Environment, irreversibility and optimal efﬂuent...

© Australian Agricultural and Resource Economics Society Inc. and Blackwell Publishing Ltd 2004

The Australian Journal of Agricultural and Resource Economics

, 48:1, pp. 127–158

Environment, irreversibility and optimal effluent standards

*

Jyh-Bang Jou

†

The present article investigates the use of performance standards to correct envi-ronmental externalities. Each firm in an industry emits waste in the productionprocess, and, in turn, the average waste emissions of the industry adversely affectthe firm’s productivity. The firm, which incurs sunk costs when employing capitalto abate waste emissions, is uncertain about the efficiency of capital. The firm willunderestimate environmental externalities and will therefore pollute more than issocially efficient. To correct this tendency, the regulator can set a limit on eitheremissions or the emission-output ratio at the socially efficient level. The firm willinvest more, produce more, and pollute less when the regulator implements theformer than when the regulator implements the latter.

1. Introduction

Manufacturing firms usually emit waste in the production process. Theregulator can induce these firms to employ more abatement capital by imple-menting command-and-control or market-based instruments. The investmentcosts on abatement capital, however, are usually sunk (e.g., see Pindyck 2000).

1

For example, massive sunk costs are needed to switch from coal burning tonatural gas burning power plants so as to reduce CO

2

emissions (Saphoresand Carr 1999). Unlike earlier articles on optimal environmental protectionpolicies (see e.g., Mohtadi 1996), the present article will account for thesesunk costs.

The present article compares the use of absolute performance standards(defined as emission limits) with the use of proportional performance

* I would like to thank two anonymous reviewers, Tan Lee, and seminar participants atthe National Taipei University and the 28th Annual Meeting of the Academy of Economicsand Finance held in February 2001.

†

Jyh-Bang Jou is a professor in the Graduate Institute of National Development, Collegeof Social Science, National Taiwan University.

1

Costly reversibility may arise because of asset specificity, the lemons problem, or govern-ment regulation (Dixit and Pindyck 1994).

128 J.-B. Jou


standards (defined as a limit on the emission-output ratio) (e.g., see Keohane

et al.

1998), both of which can correct environmental externalities.

2

Thepresent article builds a model in which an industry composed of a fixednumber of firms faces a demand function with a constant price elasticity.Each firm’s output depends positively on the amount of labour it employs,but negatively on industry’s average waste emissions. The firm can employcapital to abate waste emissions, and both the firm and the regulator sharethe same information about uncertainty regarding the firm’s abatementtechnology. The firm will underestimate environmental externalitiesbecause it will ignore the adverse effect of its emissions on other firms’ pro-ductivity. Therefore, it will pollute more than is socially efficient. To correctthis tendency, the regulator can either set an absolute or a proportionalperformance standard at the socially efficient level. The responses of a firmto these two policies, however, are divergent. The firm will invest more, pro-duce more, and pollute less when the regulator implements the former thanwhen the regulator implements the latter.

The present article is related to the published literature that applies thetheory of non-cooperative dynamic games to environmental managementsuch as Haurie and Krawzyck (1997), Haurie and Zaccour (1995), Krawzyck(1995), and Krawzyck and Zaccour (1999). Like these articles, the presentarticle models effluent management as a hierarchical game with a Cournot-Nash equilibrium at the lower level and a Stackelberg equilibrium at theupper level. That is, at the lower level of the game, firms compete for pro-duction and investment in a Cournot-Nash type environment. At the upperlevel of the game, the regulator (i.e., the leader), who anticipates thebehavior of a firm (i.e., the follower), sets an efficient level of performancestandard accordingly. However, these articles abstract from the investmentdecision of a firm, and resort to simulation analysis because they constructa more comprehensive model than that constructed in the present article.

Previous studies on performance standards either focus on different issuesor adopt assumptions that differ from those made in the present article.For example, a large volume of empirical articles (see e.g., Barbera andMcConnell 1986; Gray and Shadbegian 1995) find that performance stand-ards impact on productivity at the firm or plant level, but they ignore thefact that abatement capital may enhance productivity through reducingemissions (as emphasised by the present article). A number of other articlesassume that firms are better informed than the regulator, and then rank

2

Several studies on real options either allow government intervention while ignoringexternalities (see e.g., Dixit 1991; Dixit and Pindyck 1994; Hassett and Metcalf 1992) orfocus on pecuniary rather than production externalities (see e.g., Dixit and Rob 1994; Fatasand Metrick 1997).

Environment, irreversibility, and optimal effluent standards 129


policy instruments such as effluent taxes and subsidies, performance stand-ards, and marketable permits (see e.g., Mohtadi 1996; Montero 2002;Requate and Unold 2003; Roberts and Spence 1976; Weitzman 1974).In contrast, the present article abstracts from the issue of asymmetricinformation.

The present article is also closely related to studies that apply realoptions to investigate optimal environmental policy, such as Jou (2001).

3

Jou assumes that a firm’s emissions are independent of its output level, andinvestigates the use of effluent fees to correct environmental externalities.By contrast, the present article assumes that a firm’s emissions increase asthe firm produces more, and emphasises that the uses of absolute and pro-portional performance standards may have different impacts on the firm’sinvestment, production, and polluting behavior.

The remaining sections are organised as follows. Section 2 constructs themodel and calculates the short-run equilibrium private and social profits.Section 3 allows abatement investment to be either costlessly reversible orcompletely irreversible. The absolute and proportional performance stand-ards required to support social efficiency for both types of investments arederived. The impacts of several exogenous factors such as uncertainty,costly expandability, and competitive pressure on these two instruments areinvestigated both analytically and numerically. The last section concludesand suggests extensions for future research.

2. The Model

The present article builds a model that incorporates the main featuresappearing in both the standard real options (see e.g., Dixit 1991) and theenvironmental economics published literature (see e.g., Cropper and Oates1992). In the present article, the whole economy is represented by an indus-try composed of

N

identical risk-neutral firms facing a demand functionwith a constant price elasticity

ε

(

>

0); that is,

3

See also Pindyck (2000, 2002), Saphores and Carr (1999), and Xepapadeas (1999). Theformer three articles consider a regulator who intends to reduce stocks of environmentalpollutants once-and-for-all. In Pindyck’s two articles, the regulator controls a flow variablethat is related to these stocks. In contrast, in Saphores and Carr, the regulator directly con-trols these stocks. These three articles show that the interaction of irreversibility and uncer-tainty will affect the regulator’s choice of optimal timing to implement policy. While thesearticles allow the regulator to implement policy just once, the present article allows the reg-ulator to adopt performance standards that may vary over time. The study by Xepapadeas(1999) allows uncertainty in demand, emission tax, and abatement technology, but it doesnot explicitly model externality nor relate asset characteristics to optimal environmentalpolicies.

130 J.-B. Jou


Q

(

t

)

=

P

(

t

)

−

ε

, (1)

where

Q

(

t

) is quantity, and

P

(

t

) is price. Each firm

i

in the industry producesfinal goods according to a Cobb-Douglas technology given by

q

i

(

t

)

=

l

i

(

t

)

x

a

(

t

)

−

λ

,

λ

>

0, (2)

where

q

i

(

t

) is firm

i

’s output,

l

i

(

t

) is the amount of labour employed by firm

i

, and

x

a

(

t

), which is industry’s average waste emissions, will be strictly pos-itive in industry equilibrium. The adverse effect of

x

a

(

t

) on

q

i

(

t

) is externalwith

λ

measuring its size. This external effect indicates that as firm

i

emitsmore, its output as well as the other firms’ output will decline.

4

For example,a firm that emits hazardous pollutants will adversely affect health conditionsof the firm’s and the other firms’ employees. Consequently, the productivityof all firms in the industry will decline. Pollution may also adversely affectthe welfare of consumers (see e.g., Cropper and Oates 1992). The presentarticle, however, abstracts from this negative effect. Equation (2) also depictsa case of a uniformly distributed and diffused pollutant, and thus abstractsfrom both spatial distribution of firms (see e.g., Haurie and Krawzyck 1997)and stock externalities (see e.g., Farzin 1996). Equation (2) is restricted, yetit leads to analytically tractable solutions, and thus simplifies analysis.

By equation (2), firm

i

does not directly employ any productive capital,but the firm can use capital to abate emissions.

5

Firm

i

’s waste emissions,

x

i

(

t

),are a function of its output,

q

i

(

t

), its employed abatement capital,

k

i

(

t

), anda technology-shock factor,

Z

(

t

), in the form given by

(3)

Equation (3) indicates that abatement capital exhibits diminishing returnsbecause an increase in abatement capital reduces emissions at a decliningrate. The specification in equation (3) also indicates that the amount ofwaste emitted by firm

i

,

x

i

(

t

), is an increasing function of its output,

q

i

(

t

).

4

In reality, waste emissions from all industries will have a negative impact on a firm’soutput in one industry (Ballard and Medema 1993), yet the present article assumes that thewhole economy consists of only one industry for ease of exposition.

5

Some articles on environmental economics allow a firm to employ both productive andabatement capital (see e.g., Kort 1996). The empirical study by Gray and Shadbegian (1998)indicates that these two types of capital tend to crowd out each other. Several articles inpublished the real options literature also allow two types of capital investment (see e.g.,Dixit 1997). However, productive capital is ignored here because including it complicates theanalysis while adding little insight into the issue on which the present article focuses.

x t q t k t Z ti i i( ) ( ) ( ( ) ( )) , , .= >−γ γ γ γ1 21 2 0



The reason is obvious, for example, if steel or electricity production in-creases, pollution also increases. This specification is more generalisedthan that in the published literature. For example, Jou (2001) assumes thata firm’s emissions are uncorrelated with its output (

γ

1

=

0), and Krawzyck(1995) and Xepapadeas (1999) assume that a firm’s emissions are propor-tional to its output (i.e.,

γ

1

=

1). The present article, by contrast, allowsemissions to be increasing with output at a constant rate, an increasing rate(i.e.,

γ

1

>

1), or a decreasing rate (i.e., γ 1 < 1).In equation (3), the factor Z(t) exhibits a negative correlation with xi (t),

indicating that abatement capital is more efficient in good states of the world(i.e., when Z(t) is higher). The factor Z(t), which is required to be positiveso as to ensure that firm i’s waste emissions given by equation (3) are positive,is assumed to evolve as geometric Brownian motion with no drifts:

dZ (t ) = σZ (t )dΩ (t ), (4)

where σ (>0) is the instantaneous volatility of the growth rate of Z(t), anddΩ(t) is an increment to a standard Wiener process, with EdΩ(t) = 0 andEd Ω(t)2

= dt. By the standard theory of Brownian motion, we know thatZ(t) has a lognormal distribution with mean Z(0) and variance .6

Equation (4) indicates that information about the evolution of a firm’swaste emissions arriving in time is independent of its investment decision.This contrasts with that of Kolstad (1996) where uncertainty can beresolved over time through learning. Equation (4) also indicates a case ofsymmetric information between a firm and the regulator regarding the effi-ciency of abatement capital because both regard Z(t) as exogenously given.7

The case of asymmetric information is considered in the standard publishedenvironmental economics literature – see Jung et al. (1996), Requate and Unold(2003), and Weitzman (1974). These articles, however, focus on ranking variouspolicy instruments including effluent taxes and subsidies, performance stand-ards, and marketable permits. The published real options literature thatincorporates this issue includes Gaudet et al. (1998), which, however, does notallow environmental externalities.

6 Equation (3) indicates that I do not allow new abatement capital to be more efficientthan old capital. One alternative way to take this into consideration is to replace ki (t) bya(t)ki (t), where da(t)/dt ≥ 0, indicating that there is technological progress.

7 The present article allows a firm to adopt only one type of abatement capital. One mayassume that a firm chooses between an old and a new abatement technique, and then followsthe solution method in Dixit (1991) to solve the firm’s choice. The qualitative results of thepresent article, however, will not be affected even though such a choice is allowed.

Z e t( ) ( )0 12 2σ −

132 J.-B. Jou


Suppose that q (t ) = (q1(t), ... , qN(t)) and k(t) = (k1(t), ... , kN(t)). The oper-ating profit flow of firm i, denoted by , is equal to itsoperating revenue P (t)qi(t), minus its variable cost denoted by Z (t )) in equation (6) below, thus yielding

(5)

Denote w as a given wage rate. By equation (2), firm i ’s variable cost, wli(t),is given by

(6)

where .

Following the published literature applying non-cooperative dynamicgames to environmental management (see e.g., Haurie and Krawzyck 1997;Haurie and Zaccour 1995; Krawzyck 1995; Krawzyck and Zaccour 1999),the present article models effluent management as a hierarchical game witha Cournot-Nash equilibrium at the lower level where firms compete forproduction and investment. At the upper level of the game, the regulatoracts as the leader and a firm acts as the follower. The regulator shouldanticipate the production and investment behavior of the firm, and then setperformance standards at the socially efficient level accordingly.

In Cournot-Nash short-run equilibrium, firm i will take all the other firms’production strategies as given, while choosing an output level, denotedby , to maximise its operating profit, given by equation (5). Con-sequently, is derived by setting the derivative of with respect toqi(t) equal to zero. This yields the equality of marginal revenue with short-run marginal cost as given by

(7)

Imposing the short-run equilibrium condition qj(t ) = ( j = 1, ... , N ) in(7) yields and its corresponding P

1(t ):

(8)

(9)

π i q t k t Z t1( ( ), ( ), ( ))C q t k ti

1( ( ), ( ),

π i i iq t k t Z t P t q t C q t k t Z t1 1( ( ), ( ), ( )) ( ) ( ) ( ( ), ( ), ( )).= −

C q t k t Z t wq t x ti i a1( ( ), ( ), ( )) ( ) ( ) ,= λ

x t q t k t Z t Na j jj

N( ) ( ) ( ( ) ( ))= ( )−

=∑ γ γ1 2

1

q ti1( ) π i

1( )⋅q ti

1( ) π i1( )⋅

P tq tQ t

wx t x tx t

Ni

a ai( )

( )( )

( ) ( ) ( )

.1 1 1−

= +

−

ελγλ

q ti1( )

q ti1( )

q t whN

N Z t G G k tih

i

h1

1

11

1 1

1

1 2 2

1

( ) ( ) ( ( ) ) ,( )

= −

+

−− − − −

−+

λ λγ λ γλγ

λγ

P t Nq tih1 1( ) ( ( )) ,= −



where h = 1/ε and . Evaluating in equation (5) at qj (t ) =

( j = 1, ... , N ) yields the optimised value of firm i’s operating profit,denoted by π 1(k (t), Z (t )), as given by

(10)

In contrast, a social planner will internalise environmental externalitiesbefore producing final goods. The social planner perceives that firm i ’semissions, xi(t), are equal to the industry’s average emissions, xa(t), becauseall firms are identical. By equation (3), this also implies that the centralplanner will impose q j (t) = qi (t) and kj (t) = ki (t) ( j = 1, ... , N ) before pro-ducing final goods. The operating profit of firm i perceived by the socialplanner, denoted by , is equal to operating revenueP(t)q i (t) minus the variable cost of firm i perceived by the social planner,denoted by in equation (11) below, thus yielding

(11)

By equation (2) and the condition xa(t) = xi(t), the variable cost of firm iperceived by the social planner, wli(t), is given by

(12)

The social planner will choose an output level, denoted by , to max-imise firm i ’s operating profit, given by equation (11). Consequently,

is derived by setting the derivative with respect to qi (t) equal tozero. This yields the equality given by

(13)

where ε is required to be larger than one (and thus h = 1/ε < 1) so as toensure that marginal revenue is positive. Imposing the condition qj (t ) =

G k tjj

N

1 12 ( )= −

=∑ γ π i1( )⋅

q ti1( )

πλγ

λγ11 1 2

1

31

1

1 0( ( ), ( )) ( ) ,( )

( )k t Z t B G G G Z ta h a=− +

+

where ahh

ah

hG G k t G

hN

G

k B w

i

i

h

h

02

11

12 1 1 3 1

1 1

1

1 1 12

2 1

( )

( ),

( )( )( )

, ( ) ,

, ( )

=−+

=− −

+= + =

+ =

−

−−+

λγλγ

λλγ

λγ

λγ

γ

γ λγ 11

11

11

1 .

( )

( )( )( )

( )−

−+ − +

+ −+h

NN

h

hh

hλγ λ

λγ λλγ

π i i iq t k t Z t2( ( ), ( ), ( ))

C q t k t Z ti i i2( ( ), ( ), ( ))

π i i i i i i iq t k t Z t P t q t C q t k t Z t2 2( ( ), ( ), ( )) ( ) ( ) ( ( ), ( ), ( )).= −

C q t k t Z t wq t x ti i i i i2( ( ), ( ), ( )) ( ) ( ) .= λ

q ti2( )

π i2( )⋅

q ti2( ) π i

2( )⋅

P t w q t k t Z ti i( ) ( ) ( ) ( ( ) ( )) ,11

1 11 2−

= + −

ελγ λγ λγ

q ti2( )

134 J.-B. Jou


( j = 1, ... , N ) in equation (13) yields and its corresponding P 2(t) asgiven by

(14)

(15)

Evaluating in equation (11) at qi (t ) = yields the optimised valueof , denoted by π 2 (ki (t), Z(t)), as given by

(16)

where

3. Efficient performance standards

When industry pollution exhibits negative externalities as shown by equa-tion (2), then market outcomes will be inefficient. The policy to correct thisincludes market-based instruments or command-and-control regulations.Market-based instruments, however, are still not popular: the applicationof tradable permits appeared in few countries such as Australia, Canada,Chile, and the USA (see e.g., Gomboso et al. 1999), and effluent fees (andsubsidies) adopted by Europe have been set at very low levels, and thus areusually not considered as an effective way to control pollution (Kolstad, 2000).Command-and-control regulations, which are still the dominant form ofenvironmental regulation in the world today, have two major types: technology-and performance-based standards (Stavins 2000). Technology-based standardsspecify the method and/or the equipment (the so-called best available controltechnology) that firms must use to comply with a particular regulation. Inthe USA, the specific technologies are usually determined by states on acase-by-case basis. However, the USA has established national emissionstandards for new sources of pollutants, called the New Source PerformanceStandards (Tietenberg 1992, p. 397). In the present article I focus on performancestandards, which could specify an absolute quantity of permissible emissions,but more typically these standards establish allowable emissions in proportionalto output (Keohane et al. 1998). In what follows, I will investigate both typesof performance standards, while abstracting from the auditing and admin-istrative costs associated with implementing them.

For ease of exposition, I will abstract from capital depreciation and alsoassume that the purchasing and installation price of abatement capital, denotedby PK (t), grows at a constant rate µ; that is,

q ti2( )

q t w h N k t Z tih

ih2

11

1

1 1 2 1( ) [( ) ( ) ( ( ) ( )) ] ,( )= + − − −−

+λγ λγ λγ

P t Nq tih2 2( ) ( ( )) .= −

π i2( )⋅ q ti

2( )π i

2( )⋅

π 22

0 0( ( ), ( )) ( ) ( ) ,k t Z t B k t Z ti ia a=

B h h w Nh

h h h2 1

1

1

1

1 11

1

1 ( ) [( )/ ] [ ( )] .( )

( )

( )

( )= + − +−

+− +

+λγ λγλγλγ

λγ



dPK (t ) = µPK (t )dt. (17)

A firm is in an environment where it is costly to expand capacity later whenµ is positive.8 The following analysis, however, is suitable for both positiveas well as non-positive values of µ. Two types of investments are considered:(i) investment is costlessly reversible where the resale price of abatementcapital is equal to the purchase price of capital;9 and (ii) investment iscompletely irreversible where the resale price of abatement capital is equalto zero.

3.1 Costlessly reversible investment

Suppose that investment is costlessly reversible. Then ki(t) will be a choicerather than a state variable. Let ρ denote a given (risk-adjusted) discountrate. The abatement capital stock chosen by firm i at each instant will befound by equating the private marginal return to capital with the user costof capital (i.e., the rental cost of abatement capital net of capital gain)(Jorgenson 1963); that is,

(18)

where π 1(k (t ), Z (t )) is given by equation (16). In Cournot-Nash equilibrium,all firms will choose an equal amount of abatement capital, denoted by ,where the subscript ‘u’ denotes the situation where no regulation is imposed.Substituting this condition into equation (18) and then rearranging yields

(19)

8 This may arise because of limited land, natural resource reserves, or the need for a permitthat is in short supply (Dixit and Pindyck, 1998).

9 One may allow the installed abatement capital to have some resale value. Under such asituation, in addition to effluent standard restrictions, the regulator may add one more policyinstrument such as a tax on disinvestment. Such an extension, however, will not alter themain results of the present article. See, for example, the article by Jou and Lee (2001) whereR&D capital exhibits positive production externalities and partial irreversibilities, and theregulator gives investment tax credits when R&D capital is purchased, but imposes taxationwhen the installed R&D capital is sold.

∂ π∂

ρ µ1( ( ), ( ))

( ) ( ) ( ),

k t Z tk t

P ti

K= −

k tfu1( )

k thN N

N h HZ tP tf

uh a

K

a

11

11

1

1

1

1 1

1

1 0

0

( ) ( ) ( )

( ) ( ),

( )

= −

+

−−

−++ −

−

λγρ µ

λγλγ

136 J.-B. Jou


where .10 Here and in what follows, I willassume that ρ > µ so as to ensure that demand for abatement capital isdecreasing with the price of capital; that is, given by equation (19) isdecreasing with PK(t). Substituting into equation(8) yields the choice of output for firm i without any regulation, denoted by

:

(20)

where . Substituting and

into equation (3) yields the choice of emission for firm i withoutany regulation, denoted by :

(21)

Similarly, the capital stock chosen by a social planner at each instant,denoted by k f 2 (t ), is found by equating the social marginal return to capitalwith the user cost of capital; that is,

(22)

where π 2(·) is given by equation (16). Evaluating equation (22) at ki (t) = k f 2(t),and then rearranging yields

(23)

Substituting ki (t) = k f 2(t) into equation (14) yields the social planner’s choiceof output, denoted by q f 2(t):

10 One may rigorously show that all Cournot-Nash equilibria considered here and in whatfollows are unique. See, for example, Grenadier (2002), Haurie and Zaccour (1995), andRosen (1965).

H h w Nh

h

h

h ( ) ( )

( )

( )= −−+

− ++λγ λγ

λγλγ

2

1 1

1 1

1

1

k tfu1( )

k t k t j Nj fu( ) ( ) ( , ... , )= =1 1

q tfu

1( )

q t A k t Z tfu

fu h h

1 1 1

2

1

2

1( ) ( ) ( ) ,( ) ( )= + +λγ

λγλγ

λγ

AN

whN

N hh

11

11

1 11

( )

= +

−

−−

+λγ λγ

q t q ti fu( ) ( )= 1

k t k ti fu( ) ( )= 1

x tfu

1( )

x t A k t Z tfu

fu

h

h

h

h1 1 1

1

2

1

2

1( ) ( ) ( ) .( ) ( )=−

+−

+γγ

λγγ

λγ

∂ π∂

ρ µ2( ( ), ( ))

( ) ( ) ( ),

k t Z tk t

P ti

iK= −

k t hh HZ t

P tfh

a

K

a

2 11

1 1

1

1

1 111

10 0

( ) (( ) ( ) )( ) ( )

( ) ( ).( )= − +

−−

−++

−

−−

λγρ µ

λγλγ



(24)

where . Substituting qi (t) = qf 2(t) and ki (t)= kf 2(t ) into equation (3) yields the choice of emission by the social planner,denoted by x f 2(t ):

(25)

3.2 Irreversible investment

In the long-run, through Cournot-Nash competition, firm i will maximisethe expected discounted private profit flow net of the investment costs. Sup-pose that V1(k ( t ), Z( t ), PK( t )) denotes the value function for firm i, whichis given by

(26)

where the state equation for ki (τ ) is given by

(27)

Ii (τ ) is also the gross investment rate for firm i at time τ, π1(·) is given byequation (10), ρ is the discount rate, and Et· denotes the conditional expec-tation taken at time t. The maximisation problem faced by firm i includesN + 2 state variables: Z(t), PK(t), and an n-tuple of strategies, k (t). Firm i willtreat Z(t), PK(t) and k−i(t) = (k1(t), ... , ki−1(t), ki+1(t), ... , kN (t)) as exogenouslygiven, while controlling Ii (t). However, all elements of k−i(t) should be equalto ki (t) in Cournot-Nash equilibrium.

As is well known in the published real options literature, the interactionof the stochastic evolution of Z(t) and investment irreversibility indicatesthat firm i solves a problem of instantaneous control of Brownian motion.The optimal policy is to regulate the state variable ki (t) at a lower barrier,denoted by the desired capital stock ki*(t) (Bertola and Caballero 1994). Ifthe currently installed capital ki (t) is smaller than ki*(t), firm i should investso as to obtain ki (t) = ki*(t); otherwise no action should be taken. Firm ishould form expectations about the distant future regarding when and howmuch to invest because irreversible investment decisions will affect future

q t A k t Z tf fh h

2 2 2

2

1

2

1( ) ( ) ( ) ,( ) ( )= + +λγ

λγλγ

λγ

A w h N h h2 1

1

1

1 1 1 [( ) ( ) ] ( )= + − −−

+λγ λγ

x t A k t Z tf f

h

h

h

h2 2 2

1

2

1

2

1( ) ( ) ( ) .( ) ( )=−

+−

+γγ

λγγ

λγ

V E e k Z P I dI t tt

tK ii1

1( ) max [ ( ( ), ( )) ( ) ( )] , ( )( )⋅ = −

∞− − ρ τ π τ τ τ τ τ

Id k

dii( ) ( )

,ττ

τ=

138 J.-B. Jou


cash flows: firm i may be stuck with an excessive stock of capital in the initialperiod as pointed out by Clark et al. (1979), and Bertola and Caballero (1994).Given the abatement technology shown by equation (3), it is equivalent tosaying that firm i should regulate its pollution, xi (t) in equation (3), at an upperbarrier, denoted by . Consequently, is also the maximum amountof waste emissions tolerable by firm i.

Define .

When ki (t) > ki*(t), the private marginal gain from increasing the abatementcapital stock, v1(·) = ∂V1(·) /∂ki(t ), is given by (see Appendix)

(28)

where

F1 is a constant to be determined, β is the larger root of τ in the quadraticequation given by equation (56), and φ (1) is obtained by setting τ = 1 inequation (57). On the right-hand side of equation (28), the first term isthe private value of the option to install an additional unit of capital, whilethe second term is the private value of the last incremental unit of installedcapital. Two optimal conditions must be satisfied at ki(t) = ki*(t) (Pindyck1988). First, the private marginal gain from increasing the capital stockmust equal its marginal costs at the instant of investing (see equation (53)in Appendix); that is,

v1(·) = PK(t). (29)

This is the value-matching condition. Second, the equality given by equation(29) must be satisfied at the states both just before and just after the investment,thus yielding

(30)

This is the smooth-pasting condition.The ki (t) desired by firm i when investment is completely irreversible

without any regulation, denoted by , is solved from the proceduresas follows. First, evaluate both equation (29) and equation (30) at ki (t) =

. Second, multiply equation (30) by Z(t)/(a0β ), and then add the

x ti*( ) x ti*( )

G k t G G k t G hG N kjj

N

i i41

1 5 4 11

6 4 112 2 2= = + = +− −

=− − − −∑ ( ) , ( ) , /γ γ γλγ λγand

v F GZ t P t GZ ta

Ka

1 110 0 1( ) [ ( ) ] ( ) ( ) / ( ),⋅ = +−β β φ

G B G a G G G G G Gh

G G Ga h ( )( )

,( )

( )= − − +++

−− +

+−

γ λγλγ

λγλγ

2 1 11

2

11

1 2 3 4 1 2 61

11 3 5

1

1

11

∂∂

vZ t

1 0( )( )

.⋅

=

k tsu1( )

k tsu1( )



result into equation (29). Third, use the value of φ (1) given by equation (57).Finally, impose the equilibrium condition . Theresult is

(31)

where −α is the smaller root of τ in the quadratic equation given by equation(56), and is given by equation (19). Substituting intoequation (8) yields the maximum level of output produced by firm i, denotedby :

(32)

Substituting both and ki(t) = into equation (3) yields themaximum amount of waste emissions tolerable by firm i, denoted by :

(33)

Similarly, suppose that V2(ki (t ), Z(t), Pk (t )) denotes the value functionfor firm i perceived by the social planner, which is given by

(34)

where π 2(·) is given by equation (16). The maximisation problem faced by thesocial planner includes three state variables: Z(t), PK(t), and ki (t). The socialplanner will treat Z(t) and PK (t) as exogenously given, while controlling Ii (t).

Suppose that k i*(t) and respectively denote the desired capital stockand the maximum amount of waste emissions tolerable by the social planner.When k (t) > k i*(t), the social marginal gain from increasing the abatementcapital v2(·) = ∂V2(·)/∂ki (t), is given by (see Appendix)

(35)

where F2 is a constant to be determined. Suppose that k i*(t) chosen by thesocial planner is denoted by ks2(t). The value-matching and smooth-pastingconditions applied to v2 (·) are respectively given by

k t k t j Nj su( ) ( ) ( , , )= =1 1 L

k t m k t msu

fu a

1 1 1 11 11 0( ) ( ), [ /( )] ,/( )= = + −where α α

k tfu

1( ) k t k ti fu( ) ( )= 1

q tsu1( )

q t A k t Z t m q t msu

su h h

fu

h

1 2 1 2 1 2

1

1 112

1

2

1

1

2 21( ) ( ) ( ) ( ), [ /( )] .( ) ( )= = = ++ +− + +

λγλγ

λγλγ

γγ λγα αwhere

q t q ti su( ) ( )= 1 k ts

u1( )

x tsu1( )

x t A k t Z t m x t msu

su

h

h

h

hfu h

h

1 1 1 3 1 3

1

11

1

2

1

2

1

1

2 21 1( ) ( ) ( ) ( ), [ / ] .( ) ( )= = = +−

+−

+− +

+γ

γλγ

γλγ

λ γγ γαwhere

V E e k Z P I dI tt

tK ii2

2( ) max [ ( ( ), ( )) ( ) ( )] ( )( )⋅ = −

∞− −

τρ τ π τ τ τ τ τ

x ti*( )

v F k t Z t P t a B k t Z tia a

K ia a

2 21 1

0 210 0 0 0 1( ) [ ( ) ( ) ] ( ) ( ) ( ) / ( ),⋅ = +− − −β β φ

140 J.-B. Jou


v2(·) = PK ( t ), (36)

(37)

Evaluating both conditions (36) and (37) at ki (t ) = ks2(t ), and jointly solvingthese two equations yields

ks2(t ) = m1k f 2(t ). (38)

Substituting ki (t) = ks2 (t) into equation (14) yields the maximum outputlevel produced by the social planner, denoted by qs2 (t):

(39)

Substituting both q i (t) = qs2(t) and ki (t) = ks2(t) into equation (3) yields themaximum amount of waste emissions tolerable by the social planner, denotedby xs2(t):

(40)

3.3 Policy implications

The information in this section is summarised in the form of Proposition 1:

Proposition 1: No matter whether capital investment is costlessly reversible ornot, a firm will invest at less, produce at less, and pollute at more than thesocially efficient levels (Proof: see Appendix).

An individual firm will underestimate the adverse effect of pollution onproduction. Therefore, it will pollute more than is socially efficient as indi-cated by Proposition 1. Equivalently, the firm will invest less in abatementcapital than is socially efficient. The firm will also produce less than issocially efficient because as indicated by equation (20), investment is posi-tively correlated with the output level.

As the leader, the regulator will anticipate and thus correct the resultstated in Proposition 1 by using either an absolute or a proportional per-formance standard. While the regulator can set an absolute performancestandard at the socially efficient level, he cannot be sure whether the firmwill respond by producing at the socially efficient level. By contrast, if theregulator implements a proportional performance standard, he can set it

∂∂

vZ t

2 0( )( )

.⋅

=

q t A k t Z t m q ts sh h

f2 2 2 2 2

2

1

2

1( ) ( ) ( ) ( ).( ) ( )= =+ +λγ

λγλγ

λγ

x t A k t Z t m x ts s

h

h

h

hf2 1 2 3 2

1

2

1

2

1( ) ( ) ( ) ( ).( ) ( )= =−

+−

+γγ

λγγ

λγ



equal to the ratio of the socially efficient emission level over the sociallyefficient output level. This is precisely the efficient proportional perform-ance standard considered in what follows.

I will discuss both cases where absolute and proportional performancestandards are imposed at their respective socially efficient levels. Supposethat capital investment is costlessly reversible, and the regulator imposes oneach firm an absolute performance standard equal to x f 2(t) in equation(25). Replacing in equation (21) by x f 2(t) yields the correspondingstock of capital, denoted by :

(41)

where . Replacing

in equation (20) by yields the corresponding output level, denoted by:

(42)

Consider the case where capital investment is completely irreversible, andthe regulator imposes on each firm an absolute performance standard equalto xs2(t) in equation (40). Replacing in equation (33) by xs2(t) yieldsthe corresponding stock of capital, denoted by :

(43)

Replacing in equation (41) by yields the corresponding outputlevel, denoted by :

(44)

Given the emission function equation (3), a firm that underestimates theexternality from pollution will underestimate both the beneficial effect ofabatement capital in reducing pollution, and the adverse effect of productionin raising pollution. This yields the result stated in Proposition 2:

Proposition 2: No matter whether capital investment is costlessly reversible ornot, if the regulator restricts pollution at the socially efficient level, then afirm will over-invest and over-produce (Proof: see Appendix).

x tfu1( )

k tfa1( )

k t m k tfa

k f1 2( ) ( ),=

mhN N

hk

h

( ) ( ) = + −

+

−

>−

−1 1 1 1 111

1

1

1

2

λγ λγγ

γ

k tfu1( )

k tfa1( )

q tfa

1( )

q t m q t m mfa

q f q k1 2

2

1 1( ) ( ), .= = >whereγγ

x tsu1( )

k tsa1( )

k t m k tsa

k s1 2( ) ( ).=

k tsu1( ) k ts

a1( )

q tsa1( )

q t m q tsa

q s1 2( ) ( ).=

142 J.-B. Jou


Let us switch to the case where the regulator imposes on each firm a propor-tional performance standard, which is defined as the permissible emission perunit output (Montero 2002; Verdier 1995). Define , andyf 2(t) = x f 2(t) /q f 2(t). Suppose that capital investment is costlessly reversible,and that the regulator imposes on each firm a proportional performancestandard equal to yf 2(t). Equating y f 2(t) and yields the choice of capitalstock with regulation, denoted by , as given by

(45)

Replacing in equation (20) and equation (21) by yields thecorresponding output and emission levels, denoted by and ,respectively:

(46)

(47)

Define , and ys2(t) = xs2(t)/qs2(t). Suppose that capitalinvestment is irreversible and that the regulator imposes on each firm a pro-portional performance standard equal to ys2(t). Equating ys2(t ) and yields the desired capital stock with regulation, denoted by :

(48)

Replacing in equation (32) and equation (33) by yields thecorresponding output and emission levels, denoted by and ,respectively:

(49)

(50)

Pooling all above information yields the results stated in Propositions 3 and 4:

Proposition 3: Suppose that the regulator imposes on a firm a proportionalperformance standard at the socially efficient level. No matter whether investmentis costlessly reversible or not, the firm will over-produce and over-pollute. The firmwill also over-invest (under-invest) if emission is increasing with output at anincreasing (a decreasing) rate (Proof: see Appendix).

y t x t q tfu

fu

fu

1 1 1( ) ( )/ ( )=

y tfu

1( )k tf

p1( )

k t m k tfp

k

h

hf1

1

2

1

1( ) ( ).( )

( )=−

+γ

λ γ

k tfu1( ) k tf

p1( )

q tfp1 ( ) x tf

p1 ( )

q t m q tfp

q

h

hf1 2( ) ( ),( )= +λ

x t m x tfp

q

h

hf1 2( ) ( ).( )= +λ

y t x t q tsu

su

su

1 1 1( ) ( ) / ( )=

y tsu1( )

k tsp1( )

k t m k tsp

k

h

hs1

1

2

1

1( ) ( ).( )

( )=−

+γ

λ γ

k tsu1( ) k ts

p1( )

q tsp1( ) x ts

p1( )

q t m q tsp

q

h

hs1 2( ) ( ),( )= +λ

x t m x tsp

q

h

hs1 2( ) ( ).( )= +λ



Proposition 4: No matter whether investment is costlessly reversible or not, afirm will invest more, produce more, and pollute less if the regulator implementsan efficient absolute performance standard as compared to the case where theregulator implements an efficient proportional performance standard (Proof: seeAppendix).

The reason behind Proposition 4 follows from Proposition 2. Proposition2 indicates that a regulator who imposes on a firm an absolute performancestandard at the socially efficient level will induce the firm to produce atmore than the socially efficient level. The emission-output ratio will thusbe lower than is socially efficient. This indicates that abatement capital isemployed (and thus output is produced) at a level which is beyond the levelthat keeps the emission-output ratio at the socially efficient level. If,instead, the regulator sets a proportional performance standard at thesocially efficient level, then the firm will invest and produce less, and thuspollute more as compared to the case where the regulator sets an efficientabsolute performance standard.

Proposition 4, however, does not rank absolute and proportional per-formance standards. As indicated by Proposition 2, given environmentalexternalities a firm will not produce and pollute at the socially efficientlevel. These two inefficiencies should be corrected by implementing twoinstruments together, for example, effluent taxes and performance stand-ards. The use of either an absolute performance standard or a proportionalperformance standard will induce the firm to over-pollute as well as over-produce, as indicated by Propositions 2 and 3, respectively.

Finally, I will investigate how various factors such as uncertainty (σ ),costly expandability (µ), and competitive pressure (N ) affect the regulatedperformance standards. Some important results are stated in Proposition 5:

Proposition 5: (a) Suppose that investment is costlessly reversible and it ismore costly to purchase capital in the future (µ is increased ). The regulatorshould then implement a more stringent absolute performance standard, whichinduces a firm to invest and produce more. The regulator should then alsoimplement a more stringent proportional performance standard, which induces afirm to invest and produce more, and to pollute less. (b) Suppose that investmentis irreversible and uncertainty becomes greater (σ is increased ). The regulatorshould then implement a less stringent absolute performance standard, whichinduces a firm to invest and produce less. The regulator should then also imple-ment a less stringent proportional performance standard, which induces a firm toinvest and produce less, and to pollute more. (c) Suppose that competition be-comes more intense (N is increased) whether investment is costlessly reversibleor not. The regulator should then implement a more stringent absolute performance

144 J.-B. Jou


standard provided that γ 1 > γ 2 , which induces a firm to have ambiguous attitudestoward investment and output decisions. The regulator should then also implementa more stringent proportional performance standard provided that γ 1 > 1 + γ 2 ,which induces a firm to have ambiguous attitudes toward investment, output, andpolluting decisions (Proof: see Appendix).

The reason behind Proposition 5(a) is as follows. Suppose that it is morecostly to purchase capital at a later date. As indicated by equation (70),when investment in abatement capital is costlessly reversible, a social plannerwill then employ more capital and therefore pollute less. Consequently, theregulator should set more stringent absolute and proportional performancestandards. However, as investment becomes irreversible, an offsetting effectwill arise (see equation (75)) and, therefore, it becomes ambiguous whetherthe regulator should implement a more stringent policy.

The results of Proposition 5(a) entail policy implications. Suppose thatcapital investment is almost costlessly reversible. Proposition 5(a) then indi-cates that the regulator should set a much more stringent standard (at leastin terms of the adjustment period) when the expected cost of meeting a stand-ard rises. Consequently, older and newer firms should be treated differentlywith newer firms getting preferential treatment. However, in practice, olderfirms rather than newer firms receive preferential treatment. Such a policy,which may be justified if investment is irreversible, usually worsens pollution byencouraging firms to keep old, dirtier plants in operation (Keohane et al. 1998).

The reason behind Proposition 5(b) is as follows. As indicated by equa-tion (79), when investment is irreversible, a social planner will employ lesscapital and thus pollute more as uncertainty becomes greater. Accordingly,the regulator should be less stringent in implementing absolute and propor-tional performance standard policies, as indicated by equation (74) andequation (95), respectively. The firm will then be induced to invest and toproduce less, as indicated by equation (81) and equation (88) for the formerpolicy, and by equation (100) and equation (105) for the latter policy.

The policy implication of Proposition 5(b) is as follows. The so-calledhigh-tech industries usually invest in an environment that is more volatilethan the traditional industries. As a result, the regulator should favour high-tech industries when designing performance standard policies.

The reason behind Proposition 5(c) is as follows. Consider the situationthat competition becomes more severe. If the adverse effect of productionin raising pollution outweighs the beneficial effect of abatement capital inreducing pollution; that is, γ 1 > γ 2, then a social planner will pollute less (seeequation (71) and equation (76) no matter whether investment is irreversible ornot. Therefore, the regulator should set a more stringent absolute perform-ance standard. The regulator could also set a more stringent proportional



performance standard if the adverse effect is sufficiently larger than the bene-ficial effect; that is, γ 1 > 1 + γ 2, as indicated by equation (92).

The result of Proposition 5(c) indicates that whether the regulator shouldimplement a performance standard policy that favours certain industriesdepends on the efficiency of abatement capital employed by firms in thisindustry. For example, suppose that firms in two industries employ thesame kind of abatement capital and suppose that γ 1 > γ 2 (γ 1 > 1 + γ 2), thenthe regulator should implement a more stringent absolute (proportional)performance standard policy for the industry that is more competitive.11

3.4 Numerical examples

In this section I make the theoretical results more meaningful throughnumerical examples. The benchmark parameter values are chosen as follows:Z(t) = 1, λ = 0.1, σ = 20 per cent per year, γ 1 = 1, γ 2 = 0.5, ε = 3, N = 500, µ = 0per year, w = 0.5, ρ = 5 per cent per year, and Pk(t) = 1.12 For ease of exposi-tion, I only present the results for the case where abatement investment iscompletely irreversible. Given these benchmark parameter values, I findthat the absolute performance standard that is socially efficient is given byxs2(t) = 0.1309, and the desired capital stock, and the output level inducedby it are respectively given by = 0.0503, and = 0.0294. Theproportional performance standard that is socially efficient is given byys2(t) = 19.968, and the maximum level of pollution, the desired capital stock,and the maximum level of output induced by it are respectively given by

= 0.4149, = 0.0025, and = 0.0208. These results confirmthe results of Proposition 4 indicating that an absolute performance standardpolicy will induce a firm to invest and produce more, and pollute less ascompared to a proportional performance standard policy. I also investigatethe responses in these seven endogenous variables to a change of σ aroundthe region (0, 40 per cent), µ around the region (−1 per cent, 1 per cent),and N around the region (400, 600), holding the other parameters at theirbenchmark values. These responses are respectively shown in figures 1–3.13

11 It is interesting to investigate whether the regulator should favor new abatement capitalthat is more efficient (higher γ 2 ). However, our comparative-statics results regarding this issueyield ambiguous results (not shown in the Appendix).

12 As suggested by Dixit (1989), some capital costs arise from depreciation and are morethought of as recurrent, and some costs are recoverable when disinvestment occurs.Accordingly, a ratio of w :ρPK = 10:1 seems plausible.

13 For ease of exposition, I scale down ys2(t) by a factor of 100, and scale up all by a factor of 10.

k tsa1( ) q ts

a1( )

x tsp1( ) k ts

p1( ) q ts

p1( )

k tsa1 ( ),

k t q t q tsp

sa

sp

1 1 1( ), ( ), ( )and

146 J.-B. Jou


All results shown in figures 1–3 conform to those stated in Proposition 5.Figure 1 indicates that as capital investment is more costly to expand later,the regulator should be more stringent in both absolute and proportionalperformance standard policies toward irreversible investments. In response,a firm will invest and produce more. Figure 2 shows that as uncertaintybecomes greater, the regulator should be more lenient in both policiestoward irreversible investments, a finding that is in line with Jou (2001) that

Figure 1 The impact of a change in µ .

Figure 2 The impact of a change in σ .



investigates the optimal effluent fee policy. In response, a firm will be lesswilling both to invest and produce. Finally, figure 3 indicates that as competi-tion becomes more intense, the regulator should impose a more stringentabsolute performance standard policy because γ 1 > γ 2 (see Proposition 5(c)),but should implement a less stringent proportional performance standardpolicy because the constraint γ 1 > 1 + γ 2 stated in Proposition 5(c) is not satis-fied. In response, a firm will invest and produce less.

4. Conclusion

The present article investigates the use of performance standards to correctenvironmental externalities. Each firm in an industry emits waste in theproduction process, and, in turn, the average waste emissions of an industryadversely affect the firm’s productivity. The firm, which incurs sunk costswhen employing capital to abate waste emissions, is uncertain about theefficiency of capital. The firm will underestimate environmental externalities,and therefore, it will pollute more than is socially efficient. To correct thistendency, the regulator can set a limit on either emissions or the emission-output ratio at the socially efficient level. The firm will invest more, producemore, and pollute less when the regulator implements the former limit thanwhen the regulator implements the latter limit.

The present article has abstracted from several features that may beallowed in future research. First, it ignores both the adverse effect of pollu-tion on the welfare of consumers and spatial distribution of firms. Second,it ignores the costs associated with implementing performance standards. It

Figure 3 The impact of a change in N.

148 J.-B. Jou


is worth investigating whether the conclusion of the present article will beaffected when taking these two features into account.

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Appendix

Derivation of v1(·) and v2(·)

Consider the problem of choosing Ii (t) such that equation (26) is maximised.The Bellman-Hamiton-Jacobi for this problem can be written as

(51)

Equation (51) says that the total return on this asset, ρV1(·) has two com-ponents, the cash flow π1(k (t), Z(t)) − PK (t)Ii (t), and the expected rate of

capital gain, , where

(52)

by using both equations (4) and (11), and by applying Itô’s Lemma. Differenti-ating the right-hand side of equation (51) with respect to Ii (t), and then settingthe result equal to zero yields

(53)

Substituting given by equation (17) and in equation (52) intoequation (51), and using the condition in equation (53) yields the differentialequation

(54)

Let ∂V1(·) /∂ ki (t) = v1(·). Differentiating equation (54) term by term withrespect to k i (t) yields

ρ πV k t Z t P t I tdt

E dVI t K i ti11

1

1( ) max [ ( ), ( )] ( ) ( ) ( ) . ( )⋅ = − + ⋅

11dt

E dVt ( )⋅

EdV

dtZ t

V

Z tP t

V

P tI t

V

k tt KK

ii

1 2 22

12

1 112

( ) ( )

( )

( ) ( )

( )

( ) ( )

( )

( ),

⋅=

⋅+

⋅+

⋅σ

∂∂

µ∂∂

∂∂

∂∂

V

k tP t

iK

1( )

( ) ( ).

⋅=

π i1( )⋅

11dt

E dVt ( )⋅

12

02 22

12 1

11 1 2

1

31

1

10σ

∂∂

ρ µ∂∂

λγλγZ t

V

Z tV P t

V

P tB G G G Z tK

K

a h a( )( )

( ) ( ) ( )

( )

( ) ( ) .

( )

( )⋅

− ⋅ +⋅

+ =− +

+



(55)

where G is defined in equation (28). Based on Bertola and Caballero (1994),the term solves the homogeneous part of equation (55).Substituting this into equation (55) yields the quadratic equation

(56)

Denote β and −α respectively as the larger and smaller roots in the quadraticequation given by equation (56). Following Dixit (1991), equation (56) canbe rewritten as

(57)

where φ (τ ) > 0 if –α < τ < β. Figure 4 depicts φ (τ ) as a function of τ.One particular solution from the non-homogeneous part of equation (55)

is given by

(58)

Since the value function V1(·) must approach zero as Z(t) approacheszero, only the positive root in equation (56) should be considered. The gen-eral solution of equation (55), which is composed of solutions from boththe homogeneous and non-homogeneous parts, is shown by equation (28).

Similarly, the Bellman-Hamiton-Jacobi equation for the problem describedin equation (34) can be written as

(59)

12

02 22

12 1

1 0σ∂∂

ρ µ∂∂

Z tv

Z tv P t

v

P tGZ tK

K

a( )( )

( ) ( ) ( )

( )

( ) ( ) ,

⋅− ⋅ +

⋅+ =

Figure 4 φ (τ ) versus τ .

[ ( ) ] ( )GZ t P ta

K0 1τ τ−

φ τ σ τ τ µ τ ρ( ) ( ) ( ) .= − − − − + =12

1 1 020 0a a

φ τ σ α τ β τ ρ µ α τ β τα β

( ) ( ) ( ) ( )( )( )

,= − + − =− + −1

2 02 2a

v GZ t a1

0 1( ) ( ) / ( ).⋅ = φ

ρ πV k t Z t P t I tdt

E dVI t i K i ti22

2

1( ) max [ ( ), ( )] ( ) ( ) ( ) , ( )⋅ = − + ⋅

152 J.-B. Jou


where

(60)

Differentiating the right-hand side of equation (59) with respect to Ii (t),and then setting the result equal to zero yields

(61)

Substituting π 2(·) given by equation (16) and in equation (60) intoequation (59), and using the condition given by equation (61) yields

(62)

Let ∂V2(·) /∂ ki (t) = v2(·). Differentiating equation (62) term by term with re-spect to ki (t) yields

(63)

The solution to v2(·) in equation (63) is thus given by equation (35).

Proof of proposition 1

Dividing in equation (19) by k f 2(t) in equation (23) yields

(64)

since if N = 1 then , and differentiating with respectto N yields a negative sign.

Dividing in equation (21) by xf 2(t) in equation (25) yields

(65)

EdV

dtZ t

V

Z tP t

V

P tI t

V

k tt KK

ii

2 2 22

22

2 212

( ) ( )

( )

( ) ( )

( )

( ) ( )

( )

( ).

⋅=

⋅+

⋅+

⋅σ

∂∂

µ∂∂

∂∂

∂∂

V

k tP t

iK

2( )

( ) ( ).

⋅=

E dV dtt [ ( )/ ]2 ⋅

12

02 22

22 2

22

0 0σ∂∂

ρ µ∂∂

Z tV

Z tV P t

V

P tB k t Z tK

Ki

a a( )( )

( ) ( ) ( )

( )

( ) ( ) ( ) .

⋅− ⋅ +

⋅+ =

12

02 22

22 2

20 2

10 0σ∂∂

ρ µ∂∂

Z tv

Z tv P t

v

P ta B k t Z tK

Ki

a a( )( )

( ) ( ) ( )

( )

( ) ( ) ( ) .

⋅− ⋅ +

⋅+ =−

k tfu1( )

k t k tN h

hhN N

hfu

f

ah a

1 2

1

1

11

1

1

1

1

11 1 1 1 1

0

1

1 0

( ) / ( )

( ) ( ) ,

( )( )

=−

−

+ −

+

−

<

−−

−−

++ −

λγ λγλγ

λγ

k t k tfu

f1 2 1( )/ ( ) = k t k tfu

f1 2( )/ ( )

x tfu1( )

x t x thN N

hk t

k tfu

f

hfu

f

h

h

1 2 11

1

1 1

2

1 1 1 1 1

1

1

2

1

( ) / ( ) ( ) ( ) ( )

( ) .

( ) ( )

= + −

+

−

>

−−

+−

+

λγ λγγ

λγγ

λγ



Dividing in equation (31) by ks2(t) in equation (38) yields

(66)

Dividing in equation (33) by xs2(t) in equation (40) yields

(67)

Dividing in equation (20) by q f 2(t) in equation (24) yields

(68)

since if N = 1 then , and (for N ≥ 2) differentiating q f 2(t) with respect to N yields a negative sign.

Dividing in equation (32) by qs2(t) in equation (39) yields

(69)

This completes the proof.


Proposition 2 follows because mk > 1 and mq > 1 such that .


Because it follows that

and If γ 1 < (>)1 then and thus and . This completes the proof.


Because m k > 1 it follows that Because mq > 1,it follows that and .This completes the proof.

k tsu1( )

k t k t k t k tsu

s fu

f1 2 1 2( )/ ( ) ( ) / ( ).=

x tsu1( )

x t x t x t x tsu

s fu

f1 2 1 2( )/ ( ) ( ) / ( ).=

q tfu1( )

q t q t NhN N

hfu

fa h

h a

1 21

11

1

1

11

1

12

0 1

1

1

0

1 1 1 1 1( )/ ( ) ( ) ( ) ,( )( )

( ) ( )

= + −

+

−

<−

− +−

−+

++

−

γ λ

γ λλγ

λγ

λγ λγ

q t q tfu

f1 2 1( )/ ( ) = q tfu

1( )/

q tsu1( )

q tq t

q t

q tsu

s

fu

f

1

2

1

2

1( )( )

( )

( ) .= <

k t k tfa

f1 2( ) ( ),>q t q t k t k t q t q tf

af s

as s

as1 2 1 2 1 2( ) ( ), ( ) ( ), ( ) ( )> > >and

mq

h

h( ) λ+ > 1 q t q t x t x t q t q tft

f fp

f sp

s1 2 1 2 1 2( ) ( ), ( ) ( ), ( ) ( ),> > >

x t x tsp

s1 2( ) ( ).> mk

h

h

( )

( ) ( )γ

λ γ1

1

1

1−

+ < > k t k tfp

f1 2( ) ( ) ( ) < >k t k ts

ps1 2( ) ( ) ( )< >

k t k t k t k tfp

fa

sp

sa

1 1 1 1( ) ( ) ( ) ( ). < <andq t q t q t q t x t x tf

pfa

sp

sa

fp

f1 1 1 1 1 2( ) ( ), ( ) ( ), ( ) ( )< < < x t x tsp

s1 2( ) ( )<

154 J.-B. Jou



Differentiating xf 2(t) with respect to µ and N yields

(70)

(71)

Let X = (1 + α )/α > 1 and , then m3 = X Y.

Differentiating m3 with respect to σ and µ yields

(72)

since from figure 4;

(73)

where

Differentiating xs2(t) with respect to σ, µ and N yields

(74)

(75)

(76)

Differentiating with respect to µ and N yields

(77)

∂∂µ

γλγ

∂∂µ

∂∂µ ρ µ

x t hx t

h k t

k t k t k t

af f

f

f f f2 2 2

1 2

2 2 2

0

01

0( )

( )

( ) ( )

( ) ,

( )

( )

( )( ) ,=

−+

< =− −

>where

∂∂

γ γ λγλγ λγ λγ

∂∂

γ γx t

N

h h x t

N h h h

x t

Nf f f2 2 1 1 2

1 1 2

22 11

0( ) ( )( ) ( )

( )( ( )),

( ).=

− ++ + − −

< <where if

Yh

h = − +

+

>

−λ γ

γ γ1

101

2 2

1

∂∂σ α α

∂α∂σ

∂α∂σ

∂φ α ∂σ∂φ α ∂τ

m m Y3 3

10 0

( )

( ) /( ) /

,=−

+> =

−−

<, where

∂φ α∂σ

σ α α ∂φ α∂τ

( ) [ ( )] [ ( ) ] ,

( )

−= − − − − <

−>a a0 0 1 0 0and

∂∂ µ α α

∂α∂ µ

m m Y3 3

10

( ) ,=

−+

>

∂α∂ µ

∂φ α ∂ µ∂φ α ∂τ

∂φ α∂ µ

α ( ) /( ) /

, sin ( )

( ( )) .=−−

<−

= − − − <0 1 0ce

∂∂σ

∂∂σ

x tx t

msf

22

3 0( )

( ) ,= >

∂∂ µ

∂∂ µ

∂∂ µ

x tm

x tx ts f

fm2

32

23

( )

( ) ( ) ,

( ) ( )

= +

− +

∂∂

∂∂

∂∂

γ γx t

Nm

x t

N

x t

Ns f f2

32 2

2 10( )

( )

, ( )

.= < <where if

k tfa1( )

∂∂µ

∂∂µ

k tm

k tfa

kf1 2 0

( )

( ) ,= >



(78)

where and

> 0.Differentiating m1 with respect to σ and µ yields

(79)

(80)

Differentiating with respect to σ, µ and N yields

(81)

(82)

(83)


(84)

(85)

where

∂∂

∂∂

∂∂

k t

Nm

k t

Nk t

m

Nfa

kf

fk1 2

2

( )

( ) ( ) ,

( ) ( )

= +

− +

∂∂

λγλγ

k t

N

h k t

a h Nf f2 1 2

0 1

1

10

( )

( ) ( )

( )( ) ,=

− +− +

<∂∂

γ λγ

γ λγmN

h m

hhN

NN

k k ( )

=+

−

+

1 1

22 11 1

∂∂σ α α

∂α∂σ

m m

a1 1

01 10

( ) ( ) ,=

− +<

∂∂µ α α

∂α∂µ

m m

a1 1

01 10

( ) ( ) .=

− +<

k tsa1( )

∂∂σ

∂∂σ

k tk t

msa

fa1

11 0

( ) ( ) ,= <

∂∂µ

∂∂µ

∂∂σ

k tm

k tk t

msa

fa

fa1

11

1

( )

( ) ( ) ,

( ) ( )

= +

+ −

∂∂

∂∂

k t

Nm

k t

Nsa

fa

11

1( )

( ).=

q tfa

1( )

∂∂µ

λγλγ

∂∂µ

q t q t

h k t

k tfa

fa

f

f1 2 1

1 2

2 0( )

( )

( ) ( )

( ) ,=

+>

∂∂

∂∂

∂∂

q t

Nm

q t

Nq t

m

Nfa

qf

fq1 2

2

( )

( ) ( ) ,

( ) ( )

= +

− +

∂∂ λγ

λγ λγλγ

∂∂

γγ

∂∂

γγ

q t

N

hq t

N h a h

m

Nm

m

Nf f q

kk2 2

1

2 1

0 1

1

2

11

11

0 01

2

( )

( )

( )

( )( )( )

, .=−

++

+− +

< = >−

156 J.-B. Jou


Let , then m2 = X −Z. Differentiating m 2 with

respect to σ and µ yields

(86)

(87)


(88)

(89)

(90)

Differentiating yf 2(t) with respect to µ and N yields

(91)

(92)

Define , then ys2(t) = m ys2(t). Differentiating m4 withrespect to σ and µ yields

(93)

Z h = − + +

>

−γγ λγ

1

2 2

1

1 11

0

∂∂σ α α

∂α∂σ

m m Z2 2

10

( ) ,=

+<

∂∂µ α α

∂α∂µ

m m Z2 2

10

( ) .=

+<

q tsa1( )

∂∂σ

∂∂σ

q tq t

msa

fa11

2 0( )

( ) ,= <

∂∂µ

∂∂µ

∂∂σ

q tm

q tq t

msa

fa

fa1

21

12( )

( )

( ) ,

( ) ( )

= +

+ −

∂∂

∂∂

q t

Nm

q t

Nsa

fa

12

1( )

( ).=

∂∂µ

γ λλγ

∂∂µ

∂∂

λγ γ γ λγ γλγ

y t hh k t

k t

y t

N

h h h y t

f

f

f

f f

2 2

1 2

2

2 1 1 2 2 1 2

1

0

1 1

( )

( )( ) ( )

( ) ,

( )

[( )( ) ( )] ( )

(

=− +

+<

=− + − − + −

+ )( ( )),

h h h Nλγ λγ1 2 1+ − −

where if∂

∂γ γ

f t

Nz ( )

.< > +0 11 2

m m m m h4 2 3 3

1 / = =

+λ

∂∂σ

λ ∂∂σ

m

h

m

m

m4 4

3

31 0 ( ) ,= + >



(94)

Differentiating ys2(t ) with respect to σ, µ, and N yields

(95)

(96)

(97)


(98)

(99)

Differentiating with respect to σ, µ, and N yields

(100)

(101)

(102)


(103)

∂∂µ

λ ∂∂µ

m

h

m

m

m4 4

3

31 0 ( ) .= + >

∂∂σ

∂∂σ

y ty t

msf

22

4 0( )

( ) ,= >

∂∂µ

∂∂µ

∂∂µ

y tm

y ty t

ms ff

24

22

4( )

( ) ( )= + ,

(–) (+)

∂∂

∂∂

y t

Nm

y t

Ns f2

42( )

( )

= .

k tfp1( )

∂∂µ

∂∂µ

γλ γk t

mk tf

p

k

h

h f1

1

2

1

1 0( )

( )

,

( )

( )= >−

+

∂∂

∂∂

γλ γ

∂∂

γλ γk t

Nm

k t

Nk t

h

h m

m

Nfp

k

h

h ffp

k

k1

1

21

1

1

1

11( )

( )

( )( )

( ).

( ) ( )

( )

( )= +−

+− +

−+

k tsp1( )

∂∂σ

∂∂σ

γλ γk t

mk ts

p

k

h

h s1

1

2

1

1 0( )

( )

,

( )

( )= <−

+

∂∂µ

∂∂µ

γλ γk t

mk ts

p

k

h

h s1

1

2

1

1 0( )

( )

,

( )

( )= >−

+

∂∂

∂∂

γλ γ

∂∂

γλ γk t

Nm

k tN

k th

h m

m

Nsp

k

h

h ssp

k

k1

1

21

1

1

1

11( )

( )

( )( )

( ).

( ) ( )

( )

( )= +−

+− +

−+

q tfp1( )

∂∂µ

∂∂µ

λq tm

q tfp

q

h

h f1 2 0( )

( )

,( )= >+

158 J.-B. Jou


(104)


(105)

(106)

(107)


(108)

(109)


(110)

(111)

(112)

∂∂

∂∂ λ

∂∂

λ λq t

Nm

q t

Nq t

hh

mm

Nfp

q

h

h ff q

h

h q1 22

1( )

( ) ( )

( ).

( ) ( )

( ) ( )= ++

− +

+ +−

q tsp1( )

∂∂σ

∂∂σ

q tq t

msp

fp1

12 0

( ) ( ) ,= <

∂∂ µ

∂∂ µ

∂∂σ

q tm

q tq t

msp

fp

fp1

21

12( )

( )

( ) ,

( ) ( )

= +

+ −

∂∂

∂∂

q tN

mq t

Nsp

fp

12

1( )

( ).=

x tfp1( )

∂∂ µ

∂∂ µ

λx tm

x tfp

q

h

h f1 2 0( )

( )

,( )= <+

∂∂ λ

∂∂

∂∂

λ λx t

Nh

hm

m

Nx t m

x t

Nfp

q

h

h qf q

h

h f11

22( )

( )

( ) ( )

.

( )

( ) ( )=+

+

+

+−

+

x tsp1( )

∂∂σ

∂∂σ

λx tm

x tsp

q

h

h s1 2 0( )

( )

,( )= >+

∂∂ µ

∂∂ µ

λx tm

x tsp

q

h

h s1 2( )

( ),( )= +

∂∂ λ

∂∂

∂∂

λ λx tN

hh

mm

Nx t m

x tN

sp

q

h

h qs q

h

h s11

22( )

( )

( ) ( )

.

( )

( ) ( )=+

+

+

+−

+

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