 2, at which the marketfor pollution control services clears. Finally we prove that the equilibrium duration of job searchis shorter in the abatement sector than in the production sector.

3.1. The production sector

At a price k for each unit of pollution control, the abatement costs of a filled job in industry i

with pollution Pi are kPi. In the income Eqs. (6) and (8), the current match rent of a job inindustry i is now piðkÞ ¼ yi � kPi þ c � b. In the equilibrium of an economy where the regulator

Fig. 2. Equilibrium E.

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forces polluting firms to clean their emissions the tightness of the labor market of industry i is astrictly increasing function of the reservation wage Y and—given Y—of the cleaning price k.12 Ifk40, assumption (CX) implies the strict inequality pCðkÞ4pXðkÞ. From this inequality, and fromthe workers’ mobility between the industries C and X, it follows that the equilibrium labor markettightness of the dirty industry X is strictly larger than that of industry C, yXðY ; kÞ4yCðY ; kÞ.

Competitiveness: The process of job creation depends on the firms’ willingness to open vacanciesand search for job applicants. Opening a vacancy in industry i pays only if its permanent income(8) is non-negative. Using the labor market tightness yiðY ;kÞ in industry i we get the permanentincome of a vacancy as a function of the reservation wage and the cleaning price, rViðY ; kÞ. FromrViðY ;kÞ ¼ 0 we derive the industry-specific reservation wage Y i at which industry i loses itscompetitiveness as a function of k, Y iðkÞ, and its inverse, the ‘‘shutdown cleaning price’’ forindustry i, kiðY Þ, as a function of Y.

Lemma 1*. (1) Given the production sector’s reservation wage Y, industry i is competitive at thecleaning price k if kpkiðY Þ.(2) Due to the assumption (CX), kCðY Þ4kXðY Þ for all Y at which industry X is viable.

Equilibrium: Accounting for yi ¼ yiðY ;kÞ, we replace (9) by the following excess supply functionfor the production sector’s aggregate labor market

ElðY ;kÞ ¼ lP � lCðyCðY ; kÞÞ � lXðyXðY ;kÞÞ: ð11Þ

Lemma 2. Given the cleaning price k40 and the labor force lP staying at the economy’s production

sector, their exists a unique reservation wage Y ðk; lPÞ at which ElðY ðk; lPÞ;kÞ ¼ 0. The equilibriumreservation wage is as shown in Fig. 2 a decreasing function of k and lP.

Since yi ¼ yiðY ;kÞ and Y ¼ Y ðk; lPÞ, the following equations hold for the equilibrium flow ofemissions and rate of unemployment:

Dðk; lPÞ ¼ DCðk; lPÞ þ DXðk; lPÞ; uðk; lPÞ ¼ uCðyCðk; lPÞÞ þ uXðyXðk; lPÞÞ; ð12Þ

where Diðk; lPÞ ¼ eiðyiðk; lPÞÞPi is the emission or the demand of industry i for abatement services.Shutdown price: In view of assumption (CX) both the clean and dirty industry are competitive

at the reservation wage Y ðk; lPÞ if the cleaning price k does not exceed the shutdown cleaning pricefor industry X, kpkXðY ðk; lPÞÞ. For a given labor force lP, the shutdown price kXðY ðk; lPÞÞ is astrictly increasing contraction mapping.13 The intuition for this result is as follows. Increasing thecleaning price k reduces the reservation wage Y ðk; lPÞ which clears the labor market of theproduction sector. But the reservation wage Y ðk; lPÞ is the return on the unemployed worker’shuman capital and therefore the job seekers ‘‘threat point’’ during wage negotiations. As a highercleaning price reduces the ‘‘threat’’, the bargained wages and the wage costs decline in bothindustries. Therefore, both industries can sustain higher expenses for abatement activities, so thatthe shutdown prices move up.From the observation that the shutdown cleaning price is a contraction, it follows that there

exists a unique fix point, i.e. a cleaning price knX such that kn

X ¼ kXðY ðknX; lPÞÞ [37]. Given the labor

12See Lemma A.4*.13See Lemma A.5 in Appendix A.

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force lP staying at the economy’s production sector, despite the burden caused by the pollutioncontrol costs, the dirty industry and a fortiori the clean industry are both viable as long as kpkn

X.

Remark. 1. Notice that for the equilibrium reservation wage Y n of the unfettered marketY � ¼ Y ð0; lÞ, see Fig. 2.2. To secure kXðY ðk�X; lPÞÞ40, it is sufficient to assume that the shutdown cleaning price for

industry X at the equilibrium reservation wage of the unfettered market economy, Y ð0; lPÞ, isstrictly greater than zero, kXðY ð0; lPÞÞ40.3. The fact that a growing labor force lP in the production sector reduces the reservation wage

Y ðk; lPÞ implies that the shutdown cleaning price for the dirty industry is indeed an increasingfunction of lP, k�XðlPÞ ¼ kXðY ðk�XðlPÞ; lPÞÞ (see Lemma A.5).4. Further below in the section on the overall equilibrium it will be shown that the equilibrium

labor force staying at the economy’s production sector is a decreasing function of the cleaningprice k. Industry i is competitive at k if and only if kpk�i ðlPðkÞÞ, where like lPðkÞ, k�i ðlPðkÞÞ is adecreasing function of the price for the cleaning service. Lemma A.6* establishes that for k�i ðlPðkÞÞthere is a fixed point K�

i ¼ k�i ðlPðK�i ÞÞ, such that the industry i is viable at all k, for which kpK�

i .

3.2. The abatement sector

An abatement firm is conceptualized along the lines of the theory of equilibriumunemployment. Firms consist of one job. The job is vacant or occupied. Each occupied jobrents a capital good and produces cleaning services. Each vacancy is actively searching for jobapplicants. As soon as a worker is taken on, the firm starts production and offers its output on themarket for the abatement service. There are two conditions for the abatement sector to becomeactive. First, a vacancy in the sector must earn a non-negative income. Second, the income offeredto job seekers must induce them to move to the abatement sector.

Jobs and matching technology: Each abatement job is either vacant or filled with a worker whoproduces the quantity of pollution control a measured in units of the pollutant. For the moment,we assume the cleaning price k as given and investigate only the abatement sector’s labor market.The UI benefit, b, the separation rate, l, the discount rate, r, the hiring costs, c, and the bargainingstrength of the workers, b, are the same as in the production sector of the economy. Moreover,yA ¼ vA=uA is the tightness in the abatement sector’s labor market where uA job seekers and vAvacancies are searching for each other. The abatement sector uses the same matching technologyas the production sector, qðyAÞ is the arrival rate of job seekers at a given vacancy, and pðyAÞ ¼yAqðyAÞ is an unemployed worker’s rate of transition to employment.

Income equations: The asset equations for the value of a filled job, a vacancy, an employedworker, and a job seeker, and the equation for the distribution of the match rent have the samestructure as Eqs. (1)–(5). The Eqs. (6) and (8) for the permanent income of a job seeker and avacancy in the abatement sector, reproduced here for easier reference, are

rUAðyAÞ ¼ b þ gðyAÞpAðkÞ; ð13Þ

rVAðyAÞ ¼ �c þ dðyAÞpAðkÞ; ð14Þ

where pAðkÞ ¼ ka þ c � b is the current match rent of an occupied abatement job.

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The capacity of the abatement sector is endogenous, the inflow of vacancies continues untiltheir value is driven to zero. Using Eq. (14) and VA ¼ 0, we obtain the job creation condition forthe abatement sector

pAðkÞ � c=dðyAÞ ¼ 0: ð15Þ

For the cleaning price k, (15) gives the tightness at which the inflow of vacancies into theabatement sector ceases and the job creation process stops. Note that as a consequence of the freeentry condition the equilibrium tightness does not depend on the labor force staying in theabatement sector.If we substitute the current match rent pAðkÞ from (15) into (13), we obtain the equilibrium

reservation wage of the sector,

rUAðyAÞ ¼ b þ byAc=ð1� bÞ: ð16Þ

Supply of pollution control services: The first vacancy entering the labor market of the abatementsector faces the market tightness yA ¼ 0, and therefore can expect the return rV ð0Þ ¼ ka � b, asdð0Þ ¼ 1. From this condition for the ‘‘extensive margin’’ of the job creation process we derive theshutdown price of the abatement sector kA ¼ b=a, see Fig. 2.Given the labor force lA and the cleaning price k4kA the aggregate supply of cleaning

services is14

Sðk; lAÞ ¼ mðyAðkÞÞlAa: ð17Þ

In (17), mðyAÞ � pðyAÞ=ðlþ pðyAÞÞp1, with mð0Þ ¼ 0 and mð1Þ ¼ 1, denotes the abatementsector’s equilibrium rate of employment which is a strictly increasing function of the labor markettightness.

Equilibrium: For a given cleaning price k4kA and a given labor force lA the equilibrium in thelabor market of the abatement sector is a state ðy�A; rU�

A;S�Þ —with the tightness y�A ¼ yAðkÞ, the

reservation wage rU�A ¼ rUAðkÞ, and the supply of abatement services S� ¼ Sðk; lAÞ —which

satisfies the job creation condition (15), the income Eq. (16), and the supply function (17). Forðy�A; rU�

A;S�Þ the following lemma holds:

Lemma 3. (1) The labor market tightness, the reservation wage of the sector (see Fig. 2), and thesupply of abatement services are increasing functions of the cleaning price k.(2) The supply of abatement services is, moreover, an increasing function of the labor force lA

employed in or seeking employment in the sector.

3.3. Overall equilibrium

The market for pollution control is a Walrasian auction market with perfectly informedparticipants behaving as atomistic price takers. The production sector’s aggregate demand forpollution control services, Dðk; lPÞ, is determined by emissions (12), while the supply of theabatement sector, Sðk; lAÞ, is given by the aggregate supply function (17). The steady-stateequilibrium in the three labor markets of the economy and in the market for pollution control is a

14Use (A.1), Appendix A, to calculate the measure of filled jobs as a function of the labor force, e ¼ ml, with

m ¼ p=ðlþ pÞ; then, take into account the fact that a filled job produces the quantity of pollution control a, and obtain

S ¼ mla.

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state ðk�; l�P; l�AÞ where k

� is the cleaning price, and l�A and l�P, respectively, denote the labor force inthe abatement sector and the production sector15 which satisfies the following conditions:

Sðk; lAÞ ¼ Dðk; lPÞ ð18Þ

rUAðkÞ ¼ Y ðk; lPÞ ð19Þ

l ¼ lP þ lA: ð20Þ

Condition (18) requires that the market for pollution control clears.16 The mobility condition(19) implies that in equilibrium the job seekers are indifferent between the production sector andthe abatement sector. Finally, condition (20) requires that the aggregate labor market clears.Moreover, at the production sector’s reservation wage, Y ðk; lPÞ, the sector’s labor markets clear,and the excess supply (11) approaches zero. Is there an overall equilibrium with an activeabatement sector for a regulated market economy?Vacancies for abatement services are offered only if the cleaning price exceeds the sector’s

shutdown price kA, see Fig. 2. On the other hand, for cleaning prices exceeding k�i ðlÞ—where k�i ðlÞis the shutdown cleaning price for the industry i when a labor force of size l is staying in theproduction sector17—industry i is not viable, and its demand for pollution control falls to zero.Since k�CðlÞ4k�XðlÞ under assumption (CX), k�CðlÞ4kA is necessary for the two-sector model tohave a non-trivial solution.

Mobility rent: The mobility rent EY ðk; lPÞ ¼ rUAðkÞ � Y ðk; lPÞ is continuous and strictlyincreasing in k. Hence, for a total labor force of size l staying at the production sector, a uniquecleaning price kl with kAoklok�CðlÞ exists, for which the mobility rent is zero, EY ðkl ; lÞ ¼ 0,18 seeFig. 2 point B. Therefore, in an economy with a total labor force of size l a general equilibriumwith an active abatement sector can only develop at cleaning prices kXkl . At lower prices themobility rent is negative, and no job seeker would move to the abatement sector. Let kXkl. If thecleaning price rises, the reservation wage offered to job seekers in the abatement sector, rUAðkÞ,increases, see Fig. 2, the job creation process attracts a growing labor force lA and thecomplementary job destruction in the production sector reduces the labor force lP.

19 Recall thatthe production sector’s equilibrium reservation wage, Y ðk; lPÞ, is decreasing in both of itsarguments.20 Thus, the increasing cleaning price reduces Y ðk; lPÞ. Therefore, the decline of theproduction sector and its labor force lP must compensate first the negative effect of the increase ink and second raise Y ðk; lPÞ, such that the mobility condition (20) is fulfilled.

Equilibrium: With a rising abatement price, the labor force moves to the abatement sector,attracted by the newly opened vacancies and the growing labor income. But this reallocation ofthe labor force leads to a drop in the shutdown cleaning prices for the industries C and X.21

15Recall that the production sector consists of the polluting industries C and X implying that l�P ¼ l�C þ l�X.16Note that equilibrating supply and demand for abatement services does not necessarily mean that polluters abate all

of their emissions! The surviving jobs clean that part of their emissions, for which, for example, pollution control is

technically feasible or legally required.17See Remark 3 and Lemma A.5.18See Lemma A.6*.19See Lemma A.6*.20See Lemma 2.21See Lemma A.6*.

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Hence, an equilibrium with an active abatement sector exists only if we can find a cleaning pricekXkl at which at least the clean industry C is competitive, so that kpK�

C.22 Provided that at K�

C

the excess supply of the abatement market is positive there indeed exists a unique kn, withklok�pK�

C, at which the market for the abatement service clears.23

In the overall equilibrium ðk�; l�P; l�AÞ the clean industry is competitive and demands abatement

services. Whether the dirty industry survives depends on the industry’s shutdown price K�X. If

k�pK�X, as we will assume in the following, industry X is also viable.

For cleaning prices k, with klpkpk�, for which the equilibrium conditions (19) and (20) aresatisfied we are now prepared to prove:

Lemma 4*. At k the labor market tightness of the abatement sector is at least as high as that of the

clean and dirty industry. In particular, it is true that yAðkÞXyXðk; lPÞ4yCðk; lPÞ, where lP ¼ lPðkÞ isthe equilibrium labor force of the production sector. Moreover, the current match rents at the threelocations satisfy the inequalities pAðkÞppXðkÞopCðkÞ.

From Lemma 4* it follows that the average duration of unemployment, 1=pðyÞ, is shortest inthe abatement sector and longest in the clean industry, while the expected duration of a vacancy,1=qðyÞ, is longest in the abatement sector and shortest in C. Thus a reallocation of the mar-ginal worker from C to A would reduce the aggregate rate of unemployment, since the aggre-gate duration of unemployment declines and the job destruction rate l is the same at alllocations.What is the intuition for Lemma 4*? The result is a consequence of the model’s mobility and

entry assumptions. While job seekers are perfectly mobile, jobs are not. Therefore, in equilibrium,the expected income of an unemployed worker is the same everywhere, while the income of avacancy differs between sectors and locations. In particular, the income in the production sector isat least as high as that in the abatement sector, rViX0 ¼ rVA, i ¼ C;X.Workers have rational expectations. They anticipate that immediately after the separation

from the job, the job owner will open a new vacancy with an income equal to rViX0. Thisexpectation is reflected in the option value component of the return from the search. From (6)we know that the income of an unemployed worker in industry i is rUi ¼ b þ gðyiÞpi,while, according to (8), the income from a vacancy is rVi ¼ �c þ dðyiÞpi. Substituting forthe current match rent in the job seeker’s income equation and rearranging terms we findthat rUi ¼ b þ byiðc þ rV iÞ=ð1� bÞ. Hence, the higher the asset value of a vacancy in industryi the higher is, ceteris paribus, the income of an unemployed worker searching for a job inthat industry. The mobility rent the job seekers can expect attracts the unemployed of theother labor markets to the industry i, and the tightness at i declines until the mobility rentsvanish.Since in the steady state we have rVA ¼ 0, the unemployed in the abatement sector in fact do

not benefit from a positive asset value of the sector’s vacancies. Therefore, if the job seekerslooking for an abatement job were not compensated they would leave the sector and move to theindustries C or X. The tightness of the labor market for abatement services would increase, the

22See Remark 4. Lemma A.6* shows that for K�C the strict inequalities kloK�

CoknCðlÞ hold.

23See Lemma A.7.

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rate of transition from unemployment to employment would follow suit, and, finally, the optionvalue of the human capital of a job seeker would be driven up.

4. The emission tax

In the Pigouvian economy, the government regulates the flow of pollutants through a tax onemissions [20]. The tax rate, t, is defined in units of output per unit of the pollutant. A filled job inindustry i which does not control pollution must pay the tax tPi. Two intervals of the tax rate canbe distinguished with regard to the impact of the tax on the equilibrium rate of unemployment.

Tradeoff: The first interval is ½0;klÞ. Recall that for lP ¼ l, the upper support kl is the cleaningprice at which the mobility rent equals zero (see Fig. 2 point B). Hence, for cleaning pricesk 2 ½0; klÞ the mobility rent is negative and no job seeker will move to the abatement sector.Producers must pay the tax, and—if assumption (CX) holds—society is confronted with the wellknown trade-off between employment and environmental quality.

Proposition 1*. (1) If the government levies a tax t 2 ½0; klÞ on emissions, job seekers move from the

dirty industry X to the clean industry C, such that the labor market tightness increases in X anddecreases in C.(2) In the dirty industry the unemployment rate is a decreasing function and in the clean industry it

is an increasing function of t. The reallocation of the labor force induced by an increase in theemission tax causes the equilibrium rate of unemployment to rise and the quantity of emissions to fall.

The reason for this result is that the rising labor market tightness in X increases the rate oftransition into employment and reduces the pool of job seekers in industry X, while the decreasingtightness in C has just the opposite effects on the industry-specific transition and unemploymentrates. Since the labor market tightness at X is strictly higher and the average duration ofunemployment strictly lower than in industry C (see Lemma 4*), the reallocation of the laborforce induced by an increasing emission tax raises the aggregate rate of unemployment.

Complementarity: The second tax rate interval ½kl ; k�Þ is bounded from above by the cleaningprice k� which clears the market for pollution control. A tax rate t 2 ½kl ; k�Þ has the effect of aprice ceiling. In particular a cleaning price k4t cannot be sustained because paying the tax is theleast-cost strategy. On the other hand, at kot producers prefer to control their emissions and, dueto kotok�, an excess demand for abatement services develops. The abatement price begins torise until k ¼ t. Hence, in the equilibrium of the Pigouvian economy, producers are indifferentbetween paying taxes and controlling emissions, and we assume that they choose theenvironmentally sound strategy.If assumption (CX) holds, the employment and the environmental policy goal of emission

reduction are complementary at tax rates t 2 ½kl ; k�Þ:

Proposition 2*. At tax rates t 2 ½kl ;k�Þ the abatement sector offers vacancies to job seekers and

pollution control services to producers. As t increases, the cleaning capacity of the abatement sectorgrows and workers move from the industries C and X into the abatement sector. The unemployment

rates decrease in C and X and increase in A. Since the average duration of unemployment in theabatement sector is strictly shorter than in the production sector, this reallocation of the labor force

causes the equilibrium rate of unemployment to decrease.

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5. The emission standard

In the Pigouvian economy, the government regulates the flow of pollutants through a tax onemissions; in the command-and-control economy the regulator intervenes directly in the decisionsof the firms through binding performance standards [3]. The emission standard E is defined inunits of the pollutant and indicates the permitted quantity of emissions per job and time unit.Clean jobs for which PC � Ep0 have a free but non-tradable right to pollute. Dirty jobs withPX � E40 are obliged to control PX � E40 of the discharge volume PX and therefore demandabatement services. Thus, only the X-firms enter the market for pollution control, each occupiedjob demanding PX � E units of the service. At the cleaning price k, each firm in industry X facesthe abatement costs kðPX � EÞ. Therefore, the current match rent is pXðk;EÞ ¼ yX � kðPX �

EÞ þ c � b for a filled job in industry X, while, for a job with advanced technology, the rent is as inthe unfettered market economy pC ¼ yC þ c � b. From the assumption (CX) it follows thatpC4pXðk;EÞ if k40.In the equilibrium of the regulated sector, the income Eq. (6) and the mobility condition (7)

hold, and the excess supply (11) is zero. Consequently, the labor market tightness, the reservationwage, and the regulated sector’s demand for abatement services are functions of the cleaning pricek, the labor force located in the production sector lP, and the emission standard E. In view ofyX ¼ yXðk; lP;EÞ we write the aggregate demand for pollution control as

DXðk; lP;EÞ ¼ eXðyXðk; lP;EÞÞðPX � EÞ; ð21Þ

where eXðyXðk; lP;EÞÞ is the measure of filled jobs in industry X. Given the cleaning price k and thelabor force lP, the following lemma holds:

Lemma 5. If the regulator tightens the emission standard, workers move from the dirty industry intothe clean industry, the tightness in the labor market of industry X (C) increases (decreases), and the

production sector’s reservation wage, Y ðk; lP;EÞ, declines, while the reaction of the demand forpollution control, DXðk; lP;EÞ, is ambiguous.

Changes in the standard E cause two opposing reactions of the demand for abatement services.If the regulator cuts down the permitted quantity of emissions, demand increases, since filled jobsin the dirty industry must undertake additional efforts to control their emissions; on the otherhand, a stricter standard reduces the match rent of the X-jobs and induces job seekers to move toindustry C so that the tightness in the labor market of industry X grows. The rate of job seekersarriving at a vacancy in X declines, jobs are destroyed and the actively used part of the productioncapacity in X shrinks. How far can the regulator tighten the standard without threatening thecompetitiveness of industry X?The shutdown cleaning price k X for industry X is implicitly defined by the equation

rVXðk X; lP;EÞ ¼ 0. k X is an increasing function of the labor force lP and the standard E,k XðlP;EÞ.24

24See Lemma A.8.

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An overall equilibrium in the three labor markets of the command-and-control economy and inthe market for abatement services is a state ðk ; l P; l

AÞ which satisfies the conditions:

Sðk; lAÞ ¼ DXðk; lP;EÞ; ð22Þ

rUAðkÞ ¼ Y ðk; lP;EÞ; ð23Þ

l ¼ lP þ lA: ð24Þ

Whether an equilibrium with an active abatement sector exists depends on the viability of thedirty industry. From the mobility condition (23) and the allocation rule (24) it follows that lP andlA are functions of the cleaning price k and the standard E, where lP½lA decreases [increases] whenthe performance standard is tightened.25 As in Lemma A.6* it can be shown that for industry Xthere exists a shutdown cleaning price, K

XðEÞ, beyond which the industry is no longercompetitive. The shutdown cleaning price is an increasing function of the standard E. Therefore,in the command-and-control economy, an equilibrium with an active abatement sector exists onlyif the excess supply of the abatement sector is positive at the cleaning price K

XðEÞ.Intuitively, one would expect the equilibrium cleaning price to increase if the regulator tightens

the standard. Since the excess supply of the market for pollution control, ESðk;EÞ, is anincreasing function of k, the conjecture follows from the implicit function theorem if, and only if,ESðk;EÞ is increasing in E. But, in general, the excess supply does not adjust unambiguously tochanges in the performance standard. However, if �S, the (negative of the) elasticity of the supplyof abatement services with respect to the standard, exceeds the corresponding elasticity of thedemand, �D, ESðk;EÞ is, at least locally, an increasing function of E. The determinants of thedifference between these elasticities are spelled out in

Lemma 6. The excess supply in the market for pollution control is an increasing function of the

standard E if the supply and demand elasticities of the standard satisfy �S � �D40, where

�S � �D ¼E=PX

1� E=PX� aX�qX � �kA

: ð25Þ

The difference between the elasticities (25) depends on the fraction of free emissions, E=PXo1,on the fraction of the vacant capacity in the dirty industry, aX ¼ vX=kXo1, on the elasticity of thejob seekers’ rate of arrival at a vacancy in industry X with respect to E, �qX40, and on theelasticity of the abatement sector’s capacity with respect to E, �kA

40. Other things being equal,the difference (25) is higher, the higher the fraction of free emissions and the lower the fraction ofvacant capacity in X.If assumption (CX) holds, the policy goals of a low equilibrium rate of unemployment and a

high equilibrium quality of the environment are complementary even in a command-and-controleconomy regulated with an emission standard.

Proposition 3*. If the clearing price of the market for pollution control in the command-and-controleconomy increases when the regulator tightens the standard E, then the quantity of emissions is an

25See Lemma A.9.

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increasing function of E. Moreover, the environmental and the employment goals are complementary,

and the tightening of the standard leads to a reduction in the equilibrium rate of unemployment andan increase in the equilibrium quality of the environment.

6. Efficiency

The equilibria of the unfettered and the regulated market economies that we analyzed in thepreceding sections suffer from search and environmental externalities. In this section, we focus onthe optimal control problem and the implementation of the first-best allocation.

6.1. Social welfare function

The planner optimizes subject to the same matching technology and the same technologies ofthe production and the abatement jobs as firms and workers. Per unit time, ðki � yiuiÞ occupiedjobs in industry i produce the aggregate output ðki � yiuiÞyi and the quantity of emissionsðki � yiuiÞPi. The abatement sector removes ðkA � yAuAÞa units of the pollutant. The number ofjobs ki, i ¼ C;X, is exogenous, whereas the capacity of the abatement sector, kA, and the tightnessyi, i ¼ A;C;X, of the three labor markets are controlled by the planner. Each job seeker derivesutility of leisure b and thus the aggregate flow of utility per unit time is

Puib. Each vacancy costs

society c. Hence, in view of the yiui vacancies at location i, the aggregate hiring costs areP

yiuic.An amount NE of net emissions causes a per capita damage of ZðNEÞ. The damage function iscontinuously differentiable, convex, and monotonically increasing, Z040 and Z00

X0, and we haveZðNEÞ ¼ 0 for NEp0.The perfectly informed planner has an infinite time horizon and uses the control variables

ðy; kAÞ, with y ¼ ðyA; yC; yXÞ, to maximize the social welfare function

O ¼

Z 1

0

Xi¼C;X

ðki � yiuiÞyi þX

i¼A;C;X

uib �X

i¼A;C;X

yiuic � lZðNEÞ

" #e�rt dt ð26Þ

subject to the following four constraints. First, the net emissions satisfy

NE ¼X

i¼C;X

ðki � yiuiÞPi � ðkA � yAuAÞa: ð27Þ

Second, the planner is constrained by the transition equations of the state variables

_ui ¼ lðki � yiuiÞ � pðyiÞui; i ¼ A;C;X; ð28Þ

where _ui ¼ dui=dt. Third, the planner observes the resource constraint

l ¼X

i¼A;C;X

ki � ðyi � 1Þui½ ; ð29Þ

where li ¼ ki � ðyi � 1Þui is the labor force at location i, i ¼ A;C;X. Finally, the planner musttake into account the capacity restrictions yiuipki, i ¼ A;C;X.

Solution: We consider solutions of the optimal control problem with the following twocharacteristics: first, at each location there exists a positive number of occupied jobs, such that

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yiuioki, and, second, the planner utilizes the entire capacities ki, i ¼ A;C;X.26 Since the planneris not only restricted by the transition Eq. (28) but must also take account of the resourceconstraint (29), we need to solve an optimal control problem (26)–(29) of the Bolza-Hestenes type[38] with the Lagrangian function in present value terms

Lðu; y; kA; m; rÞ ¼X

i¼C;X

ðki � yiuiÞyi þX

i¼A;C;X

uib �X

i¼A;C;X

yiuic � lZðNEÞ

" #e�rt

þX

i¼A;C;X

mi½lðki � yiuiÞ � pðyiÞui þ r l �X

i¼A;C;X

½ki � ðyi � 1Þui

" #ð30Þ

where u ¼ ðuA; uC; uXÞ and m ¼ ðmA; mC; mXÞ. The co-state variable mi indicates the marginal socialcosts of an additional job seeker at location i ¼ A;C;X. The Lagrange multiplier r associatedwith constraint (29) is the shadow price of the labor force l. In the following, the shadow prices aredenoted in current value terms, with ~r ¼ ertr and ~mi ¼ ertmi.The FOCs of the optimal control problem are given in Appendix A. From the FOCs we get

Lemma 7. (1) The planner chooses the labor market tightness at the three locations such that the

location-specific marginal social revenue from filling an additional vacancy at location i equals theshadow price of labor:

~r ¼ b þ piðlZ0Þ þ ~miflþ qðyiÞ½1� ZðyiÞg; i ¼ A;C;X; ð31Þ

where piðlZ0Þ ¼ yi � lZ0Pi þ c � b, i ¼ C;X, and pAðlZ

0Þ ¼ lZ0a þ c � b are the current social

match rents and lZ0 with labor force l is the marginal social damage of the pollutant.(2) The capacity of the abatement sector satisfies the job creation condition:

~r ¼ lZ0a þ ~mAl ð32Þ

(3) The social revenue of the marginal job seeker at location i equals his location-specific marginal

social costs, so that

b � ~mi½r þ pðyiÞ ¼ ð1� yiÞ ~rþ yi½piðlZ0Þ þ b þ ~mil; i ¼ A;C;X: ð33Þ

Interpretation: The RHS of Eq. (31) denotes the marginal social revenue of filling an additionalvacancy at location i given the location’s capacity ki, i ¼ A;C;X. piðlZ

0Þ is the current social

match rent of the occupied job, while the term ~miflþ qðyiÞ½1� ZðyiÞg represents the expectedsocial costs of job destruction and creation. ~mi, with ~mio0, are the social costs of an additional jobseeker at location i, l is the rate with which occupied jobs are destroyed, and p0ðyiÞ ¼

qðyiÞ½1� ZðyiÞ is the search externality caused by posting or filling an additional vacancy. Thus,~milo0 are the expected social costs of job destruction, while ~miqðyiÞ½1� ZðyiÞo0 are the socialcosts of filling an additional vacancy at location i. ZðyiÞ is the elasticity of the hirings at location i,mðui; viÞ, with respect to the number of job seekers, ui. By the homogeneity of mðui; viÞ, ZðyiÞ is afunction of the labor market tightness only, and—because of Euler’s Theorem—we haveZðyiÞ 2 ð0; 1Þ.

26In view of ki ¼ ei þ vi þ I i, where I i is the idle capacity in i, the planner utilizes the entire capacity ki, if I i ¼ 0.

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The job creation condition (32) says that, for the first-best capacity of the abatement sector, thesocial revenue of the marginal occupied cleaning job equals the shadow price of labor. The socialrevenue of an additional occupied cleaning job, given the number of cleaning vacancies, is thedifference between the environmental damage avoided through the pollution control services ofthat job, lZ0a, and the expected social costs of job destruction, ~mAlo0.The social revenue of an additional job seeker at location i consists of his utility of leisure, b,

and the avoided social costs of unemployment, � ~mi½r þ pðyiÞ40, which together form the LHS ofEq. (33). The cost savings occur, first, because an additional worker added to the pool ofunemployed at the given tightness yi, will increase the number of job seekers who move fromunemployment into employment by pðyiÞ and, second, because the interest lost on the humancapital tied up in the pool of unemployed decreases by the amount r of the discount rate. On theother hand, the labor force located at i increases with the new job seeker by ð1� yiÞ. To keep thelabor market tightness constant, the planner must increase the number of vacancies by yi, which,at a given capacity, necessitates the destruction of yi occupied jobs. The increase in the labor forcecauses social costs of ð1� yiÞ ~r, and the destruction of yi occupied jobs brings about additionalcosts of yi½piðlZ

0Þ þ b þ ~mil.

We denote the steady-state solution of the optimal control problem by ðy; kAÞ, where y ¼

ðyA; yC; yXÞ is the vector of the efficient labor market tightness in the abatement sector and thepolluting industries, respectively, while kA is the first-best measure of vacant and producingabatement jobs.Given the assumption (CX) we obtain from the FOCs of the control problem:

Corollary 1*. The current social match rent of an occupied job in the clean industry is strictly largerthan in the dirty industry, pCðlZ

0Þ4pXðlZ

0Þ, while the socially efficient tightness of the labor market

in the clean industry is strictly smaller than in the dirty industry, yX4yC.

Comparing Corollary 1* with Lemma 4*, the similarity between the first-best solution and themarket solution is obvious. In both cases the tightness of the labor market in the dirty industry isstrictly larger than in the clean industry. But, as Lemma 4* shows, in addition, in the economyregulated with the emission tax, the tightness of the labor market in the abatement sector isgreater than the tightness in the producing industries. Whether the inequality yAXyi also holdsfor the first-best allocation, however, depends on the parameters of the model.Assume that the marginal environmental damage is a constant such that Z00ðNEÞ ¼ 0. Then the

comparative static analysis of the first-best allocation provides the following results.

Lemma 8. (1) An incremental change in the productivity of an abatement job, a, has the followingimpact on the first-best allocation: dyi=da40, i ¼ A;C;X, and dkA=da40.(2) A small change in the productivity of a polluting job at the industry i, yi, has the following

effects: dyi=dyio0, dkA=dyio0, i ¼ C;X, and dyj=dyi ¼ 0, where j ¼ A;C;X and jai.(3) A marginal change in the quantity of pollution per occupied job at industry i, Pi, implies:

dyi=dPi40, dkA=dPi40, i ¼ C;X, and dyj=dPi ¼ 0, where j ¼ A;C;X and jai.

Hence, an increase in the productivity of the jobs in industry i, yi, or a decrease in the quantityof pollution, Pi, will reduce the industry-specific tightness yi, without effecting the first-besttightness of the other two labor markets. Therefore, whether or not yAXyi depends in particularon the industry-specific parameters ðyi;PiÞ, i ¼ C;X.

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6.2. Implementation

We assume that the policy maker regulates the flow of emissions and the job creation anddestruction process with the policy instruments ðt;j; sÞ, where t is a tax on emissions, j ¼

ðjC;jXÞ is a vector of employment subsidies (taxes), and s ¼ ðsC;sXÞ is a vector of recruiting costallowances. A recruiting allowance is a flow si paid to the vacancies in industry i during the timeof search and recruiting job applicants. The employment subsidy is a flow ji paid to the occupiedjobs in industry i, i ¼ C;X.In the equilibrium of the abatement sector all profit opportunities from creating new jobs are

exhausted, so that VA ¼ 0. The private job creation condition of the abatement sector (15) followsfrom the income Eq. (14) and VA ¼ 0. The job creation condition (15) in turn implicitly definesthe equilibrium price for the abatement service as a strictly increasing function of the equilibriumtightness of the labor market for abatement workers, tðyAÞ. Given that the equilibrium price forthe abatement service, k, satisfies k ¼ t, the first-best tightness yA can be implemented by levying atax on emissions t at rate:

t ¼ tðyAðþÞ

Þ �b

aþ

1� dðyAÞ

dðyAÞ

c

a: ð34Þ

Next, we prove that the equilibrium price for the abatement service indeed satisfies k ¼ t.Suppose that kot were true. Then the equilibrium tightness of the labor market for abate-ment workers would be too small, yAoyA, and the net emissions of the economy would betoo high, NE4NE. But by virtue of the properties of the damage function the first-bestnet emission of the pollutant is non-negative, NEX0. Given that NE4NEX0 and kot thepolluting jobs could increase their profits through additional abatement efforts and hence anexcess demand would develop in the market for the abatement service—a contradiction whichproves that k ¼ t.In equilibrium, the job seekers who move to the abatement sector earn an income equal to

rUAðyAÞ, which is determined by the income Eq. (16). In the overall equilibrium of the regulatedeconomy, the mobility rent is zero, so that rUAðyAÞ ¼ Y , where Y is the reservation wage of theproduction sector. Thus, in equilibrium, the job seekers in the production sector earn a wageincome Y , which satisfies Y ¼ rUAðyAÞ.In the market economy firm and worker bargain over the distribution of the match rent

inclusive of the employment subsidy, piðtÞ þ ji, i ¼ C;X. Consequently for the economyregulated by the instruments ðt;j; sÞ we must modify the income Eqs. (6) and (8) of the jobseekers and the vacancies of the production sector, respectively, as follows:

rUiðt; yi;jiÞ ¼ b þ gðyiÞ½piðtÞ þ ji; ð35Þ

rViðt; yi;ji; siÞ ¼ �c þ dðyiÞ½piðtÞ þ ji þ si; i ¼ C;X: ð36Þ

Given the reservation wage Y, the tightness of the labor market in the industry i, yi,and the cleaning price t, the mobility condition for industry i follows from (35) with:Y ¼ b þ gðyiÞ½piðtÞ þ ji, i ¼ C;X. Solving for ji we get the employment subsidy ji necessary to

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implement the first-best labor market tightness of industry i:

ji ¼ jiðYðþÞ

; tðþÞ

; yið�Þ

Þ �Y � rUiðt; yi; 0Þ

gðyiÞ; i ¼ C;X: ð37Þ

In general, the policy scheme ðt; jÞ is not sufficient to implement the first-best solution as amarket equilibrium. The reason is that with ðt; jÞ the income of a vacancy in the production sectormight turn out to be negative, so that no vacancy is supplied. To guarantee the participation ofthe vacancies, the planner employs the recruiting allowances.The recruiting allowance that induces a positive supply of vacancies in industry i follows from

(36) by virtue of the fact that rViðt; yi;ji;siÞ ¼ 0 is necessary and sufficient to stimulate the profitmaximizing firms to supply vacancies at i. Assume that rViðt; yi;ji; 0Þo0. Then the minimalallowance that ensures the vacancies’ participation is

sið tðþÞ

; yiðþÞ

; jið�Þ

Þ ¼ �rViðt; yi;ji; 0Þ: ð38Þ

To implement the first-best solution as a market equilibrium the policy maker must subsidizethe vacancies of industry i with a recruiting allowance siX0, for which

si ¼ maxf0;siðt; yi; jiÞg; i ¼ C;X: ð39Þ

The policy vector ðt; j; sÞ, that satisfies the implementation conditions (34), (37) and (39),ensures that the private equilibrium is efficient and, moreover, that the participation conditions ofthe jobseekers and vacancies are fulfilled.

6.3. Evaluation

First, we show that the optimal emission tax is an increasing function of the workers’bargaining strength. Next, we discuss conditions, that ensure that the policy maker needs norecruiting allowances to induce the first-best allocation. Finally, we demonstrate that the emissiontax can generate a double dividend exactly under these conditions.From the FOCs of the optimal control problem we find that the first-best allocation fulfills the

job creation condition

pAðlZ0Þ � c=dðyAÞ ¼ 0; ð40Þ

where

dðyAÞ ¼ ½1� ZðyAÞqðyAÞ=dðyAÞ and dðyAÞ ¼ r þ lþ ZðyAÞpðyAÞ þ ½1� ZðyAÞqðyAÞ:

Comparing the private job creation condition (15) with the social job creation condition (40) weobtain:

Corollary 2. The optimal emission tax, t, that the government imposes to induce the first-best

pollution, is a strictly increasing function of the workers’ bargaining strength b. In particular, t�o

lZ0

iff b�oZðyAÞ.

Whether the government should levy the Pigouvian tax on emissions to stimulate the first-bestsolution, that is whether t ¼ lZ0, depends on the workers’ bargaining strength, as shown by

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Corollary 2. The threshold value for the bargaining strength is the first-best elasticity of the numberof hirings in the abatement sector with respect to the sector’s unemployment. In particular, for labormarkets where, as a consequence, for example, of the prevailing labor law, workers have a strongbargaining position, the optimal pollution tax may easily exceed the Pigouvian tax.Next, we state two corollaries with results concerning the recruiting allowances for the polluting

industries.

Corollary 3*. Given the assumption (CX), the optimal recruiting allowance for the dirty industry isat least as large as the allowance for the clean industry, sXXsCX0.

Whether the policy maker must employ all five instruments to reap the efficiency gains of atransition to the first best depends on the structure of the efficient allocation. In particular, therecruiting allowances are not always necessary to steer the economy into the first-best.

Corollary 4. The optimal allowance for the vacancies at industry i is equal to zero, si ¼ 0, if and onlyif yAXyi, or equivalently, if and only if pAðtÞppiðtÞ þ ji, i ¼ C;X.

Therefore, whether or not si ¼ 0 depends in particular on the industry-specific parametersðyi;PiÞ, i ¼ C;X, as Corollary 4 and Lemma 8 show.Consider the market economy from Section 4 and assume that it is regulated with an emission

tax t 2 ½kl ; k�Þ. Recall that k� is the cleaning price that clears the market for abatement services½NEðk�Þ ¼ 0, and that kl is the price at which the mobility condition (19) is fulfilled for a givenlabor force of size l being located in the economy’s production sector. From Proposition 2 weknow that an increase in the tax rate t 2 ½kl ; k�Þ raises the environmental quality and reduces theequilibrium unemployment.In an economy regulated by the policy instruments ðt;jÞ the boundaries of the above tax

interval, ½klðjÞ;k�ðjÞÞ, are—strictly increasing—functions of the employment subsidies j.Consider now an optimal policy ðt; j; sÞ, with sX ¼ 0. Moreover, suppose that the assumption(CX) holds. Then, in view of Corollary 3*, sC ¼ 0. Hence, the instruments ðt; jÞ suffice toimplement the first-best solution and in addition, we can apply Lemma 4* and Proposition 2 tothe economy regulated with the emission tax t 2 ½klðjÞ; k�ðjÞÞ. Given that the first-best netemission of the pollutant is strictly greater than zero, NE40, we have t 2 ðklðjÞ;k�ðjÞÞ, and wecan state without proof:

Corollary 5*. (1) An incremental increase in the emission tax t 2 ½klðjÞ; tÞ raises social welfare,

enhances the quality of the environment and reduces the aggregate unemployment; hence the societyreaps a double dividend.(2) In contrast, a marginal increase in the emission tax t 2 ðt;knðjÞÞ reduces social welfare,

despite the fact that the environmental quality improves and the equilibrium unemployment declines.

7. Conclusions

The paper integrates instruments of environmental policy into the theory of equilibriumunemployment. The labor markets are characterized by two-sided search. Spatial and otherheterogeneities, informational imperfections and market institutions are implicitly contained in

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the matching technology of the labor markets. Two production technologies are available. Foreach of these there is an exogenous number of jobs which are either filled or advertised asvacancies in the job market. Each job is associated with a capital good, the design of whichdetermines the emitted quantity of a pollutant produced as a by-product and in fixed proportionsof the output. Clean jobs use an advanced technology and hence enjoy a higher productivity thanjobs in the dirty industry. The environmental damage is a public bad that reduces the welfare ofjob owners and job seekers alike. Under certain conditions, an abatement sector develops whichoffers vacancies to job seekers and supplies pollution control services to the polluting industries.The analysis yields the following results. First, in an economy where a regulator imposes a tax

on the emission of pollutants, we can distinguish two intervals of the tax rate differing in theiremployment effects: in the first interval the tax is so low that an abatement sector is not viable. Inview of the sector’s low wage income, job seekers have no incentive to leave the pollutingindustries and move to the abatement sector. Society is faced with the well-known trade-offbetween employment and environmental quality. In the second interval, an abatement industrydevelops which provides vacancies for the job seekers and cleaning services for the pollutingindustries. With an increasing emission tax, gross and net emissions decline and so too does theequilibrium rate of unemployment. Aggregate employment and environmental quality arecomplementary policy goals.Second, if the market price for pollution control services increases with the strictness of the

emission standard, then the effects of a command-and-control approach are very similar to thosewe observe in a Pigouvian economy: reducing the equilibrium rate of unemployment andincreasing the equilibrium quality of the environment are complementary policy goals.Third, in general, five policy instruments are necessary for the government to internalize the

environmental and search externalities and to induce the participation of job seekers andvacancies. In our model, in addition to the pollution tax, for the production sector the policymaker uses employment subsidies for the occupied jobs and recruiting allowances for thevacancies. The optimal emission tax is a strictly increasing function of the workers’ bargainingstrength and may easily exceed the Pigouvian tax rate. The normative analysis reveals that, withrespect to the tax interval where the policy goals of a clean environment and a low unemploymentare complementary, we can distinguish two different segments. In the lower segment an increase inthe emission tax produces efficiency gains, environmental quality increases and equilibriumunemployment declines. However, in the upper segment of the tax interval an incremental taxincrease causes efficiency losses, although the quality of the environment increases and thenumber of unemployed workers declines.For future research, it seems worthwhile to analyze the effects of environmental policy within a

search model with endogenous job destruction and wage posting. Other important research topicsare the implementation of second-best environmental policy instruments and the analysis of theimpacts of tax recycling.

Acknowledgments

Especially I want to thank Rudiger Pethig for his tremendous help. Furthermore, I am gratefulto the editors and two anonymous referees, to Gerard Gaudet and Christopher Pissarides, to the

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participants at the EARE 2000 conference, Crete, the EEA 2000 Conference, Bolzano and theEALE/SOLE 2000 conference, Milan. Finally, I thank Elke J. Jahn and Peter Mottershead fortheir helpful comments.

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