I-Shou University Institutional Repository:Item 987654321/13613

ENVIRONMENTAL SUBSIDY AND AUDIT UNDER

BUDGET CONSTRAINT

RUEY-JI GUO

Department of Accounting, Soochow University

ABSTRACT

While the investment in environmental protection, such as equipment of pollution abatement, can reduce the level of environmental damage, it increases the firm’s cost burden. Subsidy on investment in environmental protection and audit on pollutant emission are often used to alleviate costly externalities generated by optimizing economic firms. Nevertheless, budget constraint can bring about a significant impact on the determination and effectiveness of those policies; e.g., the insufficiency of budget will result in idleness of de-pollution equipment. In this paper, we intend to examine the interaction between two policy measures, specifically under budget constraint, in order to enlighten some policy implications.

Keywords: environmental protection; budget constraint; subsidy policy; audit policy (Received: March, 2010; 1st revised: November, 2010; 2nd revised: April, 2011; accepted: August, 2011)

INTRODUCTION

There exist a number of pollution problems in the production of goods or services, whether in manufacturing or non-manufacturing business. Therefore, environmental pollution can be considered a necessary consequence of business activities. How to resolve or alleviate the related problems has become an unavoidable social responsibility of the business concerned. In the research of Boer, Curtin, and Hoyt (1998), they present three strategies related to pollution abatement; i.e., end of pipe

strategies, process improvements, and pollution prevention. All of them are involved with different levels of investment in personnel and equipment, which can be classified as investments in environmental protection. Among others, “end-of-pipe strategies” maintain the production process running as usual and then incurring costs to dispose of the waste produced. It is relatively easier to implement because any modification to production can be made only at the finished stage. In most cases, they need to employ de-pollution equipments, such as filtrating equipments, at some costs to reduce pollution emission. Accordingly, they are highly related to the firm’s investment in equipment of pollution abatement, and will be discussed in this paper.

Since most of investments in environmental protection result in cost incurrence, unless there are adequate incentives, the firm will choose to avoid those costs and not to make investment necessary in environmental protection. Hence, investment subsidy becomes a prominent policy measure often used by regulatory agencies. In the previous literature, there exist a few discussions on investment subsidy. For instance, Knesse and Bower (1968) argue that investment in de-pollution equipment will bring about a negative effect on the firm’s profit since investment tax credit, unless up to a 100% of subsidy rate, can only lead to a decrease in cost burden rather than an increase in revenue. Also, Mills (1972) claims that investment tax credit can lead to a bias of the firm’s decision on how to treat pollution. It is highly possible the firm will purchase some de-pollution equipments for satisfying the regulation of investment tax credit and forgo its original efficient method of pollution treatment. On the other hand, Harberger (1980) points out the investment incentive on tax should follow tax neutrality, and considers neither investment tax credit nor accelerated depreciation satisfying the criteria since the former tends to induce a firm to invest in the asset with a shorter

useful life than the latter. More specifically, Toshimitsu (2010) shows that,

paradoxically, the subsidy policy degrades the environment, and that the optimal policy depends on the degree of marginal social valuation of environmental damage. That is, if the marginal social valuation of environmental damage is larger than a certain value, a consumer-based environmental subsidy policy is not socially optimal. In addition, Slitor (1976), Baumol and Oates (1979), Fisher (1983), and Fredriksson (1998) also

express a few negative opinions on investment tax credit or subsidy from government. Nevertheless, there remain a few different viewpoints on subsidy policy. Laplante (1990) finds, in Cournot oligopoly market model, if government offers some kind of subsidy (e.g., investment tax credit) on de-pollution equipments, it will avoid the collusion between the firm and its competitor for reducing production volume and then lowering pollution level. Meanwhile, the firm will undertake the investment in de-pollution equipment and make its output to fulfill the optimal demand of society. Kort, van Loon, and Luptacik (1991) suggest, while an increase in subsidy rate on investment in de-pollution equipments enhances governmental expenditures, more investment in those equipments will contribute to lower pollution level and higher economic growth. Thus, it will benefit the future tax revenues and the employment opportunities. Specifically, subsidy policy has a much more prominent effect on the investment in very expensive capital goods. Rajah and Smith (1993) argue even if the subsidy on investment in pollution abatement may increase public sector’s expenditures and become a hidden protection, it is still one of the important policy measures used to be coupled with environmental taxes. Bansal and Gangopadhyay (2003) find that while a uniform subsidy policy improves average environmental quality, a uniform tax policy worsens it. Further while a discriminatory subsidy policy reduces total pollution and enhances aggregate welfare, a discriminatory tax policy may increase total pollution and may reduce aggregate welfare. Chau, Wong, and Yiu (2005) state that the government subsidy can be used to mitigate negative externalities, and to improve the environment by indirectly making use of market information in Hong Kong.

Furthermore, even if the firm makes the necessary investment in equipment of pollution abatement, there remains some incentive for it not to operate the equipment and to leave an increase in pollution. Hence, the audit of pollution abatement can be regarded as an important measure to induce a firm to operate de-pollution equipment properly. With regard to the audit system, there exist a number of researches in the past literature (e.g., Antle, 1982, 1984; Baron & Besanko, 1984; Demski & Sappington, 1987; Penno, 1990; Baiman, Evans, & Nagarajan, 1991; Kofman & Lawarree, 1993, 1996). As for the audit system concerning environmental protection, we can find some

related discussions in Doyle (1992), Morelli (1994), Campbell and Byington (1995), and Friesen (2006).

As shown in literature, budget constraint has been playing a key role in the variety of decisions (e.g. Cook & Poremba, 1985; Nanthavanu & Yenradee, 1998; Wang & Lin, 2002, Chang, Wei, & Huang, 2006). Hence, under budget constraint, this paper intends to examine the possible interactions between two policy measures (audit and subsidy) to enlighten some policy implications. In next section, we’ll characterize the basic model used in this paper. The related analyses and results will be presented in section 3. Finally, in the concluding section, we’ll summarize the results of this research as well as present the related policy implications.

THE MODEL

In this paper, the regulator can manipulate two policy measures, subsidy and audit, to induce the pollution-producing firm to make a higher investment in equipment of pollution abatement and to operate the equipment properly. The decision process can be regarded as a one-period game played by a firm and a regulator. Also, it is assumed the

firm is required to make at least a low investment (I ) in de-pollution equipment l

without subsidy from the regulator since the related business is considered a pollution-producing one.

In the beginning of the game, the regulator decides and announces a subsidy rate

s (and s[0,1]) for the portion of increased investment (I I_h I_l) if the firm

make a high investment (I ) rather than a low investment (_h I ) in de-pollution _l

equipment; i.e., the amount of subsidy is sI. The firm will then choose a high or low

investment in de-pollution equipment according to the regulator’s subsidy policy and the anticipated audit policy. After the firm makes investment, the nature will determine

either a good state or a bad state with corresponding probability of “g ” and “1g,”

respectively. Since the firm is required to periodically present financial reports, the realized state of nature is assumed to be common information. If the state of nature is

high production volume. In contrast, the pollution level (under no de-pollution

equipment) will become only P (l Ph) if the state of nature is bad. However, if the

firm makes a low (high) investment and properly operates de-pollution equipment, the pollution level can be reduced to d₁P_h (d₂P_h) in good state of nature or d₁P_l (d₂P_l)

in bad state of nature, where 0d₂ d₁ 1. Whatever pollution level happens, social damage cost (external cost) per unit of pollution level is constant and assumed to bex.

As the state of nature becomes realized, the firm will decide whether to operate

de-pollution equipment or not. Under low investment, e (_lg e ) is used to denote the _lb

probability of using de-pollution equipment in good (bad) state of nature. Similarly, under high investment, e (_hg e ) denotes the probability of using the equipment in good _hb

(bad) state of nature. It is further assumed the cost of operating the equipment is dependent on investment level and pollution level. That is, under low (high) investment, the operating cost will be k1Ph (k2Ph) if the state of nature is good, but k1Pl (k2Pl) if

the state of nature is bad, where k1 k2. Next, according to the actual investment level

and the realized state of nature, the regulator will determine an audit probability to

verify if the firm operated de-pollution equipment or not.1 Specifically, a (lg a ) lb

denotes the audit probability under low investment and good (bad) state of nature, and

hg

a (a ) denotes the audit probability under high investment and good (bad) state of _hb

nature. In the paper, C is the cost of a complete (100%) audit and audit quality is _a

assumed to be perfect.

Furthermore, the firm has to pay some pollution taxes. The pollution taxes include a preliminary pollution tax and an additional pollution tax (penalty). The preliminary pollution tax is determined by the investment level and the state of nature. Under low (high) investment, the preliminary pollution tax will be td1Ph (td2Ph) if the state of

nature is good, and td1Pl (td2Pl) if the state of nature is bad. After the environmental

audit, if the firm was found not using the de-pollution equipment and the audited

1

It is assumed the pollution level will become Ph(Pl) under good (bad) nature state if the firm did not

pollution level (P where _j jhor l) is higher than the originally assumed level (d_iP_j

where i1or 2), the firm will be required to pay an additional pollution tax (penalty),

j i P

d

t(1 ) . Conceivably, t is assumed to be larger than t but less than x.2 Since this paper intends to deal with the problem of budget constraint, in the latter analysis of the paper, the regulator will be required to expend the costs of subsidy and audit subject to a certain budget constraint, the amount of which is denoted by B .

The timing of the related events in the model is summarized as follows:

(1) At the beginning of period concerned, the regulator announces a subsidy policy of investment in environmental protection. The subsidy rate for the portion of increased investment (I I_h I_l) is s (and s[0,1]).

(2) The firm will then choose a high or low investment (I or h I ) in de-pollution l

equipment according to the regulator’s subsidy policy and subsequent audit policy. (3) Nature determines the state is “good” (with probability of g ) or “bad” (with

probability of 1g).

(4) Depending on the actual investment level and the realized state of nature, the firm

decides to properly operate de-pollution equipment by a probability of e , where ij

subscripts i and j correspond to the investment level and the state of nature, respectively.

(5) According to the actual investment level and the realized state of nature, the regulator decides to verify whether the firm operated de-pollution equipment or not

by an audit probability, a , where subscripts i and j correspond to the _ij

investment level and the state of nature, respectively. (6) Transfers are realized.

THE RESULTS

In the model of this paper, the objective function of the firm is to minimize the

2

expected net environmental costs, and the objective function of the regulator is to

minimize the expected net external costs derived from pollution.3 Basically, the

analyses in this section can be classified into two parts. The first one is discussing various strategy equilibriums without budget constraint, and the second one is presenting possible strategy equilibriums subject to budget constraint. In the part one, we first need to understand the possible strategy interaction between the firm’s equipment-operating decision and the regulator’s environmental audit policy. Faced with various possible audit policies, the optimal equipment-operating decision made by firm are summarized in lemma 1.

Lemma 1.

(1) If a_lb k₁/t(1d₁)a₁, then e_lb* 1; otherwise, e_lb* 0. (2) If a_lg k₁/t(1d₁)a₁, then e_lg* 1; otherwise, e_lg* 0. (3) If a_hb k₂/t(1d₂)a₂, then e_hb* 1; otherwise, e_hb* 0. (4) If a_hg k₂/t(1d₂)a₂, then e_hg* 1; otherwise, e_hg* 0. Proof. See appendix.

By the results of lemma 1, it is shown that in the model concerned the firm’s optimal strategy is either to operate de-pollution equipment or not to operate it. There does not exist any mixed strategy. For simplification, in the latter discussion, it is assumed that k1 t(1d1) and k1



k2 x(1d1)



/2 .

4

On the other hand, in response to the firm’s possible strategies, the regulator will set up optimal audit policy depending on the condition of audit cost. The related results are summarized in lemma 2.

3

In the model, expected net environmental costs ＝ investment in de-pollution equipment －investment subsidy revenue ＋ expected equipment-operating cost ＋ expected pollution taxes, and expected net external costs ＝ expected social damage cost － expected pollution tax revenue ＋ investment subsidy expenditure ＋ expected audit cost.

4

Lemma 2. (1) If 2 ₁ 1) / 1 ( d xP k t Ca    l , then 1 * a alb  ; otherwise, 0 *  lb a . (2) If 1 2 1) / 1 ( d xP k t Ca    h , then 1 * a alg  ; otherwise, 0 *  lg a . (3) If 2 2 2) / 1 ( d xP k t Ca    l , then 2 * a ahb  ; otherwise, 0 *  hb a . (4) If 2 2 2) / 1 ( d xP k t Ca    h , then 2 * a ahg  ; otherwise, 0 *  hg a .

Proof. See appendix.

In lemma 2, it is shown there can exist some mixed strategy since the

regulator can use a random audit policy, in which 0 a11 or 0 a2 1. For

avoiding the complexity, it is assumed C_a t(1d₁)2xP_l/k₁ in the analyses related to no budget constraint, but further assumed C_a t(1d₁)P_l in the analyses involved with budget constraint. The reason for the assumption of C_a t(1d₁)2xP_l/k₁ is presented in lemma 3.

Lemma 3.

For inducing the firm to make a required low investment in environmental protection and to properly operate the related equipments, it is a necessary condition that C_a t(1d₁)2xP_l/k₁ . In that situation, a_l* a₁ k₁ t(1d₁) , a*_h a₂

) 1

( ₂

2 t d

k   , and then e_l* e_h* 1. Meanwhile, a and *_l a denote the regulator’s *_h

optimal audit policy contingent on I and _l I (whether the state of nature is good or _h

bad). Also, e and *_l e denote the firm’s optimal operating policy contingent on *_h I _l

and I (whether the state of nature is good or bad). _h

Proof. See appendix.

After learning possible interaction between equipment-operating decision and environmental audit policy, we move one more step to consider the possible strategy interaction between the firm’s investment decision and the regulator’s subsidy policy.

First of all, in lemma 4, the regulator’s optimal subsidy policy will be presented for inducing the firm to make a high investment in environmental protection and to properly operate de-pollution equipment.

Lemma 4.

To induce the firm to make a high investment in environmental protection and to properly operate de-pollution equipment, it can be necessary in some situation for the regulator to offer a subsidy rate,s, and

(1) 0s*  if I (tdk)P (2)





1 * / ) ( 1 t d k P I s s       if I (tdk)P where I I_hI_l, d d₁d₂, k k₁k₂ and P gP_h(1g)P_l. Proof. See appendix.

By the result of lemma 4, whether the regulator should offer the firm some investment subsidy is dependent on the increased investment level. If the increased investment does not exceed the savings in preliminary pollution tax and in equipment-operating cost, it will be unnecessary for the regulator to take investment subsidy into account. Meanwhile, under the assumption of C_a t(1d₁)2xP_l /k₁, even if the firm is faced with a random audit by the regulator, the former will make a high investment in de-pollution equipment and normally operate the equipment as long as I (td k)P. The related inference is summarized in proposition 1.

Proposition 1: Under C_a t(1d₁)2xP_l/k₁ , if I (tdk)P , then s* 0 ,

h I I* , 1e_h*  , a_l* a₁ , and a_h* a₂ ; where I I_hI_l , d d₁d₂ , 2 1 k k k   , a₁ k₁ t(1d₁), a₂  k₂ t(1d₂) and P gP_h (1g)P_l. Proof. See appendix.

On the other hand, even if the increased investment exceeds the savings in both preliminary pollution tax and equipment-operating cost, the regulator still needs to consider whether the subsidy policy is better than the policy of no subsidy. In

proposition 2, it is shown that the former will dominate the latter only if the increased investment does not exceed the total savings in social damage cost (external cost), equipment-operating cost, and expected audit cost. In that case, the regulator will use both subsidy and audit measures simultaneously to induce the firm to make a high investment and to properly operate the equipment.

Proposition 2: Under 1 2 1) / 1 ( d xP k t Ca    l , if (td k)P I (xd k)P(a1a2)Ca , then 1 * s s  , I Ih * , 1e*h  , 1 * a al  , and 2 * a ah  ; where I Ih Il , 2 1 d d d    _, k k1k2_, s1 1



(tdk)P/I



_, a1 k1 t(1d1)_, a2  ) 1 ( 2 2 t d k   , and P gPh (1g)Pl.

Proof. See appendix.

However, if the increased investment exceeds the total savings in social damage cost (external cost), equipment-operating cost, and expected audit cost, there will be no economic benefit for the regulator to use subsidy policy. That implies the regulator will leave the firm to make a low investment in de-pollution equipment, but the former will undertake necessary audit to induce the latter to operate de-pollution equipment. The related inference is shown in proposition 3.

Proposition 3: Under 1 2 1) / 1 ( d xP k t Ca    l , if I (xdk)P(a1a2)Ca, then 0 *  s , I Il * , 1el*  , 1 * a al  , and 2 * a ah  ; where I  Ih Il , 2 1 d d d    , k k₁k₂ , a₁ k₁ t(1d₁) , a₂ k₂ t(1d₂) and l h g P gP P  (1 ) . Proof. See appendix.

The aforementioned propositions are set up on the basis of no budget constraint. If the budget is limited, the regulator must decide the optimal subsidy and audit policies in consideration of the budget constraint. Before formal introduction of propositions

concerned, let’s first look at the following lemma 5. Lemma 5. (1) If Ca t(1d1)Pl, then C



elb 0,alb a1



/ a1 0 a1 [0, a1) R lb           . (2) If Ca t(1d1)Ph, then C



elg 0,alg a1



/ a1 0 a1 [0, a1) R lg           . (3) If Ca t(1d2)Pl, then C



s,ehb 0,ahb a2



/ a2 0 a2 [0 ,a2) R hb           . (4) If Ca t(1d2)Ph, then C



s,ehg 0,ahg a2



/ a2 0 a2 [0,a2) R hg           . Proof. See appendix.

In fact, lemma 5 implies the regulator will never use the strategy of no audit as long as audit cost is low enough. This lemma will be useful in the analyses of subsequent propositions. Furthermore, the assumption of C_a t(1d₁)P_l will also be used to replace the prior assumption of C_a t(1d₁)2xP_l/k₁ for simplification. In proposition 4, the possible strategies equilibriums will be presented in response to a stricter budget constraint.

Proposition 4: Provided C_a t(1d₁)P_l and Ba₂C_a, (1) if I Max



0,



t



Bt/Ca





dP



, then 0 * s , I Ih * , al ah B/Ca * *   , and 0 *  h e ;

(2) if I Max



0,



t



Bt/C_a





dP



, then s* 0, I* I_l , a_l*a*_h B/C_a, and 0

*

l

e ;

where I Ih Il, d d1d2, a2 k2 t(1d2) and P gPh (1g)Pl.

Proof. See appendix.

By proposition 4, if budget is lower than the necessary audit cost for inducing the firm to operate the equipment under high investment, the regulator will not afford an effective subsidy policy. Hence, the firm will not make a high investment unless the increased investment is economically favorable to itself. Meanwhile, even if the firm

makes a high investment, it still will not operate the de-pollution equipment in absence of effective audit. It is noteworthy that scarcity of budget can result in some kind of waste of resources. However, if the regulator’s budget is not so limited, e.g. budget is sufficient for the necessary audit cost of inducing the firm to operate the equipment under high investment, the waste of resources can be alleviated. That will be shown in proposition 5.

Proposition 5: Provided C_a t(1d₁)P_l and a₂C_a Ba₁C_a, (1) if I 



td 



tB(1d1)/Ca



k2



P, then 0 *  s , I Ih * , al B Ca * , 2 * a ah  , and 1e*h  ; (2) if



td 



tB(1d1)/Ca



k2



P I (td k)P (a1a2)Ca, then (i) s*0, I Il * , al B Ca * , 2 * a ah  , and 0 *  l e supposing



I t d k P aCa







t d P Ca





B  (  2)  2 /1 (1 1) / , and (ii) 2 * s s  , I Ih * , al B/Ca *  , 2 * a ah  , and 1 *  h e supposing



I t d k P aCa







t d P Ca





B  (  2)  2 /1 (1 1) / ; (3) if I (td k)P(a₁a₂)C_a, then s* 0, I* I_l, a_l* B/C_a, a*_h a₂, and 0el* ; where I IhIl, d d1d2, k k1k2, a1k1 t(1d1), ) 1 ( 2 2 2 k t d a    , s2 1





td 



tB(1d1)/Ca



k2



P/I



and l h g P gP P  (1 ) . Proof. See appendix.

From the results of proposition 5, if the regulator’s budget can afford the necessary audit cost for inducing the firm to operate the equipment under high investment, it will not happen that the firm makes a high investment in de-pollution equipment but does not use it. In other words, as long as there is a sufficient incentive

for the firm to make a high investment, it will normally operate the equipment and will not result in waste of resources. Nevertheless, since budget is insufficient for inducing the firm to operate the equipment under low investment, the firm will choose not to use the equipment once it has made a low investment.

Finally, if the regulator’s budget is sufficient for inducing the firm to operate the equipment under low investment, the phenomenon of idle equipment (waste of resources) can be completely eliminated, as illustrated in proposition 6.

Proposition 6: Provided C_a t(1d₁)P_l and Ba₁C_a,

(1) if I (td k₂)P, then s* 0, I*I_h, a_l* , a₁ a_h* a₂, and e*_h 1; (2) if (tdk)P I (tdk)P(a1a2)Ca, then 1 * s s  , I Ih * , 1 * a al  , 2 * a ah  , and 1 * h e ; (3) if (tdk)P(a1a2)Ca I (xdk)P(a1a2)Ca, then (i) s*0, I Il * , 1 * a al  , 2 * a ah  , and 1 *  l e supposing a a B I t d k P a C C a1   (  )  2 _, and (ii) 1 * s s  , I Ih * , 1 * a al  , 2 * a ah  , and 1 * h e supposing a C a P k d t I B (  )  2 _; (4) if I (xdk)P (a1a2)Ca, then 0 *  s , I Il * , 1 * a al  , 2 * a ah  , and 1 * l e ; where I IhIl, d d1d2, k k1k2, a1k1 t(1d1), ) 1 ( 2 2 2 k t d a    , s11



(tdk)P/I



and P gPh(1g)Pl.

Proof. See appendix.

As shown in proposition 6, if budget is increased up to a certain level, i.e. enough for inducing the firm to operate the equipment under low investment, the firm will necessarily use de-pollution equipment whether it has made a high or low investment.

As for whether the regulator needs to employ subsidy policy or not will depend on if the subsidy is necessary for inducing high investment and if budget is sufficient for enforcing both subsidy and audit policies. Certainly, a much more important precondition is that the subsidy policy should economically dominate the policy of no subsidy.

CONCLUSIONS

While the investment in environmental protection, such as equipment of pollution abatement, can reduce the level of environmental damage, it increases the firm’s cost burden. If there is no adequate incentive, the firm can choose to avoid those costs and not to make an investment necessary in environmental protection. Even if the firm makes a necessary investment in equipment of pollution abatement, there remain some incentives for it not to operate the equipment, and that will result in an increase in pollution. Hence, subsidy on investment in environmental protection and audit on pollutant emission are often used as important measures for alleviating costly externalities generated by optimizing economic firms. However, budget constraint can bring about a significant impact on the determination and effectiveness of those policies. This paper presents a model to examine the interaction between these two policy measures, especially under budget constraint.

The results reveal that if the audit cost budget is insufficient for inducing the firm to operate the equipment under high investment, the firm will not make a high investment unless the increased investment is economically favorable to itself. Even if having made a high investment, the firm will not properly operate the equipment of pollution abatement in absence of effective audit. Accordingly, it is highly possible that the insufficiency of budget will lead to the consequence of resource idleness.

In contrast, the situation of resource idleness can be mitigated if the regulator’s budget is not so constrained, or the budget is sufficient to cover the audit cost necessary for inducing the firm to operate the equipment under high investment. In other words, as long as there is a sufficient incentive for a firm to make a high investment, the firm

will tend to properly operate the invested equipment. However, the firm still will not properly operate the related equipment after making a low environmental investment in that audit cost budget is not enough for inducing the firm to operate the equipment under low investment.

Ideally, the phenomenon of equipment idleness (resource waste) can be eliminated if the regulator’s budget is sufficient for inducing the firm to operate the equipment under low investment. In that case, the firm will choose to use the equipment of pollution abatement no matter if it has made a high or low investment, and the regulator’s subsidy policy will depend on if subsidy is necessary for inducing high investment and if budget is sufficient for enforcing an effective subsidy and audit policies, provided the subsidy policy is economically better than the policy of no subsidy.

In Taiwan, according to the current tax law and the ITC regulation for encouraging an enterprise to participate in public constructions, the government essentially adopts both accelerated depreciation and tax credit to subsidize the firm’s investment in pollution abatement equipment and technology. Nevertheless, the effectiveness of environmental management seems to be not satisfactory and even discouraging. The reason may partly come from the limited budget. As mentioned in propositions 4 and 5 of this paper, the budget constraint can result in either the insufficiency of environmental investment or the idleness of invested equipment, both of which will enhance the social environment burden. Hence, the legislative sector should pay much more attention to consider the effect of budget allocation on environment.

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APPENDIX

Proof of Lemma 1.

(1) Given that the firm has made a low investment in environmental protection, if the state of nature is “bad,” the firm’s expected costs will become



lb lb



l l lb l lb lb l

F

lb e a I td P e k P e a t d P

C ,   1  1 (1 ) (1 1) .

To minimize the expected costs, the firm will choose e_lb 1 only if



lb lb



lb F

lb e a e

C 

 , / k₁P_l a_lbt(1d₁)P_l 0 or a_lb k₁/t(1d₁)a₁; otherwise, 0e_lb  will be optimal.

(2) By the same token as (1), the firm will choose elg 1 only if



lg lg



lg

F lg e a e C   , / 0 ) 1 ( ₁ 1     k P_h a_lgt d P_h or a_lg k₁/t(1d₁)a₁; otherwise, e_lg 0 will be optimal.

(3) Given that the firm has made a high investment in environmental protection, if the state of nature is “bad,” the firm’s expected costs will become



hb hb



h l hb l hb hb l

F

hb s e a I s I td P e k P e a t d P

C , ,     ₂  ₂ (1 ) (1 ₂) .

To minimize the expected costs, the firm will choose e_hb 1 only if



hb hb



hb

F

hb s e a e

C 

 , , / k2Pl ahbt(1d2)Pl 0 or ahb k2 /t(1d2)a2;

otherwise, 0e_hb  will be optimal.

(4) By the same token as (3), the firm will take e_hg 1only if C_hgF



s,e_hg,a_hg



/e_hg

0 ) 1 ( ₂ 2     k Ph ahgt d Ph or ahg k2/t(1d2)a2; otherwise, ehg 0 will be optimal. Proof of Lemma 2.

(1) Given that the firm has made a low investment in environmental protection, if the state of nature is “bad,” the regulator’s expected costs will become



e 1,a a₁



(x t)d₁P a₁C a₁ [a₁ ,1]

C_lbR _lb  _lb     _l   _a    for inducing e_lb 1 and



e 0,a a₁



xP td₁P a₁[C t (1 d₁)P] a₁ [0, a₁)

C_lbR _lb  _lb    _l  _l   _a    _l   for

inducing 0e_lb  .

To minimize the expected costs, the regulator will induce e_lb 1 only if











    1 1] , [ ₁ 1, 1 a a e C Min lb lb R lb a a









     _ 1 ) , 0 [ ₁ 0, 1 a a e C Min lb lb R lb a a .

Let  0 and  0, then











    1 ) , 0 [ , 0 1 1 a a e C Min _lbR _{lb lb} a a





    C_lbR e_lb 0,a_lb a₁ if l a t d P C  (1 ₁) ; otherwise,











    1 ) , 0 [ ₁ 0, 1 a a e C Min lb lb R lb a a 



lb 0, lb 0



R lb e a C .

On the other hand,











    1 1] , [ ₁ 1, 1 a a e C Min lb lb R lb a a C



elb 1,alb a1



R lb    .

Hence, if C_a t(1d₁)P_l, it holds that











    1 1] , [ , 1 1 1 a a e C Min _lbR _{lb lb} a a









     _{ 1} ) , 0 [ , 0 1 1 a a e C Min _lbR _{lb lb} a a since C



elb 1,alb a1



R lb   C



elb 0 ,alb a1



R lb as  0 and tx. However, if C_a t(1d₁)P_l, then











    1 1] , [ ₁ 1, 1 a a e C Min lb lb R lb a a









     _ 1 ) , 0 [ ₁ 0, 1 a a e C Min lb lb R lb a a C_lbR



e_lb 1,a_lb a₁



C_lbR



e_lb 0,a_lb 0



only if C_a t(1d₁)2xP_l /k₁ also holds. Since t(1d₁)2xP_l /k₁ t(1d₁)P_l, we have











    1 1] , [ ₁ 1, 1 a a e C Min lb lb R lb a a









     _ 1 ) , 0 [ ₁ 0, 1 a a e C Min lb lb R lb a a if t(1d₁)P_l C_a t(1d₁)2xP_l /k₁ and











    1 1] , [ ₁ 1, 1 a a e C Min lb lb R lb a a









     _ 1 ) , 0 [ ₁ 0, 1 a a e C Min lb lb R lb a a if

C_a t(1d₁)2xP_l /k₁.

Therefore, only if C_a t(1d₁)2xP_l/k₁, the regulator will choose a_lb  to a₁

induce 1e_lb  ; otherwise, the regulator will choose a_lb 0 and leave e_lb 0.

(2) By the same token as (1), the regulator will choose a_lg  to induce a₁ e_lg 1

only if C_a t(1d₁)2xP_h/k₁; otherwise, the regulator will choose a_lg 0 and leave 0e_lg  .

(3) Given that the firm has made a high investment in environmental protection, if the state of nature is “bad,” the regulator’s expected costs will become



s ,e 1,a a₂



(x t)d₂P s I a₂C a₂ [a₂,1] C_hbR _hb  _hb     _l    _a   for inducing 1  hb e and C_hbR



s,e_hb 0,a_hb a₂



xP_l td₂P_l sIa₂[C_a t(1d₂)P_l] ) , 0 [ ₂ 2 a a    for inducing e_hb 0.

To minimize the expected costs, the regulator will induce e_hb 1 only if











    2 1] , [ , 1 , 2 2 a a e s C Min _hbR _{hb hb} a a









     _{ 2} ) , 0 [ , 0 , 2 2 a a e s C Min _hbR _{hb hb} a a .

Let  0 and  0, then











    2 ) , 0 [ ₂ , 0, 2 a a e s C Min hb hb R hb a a C



s,ehb 0,ahb a2 



R hb if l a t d P C  (1 ₂) ; otherwise,











    2 ) , 0 [ , 0 , 2 2 a a e s C Min _hbR _{hb hb} a a





0 , 0 ,    R _{hb hb} hb s e a C .

On the other hand,











    2 1] , [ , 1 , 2 2 a a e s C Min _hbR _{hb hb} a a



2



, 1 ,e a a s C_hbR _hb  _hb   .

Hence, if C_a t(1d₂)P_l, it holds that











    2 1] , [ ₂ , 1, 2 a a e s C Min hb hb R hb a a









     _ 2 ) , 0 [ ₂ , 0, 2 a a e s C Min hb hb R hb a a since C_hbR



s,e_hb 1,a_hb a₂



C_hbR



s,e_hb 0,a_hb a₂ 



as  0 and t x.

However, if C_a t(1d₂)P_l, then











    2 1] , [ , 1 , 2 2 a a e s C Min _hbR _{hb hb} a a









     _{ 2} ) , 0 [ , 0 , 2 2 a a e s C Min _hbR _{hb hb} a a



s,e 1,a a2



C hb hb R hb    



, hb 0, hb 0



R hb s e a C only if 2 2 2) / 1 ( d xP k t Ca    l also holds. Since t (1 d ) xPl /k2 t(1 d2)Pl 2 2      , we have











    2 1] , [ , 1 , 2 2 a a e s C Min _hbR _{hb hb} a a









     _{ 2} ) , 0 [ , 0 , 2 2 a a e s C Min _hbR _{hb hb} a a if 2 2 2 2) (1 ) / 1 ( d P C t d xP k t  l  a    l and











    2 1] , [ , 1 , 2 2 a a e s C Min _hbR _{hb hb} a a









     _{ 2} ) , 0 [ , 0 , 2 2 a a e s C Min _hbR _{hb hb} a a if 2 2 2) / 1 ( d xP k t Ca    l . Therefore, only if 2 2 2) / 1 ( d xP k t

Ca    l , the regulator will choose ahb a2 to

induce 1e_hb  ; otherwise, the regulator will choose a_hb 0 and leave e_hb 0.

(4) By the same token as (3), the regulator will choose a_hg a₂ to induce e_hg 1

only if C_a t(1d₂)2xP_h/k₂; otherwise, the regulator will choose a_hg 0 and leave 0e_hg  .

Proof of Lemma 3.

By the assumption of k₂ k₁ t(1d₁)t(1d₂)and P_l  P_h, it can be inferred that t(1d₁)2xP_l/k₁ t(1d₂)2xP_l /k₂ t(1d₂)2xP_h/k₂ and 1 2 1 1 2 1) / (1 ) / 1 ( d xP k t d xP k

t  _l    _h . Hence, according to the results of lemma 1 & 2,

we have a_lb* a_lg* a₁ , a_hb* a_hg* a₂ , and then e_lb* e_lg* 1, e_hb* e_hg* 1 if

1 2 1) / 1 ( d xP k t C_a    _l .

Proof of Lemma 4.

By the result of lemma 3, if 1

2 1) /

1

( d xP k

t

Ca    l , the firm will normally operate

the de-pollution equipment whether the firm makes a high or low investment and whether the state of nature is good or bad. Thus, to induce the firm to make a high investment in environmental protection, the necessary condition is







C I ,e e 1



E



C



I ,e e_lg 1





E _hF _{h hb hg l}F _{l lb} l h h s I g td k P g td k P I    ( ₂  ₂) (1 )( ₂  ₂)  l h l g td k P g td k P I  ( ₁ ₁) (1 )( ₁ ₁)  l h g t d k P P k d t g I I s   (  ) (1 )(  ) 



( ) /



1 1 t d k P I s s      

Hence, the optimal subsidy policy of the regulator will bes* Max

 

0 s, ₁ . In other

words, if I (td k)P , * ₁

s

s  ; otherwise, * 0

s .

Proof of Proposition 1.

By the result of lemma 3, as 1

2 1) / 1 ( d xP k t Ca    l , we have 1 * a al  , 2 * a ah  ,

and 1el* eh*  . Meanwhile, according to the part (1) of lemma 4, if

P k d t I (  )  , then * 0

s is enough for inducing I Ih

*

.

Proof of Proposition 2.

According to lemma 3, C_a t(1d₁)2xP_l/k₁ will induce a*_l  , a₁ a_h* a₂, and 1

* *  

h

l e

e . In addition, by the part (2) of lemma 4, if I (tdk)P, then the

regulator need to offer s*  to induce s₁ I*  I_h. However, to ensure the subsidy

policy is better than the policy of no subsidy, the following condition also needs to be satisfied; i.e.

a a x t d P aC C a I s P d t x ) _{2 1 2} ( ) _{1 1} (         a C a a P d t x P k d t I (  ) (  ) ( ₁  ₂)   a C a a P k d x I (   ) ( ₁ ₂)   . Proof of Proposition 3.

By the proof of proposition 2, it is straightforward that the regulator will not induce the firm to make a high investment if I (xdk)P (a₁a₂)C_a. Hence, the optimal subsidy policy will become s* 0 and leave I* I_l.

Proof of Lemma 5. Since C



elb 0,alb a1



xPl td1Pl a1[Ca t (1 d1)Pl] a1 [0, a1) R lb              , we have 



0,  1



/ 1 0   a a a e C lb lb R

lb if Ca t(1d1)Pl. By the same token, we can

obtain the other results.

Proof of Proposition 4.

By lemma 5, if C_a t(1d₁)P_l and Ba₂C_a , the regulator’s optimal audit policy will be a h l C B a

a*  *  and leave el* eh* 0. Meanwhile, since Ba2Ca, the

regulator can not afford an effective subsidy policy. Hence, the firm will still make a high investment only if

E



C_hF(s0,e_h 0,a_h B/C_a)

 

E C_lF(s0,e_l 0,a_l B/C_a)



I_h td₂P 



Bt(1d₂)P/C_a



I_l td₁P 



Bt(1d₁)P/C_a



I tdP



BtdP/C_a







t



Bt/C_a





dP, which can be larger or less than 0.

Proof of Proposition 5.

audit, a_l  , to induce a₁ e_l 1 as I  . Thus, if I_l C_a t(1d₁)P_l, the following

condition must be satisfied for inducing the firm to take the action of I I_h and

1  h e ; i.e.



( , 1, 2)

 

( 0, l 0, l / a)



F l h h F h s e a a EC s e a B C C E       Ih sI (td2 k2)P Il td1P



Bt(1d1)P/Ca



sI I(td k2)P 



Bt(1d1)P/Ca



s1





td 



tB(1d1)/Ca



k2



P/I



s2

Hence, if I 



td



tB(1d₁)/C_a



k₂



P , s 0 will be enough for inducing

h

I

I  , and the budget also allows the regulator to enforce a_h a₂ to induce e_h 1. We have the result of part (1).

However, if I 



td 



tB(1d₁)/C_a



k₂



P , ss2 will become a necessary

subsidy policy to induce I I_h. In that situation, the regulator needs to consider if the budget is enough for enforcing both subsidy and audit policies. That is, the following condition must be satisfied for enforcing both subsidy and audit policies.

sIa₂C_a  B I 



td 



tB(1d₁)/C_a



k₂



P a₂C_a B I (tdk2)P a2C_a 



1



t(1d1)P/C_a





B B



I (tdk2)Pa2Ca



/



1



t(1d1)P/Ca





. In addition, since



I (tdk2)Pa2Ca



/



1



t(1d1)P/Ca





a1Ca P k C a C a P k d t I(  ₂)  _{2 a}  _{1 a}  ₁   a C a a P k d t I (  ) ( ₁ ₂)   ,

there exists the possibility that budget is enough for enforcing both subsidy and audit policies only if I (td k)P(a1a2)Ca.

Hence, under



td



tB(1d1)/Ca



k2



P I (td k)P (a1 a2)Ca , it

can be inferred that a2Ca 



I (tdk2)Pa2Ca



/



1



t(1d1)P/Ca





a1Ca. To

ensure the subsidy policy is better than the policy of no subsidy, the following condition also needs to be satisfied; i.e.













l l a





R l h h R h s s e a a EC s e a B C C E  2, 1,  2  0, 0,  / . Under C_a t(1d₁)P_l, by lemma 5, we have













l l a





R l l l R l s e a a EC s e a B C C E 0, 0,  ₁   0, 0,  / , where 0 and 0   . Hence, E



C_hR



ss₂,e_h 1 ,a_h a₂





E



C_lR



s0 ,e_l 0 ,a_l  B/C_a





is satisfied provided E



C_hR



ss₂,e_h 1,a_h a₂





E



C_lR



s0,e_l 0,a_l a₁





. Moreover,







C s s2,e 1,a a2



E



C



s0,e 0,a a1 





E R _{l l} l h h R h



C t d P



a P td P x C a I s P d t x ) _a ( ) _a (1 ) (  2  2  2   1  1     1  









P d t a C a P d t x P x d C a P k C d B t d t I P d t x a a a ) 1 ( ) ( ) ( ) ( ) 1 ( / ) 1 ( ) ( 1 1 1 1 1 2 2 1 2                       









) 0 ( ) 1 ( ) 1 ( ) ( ) ( / ) 1 ( ) ( 1 1 1 2 1 1 1                        P d t a P x d C a a P d t x P C d B t k P k d t I a a









) 0 ( ) 1 ( ) 1 ( / ) 1 ( ) ( ) ( 1 1 1 1 1 2 1                     P d t a P d x P C d B t k C a a P k d x I a a





)) 1 ( / ( ) 1 ( ) ( ) ( 2 2 1 1 2 1 2 1 d t C k B P k P d x P k k C a a P k d x I a a                 