Regulating Pollution with Endogenous Monitoring - Agricultural and ...

Journal of Environmental Economics and Management 44, 221᎐241 Ž2002. doi:10.1006rjeem.2001.1208

Regulating Pollution with Endogenous Monitoring1 Katrin Millock Centre International de Recherche sur l’En¨ ironnement et le De´¨ eloppement (CIRED), 45 bis, A¨ enue de la Belle GabrielleᎏJardin Tropical, 94736 Nogent sur Marne Cedex, France E-mail: [email protected]

and David Sunding and David Zilberman Department of Agricultural and Resource Economics, 207 Giannini Hall, Uni¨ ersity of California at Berkeley, Berkeley, California 94720-3310 E-mail: [email protected], [email protected] Received June 25, 2000; revised June 12, 2001; published online January 23, 2002 The paper offers a new perspective on nonpoint source pollution by explicitly considering the cost of monitoring individual emissions. The distinction between point and nonpoint source pollution is shown to depend on the cost of monitoring, the environmental cost of pollution, and the impact of monitoring on profits. A regulatory scheme of differential taxation is proposed, wherein taxes are predicated on whether the agent has installed an emissions monitoring device. The optimal degree of monitoring as well as conditions for optimal regulation in the extreme cases of no monitoring and full monitoring, is identified. 䊚 2002 Elsevier Science ŽUSA .

1. INTRODUCTION The categorization of some pollution sources as ‘‘point’’ sources and others as ‘‘nonpoint’’ sources reflects the capability of existing technology for emissions monitoring and other factors such as the cost of the technology. The distinction between point and nonpoint pollution is ultimately an economic decision rather than a purely technical one, and the classification of an individual source of pollution may change over time as monitoring technology advances and the costs of monitoring decline. In this paper, we consider a situation in which a new, costly monitoring technology is introduced. Regulators who previously could only observe whether firms operate are now able to know with certainty the pollution levels of individual firms by using the technology. Policymakers must then decide whether to require 1

This research was funded by Cooperative Agreements with the U.S. Environmental Protection Agency and the California Department of Food and Agriculture. Katrin Millock also thanks the Royal Swedish Academy of Sciences and Tore Browaldhs Stiftelse for financial support. The authors also thank J.-J. Laffont, S. Scotchmer, and F. Salanie ´ for helpful comments and advice. Any errors and omissions are ours. We also acknowledge participants in the 3rd Ulvon ¨ Conference of Environmental Economics, Sweden, the 8th Meeting of the EAERE in Tilburg, and the 12th Annual Congress of the EEA in Toulouse where precursors to this paper were presented. 221 0095-0696r02 $35.00 䊚 2002 Elsevier Science ŽUSA . All rights reserved.

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MILLOCK, SUNDING, AND ZILBERMAN

adoption of the technology by all firms in the industry, whether to provide incentives for voluntary adoption of the monitoring technology Žrecognizing that in some cases not all firms will adopt the technology., or whether to ignore the new technology altogether and continue regulating as before. Thus, this paper addresses the different types of pollution regulations that may occur at various points along the spectrum from extreme nonpoint source pollution to the case where a regulatory agency can observe individual pollution. In particular, we consider pollution regulation in three basic situations: 1. Monitoring technology is unavailable and regulators only know whether firms operate Ži.e., they do not know firms’ input or output levels.. In that case, we will argue that the only means to regulate pollution is through a uniform license fee on each enterprise that will result in the withdrawal of the least profitable units. 2. Monitoring technology is mandatory. A special case of this situation is the full information, first-best solution. 3. Monitoring technology is voluntary. We will investigate this type of solution and argue that, given information, availability, and the cost of monitoring technology, it is possible to identify conditions for optimal outcomes associated with voluntary adoption of monitoring technology Žor endogenizing the distinction between point and nonpoint sources.. Previous analyses use two approaches to regulate nonpoint source pollution. The first approach is to design regulations like input or output taxes that are based on observable data. For example, Larson et al. w15x analyze how closely uniform water and nitrogen input taxes approximate the optimal pollution reduction. In another paper, Green and Sunding w6x use a putty-clay framework to compare the long-run efficiency of firm-specific pollution taxes with uniform input and fixed factor taxes to control seawater intrusion. The second approach is to consider incentive schemes from the mechanism design literature applied to the moral hazard problem of nonobservable emissions. These schemes use incentive-compatible mechanisms that rely either upon observation of data on individual firm input or output ŽLaffont w14x, Smith and Tsur w21x. or upon observations of the ambient quality resulting from aggregate pollution ŽSegerson w20x, Cabe and Herriges w2x, Xepapadeas w25, 26x, Herriges et al. w10x, Hansen w7x..2 They all share the common assumption that monitoring of individual emissions is impossible, or rather, its cost infinite. Xepapadeas w28x provides to our knowledge the first model attempting to endogenize monitoring.3 When ambient pollution concentration is stochastic, Xepapadeas w28x shows that a risk-averse agent prefers to have an effluent charge imposed on his observed individual emissions in exchange for a reduction in the level of the ambient charge. The current paper abandons the assumption that monitoring is exogenous and of infinite cost, and instead incorporates monitoring costs in the analysis. To some extent, this assumption is similar to the assumption of self-reporting in Innes w12x. We show that the distinction between point source and nonpoint source pollution is based on the cost of monitoring technology. This implies that as technology 2 3

Lewis w16x provides a useful survey of environmental regulation under asymmetric information. See also Xepapadeas w27x.

REGULATING POLLUTION WITH ENDOGENOUS MONITORING

223

improves, many nonpoint problems may become point source problems. We will show how a simple scheme can induce agents to invest in monitoring, and implement a second-best allocation of individual emissions under asymmetric information. The original incentive scheme that we propose enables partial adoption of monitoring rather than full monitoring of all individual pollution sources or no monitoring at all. The central tradeoff that we analyze is accuracy of information, permitting targeted regulation, versus the cost of obtaining information. By investing in a costly monitoring system, the regulator will obtain information on individual pollution levels. The monitoring system could be some physical equipment or it could be a system of random audits ŽLinder and McBride w17x, Harford w8, 9x, Malik w18x..4 We do not model the details of the monitoring system in order to focus on the regulator’s problem of when to improve information through the investment in costly monitoring. The paper is structured as follows. In Section 2, we outline the basic elements of the model, including our specification of aggregate pollution and producer heterogeneity. In Section 3, we consider optimal regulation in the case where there is no monitoring technology and the regulator must rely on a system of licensing fees. Section 4 outlines the case of mandatory monitoring in which all producers can be regulated as point sources. In this case, a Pigouvian tax is the full information optimum. Next, in Section 5, we discuss the more general case of voluntary monitoring in which some operating units may be point sources and other units are nonpoint sources. In this environment, we consider the merits of a policy that provides incentives for low-polluting firms to reveal themselves to the regulator by adopting monitoring technology. Section 6 presents a welfare analysis of these three cases, and concluding comments are contained in Section 7. Unless otherwise noted in the text, proofs of all major propositions are given in the Appendix.

2. THE BASIC MODEL: PRODUCTION, POLLUTION AND HETEROGENEITY We model a continuum of agents that differ with regard to a scalar parameter ␪ . This parameter may be interpreted as an index of efficiency in input use, for instance, the soil retention capacity of farmland, or the energy efficiency of a car. The important aspect of the model is that the regulator does not know the values of ␪ for individual agents but does know the distribution of ␪ . ␪ is defined on a support w ␪ , ␪ x with a known, continuous, strictly positive density function f Ž ␪ . and a distribution function F Ž ␪ .. We use a general specification of the production function, y s g Ž x, ␪ ., which we assume to be twice differentiable and concave in input use, x. The specification of g Ž x, ␪ . plays an important role in the interpretation of ␪ and the results obtained later on, so we will explain the status quo situation in some detail. Production is

4 See Stranlund and Dhanda w22x for an analysis of the optimal inspection probability in a tradeable permit system. Florens and Foucher w4x compare the welfare gains from two different monitoring technologies for probabilistic enforcement policies to counteract illegal oil dumping.

224


assumed to increase with ␪ ,

⭸g

) 0, but there are two alternative plausible assump-

⭸␪

⭸ g 2

tions about the behavior of ⭸ x ⭸␪ . Under one scenario, it is assumed that for every ␪ , there exists an x s ˜ x Ž ␪ . such that

⭸ 2g ⭸ x ⭸␪ ⭸ 2g ⭸ x ⭸␪

)0

for

x-˜ x ᭙␪

-0

for

x)˜ x ᭙␪ .

One interpretation of this scenario is that there is an upper bound on output and that units with higher ␪ reach close to this with fewer units of input x, and then ⭸g the marginal productivity of the input MPx s ⭸ x becomes small. Thus, for relatively low input levels Ž x F ˜ x ., as in Fig. 1a, ␪ 2 has a higher marginal Ž . productivity of input MPx , but the marginal productivity of input of ␪ 1 is higher for relatively larger input levels.

ž

∂g ∂x

/

∂g(x(θ 2 ),θ 2 ) ∂x(θ 2 )

∂g(x(θ 1 ),θ 1) ∂x(θ1 )

x˜ (θ )

x (θ ) (a)

∂g(x(θ 2 ),θ 2 ) ∂x(θ 2 )

∂g ∂x

∂g(x(θ1 ),θ 1 ) ∂x (θ1 )

x (θ ) (b) x Ž ␪ .. Then agents FIG. 1. Ža. The analysis assumes input use x Ž ␪ . is in the region where x Ž ␪ . ) ˜ with high ␪ use input more efficiently. Žb. In this case, agents with high ␪ are always more productive. The figures assume only two values of ␪ , ␪ 2 ) ␪ 1.

225


A specification that corresponds to Fig. 1a is y s g 1Ž x ␪ ., with g ⬘ ) 0.5 In this case, output is a function of efficient input units and ␪ is an indicator of input use efficiency. Higher ␪ indicates higher utilization of applied input. In this case, ␪ can be an indicator of vintage where newer machines utilize energy better, or soil quality, where higher quality soils utilize chemicals better. ⭸ 2g

Figure 1b corresponds to the alternative assumption that ⭸ x ⭸␪ ) 0 ᭙␪ , x, and the marginal productivity of ␪ 2 always exceeds that of ␪ 1. A plausible specification under this assumption is y s ␪ g 2 Ž x . when ␪ is a multiplicative productivity parameter ŽFig. 1b.. Suppose input is used by price-taking, profit-maximizing firms, when output price is p y and input price is p x . The maximum profits of a unit of quality ␪ without regulation are

␲ U0 Ž ␪ . s p y g Ž xU0 Ž ␪ . , ␪ . y p x xU0 Ž ␪ . , where optimal input use xU0 Ž ␪ . is determined solving the first-order condition ⭸g p y Ž xU0 , ␪ . s p x . This optimal input use will vary with ␪ according to the sign of ⭸x

U

⭸ 2g

dx ⭸ g , e.g., when d ␪0 ) 0 Ž- 0. when ⭸ x ⭸␪ ) 0 Ž- 0.. Figure 1 can be used to help explain the relationship between an agent’s type and optimal input use. In Fig. 1a, when the price of input relative to output is low, then the MPx of type ␪ 2 is below that of ␪ 1. In this case, higher ␪ implies lower input use, or Ž xU0 L Ž ␪ 2 . - xU0 L Ž ␪ 1 ... When the price of input relative to output is high, agents with higher ␪ will use more input, or Ž xU0 H Ž ␪ 2 . ) xU0 H Ž ␪ 1 ... It is assumed that pollution z depends on input use x, and the heterogeneity parameter ␪ : z s z Ž x, ␪ .. It is plausible that pollution increases with input use ⭸z ) 0 . On the other hand, there are plausible arguments for both a positive and ⭸x negative relation between the heterogeneity parameter and z. We use the same quality parameter ␪ in both the production and pollution function. This specification simplifies the analysis, but it also implies two different interpretations: the output externality case and the residue externality case. Sometimes pollution is ⭸z seen as an inevitable effect of production: z s hŽ ␪ x ., with h⬘ ) 0. Hence ) 0. 2

⭸ x ⭸␪

ž

/

⭸␪

We will label this situation the output externality case. Khanna and Zilberman w13x documented several production processes where waste of input results in pollution: energy use, pesticide application, etc. In such cases, for example, wasted pesticide or fertilizer will end up as run-off and pollute water bodies, or fuel residues will evaporate to pollute the air. If pollution is equal ⭸z to residue, z s Ž1 y ␪ . x, - 0. We will refer to this case as the residue externality ⭸␪

case. Assumptions on both the pollution function and the production function also ⭸z

⭸ z dx

determine how pollution varies with ␪ : d␪ s ⭸␪ q ⭸ x d ␪ . Defining zU0 as the pollution level arising from input use xU0 , the sufficient conditions for a monotonic dz

5

g ⬘ indicates partial derivative.

226


relationship between zU0 and ␪ are the following:

If

⭸z ⭸␪ )

0

and

⭸ 2g ⭸ x ⭸␪ )

0,

then

dzU0 d␪ )

0.

Ž 1.

Both sets of conditions are plausible cases. For example, the sufficient conditions for negative correlation between ␪ and zU0 hold in the case when ␪ is an indicator p of input use efficiency Ž y s g 1Ž ␪ x .., p x is small so xU0 ) ˜ x, as in Fig. 1a, and when y

pollution is unutilized residue Ž z s Ž1 y ␪ . x .. One example occurs when pollution results from unconsumed liquid or gas and production units vary in their consumption efficiency. The necessary conditions for positive correlation between ␪ and zU0 are verified, for example, when ␪ is a multiplicative productivity shifter y s ␪ g 2 Ž x . and the output externality case holds Ž z s hŽ ␪ x ... d␲ U

⭸g

The marginal impact of ␪ on ␲ U0 is d␪0 s p y ⭸␪ ) 0. Figures 2a and 2b summarize the situation in status quo before any nonpoint pollution policy is implemented. In Case 1 ŽFig. 2a., profits and pollution are negatively correlated. In Case 2, corresponding to Fig. 2b, profits and pollution are positively correlated.

π (θ ),z (θ )

π (θ )

z(θ )

θ (a)

π (θ ),z (θ )

π (θ ) z(θ )

θ (b) FIG. 2. Ža. Negative correlation between profits and pollution. Žb. Positive correlation between profits and pollution.

227


Suppose that the social cost of pollution is a convex function, C Ž Z ., Ž C⬘ ) 0, C⬙ ) 0. of aggregate pollution where aggregate pollution Z is Zs

H␪␪

z Ž x Ž ␪ . , ␪ . f Ž ␪ . d␪ .

Ž 2.

Suppose that the regulator has limited knowledge about the industry. She knows the production function g Ž x, ␪ ., the pollution function z Ž x, ␪ ., and the distribution of ␪ . She cannot observe the heterogeneity parameter or input or output levels of individual firms, but can observe whether an individual firm operates. The objective of the regulator is to maximize gross benefits of output less costs of production and environmental damage costs C Ž Z .. If monitoring technology exists, we assume that its cost is ␯ dollars per firm. Below we will evaluate the outcome under three scenarios: Ž1. no monitoring, Ž2. mandatory monitoring, and Ž3. voluntary monitoring. 3. POLLUTION CONTROL WITHOUT MONITORING TECHNOLOGY We start by considering the case where no emissions monitoring technology is feasible and the regulator cannot observe x or y, only whether a production unit is in operation. This is the most extreme case of nonpoint source pollution. The regulator knows only the functions g Ž⭈, ⭈ ., z Ž⭈, ⭈ ., and C Ž Z . and the distribution of ␪ . With this information, the regulator can only control pollution by using a per unit tax, denoted by T0 , on operating units. This may be a per acre land tax in the case of agricultural pollution, or a license for dry cleaners or motor vehicles. The maximum profits of units of quality ␪ when x, ␪ , y, and z are not monitored is ␲ U0 Ž ␪ . y T, and the optimal input is xU0 Ž ␪ . when the operation is d␲ U ⭸g profitable. Since d␪0 s p y ⭸␪ ) 0, only units with ␪ G ␪m0 ŽT . will operate under the operation fee, where the marginal quality ␪m0 ŽT . is determined solving

␲ U0 Ž ␪m0 Ž T . . s T .

Ž 3.

The fixed license fee does not affect input use, but instead operates on the extensive margin, raising the lower unit for which operation is profitable. Formally, the regulator’s problem is L0 Ž T . s max T

H␪␪ ŽT .␲ 0 m

U 0

Ž xU0 Ž ␪ . , ␪ . f Ž ␪ . d␪ y C

␪

žH

␪m0 Ž T .

market surplus

z Ž xU0 Ž ␪ . , ␪ . f Ž ␪ . d ␪ .

/

cost of pollution

Ž 4.

PROPOSITION 1. The optimal operating unit fee T * is determined sol¨ ing C⬘ Ž Z0 Ž T * . . zU0 Ž ␪m0 Ž T . . s ␲ U0 Ž ␪m0 Ž T * . . s T *, cost of pollution of marginal unit

surplus generated by marginal unit

tax

Ž 5.

228


π 0 , C ′(Z ) z0 *

*

(

)

C ′ Z 0 (θ m0) z0 (θ m0 ) *

A ∗

∗ 0 T = π 0 (θ m )

θ

θ

FIG. 3. The determination of the optimal per firm license fee T *.

where Z0 Ž T . s

H␪␪ ŽT .z 0 m

U 0

Ž x 0 Ž ␪ . , ␪ . f Ž ␪ . d␪ .

Figure 3 depicts the determination and existence of the optimal license fee ŽT *.. The marginal quality Ž ␪m0 . that may assume values between ␪ to ␪ is on the horizontal axis; market surplus Ž␲ U0 . and the cost of pollution of the marginal unit Ž C*Ž Z0 . zU0 . are depicted in the function of the marginal unit Ž ␪m0 .. ␲ U0 is increasing with ␪m0 . When ␪m0 s ␪ , aggregate pollution Ž Z0 . is at its maximum level, zU0 ) 0, and since C⬙ ) 0, C⬘Ž Z0U . is at its highest level so that C⬘Ž Z0U . ⭈ zU0 is positive. Assuming that both ␲ U0 and C⬘Ž Z0U . ⭈ zU0 are a continuous function of ␪m0 , they intersect, and the optimal fee on operation unit T * is equal to the value of the two functions at the intersection point A.6 The optimality condition Ž5. suggests that gross profit per unit of pollution Ži.e., ␲U profit not including pollution costs. zU0 at the marginal quality Ž ␪m0 . is equal to 0 the marginal pollution cost Ž C⬘Ž Z0 ... The optimal tax on each operating unit ŽT *. is equal to the marginal pollution cost times the pollution of the marginal unit. The operating unit fee is the optimal policy, given the assumed informational constraints. Define welfare under complete information as L1Ž ␯ s 0.. The extent of loss in social welfare because of the informational constraint is then L1Ž ␯ s 0. y L0 . This difference depends on the production and pollution functions, and in particular, on the correlation between profits and pollution, and the rigidity of the production and pollution functions. Consider the case when profits increase but pollution decreases with ␪ ŽFig. 2a.. In this case, the units closed down due to the license fee are the units with the highest pollution but also are the lowest contribution to output. Compare this with the case where profits and pollution levels are positively correlated. In that case, the instrument leaves the most polluting units unaffected. The license fee thus works best when the least profitable firms also have the highest pollution.

ž /

. U In the unlikely event of more than one intersection point, optimal T is at a point where C⬘Ž ZU 0 z0 U intersects ␲ 0 from above and market surplus minus cost of pollution is maximized. 6


229

There will not be any difference between the full information and constrained information optima if both production and pollution functions are fixed proportions. In these cases, control of pollution can only be done through changes in the extensive margin and they are achieved by the fixed operating fee, T *. The inability to use pollution taxes is more costly the more responsive the pollutionroutput ratio of firms is to changes in tax rates ŽVerhoef and Nijkamp w24x.. 4. MANDATORY MONITORING UNDER HETEROGENEITY Historically, the policy used in practice has been to impose monitoring of all sources as pollution problems grow more severe.7 The nonpoint problem is then effectively changed into point source pollution. This extreme policy can take two forms: the regulator buying monitoring equipment for installation at all firms, or ordering all firms to invest in monitoring. If monitoring is very costly, mandatory monitoring will force some firms to close down. As a simplifying assumption, we assume the monitoring technology enables the regulator to perfectly assess the firm’s pollution level. This simplifying assumption does, however, capture the fundamental tradeoff between investing in costly monitoring and obtaining better information about individual pollution levels. We will reassess the implications of this assumption and suggest some extensions in the conclusions of the paper. With mandatory monitoring, the welfare maximization problem is then ␪ ␺ Ž␪ . H ␺ Ž␪ . x Ž␪ . ␪

L1 Ž ␯ . s max yC

p y g Ž x Ž ␪ . , ␪ . y px x Ž ␪ . y ␯ f Ž ␪ . d␪

H␪␪␺ Ž ␪ . z Ž x Ž ␪ . , ␪ . f Ž ␪ . d␪

,

subject to

␺ Ž ␪ . g 0, 1 4 x Ž ␪ . G 0, Ž . where ␺ ␪ is an indicator designating whether a unit ␪ operates. Using Lagrange multiplier techniques, the necessary conditions holding at the optimal solution of x*Ž ␪ . and ␺ *Ž ␪ . Žand the resulting y*Ž ␪ ., z*Ž ␪ .. at Z* are

⭸ g Ž xU1 Ž ␪ . , ␪ . ⭸x

p y s p x q C⬘ Ž Z* .

⭸ z Ž xU1 Ž ␪ . , ␪ . ⭸x

᭙␪ ,

␺ *Ž ␪ . s 1

if

p y yU Ž ␪ . y p x xU1 Ž ␪ . y C⬘ Ž Z* . z Ž xU1 Ž ␪ . , ␪ . y ␯ ) 0

␺ *Ž ␪ . s 0

if

p y y* Ž ␪ . y p x xU1 Ž ␪ . y C⬘ Ž Z* . z Ž xU1 Ž ␪ . , ␪ . y ␯ - 0

Ž 6. ᭙␪ .

Ž 7. The marginal unit of operation, ␪m , is defined by p y g Ž xU1 Ž ␪m . , ␪m . y p x xU1 Ž ␪m . y ␯ y tz Ž xU1 Ž ␪m . , ␪m . s 0. 7

Ž 8.

One example is sewerage, where compulsory connection to municipal systems involves both monitoring and joint abatement.

230


Condition Ž6. states that the value of marginal product at the optimal input use level is equal to the input price and its marginal environmental cost.8 Condition Ž7. states that firms whose revenue cannot cover their monitoring cost, input cost Ž p x x*., and externality cost Ž C⬘Ž Z . z Ž x Ž ␪ ., ␪ .. should not operate in an optimal resource allocation. These conditions can be attained by the following policy: PROPOSITION 2. Ža. With mandatory monitoring, the optimal policy is a Pigo¨ ian tax, t s C⬘Ž Z ., and a requirement that the polluter pays the monitoring costs. Žb. The more costly the monitoring technology, the lower will be the Pigo¨ ian tax, that is,

dt d␯

- 0.

The full information optimum is a special case corresponding to ␯ s 0, and it corresponds to the traditional use of a Pigovian tax. In this sense, Proposition 2 constitutes a benchmark that is well established in the literature. However, the novelty here is the introduction of costly monitoring of individual emissions. Proposition 2 first establishes the criterion: under mandatory monitoring, who actually pays for monitoring is important. Subsidizing agents to install monitoring equipment would lead to too much monitoring compared to what would be optimal. Because of free exit and entry, having the government subsidize the monitoring cost would lower the marginal unit, ␪m , below the optimally defined ␪mU in Eq. Ž8.. Second, since the cost of monitoring decreases production surplus, production is at a lower level with high monitoring costs and consequently, pollution is at a lower level, and the Pigouvian tax rate should be lower ŽProposition 2b.. The introduction of mandatory monitoring and a pollution tax will reduce production and pollution, relative to no intervention, through its impact on the intensive margin Žhigher cost of input will reduce its use. and the extensive margin Žhigher cost will lead to the exit of some firms.. Mandatory monitoring and taxation may lead to exit as profits decline because of the cost of monitoring equipment, taxation, and change in input. For units in production, profits change with ␪ according to d␲ 1U d␪

s py

⭸g ⭸␪

yt

⭸z ⭸␪

.

Given the assumption of output increasing in type ⭸z

Ž 9.

ž

⭸g ⭸␪

) 0 , when pollution

/

decreases in type ⭸␪ - 0 , profits increase with ␪ and exiting agents are at the lower end of ␪ , with ␪ F ␪m . This pattern will also occur when the firm’s pollution

ž

is increasing in type,

/

⭸z ⭸␪

) 0, as long as the tax rate is sufficiently small.9

8 The second-order condition holds because of the assumptions of a concave production function and a pollution production function that is convex in input use. 9

When the firm’s pollution is increasing in type

that producers with larger ␪ will exit. When ᭙␪ ,

⭸z ⭸␪

⭸z ⭸␪

) 0 , it is possible, even though not very likely,

/

) 0 and substantial, and

d␲ U 1

t py

is sufficiently high,

- 0 / , and the marginal ␪ Ž ␪ . is an upper bound on profitable firms and ž F ␪ F ␪.

profits decrease in type exit will occur for ␪m

ž

d␪

m


231

The assumptions on the production and the pollution functions that were detailed in Section 2 thus determine the characteristics of agents that exit following imposition of a Pigovian charge. If there is a negative correlation between production and pollution ŽFig. 2a., low-quality agents exit, while remaining firms have high productivity and low pollution. On the other hand, if production and emissions are positively correlated, exiting units may have both high pollution and high output levels, and remaining units have both lower output and pollution levels ŽFig. 2b.. This holds only in the rare cases specified above. 5. VOLUNTARY MONITORING After analyzing the two extreme regulationsᎏtaxation of operating units with no monitoring of pollution, and taxation of pollution with mandatory monitoringᎏwe analyze an intermediate policy of voluntary monitoring. Under our assumptions, the only information available to policymakers about specific nonmonitored units is whether they operate, but the policymaker knows the actual total pollution of monitored units. The policy we present provides incentives to low-polluting firms to reveal themselves by adopting monitoring.10 Later, we will show the relation between our results and a policy of self-reporting as analyzed in Innes w12x. Since the policymaker cannot distinguish nonmonitored units, they will all pay Žreceive. the same fixed fee ŽT0 .. The incentives for adopters of monitoring include an emission tax t per unit of pollution, and a fixed fee ŽT1 . per operating unit, which is a tax when T1 ) 0, and a subsidy when T1 - 0. We selected this scheme because of its ease of implementation. The optimal linear incentive will be selected by maximizing aggregate revenue minus production, pollution, and monitoring cost subject to incentive compatibility constraints. We start by characterizing the situation under voluntary monitoring. Then we specify when voluntary monitoring is optimal compared to mandatory and no monitoring policies ŽSection 6.. To derive incentive compatibility constraints that take into account individual rationality and profit maximization, let xU1 Ž ␪ . and ␲ ˜ 1U Ž ␪ . define optimal variable input use and maximum profit for units with a given ␪ if they adopt monitoring. At ⭸g ⭸z xU1 Ž ␪ ., p y ⭸ x y p x y t ⭸ x s 0, and ␲ ˜ 1U Ž ␪ . s p y g Ž xU1 Ž ␪ ., ␪ . y px xU1 y tz Ž xU1 Ž ␪ ., ␪ . y ␯ y T1. Similarly, xU0 and ␲ ˜ U0 Ž ␪ . define optimal variable input and maximum ⭸g profits if units with ␪ operate but are not monitored. At xU0 Ž ␪ ., p y ⭸ x y p x s 0, U U U and ␲ ˜ 0 Ž ␪ . s p y g Ž x 0 Ž ␪ ., ␪ . y p x x 0 y T0 . Under the voluntary monitoring policy, the spectrum of agent types is separated into three sets: Ž1. ⌰N , which includes all of the units that do not operate; Ž2. ⌰ 0 , which includes all of the units that operate and do not adopt monitoring; and Ž3. ⌰ 1 for all of the units that operate and adopt monitoring:

0)␲ ˜ 1U Ž ␪ . , ␲˜ U0 Ž ␪ .

᭙␪ g ⌰N

␲ U0 ␲ 1U

᭙␪ g ⌰ 0

␲ 1U ␲ U0

␲ U0 ␲ 1U

˜ Ž␪ . G ˜ Ž␪ ., ˜ Ž␪ . G 0 ˜ Ž␪ . G ˜ Ž␪ ., ˜ Ž␪ . G 0

᭙␪ g ⌰ 1 .

10 Truth-revealing mechanisms as in Laffont w14x and Smith and Tsur w21x cannot be used in our case since the regulator cannot observe input or output levels and implement individual input or output taxes.

232


For tractability and simplicity, we will identify and analyze plausible outcomes where the sets are connected, and each is a segment of ␪ ’s. PROPOSITION 3. Ža. Ž1. Ž2. Ž3.

If

⭸ 2g ⭸ x ⭸␪

- 0 for xU1 Ž ␪ . with positi¨ e production, and

⭸z ⭸␪

- 0:

an optimal linear incenti¨ e that leads to ¨ oluntary monitoring will result in ⌰N with ␪ - ␪m0 ; ⌰ 0 with ␪m0 F ␪ F ␪c ; ⌰1 with ␪c F ␪ F 1,

where p y yU0 y p x xU0 s T0 at ␪m0 , and p y yU1 y p x xU1 y tz Ž xU1 , ␪c . y ␯ y T1 s p y yU0 y p x xU0 y T0 at ␪c . Žb. The parameter of the optimal linear incenti¨ e scheme is T0 s C⬘ Ž Z* . zU0 Ž ␪m0 . t s C⬘ Ž Z* . T1 s C⬘ Ž Z* . zU0 Ž ␪m0 . y zU0 Ž ␪c . . The optimal linear policy for the case of negative correlation between ␪ and pollution can be interpreted as a fixed tax, T0 s C⬘Ž Z*. zU0 Ž ␪m0 ., on all operating units and a subsidy of C⬘Ž Z . zU0 Ž ␪c . on units that adopt monitoring. The monitored unit also pays a variable pollution tax, C⬘Ž Z*. zU1 Ž ␪ .. The fixed tax, T0 , is equal to the tax in the case with no monitoring. Monitored units are taxed according to the pollution they generate, but subsidized for part of the overestimate of pollution before they installed monitoring. This tax formula assures that at ␪c , the per unit bound quasi-rents of the two monitoring options are the same Ž p y Ž yU0 y yU1 . y p x Ž xU0 y xU1 . y C⬘Ž Z 0 .Ž zU0 y zU1 . q ␯ s 0.. Beyond this point the quasi-rent of monitored units is higher and monitoring is optimal. The proposed policy of voluntary monitoring is related to Innes’ analysis w12x of self-reporting. In fact, a polluter that invests in monitoring equipment is equivalent to a polluter who voluntarily reports his emissions to the regulator rather than paying a fixed penalty. The alternative choice of making a fixed tax payment, which was shown to be dominated by the installation of monitoring equipment for a subsample of the polluters Žunder certain restrictions on the level of monitoring costs., is equivalent to a random inspection policy in which the monitoring probability is invariant with the amount of pollution of the individual source, and under which the fine is constant regardless of the exceedance. These are quite strict conditions, however, and it is likely that the fine will vary with the level of pollution exceeding the standard. Figure 4a depicts the relationship between ␲ 1U and ␲ U0 that results in an optimal solution with voluntary monitoring. While private quasi-rent ␲ U0 is positive and increases with ␪ , social quasi-rent without monitoring Ž␲ U0 y C⬘Ž Z*. zU0 . is negative for small ␪ , and equal to 0 at ␪m0 . The social quasi-rent with monitoring, ␲ 1U Ž ␪ ., is negative at ␪m0 , but its marginal increase with ␪ is larger than the marginal increase of ␲ U0 y C⬘Ž Z*. zU0 , and the social quasi-rents with and without monitoring intersect at ␪c , and from that point on ␲ 1U ) ␲ U0 y C⬘Ž Z*. zU0 .


233

π 1∗ (θ )

π 0∗ (θ ) π 0 (θ) – C ′(Z )z 0 (θ ) *

*

*

T

θm 0

θ θ

0 c

-v (a)

f (θ )

θ θ m0

θ 1c

θc

0

(b) FIG. 4. Determination of the cutoff level ␪c0 under voluntary monitoring.

No one will adopt monitoring when ␲ 1U and ␲ U0 y C⬘Ž Z*. zU0 do not intersect at any ␪c ) ␪m0 . The case with voluntary Žand not full. monitoring is optimal when for a range of values between ␪m and ␪c , the direct cost of monitoring Ž ␯ . and the cost of reduced market profit, p y Ž yU0 y yU1 . y p x Ž xU0 y xU1 ., are greater than the gain from monitoring C⬘Ž Z*.Ž zU0 y zU1 .. If the distribution of ␪ is that depicted in Fig. 4b, much of the population Žunits in the range w ␪m0 , ␪c0 x. are not monitored. In other cases, the size of the nonmonitored population may be smaller. In the case of ␯ declining over time, the switch quality will move from ␪c0 to ␪c1, and the size of the nonmonitored population will decline. Future research will investigate the diffusion process and social welfare gains from subsidizing the development and adoption of monitoring technology. 6. WELFARE ANALYSIS We have now characterized the situation under three possible regimes: no monitoring, mandatory monitoring, and voluntary monitoring. We also defined social welfare under no monitoring and mandatory monitoring as L0 , and L1 ,

234


respectively. The difference in welfare depends upon three factors: the direct cost of monitoring, the difference in environmental quality, and the change in the private profit differential from constraining use of the polluting input. We argued earlier that it is likely that as monitoring costs decrease, there will exist a situation where mandatory monitoring will dominate a situation of no monitoring, which is likely to be optimal when monitoring costs are very high. A critical cost of monitoring, ␯ c , could be defined where ␪c s ␪m0 s ␪m1 , and for this ␯ and smaller ␯ values, all producers will adopt monitoring. Once monitoring costs are at a level such that either mandatory monitoring or voluntary monitoring are feasible, social welfare under either option can easily be defined. Nevertheless, a welfare assessment between these two forms of policies is not straightforward, since the extensive margin differs between the two cases. The choice will depend upon the same three factors identified earlier: the direct cost of monitoring, the private profit differential from voluntary monitoring, and the environmental cost. The monitoring cost factor is positive, since voluntary monitoring saves on monitoring costs compared to compulsory monitoring of all sources. The private profit differential is positive, since decentralized monitoring allows more output to be produced for a given amount of pollution. The environmental cost effect is negative, however, since more pollution is created under decentralized monitoring than under compulsory monitoring. The regulator must decide on the optimal tradeoff between the two positive effects of voluntary monitoring compared to its social cost. In deriving the situation under voluntary monitoring ŽProposition 3., we showed how the situation of voluntary monitoring is characterized by an interior cutoff level ␪c . The analysis of voluntary monitoring is however general enough to encompass the cases of no monitoring and mandatory monitoring as special cases, as we now formulate in a corollary to Proposition 3: COROLLARY. A policy of ¨ oluntary monitoring encompasses as special cases the regimes of no monitoring and mandatory monitoring. If ␪c s ␪m0 , then mandatory monitoring is optimal. If ␪c s ␪ , then social welfare is higher if there is no monitoring at all. The proof is immediate from the definition of social welfare under voluntary monitoring, Ls : ␪c

Ls s max

H t, T , T ␪ 1

0

0 m

1

H␪ w p

q

c

y

w p y yU0 y px xU0 x f Ž ␪ . d␪

yU1 y p x xU1 y ␯ x f Ž ␪ . d ␪ y C

␪c U

H␪

z 0 f Ž ␪ . d␪ q

m

1 U

H␪ z

1

f Ž ␪ . d␪ .

c

It follows from the specification of the optimization problem that when the positive effect of monitoring in terms of quality exceeds the loss in private surplus and the monitoring cost ␯ for any value of ␪ , ␪c will equal ␪m0 and it will be optimal to monitor all sources. Similarly, there could exist levels of parameters Ž ␯ , p y , p x . when the positive effect of monitoring in terms of environmental quality does not outweigh its cost for all values of ␪ , and then ␪c s ␪ . As ␯ declines to 0, the optimization problem approaches the full information case where standard arguments about Pigouvian taxation apply. It is thus possible


235

to consider how our model of nonpoint pollution regulation evolves in the reasonable case where ␯ declines over time. In the early period, ␯ is large and the case of voluntary adoption of monitoring is most relevant. As ␯ declines, the case where all firms adopt monitoring technology is most likely. Finally, as ␯ approaches zero, the regulation problem approximates the full information solution and Pigouvian taxes are optimal. Given that the magnitude of ␯ can be influenced by research and development activity, this discussion also suggests a framework for calculating the rate of return from public and private investments to improve monitoring technology.

7. CONCLUSIONS The paper presents a model of pollution regulation that endogenizes the choice of emissions monitoring technology. In this sense, we go beyond the usual distinction between point and nonpoint source pollution. We show that some extent of nonpoint source pollution is optimal and that the regulator trades off monitoring costs with the loss in efficiency in input allocation. Much of the literature assumes that the regulator can observe a proxy variable that pollution regulation can be based upon. In many difficult pollution problems, however, the regulator cannot observe individual input or output. Examples are plenty: emissions from stoves or household burners adding to global carbon emissions, farmers contributing to local pollution problems, etc. We propose a regulation to deal with this extreme form of nonpoint source pollution. The proposed regulatory scheme differentiates taxation according to the installation of individual monitoring equipment. The two polar cases of mandatory monitoring and no monitoring at all were identified as special cases of the proposed regulation. We identified conditions for an intermediary solution with partial adoption of monitoring equipment. The proposed regulation is particularly useful to deal with the transition phase when a pollution problem is identified but there is no information at all on individual parameters and monitoring all sources is too costly. As a simplifying starting point, we assumed that investment in the monitoring cost ␯ enabled the regulator to know individual pollution levels perfectly. In future work, it would be relevant to allow for imperfect monitoring equipment. One possibility to model this is to posit that observable pollution Žwhich is taxed and thus internalized by the polluter. increases with the investment ␯ : z Ž x Ž ␪ ., ␪ ; ␯ . where ⭸ Ž z .r⭸␯ ) 0, and suitable bounds are defined. Alternatively, two different monitoring technologies could be specified and the index 0, 1, 2 would indicate whether the agent adopted 0 s no monitoring, 1 s cheap but unreliable monitoring technology, or 2 s the perfect technology we have assumed. The reduction in monitoring costs obtained by investment in a cheap but less reliable monitoring technology will then be counterbalanced by the social cost incurred by not obtaining the socially optimal level of pollution for monitored sources. We suspect that there could be cases where the optimal policy would involve investment only in the imperfect monitoring technology, depending upon the cost of technology and the social damage cost of pollution. The general analysis of the basic tradeoff between obtaining information versus incurring monitoring costs follows through

236


though. However, the intervals of optimal adoption of different technologies would change, depending on the relevant monitoring cost levels and parameters. The assumption of an exogenous monitoring cost could be extended in yet another manner. The cost of monitoring pollution could depend upon the type of agent. If monitoring costs vary negatively with type, the qualitative conclusions of our analysis follow through, e.g., agents with high ␪ adopt monitoring, if adoption occurs. However, the profit differential between adoption of monitoring and no monitoring may not be monotonic if ␯ ⬘Ž ␪ . ) 0. The proposed monitoring scheme works through self-selection, in that the choice of installation of monitoring equipment informs the regulator about agents’ types. The scheme was designed to deal with nonpoint pollution problems, but a similar idea of transferring the burden of proof to agents can be used for other cases when agents’ actions are not easily observable. A particularly relevant example in environmental regulation is payment to the agricultural sector for the supply of environmental services. In addition, practical examples of the theoretical idea outlined here already exist. French water law, for instance, leaves open the possibility for some polluters to opt for installation of monitoring equipment rather than paying precalculated taxes levied on the basis of production᎐pollution coefficients ŽFrench Ministry of Environment w5x.. An incentive scheme delegating the choice of monitoring to the regulated agent, like that analyzed here, has broader applications than the problem of nonpoint source pollution. In many cases, charges must be based on some imperfectly observable characteristic that is private information. Metering of energy use is one case in point: some consumers would prefer switching over to charges based on individual monitoring of use rather than being charged according to an average consumption figure. By offering agents a choice between two different forms of payments, the regulator bypasses the problem of inaccessible private information through an equilibrium that separates agents into two categories. Taxation of vehicles based on actual emissions is yet another example. Our model suggests how Innes’ w11x analysis of policy instrument choice regulating automobile emissions can be extended to include actual emissions monitoring. The adoption of technological monitoring devices would in that case provide a useful alternative to flat rate taxation of vehicles or regulation based on input taxes.11 The analysis addressed one aspect of heterogeneity and assumed for simplicity that the type of externality only required a uniform tax on all agents of similar characteristics regardless of location. Nonpoint source pollution problems can have two aspects: difficulties in observing individual emissions and stochastic damage ŽBraden and Segerson w1x.. This paper assumes damages are deterministic. If instead damages are stochastic due to the influence of random variables, complications may arise with an instrument based on emissions and not actual damage. A particularly relevant extension of the model would be to incorporate and distinguish between two forms of uncertainty, one representing randomness in weather and pollution transport, and one representing asymmetric information on production parameters. Here, the regulator did not know the parameter ␪ , which was private information of each agent. Incentive schemes can in certain circumstances be used to elicit such information, however, whereas uncertainty regarding pollu11

The installation of on-board diagnostic systems to new cars is indeed part of a proposal by the European Parliament regarding the reduction of car pollution.

237


tion transport and diffusion patterns can be gained from scientific expertise. It would be clarifying to distinguish between different forms of incomplete information and model their different implications for policy. APPENDIX Proof of Proposition 1. The regulator aims to maximize the market surplus minus environmental costs. The market surplus in this case is producer surplus Žwhich is the sum of ␲ U0 Ž ␪ . y T for all units that operate. plus government revenue Žwe abstract from any cost of transfer .. Thus, the regulator maximization problem is L0 Ž T . s max T

H␪␪ ŽT .␲ 0 m

U 0

Ž xU0 Ž ␪ . , ␪ . f Ž ␪ . d␪ y C

␪

žH

␪m0 Ž T .

market surplus

z Ž xU0 Ž ␪ . , ␪ . f Ž ␪ . d ␪ .

/

cost of pollution

␲ U0

Using Leibniz’ rule, rearranging terms and replacing with first-order condition to determine T * becomes y

⭸␪m0 Ž T . ⭸T

Ž A-1.

with T *, the

␲ U0 Ž xU0 Ž ␪m0 Ž T . . . y C⬘ Ž Z0U . ⭈ zU0 Ž ␪m0 Ž T . . f Ž ␪m0 Ž T . . s 0. Ž A-2. y1

0 ⭸␪ m ŽT .

⭸␲ s ⭸␪0 Ž ␪m0 . ) 0 and ␪m0 are meaningful only when f Ž ␪m0 . ) 0. Thus ŽA-2. implies that T * is determined solving ␲ U0 s C⬘Ž Z0 . zU0 at ␪ s ␪m0 ŽT ., where ␲ U0 s T *. Q.E.D. ⭸T

Differentiation of ŽA-2. with respect to T and rearranging terms yields a sufficient condition for the optimal T y

⭸␪m0

2

ž / ⭸T

⭸␲ U0 ⭸␪

y C⬘ Ž Z0U .

⭸ zU0 ⭸␪

2

q C⬙ Ž Z0U . z 0 Ž ␪m0 Ž T . . f Ž ␪m0 Ž T . . f Ž ␪m0 Ž T . . F 0.

Ž A-3. This condition is likely to hold in most circumstances. Since and C⬙ Ž Z0U . ) 0, it will always hold when as long as

⭸ zU0 ⭸␪

⭸␲ U0 -

⭸␪

⭸ zU 0 ⭸␪

- 0, and even when

⭸ zU 0 ⭸␪

⭸␲ U 0 ⭸␪

) 0, ŽA-3. holds

2

q C⬙ Ž Z0U . z 0 Ž ␪m0 Ž T . . f Ž ␪m0 Ž T . . C⬘ Ž Z0U .

.

Proof of Proposition 2. Ža. Each profit-maximizing agent solves

␲ 1U Ž ␪ . s max p y g Ž x Ž ␪ . , ␪ . y p x x Ž ␪ . y tz Ž x Ž ␪ . , ␪ . y ␯ , x

subject to

␲ 1U Ž ␪ . G 0,

) 0 and

238


and sets input use at xU1 Ž ␪ . where

⭸ g Ž xU1 Ž ␪ . , ␪ .

py

⭸x

s px q t

⭸ z Ž xU1 Ž ␪ . , ␪ .

,

⭸x

Ž A-4.

and the second-order condition for xU1 Ž ␪ . is Ds

⭸ 2 g Ž xU1 Ž ␪ . , ␪ .

y

⭸ x2

t ⭸ 2 z Ž xU1 Ž ␪ . , ␪ .

⭸ x2

py

-0

by our assumptions about g Ž x, ␪ . and z Ž x, ␪ .. If social damage costs of aggregate pollution can be measured, optimal input use can be implemented by setting a charge per unit pollution denoted by t, where t s C⬘Ž Z .. Requiring polluters to pay the monitoring costs and ␲ 1U Ž ␪ . G 0 will lead to outcomes that meet condition Ž7.. Žb. is proved by negation. Suppose an increase in ␯ increases t. Since t s C⬘Ž Z . and C⬙ Ž Z . ) 0, it will also increase Z, but higher ␯ and t will reduce the range of ␪ values where ␲ 1U Ž ␪ . G 0 and reduce xU1 Ž ␪ . when it is positive, and thus reduce Z dt

and we have a contradiction; thus

d␯

F 0.

Q.E.D.

Proof of Proposition 3. Ža.. The proof of Proposition 3Ža. is in three steps: lim ␲ ˜ 0 Ž ␪ . - 0 since

␪ª0

Ž i.

lim p y yU0 Ž ␪ . y p x xU0 Ž ␪ . y T - 0.

␪ª0

d␲ U

␲ ˜ 0 is increasing in ␪ since d␪0 s p y p y y*Ž ␪ . y p x xU0 Ž ␪ . s T, and ␪ s ␪m0 . Žii. For every ␪ , d Ž␲ ˜ 1U y ␲˜ 0 . d␪

s py

⭸g ⭸␪

) 0. Thus, for a sufficiently large ␪ ,

⭸ yU1 ⭸␪

y

⭸ yU0 ⭸␪

yt

⭸ z1 ⭸␪

.

Ž A-5.

ŽFor simplicity, yU1 s g Ž xU1 , Ž ␪ ., ␪ ., yU0 s g Ž xU0 , Ž ␪ ., ␪ ... By the Mean Value Theorem,

⭸ yU1 ⭸␪

y

⭸ yU0 ⭸␪

s

⭸ 2gŽ ˆ x, ␪ . ⭸ x ⭸␪

Ž ˆx, ␪ . Ž xU1 y xU0 . ,

where ˆ x is between xU1 and xU0 . By our assumption,

⭸ 2g ⭸ x ⭸␪

- 0. Thus,

⭸z xU1 y xU0 y t

⭸␪ -0 ⭸ 2g ⭸ x2

since

⭸ 2g ⭸ x2

- 0. Since both

By assumption

⭸z ⭸␪

⭸ 2g ⭸ x ⭸␪

- 0 and xU1 y xU0 - 0, it follows that

⭸z d ) 0 and yt ⭸␪ ) 0, and thus ŽA-5. implies that Ž

⭸ yU 1 ⭸␪ ␲ ˜ U1

y

y␲ ˜ U0

d␪

⭸ yU0 ⭸␪

.

) 0.

) 0.

239


Žiii. If at ␪m0 ␲ ˜ 1U ) ␲˜ U0 , mandatory monitoring is optimal since ␲˜ 1U ) ␲˜ U0 for U

U

⭸ ␲ y␲ ␪ ) ␪m0 , but if ␲ ˜ 1U Ž ␪m0 . - 0, since Ž ˜ 1 ⭸␪ ˜ 0 . ) 0, there might be 1 ) ␪c ) ␪m0 when ␲ ˜ 1U s ␲˜ U0 and ␲˜ 1U ) ␲˜ U0 for ␪ ) ␪c . ŽIf at ␪c s 1, ␲˜ U0 ) ␲˜ 1U , no monitoring is optimal.. Žb. To prove 3Žb., first note that the relationships dictated by incentive compatibility conditions, xU1 Ž ␪ ., xU0 Ž ␪ ., ␪m0 , and ␪c , are functions of the policy instruments t, T1 , and T0 . Optimal values of these parameters are derived by solving

Ls s max

␪c

H

t , T 1 , T 0 ␪m0 1

H␪ w p

q

y

w p y yU0 y px xU0 x f Ž ␪ . d␪

yU1 y p x xU1 y ␯ x f Ž ␪ . d ␪ y C

c

␪c U

H␪

0 m

z 0 f Ž ␪ . d␪ q

1 U

H␪ z

1

f Ž ␪ . d␪ .

c

Ž ␪ has been set equal to 1 for simplification.. The F.O.C. are

⭸ Ls ⭸ T0

sy q

⭸␪m0

p y yU0 Ž ␪m0 . y p x xU0 Ž ␪m0 . y C⬘ Ž Z . zU0 Ž ␪m0 .

⭸ T0 ⭸␪c

p y yU0 Ž ␪c . y yU1 Ž ␪c . y p x xU0 Ž ␪c . y xU1 Ž ␪cU .

⭸ T0

q␯ y C⬘ Ž Z . zU0 Ž ␪c . y zU1 Ž ␪c .

⭸ Ls ⭸ T1

sy

⭸␪c

p y yU0 Ž ␪c . y yU1 Ž ␪c . y p x xU0 Ž ␪c . y xU1 Ž ␪c .

⭸ T1

q␯ y C⬘ Ž Z . zU0 Ž ␪c . y zU1 Ž ␪c .

⭸ Ls ⭸t

⭸␪c

sy

s0, Ž A-6.

⭸t

py

s0, Ž A-7.

yU0 Ž ␪c . y yU1 Ž ␪c . y p x xU0 Ž ␪c . y xU1 Ž ␪c . q␯ y C⬘ Ž Z . zU0 Ž ␪c . y zU1 Ž ␪c .

q

1

H␪

c

py

⭸ yU1 ⭸x

Earlier arguments show that

y p x y C⬘ Ž Z . ⭸␪ c

⭸ T 1U

⭸ zU1

dxU1

⭸x

dt

f Ž ␪ . d ␪s0.

4 Ž A-8.

) 0. Thus for ŽA-7. to hold at ␪c ,

p y Ž yU0 y yU1 . y p x Ž xU0 y xU1 . q ␯ y C⬘ Ž Z . w zU0 y zU1 x s 0.

Ž A-9.

By definition of ␪c , p y Ž yU0 y yU1 . y p x Ž xU0 y xU1 . q ␯ y T0 q T1 q tz1 s 0 at ␪c , which implies that T0 y T1 y tzU1 Ž ␪c . s C⬘ Ž Z . zU0 Ž ␪c . y zU1 Ž ␪c . .

Ž A-10.

Introducing ŽA-9. into ŽA-6. and using the fact that p y yU0 y p x xU0 y T0 s 0 at it follows that

␪m0 ,

T0 s C⬘ Ž Z . zU0 Ž ␪m0 . .

Ž A-11.

240 Finally, since

MILLOCK, SUNDING, AND ZILBERMAN dxU 1 dt

- 0 for ␪c F ␪ - 1, for ŽA-8. to hold it must be true that py

⭸ yU1 ⭸x

y p x y C⬘ Ž Z* .

⭸ zU1 ⭸x

s0

for ␪c F ␪ F 1, but from profit maximizing behavior py

⭸ yU1 ⭸x

y px y t

⭸ zU1 ⭸x

s0

and hence t s C⬘ Ž Z* . .

Ž A-12.

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