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 production process, and, in turn, the average waste emissions of the industry adversely affect the firm’s productivity. The firm, which incurs sunk costs when employing capital to abate waste emissions, is uncertain about the efficiency of capital. The firm will underestimate environmental externalities and will therefore pollute more than is socially efficient. To correct this tendency, the regulator can set a limit on either emissions or the emission-output ratio at the socially efficient level. The firm will invest more, produce more, and pollute less when the regulator implements the former than when the regulator implements the latter.
1. Introduction
Manufacturing firms usually emit waste in the production process. The regulator can induce these firms to employ more abatement capital by imple-menting command-and-control or market-based instruments. The investment
costs 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 to
natural gas burning power plants so as to reduce CO2 emissions (Saphores
and Carr 1999). Unlike earlier articles on optimal environmental protection policies (see e.g., Mohtadi 1996), the present article will account for these sunk 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 at the National Taipei University and the 28th Annual Meeting of the Academy of Economics and Finance held in February 2001.
† Jyh-Bang Jou is a professor in the Graduate Institute of National Development, College of 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 The
present article builds a model in which an industry composed of a fixed number 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 employ capital to abate waste emissions, and both the firm and the regulator share the same information about uncertainty regarding the firm’s abatement technology. The firm will underestimate environmental externalities because it will ignore the adverse effect of its emissions on other firms’ pro-ductivity. Therefore, it will pollute more than is socially efficient. To correct this tendency, the regulator can either set an absolute or a proportional performance standard at the socially efficient level. The responses of a firm to these two policies, however, are divergent. The firm will invest more, pro-duce more, and pollute less when the regulator implements the former than when the regulator implements the latter.
The present article is related to the published literature that applies the theory of non-cooperative dynamic games to environmental management such as Haurie and Krawzyck (1997), Haurie and Zaccour (1995), Krawzyck (1995), and Krawzyck and Zaccour (1999). Like these articles, the present article models effluent management as a hierarchical game with a Cournot-Nash equilibrium at the lower level and a Stackelberg equilibrium at the upper 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 upper level of the game, the regulator (i.e., the leader), who anticipates the behavior of a firm (i.e., the follower), sets an efficient level of performance standard accordingly. However, these articles abstract from the investment decision of a firm, and resort to simulation analysis because they construct a more comprehensive model than that constructed in the present article.
Previous studies on performance standards either focus on different issues or adopt assumptions that differ from those made in the present article. For example, a large volume of empirical articles (see e.g., Barbera and McConnell 1986; Gray and Shadbegian 1995) find that performance stand-ards impact on productivity at the firm or plant level, but they ignore the fact that abatement capital may enhance productivity through reducing emissions (as emphasised by the present article). A number of other articles assume that firms are better informed than the regulator, and then rank
2 Several studies on real options either allow government intervention while ignoring externalities (see e.g., Dixit 1991; Dixit and Pindyck 1994; Hassett and Metcalf 1992) or focus on pecuniary rather than production externalities (see e.g., Dixit and Rob 1994; Fatas and 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 asymmetric information.
The present article is also closely related to studies that apply real
options to investigate optimal environmental policy, such as Jou (2001).3
Jou assumes that a firm’s emissions are independent of its output level, and investigates the use of effluent fees to correct environmental externalities. By contrast, the present article assumes that a firm’s emissions increase as the firm produces more, and emphasises that the uses of absolute and pro-portional performance standards may have different impacts on the firm’s investment, production, and polluting behavior.
The remaining sections are organised as follows. Section 2 constructs the model and calculates the short-run equilibrium private and social profits. Section 3 allows abatement investment to be either costlessly reversible or completely irreversible. The absolute and proportional performance stand-ards required to support social efficiency for both types of investments are derived. The impacts of several exogenous factors such as uncertainty, costly expandability, and competitive pressure on these two instruments are investigated both analytically and numerically. The last section concludes and suggests extensions for future research.
2. The Model
The present article builds a model that incorporates the main features appearing in both the standard real options (see e.g., Dixit 1991) and the environmental economics published literature (see e.g., Cropper and Oates 1992). In the present article, the whole economy is represented by an
indus-try composed of N identical risk-neutral firms facing a demand function
with a constant price elasticity ε (>0); that is,
3 See also Pindyck (2000, 2002), Saphores and Carr (1999), and Xepapadeas (1999). The former three articles consider a regulator who intends to reduce stocks of environmental pollutants once-and-for-all. In Pindyck’s two articles, the regulator controls a flow variable that 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 these articles 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 does not explicitly model externality nor relate asset characteristics to optimal environmental policies.
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 produces
final goods according to a Cobb-Douglas technology given by
qi(t) =li(t)xa(t)−λ, λ> 0, (2)
where qi(t) is firm i’s output, li(t) is the amount of labour employed by firm
i, and xa(t), which is industry’s average waste emissions, will be strictly
pos-itive in industry equilibrium. The adverse effect of xa(t) on qi(t) is external
with λ measuring its size. This external effect indicates that as firm i emits
more, its output as well as the other firms’ output will decline.4 For example,
a firm that emits hazardous pollutants will adversely affect health conditions of the firm’s and the other firms’ employees. Consequently, the productivity of all firms in the industry will decline. Pollution may also adversely affect the welfare of consumers (see e.g., Cropper and Oates 1992). The present article, however, abstracts from this negative effect. Equation (2) also depicts a case of a uniformly distributed and diffused pollutant, and thus abstracts from both spatial distribution of firms (see e.g., Haurie and Krawzyck 1997) and stock externalities (see e.g., Farzin 1996). Equation (2) is restricted, yet it 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, qi(t), its employed abatement capital, ki(t), and
a technology-shock factor, Z(t), in the form given by
(3) Equation (3) indicates that abatement capital exhibits diminishing returns because an increase in abatement capital reduces emissions at a declining rate. The specification in equation (3) also indicates that the amount of
waste emitted by firm i, xi(t), is an increasing function of its output, qi(t).
4 In reality, waste emissions from all industries will have a negative impact on a firm’s output in one industry (Ballard and Medema 1993), yet the present article assumes that the whole economy consists of only one industry for ease of exposition.
5 Some articles on environmental economics allow a firm to employ both productive and abatement 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 in published 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 the analysis while adding little insight into the issue on which the present article focuses.
x ti( ) ( ) (=q ti k t Z ti( ) ( )) , , .>
−
γ1 γ2 γ γ
Environment, irreversibility, and optimal effluent standards 131 The reason is obvious, for example, if steel or electricity production in-creases, pollution also increases. This specification is more generalised than that in the published literature. For example, Jou (2001) assumes that
a firm’s emissions are uncorrelated with its output (γ1 = 0), and Krawzyck
(1995) and Xepapadeas (1999) assume that a firm’s emissions are
propor-tional to its output (i.e., γ1 = 1). The present article, by contrast, allows
emissions to be increasing with output at a constant rate, an increasing rate (i.e., γ1> 1), or a decreasing rate (i.e., γ1 < 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 positive so 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:
d Z (t ) = σZ(t)dΩ(t), (4)
where σ (>0) is the instantaneous volatility of the growth rate ofZ(t), and
dΩ(t) is an increment to a standard Wiener process, with E{dΩ(t)} = 0 and
E{dΩ(t)}2= dt. By the standard theory of Brownian motion, we know that
Z(t) has a lognormal distribution with mean Z(0) and variance .6
Equation (4) indicates that information about the evolution of a firm’s waste emissions arriving in time is independent of its investment decision. This contrasts with that of Kolstad (1996) where uncertainty can be resolved over time through learning. Equation (4) also indicates a case of symmetric 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 published environmental economics literature – see Jung et al. (1996), Requate and Unold (2003), and Weitzman (1974). These articles, however, focus on ranking various policy instruments including effluent taxes and subsidies, performance stand-ards, and marketable permits. The published real options literature that incorporates this issue includes Gaudet et al. (1998), which, however, does not allow environmental externalities.
6 Equation (3) indicates that I do not allow new abatement capital to be more efficient than old capital. One alternative way to take this into consideration is to replace ki(t) by
a(t)ki(t), where d a (t)/d t ≥ 0, indicating that there is technological progress.
7 The present article allows a firm to adopt only one type of abatement capital. One may assume that a firm chooses between an old and a new abatement technique, and then follows the solution method in Dixit (1991) to solve the firm’s choice. The qualitative results of the present article, however, will not be affected even though such a choice is allowed.
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 its
operating 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 dynamic games 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 with a Cournot-Nash equilibrium at the lower level where firms compete for production and investment. At the upper level of the game, the regulator acts as the leader and a firm acts as the follower. The regulator should anticipate the production and investment behavior of the firm, and then set performance 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, denoted
by , to maximise its operating profit, given by equation (5).
Con-sequently, is derived by setting the derivative of with respect to
qi(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 correspondingP1(t ):
(8) (9) πi q t k t Z t 1 ( ( ), ( ), ( )) C q t k ti 1 ( ( ), ( ), πi q t k t Z t P t q ti C q t k t Z ti 1 1 ( ( ), ( ), ( )) ( ) ( ) ( ( ), ( ), ( )).= − C q t k t Z ti wq t x ti a 1 ( ( ), ( ), ( )) ( )= ( ) ,λ x ta q tj k t Z tj N j N ( ) =
(
( ) ( ( ) ( ))−)
=∑
γ1 γ2 1 q ti 1 ( ) πi 1 ( )⋅ q ti1( ) π1i( )⋅ P t q t Q t wx t x t x t N i a a i ( ) ( ) ( ) ( ) ( ) ( ) . 1− 1 1 = + − ε λγ λ q ti 1 ( ) q ti1( ) q t w h N N Z t G G k t i h i h 1 1 1 1 1 1 1 1 2 2 1 ( ) ( ) ( ( ) ) , ( ) = − + − − − − − − + λ λγ λ λγ γ λγ P t1( ) (= Nq t1i( )) ,−hwhere 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 byπ1(k (t), Z (t )), as given by
(10)
In contrast, a social planner will internalise environmental externalities before producing final goods. The social planner perceives that firm i ’s
emissions, xi(t), are equal to the industry’s average emissions, xa(t), because
all firms are identical. By equation (3), this also implies that the central
planner will impose qj(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 social
planner, denoted by , is equal to operating revenue
P(t)qi(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 conditionxa(t) = xi(t), the variable cost of firm i
perceived 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 to
zero. This yields the equality given by
(13)
where ε is required to be larger than one (and thus h = 1/ε < 1) so as to
ensure that marginal revenue is positive. Imposing the condition qj(t ) =
G j k tj N 1 1 2 =
∑
= ( )−γ πi 1 ( )⋅ q t1i( ) π λγ λγ 1 1 1 2 1 3 1 1 1 0 ( ( ), ( )) ( ) , ( ) ( ) k t Z t =B G Ga h G Z t a − + + where a h h a h h G G k t G h NG k B w i i h h 0 2 1 1 1 2 1 1 3 1 1 1 1 1 1 1 2 2 1 ( ) ( ), ( )( ) ( ) , ( ) , , ( ) = − + = − − + = + = + = − − − + λγ λγ λ λγ λγ λγ γ γ λγ 11 1 1 1 1 1 . ( ) ( ) ( )( ) ( ) − − + − + + − + h N N h h h h λγ λ λγ λ λγ πi q t k t Z ti i 2 ( ( ), ( ), ( )) Ci2( ( ), ( ), ( ))q t k t Z ti i πi q t k t Z ti i P t q ti Ci q t k t Z ti i 2 2 ( ( ), ( ), ( )) ( ) ( ) = − ( ( ), ( ), ( )). Ci q t k t Z ti i wq t x ti i 2 ( ( ), ( ), ( )) ( ) ( ) .= λ q ti 2 ( ) πi 2 ( )⋅ q ti 2 ( ) πi 2 ( )⋅ P t( ) ( 1− 1 1 1)w q ti( ) 1( ( ) ( ))k t Z ti 2, = + − ε λγ λγ λγ q ti2( )( j = 1, ... , N) in equation (13) yields and its corresponding P2(t) as given by
(14) (15)
Evaluating in equation (11) at qi(t ) = yields the optimised value
of , denoted by π2(k
i(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 this includes market-based instruments or command-and-control regulations. Market-based instruments, however, are still not popular: the application of tradable permits appeared in few countries such as Australia, Canada, Chile, and the USA (see e.g., Gomboso et al. 1999), and effluent fees (and subsidies) adopted by Europe have been set at very low levels, and thus are usually not considered as an effective way to control pollution (Kolstad, 2000). Command-and-control regulations, which are still the dominant form of environmental regulation in the world today, have two major types: technology-and performance-based sttechnology-andards (Stavins 2000). Technology-based sttechnology-andards specify the method and/or the equipment (the so-called best available control technology) that firms must use to comply with a particular regulation. In the USA, the specific technologies are usually determined by states on a case-by-case basis. However, the USA has established national emission standards for new sources of pollutants, called the New Source Performance Standards (Tietenberg 1992, p. 397). In the present article I focus on performance standards, which could specify an absolute quantity of permissible emissions, but more typically these standards establish allowable emissions in proportional to output (Keohane et al. 1998). In what follows, I will investigate both types of 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 also assume that the purchasing and installation price of abatement capital, denoted
by PK(t), grows at a constant rate µ; that is,
q ti 2 ( ) q ti2 1 w h 1N k t Z th i h 1 1 1 2 1 ( ) [( = + ) ( )− − ( ( ) ( ))− ]( ), − + λγ λγ λγ P t2( ) (= Nq ti2( )) .−h πi 2 ( )⋅ q ti2( ) πi 2 ( )⋅ π2 2 0 0 (k t Z ti( ), ( )) =B k ti( )a Z t( ) ,a B h h w N h h h h 2 1 1 1 1 1 1 1 1 1 ( ) [( )/ ] [ ( )] . ( ) ( ) ( ) ( ) = + − + − + − + + λγ λγ λγ λγ λγ
dPK(t ) = µPK(t ) dt. (17)
A firm is in an environment where it is costly to expand capacity later when
µ is positive.8 The following analysis, however, is suitable for both positive
as well as non-positive values of µ. Two types of investments are considered:
(i) investment is costlessly reversible where the resale price of abatement
capital is equal to the purchase price of capital;9 and (ii) investment is
completely irreversible where the resale price of abatement capital is equal to zero.
3.1 Costlessly reversible investment
Suppose that investment is costlessly reversible. Then ki(t) will be a choice
rather than a state variable. Let ρ denote a given (risk-adjusted) discount
rate. The abatement capital stock chosen by firm i at each instant will be found by equating the private marginal return to capital with the user cost of capital (i.e., the rental cost of abatement capital net of capital gain) (Jorgenson 1963); that is,
(18)
where π1( 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 permit that is in short supply (Dixit and Pindyck, 1998).
9 One may allow the installed abatement capital to have some resale value. Under such a situation, in addition to effluent standard restrictions, the regulator may add one more policy instrument such as a tax on disinvestment. Such an extension, however, will not alter the main results of the present article. See, for example, the article by Jou and Lee (2001) where R&D capital exhibits positive production externalities and partial irreversibilities, and the regulator gives investment tax credits when R&D capital is purchased, but imposes taxation when the installed R&D capital is sold.
∂ π ∂ ρ µ 1 ( ( ), ( )) ( ) ( ) ( ), k t Z t k ti P t K = − kf t u 1( ) k t h N N N h HZ t P t f u h a K a 1 1 1 1 1 1 1 1 1 1 1 0 0 ( ) ( ) ( ) ( ) ( ) , ( ) = − + −− − ++ − − λγ ρ µ λγ λγ
where .10 Here and in what follows, I will
assume that ρ > µ so as to ensure that demand for abatement capital is
decreasing with the price of capital; that is, given by equation (19) is
decreasing 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 without
any regulation, denoted by :
(21) Similarly, the capital stock chosen by a social planner at each instant,
denoted by kf 2(t ), is found by equating the social marginal return to capital
with the user cost of capital; that is,
(22)
where π2(·) is given by equation (16). Evaluating equation (22) at k
i(t) = kf 2(t),
and then rearranging yields
(23)
Substituting ki(t) = kf 2(t) into equation (14) yields the social planner’s choice
of output, denoted by qf 2(t):
10 One may rigorously show that all Cournot-Nash equilibria considered here and in what follows are unique. See, for example, Grenadier (2002), Haurie and Zaccour (1995), and Rosen (1965). H h w N h h h h ( ) ( ) ( ) ( ) = − − + − + + λγ λγ λγ λγ 2 1 1 1 1 1 1 kfu1( )t k tj kf t j N u ( ) = 1( ) ( , ... , =1 ) quf1( )t quf1t A k1 fu1 t h Z t h 2 1 2 1 ( ) = ( )( + ) ( )( + ), λγ λγ λγ λγ A N w h N N h h 1 1 1 1 1 1 1 ( ) = + − − −+ λγ λγ q t q t i f u ( ) ( )= 1 k ti kf t u ( ) = 1( ) xfu1( )t xfu t A kfu t Z t h h h h 1 1 1 1 2 1 2 1 ( ) = ( )( ) ( )( ). − + − + γ γ λγ γ λγ ∂ π ∂ ρ µ 2 ( ( ), ( )) ( ) ( ) ( ), k t Z t k t P t i i K = − k t h h HZ t P t f h a K a 2 1 1 1 1 1 1 1 1 1 1 1 0 0 ( ) (( ) ( ) ) ( ) ( ) ( ) ( ) . ( ) = − + − − − + + − − − λγ ρ µ λγ λγ
(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 xf 2(t ):
(25)
3.2 Irreversible investment
In the long-run, through Cournot-Nash competition, firm i will maximise the 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, which
is 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, π
1(·) is given by
equation (10), ρ is the discount rate, and Et{·} denotes the conditional
expec-tation taken at time t. The maximisation problem faced by firm i includes
N + 2 state variables: Z(t), PK(t), and an n-tuple of strategies, k (t). Firm i will
treat Z(t), PK(t) and k−i(t) = (k1(t), ... , ki−1(t), ki+1(t), ... , kN(t)) as exogenously
given, while controlling Ii(t). However, all elements of k−i(t) should be equal
to ki(t) in Cournot-Nash equilibrium.
As is well known in the published real options literature, the interaction of the stochastic evolution of Z(t) and investment irreversibility indicates that 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). If
the currently installed capital ki(t) is smaller than ki*(t), firm i should invest
so as to obtain ki(t) = ki*(t); otherwise no action should be taken. Firm i
should form expectations about the distant future regarding when and how much to invest because irreversible investment decisions will affect future
qf2 t A k2 f2 t hZ t h 2 1 2 1 ( ) = ( )( + ) ( )( + ), λγ λγ λγ λγ A2 1 w h 1Nh h 1 1 1 1 [( ) ( )= + − − ]( ) − + λγ λγ xf t A kf t Z t h h h h 2 2 2 1 2 1 2 1 ( ) = ( )( ) ( )( ). − + − + γ γ λγ γ λγ V I t Et e k Z P I d t t K i i 1 1 ( ) max⋅ = { ( )} ( )[ ( ( ), ( )) − ( ) ( )] , ∞ − −
ρ τ π τ τ τ τ τ I d k d i i ( ) τ ( )τ , τ =cash flows: firm i may be stuck with an excessive stock of capital in the initial period as pointed out by Clark et al. (1979), and Bertola and Caballero (1994). Given the abatement technology shown by equation (3), it is equivalent to
saying that firm i should regulate its pollution, xi(t) in equation (3), at an upper
barrier, denoted by . Consequently, is also the maximum amount
of waste emissions tolerable by firm i.
Define .
When ki(t) > ki*(t), the private marginal gain from increasing the abatement
capital 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 quadratic
equation given by equation (56), and φ(1) is obtained by setting τ = 1 in
equation (57). On the right-hand side of equation (28), the first term is the private value of the option to install an additional unit of capital, while the second term is the private value of the last incremental unit of installed
capital. Two optimal conditions must be satisfied at ki(t) = ki *(t) (Pindyck
1988). First, the private marginal gain from increasing the capital stock must 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 procedures
as 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*( )i t x*( )i t G k tj G G k t G hG N k j N i i 4 1 1 5 4 1 1 6 4 1 1 2 2 2 = − − = + = + = − − − −
∑
( )γ , λγ ( )γ ,and / λγ γ v F GZ t a P tK GZ t a 1 1 1 0 0 1 ( ) [⋅ = ( ) ]β ( )−β + ( ) / ( ),φ G B G a G G G G G G h G G G a h ( ) ( ) , ( ) ( ) = − − + + + − − + + − γ λγ λγ λγ λγ 2 1 1 1 2 1 1 1 2 3 4 1 2 6 1 1 1 3 5 1 1 1 1 ∂ ∂ v Z t 1( ) 0 ( ) . ⋅ = ksu1( )t ksu1( )tresult into equation (29). Third, use the value of φ(1) given by equation (57).
Finally, impose the equilibrium condition . The
result 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 into
equation (8) yields the maximum level of output produced by firm i, denoted by :
(32)
Substituting both and ki(t) = into equation (3) yields the
maximum amount of waste emissions tolerable by firm i, denoted by :
(33)
Similarly, suppose that V2( ki(t ), Z(t), Pk(t )) denotes the value function
for firm i perceived by the social planner, which is given by
(34)
where π2(·) is given by equation (16). The maximisation problem faced by the
social planner includes three state variables: Z(t), PK(t), and ki(t). The social
planner will treat Z(t) and PK(t) as exogenously given, while controlling Ii(t).
Suppose that ki *(t) and respectively denote the desired capital stock
and the maximum amount of waste emissions tolerable by the social planner.
When k (t) > ki *(t), the social marginal gain from increasing the abatement
capital v2(·) = ∂V2(·)/∂ki(t), is given by (see Appendix)
(35)
where F2 is a constant to be determined. Suppose that ki *(t) chosen by the
social planner is denoted by ks2(t). The value-matching and smooth-pasting
conditions applied to v2(·) are respectively given by
k tj ks t j N u ( ) = 1( ) ( , =1L, ) ks t m k t m u f u a 1 1 1 1 1 1 1 0 ( ) = ( ), where [ /( )]= α +α /(− ), kuf1( )t k ti kf t u ( ) = 1( ) qsu1( )t q tsu A ksu t h Z t h m quf t m h 1 2 1 2 1 2 1 1 1 1 2 1 2 1 1 2 2 1 ( ) = ( )( + ) ( )( + ) = ( ), [ /( )]= + − + + . λγ λγ λγ λγ γ γ λγ α α where q ti( ) ( )=qsu1t ksu1( )t xsu1( )t x ts A k t Z t m x t m u s u h h h h f u h h 1 1 1 3 1 3 1 1 1 1 2 1 2 1 1 2 2 1 1 ( ) = ( )( ) ( )( ) = ( ), [ / ]= + . − + − + − + + γ γ λγ γ λγ λ γ γ γ α where V I Et e k Z P I d t t K i i 2 2 ( ) max⋅ = { ( )} ( )[ ( ( ), ( )) − ( ) ( )] ∞ − − τ
ρ τ π τ τ τ τ τ xi*( )t v2( ) [ ( )⋅ = F k t2 i a0−1Z t( ) ]a0 βP tK( )1−β +a B k t0 2 i( )a0−1Z t( ) / ( ),a0 φ 1v2(·) = PK( t ), (36)
(37)
Evaluating both conditions (36) and (37) at ki(t ) = ks2(t ), and jointly solving
these two equations yields
ks2(t ) = m1kf 2(t ). (38)
Substituting ki(t) = ks 2(t) into equation (14) yields the maximum output
level produced by the social planner, denoted by qs2(t):
(39)
Substituting both qi(t) = qs2(t) and ki(t) = ks2(t) into equation (3) yields the
maximum amount of waste emissions tolerable by the social planner, denoted by 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 or not, a firm will invest at less, produce at less, and pollute at more than the socially efficient levels (Proof: see Appendix).
An individual firm will underestimate the adverse effect of pollution on production. Therefore, it will pollute more than is socially efficient as indi-cated by Proposition 1. Equivalently, the firm will invest less in abatement capital than is socially efficient. The firm will also produce less than is socially 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 result stated in Proposition 1 by using either an absolute or a proportional per-formance standard. While the regulator can set an absolute perper-formance standard at the socially efficient level, he cannot be sure whether the firm will respond by producing at the socially efficient level. By contrast, if the regulator implements a proportional performance standard, he can set it
∂ ∂ v Z t 2( ) 0 ( ) . ⋅ = qs t A ks t Z t m q t h h f 2 2 2 2 2 2 1 2 1 ( ) = ( )( + ) ( )( + ) = ( ). λγ λγ λγ λγ x ts A ks t Z t m x t h h h h f 2 1 2 3 2 1 2 1 2 1 ( ) = ( )( ) ( )( ) = ( ). − + − + γ γ λγ γ λγ
equal to the ratio of the socially efficient emission level over the socially efficient output level. This is precisely the efficient proportional perform-ance standard considered in what follows.
I will discuss both cases where absolute and proportional performance standards are imposed at their respective socially efficient levels. Suppose that capital investment is costlessly reversible, and the regulator imposes on
each firm an absolute performance standard equal to xf 2(t) in equation
(25). Replacing in equation (21) by xf 2(t) yields the corresponding
stock 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, and the regulator imposes on each firm an absolute performance standard equal
to xs 2(t) in equation (40). Replacing in equation (33) by xs 2(t) yields
the corresponding stock of capital, denoted by :
(43)
Replacing in equation (41) by yields the corresponding output
level, denoted by :
(44) Given the emission function equation (3), a firm that underestimates the externality from pollution will underestimate both the beneficial effect of abatement capital in reducing pollution, and the adverse effect of production in raising pollution. This yields the result stated in Proposition 2:
Proposition 2: No matter whether capital investment is costlessly reversible or not, if the regulator restricts pollution at the socially efficient level, then a firm will over-invest and over-produce (Proof: see Appendix).
xuf1( )t kf t a 1( ) kf t m k t a k f 1( ) = 2( ), m h N N h k h ( = + ) − ( ) + − > − − 1 1 1 1 1 1 1 1 1 1 2 λγ λγ γ γ kf t u 1( ) kfa1( )t qf t a 1( ) qf t m q t m m a q f q k 1 2 2 1 1 ( ) = ( ), where = .> γ γ xs t u 1( ) ksa1( )t ksa1( ) t = m k tk s2( ). ksu1( )t ksa1( )t qs t a 1( ) qas1( ) t =m qq s2( ).t
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 per
unit output (Montero 2002; Verdier 1995). Define , and
yf 2(t) = xf 2(t) /qf 2(t). Suppose that capital investment is costlessly reversible,
and that the regulator imposes on each firm a proportional performance
standard equal to yf 2(t). Equating yf 2(t) and yields the choice of capital
stock with regulation, denoted by , as given by
(45)
Replacing in equation (20) and equation (21) by yields the
corresponding output and emission levels, denoted by and ,
respectively:
(46)
(47)
Define , and ys 2(t) = xs 2(t)/qs 2(t). Suppose that capital
investment is irreversible and that the regulator imposes on each firm a
pro-portional performance standard equal to ys 2(t). Equating ys 2(t ) and
yields the desired capital stock with regulation, denoted by :
(48)
Replacing in equation (32) and equation (33) by yields the
corresponding 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 proportional performance standard at the socially efficient level. No matter whether investment is costlessly reversible or not, the firm will over-produce and over-pollute. The firm will also over-invest (under-invest) if emission is increasing with output at an increasing (a decreasing) rate (Proof: see Appendix).
yf t x t q t u f u f u 1( ) = 1( )/ 1( ) yuf1( )t kf t p 1( ) kfp t mk k t h h f 1 1 2 1 1 ( ) ( ). ( ) ( ) = − + γ λ γ kfu1( )t kfp1( )t qfp1( )t xfp1( )t qfp t mq q t h h f 1( ) 2( ), ( ) = λ+ xf t m x t p q h h f 1( ) 2( ). ( ) = λ+ ys t x t q t u s u s u 1( ) = 1( ) / 1( ) ys t u 1( ) ks t p 1( ) ksp t mk k t h h s 1 1 2 1 1 ( ) ( ). ( ) ( ) = − + γ λ γ ksu1( )t ksp1( )t qsp1( )t xsp1( )t qsp t mq q t h h s 1( ) 2( ), ( ) = λ+ xsp t mq x t h h s 1( ) 2( ). ( ) = λ+
Proposition 4: No matter whether investment is costlessly reversible or not, a firm will invest more, produce more, and pollute less if the regulator implements an efficient absolute performance standard as compared to the case where the regulator implements an efficient proportional performance standard (Proof: see
Appendix).
The reason behind Proposition 4 follows from Proposition 2. Proposition 2 indicates that a regulator who imposes on a firm an absolute performance standard at the socially efficient level will induce the firm to produce at more than the socially efficient level. The emission-output ratio will thus be lower than is socially efficient. This indicates that abatement capital is employed (and thus output is produced) at a level which is beyond the level that keeps the emission-output ratio at the socially efficient level. If, instead, the regulator sets a proportional performance standard at the socially efficient level, then the firm will invest and produce less, and thus pollute more as compared to the case where the regulator sets an efficient absolute performance standard.
Proposition 4, however, does not rank absolute and proportional per-formance standards. As indicated by Proposition 2, given environmental externalities a firm will not produce and pollute at the socially efficient level. These two inefficiencies should be corrected by implementing two instruments together, for example, effluent taxes and performance stand-ards. The use of either an absolute performance standard or a proportional performance standard will induce the firm to 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 regulated
performance standards. Some important results are stated in Proposition 5:
Proposition 5: (a) Suppose that investment is costlessly reversible and it is
more costly to purchase capital in the future (µ is increased). The regulator
should then implement a more stringent absolute performance standard, which induces a firm to invest and produce more. The regulator should then also implement a more stringent proportional performance standard, which induces a firm to invest and produce more, and to pollute less. (b) Suppose that investment
is irreversible and uncertainty becomes greater (σ is increased ). The regulator
should then implement a less stringent absolute performance standard, which induces a firm to invest and produce less. The regulator should then also imple-ment a less stringent proportional performance standard, which induces a firm to invest and produce less, and to pollute more. (c) Suppose that competition be-comes more intense (N is increased) whether investment is costlessly reversible or not. The regulator should then implement a more stringent absolute performance
standard provided that γ1> γ2, which induces a firm to have ambiguous attitudes
toward investment and output decisions. The regulator should then also implement
a more stringent proportional performance standard provided that γ1> 1 + γ2,
which induces a firm to have ambiguous attitudes toward investment, output, and polluting decisions (Proof: see Appendix).
The reason behind Proposition 5(a) is as follows. Suppose that it is more costly to purchase capital at a later date. As indicated by equation (70), when investment in abatement capital is costlessly reversible, a social planner will then employ more capital and therefore pollute less. Consequently, the regulator should set more stringent absolute and proportional performance standards. However, as investment becomes irreversible, an offsetting effect will arise (see equation (75)) and, therefore, it becomes ambiguous whether the regulator should implement a more stringent policy.
The results of Proposition 5(a) entail policy implications. Suppose that capital investment is almost costlessly reversible. Proposition 5(a) then indi-cates that the regulator should set a much more stringent standard (at least in terms of the adjustment period) when the expected cost of meeting a stand-ard rises. Consequently, older and newer firms should be treated differently with newer firms getting preferential treatment. However, in practice, older firms rather than newer firms receive preferential treatment. Such a policy, which may be justified if investment is irreversible, usually worsens pollution by encouraging 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 less capital 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) and equation (95), respectively. The firm will then be induced to invest and to produce less, as indicated by equation (81) and equation (88) for the former policy, and by equation (100) and equation (105) for the latter policy.
The policy implication of Proposition 5(b) is as follows. The so-called high-tech industries usually invest in an environment that is more volatile than 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 situation that competition becomes more severe. If the adverse effect of production in raising pollution outweighs the beneficial effect of abatement capital in
reducing pollution; that is, γ1> γ2, then a social planner will pollute less (see
equation (71) and equation (76) no matter whether investment is irreversible or not. 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, γ1> 1 + γ2, as indicated by equation (92).
The result of Proposition 5(c) indicates that whether the regulator should implement a performance standard policy that favours certain industries depends on the efficiency of abatement capital employed by firms in this industry. For example, suppose that firms in two industries employ the
same kind of abatement capital and suppose that γ1> γ2 (γ1 > 1 + γ2), then
the 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 through numerical examples. The benchmark parameter values are chosen as follows:
Z(t) = 1, λ = 0.1, σ = 20 per cent per year, γ1= 1, γ2= 0.5, ε = 3, N = 500, µ = 0
per 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 is completely irreversible. Given these benchmark parameter values, I find that the absolute performance standard that is socially efficient is given by
xs 2(t) = 0.1309, and the desired capital stock, and the output level induced
by it are respectively given by = 0.0503, and = 0.0294. The
proportional performance standard that is socially efficient is given by
ys 2(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 confirm
the results of Proposition 4 indicating that an absolute performance standard policy will induce a firm to invest and produce more, and pollute less as compared to a proportional performance standard policy. I also investigate
the responses in these seven endogenous variables to a change of σ around
the 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 their
benchmark values. These responses are respectively shown in figures 1–3.13
11 It is interesting to investigate whether the regulator should favor new abatement capital that is more efficient (higher γ2). However, our comparative-statics results regarding this issue yield ambiguous results (not shown in the Appendix).
12 As suggested by Dixit (1989), some capital costs arise from depreciation and are more thought 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 y
s 2(t) by a factor of 100, and scale up all
by a factor of 10. ks t a 1( ) qs t a 1( ) xs t p 1( ) ks t p 1( ) qs t p 1( ) ksa1( ),t ks t q t q t p s a s p 1( ), ( ), ( )1 and 1
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 proportional performance standard policies toward irreversible investments. In response, a firm will invest and produce more. Figure 2 shows that as uncertainty becomes greater, the regulator should be more lenient in both policies toward irreversible investments, a finding that is in line with Jou (2001) that Figure 1 The impact of a change in µ .
investigates the optimal effluent fee policy. In response, a firm will be less willing both to invest and produce. Finally, figure 3 indicates that as competi-tion becomes more intense, the regulator should impose a more stringent
absolute performance standard policy because γ1> γ2 (see Proposition 5(c)),
but should implement a less stringent proportional performance standard
policy because the constraint γ1> 1 + γ2 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 correct environmental externalities. Each firm in an industry emits waste in the production process, and, in turn, the average waste emissions of an industry adversely affect the firm’s productivity. The firm, which incurs sunk costs when employing capital to abate waste emissions, is uncertain about the efficiency of capital. The firm will underestimate environmental externalities, and therefore, it will pollute more than is socially efficient. To correct this tendency, the regulator can set a limit on either emissions or the emission-output ratio at the socially efficient level. The firm will invest more, produce more, and pollute less when the regulator implements the former limit than when the regulator implements the latter limit.
The present article has abstracted from several features that may be allowed in future research. First, it ignores both the adverse effect of pollu-tion on the welfare of consumers and spatial distribupollu-tion of firms. Second, it ignores the costs associated with implementing performance standards. It Figure 3 The impact of a change in N.
is worth investigating whether the conclusion of the present article will be affected 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)) − P
K(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 setting
the result equal to zero yields
(53)
Substituting given by equation (17) and in equation (52) into
equation (51), and using the condition in equation (53) yields the differential equation
(54)
Let ∂V1(·) /∂ ki(t) = v1(·). Differentiating equation (54) term by term with
respect to ki(t) yields ρV π k t Z t P t I t dtE dV I ti K i t 1 1 1 1 ( ) max⋅ = { ( )} [ ( ), ( )] ( ) ( ) − + ( ) .⋅ 1 1 dtE dVt ( )⋅ E dV dt Z t V Z t P t V P t I t V k t t K K i i 1 2 2 2 1 2 1 1 1 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ), ⋅ = σ ∂ ⋅ + ⋅ + ⋅ ∂ µ ∂ ∂ ∂ ∂ ∂ ∂ V k ti P t K 1( ) ( ) ( ). ⋅ = πi 1 ( )⋅ 1 1 dtE dVt ( )⋅ 1 2 0 2 2 2 1 2 1 1 1 1 2 1 3 1 1 1 0 σ ∂ ∂ ρ µ ∂ ∂ λγ λγ Z t V Z t V P t V P t B G G G Z t K 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 quadratic
equation given by equation (56). Following Dixit (1991), equation (56) can be 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) approaches
zero, only the positive root in equation (56) should be considered. The gen-eral solution of equation (55), which is composed of solutions from both the homogeneous and non-homogeneous parts, is shown by equation (28). Similarly, the Bellman-Hamiton-Jacobi equation for the problem described in equation (34) can be written as
(59) 1 2 0 2 2 2 1 2 1 1 0 σ ∂ ∂ ρ µ ∂ ∂ Z t v Z t v P t v P t G Z t K K a ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , ⋅ − ⋅ + ⋅ + = Figure 4 φ (τ ) versus τ . [G Z t( ) ]a0 τP tK( )1−τ φ τ( ) = −1σ τ τ( ) ( ) .− − µ −τ + =ρ 2 1 1 0 2 0 0 a a φ τ σ α τ β τ ρ µ α τ β τ α β ( ) = −1 ( ) ( ) + − = ( )( )( )− + − , 2 0 2 2 a v1( ) ⋅ =G Z t( ) / ( ).a0φ 1 ρV π k t Z t P t I t dtE dV I ti i K i t 2 2 2 1 ( ) max⋅ = { ( )} [ ( ), ( )] ( ) ( ) − + ( ) ,⋅
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 π2(·) given by equation (16) and in equation (60) into
equation (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 kf 2(t) in equation (23) yields
(64)
since if N = 1 then , and differentiating with respect
to N yields a negative sign.
Dividing in equation (21) by xf 2(t) in equation (25) yields
(65) E dV dt Z t V Z t P t V P t I t V k t t K K i i 2 2 2 2 2 2 2 2 1 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ). ⋅ = σ ∂ ⋅ + ⋅ + ⋅ ∂ µ ∂ ∂ ∂ ∂ ∂ ∂ V k ti P tK 2( ) ( ) ( ). ⋅ = E dVt[ 2( )/⋅ dt] 1 2 0 2 2 2 2 2 2 2 2 0 0 σ ∂ ∂ ρ µ ∂ ∂ Z t V Z t V P t V P t B k t Z t K K i a a ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) . ⋅ − ⋅ + ⋅ + = 1 2 0 2 2 2 2 2 2 2 0 2 1 0 0 σ ∂ ∂ ρ µ ∂ ∂ Z t v Z t v P t v P t a B k t Z t K K i a a ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) . ⋅ − ⋅ + ⋅ + − = kf t u 1( ) k t k t N h h h N N h f u f a h a 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 ( ) / ( ) ( ) ( ) , ( )( ) = − − + − + − < − − − − + + − λγ λγ λγ λγ kfu1( )/t kf2( ) t =1 kf t k t u f 1( )/ 2( ) xf t u 1( ) x t x t h N N h k t k t f u f h f u f h h 1 2 1 1 1 1 1 2 1 1 1 1 1 1 1 2 1 ( ) / ( ) ( ) ( ) ( ) ( ) . ( ) ( ) = + − + − > − − + − + λγ λγ γ λγ λγγ
