Emission taxes and standards in a general equilibrium with entry and exit

Journal of Economic Dynamics & Control 61 (2015) 34–60

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Journal of Economic Dynamics & Control journal homepage: www.elsevier.com/locate/jedc

Emission taxes and standards in a general equilibrium with entry and exit Zhe Li a, Jianfei Sun b,n a School of Economics and Key Laboratory of Mathematical Economics, Shanghai University of Finance and Economics, 777 Guoding Road, Shanghai 200433, China b Antai College of Economics and Management, Shanghai Jiao Tong University, 1954 Huashan Road, Shanghai 200030, China

a r t i c l e in f o

abstract

Article history: Received 7 February 2015 Received in revised form 15 August 2015 Accepted 3 September 2015 Available online 11 September 2015

This paper studies and compares the welfare effects of emission taxes and emission standards in a general equilibrium model with two sectors in which plants can freely enter and exit. In one of the sectors plants differ in their productivity, produce differentiated goods, and generate emissions that can be reduced using an abatement technology. An emission reduction policy causes resource reallocation among plants and across sectors in two ways: a static way due to the dispersion of productivity and a dynamic way due to entry and exit. The model shows that the static distributional effect favors the emission tax, while the dynamic distributional effect favors the emission standard. Calibrated to Canadian data, the model shows that the dynamic effect dominates the static one and hence the emission standard dominates the emission tax in terms of welfare. This is the case not only in the baseline model, but also in a model with a large variation of parameter values for productivity dispersion, market power, and abatement efficiency. & 2015 Elsevier B.V. All rights reserved.

JEL classification: E69 Keywords: General equilibrium Productivity dispersion Entry and exit Abatement technology Emission tax Emission standard

1. Introduction There has been a long debate on whether market-based environmental policy instruments outperform “command and control” policy instruments. Traditionally, “command and control” policies were predominant. Starting from the 1970s, market-based instruments started to be favored (see Nordhaus, 2007 and Stern, 2006, part IV, p. 310). However, a large literature argues that regulatory intensity standards can dominate market instruments when market power and leakage exist (see Buchanan, 1969 and Holland, 2009). We contribute to this literature by analyzing how productivity dispersion and plants' entry and exit affect the comparison between a market-base instrument (an emission tax) and a “command and control” policy (an emission intensity standard). This paper studies the welfare effects of emission taxes and emission standards in a general equilibrium model with two sectors in which plants can freely enter and exit. In one of the sectors, which we will call the dirty sector, plants generate emissions in their production process and adopt an identical abatement technology to reduce their emissions. Since these plants differ in their productivity and produce differentiated goods, their use of the abatement technology as well as the amount of abatement input that they apply to emission reduction efforts may differ across plants and vary under different n

Corresponding author. Tel.: þ86 21 52301597. E-mail addresses: [email protected] (Z. Li), [email protected] (J. Sun).

http://dx.doi.org/10.1016/j.jedc.2015.09.001 0165-1889/& 2015 Elsevier B.V. All rights reserved.

Z. Li, J. Sun / Journal of Economic Dynamics & Control 61 (2015) 34–60

35

policies. The different reactions of plants to an emission reduction policy will cause resource reallocation among plants and across sectors. Emission taxes and emission standards lead to different patterns of resource reallocation. We identify a static distributional effect—the reallocation among existing plants—and a dynamic distributional effect—the reallocation caused by plants' entry and exit. The static distributional effect favors the emission tax, since the value-weighted aggregate productivity remains unchanged after imposing the tax, but decreases with the imposition of an emission standard. The dynamic distributional effect favors the emission standard, since in the long run that policy leads to a smaller number of plants and larger average size than under the emission tax. The model shows that the choice between emission taxes and emission standards will depend on which distributional effect dominates. Calibrated to Canadian data, the model shows that the dynamic distributional effect dominates the static distributional effect and, hence, the emission standard dominates the emission tax. Moreover, if we vary the parameter values governing productivity dispersion, market power, and abatement efficiency around the baseline model, the emission standard still dominates the emission tax. However, when the abatement technology becomes very efficient, the static distributional effect due to productivity dispersion can dominate the dynamic effect due to plants' entry and exit, thus turning the tax into the favored policy instrument. Using a numerical model, this paper captures many elements shown in the literature to be important for evaluating the welfare effects of emission reduction policies, and it examines the compound effects of these elements in a unified framework. The first of these elements is the use of an optimal emission tax to correct externalities, known as Pigovian tax, which was first proposed by Pigou (1954). Buchanan (1969) introduced the idea that market power influences the optimal level of an emission tax. The inefficiency of emission standards was established by Helfand (1991) and extended to study leakage by Holland (2009). The effect of productivity dispersion and the efficiency of abatement technology on the comparison between an emission tax and an emission standard was shown in Li and Shi (2015). However, a comprehensive model with all the elements mentioned above and with free entry and exit has not been analyzed or quantitatively examined yet. The current paper introduces free entry and exit, and puts it into a general equilibrium model with market power, productivity dispersion, and efficiency of abatement technology in order to compare optimal emission taxes and optimal emission standards. The model framework used in this paper is a Macroeconomics general equilibrium model with heterogeneous plants as in Ghironi and Melitz (2005) and Alvarez and Lucas (2007) that adopts the technique developed in the international trade models by Eaton and Kortum (2002) and Melitz (2003). The only diversion of our model from the standard Macroeconomics model with heterogeneous plants is that we include an externality in the production of dirty goods and we allow plants to abate their emissions. The volume of papers that share this type of diversion, i.e., explore the environmental issues in a Macroeconomics model, has been growing in recent years (see, for example, Tang et al., 2014; Adao et al., 2014, and Li et al., 2014). There are also some classical papers that address international trade and the emission control problem in an open economy general equilibrium model (see Copeland and Taylor, 2005 among others). The current paper extends the general theory of Li and Shi (2015) in two ways. First we incorporate plants' free entry and exit into the model to make the model a dynamic one. Second, we calibrate the model with Canadian data and compare the welfare effects of an emission tax and an emission standard numerically. To our knowledge, the current paper is the first one that addresses the role of plants' entry and exit in the comparison between emission taxes and standards. It is also the first one that quantifies the importance of the dynamic distributional effects of plants' entry and exit for the welfare evaluation of emission taxes and emission standards in the long run. The current paper also shares some features of a Melitz-type model with Konishi and Tarui (2015), who qualitatively study how different emissions trading mechanisms affect intraindustry reallocation. However, their paper's key predictions depend to a large extent on some specific assumptions that were used to gain model tractability. For example, they assume that the firm's entry expenditure discharges the same proportion of emissions as the production input does. This paper uses an additive disutility function from emission stock that makes it possible to study the uncertainty of the damage from emission stock, although that is not the focus of the current paper. Potentially, the model can provide new insight on the comparison between price and quantity instruments regarding plants' reaction to policies and their associated distributional and aggregate costs. This could be compared to the classical work by Weitzman (1974), who found that the shapes of the marginal abatement benefit and the marginal abatement cost are crucial for the choice between a price and a quantity instrument. With heterogenous plants and plants' entry and exit, the marginal cost from abating emissions consists not only of the marginal abatement cost, but also of the cost caused by misallocation of resources across plants and across sectors due to productivity dispersion and entry and exit. Therefore, the comparison between different policies will also have to consider both the static and the dynamic distributional effects. Our quantitative results show that if there is damage uncertainty, the emission standard is more reliable, while the emission tax may cause the realized level of emissions to largely deviate from the optimal level. The rest of this paper proceeds as follows. Section 2 describes the model setup; Section 3 characterizes the optimal choices; Section 4 provides aggregation of variables, plants' entry and exit conditions, and market clearing conditions; it also defines the equilibrium and solves an equilibrium without emission reduction; Section 5 solves for the social planner's problem; Section 6 solves for the equilibrium under emission taxes and the equilibrium under emission standards, and it also compares the static and dynamic distributional effects under different policies; Section 7 parameterizes the model; Section 8 conducts

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Z. Li, J. Sun / Journal of Economic Dynamics & Control 61 (2015) 34–60

numerical experiments; Section 9 concludes. Finally, Appendix A provides proofs of the results in Section 6 and Appendix B provides data description.

2. The model 2.1. The households The economy is populated by a unit measure of identical households. The representative household is infinitely lived and has preferences over streams of consumption goods and pollutant stocks (pollution) at each date. The expected discounted life time utility is 1 n o X 1=ρ U 0 ¼ E0 βt ð1  αÞmρt þ αqρt g ðDt Þ : ð2:1Þ t¼0

Here, mt is the consumption of clean goods, qt is the consumption of an aggregate of dirty goods, and Dt is the level of the pollutant stock. Restrict  1 o ρo 0, so that the clean goods and the dirty aggregate are poor substitutes. The elasticity of substitution between mt and qt is 1=ð1 ρÞ, and 0 o αo 1 parameterizes the relative importance of qt : The disutility from  Ψ pollution is defined by a function g ðDt Þ ¼ λ Dt =D , where the parameters λ 40 and Ψ 40. Here, D 40 is a number, referred to as a reference level of the pollutant stock. The subjective discount factor is β A ð0; 1Þ. Every household is endowed with l units of resource per period. This resource is the only input required for production. The empirical counterpart of this resource is a Cobb–Douglas combination of capital, labor, and energy. The household supplies l inelastically. The production of dirty goods generates emissions. The total amount of emissions generated in period t is denoted as Et. Emissions accumulate according to Dt ¼ ð1  δÞDt  1 þ Et ;

ð2:2Þ

where δA ð0; 1Þ is a natural decay factor for the pollutant stock. Let Dt  1 be the pollutant stock at the beginning of the period. Different dirty goods appear in the utility function through the following aggregator: Z σ=ðσ  1Þ ðσ  1Þ=σ qt ¼ qi;t di : ð2:3Þ i A Ωt

Here, σ 4 1 is the elasticity of substitution across dirty goods, and Ωt is the set of dirty goods available at period t. Let Ω ¼ [ t Ωt be the entire set of dirty goods available over time, where Ω is assumed to be a continuum. 2.2. The producers Potential plants can choose to enter either the clean sector (with no entry cost) or the dirty sector (with a sunk entry cost fe in units of resource). For simplicity, we assume f e 40 is fixed over time. The plants enter the dirty sector if the present value of the expected profit stream can cover this entry cost. If a plant chooses to enter the dirty sector at time t, it can start producing at time t þ 1. After entry, the plant draws a productivity level x from a common distribution G(x) with support on ½xmin ; 1Þ, where xmin 40 is the lower bound of productivity level. As shown later, the plant's profit is an increasing function of the level of productivity. Because every plant needs to pay a positive fixed cost in order to produce, some low productivity plants do not produce and exit immediately after entry; only plants with a productivity level above a threshold value xe;t will stay. We define the set of producing plants by Φt ; Φt D½xe;t ; 1Þ. The producing plants keep their productivity levels until they are hit by an exogenous exit-inducing shock with probability μ. This exit-inducing shock is independent of the plants' productivity levels, so Gðxe;t Þ also represents the productivity distribution of all producing plants.1 2.2.1. The clean sector The clean good Mt is produced by a linear production technology: M t ¼ XLm;t , where X 4 0 represents the level of productivity and Lm;t Z 0 is the quantity of the resources used to produce clean good M t in period t. The competitive feature of this market ensures that the factor price wt equals the level of productivity, i.e., wt ¼ X. 2.2.2. The dirty sector Potential plants are identical before they enter the dirty sector. Upon entry, each plant draws a productivity level x from the common distribution G(x). This productivity level remains constant for the plant thereafter. Thus, a plant with productivity x is referred to as plant x . Plants are monopolistically competitive, with each one producing a variety of goods. So the good produced by plant x can also be indexed by x. Under the specification of the aggregator in (2.3), the elasticity of the demand for each good is σ. 1 In order to simplify the analysis, we will only look at the equilibria in which the tax rate or the standard stays the same in every period. A change in pollution regulation can be seen as a permanent shock which can move one stationary equilibrium to another.

Z. Li, J. Sun / Journal of Economic Dynamics & Control 61 (2015) 34–60

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In order to produce, a plant needs to pay a fixed cost f 40 in units of resource. For simplicity, assume that this fixed cost is the same for all the plants in all the periods. Hence, some plants never produce if they draw a low level of productivity after entry. Producing the dirty goods generates air emissions. The production technology of a plant that draws productivity x and its emission generation function are qt ðxÞ ¼ xLg;t ðxÞ

and

et ðxÞ ¼ zt ðxÞ  1=γ bLg;t ðxÞ;

ð2:4Þ

where Lg;t ðxÞ is the variable input required for producing goods x. The parameter b4 0; referred to as an emission factor, captures the intensity of emissions generated from production. We allow the plant to reduce emissions by incurring abatement input La;t ðxÞ: For tractability, and without loss of generality, we normalize La;t ðxÞ by the amount of resource used in production, Lg;t ðxÞ; and define a cleanliness index as zt ðxÞ ¼ 1 þLa;t ðxÞ=Lg;t ðxÞ. The parameter γ 4 0 measures the efficiency of abatement technology. 2.3. The government policy As clear from (2.1), pollution is an externality that is not internalized in a laissez-faire economy. This creates role for government policy. The government monitors and regulates emissions on behalf of consumers. Two instruments are available for that: one is an ad valorem tax on emissions, the other is a uniform standard on the emission-output ratio. When the tax instrument is applied, the tax revenue is returned to consumers as a lump sum transfer. The government's budget constraint is T t ¼ τ t Et ;

ð2:5Þ

where Tt is a lump sum transfer to consumers, and τt is the tax rate. When a standard st on the emission-output ratio is set, all the plants are required to satisfy et ðxÞ=qt ðxÞ rst : We study an emission intensity standard, rather than a cap on emission levels, for its theoretical and practical relevance. The intensity standard increases output relative to the cap on the emission levels (e.g., Fischer and Springborn, 2011). Since increasing output relaxes the intensity standard, the intensity standard acts as a subsidy to output, thereby reducing the inefficiency of under-production caused by market power. Because of this feature, some countries, including Canada, China, and India, have announced plans to pursue intensity targets to reduce greenhouse gas emissions (Herzog et al., 2006; Pizer, 2005).

3. The optimal choices 3.1. The households' problem The representative household enters period t with an endowment of the resource l 4 0 and mutual fund share holdings At ; which finance the continuing operation of all pre-existing dirty plants and all new entrants in the dirty sector during period t. During period t; a mass Nt of dirty plants is in operation and pays dividend, and a mass N e;t of new dirty plants enters. The average value of the operating dirty plants is denoted as v~ t and the average value of the new dirty plants is denoted as v~ e;t : v~ t and v~ e;t will be defined later. In each period the mutual fund pays a total profit that is equal to the total profit of all the dirty plants producing in that period. The average profit of a share is denoted as π~ t : Let T t be the lump sum transfer from the government. The period budget constraint of the representative household (in units of the clean goods) is P t qt þ mt þ ðv~ t Nt þ v~ e;t Ne;t ÞAt þ 1 ¼ wt l þ ðv~ t þ π~ t ÞN t At þT t ;

ð3:1Þ

where wt is the resource price denominated in clean good and Pt is the price of the aggregate of the dirty goods. The mass of dirty plants evolves according to Nt þ 1 ¼ ð1  μÞNt þ ð1 μÞNe;t ð1  Gðxe;t ÞÞ:

ð3:2Þ

A proportion μ of the remaining plants will be hit by the exogenous exit shock at the end of period t, so only a proportion 1  μ of them continues to the next period. The plants with productivity levels lower than xe;t exit immediately after entry, so only a proportion 1  Gðxe;t Þ of new entrants will stay to the end of period t, and a proportion ð1  μÞ of them will actually produce in period t þ 1.2 Given the budget constraint (3.1), the household chooses mt , qt ; and the share At þ 1 to maximize expected intertemporal utility (2.1). The first order conditions give that  U m 1  α mt ρ  1 1 ¼ ¼ : α Pt Uq qt

ð3:3Þ

2 Without aggregate shock, Gðxe;t Þ ¼ Gðxe;t þ 1 Þ: Both an emission tax and an emission standard are considered as a one-time aggregate shock. Since we will only solve the steady state equilibrium, the case where Gðxe;t Þa Gðxe;t þ 1 Þ is ignored. Otherwise, after an emission tax or an emission standard is      imposed, we have N t þ 1 ¼ ð1 μÞ N t þ N e;t ð1 Gðxe;t ÞÞ ð1  Gðxe;t þ 1 ÞÞ = ð1  Gðxe;t ÞÞ :

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The Euler equation for the share is   v~ t ¼ Et Rt þ 1 ðv~ t þ 1 þ π~ t þ 1 Þ ;

ð3:4Þ

where Rt þ 1 ¼ βð1 μÞ is the discount factor. Let pi;t denote the price of dirty good i A Ωt . Given a budget on the dirty goods P t qt , the household will choose the optimal demand for each variety of goods according to the following maximization problem: Z maxqt s:t: pi;t qi;t di ¼ P t qt : qi;t

i A Ωt

The optimal demand for good i is  σ Pt : ð3:5Þ qi;t ¼ qt pi;t hR i1=ð1  σÞ ; derived by multiplying each side of the Eq. (3.5) by pi;t and integrating the new The relative price is P t ¼ i A Ωt p1i;t σ di equation over i. 3.2. The producers' problem In this subsection we solve the producing producers' optimal problem, so we restrict plants' productivity levels to x A Φt . We leave producers' entry and exit decision to the next section. 3.2.1. Plants' optimal decision under ad valorem emission tax If an emission tax is imposed, a producing plant can either emit and pay the emission tax, or abate its emissions. The objective of the plant is to choose the optimal La;t ðxÞ to minimize the total cost consisting of τt et ðxÞ and wt ðLa;t ðxÞ þ Lg;t ðxÞÞ, taking as given the resource price wt and the tax rate τt ; subject to the production input not being less than Lg;t ðxÞ. The problem is equivalent to choosing the optimal cleanliness index zt ðxÞ; zt ðxÞ ¼ 1 þ La;t ðxÞ=Lg;t ðxÞ; to minimize the cost per unit of production input, that is, minτt bzt ðxÞ  1=γ þ wt zt ðxÞ: zt ðxÞ

Solving the above optimization problem, the optimal cleanliness index is 8 γ=ðγ þ 1Þ > < τt b if τt 4 γwt =b; zt ¼ γwt > : 1 otherwise: Note that zt is identical for all plants. Moreover, if the tax rate τt is lower than γwt =b; the plants do not abate their emissions but rather pay the tax. This is because the marginal abatement cost is an increasing function of La;t ðxÞ. We can derive the marginal abatement cost from the emission generation function in (2.4)   dwt La;t ðxÞ γwt La;t ðxÞ 1=γ þ 1 ¼ 1þ : det ðxÞ Lg;t ðxÞ b     It is obvious that d wt La;t ðxÞ =dðet ðxÞÞ increases in La;t ðxÞ and d wt La;t ðxÞ =dðet ðxÞÞ ¼ γwt =b when La;t ðxÞ ¼ 0. As a result, only if the tax rate τt 4 γwt =b; the plant will choose to abate its emissions. The profit maximizing plants will set prices according to a fixed markup over variable costs, including the marginal cost of producing goods, the marginal cost of reducing emissions, and the marginal tax payment. It is easy to show that this pricing strategy is optimal for the plant. The optimal price is pt ðxÞ ¼ ϕt x  1 ; where ϕt is defined as follows: 8  γ=ðγ þ 1Þ > < σ=ðσ  1Þw ðγ þ 1Þ τt b if τt 4γwt =b; t γwt ð3:6Þ ϕt ¼ > : σ=ðσ  1Þw ð1 þ τ b=w Þ otherwise: t t t It is obvious that pt ðxÞ decreases in x and pt ðxÞ increases if τt increases. The aggregate price is " #1=ð1  σÞ Z 1  1  σ Nt pt ðxÞ dGðxÞ ; Pt ¼ 1  Gðxe;t Þ xe;t and the aggregate quantity is ( )σ=ðσ  1Þ Z 1 Nt ½qt ðxÞðσ  1Þ=σ dGðxÞ : Qt ¼ 1  Gðxe;t Þ xe;t The producers do not choose the quantity of supply, which is determined by equilibrium demand given the prices.

Z. Li, J. Sun / Journal of Economic Dynamics & Control 61 (2015) 34–60

39

 σ According to the demand function, the quantity of x is qt ðxÞ ¼ P σt Q t pt ðxÞ . The corresponding emission level is  1=γ σ σ σ 1 bP t Q t ϕt x . A notable feature is that the elasticity of substitution among dirty goods, σ, influences the diset ðxÞ ¼ zt persion of qt(x) and et(x) across plants. The higher σ is, the easier the goods can be substituted by others, and the larger the dispersion of qt(x) and et(x) is. The profit of plant x is π t ðxÞ ¼ 1=σP σt Q t ½pðxÞ1  σ  wt f :

ð3:7Þ

Since pt(x) is a decreasing function of x; π t ðxÞ increases in x. 3.2.2. Plants' optimal decision under emission standard If an emission standard st is imposed, et ðxÞ=qt ðxÞ rst has to be satisfied. Recall that the amount of emissions is determined by both the production and the abatement, i.e. et ðxÞ ¼ zt ðxÞ  1=γ bqt ðxÞx  1 ; so the emission standard requires zt ðxÞ  1=γ bx

1

r st :

This condition is less restrictive if x is high. If x is higher than b=st ; then the plant x does not need to abate its emissions. Let xn;t ¼ b=st , then xn;t is the threshold value of productivity, below which plants need to abate their emissions by raising the value of zt ðxÞ above 1. The choice of abatement and the value of zt(x) depend on the plants' productivity level x, 8 > <  1 if x 4xn;t ; γ b zt ðxÞ ¼ if x r xn;t : > : st x The profit maximizing plants will set prices according to a fixed markup over variable costs, including the marginal cost of producing goods and the marginal cost of reducing emissions, given the resource price wt. The optimal price is pt ðxÞ ¼ σ=ðσ  1Þwt zt ðxÞx  1 : It is obvious that pt ðxÞ decreases in x: The emission standard affects price level through zt(x): if standard st becomes more stringent, then zt(x) (weakly) increases and pt(x) (weakly) increases.  σ According to the demand function, the quantity of x is qt ðxÞ ¼ P σt Q t pt ðxÞ . Depending on whether the plants abate their emissions, their emission level will be σ 1 ( σ  x if x 4xn;t ; bP t Q t pt ðxÞ  σ et ðxÞ ¼ if x r xn;t : st P σt Q t pt ðxÞ The profit of plant x is  1  σ wt f : π t ðxÞ ¼ 1=σP σt Q t pt ðxÞ

ð3:8Þ

Again, π t ðxÞ increases in x.

4. Aggregation and equilibrium 4.1. Plant average and aggregation In order to aggregate the variables, we parameterize the distribution of productivity draws GðxÞ: As in Melitz (2003) we use a Pareto distribution for x. Define GðxÞ ¼ 1  ðx=xmin Þ  k ; for x Zxmin ; where k 4 σ ðγ þ 1Þ is the shape parameter that indexes the dispersion of productivity draws.3 Dispersion decreases as k increases, and the plant productivity levels are increasingly concentrated toward their lower bound xmin : The standard deviation of ln x is equal to 1=k. 4.1.1. Emission tax Under the emission tax, all the plants with different x reduce their emissions in the same proportion. Let us define a special “average” productivity levels, x~ t ; for all the producing plants  1=ðσ  1Þ  1=ðσ  1Þ Z 1 k xσ  1 dGðxÞ ¼ υxe;t ; where υ ¼ x~ t ¼ 1  Gðxe;t Þ x A Φt k þ 1 σ   and Φt D xe;t ; 1 . Note that the integration requires k þ 1  σ 4 0 for σ 4 1: It is easy to show that x~ t completely summarizes the information in the distribution of productivity levels G(x) relevant to all aggregate variables. Thus, this economy is isomorphic, in terms of all aggregate outcomes, to one where Nt plants have a productivity x~ t . Accordingly, p~ t  pðx~ t Þ 3

k 4 σ ðγ þ 1Þ is required to ensure that the variance of the plants' effective productivity is finite under the emission standard.

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Z. Li, J. Sun / Journal of Economic Dynamics & Control 61 (2015) 34–60

1=ð1  σÞ represents the average price and the price of the dirty aggregate is P t ¼ N t p~ t ; q~ t  qðx~ t Þ represents the average quantity σ=ðσ  1Þ 4 q~ t . and the quantity of the dirty aggregate is Q t ¼ N t Similarly, denote π~ t  πðx~ t Þ as the average profit, and e~ t  et ðx~ t Þ as the average level of emissions. It is easy to show that

π~ t ¼ 1=σP σt Q t ½pðx~ t Þ1  σ  wt f ;

ð4:1Þ

 1=γ σ and e~ t ¼ zt bP t Q t ϕt σ x~ σt  1 . The aggregate level of emissions is Et ¼ Nt e~ t .

4.1.2. Emission standard Under the emission standard, plants are divided into two groups. Denote Na;t and N n;t as the mass of plants that abate  k and do not abate, respectively. Thus, the proportion of plants that do not abate is nn;t ¼ Nn;t =N t ¼ xe;t =xn;t ; while the  k proportion of plants that abate is na;t ¼ N a;t =Nt ¼ 1  xe;t =xn;t : The average productivity of the plants that do not abate is " x~ n;t ¼

1 1  Gðxn;t Þ

Z

1

#1=ðσ  1Þ σ1

x

dGðxÞ

¼ υxn;t :

xn;t

The average productivity of the plants that abate is " x~ a;t ¼

1 Gðxn;t Þ  Gðxe;t Þ

Z

xn ;t

xðγ þ 1Þðσ  1Þ dGðxÞ

xe;t

 ϖ¼

where

#1=½ðσ  1Þðγ þ 1Þ

k k  ðγ þ 1Þðσ  1Þ

1=½ðσ  1Þðγ þ 1Þ

2

31=½ðσ  1Þðγ þ 1Þ  xe;t k  ðγ þ 1Þðσ  1Þ 61  x 7 6 7 n;t ¼ ϖxe;t 6 7  4 5 xe;t k 1 xn;t

:

The average price ðp~ a;t ; p~ n;t Þ and quantity ðq~ a;t ; q~ n;t Þ can be calculated according to the average productivity. The aggregate price of the dirty goods is defined as h i1=ð1  σÞ : P t ¼ Na;t ðp~ a;t Þ1  σ þ Nn;t ðp~ n;t Þ1  σ The average profit is defined as ð4:2Þ π~ t ¼ na;t π~ a;t þnn;t π~ n;t ;  1  σ  1  σ σ σ  wt f and π~ n;t ¼ 1=σP t Q t pðx~ n;t Þ wt f are the average profit of the plants that abate and where π~ a;t ¼ 1=σP t Q t pðx~ a;t Þ do not abate, respectively. Define " x^ a;t ¼

1 Gðxn;t Þ  Gðxe;t Þ

Z

xn;t

#1=ðσðγ þ 1ÞÞ xσ ðγ þ 1Þ dGðxÞ

xe;t

;

then 2

x^ a;t

31=ðσðγ þ 1ÞÞ  xe;t k  σðγ þ 1Þ 1  6 7 xn;t 6 7 ¼ ωxe;t 6 ; 7

k 4 5 xe;t 1  xn;t

 where

ω¼

1=ðσðγ þ 1ÞÞ k : k σ ðγ þ 1Þ

 σ  σ 1 σ The aggregate emission level is Et ¼ N a;t e^ a;t þ Nn;t e~ n;t ; where e^ a;t ¼ st P σt Q t pt ðx^ a;t Þ and e~ n;t ¼ bP t Q t pt ðx~ n;t Þ x~ n;t are the average amount of emissions from plants that abate emissions and plants that do not abate emissions, respectively. 4.2. Entry and exit P1  stream π s ðxÞ and the discount A producing plant x has the value vx;t ¼ Et s ¼ t þ 1 Rs π s ðxÞ ; given its expectation of profit P1  factor Rs ¼ ½βð1 μÞs  t for all sZ t þ 1. The average value of incumbent plants is v~ t ¼ Et ~ s ; and the ex ante s ¼ t þ 1 Rs π   expected value of entrants is v~ e;t ¼ 1 Gðxe;t Þ v~ t . Plants enter the dirty sector if v~ e;t Zwt f e ; i.e., if the following free entry i1=ð1  σÞ h i1=ð1  σÞ  1  σ R 1 σ 1 1=ð1  σÞ 1 pt ðxÞ dGðxÞ : Substitute pt ðxÞ ¼ ϕt x  1 into P t ; we get P t ¼ N t ϕt 1  Gðx x dGðxÞ : Use the e;t Þ xe;t n o σ=ðσ  1Þ  σ R1 1=ð1  σÞ 1=ð1  σÞ σ ðσ  1Þ=σ t definition of x~ t ; we get P t ¼ N t ϕt x~ t 1 ¼ N t ½q ðxÞ dG ð x Þ . Substituting q ðxÞ ¼ P Q p ðxÞ into Q t ; we p~ t . Similarly, Q t ¼ 1  NGðx t t t t t x Þ e;t e;t 4

Recall that P t ¼

σ=ðσ  1Þ

get Q t ¼ N t

q~ t :

h

R1 Nt 1  Gðxe;t Þ xe;t

Z. Li, J. Sun / Journal of Economic Dynamics & Control 61 (2015) 34–60

41

condition is satisfied:   1  Gðxe;t Þ Et

1 X

! Rs π~ s Z wt f e ;

ð4:3Þ

s ¼ tþ1

where π~ s is from (4.1) (under tax) or from (4.2) under standard. Since π t ðxÞ is an increasing function of x; vx;t is also an increasing function of x. After entering the dirty sector and drawing a productivity level x; a plant exits if its present value of profit stream vx;t is negative. Let xe;t be the productivity level such that vxe;t ¼ 0; then all the plants that draw a productivity level x Z xe;t will earn non-negative profit and have a non-negative value vx;t . Hence, the plant's exit condition is ! 1 X Rs π s ðxe;t Þ ¼ 0: ð4:4Þ x oxe;t and vxe;t ¼ Et s ¼ tþ1

4.3. Markets clear In equilibrium, all the goods markets, the resource markets, and the share markets clear. That is, for all t M t ¼ mt ;

Q t ¼ qt ;

Lm;t þ Ne;t f e þ Nt f þ

Nt 1  Gðxe;t Þ

ð4:5Þ Z

1

  Lg;t ðxÞ þ La;t ðxÞ dGðxÞ ¼ l;

ð4:6Þ

xe;t

and At ¼ 1:

ð4:7Þ

4.4. A steady state equilibrium We here define a steady state equilibrium given an arbitrary emission tax (standard). It follows that a second best steady state equilibrium under an emission tax (standard) is one of those steady state equilibria that maximizes the representative household's utility. Given a time-invariant tax policy ðτ; TÞ (or a time-invariant emission standard policy s), the Steady State Equilibrium is defined as follows. Definition 1. An allocation is comprised of quantities of ðm; qi i A Ω ; DÞ for consumers, ðLm ; MÞ for producers in the clean

sector, and Lg ðxÞ; La ðxÞ; qðxÞ; eðxÞ x A Φ for producers in the dirty sector.

Definition 2. A price system is comprised of ðw; P; pðxÞ x A Φ Þ.

Definition 3. A steady state equilibrium is a time-invariant allocation, a time-invariant price system, and a law of motion of the aggregate level of emission stock with emission level constant over time, i.e., δD ¼ E, such that: (a) Given the government policy, the law of motion of the aggregate level of emission stock, the price of resource w; the relative price of the dirty

aggregate P; the prices pðxÞ x A Φ ; (i) the quantities ðLm ; MÞ solve the plant's problem in the clean sector; (ii) the quantities

Lg ðxÞ; La ðxÞ; qðxÞ; eðxÞ x A Φ solve the plant's problem in the dirty sector; and (iii) the quantities ðm; qi i A Ω ; DÞ solve the consumers' problem. (b) Given the allocation, the price system, and the law of motion of the aggregate level of emission stock, the government policy satisfies the budget constraint (2.5). (c) The market clearing conditions from (4.5) to (4.7) are satisfied. (d) The free entry condition (4.3) and exit condition (4.4) hold. (e) The distributions for plants' output, emissions, profit, and value are stationary. (f) There is consistency between the individual plants' behavior and aggregate variables. 4.5. A steady state equilibrium without emission reduction Consider the steady state equilibrium without imposing any emission reduction policy and, therefore, without the producers incorporating the externality of emissions from their production. In this case, pðxÞ ¼ σ=ðσ  1ÞXx  1 ; qðxÞ ¼ P σ q½pðxÞ  σ ; and the profit is πðxÞ ¼ 1=σP σ q½pðxÞ1  σ Xf : First, we use the entry and exit conditions to solve for xe : In the steady state, the exit condition (4.4) is equivalent to a zero-profit condition, i.e., πðxe Þ ¼ 0. Accordingly, we get the cut-off productivity level  1=ðσ  1Þ 1 f ðσX Þσ=ðσ  1Þ σ xe ¼ ; ð4:8Þ σ 1 P q which depends on P σ q.

42

Z. Li, J. Sun / Journal of Economic Dynamics & Control 61 (2015) 34–60

The steady state version of free entry condition (4.3) is βð1  μÞ ½1  Gðxe Þ π~ ¼ Xf e : ð4:9Þ 1 βð1  μÞ h i1  σ  Xf . Solving P σ q from (4.8) in terms of xe and substituting P σ q into π~ , we get Recall that π~ ¼ 1=σP σ q σ=ðσ  1ÞX ðυxe Þ  1 π~ ¼

σ 1 Xf : k þ1 σ

ð4:10Þ

Substituting π~ into the free entry condition (4.9), we get  1=k βð1  μÞ σ 1 f xmin : xe ¼ 1  βð1 μÞ k þ 1  σ f e

ð4:11Þ

Next, we characterize the equilibrium m; q; D; and N: Using the steady state version of budget constraint and v~ e ¼ f e , we get Pq þ m þ Ne Xf e ¼ Xl þ

σ 1 NXf : k þ1 σ

ð4:12Þ

Using the steady state version of resource constraint Lm þ Ne f e þ Nf þ Lg ¼ l, where Lg is Z 1 N Lg ðxÞ dGðxÞ ¼ ðσ  1Þυσ  1 XfN; Lg ¼ 1  Gðxe Þ xe we get m þ Ne Xf e þ NXf þ

ðσ  1Þk NXf ¼ Xl: k þ1 σ

ð4:13Þ

Combining Eqs. (4.12) and (4.13), we get Pq  συσ  1 NXf ¼ 0: Using P ¼ N P¼N

ð4:14Þ

1=ð1  σÞ ~

p; we get P in terms of N;

1=ð1  σÞ

σ X : σ  1 υxe

ð4:15Þ

Substituting P from Eq. (4.15) into (4.14), we can solve q in terms of N; q ¼ xe ðσ  1Þf υσ Nσ=ðσ  1Þ : Using the household's first order condition (3.3), we get   ρ=ð1  ρÞ  σ  1 ½σ  1=ð1  ρÞ=ðσ  1Þ 1  α 1=ð1  ρÞ σX m ¼ σXf υ N : α ðσ  1Þxe

ð4:16Þ

ð4:17Þ

The total emission stock is b D ¼ ðσ  1Þf υσ  1 N: δ Finally, substituting P; q; and m into Eq. (4.12), we can solve for N using the following Eq. (4.18), in which N is the only endogenous variable (except xe ; which is pinned down by (4.11)),   ρ=ð1  ρÞ  σ  1 ðσ  1=ð1  ρÞÞ=ðσ  1Þ ð1  βÞðkσ σ þ1Þ þ kσβμ σ  1 1  α 1=ð1  ρÞ σX l υ ð4:18Þ υ N þ N¼ : σ ½1  βð1 μÞk α ðσ  1Þxe f

5. Social planner's solution

Consider a social planner who chooses consumption, ðm; q; DÞ, production input and output, Lg ðxÞ; qðxÞ x A Φ ; abatement,

La ðxÞ x A Φ ; number of entrants, N e ; and threshold value of productivity xe. The social planner's allocation maximizes  Ψ U ¼ ½ð1 αÞmρ þ αqρ 1=ρ  λ D=D ; subject to the following constraints:  σ=ðσ  1Þ Z 1 N q¼ ½qðxÞðσ  1Þ=σ dGðxÞ ; qðxÞ ¼ xLg ðxÞ; 1  Gðxe Þ xe " #    1=γ Z 1 1 N La ðxÞ bLg ðxÞ 1 þ dGðxÞ ; D¼ δ 1 Gðxe Þ xe Lg ðxÞ Z 1   m N þ Lg ðxÞ þ La ðxÞ dGðxÞ þNf þ Ne f e ¼ l; X 1  Gðxe Þ xe

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43

and Ne μ ¼ : N ð1  μÞð1 Gðxe ÞÞ Rearrange the above constraints to get m; q; and D in terms of Lg ðxÞ; La ðxÞ; Ne ; and xe (q(x) and N are canceled out) Z 1   1μ XN e Lg ðxÞ þ La ðxÞ dGðxÞ þ ð1  Gðxe ÞÞf  XNe f e ; m ¼ Xl  μ xe q¼

 σ=ðσ  1Þ Z ð1  μÞNe 1 ½xLg ðxÞðσ  1Þ=σ dGðxÞ ; μ xe

D¼

ð1 μÞNe δμ

Z

1

xe

ð5:1Þ

ð5:2Þ

  La ðxÞ  1=γ bLg ðxÞ 1 þ dGðxÞ: Lg ðxÞ

ð5:3Þ

Substitute m; q; and D into the objective function. The first order conditions with respect to Lg ðxÞ; La ðxÞ; Ne ; and xe are 

q xLg ðxÞ

 XMU m þx

 XMU m   

1=σ MU q þ

  b La ðxÞ ½zðxÞ  ðγ þ 1Þ=γ γ þ ð1 þ γ Þ MU D ¼ 0; γδ Lg ðxÞ

b ½zðxÞ  ðγ þ 1Þ=γ MU D Z 0 γδ

equality if

ð5:4Þ

La ðxÞ 40;

ð5:5Þ

 ð1  μÞð1 Gðxe ÞÞX Lg þ La σ q D þf þ Xf e MU m þ MU q þ MU D ¼ 0; μ σ 1 Ne Ne N

ð5:6Þ

and  X zðxe Þ þ

   1=σ f σq1=σ ðσ  1Þ=σ  b MU m  x Lg ðxe Þ MU q  zðxe Þ  1=γ MU D ¼ 0: Lg ðxe Þ δ ðσ  1Þ e

ð5:7Þ

Here, MU m ; MU q ; and MUD are the marginal utility with respect to m; q; and D; respectively MU m ¼ ð1  αÞ½ð1  αÞmρ þαqρ 1=ρ  1 mρ  1 ;

MU q ¼ α½ð1 αÞmρ þ αqρ 1=ρ  1 qρ  1 ;

Ψ

and MU D ¼  λΨ DΨ  1 =D :

It follows from (5.5) that the social optimal cleanliness index zo ðxÞ ¼ 1 þ LLga ðxÞ ðxÞ is equalized across all the plants 8 γ=ðγ þ 1Þ > b MU D MU D < if  4 γX=b;  zo ¼ γX δMU m δMU m > : 1 otherwise: It is optimal to abate emissions if MU D =ðδMU m Þ is large and γX=b is small: that is, relatively speaking, when the clean good sector is not productive (X small), the abatement technology is efficient (γ small), the emission decay rate is low (δ small), the emission factor is large (b large), the marginal utility of lean good is small (MUm small), and the marginal disutility from emissions is large ( MU D large), then the plants should abate their emissions. In the following solution we focus on the case where zo 41: In this case (5.5) gives MU D ¼ 

δγX  o ðγ þ 1Þ=γ z MU m : b

Substituting MUD from (5.8) into Eq. (5.4), we get   σ MU q σ  qðxÞ ¼ q X ð1 þ γ Þzo =x : MU m Integrating Lg ðxÞ ¼ qðxÞ=x over x; we get   σ σ1 Lg MU q σ  ¼ q X ð1 þ γ Þzo ; x~ N MU m where x~ ¼

h

R

1 σ 1 dGðxÞ 1  Gðxe Þ x A Φ x

n

R1

ð5:8Þ

ð5:9Þ

ð5:10Þ

i1=ðσ  1Þ

¼ υxe . The average productivity in the social planner's solution is the same as under oσ=ðσ  1Þ ½qðxÞðσ  1Þ=σ dGðxÞ and Eq. (5.9), we get

the tax. Using q ¼ 1  NGðxe Þ xe   σ σ MU q σ  q ¼ Nσ=ðσ  1Þ q X ð1 þ γ Þzo x~ : MU m

ð5:11Þ

44

Z. Li, J. Sun / Journal of Economic Dynamics & Control 61 (2015) 34–60

Combine Eqs. (5.10) and (5.11), and we get q ¼ N σ=ðσ  1Þ q~ ¼ Nσ=ðσ  1Þ x~

Lg : N

ð5:12Þ

~ Eq. (5.12) says that the optimal production plan is isomorphic to an economy with N plants producing with productivity x. From (5.11), we get   σ σ MU q σ  q~ ¼ q X ð1 þ γ Þzo x~ : MU m MU q ~   σ . The counterpart of MU in the market equilibrium is P, and the Recall that in the market equilibrium q~ ¼ P σ q½pðxÞ m o counterpart of X ð1 þ γ Þz =x in the market equilibrium is p(x). Similar to P ¼ N 1=ð1  σÞ p~ in the market equilibrium, the MU aggregate shadow price MU mq here integrates the individual shadow prices

MU q X ð1 þ γ Þzo : ¼ N 1=ð1  σÞ x~ MU m

ð5:13Þ

Substitute MUD from (5.8) into (5.7), and we get 

MU q MU m

qðxe Þ ¼

   σ σ  σ 1 Xf X ð1 þ γ Þzo =xe þ q : σ qðxe Þ

ð5:14Þ

Use (5.9) at xe ; 

MU q MU m

qðxe Þ ¼

σ

 σ q X ð1 þ γ Þzo =xe ;

ð5:15Þ

and combine (5.14) and (5.15), then we get qðxe Þ ¼

σf xe : ð1 þγ Þzo

ð5:16Þ

Eq. (5.16) gives the condition that determines xe ; which is similar to the condition in a market equilibrium that πðxe Þ ¼ 1=σpðxe Þqðxe Þ ¼ Xf . MU Substituting MU mq from (5.13) and qðxe Þ from (5.16) into (5.15), we get q¼

xe σf υσ Nσ=ðσ  1Þ : ð1 þ γ Þzo

We can also get zo Lg ¼

σf σ  1 υ N: 1þγ

ð5:17Þ

Substitute MUD from (5.8) into the first order condition for Ne , (5.6), together with (5.13) and zo Lg from (5.17), then we solve for xoe ; xoe ¼

 1=k  ð1  μÞ k σ f 1 xmin : μ k þ 1  σ σ 1 fe

ð5:18Þ

The optimal m is  m ¼ σXf

 1  α 1=ð1  ρÞ X ð1 þ γ Þzo ρ=ð1  ρÞ  σ  1 ðσ  1=ð1  ρÞÞ=ðσ  1Þ υ N : α xe

The optimal level of pollution stock is D¼

b σ ðzo Þ  1=γ σ  1 fυ N: δ 1 þγ

ð5:19Þ

In order to derive the equation parallel to (4.18), we use the resource constraint. The optimal No is characterized by the following equation: σ

  1  α 1=ð1  ρÞ X ð1 þ γ Þzo ρ=ð1  ρÞ  σ  1 ðσ  1=ð1  ρÞÞ=ðσ  1Þ σ ðσ þγ Þ l υσ  1 N ¼ : υ N þ α ðσ 1Þð1 þγ Þ f xoe

ð5:20Þ

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45

6. Compare emission tax and emission standard 6.1. Emission tax We solve the equilibrium under emission tax in two steps. In the first step, we solve an equilibrium with an arbitrary tax rate τ; i.e., an equilibrium in which both the households and the plants take τ as given. In the second step, we solve for the optimal τ that gives the households highest utility. First, we consider a steady state equilibrium with tax rate τ. Recall that, qðxÞ ¼ P σ q½pðxÞ  σ ; πðxÞ ¼ 1=σP σ q½pðxÞ1  σ Xf , and pðxÞ ¼ ϕx  1 ; where 8  γ=ðγ þ 1Þ > < σ=ðσ 1ÞX ðγ þ 1Þ τb if τ 4 γX=b; γX ϕ¼ > : σ=ðσ 1ÞðX þτbÞ otherwise: As in the equilibrium without emission control, we use the entry and exit conditions to find xe : In the steady state, the exit condition (4.4) is equivalent to a zero-profit condition, i.e. πðxe Þ ¼ 0. Accordingly, we get the cut-off productivity level 8  γ=ðγ þ 1Þ  1=ðσ  1Þ > 1 τb f > σ=ðσ  1Þ > ðγ þ 1Þ if τ 4 γX=b; > σ < σ  1 ðσX Þ γX P q ð6:1Þ xe ¼  1=ðσ  1Þ >   f 1 > σ=ðσ  1Þ > > ð σX Þ 1 þ τb=X otherwise: : σ 1 Pσ q The cleanliness index is 8  < τb γ=ðγ þ 1Þ if τ 4γX=b; t γX z ¼ : 1 otherwise: The steady state version of free entry condition (4.3) is βð1 μÞ ½1 Gðxe Þ π~ ¼ Xf e : 1  βð1  μÞ Recall that 8  1  σ

γ=ðγ þ 1Þ > > σ τb > ðυxe Þ  1  Xf < 1=σP q σ=ðσ 1ÞX ðγ þ 1Þ γX π~ ¼ h i 1σ > > > 1=σP σ q σ=ðσ  1ÞðX þ τbÞðυxe Þ  1  Xf :

ð6:2Þ

if τ 4γX=b; otherwise:

Solve P σ q from (6.1) in terms of xe and substitute P σ q into π; ~ we get π~ ¼

σ 1 Xf : kþ1σ

ð6:3Þ

Substituting π~ into the free entry condition (6.2), we get  1=k βð1  μÞ σ 1 f xmin : xte ¼ 1  βð1  μÞ k þ 1  σ f e

ð6:4Þ

As in the equilibrium without emission control, next, we characterize the equilibrium m; q; D; and N. Using the steady state version of budget constraint Pq þ m þ v~ e Ne ¼ Xl þ π~ N þT, v~ e ¼ f e , and T ¼ τE ¼ τz  1=γ bLg ; we get Pq þm þN e f e ¼ Xl þ

σ 1 XNf þ τz  1=γ bLg : kþ1σ

ð6:5Þ

Using the steady state version of resource constraint, we get Lm þ Ne f e þ Nf þ zLg ¼ l;  R1  where zLg ¼ 1  NGðxe Þ xe Lg ðxÞ þ La ðxÞ dGðxÞ, and 8 σ 1 k > >  γ=ðγ þ 1Þ k þ1 σ f if τ 4 γX=b; > > > τb Lg < ðγ þ 1Þ ¼ γX N > > > σ 1 k > > : Xf otherwise: X þ τb k þ 1  σ

ð6:6Þ

Combining Eqs. (6.6) and (6.5), we get Pq συσ  1 NXf ¼ 0:

ð6:7Þ

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Z. Li, J. Sun / Journal of Economic Dynamics & Control 61 (2015) 34–60

~ we get P in terms of N Using P ¼ N1=ð1  σÞ p; P ¼ N 1=ð1  σÞ

ϕ : υxe

ð6:8Þ

Substituting P from Eqs. (6.8) into (6.7), we can solve q in terms of N and xe ; 8 xe ðσ  1Þf σ σ=ðσ  1Þ > >  γ=ðγ þ 1Þ υ N > > > < ðγ þ 1Þ τb q¼ γX > > > > xe ðσ  1ÞXf σ σ=ðσ  1Þ > : υ N X þ τb

if τ 4γX=b; ð6:9Þ otherwise:

Using the household's first order condition (3.3), we get

m¼

8  γ=ðγ þ 1Þ 1ρ=ð1  ρÞ 0 > τb > > σX ðγ þ 1Þ > > B C  ðσ  1=ð1  ρÞÞ=ðσ  1Þ   > 1=ð1  ρÞ γX 1  α > B C > Xf υσ  1 N <σ α ðσ  1Þxe @ A > > >   ρ=ð1  ρÞ > >  ðσ  1=ð1  ρÞÞ=ðσ  1Þ > 1  α 1=ð1  ρÞ σ ðX þ τbÞ > > Xf υσ  1 N :σ α ðσ  1Þxe

The total pollution stock is 8 γðσ 1Þ > σ 1 > N > < τδðγ þ1Þ Xf υ D¼ bðσ  1Þ > σ1 > > N : δðX þτbÞ Xf υ

if τ 4γX=b; ð6:10Þ otherwise:

if τ 4 γX=b; ð6:11Þ otherwise:

Substituting P; q; and m into Eq. (6.5), we can solve for N using the following Eqs. (6.12) and (6.13), in which N is the only endogenous variable (except xe ; which is pinned down by (6.4)), m l ð1 βÞðkσ  σ þ 1Þ þ kσβμ σ  1 γðσ 1Þ σ  1 ¼  υ υ Nþ N; Xf f ð1  β þ βμÞk γ þ1

if τ 4γX=b;

ð6:12Þ

otherwise:

ð6:13Þ

m l ð1 βÞðkσ  σ þ 1Þ þ kσβμ σ  1 τbðσ 1Þ σ  1 ¼  υ υ Nþ N; Xf f ð1  β þ βμÞk X þ τb

Next, we solve for the optimal standard τ . According to the above equations, we can solve m, q, and D in terms of tax rate τ . Substituting m, q, and D into the utility function, we get the welfare function in terms of τ. To find the optimal tax rate, we differentiate the household's utility with respect to τ. The first order condition is MU m

dm dq dD þMU q þ MU D ¼ 0: dτ dτ dτ

6.2. Emission standard We solve the equilibrium under emission standard in two steps. In the first step, we solve an equilibrium with an arbitrary standard s, i.e., an equilibrium in which both the households and the plants take s as given. In the second step, we solve for the optimal s that gives the households highest utility. First, we consider a steady state equilibrium with an arbitrary standard s. Recall that, qðxÞ ¼ P σ q½pðxÞ  σ , πðxÞ ¼ 1=σP σ q½pðxÞ1  σ  Xf , pðxÞ ¼ σ=ðσ  1ÞXzðxÞx  1 , and the cleanliness index is 8 > <  1 γ ifx 4xn ; b zðxÞ ¼ ifx r xn : > : sx In order to characterize xe ; we use the entry and exit conditions. In the steady state, the exit condition (4.4) is equivalent to a zero-profit condition, i.e., πðxe Þ ¼ 0. Accordingly, we get the cut-off productivity level,  xe ¼

1 ðσX Þσ=ðσ  1Þ σ 1

1=½ðσ  1Þðγ þ 1Þ 1=ðγ þ 1Þ  γ=ðγ þ 1Þ  b f ; σ s P q

ð6:14Þ

Z. Li, J. Sun / Journal of Economic Dynamics & Control 61 (2015) 34–60

47

which depends on P σ q.5 We can also solve P σ q in terms of xe 

1 σ 1

Pσ q ¼

ðσ  1Þ

ðσX Þσ

 γðσ  1Þ b f : s xeðσ  1Þðγ þ 1Þ

The steady state version of free entry condition (4.3) is βð1 μÞ ½1 Gðxe Þ π~ ¼ Xf e : 1  βð1  μÞ

ð6:15Þ

 k Recall that π~ ¼ na π~ a þ nn π~ n , π~ a ¼ 1=σP σ Q ½pðx~ a Þ1  σ Xf , π~ n ¼ 1=σP σ Q ½pðx~ n Þ1  σ  Xf , x~ n ¼ υxn , xn ¼ b=s, nn ¼ xe =xn , and  k na ¼ 1  xe =xn . For the plants that do not abate, their average price is  1=ð1  σÞ Z 1 1 σ X x~  1 : pðxÞ1  σ dGðxÞ ¼ pðx~ n Þ ¼ 1  Gðxn Þ xn σ 1 n For the plants that do abate their emissions, their average price is  γ  1=ð1  σÞ Z xn 1 σ b X x~ a 1  γ : pðxÞ1  σ dGðxÞ ¼ pðx~ a Þ ¼ Gðxn Þ  Gðxe Þ xe σ 1 s The average profit is  γ ðσ  1Þ  i1  σ 1h σ s X π~ ¼ P σ q na x~ ðaσ  1Þðγ þ 1Þ þ nn x~ σn  1  Xf : σ σ 1 b Substituting P σ q into π~ , we get π~ in terms of xe 2  3

sx k  ðσ  1Þðγ þ 1Þ e sxe k  ðσ  1Þðγ þ 1Þ k 1 6 7 k b 7 Xf : þ b π~ ¼ Xf 6 4 5 k  ðγ þ 1Þðσ  1Þ kþ1σ

ð6:16Þ

Substituting π~ from (6.16) into the free entry condition (6.15), we get the following equation that determines xe implicitly, 8 2  3 91=k

sx k  ðσ  1Þðγ þ 1Þ

sx k  ðσ  1Þðγ þ 1Þ e > > e > > k 1  < βð1  μÞ 6 k 7f = b b 6 7 þ  1 xe ¼ 5f > xmin : > 1  βð1 μÞ4 k  ðγ þ 1ðσ  1Þ kþ1σ e> > ; :

ð6:17Þ

To characterize m; q; D; and N; we use the budget constraint and the resource market clearing constraint. Using the steady state version of budget constraint and v~ e ¼ f e , we get Pq þm þf e N e ¼ Xl þ π~ N:

ð6:18Þ

Using the steady state version of resource constraint, we get m þ f e Ne þσXNf þ ðσ  1Þπ~ N ¼ Xl

ð6:19Þ

Combine Eqs. (6.19) and (6.18), substitute away π~ ; and we get   Pq σ ð1  εÞϖ ðσ  1Þðγ þ 1Þ þευσ  1 NXf ¼ 0

where

ε¼

sx k  ðσ  1Þðγ þ 1Þ e : b

ð6:20Þ

The aggregate price index is  1=ð1  σÞ σX ð1 εÞϖ ðσ  1Þðγ þ 1Þ þ ευσ  1 : P ¼ N1=ð1  σÞ

s γ σ 1 xγe þ 1 b Substituting P into (6.20), we can solve q in terms of N and xe ;

sx γ  σ=ðσ  1Þ e q ¼ ðσ 1Þf ð1 εÞϖ ðσ  1Þðγ þ 1Þ þευσ  1 xe Nσ=ðσ  1Þ : b Define x a ¼

5



1 Gðxn Þ  Gðxe Þ

Z

xn

xe

1=ðσðγ þ 1Þ  1Þ xσ ðγ þ 1Þ  1 dGðxÞ ;

Without loss of generality, here we have assumed xe r xn .

ð6:21Þ

48

Z. Li, J. Sun / Journal of Economic Dynamics & Control 61 (2015) 34–60

then 2

 k þ 1  σðγ þ 1Þ 31=ðσ ðγ þ 1Þ  1Þ xe 1  6 7 xn 6 7 ; x a ¼ ϱxe 6 7 xe k 4 5 1  ðxn Þ

 where

The total production cost is

sx γ   e ½1  ϵϱσðγ þ 1Þ  1 þ ϵυσ  1 N; Lg ¼ ðσ  1Þf b

ϱ¼

where

k k þ 1  σ ðγ þ 1Þ

ϵ¼

1=ðσðγ þ 1Þ  1Þ

:

sx k þ 1  σðγ þ 1Þ e : b

Using the household's first order condition (3.3), we get m¼σ

 1   ðσ  1=ð1  ρÞÞ=ðσ  1Þ ρ=ð1  ρÞ  1 α 1  ρ σXz ϖ ðγ þ 1Þðσ  1Þ σ 1 Xf ð1  εÞ þε υ N : α ðσ  1Þxe υσ  1

The total pollution stock is

sx γ þ 1

b e ð1  ιÞωσðγ þ 1Þ þ ιυσ  1 N; D ¼ ðσ  1Þf δ b

where

ι¼

sx k  σðγ þ 1Þ e

b

ð6:22Þ

:

Substitute m into the resource constraint (6.19), and we get an implicit function for N     1  β þ βμ ϖ ðγ þ 1Þðσ  1Þ k þ1 σ σ  1 ð1 εÞ ð1  β Þ þε þ σ  1 m l 1 β k υ υσ  1 N ¼ : þ 1 β þ βμ Xf f

ð6:23Þ

Next, we solve for the optimal standard s . According to the above equations, we can solve m, q, and D in terms of standard s. Substituting m, q, and D into the utility function, we get the welfare function in terms of s. To find the optimal tax rate, we differentiate the household's utility with respect to s. The first order condition is MU m

dm dq dD þMU q þ MU D ¼ 0: ds ds ds

6.3. Comparison between tax and standard By incorporating productivity dispersion into the model, we can study the allocation and dislocation of resources across plants under emission taxes and standards. By incorporating plants' free entry and exit, we can study the dynamic reallocation of resources under both conditions. Both the static and the dynamic effects are different under different environmental policies. We establish the static and dynamic differences in the distributional effects under emission taxes and standards in Propositions 1 and 2 , respectively. Proposition 1. The static distributional effect due to productivity dispersion favors the emission tax over the emission standard. The average productivity is higher under the emission tax than under the emission standard. This result from Proposition 1 is valid no matter we compare two equilibria with an optimal tax and an optimal standard, respectively, or two equilibria with an emission tax and an emission standard that are implementing the same aggregate  o target of emissions. To separate the static distributional effect, here we assume that xe and N are fixed. Let ϰ o ¼ q=Lg ,  t  s t s ϰ ¼ q=Lg , and ϰ ¼ q=Lg be the average productivity in the social planner's problem, in the equilibrium under emission taxes, and in the equilibrium under emission standards, respectively. We can show that ϰ o ¼ ϰ t Zϰ s . The proof is shown in Appendix A.1. The reason for this result is that the tax does not change the distribution of output across plants, while the standard shifts the input in production to high-output plants and tilts abatement toward low-output plants. The message in Proposition 1 is the same as in Lemma 4.1 of Li and Shi (2015) in a static model without plants' entry and exit, while the message in Proposition 2 below is about the dynamic distributional effect due to plants' entry and exit. Proposition 2. The dynamic distributional effect due to plants' entry and exit favors the emission standard over the emission tax. The cut-off value of productivity is higher under the emission standard than under the emission tax, i.e., xse Z xte . The average size s t of plants is larger under the emission standard than under the emission tax, i.e., ðp~ q~ Þ 4 ðp~ q~ Þ . The number of plants is larger s under the emission tax than under the emission standard, i.e., N t Z Ns ; for all s Z s^ . Let x^ e be the equilibrium cutoff value of productivity corresponding to s^ ; then s^ is defined by    1=ðk  ðσ  1Þðγ þ 1ÞÞ s ðρ  1Þðσ 1Þ k  ðσ 1Þðγ þ 1Þ s^ x^ e ¼ 1þ : ρ k  ðσ  1Þ b

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49

This result from Proposition 2 is valid no matter we compare two equilibria with an optimal tax and an optimal standard, respectively, or two equilibria with an emission tax and an emission standard that are implementing the same aggregate target of emissions. Corollary 1. Increasing the tax rate τ does not change xte ; but it causes the number of plants Nt to increase. Corollary 2. The cutoff value of productivity under emission taxes is lower than that in the social planner's solution, i.e., xte o xoe . s o The lower bound of xse is lower than xoe ; but the upper bound of xe may be higher or lower than xe . Corollary 3. A more stringent standard s (smaller s) increases xse ; and it causes the number of plants Ns to decrease, for all s Z s^ . Let h ¼ sxse =b; h increases in s. The proof for Proposition 2 and Corollaries 1–3 is shown in Appendix A.2. A lower threshold value of productivity and a larger number of plants under the emission tax than under the emission standard implies that the average size of plants is smaller under t the emission tax than under the emission standard. According to Eq. (6.4), the change in τ does not affect the threshold value xe. In response to an increase in emission tax, plants tend to increase their prices, causing a decline in q(x). The rise of price increases the profit and attracts plants to enter, leading to a higher Nt. The size of plants becomes smaller in general. Under the emission s standard, xe increases as the standard becomes more stringent, according to Eq. (6.17). The smallest plants have to exit from the industry. The number of plants Ns becomes smaller. As a result, the average size of plants becomes larger. Although we can characterize the distributional effects under emission taxes and standards and compare the relative advantages and disadvantages of different policies, we cannot compare the welfare under the two policies. In order to compare the welfare effects of different policies quantitatively, we calibrate the model to the Canadian data.

7. Calibration We first calibrate the parameters that can be identified using conventional empirical targets in the literature. Given the value of those parameters we calibrate the model to the Canadian data. These parameters are the time preference β, the exit rate μ, the emission decay rate δ , the reference emission stock D, and the initial emission stock D  1. The time preference β ¼ 0:96 is targeted at an annual real interest rate of 4%. The exogenous exit rate μ ¼ 0:08 is calibrated to the annual failure rate of U.S. manufacturing plants (Dunne et al., 1989). The annual decay rate of GHG emissions is δ ¼ 0:008 (Kolstad, 1996). The reference value D is taken as the 1965 stock level of GHG (1965 is usually taken as a reference point; see Nordhaus, 1993 and Kolstad, 1996). D is set to equal 32 Gt CO2 equivalent.6 Given D; D  1 is calculated as 32:1 Gt at the beginning of 1990. The preference parameters depend on the definition of the dirty sector and the clean sector. The definition of the dirty sector is provided in the appendix B. The clean to dirty goods sales ratio Y m;t =Y Q ;t and the relative price Pt during 1990 and 2006 are used to calibrate α and ρ. The relative price of dirty goods to clean goods is constructed (see the appendix B (Table B1) for detail). In order to identify α and ρ; we rewrite Eq. (3.3) from the households' problem,  Y m;t 1 1α ρ ln ln þ ln P t : ¼ ð7:1Þ α 1 ρ Y Q ;t 1  ρ Given the time series data on Pt and Y m;t =Y Q ;t in the sample period, the preference parameters can be estimated by using Eq. (7.1). Eq. (7.1) predicts that Y m;t =Y Q ;t decreases in the relative price Pt since the dirty goods and the clean goods are complementary, 1 o ρ o0. The coefficient ρ determines the magnitude of this effect. The higher the absolute value of ρ, the lower the substitutability, and the larger the effect of the price change on Y m;t =Y Q ;t . The value of the exogenous share of dirty goods α in the model should not be very different from the dirty goods share Y Q ;t =Y t in the data, 0.38. Given these restrictions, parameters α and ρ are estimated by searching locally the estimate that minimizes the divergence between the model and the data. We find the best fit when α ¼ 0:35 and ρ ¼ 0:5. The disutility from pollution is separable from the utility from goods in the model; we set λ ¼ 0:025 and Ψ ¼ 0:5. The sensitivity of the magnitude of disutility parameter λ will be checked in the numerical exercise. We calibrate the model without emission reduction (Section 4.5) using Canadian data from 1990 to 2000. Before 2000, reduction activities in Greenhouse Gas (GHG) emissions were rarely reported. In 1997 Canada signed the Kyoto Protocol and made a commitment to reduce GHG emissions for the period 2008–2012 to a level of 6% below the 1990 level.7 Accordingly, after 2000 the government and industries signed agreements that focused on immediate reduction of emissions with some plants reportedly adopting new abatement technologies. We thus assume no abatement in the economy before 2000, and use the data before that year to calibrate the model without emission reduction (Section 4.5). We calibrate the parameters related to abatement technology using data on the expenditures on abatement and effectiveness of abatement technologies from a report by Statistics Canada, Environment (2004). 6 The literature (e.g., Nordhaus, 1993 and Kolstad, 1996) uses 667 Gt as the stock of GHG for U.S. in 1965. Since Canada emits roughly 10% of what US emits and this paper cuts off the emissions other than industrial emissions (about 52% of total GHG), 4.8% of 667 is used as D. 7 Canada resigned from the Kyoto Protocol in December of 2011.

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Both the average productivity in the clean sector X and the prior average productivity in the dirty sector are normalized R1 to 1. Given that xmin xdGðxÞ ¼ 1, xmin ¼ 0:7222. In our model, the parameter value σ measures the elasticity of substitution among 16 industries. We set σ ¼ 2:4. The sensitivity of this parameter σ will be checked in the next section. Other parameters are identified simultaneously by simulating the model to match the targets in the data. The fixed production cost f¼ 0.04 and the entry cost f e ¼ 0:1771 are targeted at Gðxe Þ ¼ 0:08 and N ¼1, since the value of f and fe determine xe and N through Eqs. (4.11) and (4.18) in the steady state equilibrium without emission reduction, as shown in section 4.5. We let Gðxe Þ ¼ 0:08, given that the annual new entrants rate Ne =N is approximately 0.095 in the data (Dunne et al., 1989) and the stationary distribution of plants requires that Ne μ ¼ : N ð1 μÞð1  Gðxe ÞÞ

ð7:2Þ

The emission intensity b¼3.6 kt CO2 emissions equivalent per million dollars is obtained according to the relationship b ¼ σ=ðσ  1ÞE=Y Q from integrating emissions across plants and the moment of the emission-sales ratio E=Y Q ¼ 2:10 in the data. The parameter k ¼3.6 is calibrated to the average dirty goods sales share, 0.42. The endowment is set to be 0.35 trillion dollars in order to equate the level of emissions generated in the model to the average level of emissions in the data, 0.3 Gt. In order to calibrate the abatement technology, we use data from Statistics Canada, Environment Accounts and Statistics Division, System of National Accounts, 2004, “Environmental Protection Expenditures in the Business Sector.” The total operating and capital expenditures on environmental processes and technologies to reduce greenhouse gas emissions in 2002 were reported as 523 and 583.3 million dollars, respectively. The total expenditure accounts for about 0.6% of total sales in the dirty sector. As Table C3 in Statistics Canada, Environment (2004) shows, on average, 24% of establishments adopted new or significantly improved systems or equipment to reduce greenhouse gas emissions within a three year period, 2000–2002. Respondents who reported having adopted new or significantly improved systems or equipment ranked their impact on greenhouse gas emission reductions as being small (44%) or medium (44%), rather than large (13%). According to these numbers, we suppose that the establishments that adopted new or significantly improved systems or equipment reduced on average 12% of their emissions. The abatement cost of an average plant is 0:6%=24% ¼ 2:5% of its sales, and approximately 3.2% of production input, i.e., z¼1.032. Accordingly, z  1=γ ¼ 0:88, so we have γ ¼ 0:25. For the model to have an equilibrium, we need k  ð1 þ γÞσ 40. This condition restricts the value of σ and γ.

8. Quantitative results The quantitative experiment reveals several interesting results. The introduction of free entry of plants into the dirty sector in a general equilibrium is crucial to these results. We first discuss the aggregate predictions of the model, and then analyze the micro foundation of these aggregate results. All the results discussed here are in the long run steady state equilibrium, abstracting from the transition dynamics. 8.1. Aggregate predictions First, the optimal level of emissions is quite different under the optimal emission tax and the optimal emission standard, and the welfare is higher under the optimal emission standard than under the optimal emission tax, in the baseline model where the disutility parameter λ ¼ 0:025. The optimal level of emission stock is 40.3677 Gt under the emission tax and 0.124 0.1235 0.123

welfare

0.1225 0.122 0.1215 0.121 0.1205 0.12 10

15

20

25

30

35

40

45

Emission stock

Fig. 1. Emission stock and welfare: This figure plots how the welfare changes in the aggregate level of emissions in the equilibrium under emission taxes (solid line) and in the equilibrium under emission standards (dotted line). Parameter values are from the calibrated baseline model.

Z. Li, J. Sun / Journal of Economic Dynamics & Control 61 (2015) 34–60

0.124

0.124

0.122 welfare

0.123 welfare

51

0.122 0.121

0.12 0.118 0.116

0.12 10

0.114

20 30 40 50 Emission stock (λ = 0.025)

0

20 40 Emission stock (λ = 0.03)

60

Fig. 2. Emission stock and welfare under emission taxes: This figure depicts the kinked welfare function under the emission tax. As the value of λ increases, the optimal level of emission taxes increases and the corresponding emission stock decreases, leading the equilibrium moves from the right of the inflection point to the left of it. Parameter values are λ ¼ 0:025 (left panel) and λ ¼ 0:03 (right panel).

0.116

welfare

0.114 0.112 0.11 0.108 0.106 0.104

0

5

10

15

20

25

30

35

40

45

Emission stock (λ = 0.04)

Fig. 3. Emission stock and welfare: This figure plots the welfare as the value of λ increases to 0.04 under emission taxes (solid line) and under emission standards (dotted line).

36.9440 Gt under the emission standard. The welfare curves in terms of aggregate emission stocks under an emission tax or an emission standard are shown in Fig. 1. From Fig. 1 we observe that the welfare function in emission stock under the emission standard is globally concave, while the welfare function in emission stock under the emission tax has an inflexion point. The inflexion point corresponds to the point where τ ¼ Xγ=b, i.e., where the marginal abatement cost of plants equals the tax rate. When the tax rate is smaller than the marginal abatement cost, τ o Xγ=b; the plants do not abate their emissions at all. The emission reduction solely comes from the reduction of the dirty goods production. When the tax rate is large, τ ZXγ=b; the plants start to introduce abatement input to abate their emissions. As shown in Fig. 2, when λ ¼ 0:025, the optimal emission tax is still in the region where τ o Xγ=b and the optimal emission stock is to the right of the inflexion point. However, as λ increases to λ ¼ 0:03, the optimal emission tax satisfies τ Z Xγ=b and the optimal emission stock is to the left of the inflexion point.8 Second, the optimal emission tax dominates the optimal emission standard in welfare when disutility from emissions is large, λ ¼ 0:04; as shown in Fig. 3. As the value of the disutility parameter λ increases from 0.025 to 0.04 (compare Fig. 1 with Fig. 3), the optimal level of emission stock decreases under both policy instruments, but the magnitude of the decline is different. The optimal level of emission stock under the emission tax shifts from the right of the inflexion point to the left of the inflexion point. Therefore, there is a large drop in the level of emission stock under the emission tax. In contrast, the change in the optimal level of emission stock is smaller under the emission standard. This result implies that the emission tax is more likely to dominate the emission standard when a large proportion of emissions generated by the plants needs to be reduced. Third, if there is uncertainty on the real damage from emission stocks, i.e., if the value of λ is uncertain, then a tax instrument may not be reliable since a wrong evaluation of λ may cause a large deviation from the optimal level of emission 8 Here we have used an abatement technology that features constant returns to scale by normalizing La(x) by Lg ðxÞ: If we use an abatement technology with decreasing returns to scale instead, then the larger plants (with high x) who have to reduce more emissions will be negatively affected. The average productivity in the dirty sector is also distorted under the emission tax. This is negative to the welfare under emission taxes. Any way, the inflexion point may still exist as long as the marginal cost of reducing the first unit of emissions is strictly positive.

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stock. As shown in Fig. 2, if λ ¼ 0:025; it is optimal not to abate emissions under the emission tax and the optimal level of emission stock is 40.3677 Gt. But if λ ¼ 0:03, it is optimal to abate emissions and the optimal level of emission stock is 18.2737 Gt. On the other hand, there is a smaller shift in the optimal level of emission stock under the emission standard; it is 36.9440 Gt and 29.0893 Gt for λ ¼ 0:025 and λ ¼ 0:03, respectively. Fourth, in the benchmark model where the disutility parameter λ ¼ 0:025; to reduce the aggregate emissions to a level that is 6% below the 1990 level, the emission standard is better than the emission tax in terms of welfare. Moreover, as long as the aggregate target of emissions is above the optimal level of emissions, a more stringent policy improves welfare under the emission standard, while it may decrease welfare under the emission tax before the inflexion point. In practice, the policy maker may not search for an optimal level of emissions, but set an aggregate emission target. In this case, if the target is a modest reduction of emissions, then the emission standard is likely to dominate the emission tax. If the target is an ambitious reduction of emissions, then an emission tax may dominate the emission standard. These results are important for making decisions on environmental policies, including the setup of an aggregate emission target and the choice of different policy instruments to achieve the target. To interpret the above results, we need to analyze the plants' choices and the resulting distribution of plants under different emission reduction policies. 8.2. Plants' choices and industrial distribution To understand the above aggregate results, we analyze the plants' behavior under each policy. First, under the emission tax, all the plants respond to the tax rate in the same way and their behavior changes at the threshold level of the tax rate. Suppose we increased the tax rate from zero; the plants would not start to abate emissions as long as the tax rate remained below the marginal cost of reducing emissions, i.e., τ o wγ=b. The plants would rather pay the tax. Moreover, they could transfer part of the tax burden to the consumers in setting the price of their goods. As a result, the equilibrium output of dirty goods would decline in accordance with the emission reduction. When the price instrument - i.e., the emission tax – is used, the plants first reduce their output to reduce emissions. Only when τ 4 wγ=b would all the plants start to abate their emissions. But still, the plants depend on both reduction of their output and abatement as a means to reduce their emissions. It is because of this change in the plants' behavior as the tax rate increases that we observe the kinked welfare curves under the emission tax in Fig. 2. In the next step under the emission standard, we divide plants into two groups according to their emission intensity, which is determined by their x. One group of plants (high x) satisfies the emission standard automatically and does not need to reduce emissions, while the other group of plants (low x) has to abate their emissions to satisfy the standard. The emission standard will affect the industrial distribution of plants in two ways. One is a static way through productivity dispersion; the other is a dynamic way through plants' entry and exit. In the static way the standard moves the market share from the small x variety of goods to the high x variety of goods, causing the value-weighted aggregate productivity in the dirty sector to decrease (see Proposition 1). In the dynamic way the standard lets the average size of the plants grow larger while the number of plants declines in the dirty sector, thus saving some fixed costs in the dirty sector (see Proposition 2). In contrast with the emission standard, the emission tax does not effect a shift of market share from small (low x) to large (high x) plants, nor does it make the average size of plants larger, since all plants respond in the same way to a certain tax rate. As a result, the static effect favors the emission tax, while the dynamic effect favors the emission standard. The comparison between the equilibrium under the emission tax and the equilibrium under the emission standard in the baseline model is shown in Table 1. As shown in Table 1, under the emission standard the static distortion effect causes a lower average productivity in the dirty sector: the output - variable cost ratio is 1.0260, while it is 1.0613 under the emission tax. The dynamic effect through Table 1 Compare optimal tax and standard — baseline model. Variable

Tax τ ¼ $1=per ton

Standard s ¼3.22

m q D Gt P N N a =N (%) Gðxe Þ Lg La Nf Ne f e q/variable cost q/total cost Disutility from emissions Utility from goods Value of utility function

0.2028 0.0952 40.3677 1.6734 1.0142 0 0.08 0.0897 0 0.0406 0.0144 1.0613 0.7306 0.0281 0.1505 0.1224

0.2016 0.0947 36.9440 1.6722 0.9239 72.18 0.2546 0.0899 0.0024 0.0370 0.0106 1.0260 0.7324 0.0269 0.1496 0.1228

Z. Li, J. Sun / Journal of Economic Dynamics & Control 61 (2015) 34–60

53

0.135

0.112

welfare

welfare

0.114

0.13

0.11

0.108

0

10

20

30

0.125

40

0

10

Emission stock (σ = 2)

20

30

40

50

Emission stock (σ = 2.8)

Fig. 4. Degree of productivity dispersion: This figure depicts how the welfare changes as the value of k decreases to 3.2 (left panel) or increases to 4 (right panel) under emission taxes (solid line) and under emission standards (dotted line).

0.135

0.112

welfare

welfare

0.114

0.13

0.11

0.108

0

10

20

30

0.125

40

0

10

Emission stock (σ = 2)

20

30

40

50

Emission stock (σ = 2.8)

0.124

0.127

0.1235

0.126

0.123

0.125 welfare

welfare

Fig. 5. Market power: This figure depicts how the welfare changes as the value of σ decreases to 2 (left panel) or increases to 2.8 (right panel) under emission taxes (solid line) and under emission standards (dotted line).

0.1225

0.124

0.122

0.123

0.1215

0.122

0.121

0

10

20 30 40 Emission stock (γ = 0.2)

50

0.121

0

10

20 30 40 Emission stock (γ = 0.15)

50

Fig. 6. Efficiency of abatement technology: This figure depicts how the welfare changes as the value of γ decreases to 0.2 (left panel) and 0.15 (right panel) under emission taxes (solid line) and under emission standards (dotted line).

plants' entry and exit leads to a smaller number of plants with a larger size: the number of plants N is 0.9239 under the standard, while it is 1.0142 under the tax. This will help the dirty sector save some fixed cost Nf ; leading to a lower output – total cost ratio under the emission standard than under the emission tax. In the baseline model, the dynamic effect dominates the static effect and the emission standard dominates the emission tax. 8.3. Productivity dispersion and substitution elasticity Introducing free entry and exit into the model does not change the static distributional effect due to productivity dispersion and substitution effect predicted in a static model as in Li and Shi (2015). A higher degree of productivity dispersion (smaller kÞ still favors the emission tax, and a lower value of substitution elasticity σ still favors the emission standard.9 9 In this subsection, when we change the value of k and σ, the value of f and fe are re-calibrated according to the calibration procedure in Section 7, in order to maintain the entry and exit conditions.

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Table 2 Compare tax and standard with same aggregate target. Variable

Tax τ ¼ $7:09=per ton

Standard s ¼2.94

m q D Gt P N N a =N (%) Gðxe Þ Lg La Nf Ne f e q/variable cost q/total cost Disutility from emissions Utility from goods Value of utility function

0.2081 0.0890 35.2448 1.9246 1.0908 100 0.08 0.0796 0.0003 0.0436 0.0155 1.1134 0.7204 0.0262 0.1482 0.1220

0.2014 0.0939 35.2448 1.6906 0.9145 79.38 0.2757 0.0891 0.0035 0.0366 0.0102 1.1040 0.7269 0.0262 0.1490 0.1228

A higher degree of productivity dispersion increases the disadvantages of the emission standard, since it lowers the value-weighted aggregate productivity under the standard by distorting the distribution of varieties from low x to high x. This reallocation is absent under the emission tax, since all the plants react to the policy in the same way. As shown in Fig. 4, for all levels of aggregate emissions, the welfare under emission standards relative to the welfare under emission taxes increases as k increases from k ¼3.2 to k ¼4. Take an example, if the aggregate target of emissions is 25 Gt, then emission tax dominates emission standards when k ¼ 3:2; but emission standard dominates emission tax when k ¼4. However, in both cases with k ¼3.2 (higher degree of productivity dispersion) and k ¼ 4 (lower degree of productivity dispersion), the optimal emission standard dominates the optimal emission tax, due to other effects embodied in the model. The elasticity of substitution parameter σ has two effects. First, it determines market power. A smaller value of σ (a larger market power) makes the tax worse as in a static model (see Li and Shi, 2015). Second, it affects the dynamic allocation of resources across plants and therefore plants' size. A smaller value of σ makes the varieties of goods difficult to be substituted by others, preventing high productivity plants to grow too large. The general effect depends on the combination of the above two effects. As shown in Fig. 5, when the value of σ is large, σ ¼ 2:8, the welfare under emission taxes relative to the welfare under emission standards is higher, compared to the case when σ ¼ 2. Take an example, if the aggregate target of emissions is 25 Gt, then emission tax dominates emission standards when σ ¼ 2:8; but emission standard dominates emission tax when σ ¼ 2. However, in both cases with σ ¼ 2 and σ ¼ 2:8, the optimal emission standard dominates the optimal emission tax, due to other effects embodied in the model. 8.4. Abatement technology In the baseline model, changing γ from 0.25 to 0.2, the welfare becomes identical under the emission standard and under the emission tax. When γ 4 0:2, the emission standard dominates the emission tax. However, when γ becomes even smaller (very efficient), say γ ¼ 0:15, the optimal emission tax will dominate the optimal emission standard, as shown in Fig. 6. 8.5. Aggregate target of emissions In practice, a country that plans to control emissions often sets first its aggregate target of emission reduction and then chooses a policy instrument to achieve that target. The Kyoto Protocol is such an example that requires the signatory countries to set an aggregate target of emission reduction and to find ways to achieve it. This is mainly because an optimal level of emission reduction is hard to know, given the many kinds of uncertainties in the real world, including the damage uncertainty that we mentioned above and the technology uncertainty in abatement that is crucial for determining the optimal level of emission reduction as shown above in Fig. 6. We here compare the effects of an emission tax and an emission standard under an identical aggregate emission reduction target in order to make practical policy suggestions. This comparison is shown in Table 2. We set a target that is about 6% below the 1990 level of aggregate emissions. In this case, the emission standard again achieves a higher welfare than the emission tax, as in the baseline model. In achieving the same aggregate emission target, the two policy instruments lead to different plant distributions. Under the emission standard, Gðxe Þ ¼ 0:2757, which means that about one third of the plants drop off. In contrast, under the emission tax, Gðxe Þ ¼ 0:08; which implies that many small plants are still producing. The number of plants is N ¼0.9145 under the emission standard and N ¼1.0908 under the emission tax. This is the dynamic distributional effect through plants' entry and exit. This dynamic distributional effect results in an output – total cost ratio that is higher under the emission standard than under the emission tax, 0.7269 versus

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55

0.7204. On the contrary, the static distributional effect results in the output-variable cost ratio being higher under the emission tax than under the emission standard, 1.1134 versus 1.1040. In achieving the same aggregate emission target, the emission tax and the emission standard also cause a different allocation of resources across the two sectors. Under the emission tax, the total abatement cost is low ðLa ¼ 0:0003Þ, although all the plants abate a little bit of their emissions. Most of the reduction of emissions comes from a reduction of dirty goods output. Under the emission standard, 79.38% of plants abate their emissions and they put a lot of resources to do so in order to produce more goods ðLa ¼ 0:0035Þ. As a result, more resources are allocated into the dirty sector under the emission standard than under the emission tax: Lg ¼ 0:0796 under the emission tax and Lg ¼ 0:0891 under the emission standard.

9. Conclusion This paper uses a two sector general equilibrium model with free entry and exit to evaluate the welfare effects of an emission tax and an emission standard. Calibrated to Canadian data, the model predicts that the welfare under the emission standard is higher than the welfare under the emission tax. The main reason is that the positive dynamic distributional effect dominates the negative static distributional effect under the emission standard. The paper has also tested the sensitivity of the results to a variation of the parameter values governing productivity dispersion, market power, and efficiency of abatement technologies. The results, however, come from the comparison between steady states under emission taxes and standards. If we considered transition dynamics after imposing different policies, the results might be different and more complex.

Acknowledgments We are grateful to two referees for valuable comments. We thank Shouyong Shi, Diego Restuccia, and Miquel Faig for guidance and inspiring suggestions. We have also benefited from comments and suggestions from many seminar participants at University of Toronto, the NASM 2009, the 9th Conference of the SAET 2009, the 18th CREE Annual Meeting 2008, and the 41th CEA Annual Meeting 2007. All remaining errors are our own. Li would like to acknowledge financial support by the Shanghai Pujiang Program (11PJC 060) and the National Natural Science Foundation of China (71203127).

Appendix A A.1. Proof of Proposition 1 We compute the average productivity in the social planner's problem, in the equilibrium with emission taxes, and in the equilibrium with emission standards, respectively,  o q ¼ υxoe No1=ðσ  1Þ ; ϰo ¼ Lg ϰt ¼

 t q ¼ υxte N t Lg

ϰs ¼

σ=ðσ  1Þ  s  ð1  εÞϖ ðσ  1Þðγ þ 1Þ þ ευσ  1 q ¼ xse N s1=ðσ  1Þ : Lg ½1  ϵϱσðγ þ 1Þ  1 þ ϵυσ  1

1=ðσ  1Þ

;

and

It is obvious that the average productivity under the tax is equal to the optimal one, i.e., ϰ t ¼ ϰ o ; if we fix xe and N. It can also be shown that ϰ t Z ϰ s . The results imply that the allocation of resources is more efficient under the emission tax than under the emission standard, if we only consider the static effect. The static distributional effect favors the emission tax over the emission standard. Now we prove that ϰ s r ϰ t . First, we look at the extreme case where xe ¼ xn . In this case, sxe =b ¼ 1, and ε ¼ ϵ ¼ 1. The average productivity ϰs is identical to the one under emission taxes, i.e., ϰ s ¼ ϰ t . Second, we look at another extreme where b=s ¼ xn -1, then sxe =b-0, and ε ¼ ϵ ¼ 0. The average productivity ϰs becomes ϰs ¼

ϖ σðγ þ 1Þ s s1=ðσ  1Þ x N : ϱσðγ þ 1Þ  1 e

56

Z. Li, J. Sun / Journal of Economic Dynamics & Control 61 (2015) 34–60

We still fix xe and N, and rewrite 

σ=ðσ  1Þ R 1 ðσ  1Þðγ þ 1Þ 1 x dGðxÞ x e ϰ 1  Gðxe Þ ¼ : R1 1 Nsð1=σ  1Þ σ ðγ þ 1Þ  1 dGðxÞ x x 1  Gðxe Þ e s

In order to show that ϰ s o ϰ t , we need to show that 8hR iðσ  1Þ=σ hR i1=σ 9σ=ðσ  1Þ 1 σ1 > > < x1 xσ ðγ þ 1Þ  1 dGðxÞ = x dGðxÞ xe e R1 41: ðσ  1Þðγ þ 1Þ dGðxÞ > > x : ; xe We prove that the expression inside fg of the above expression is strictly greater than one, and so ϰ s oϰ t if xe and N are fixed. The proof is similar to the proof of Lemma 4.1 in Li and Shi (2015), using Holder's inequality (see Royden, 1988, p. 121). The detail of the proof is available upon request. Now we prove that ϰs decreases as s becomes more stringent, given xe and N fixed. Let h ¼ sxe =b, then dh 4 0 as proved in ds Corollary 3 below. We rewrite the term ϰs as a function of h; h

s

ϰ ðhÞ ¼

 iσ=ðσ  1Þ k  ðσ  1Þðγ þ 1Þ k  ðσ  1Þðγ þ 1Þ σ  1 ϖ ðσ  1Þðγ þ 1Þ þ h 1h υ

 : k þ 1  σ ðγ þ 1Þ k þ 1  σ ðγ þ 1Þ σ  1 ϱσðγ þ 1Þ  1 þ h 1 h υ

We need to prove that dϰdhðhÞ 40. Differentiating ϰ s ðhÞ with respect to h; we get ( ) γ k  ðσ  1Þðγ þ 1Þ dϰ s ðhÞ 1h 1h ¼ fg  ; dh γ k  ðσ  1Þðγ þ1Þ s

where fg stands for the terms that do not affect the sign of dϰdhðhÞ.  γ 0 0 Define a function f ðγÞ ¼ 1  h =γ. This function f ðÞ decreases in γ, i.e., f ðγÞ o 0. To show f ðγÞ o0; we derive  0 γ γ  2 f ðγÞ ¼  γ ln hh ð1  h Þ =γ . In the limit, s

γ

lim

γ-0

γ

 γ ln hh  ð1  h Þ 1 2 ¼  ðln hÞ o 0: 2 γ2

Since γ

γ

dð  γ ln hh  ð1  h ÞÞ 2 γ ¼ γ ðln hÞ h o 0; dγ 0

f ðγÞ o 0 for all γ Z0. As a result, γ

k  ðσ  1Þðγ þ 1Þ

1h 1h  4 0; γ k ðσ  1Þðγ þ 1Þ given that k ðσ  1Þðγ þ1Þ 4 γ 4 0. Therefore, dϰdhðhÞ 40, meaning that ϰs decreases as the standard becomes more stringent. As a result, ϰ s rϰ s ðh ¼ 1Þ. Since ϰ s ðh ¼ 1Þ ¼ ϰ t (keeping xe and N fixed), we have ϰ s r ϰ t . s

A.2. Proof of Proposition 2 and Corollaries 1–3 A.2.1. Proof of Corollary 1 In order to show xte o xoe , recall xoe ¼

 1=k  1μ k σ f 1 xmin ; μ k þ1 σ σ  1 fe

and  xte ¼

βð1  μÞ σ 1 f 1  βð1 μÞ k þ 1  σ f e

1=k xmin :

Given βo1 and 0 o μ o1, ð1  μÞ=μ 4 βð1 μÞ=½1 βð1  μÞ. k=ðk þ1 σ Þσ=ðσ 1Þ  1 4 ðσ 1Þ=ðk þ1 σ Þ. As a result, xte o xoe .

Given

σ 41

and

k 4 σ 1,

Z. Li, J. Sun / Journal of Economic Dynamics & Control 61 (2015) 34–60

57

A.2.2. Proof of Corollary 2 t According to (6.4), xe does not change in τ: According to Eqs. (6.10), (6.12) and (6.13), an increase in tax rate τ; keeping other parameters constant, can cause an increase in the number of plants Nt. That is m ρ γ N t dN Xf 1  ργ þ1 τ ¼ 40 m ρ 1 l dτ  Xf 1  ρ σ  1 f

if

τ 4 γX=b;

m ρ b N t dN Xf 1 ρ X þτb 40 ¼ m ρ 1 l dτ  Xf 1  ρ σ  1 f

if

τ r γX=b:

and

A.2.3. Proof of Corollary 3 Recall that the cut-off value of productivity under emission standards is implicitly given in the following equation: 8 2 1 91=k  s k  ðσ  1Þðγ þ 1Þ !  s k  ðσ  1Þðγ þ 1Þ sxe > > > > sxe > k 1  > > 6 C > k < βð1  μÞ 6 b Cf = b s 6 C xe ¼ þ  1 Cf > xmin : > 1  βð1 μÞ6 k  ðγ þ1ðσ  1Þ k þ1 σ > 4 A e> > > > > ; :  k  ðσ  1Þðγ þ 1Þ r1, given the assumption that xe rxn o1. The first two terms in ½ in the Since b=s ¼ xn , we have 0 o sxse =b  k  ðσ  1Þðγ þ 1Þ above equation is a weighted average: k=ðk þ1 σÞ weighted by ε ¼ sxse =b , and k=ðk  ðγ þ 1Þðσ 1ÞÞ weighted s 1  ε. The value of ε increases as s increases if xe is kept unchanged. An increase in ε lowers the right hand side of the above s equation given that k=ðk  ðγ þ 1Þðσ  1ÞÞ 4 k=ðk þ 1 σÞ, causing xe to decrease. This decrease in xse ; however, will not make s s sxe =b to decrease. The value of ε will still increase after we allow xe to decrease as s increases. s The opposite cannot be true. Suppose the opposite is true so that an increase in s cause xe to increase, then sxse =b must also increases and ε increases; as a result, the right hand side of the above equation decreases, but the left hand side of the dxs above equation increases. This is a contradiction. So we can only have dse o0, meaning that the cut-off value of productivity under emission standards increases as the standard becomes more stringent.  k  ðσ  1Þðγ þ 1Þ In order to show xte rxse , we first look at the extreme case that xe ¼ xn . In this case, sxse =b ¼ 1, the cut-off s s value xe is identical to the one under emission taxes. So the lower bound of xe is  1=k βð1  μÞ σ 1 f xsle ¼ xmin : 1  βð1 μÞ k þ 1  σ f e  k  ðσ  1Þðγ þ 1Þ s If b=s ¼ xn -1; then sxse =b -0, and the higher bound of xe is  1=k βð1  μÞ ðγ þ1Þðσ  1Þ f xsh xmin : e ¼ 1  βð1 μÞ k  ðγ þ 1Þðσ  1Þ f e Since xsle ¼ xte , xte r xse . o The sufficient condition for xse r xoe is that xsh e r xe , that requires  β ðγ þ1Þðσ  1Þ 1 k σ r 1 : 1  βð1 μÞ k  ðγ þ1Þðσ  1Þ μ k þ 1 σ σ  1 This condition can be satisfied when γ is small enough, i.e., the abatement technology is efficient enough. As s decreases (more stringent policy), h ¼ sxse =b also decreases according to the above reasoning, i.e. dh 4 0. ds s As for the proof of dN 40, see the end of the proof of Proposition 2. ds A.2.4. Proof of Proposition 2 According to the proof for Corollary 3, xte ¼ xsle r xse . t According to Eq. (6.7), the average sales under emission taxes is ðp~ q~ Þ ¼ συσ  1 Xf , and according to Eq. (6.20), the average s ðσ  1Þ ð γ þ 1 Þ σ  1 sales under emission standards is ðp~ q~ Þ ¼ σ ð1  εÞϖ þευ Xf . So in general, ðp~ q~ Þs Z ðp~ q~ Þt . t s Next, we compare N, N , and N . According to Eqs. (4.18), (6.12), (6.13), and (6.23), the number of plants N is determined in  an implicit function with terms N½σ  1=ð1  ρÞ=ðσ  1Þ and N, where σ  1=ð1  ρÞ =ðσ  1Þ 41 given that 1 oρ o0. We define the term of parameters in front of N½σ  1=ð1  ρÞ=ðσ  1Þ by ψ1, and the term of parameters in front of N by ψ2, then the equation that determines N becomes ψ 1 N ½σ  1=ð1  ρÞ=ðσ  1Þ ¼ fl ψ 2 N. In the equilibrium without emission reduction   ρ=ð1  ρÞ 1 α 1=ð1  ρÞ σX ψ1 ¼ σ υσ  1=ð1  ρÞ ; α ðσ  1Þxe

58

Z. Li, J. Sun / Journal of Economic Dynamics & Control 61 (2015) 34–60

and ψ2 ¼

ð1 βÞðkσ  σ þ 1Þ þkσβμ σ  1 υ : ð1  β þ βμÞk

In the equilibrium under emission taxes 0  γ=ðγ þ 1Þ 1ρ=ð1  ρÞ τb  1=ð1  ρÞ BσX ðγ þ 1Þ C 1α γX B C t ψ1 ¼ σ υσ  1=ð1  ρÞ B C @ A α ðσ 1Þxte  ψ t1 ¼ σ and ψ t2 ¼

 ρ=ð1  ρÞ 1  α 1=ð1  ρÞ σ ðX þτbÞ υσ  1=ð1  ρÞ α ðσ  1Þxte

if

if

τ 4 γX=b;

τ r γX=b;

  ð1 βÞðkσ  σ þ 1Þ þ kσβμ γðσ  1Þ σ  1  υ : ð1  β þ βμÞk γ þ1

In the equilibrium under emission standards γ 1ρ=ð1  ρÞ 0  b  1=ð1  ρÞ σX  ðσ  1=ð1  ρÞÞ=ðσ  1Þ B 1α ϖ ðγ þ 1Þðσ  1Þ sxse C B C ð 1  εÞ þε υσ  1=ð1  ρÞ ; ψ s1 ¼ σ @ A s σ  1 α ðσ  1Þxe υ and ð1 βÞ ψ s2 ¼

    1  β þ βμ ϖ ðγ þ 1Þðσ  1Þ kþ1σ σ  1 ð1  εÞ þ ε þ 1 β k υσ  1 υσ  1 : 1 β þ βμ

According to the above equations, ψ t2 o ψ 2 for τ 4 0. In the extreme case where ε ¼ 1, ψ s2 ¼ ψ 2 ; in general ψ s2 Zψ 2 , since ϖ ðγ þ 1Þðσ  1Þ =υσ  1 4 1. We draw function ψ 1 N ½σ  1=ð1  ρÞ=ðσ  1Þ and function fl  ψ 2 N in Fig. A1. The intersection of ψ 1 N½σ  1=ð1  ρÞ=ðσ  1Þ and fl  ψ 2 N indicates the equilibrium N. As shown in Fig. A1, the function l=f ψ 2 N slopes down. An increase in the value of ψ2 rotates the curve downward and a decrease in the value of ψ2 rotates the curve upward. As shown in Fig. A1, the function ψ 1 N ½σ  1=ð1  ρÞ=ðσ  1Þ slopes up, and a decrease in the value of ψ1 rotates the curve downward and an increase in the value of ψ1 rotates the curve upward. It is obvious that ψ t1 o ψ 1 , if τ 4 0, given that xte ¼ xe . As a result, the equilibrium Nt, determined by the intersection of the solid lines (equilibrium under emission taxes), is larger than N, determined by the intersection of the dotted lines (equilibrium without emission reduction). If ψ s1 Zψ 1 and ψ s2 Zψ 2 , then ψ s1 N½σ  1=ð1  ρÞ=ðσ  1Þ rotates up and l=f  ψ s2 N rotates down, compared to the equilibrium without emission reduction. As a result, the equilibrium Ns , determined by the intersection of the dashed lines, is smaller than N. Therefore, Ns r N rN t . 16 14 12 ψs1Na

10

ψ1Na

8

ψt1Na

6

Ns

4

Nt

N

2

l/f − ψt2N l/f − ψ2N

l/f − ψs2N

0 −2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

N Fig. A1. Equilibrium N; N t ; and Ns. For illustration purpose, we let N a ¼ N ½σ  1=ð1  ρÞ=ðσ  1Þ . The crossing points of function ψ 1 N ½σ  1=ð1  ρÞ=ðσ  1Þ and function l  ψ 2 N are the equilibrium N without emission reduction (dotted lines), Nt under an emission tax (solid lines),and Ns under an emission standard (dashed f lines).

Z. Li, J. Sun / Journal of Economic Dynamics & Control 61 (2015) 34–60

59

Now we show that ψ s1 Zψ 1 under some sufficient conditions. There are two differences between ψ s1 and ψ1. First, the    ρ=ð1  ρÞ terms xse 4 ðxe Þ  ρ=ð1  ρÞ , which is a force to make ψ s1 4ψ 1 . Second, the term ðσ  1=ð1  ρÞÞ=ðσ  1Þ  γρ=ð1  ρÞ  b ϖ ðγ þ 1Þðσ  1Þ ð1  εÞ þε s σ  1 sxe υ s

also tends to increase the value of ψ1 to be above ψ1, if ðσ  1=ð1  ρÞÞ=ðσ  1Þ  γρ=ð1  ρÞ  b ϖ ðγ þ 1Þðσ  1Þ ð 1  ε Þ þε Z1: sxse υσ  1

ðA:1Þ

We show that (A.1) holds under some sufficient conditions. Let h ¼ sxse =b, then the above term is a function of h. Let 

  ðγ þ 1Þðσ  1Þ σ 1  γρ k  ðσ  1Þðγ þ 1Þ ϖ k  ðσ  1Þðγ þ 1Þ þ h Ψ ðhÞ ¼ h ð1  ρÞ σ  1=ð1  ρÞ 1 h : υσ  1 In the extreme case sxse =b ¼ 1, the equilibrium goes back the one without emission reduction and Ψ ðhÞ ¼ 1. As s decreases (more stringent policy), h also decreases according to Corollary 3. Let Ψ 0 ðhÞ be the first order derivative of Ψ ðhÞ with respect to h; we show that Ψ 0 ðhÞ o 0 with some sufficient conditions. Rearrange terms, we get    γρ σ  1 σ 1 k  ðσ  1Þðγ þ 1Þ  1 γρ  ρ σ  1=ð1  ρÞ k  ðσ 1Þ h 1  ρ σ  1=ð1  ρÞ  γ ðσ 1Þh : Ψ ðhÞ ¼ k  ðσ 1Þðγ þ 1Þ Derive Ψ 0 ðhÞ Ψ 0 ðhÞ ¼  fg

      ρ k  ðσ 1Þ γρðσ 1Þ k  ðσ  1Þðγ þ 1Þ h þ k  ðσ  1Þðγ þ 1Þ  ; σð1  ρÞ  1 σð1 ρÞ 1

where fg stands for terms that do not affect the sign of Ψ 0 ðhÞ. When h is close to 1, the condition for Ψ 0 ðhÞ o 0 is     ρ k ðσ  1Þ γρðσ  1Þ o0:  k  ðσ  1Þðγ þ 1Þ   σð1 ρÞ 1 σð1  ρÞ  1

ðA:2Þ

ðA:3Þ

This condition is automatically satisfied given that  1 r ρo 0 and σ 4 1. So Ψ 0 ðhÞ o 0 when h is close to 1. Since the first term in the right hand side of (A.3) is positive, and the second term is negative, the larger the value of h, the more likely Ψ 0 ðhÞ o 0. As s decreases, h decreases. There is an h^ such that     ρ k ðσ  1Þ γρðσ  1Þ ^ k  ðσ  1Þðγ þ 1Þ h  k  ðσ  1Þðγ þ 1Þ   ¼ 0; σð1 ρÞ 1 σð1  ρÞ  1 ^ Ψ 0 ðhÞ o 0. Solve the above equation and for all h 4 h,    ð1=ðk  ðσ  1Þðγ þ 1ÞÞÞ ðρ 1Þðσ  1Þ k ðσ  1Þðγ þ1Þ h^ ¼ 1 þ : ρ k  ðσ  1Þ Since the term inside the brackets ½ in the above equation is larger than 1, h^ o 1. The condition for Ψ 0 ðhÞ o0 is s Z s^ , where    ð1=ðk  ðσ  1Þðγ þ 1ÞÞÞ s ðρ 1Þðσ 1Þ k ðσ  1Þðγ þ 1Þ s^ x^ e ¼ 1þ : ρ k ðσ  1Þ b This is also a sufficient condition for ψ s1 Z ψ 1 .

Appendix B. Data description 1. Define the dirty sector and the clean sector. According to the Environment Canada, there are 16 industries whose abatement costs per employee are more than $1000. The emissions from these 16 industries account for about 90% of all industrial emissions. These 16 industries are defined as the empirical counterpart of the dirty sector in the model. These 16 industries and their NAICS codes are as follows: Forestry and Logging (113,000), Oil and Gas Extraction (211,000), Mining (212,000), Electric Power Generation, Transmission and Distribution (221,110), Natural Gas Distribution (221,200), Food Manufacturing (311,000), Beverage and Tobacco Products (312,000), Wood Products (321,000), Pulp, Paper, and Paperboard Mills (322,100), Petroleum and Coal Products (324,000), Chemicals (325,000), Non-Metallic Mineral Products (327,000), Primary Metals (331,000), Fabricated Metal Products (332,000), Transportation Equipment (336,000), and Pipeline Transportation (486,000). 2. The aggregate emission data in Canada from 1990 to 2006 come from the Greenhouse Gas Emissions National Inventory Report (NIR) by Environment Canada. The GDP data come from Statistics Canada, CANSIM II. The industrial level emission and GDP data come from the Canadian Industrial End-use Energy Data and Analysis Centre and CANSIM II. The emissions

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Z. Li, J. Sun / Journal of Economic Dynamics & Control 61 (2015) 34–60

Table B1 Price indices for clean and dirty goods. Year

1990

91

92

93

94

95

96

97

98

99

Clean Dirty

100 100

94 98

93 99

95 100

97 104

98 112

98 111

97 111

98 104

98 108

Year

2000

01

02

03

04

05

06

Clean Dirty

99 129

102 130

106 125

108 128

113 150

114 162

122 181

from these 16 industries account for about 50% of the total emissions in Canada. Since this paper focuses only on industrial emissions, the emissions from transportation, agriculture, residence, and other sources are excluded. The aggregate GDP in this paper is therefore the GDP from the 16 dirty industries plus a half of the total GDP of the sectors that do not generate emissions. It is approximately 50% of the aggregate GDP in Canada. The series of GDP of the clean sector and the dirty sector used in estimating the parameters in the relationship between GDP ratio and price ratio in Eq. (7.1) are nominal GDP adjusted by the price indices constructed below. 3. Construct the relative price. The relative price is the ratio between the price of the dirty aggregate and the price of the clean goods. The dirty goods price is constructed as a GDP-weighted average of 12 dirty goods (out of 16 industries whose price is available): Electric Power Generation, Petroleum and Coal Products, Fabricated Metal Products, Food Manufacturing, Beverage and Tobacco Products, Wood Products, Pulp, Paper, and Paperboard Mills, Primary Metals, Fabricated Metal Products, Gasoline, Chemical and Chemical Products, and Transportation Equipment. The clean goods price is constructed as a weighted average of 3 clean goods: new houses, electrical and communication products, and farm product, with weights of 54%, 30% and 16%, respectively. The relative price at the initial date, i.e. 1990, is normalized to 100 (Table B1). 4. The total operating and capital expenditures on environmental processes and technologies to reduce greenhouse gas emissions by industry are reported by the Environment Accounts and Statistics Division of Statistics Canada, Environment (2004). These data are the empirical counterpart of the variable costs of emission abatement and the investment in new abatement technology. The Environment Accounts and Statistics Division has also reported the adoption of new or significantly improved systems or equipment to reduce GHG emissions by industry during 2000–2002. Respondents who answered Yes to the adoption of new or significantly improved systems or equipment were asked to rank the impact on greenhouse gas emission reductions as being small, medium, or large.

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