UNIVERSITÀ DEGLI STUDI DI PADOVA. Dipartimento di Scienze Economiche Marco Fanno

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1 UNIVERSITÀ DEGLI STUDI DI PADOVA Dipartimnto di Scinz Economich Marco Fanno OPTIMAL MONITORING TO IMPLEMENT CLEAN TECHNOLOGIES WHEN POLLUTION IS RANDOM INÉS MACHO-STADLER Univrsitat Autònoma d Barclona DAVID PÉREZ-CASTRILLO Univrsitat Autònoma d Barclona Visiting Univrsità di Padova Novmbr 2007 MARCO FANNO WORKING PAPER N.60

2 Optimal monitoring to implmnt clan tchnologis whn pollution is random Inés Macho-Stadlr David Pérz-Castrillo Novmbr 15, 2007 W thank Estr Camiña, Fahad Khalil, Liguo Lin, Pau Olivlla, and Pdro Ry-Bil for hlpful rmarks. W also thank commnts rcivd from participants in prsntations at Munich (CESifo Ara Confrnc on Applid Microconomics, 2007), Budapst (EEA, 2007), Valncia (EARIE, 2007) and Padova (ASSET, 2007). W gratfully acknowldg th financial support from projcts SEJ ECON, 2005SGR-00836, Consolidr-Ingnio CSD and Barclona Economics-Xarxa CREA. Th prsnt papr is a rvisd vrsion of th working papr that appard as WP # 289 at Xarxa CREA. Dpt. of Economics & CODE, Univrsitat Autònoma d Barclona. ins.macho@uab.s. Corrsponding author: Dpt. of Economics & CODE, Univrsitat Autònoma d Barclona, Bllatrra (Barclona), Spain. david.prz@uab.s. Ph: Fax:

3 Abstract W analyz nvironmnts whr firms chos a production tchnology which, togthr with random vnts, dtrmins th final mission lvl. W considr th coxistnc of two altrnativ tchnologis. Th cost of th adoption of th clan tchnology and th actual missions ar firms privat information. Th nvironmntal rgulation is basd on taxs ovr rportd missions, and on monitoring and pnaltis ovr unrportd missions. W show that th optimal monitoring is a cut-off policy, whr all rports blow a thrshold ar inspctd with th sam probability, whil rports abov th thrshold ar notmonitord. Wshowthatifthadoptionofthtchnologyisfirms privat information, too fw firms will adopt th clan tchnology undr th optimal monitoring policy. Howvr, whn th EA can chck th tchnology adoptd by th firms, th optimal policy may induc ovrswitching or undrswitching to th clan tchnology. JEL Classification numbrs: K32, K42, D82. Kywords: Production tchnology, random missions, nvironmntal taxs, optimal monitoring policy. 2

4 1 Introduction Pollution prvntion and clan tchnologis hav com to th forfront in rducing and controlling th nvironmntal ffcts cratd by firms. Environmntal Agncis (EAs) fac th important challng of ncouraging th adoption of such masurs and complling complianc with nvironmntal laws and rgulations. For this aim, thy oftn dsign a dtrrnc policy basd on inspctions. This papr contributs to th litratur that analyzs th optimal inspction policy taking into account firms stratgic bhavior. 1 W build and analyz a modl whr firms choos a production tchnology which, togthr with som random vnt, dtrmins th final mission lvl. That is, w xplicitly tak into account th random natur of pollution and its ffcts on th optimal inspction policy. W considr th coxistnc of two altrnativ tchnologis: a clan tchnology and a dirty tchnology. A clan tchnology is a manufacturing procss or product tchnology that rducs pollution or wast nrgy us, or matrial us in comparison with th dirty tchnology. That is, xpctd lvl of missions whn production is carrid out with th clan tchnology is lowr than if th firm uss th dirty tchnology. For both tchnologis, th ralizd mission lvl is random and it is privatly obsrvd by th firm. Indd, although firms can limit missions of pollutants by dciding th production tchnology, by adjusting th mix of outputs and inputs, and through th us of abating tchnologis, this control is oftn not prcis. Many factors such as wathr, quipmnt failurs, and human rror may caus ralizd missions to diffr from intndd missions. Also, input rlativ pric changs may affct th lvl of polluting input usd. In our framwork, th nvironmntal rgulation is basd on taxs ovr rportd missions, monitoring, and pnaltis ovr unrportd missions. Firms rport thir mission lvl and pay th taxs associat to thm. Th tru mission lvl can only b obsrvd (and mad vrifiabl) by th EA aftr an inspction. W considr situations whr th EA wants to induc firms to mak a major discrt invstmnt to hlp th nvironmnt. Firms fac diffrnt costs to adopt th clan tchnology, which usually cannot b obsrvd by th EA. In this papr, w study how to provid 1 Cohn (1999) and Sandmo (2000) provid two rcnt and xtnsiv rviws of th litratur. 3

5 incntivs to th adoption of th clan tchnology at th lowst costs and which firms should b ncouragd to do so. In particular, w analyz th optimal monitoring stratgy whn th EA taks into account th random natur of pollution: bad luck may caus a high lvl of missions vn whn th firm adopts th clan tchnology whil good luck may diminish missions lvl of a firm that uss th dirty tchnology. W distinguish thr cass. First, w analyz a rfrnc framwork whr w assum that th EA facs a singl firm and knows th firm s cost of adopting th tchnologis but th tchnology chosn is not vrifiabl. W show that th inspction policy on th mission lvl that inducs a firm to adopt th clan tchnology at th lowst cost is a cutoff stratgy whr all th rports undr th cut-off ar inspctd with th sam probability and rports ovr this cut-off ar not auditd. Scond, w considr th cas whr th EA facs a population of firms that diffr in th cost of adopting th clan tchnology and th tchnology adoptd by ach firm is obsrvabl, although th cost ncountrd by th firm is not. In this situation, only thos firms producing with th dirty tchnology will b inspctd through th cut-off rul corrsponding to th marginal firm. Third, w analyz situations in which both th tchnology adoptd by th firms and thir costs ar non-vrifiabl. In this cas, th EA is forcd to us th sam monitoring policy for all typs of firm. Th optimal monitoring is again a cut-off policy consisting on th on that would b dsignd for th marginal firm as if its missions distribution was an avrag btwn th clan and th dirty tchnology. In all cass, firms with low adoption costs will b inducd to switch to th clan tchnology whil high-cost firms will kp th dirty on. W compar th conditions undr which firms ar pushd to adopt th clan tchnology with th cas whr th EA has all th information (first-bst). Whn th tchnology adoptd is privat information for th firms, th optimal monitoring policy inducs to fw firms to choos th clan tchnology as compard to th first bst. In contrast, whn th cost is firms privat information, whil th tchnology adoptd is vrifiabl, th EA may want to push firms to adopt th clan tchnology too oftn to sav monitoring costs. Svral paprs hav considrd that pollution missions frquntly produc stochastic nvironmntal damags. 2 But thy hav studid diffrnt aspcts from our papr. Som 2 For xampl, th damag from a givn amount of fflunt rlasd in a rivr dpnds on faturs 4

6 authors hav analyzd th advantags and disadvantags of introducing slf-rporting (whras in our papr is assumd to b in plac) on th mission lvl in situations whr missions ar random. In particular, Inns (1999) analyzs a modl whr thr ar xpost bnfits of claning-up if an nvironmntal accidnt (high lvl of pollution) occurs. In his modl, firms choos th lvl of car (that can b intrprtd as th choic of a tchnology), and this car affcts th probability of an accidnt. Inns shows that whn thr is no slf-rporting a firm will ngag in clan-up only if auditd, whil th firm always clans-up whn slf-rporting is in plac. Malik (1993) compars th cas with and without slf-rporting in a situation whr collcting pnaltis and taxs is costly and th monitoring tchnology is imprfct (including both typs I and II of rrors). In this framwork, slf-rporting dos not ncssarily rduc rgulation costs bcaus of costly sanction. 3 Hamilton and Rquat (2006) analyz th choic btwn mission caps and nvironmntal quality standards whn missions ar random. Thy show that whn firms invst in abatmnt quipmnt, an mission standard inducs ovr-invstmnt rlativ to th socially optimal rsourc allocation, whil undr-invstmnt tnds to occur undr an ambint nvironmntal policy. Th modl analyzd in this papr also contrasts with most of th modls that study th optimal inspction policy, sinc thy assum that th firm dcids dirctly its (nonrandom) mission lvl (s, for xampl, Harford, 1978 and 1987, Sandmo, 2000, and Macho-Stadlr and Pérz-Castrillo, 2006). In Macho-Stadlr and Pérz-Castrillo (2006), w show that th EA optimal stratgy inducs a cornr solution, in th sns that thr ar always firmsthatdonotcomplywiththnvironmntalobjctivandothrsthatdo comply but all of thm vad th nvironmntal taxs. Concrning th optimality of th us of nvironmntal taxs, Macho-Stadlr (2007) shows that it is lss costly to achiv any lvl of complianc through taxs than using standards or tradabl prmits. In trms of mthodology, analyzing th audit policy to induc complianc with th nvironmntal policy is clos to th litratur on optimal auditing in tax vasion. 4 Howwhich vary tmporally, such as sasonal fluctuations in watr volum, tmpratur and turbidity. Th ffct of airborn missions on air quality dpnds on prvailing atmosphric conditions, such as thrmal structur, circulation, prssur, and humidity. 3 S also Kaplow and Shavll (1994) and Livrnois and McKnna (1999). 4 S footnot 8 for rfrncs on th optimal auditing in tax vasion modls. 5

7 vr, in this litratur th taxpayr is assumd to hav no choic othr than rporting an incom lvl. In contrast, th problm addrssd in th prsnt papr is mor complx and has not bn considrd bfor. In our modl, th agnt has to dcid first on th tchnology of production and scond on th rport about missions onc th (random) mission lvl is ralizd. Our modl combins two informational dimnsions: a moral hazard problm with rspct to th choic of th production tchnology and an advrs slction problm concrning th rport of missions. Finally, som prvious paprs hav analyzd how th rgulatory rgim via missions taxs or standards may affct firms adoption of missions abatmnt tchnology (s, for xampl, Downing and Whit, 1986, Milliman, 1989, Grsbach and Rquat, 2004, and Tarui and Polasky, 2005). Our papr is complmntary to ths contributions as w show how th monitoring policy, in nvironmnts whr missions cannot b idntifid without inspction, can b dsignd to optimiz firms adoption at th lowst cost. Th papr is organizd as follows. In Sction 2, w introduc th modl and analyz a firm s rport givn its tchnology and th inspction policy. Sction 3 introducs th scnario with two diffrnt tchnologis. In Sction 4, w analyz th rfrnc framwork whr thr is a singl firm and both th firm and th EA know th cost of adopting th clan tchnology. W charactriz th policy that th EA puts in plac if it wants to induc this firm to adopt th clan tchnology. Sctions 5 and 6 constitut th cntral part of th papr. W charactriz th optimal monitoring policy for EA that facs a family of firms whn thir adoption cost is not obsrvabl assuming ithr that th tchnology adoptd is vrifiabl (Sction 5), or that it is not (Sction 6). In Sction 7, wconclud. All proofs ar in th Appndix. 2 Th Firm s rport undr mission taxs W modl situations whr a firm s missions ar random, but thy ar influncd by th firm s choic of tchnology. A firm s lvl of missions (or damags) is distributd in th intrval [, ] according to th distribution function F (; E), whr E dnots th production tchnology chosn by th firm. W assum that F ( ; E) is continuously diffrntiabl and that f(; E) = F(; E)/ > 0 on [, ]. Th cost of th tchnology E 6

8 is sunk. W assum that missions ar taxd according to a linar schdul, with marginal tax rat t. Howvr, missions lvls ar firm s privat information. Emissions can b assssd if th firm is monitord by th EA. Th firm is askd to snd a rport z [, ] on its missions, onc thy ar ralizd. Th firm may choos a rport z that dos not coincid with th tru missions lvl. Th EA has two instrumnts to control firm s missions: monitoring and pnaltis. W dnot by α(z) th probability that th EA will audit th missions of th firm whn it rports a lvl of missions z. Thstratgyα( ) followd by th EA is dcidd prvious tothchoicofthtchnologye, that is, w assum that th EA is abl to commit to its monitoring stratgy. If th firm is monitord and its lvl of missions is found to b highr than its rport, thn a pnalty is imposd to th firm. For simplicity, w assum that th pnalty is linar in th undrrportd missions. W also assum that th marginal pnalty rat, dnotd θ, is xognous. Paramtr θ includs th taxs du to th EA, hnc θ>t. Thr is no bonus for ovrrporting. Th firm s xpctd costs whn th missions ar, th rport is z and th monitoring stratgy is α( ) ar: c(, z; α( )) = tz + α(z)θ[ z] if z, c(, z; α( )) = tz + α(z)t[ z] if z>. Th timing of th dcisions is as follows. First, th EA dcids on th monitoring stratgy α( ). Scond, th firm chooss th tchnology E at a crtain cost. Emissions ar ralizd according to th dnsity function f(; E). Third, aftr having obsrvd th ralizd missions, thfirm dcids on th rport z and pays th taxs tz. Thfirm is monitord with probability α(z). If it is auditd and it has undrrportd, thn th firm pays th pnalty θ[ z]. Th firm chooss z to minimiz its costs c(, z; α( )), as a function of th ralizd missions. That is, at th last stag, th firm chooss z(). W dnot c(; α( )) = c(, z(); α( )) firm s xpctd costs whn its missions lvl is and it maks th rport that minimizs its costs. W start with two rsults that provid usful information concrning firm s bhavior 7

9 with rspct to th rport. Lmma 1 A firm whos mission lvl is : (i) nvr rports mor than thir missions: z ; (ii) nvr rports z<if α(z) >t/θ; (iii) rports honstly, i.., z =, onlyifα(z) t/θ for all z [,). Th intuition bhind Lmma 1 is th following. Givn th tax rat t and th pnalty rat θ, a monitoring probability of t/θ is nough to spur honst bhavior. Thrfor, a firm nvr submits a rport z lowr than its ral mission if rporting z lads to inspction with a probability highr than t/θ. On th othr hand, th firm will not rport honstly if it can submit a rport z<that is monitord with a probability lowr than t/θ. According to Lmma 1, th EA will not hav incntivs to inspct any rport with a probability highr than t/θ, sinc monitoring is costly. Thrfor, t/θ is an uppr bound for th optimal monitoring probability. Proposition 1 Givn th monitoring policy α( ), ifthrport z() minimizs firm s costs whn th missions lvl is, thn: α(z()) is nonincrasing in, and (1) c(; α( )) = c(; α( )) + θ Z α(z(x))dx. (2) Morovr, if (1) and (2) hold, thn z() minimizs firm s xpctd costs ovr th st of all possibl quilibrium rports, i.., {z z = z( o ) for som o [, ]} whn th missions lvl is. W now xplain th main insights of Proposition 1, which is a classic rsult in continuoustyp advrs slction modls. For any givn rport, th pnalty that th firm pays if it is caught undrrporting incrass with its ralizd pollution lvl. Thrfor, th highr thmissionlvl,th morincntivs thfirm has to chos rports with low monitoring probability. This xplains that α(z()) is nonincrasing in. As to th xpctd costs, quation (2) stats that th cost born by th firm whn its missions ar is th intgral of th monitoring probability of vry lvl blow. This quation is also xplaind by 8

10 th firm s possibility of undrrporting. By inspcting with probability α(z(x)), thea maks th firmpayanxpctdpnaltyofθα(z(x)) whn its mission lvl is x. Butthis similarly affcts th firm s xpctd costs whn it missions ar highr than x, sincz(x) is always a possibl rport for this firm. Hnc, quation (2) provids th xpctd cost born by th firm whn its mission lvl is. Not that, although th tax rat t dos not xplicitly appar in quation (2), it plays a rol as it sts th uppr bound for th probability α(.). Th rat t is only important for thos mission lvls for which th firm rports honstly. For xampl, if th rport z() is such that α(z()) = t/θ for all ê and α(z()) <t/θothrwis, thn w can writ c(; α( )) = c(; α( )) + t [ê ]+θ Z ê α(z(x))dx. W can us Proposition 1 to comput firm s xpctd costs of using th tchnology E: Proposition 2 Givn th monitoring policy α( ), ifthrportstratgy z( ) minimizs firm s costs for all missions lvls, thn: C(E; α( )) = c(; α( )) + θ Z α(z()) [1 F (; E)] d. (3) In this sction, w hav analyzd th firm s stratgic bhavior concrning its rport, onc it knows th pollution lvl. W hav computd th firm s xpctd cost du to th nvironmntal policy of taxs, inspction, and pnaltis. W hav dvlopd th analysis for an xognous monitoring policy. In th nxt sction, w charactriz th optimal monitoring policy from th EA s point of viw. 3 Two production tchnologis W analyz a situation whr two production tchnologis ar possibl: E D and E C. Tchnology E C is a clanr but also mor xpnsiv tchnology than E D (subscript C stands for clan and D for dirty ). W assum that th firm is initially producing according to E D and w dnot by th cost of switching from th dirty tchnology to 9

11 th clan on. 5 On th othr hand, th clan tchnology has lowr avrag missions, i.., R df (; EC ) < R df (; ED ). 6 Givn th policy announcd by th Govrnmnt and th EA involving taxs ovr rportd missions, monitoring, and pnaltis ovr unrportd missions, th firm will choos th clan tchnology if and only if its total xpctd costs ar lowr than using th dirty tchnology, that is, if C(E C ; α( )) + C(E D ; α( )). This inquality can b writtn as th following incntiv constraint: θ Z α(z()) F (; E C ) F (; E D ) d. (4) It might b th cas that th firm chooss tchnology E D for any possibl monitoring stratgy. Indd, if th diffrnc in cost is vry larg, th firm may prfr paying all th xpctd taxs corrsponding to th missions inducd by E D rathr than adopting th clan tchnology. In what follows, w will assum that th st of functions α( ) that lad th firm to choos E C is not mpty, which is quivalnt to stat that th toughst policy (α(z) =t/θ for all z) ladsthfirm to us th clan tchnology. Assumption 1: <t R F (; E C ) F (; E D ) d. Although part of th analysis of th optimal policy is dvlopd without any additional assumption concrning th distribution functions F (; E C ) and F (; E D ), th complt charactrization of th policis will rquir furthr assumptions. In particular, w will assum that th dnsity functions f(; E C ) and f(; E D ) ar linar. Also, to hlp notation, w will normaliz [, ] =[0, 1]. Assumption 2: f(; E C )=a +2[1 a], f(; E D )=b +2[1 b], for all [0, 1], whr a, b (0, 2), and a>b. 5 W can also considr situations whr th firm is not using any of th two tchnologis and it has to chos on of thm. In this cas, is intrprtd as th diffrnc in costs of th tchnologis, i.., th cost to adopt th formr instad of th latr. 6 In our framwork, th missions from both tchnologis ar qually difficult to inspct. Som authors hav analyzd tchnologis that can affct th obsrvability of firms missions. Hys (1993) considrs amodlwhrfirms may invst in dcrasing inspctability. Millock t al. (2002) studis a choic of tchnology that affcts th vrifiability of mission: adopting th tchnology allows nonpoint sourcs to bcom point sourcs. 10

12 Not that th proprty F (1; E C )=F(1; E D )=1charactrizs th slop of th linar functions f(1; E C ) and f(1; E D ), onc w choos th indpndnt trms a and b. Morovr, th ida that E C is a clanr tchnology than E D is rflctd in th inquality a>b.also not that although Assumption 2 is rstrictiv, it allows th flxibility of daling with distribution functions F (; E C ) and F (; E D ) that may b linar (a =1or b =1)concav (a >1 or b>1), or convx (a <1 or b<1). On th othr hand, it is a strong assumption that is hlpful to idntify a simpl monitoring policy. W will commnt latr on th proprtis of th optimal monitoring policy in mor gnral stups. Th monitoring policy dcidd by th EA strongly influncs th choic btwn E C and E D. In th rmaining of th papr, w normaliz th cost of an inspction to 1, and w look for th optimal monitoring policy. 4 A rfrnc framwork: optimal monitoring whn th cost is public information In this sction, w charactriz th optimal monitoring policy if th EA wants a firm to adopt tchnology E C. W considr a situation whr th EA obsrvs th cost, butis uninformd about th tchnology that th firm adopts and about th ralizd mission lvl. This informational st-up may not b ralistic in most scnarios, but th analysis will b a rfrnc to th two othr cass, studid in sctions 5 and 6. Th EA rcivs th rport z from th firm. Th optimization problm of th EA, that minimizs monitoring costs, is program [P ] blow: Min (α(z)) z [,] Z α (z()) df (; E C ) s.t.: α(z()) is nonincrasing in α(z()) [0,t/θ] for all [, ] z() minimizs c(, z; α( )) for all [, ] θ Z α(z()) F (; E C ) F (; E D ) d. W can simplify program [P ] as follows. W do not tak into account th constraint 11

13 that z() minimizs c(, z; E C ; α( )), and w dnot th function α(z()) as β(). Onc w idntify β(), wwillusproposition1todcomposthfunctionβ() into th optimal monitoring function α(z) and th rport function z(). Th optimal β( ) solvs th following program, that w will dnot [P 0 ]: θ Min (β()) [,] Z β()df (; E C ) s.t.: β() is nonincrasing in β() [0,t/θ] for all [, ] Z β() F (; E C ) F (; E D ) d. Nxt Proposition stats an important gnral proprty of th solution to program [P 0 ]: Proposition 3 Undr Assumption 1 and for any distribution function F (.), thrxists asolutionβ( ) to [P 0 ] that taks on at most on valu diffrnt from 0 and t/θ. Givn Proposition 3 and β() nonincrasing in, thrxistγ (0,t/θ), 1 and 2, with 1 2, such that th optimal function β() has th following shap: β() = t/θ for all [, 1 ], β() = γ for all ( 1, 2 ), β() = 0 for all [ 2, ]. Proposition 3 shows that th optimal monitoring policy is vry simpl indpndntly on th shap of th distribution functions. Proposition 4 gos a stp forward and shows that, undr Assumptions 1 and 2, th optimal policy is vn simplr. To stat this Proposition, lt us dfin th function h() as follows: h() f(; E C F (; E C ) F (; E D ) ) R [F (x; EC ) F (x; E D )] dx F (; EC ). Th function h() plays an important rol in th proof of Proposition 4, and allows us to dfin a cut-off lvl. It is asy to chck that, undr Assumption 2, h() is first ngativ and thn positiv. W dnot by th cut-off lvl such that h() < 0 if < and h() > 0 if >, that is, is dfind by h( )=0. Notthatthcut-off lvl only 12

14 dpnds on th shap of th distribution functions, and in particular is indpndnt of th cost. Proposition 4 Suppos Assumptions 1 and 2 hold. (a) If <t R F (; E C ) F (; E D ) d, thn a solution β() to [P 0 ] is β() = bγ for all [, ), β() = 0 for all [, ], whr bγ <t/θis dfind by: Z bγθ F (; E C ) F (; E D ) d =. (5) (b) If t R F (; E C ) F (; E D ) d, thn a solution β() to [P 0 ] is whr b is dfind by: t β() = t/θ for all [, b), β() = 0 for all [b, ], Z F (; E C ) F (; E D ) d =. Th optimal monitoring policy is vry simpl. W hr highlight its main charactristics. First, for any cost, th EA will always monitor, at last, th rports corrsponding to all th mission lvls lowr than th cut-off valu. Not that th cut-off is usually high; undr assumption 2 for th intrmdiat cas a =1,whav =3/4. 7 Scond, th probability of monitoring is th sam for all th rports subjct to audit. Third, as long as th incntiv problm is not vry acut, in th sns that adopting th clan tchnology is not vry costly, th EA will only monitor whn th ralizd mission lvl is lowr than. Th probability of audit bγ is incrasing with until it rachs th maximum valu t/θ. Thisdfins th bordrlin btwn cass (a) and (b). Fromthnon,thEA audits additional rports. That is why, whn th incntiv problm is vry svr, th 7 It can b shown that = 2a+ 4a 2 +6a(1 a) 2(1 a) function of a. (0, 1) whn a 6= 1, and that is an incrasing 13

15 monitoring probability is th highst possibl, among th snsibl ons, (i.., β = t/θ) for all th rports subjct to audit. Onc w know th optimal function β(), w can us Proposition 1 to stat th optimal monitoring policy as a function of th rport, α(z), aswllasfirms rporting bhavior givn th optimal monitoring policy, z(). Proposition 5 charactrizs ths functions. Proposition 5 Suppos Assumptions 1 and 2 hold. (a) If <t R F (; E C ) F (; E D ) d, thn th following policy α (z) is optimal: α (z) = bγ for all z [,z ), α (z) = 0 for all z [z, ],whr z = + ( ) R t [F (; E C ) F (; E D )] d. Facing th monitoring policy α (z), thfirm s rporting stratgy is th following: z() = for all [, ), z() = z for all [, ]. (b) If t R F (; E C ) F (; E D ) d, thn th following policy α (z) is optimal: α (z) = t/θ for all z [, b), α (z) = 0 for all z [b, ]. Facing th monitoring policy α (z), thfirm s rporting stratgy is th following: z() = for all [, b), z() = b for all [b, ]. W now xplain th intuitions bhind Propositions 4 and 5. Th EA s objctiv is to dissuad th firm from using th dirty tchnology at th lowst (monitoring) cost. To convinc th firm, th EA must choos a monitoring stratgy that maks th firm bar high xpctd nvironmntal costs (also taking into account th pnaltis) if it uss th dirty tchnology, and low xpctd costs if it producs according to th clan on. 14

16 A dirty tchnology has a highr probability to produchighmissionlvlsthanaclan tchnology. For th cas of linar dnsity functions ovr th intrval [0, 1] (Assumption 2), th clan tchnology has highr dnsity for [0, 1/2) and lowr dnsity for (1/2, 1]. Thrfor, in trms of dissuasion, th EA would find it bnficial to mak th firm pay as much as possibl (and that can b achivd by monitoring with high probability) whn ralizd missions ar high and as littl as possibl whn ralizd missions ar low. Howvr, th EA dos not obsrv th ralizd mission lvl, it only rcivs th firm s rport. Whn missions ar not public information, quation (2) in Proposition 1 stats that th cost born by th firm whn th mission lvl is o is th intgral of th monitoring probability of vry lvl blow o. That is, incrasing th probability of monitoring th rport corrsponding to a lvl affcts in th sam way th cost suffrd for vry mission lvl highr than. Hnc, monitoring th rport corrsponding to a high mission lvl, say 0 > 1/2, has good incntiv consquncs concrning th dcision to us a clan tchnology, as it affcts th cost born for vry ralizd mission 0. On th othr hand, monitoring th rport corrsponding to a low mission lvl, say 00 < 1/2,hasmixd incntiv consquncs sinc it affcts th cost associatd to both high (vry >1/2) and low (vry [ 0, 1/2)) mission lvls. Th difficulty is that, from quation (1) in Proposition 1, th EA is constraint to us a monitoring probability nonincrasing in th mission lvl. That is, if th EA wants to monitor th (firm s optimal) rport corrsponding to a crtain lvl of missions o,thn it is forcd to monitor th rports corrsponding to all th lvls < o with, at last, th sam frquncy. To undrstand how th EA solvs th prvious difficulty, considr also that is small in such a way that inducing th firm to switch to th clan tchnology is asy (Rgion (a) in Proposition 4). Could it mak sns for th EA to monitor only th rports corrsponding to low mission lvls? Th answr is no. Th EA dos bttr monitoring rports chosn by a largr rang of mission lvls (including lvls highr than 1/2) with lowr probability. Th cost paid by th highr mission lvls will b th sam, whil th cost born by th lowr mission lvls will b lowr, which givs th firm mor incntivs to adopt th clan tchnology. Is it optimal for th EA to st a full flat 15

17 policy (i.., = )? Th answr to this qustion is also ngativ bcaus monitoring th rport corrsponding to mission lvls vry clos to only affcts th paymnt of a vry small intrval of missions. Inthcaswhrthdnsityfunctionf(; E C ) is uniform, i.., a =1,thtrad-off lads to a flat policy consisting in auditing th rports corrsponding to vry <3/4 = with th sam probability. Whn f(; E C ) is not uniform, th argumnt is mor complx, as switching monitoring probabilitis from on lvl to th othr has consquncs in trms of monitoring costs. This is why whn th distribution function f(; E C ) is dcrasing, it is optimal to stat an vn flattr tchnology ( > 3/4), whil th opposit happns whn f(; E C ) is incrasing. 8 Th prvious discussion also allows to commnt on th gnrality of th rsults with rspct to th shap of th distribution functions. First, according to our argumnts, monitoring vry singl mission with som probability (i.., 2 = ) is not optimal for gnral distribution functions. Scond, th proprty that th monitoring policy is flat for quit a wid rang of missions can b statd undr quit rasonabl hypothss. For xampl, assum that F (; E C ) >F(; E D ) for all (, ), F (; E C ) F (; E D ) is first incrasing and thn dcrasing in, andf (; E C ) is concav in. Undr ths ncssary conditions, it is possibl to prov that thr xists a cut-off valu # that lis in th rgion of missions whr F (; E C ) F (; E D ) is dcrasing such that β() is constant for all < #. In particular, th rports corrsponding to all mission lvls < # ar 8 It is worth comparing our contxt with situations in which th objctiv of th agncy is to rais th largst amount of taxs, for a givn tchnology. In such latst situations, th agncy is much lss intrstd in focusing in high-mission lvls. For xampl, in th tax vasion litratur it is assumd that th distribution of incom is givn (i.., thr is no choic of tchnology to arn incom ) and th objctiv of th nforcmnt agncy is to maximiz th collctd rvnus (taxs plus pnaltis). In this cas, th optimal policy consists in auditing all th taxpayrs rporting incoms lowr than a crtain cut-off incom with a probability high nough so that thos rports will happn to b truthful, whil th taxpayrs arning highr incoms will rport th cut-off incom and will not b subjt to audit. Th main intuition for this rsult is similar to th on w hav providd in th main txt: putting prssur ovr th rport corrsponding to an mission lvl incrass th rvnu collctd from vry highr lvl. That is, it is bnficial to concntrat th monitoring in th lowst lvls of incom (with th maximum probability t/θ). Som paprs in th tax vasion litratur ar Ringanum and Wild (1985), Scotchmr (1987), Sánchz and Sobl (1993), and Macho-Stadlr and Pérz-Castrillo (1997). 16

18 monitord with a low probability whn th cost of adopting th clan tchnology is low. On th othr hand, it sms mor difficult to propos gnral ncssary conditions to stablish th prcis form of th optimal monitoring stratgy for highr mission lvls. Although w know that th highst lvls ar nvr monitord, it is difficult to prov mor gnral rsults. Nxt, Corollary 1 stats th monitoring cost ECost( ) of th implmntation of th clan tchnology as a function of th paramtrs of th modl. Corollary 1 Suppos Assumptions 1 and 2 hold. (I) Expctd monitoring costs ECost ar th following: (Ia) If <t R F (; E C ) F (; E D ) d, thn: ECost( ) = θ F ( ; E C ) R [F (; E C ) F (; E D )] d. (Ib) If t R F (; E C ) F (; E D ) d, thn: ECost( ) = t θ F (b; EC ). (II) Expctd monitoring costs ar incrasing in th diffrnc and thy ar dcrasing with th pnalty rat θ; thy ar highr th lss clan is tchnology E C and th lss dirty is tchnology E D. Finally, xpctd costs ar incrasing in th ratio t/θ in Rgion (b). W now xplain th comparativ statics in Corollary 1. First, th highr th cost for th firm to switch to th clan tchnology, th highr th monitoring cost rquird to giv it incntivs to adopt E C. W can asily chck that: ECost = f( 2 ; E C ) θ [F ( 2 ; E C ) F ( 2 ; E D )], whr 2 = in Rgion (a) and 2 = b in Rgion (b). Scond, a highr pnalty rat θ maks it asir to convinc th firm, hnc it dcrass th EA s cost. Third, th largr (in trms of xpctd missions) th diffrnc btwn th two tchnologis, th mor th EA s monitoring can targt th dirty tchnology, which also dcrass monitoring costs. Finally, an incras in th tax rat t forcs th EA to incras th monitoring probability if it wants th firm to b honst whn th lvl of pollution is low (which is th optimal 17

19 policy in Rgion (b)). Thrfor, th monitoring costs incras with t. That is, a though policy in trms of pnalty rat and (in Rgion (b)) a soft policy in trms of tax rat hlp in kping low monitoring costs. 9 5 Optimal monitoring whn th tchnology adoptd by th firmsisvrifiabl but th cost is not In this sction, w study th nvironmnts whr th EA can asily vrify th tchnology adoptd by ach firm. Howvr,itdosnotknowthadoptioncosts. W modl this situation as follows. Th EA facs a family of firms charactrizd by th cost paramtr. Each firm knows its paramtr. Th EA dos not know th particular cost of a firm, but it knows that th paramtr is distributd in th family of firms according to th dnsity function g( ) ovr th intrval 0, ;wdnotbyg( ) th distribution function of. 10 Th framwork considrd in this sction is a plausibl on whn th tchnologis rprsnt diffrnt typs of physical capital. It may thn b asy to chck whthr a firm has indd adoptd a givn missions-rducting tchnology, but it is not asy for th EA to assss ach firm s cost of th adoption. W can hav in mind a st of firms or industris that rly on intrnal combustion ngins for production. Som of ths industris may b transportation, som manufacturing, som gas, disl, coal. All mit carbon. All can purchas an abatmnt tchnology, but th cost of th tchnology is unknown by th EA and can diffr across industris. Th EA cars about total pollution, hnc its concrn is whthr th firms choos th clan or th dirty tchnology. It wights th bnfits of th xpctd rduction of th 9 Th comparativ statics with rspct to th tax rat t mustbtaknwithcaution. Vryoftn,th pnalty rat is proportional to th tax rat, say θ =(1+π)t. In this cas, an incras in t dcrass xpctd costs in rgion (a) and an incras in π dcrass xpctd costs in both rgions. 10 W can also s th analysis dvlopd in this and nxt sction as th study of th optimal monitoring policy whn th EA monitors only on firm whos paramtr is unknown and distributd according to th function G( ). Nxt propositions and corollaris hav an immdiat intrprtation in this altrnativ contxt. 18

20 lvl of missions against th monitoring costs and th firms cost to implmnt th clan tchnology. Givn that th EA is not concrnd about th nvironmntal taxs raisd, th optimal policy in this cas involvs not monitoring at all a firm that dcids to switch to E C. Thrfor, a firm can buy immunity from nvironmntal taxs by adopting th clan tchnology. This is th first charactristic of th optimal policy. Scond, inspction of th incntiv compatibility constraint (4) maks it clar that incntivs to switch to th clan tchnology ar strictly dcrasing with th switching cost. That is, for a givn monitoring policy, if a firm with paramtr adopts th clan tchnology, a firm with paramtr 0 < will also adopt it. Thrfor, any policy α(.) will induc a firm to adopt E C if its paramtr lis in an intrval [0, v ],forsom v 0,. 11 What is th optimal monitoring policy for th firm whn it adopts E D? It nds to giv incntivs for th firm to switch to E C vn whn its costs ar v and th distribution of missions of thos firms that ar monitord is F (; E D ). Thrfor: Proposition 6 Suppos th firms cost paramtr is distributd according to G( ), it is firms privat information, th EA can obsrv th tchnology choic, and assumptions 1 and 2 hold. Thn, th optimal policy whn th EA wants that firms with [0, v ] adopt E C is: (i) A firm that adopts E C is not monitord. (ii) A firm that adopts E D is auditd according to th policy found in Proposition 5 for a firm with adoption costs qual to v. From Proposition 6, w s that th monitoring policy will only b applid to firms that us E D, which happns whn thir paramtr li in th intrval v,.morovr,th policyapplidisthonthatwouldboptimaliftheawouldfacafirm with known adoption cost of v. As w alrady dscribd in Sction 4, th optimal monitoring policy is a simpl cut-off policy: rports lowr than a crtain thrshold ar inspctd with a constant probability whil any firm can avoid inspction by rporting that thrshold mission lvl. 11 Th lttr v in v stands for (tchnology adoption) vrifiabl. In nxt sction, th adoption is supposd non vrifiabl and w will us n. 19

21 How is th optimal v dcidd? If a firm s cost was public information (and th firm s tchnology vrifiabl),th Govrnmnt(orthEA)wouldwightbnfits of adopting tchnology E C du to th rduction in pollution against costs of adoption,. This balanc would dtrmin th optimal blow which a firm in th population should (from a social point of viw) adopt E C. Whn firms hav privat information about, thn th Govrnmnt also taks into account th monitoring cost ndd to induc thm to switch. On natural form for th Govrnmnt s wlfar function is: B(G( v )) ECost v ([0, v ]) κ Z v 0 g( )d, whr B(G( v ) is an incrasing and concav function masuring th bnfits du to th firms adoption of E C whnthcostislowrthan v, ECost v ([0, v ]) is th xpctd monitoring cost to achiv firms adoption of E C for adoption costs in [0, v ],andκ (that is oftn considrd to b qual to 1) isthwightth EAgivstofirms profits. Th xpctd monitoring costs ECost v ([0, v ]) whn th tchnology usd by th firms is vrifiabl, ar: ECost v ([0, v ]) = [1 G( v )] Z bγf(; E D )=[1 G( v v F ( ; E D ) )] θ R [F (; E C ) F (; E D )] d, whn th paramtrs li in Rgion (a) of Proposition 5, i.., <t R F (; E C ) F (; E D ) d. In Rgion (b): ECost v ([0, v ]) = [1 G( v )] tf (b; ED ). θ Considr Rgion (a) (th qualitativ proprtis in Rgion (b) ar similar). It is immdiat that: ECost v ([0, v ]) v = [[1 G( v )] g( v ) v ] F ( ; E D ) θ R [F (; E C ) F (; E D )] d. An incras in v has two ffcts on th monitoring costs. On th on hand, for firms with a highr switching cost to adopt E C, th monitoring probability must incras to convinc thos firms to adopt th clan tchnology. On th othr hand, th population of firms that ar monitord is smallr, as mor firms switch to E C. Thatis,thrisan ffct (th positiv trm in th prvious quation) that maks th monitoring cost incras, whil anothr ffct (th ngativ trm) gos in th sns of dcrasing monitoring costs. 20

22 Corollary 2 highlights th main implication of th prvious discussion: thr ar nvironmnts whr thr is too much adoption of clan tchnology compard with th first-bst situation. Corollary 2 Suppos th cost paramtr is th firm s own privat information and that th adoption of th tchnology is vrifiabl. Thn, th optimal monitoring policy inducs th firm to adopt tchnology E C for an intrval of paramtrs [0, v ] that may b largr or shortr than th first-bst intrval [0, ]. Typically, w should xpct too much adoption of th clan tchnology, as compard to th first bst situation, prcisly in thos nvironmnts whr th first bst rquirs adoption for a larg rang of paramtrs ( is high), whil th informational should caustoolittladoptionwhn is low. For xampl, if g( ) is uniform, thn too many firms switch to th clan tchnology whn > /2. 12 Th rason for this rsult is th following. On th on hand, if th EA dcids to incras adoption from to + d, it savs on monitoring costs bcaus a total of g( )d firms switch to E C and thy do not nd to b monitord any longr. On th othr hand, th incras in th cost du to using a marginally toughr monitoring dpnds on th amount of firms that still chos E D,whichisqualto(1 G( )). Thrfor, th largr thmorliklyitisthatit pays th EA to (marginally) induc mor firm in th population to switch to E C. For xampl, considr firms dcision whthr to adopt rnwabl nrgy procsss (burning biomass) instad of procsss basd on fossil nrgy. Th adoption of ithr procss is asy to chck, whil th actual xtra cost du to switching to rnwabl nrgy us may b difficult to asss by th EA. To giv th firms incntivs to adopt clan procsss, th EA will monitor th pollution of fossil nrgy plants. Will th optimal monitoring policy lad to too many or too fw rnwabl plants? On th on hand, th cost of th monitoring should imply a lowr-than-optimal firms ffort, that is, too littl adoption of th clan plants. Howvr, on th othr hand, monitoring is only applid to thos firms that still us fossil nrgy. This givs th EA an xtra motivation to monitor, as toughr monitoring maks th numbr of monitord firms dcras. As th 12 In this discussion, w ar implicitly assuming that th scond-ordr condition for th EA s objctiv function with rspct to is concav, which happns, for xampl, if B() is concav nough. 21

23 prvious corollary shows, th optimal policy may imply ovrswitching or undrswitching to rnwabl nrgy procsss. 6 Optimal policy whn both th tchnology adoptd by th firms and th cost ar not obsrvabl by th EA W now addrss th EA s optimal policy whn both th cost of adopting th clan tchnology and th tchnology usd by a firm ar firms privat information. Th EA s policy is anonymous, i.., vry typ of firm is subjct to th sam monitoring policy. This nvironmnt corrsponds to situations whr tchnologis rprsnt diffrnt lvls of car, or ffort, xrcisd by th firm, for xampl in trms of protocols or th organization of intrnal activitis. In othr words, clan or dirty rfr to th car that firms tak with rspct to th maintnanc of th xisting tchnology or to avoiding mistaks. W intrprt that a firm uss a clan tchnology whn it dvots (montary and human) rsourcs to th good functioning of its quipmnt, whil a firm producs according to a dirty tchnology whn it dos not car much about th corrct running of th quipmnt, thus lading to highr xpctd lvl of missions. For similar rasons as in th prvious sction, for any givn monitoring policy (that will only b applid to th firm if it kps E D )thfirm adopts E C if its paramtr lis in an intrval [0, n ]. Nxt proposition charactrizs th policy that minimizs monitoring costs whn th EA wants all firms with in th intrval [0, n ] to switch to E C. Th policy is qualitativ thsamasthonstatdinproposition 5, althoughthcut-off lvls ar diffrnt. Th prcis valu for th paramtrs n,z n, b n, and bγ n that appar in Proposition 7 ar givn in th Appndix. Thy do not corrspond to th optimal cut-off lvls whnvr th EA would lik to giv incntivs to switch tchnology to a firm with paramtr n. That is, th homognous monitoring policy dos not coincid with th optimal policy for th marginal firm n. It would corrspond to a firm with adoption costs of n,whos incntivs ar givn by th diffrnc btwn th distribution functions F (; E C ) and 22

24 F (; E D ), but whos actual missions ar givn by th (avrag) distribution function G( n )F (; E C )+[1 G( n )] F (; E D ) instad of F (; E C ). Proposition 7 Suppos th firms cost paramtr is distributd according to G( ), itis firm s privat information, th EA cannot obsrv th tchnology choic, and assumptions 1 and 2 hold. Thn, th optimal policy whn th EA wants firms with [0, n ] to adopt E C is: (a) If n <t R n F (; E C ) F (; E D ) d, thn th following policy α n (z) is optimal: α n (z) = bγ n for all z [,z n ), α n (z) = 0 for all z [z n, ]. (b) If n t R n F (; E C ) F (; E D ) d, thn th following policy α n (z) is optimal: α n (z) = t/θ for all z [, b n ), α n (z) = 0 for all z [b n, ]. Th policy α n (z) statd in Proposition 7 rquirs monitoring all rports blow a cut-off valu ( n or b n dpnding on th rgion) with th sam probability, that is, a larg rang of (low) rports ar monitord with a uniform probability, whil high rports ar nvr monitord. Th discussion aftr Propositions 4 and 5 provids th main intuitions bhind th optimality of th policy proposd in Proposition 7. Th xpctd monitoring cost of th policy α n (z) dpnds on th intrval [0, n ] of typs of firms that th EA wants to adopt E C. Th largr th intrval, i.., th highr n, th highr th xpctd cost ECost n ([0, n ]) whn th adoption of th tchnology is not obsrvabl. Using th nvlop thorm in program [P M ] in th proof of Proposition 7, w can dduc that: 13 ECost n ([0, n ]) = γg( n ) F ( n 2 ; E C ) F ( 2 ; E D ) G( n )f( 2 ; E C )+[1 G( n )] f( 2 ; E D ) +, (6) [F ( 2 ; E C ) F ( 2 ; E D )] θ whr 2 = n and γ = bγ n in Rgion (a) and 2 = b n and γ = t/θ in Rgion (b). As it was th cas in th prvious sction, an incras in th cut-off lvl n has two ffcts on 13 Th optimal solution of program [P M ] always involvs 1 =. 23

25 th monitoring costs. First, to induc firms with a highr switching cost to adopt E C, a highr monitoring probability is ncssary. This affcts a firm indpndntly on its typ and is rflctd in th scond trm in th right-hand sid of (6). Scond, thr ar typs of firms that wr kping E D bfor th incras in th cut-off and ar adopting E C aftr thchang.firmsusinge C ar monitord mor oftn (although thir xpctd paymnt is lowr) than if thy kp E D (this is du to th proprty that th monitoring probability should b non-dcrasing in ralizd mission, s Proposition 1). Both ffcts go in th sam dirction: inducing mor firms to adopt E C incrass th monitoring costs. Givn that ECost n ([0, n ]) is incrasing in n, it is immdiat that th optimal dcision in this cas will involv a lvl n <, that is, th xpctd lvl of pollution will b highr than th first-bst lvl of pollution: Corollary 3 Suppos th cost paramtr and th tchnology adoptd ar th firms privat information. Thn, th optimal monitoring policy inducs firms to adopt tchnology E C for an intrval of paramtrs [0, n ] that is smallr than th first-bst intrval [0, ]. 7 Conclusion W hav considrd a situation whr th nvironmntal policy is basd on taxs ovr rportd missions, monitoring, and pnaltis. W hav assumd that th EA facs a population of firms. Firms missions dpnd on a dcision (adopting th clan or th dirty tchnology) and random vnts. In addition ach firm has privat information concrning its ralizd mission lvl. W analyz th optimal monitoring stratgy for th rgulator and which firms in th population ar inducd to adopt th clan tchnology. Th addd valu of our papr lis in th charactrization of this monitoring policy whn th firms cannot fully control thir missions, thy just dcid its distribution. This random charactristic is not prsnt in prvious paprs considring optimal auditing. W hav dvlopd th analysis in two diffrnt scnarios dpnding on whthr th tchnology adoptd by th firm is vrifiabl or not. In both cass, th optimal policy is a cut-off policy, whr all rports blow a thrshold ar inspctd with th sam probability, whil rports abov th thrshold ar not monitord. W hav also shown that if th adoption of th 24

26 tchnology is firms privat information, too fw firms will adopt th clan tchnology undr th optimal monitoring policy. Howvr, whn th EA can chck th tchnology adoptd by th firms, th optimal policy may induc ovrswitching or undrswitching to th clan tchnology. In this papr, w hav assumd that th nvironmntal policy is basd on taxs ovr rportd missions, monitoring, and pnaltis. W hav not considrd th possibility thatthgovrnmntortheamightgivafirm a subsidy if it switchs to th clan tchnology,orthatitimpossafixd pnalty to firms kping th dirty tchnology. Whn th tchnology adoptd by th firm is not vrifiabl (i.., only th firm knows th xpctd lvl of pollution of th tchnologis), th prvious policis basd on subsidis or pnaltis cannot b implmntd, as thy rquir th EA to b abl to chck whthr a chang to a clan tchnology has takn plac. On th othr hand, whn th EA can asily chck whthr a firm has adoptd a mor nvironmntally frindly tchnology (or whthr it is using th tchnology trying to minimiz missions), a fixdrwardorpnaltycan b optimal. Thrfor, our analysis applis to thos situations whr, du to political, tchnical, or moral hazard constraints, a policy basd on fixd subsidis or pnaltis is not possibl. 8 Appndix ProofofLmma1. First, rporting mor than th tru missions is nvr optimal, sinc th xpctd paymnt is always highr. Scond, if >zand α(z) >t/θ,thn c(, z; α( )) = tz + α(z)θ[ z] >tz+ t[ z], which is th paymnt th firm would mak if it would rport. Thrfor, rporting z is not optimal. Finally, by similar rasons, rporting is not optimal whn α(z) <t/θfor som z [,). ProofofProposition1. Considr two missions lvls 1 and 2 with 1 > 2 and th optimal rports corrsponding to ths lvls, z( 1 ) and z( 2 ). Givn that th firm prfrs rporting z( 1 ) than z( 2 ) whn th missions lvl is 1, and vicvrsa, w hav: c( 1 ; α( )) = tz( 1 )+α(z( 1 ))θ[ 1 z( 1 )] tz( 2 )+α(z( 2 ))θ[ 1 z( 2 )], c( 2 ; α( )) = tz( 2 )+α(z( 2 ))θ[ 2 z( 2 )] tz( 1 )+α(z( 1 ))θ[ 2 z( 1 )]. 25

27 Ths quations imply: α(z( 1 ))θ[ 1 2 ] c( 1 ; α( )) c( 2 ; α( )) α(z( 2 ))θ[ 1 2 ]. (7) First, sinc 1 2 > 0, (7) rquirs that α(z( 1 )) α(z( 2 )), i.., α(z()) is nonincrasing in. Scond, α(z()) nonincrasing and (7) imply that c(; α( )) is diffrntiabl in almost vrywhr, with dc(; α( )) d = α(z())θ almost vrywhr. Equation (2) immdiatly follows. Finally, assum (1) and (2) hold. Thn, a firm with missions lvl rporting z( o ) has a xpctd cost of: tz( o )+α(z( o ))θ[ z( o )] = c( o ; α( )) + α(z( o ))θ[ o ]= c(; α( ))+θ Z o α(z(x))dx+α(z( o ))θ[ o ]=c(; α( ))+θ Z o Givn (1), R o [α(z(x)) α(z( o ))] dx 0. Thrfor, z() is optimal in {z z = z( o ) for som o [, ]}. ProofofProposition2. According to quation (2): Z Z C(E; α( )) = c(; α( ))df (; E) =c(; α( )) + θ Z [α(z(x)) α(z( o ))] dx. α(z(x))dx df (; E). Intgrating by parts, w obtain: Z Z α(z(x))dx df (; E) = = Z Z = α(z(x))dx F (; E) α(z(x))dx Z = Z α(z())f (; E)d. α(z())f (; E)d Equation (3) immdiatly follows. ProofofProposition3. Considr a solution β ( ) to program [P 0 ] and B th 26

28 optimal budgt. W claim that β ( ) is also th solution to th program [P 00 ] blow: Max (β()) [,] Z β() F (; E C ) F (; E D ) d s.t.: β() is nonincrasing in β() [0,t/θ] for all [, ] Z β()df (; E C ) B. Indd, if a function β 0 ( ) would xist involving a highr valu for th solution, β ( ) would not b th solution to [P 0 ]: th EA could us β 00 ( ) that coincids with β 0 ( ) until th lowst missions lvl o that satisfis = θ Z o β 0 () F (; E C ) F (; E D ) d and β 00 () =0for all > o. This policy would b chapr than β 0 ( ), hnc it would cost lss than B, which is not possibl. W can now us known rsults (s, for xampl, Stp 4 in th proof of Proposition 1 in Sánchz and Sobl, 1991) to stat that thr xists a solution to [P 00 ] that taks on at most on valu diffrnt from 0 and t/θ. ProofofProposition4. According to Proposition 3, wcanrwrit[p 0 ] as [P 00 ] : ½ t Min (γ, 1, 2 ) θ F 1 ; E C + γ F 2 ; E C F 1 ; E C ¾ s.t.: θ = t Z 1 F (; E C ) F (; E D ) Z 2 d + γ F (; E C ) F (; E D ) d. (8) θ 1 W start by proving som claims. Claim 1:W can rstrict attntion to policis whr 2 <. To prov Claim 1, considr th st of policis charactrizd by ( 1, 2,γ), with 1 <. W do th analysis fixing th lvl of 1.Thparamtrγ is givn by (8), that is, 1 γ = R 2 [F (; E C ) F (; E D )] d θ t Z 1 F (; E C ) F (; E D ) d. θ 1 Thrfor, th cost of th policy as a function of 2 is givn by th function m( 2 ): m( 2 ) t θ F ( 1; E C )+A F ( 2; E C ) F ( 1 ; E C ) R 2 1 [F (; E C ) F (; E D )] d, 27

29 whr A is a positiv constant that dos not dpnd on 2 (it is th scond factor in th xprssion for γ). m 0 ( 2 = ) is proportional to f( 2 ; E C ) R 2 1 F (; E C ) F (; E D ) d. Hnc, m 0 ( 2 = ) > 0 givn Assumption 2. This implis that, at th optimum, it is always th cas that th cost is minimizd for a valu of 2 lowr than. Claim 2:A policy such that 1 = 2 < is not optimal. W considr th policis of th form β() =γ for all [, ) and β() =0for all [, ], for which (8) holds. In this class of policis, w considr a marginal chang in, accompanid by th corrsponding chang in γ so that (8) still holds, i.., γ = γ F (; E C ) F (; E D ) R [F (; EC ) F (; E D )] d. Th cost of any policy in this class is γf(; E C ). Hnc, th chang in cost du to th proposd marginal chang is F (; E C ) γ + γf(; E C ) = h()γ. By Assumption 2, h() < 0 givn that <. Thrfor, a marginal incras in would rduc th cost. Consquntly, a policy with γ = t/θ (i.., 1 = 2 ) cannot b optimal sinc thr is room to incras and dcras γ in a profitabl way, which provs Claim 2. Claim 3:A policy such that 1 < 2 is not optimal whn 1 <. W follow a similar stratgy of proof as in Claim 2. Considr th class of policis of th form β() =γ 0 for all [, 1 ),β() =γ for all [ 1, 2 ),andβ() =0for all [ 2, ], withγ 0 >γ, for which quation (8) holds (whr w substitut t/θ by γ 0 ). W want to show that γ 0 = t/θ cannot b optimal within this class of policis (hnc, it cannot b optimal in gnral). A marginal chang in 1 accompanid by th corrsponding chang in γ 0 so that quation (8) holds, must satisfy: γ 0 = (γ0 γ) F (1 ; E C ) F ( 1 ; E D ) R 1. 1 [F (; E C ) F (; E D )] d Givn that th cost of th policy is γ 0 F 1 ; E C + γ F 2 ; E C F 1 ; E C, th proposd marginal chang in 1 will rsult in a chang in costs of h( 1 )(γ 0 γ) 1. By th sam rasons as in Claim 2, a marginal incras in 1 would dcras th costs whnvr 1 < and γ 0 >γ.in particular, th policy whr γ 0 = t/θ cannot b optimal, sinc thr is room to dcras γ 0 and incras 1, which lowrs th cost of th monitoring. Claim 4:A policy such that 1 = 2 > and γ<t/θis not optimal. 28