Does Inequality Lead to Greater Efficiency in the Use of Local Commons? The Role of Strategic Investments in Capacity

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1 Do nqualiy Lad o Grar fficincy in h U of Local Common? Th Rol of Sragic nvmn in Capaciy by Rimjhim Aggaral and Tulika A. Narayan Working Papr No Da of Submiion: March by Rimjhim Aggaral and Tulika A. Narayan All righ rrvd. Radr may mak vrbaim copi of hi documn for non-commrcial purpo by any man, providd ha hi copyrigh noic appar on all uch copi. Dparmn of Agriculural and Rourc conomic 00 Symon Hall Univriy of Maryland Collg Park, MD USA Tl

2 Fbruary, 000 Do nqualiy Lad o Grar fficincy in h U of Local Common? Th Rol of Sragic nvmn in Capaciy Rimjhim Aggaral and Tulika A. Narayan* Abrac: Thi papr xamin h impac of inqualiy in acc o di on fficincy in xracion from a common rourc. A dynamic modl i dvlopd, hr agn ragically choo h lvl of unk capaciy and h conqun xracion pah. Sunk capaciy i a funcion of co of di and rv a a commimn dvic o dr nry or forc xi. Conrary o prviou udi bad on aic ing, our rul ho ha grar inqualiy do no ncarily lad o grar fficincy in xracion. n paricular, ho ha undr modra inqualiy, h rourc ock i lor han ha undr prfc qualiy. * Dp. of Agriculural and Rourc conomic, Univriy of Maryland, Collg Park, MD 074. mail: aggaral@arc.umd.du, ulika@arc.umd.du 1

3 . nroducion A larg par of h horical liraur on common pool rourc CPR aum agn o b homognou Gordon, 1954; Clark, 1976; Dagupa and Hal, 1979; Lvhari and Mirman, 1980; Sandlr, 199; Orom, Ovr h pa f yar, hovr, hr ha bn a groing dba on h ffc of hrogniy on h u of CPR in boh coopraiv and non-coopraiv ing. Thi lin of ork can b racd back o Olon 1965 ho hypohizd ha h grar i h alh inqualiy among mmbr of a group, h largr i h probabiliy of collciv acion. Alhough Olon ork ha bn vry influnial in h CPR liraur, i i backd by limid mpirical uppor. Fild udi on CPR managmn prn a vry varid picur ih a f upporing Olon argumn bu a majoriy finding ihr an oppoi or an ambiguou rul. 1 n a rcn ynhi of hi liraur, Baland and Plaau 1995 ugg ha o mak n of h divr finding i i hlpful o diinguih bn h diffrn yp of hrogniy, uch a ho ariing from diffrnc in ndomn, objciv or culural background of h agn. Thy argu ha hil homogniy in objciv and culural background ar abolu prrquii for collciv acion, hi do no ncarily hold ru for hrogniy in priva alh of agn. n a of papr ha appard in hi journal, Baland and Plaau BP 1997, 1998 xplor hi ffc of hrogniy in priva alh of agn on fficincy in u of common undr vral diffrn ca of rgulad and unrgulad common. On of h riking rul from hir papr ari in a conx hr agn appropria a common rourc in h abnc of any xrnal rgulaion. i ll knon ha in uch a ing, agn ar likly o xrac in xc of ocial opimum. No, conidr a iuaion hr agn hav diffrnial acc o an imporan inpu ud in h xracion of hi common rourc. To fix ida, hink of hi inpu a di. Du o informaion problm, paricularly in lo-incom counri, h 1 Thu for inanc, Johnon and Libcap 198 in hir udy of h Txa hrimp fihry found ha hn fihrmn diffr in hir inhrn kill in fihing, coopraiv agrmn uch a cach rricion ar unlikly o uccd. S alo Bardhan 1993 and Aggaral 000 for xampl from Aian irrigaion ym. Kanbur 1

4 amoun of di availabl i ofn cloly linkd o onrhip of a uch a priva land, hich can b offrd a collaral. Thrfor, inquii in a onrhip ofn ranla ino inquii in acc o di. Th folloing quion hu bcom imporan: Ho, if a all, ould h fficincy in u of CPR b affcd if h diribuion of di r changd, kping h oal amoun of availabl di conan? To xamin hi quion, BP conidr a impl aic ing, hr xracion i a concav funcion of ffor and on uni of di i rquird o xr on uni of ffor. Saring from a iuaion hr all agn hav qual acc o di hy conidr a di-qualizing chang in acc o di kping h oal amoun availabl conan. Agn ho no bcom di conraind ar likly o rduc hir xracion lvl, and a a rpon, h unconraind agn ar likly o ina hir lvl of xracion. Hovr, bcau of h aumpion of concaviy of h ffor funcion, i follo ha h ina l han compna for h da. Thu oal xracion i likly o fall, lading o a mor fficin oucom. Thy hu conclud ha h mor unqual i h diribuion of di conrain, h mor fficin i h appropriaion from CPR. Thi unambiguou rul on h ffc of inqualiy on fficincy ha imporan policy implicaion. Givn h idprad dgradaion of naural rourc undr condiion of poorly dfind propry righ, govrnmn ar ruggling o find alrnaiv ay o hal hi proc. BP claim ha hir cnral argumn i applicabl in a id array of conx in hich conrain or facor mark imprfcion limi h acc of om ur o imporan inpu ud in xracion of CPR. Thu, for inanc, if di i an imporan inpu ha i adminiraivly diribud, hn hir analyi ugg ha an unqual diribuion of availabl di among ur of a CPR ould a a 199 and Baland and Plaau 1998 provid a urvy draing upon udi from variou diffrn CPR conx. Clark 1980 alo ablih a poiiv rlaionhip bn inqualiy and fficincy in h conx of common acc fihri. Hovr, h dfin inqualiy in rm of h kill diffrnial of h agn.

5 iuaion hr h larg holdr lf-limi hir xracion, hu lading o iuaion hr xracion i lor han in h ca hr h availabl di r diribud in an qual ay Th prn papr i moivad by h obrvaion ha BP rul, drivd ihin a aic ing, may no hold in h mor raliic dynamic ing in hich mo CPR xracion including h ca of fihri xamind by BP ak plac. 3 n a dynamic rourc ing, h folloing conidraion bcom imporan. Fir, i i ll knon ha in a dynamic ing h inr mporal radoff bn conrvaion and dplion ar imporan, and hi opn up h poibiliy of a far richr of oucom including h poibiliy of lf-nforcing coopraiv quilibria, hich could ponially alr h concluion rachd in a aic ing. Scond, no ha acc o di hich i h bai of hrogniy in h BP papr, bcom pcially ignifican in iuaion hr larg invmn, ihr in h form of inalling capaciy or adopion of n chnology hav o b mad prior o xracion from h CPR. Sinc h invmn influnc xracion choic in h fuur, a a on-ag gam a ud by BP i inadqua o capur h complxiy of h ragic nvironmn ha ari du o h iming of h diffrn mov. Prior invmn in capaciy influnc fuur xracion choic in a numbr of dynamic CPR ing. A prim xampl i ha of groundar for hich propry righ ar gnrally poorly dfind in mo counri. n h pcific ca of ndia, landonr hav h righ o drill ll on hir on land and pump ou a much ar a hy dir. Hovr, h fixd co of drilling a ll and buying h pumping quipmn ar vry high, paricularly in mi-arid ara hr ll nd o b vry dp in ordr o inrcp ar-baring fracur in h ub-raa. Th dpr i h ll, h lor i h probabiliy ha i ould bcom dry a any givn im. Du o limid acc o di, a majoriy of mall and marginal landonr in h ara hav no invd in ll, hil larg farmr hav invd in mulipl ll ih vry high pumping capacii Shah, 1993; Aggaral 000. nringly, givn h hug ubidi on liciy upply, h marginal co of pumping ar 3

6 vry lo and hi ha ld o a rapid dclin in h ar abl in hi rgion. n h compiiv pumping rac ha ha nud, ho ho hav dpr ll hav urvivd hil ho ih hallor ll gnrally h mallr landonr hav bn drivn ou ovr im Bhaia, 199. An imporan policy quion bing pod in hi conx i rgarding h ffc ha govrnmn polici on diribuion of di can hav on groundar u, givn h fac ha dirc rgulaion of groundar i no adminiraivly faibl in h hor run. Similarly in h conx of many fihri, h choic of capaciy a maurd by vl iz and yp of quipmn ud i iical in drmining h yp and iz of cach. Kurin 199 in hi ca udy of coaal fihri in ouh ndia dib ho ih h xpanion of xpor mark in pran in h mid 1960, mrchan from urban ara ard o havily inv in vl capabl of dp a fihing. Tradiional fihrmn ih rlaivly poor acc o di r no abl o ak advanag of h opporunii and r loly diplacd a ock dpld and hir radiional chnologi bcam rdundan. To modl h ragic inrplay bn agn in uch cnario, hr dciion rgarding invmn in unk capaciy ar iical and hav o b mad prior o xracion, a o-ag formulaion of h dynamic gam i ndd. n hi papr dvlop uch a gam in hich agn fir choo h lvl of unk invmn in capaciy and hn h xracion pah ovr h infini horizon. W aum agn o b homognou in all rpc xcp in rm of hir acc o di. To modl hrogniy in acc o di dra upon h idly obrvd fac ha h xn of di availabl hrough h formal di mark i gnrally raiond in mo lo incom counri and i vry cloly drmind by h amoun of collaral.g. land or livock ha can b offrd. Th ridual dmand i m hrough h informal mark hr inr ra ar much highr. Thu agn fac diffrn co of di dpnding on hir xognouly givn a ndomn, and hi diffrnc in co of di coniu h bai of hrogniy among agn in h modl. 3 n hi papr only analyz h ca of xracion from a CPR in a non-coopraiv framork. Baland and Plaau 1997, 1998 alo dicu h ca of volunary conribuion oard h aion and mainnanc of a 4

7 Givn hi dynamic ing, driv a rlaionhip bn inqualiy maurd in rm of diffrnc in co of di and ady a rourc ock. Conrary o h BP rul, ho ha grar inqualiy do no ncarily lad o grar fficincy in u of CPR. n paricular, ho ha for modra lvl of inqualiy, h rourc ock i lor han ha undr prfc qualiy. i only for fairly high lvl of inqualiy ha h rourc ock approach h ocially opimal lvl. A rcn orking papr by Dayon-Johnon and Bardhan 1996 alo ugg ha h rlaion bn hrogniy in a ndomn and fficincy in rourc u may b non-monoonic. Hovr, h rucur of hir modl i vry diffrn from our. n paricular, hy hav a o priod ing in hich h capaciy lvl of agn i uniquly drmind by hir xognouly givn a ndomn and h ragy of agn i dfind by h ingl ffor lvl ha hy imulanouly choo in h fir priod in h cond priod, i i alay ru ha agn ould apply maximum ffor. A oppod o hi, in our modl, agn ragically choo h lvl of unk invmn in capaciy a ll a h conqun xracion pah ovr h infini horizon. Thu our focu li mor on h choic of unk capaciy in h fir ag of h gam and i rol a a commimn dvic o dr nry or forc xi of ohr agn from h xracion gam in h cond ag. A numbr of papr in h indurial organizaion liraur hav xplord h rol of ragic invmn for inanc, Dixi, 1980; Fudnbrg and Tirol, 1983; Spnc, Hovr, hr hav bn vry f applicaion in CPR conx. Copland 1990 xplord h ragic ffc of invmn ha nhanc or droy a common rourc in h conx of inrnaional xrnalii. Hovr, h purpo of hi papr a o ablih h condiion ha lad o undr and ovrinvmn. H did no pcifically analyz h ffc of hrogniy among agn on h choic of h invmn and on h fficincy in rourc u. n our analyi, on h ohr hand, hrogniy among agn in rm of hir co of di bcom an imporan facor in drmining boh ady a ock lvl and h oal invmn in capaciy. To h b of our CPR and find h rul o b omha diffrn from h xracion ca. 5

8 knoldg, h ffc of ragic invmn in a dynamic CPR xracion gam ih hrognou agn hav no bn xamind bfor. Th r of h papr i organizd a follo. To fix ida dvlop our modl in h conx of groundar xracion, alhough a dicud abov, h cnral ida bhind i hav much idr applicabiliy. n Scion, prn h bnchmark ca of a ingl ll onr ihin an aquifr, ho fac a compiiv mark for ar. Thi ca alo dfin h ocial opimum in our ing. Thn, in Scion, xnd hi analyi o h ca of o homognou agn ho xrac from h am groundar aquifr. n Scion V, inroduc hrogniy among h agn in rm of h co of di ha hy fac. n Scion V, u h rul from Scion V o map a rlaion bn inqualiy on h on hand and ady a ock lvl and invmn on h ohr hand. Finally, in Scion V, conclud.. Sol Onrhip For compln, bgin ih h ca of a ingl agn ho ha ol xracion righ o a groundar aquifr. Conidr h folloing o-ag modl. n h fir ag, h agn mak a dciion rgarding ho dp o drill h ll. Dph of h ll i an imporan drminan of i capaciy bcau ar canno b xracd from h ll hnvr h ar lvl in h aquifr fall blo h ba of h ll. n ohr ord, h dph of h ll dfin a lor bound, uch ha hnvr h ar ock in h aquifr fall blo, h ll bcom dry. For racabiliy, aum ha hr i on o on rlaionhip bn h invmn,, mad in h dph of h ll and hich i givn a [1] 6

9 hr ' < 0 and '' 0. Th abov invmn i rgardd a a unk invmn hich ha o b mad onc and for all, prior o xracion. 4 Th marginal co of invmn i aumd o b a conan, dnod by φ, hich dpnd upon h ra of inr facd by h agn in h di mark. n h cond ag, h agn choo an xracion pah ha maximiz h prn valu of n rurn from xracion. Folloing Gir 1983, aum ha h co of xracing ar i an inaing funcion of h xn of lif, 5 hon in figur1 a AB. Th xn of lif a any im, in urn, dpnd on h ar ock in h aquifr and hu h co funcion for xracion can b rin a [] C c hr c i a conan and i h amoun of ar xracd a im. Th ll onr i aumd o b a pric akr in h mark for ar, ih h pric of ar givn by h conan p. Th agn opimizaion problm can b olvd hrough backard inducion by fir olving h cond-ag problm condiional on h invmn dciion in h fir-ag. Th cond-ag opimizaion problm i givn a c [3] Max p d 0.. [A] & r [B] 0 [C] 0 for < hr i h dicoun ra and r i h naural rcharg ra of ar. quaion [A] govrn h ock raniion ovr im. Conrain [B] impli ha a ach inananou poin in im hr i an 4 Th horpor of h pumping quipmn may alo b an imporan drminan of capaciy. n mo miarid rgion, ubmribl pump ar ud. nvmn in uch pump, for all pracical purpo, may alo b rgardd a unk invmn. 7

10 uppr bound,, on h amoun of ar ha can b xracd. To allo for h poibiliy of compl xhauion, aum ha >r. Conrain [C] nur ha hr canno b any xracion hnvr h ock of ar fall blo. Sinc h maximand in h abov problm i linar in h conrol variabl,, h quilibrium i a bang-bang oluion. Th opimal xracion pah i givn by h folloing mo rapid approach pah MRAP [4] S r 0 if if if > < Max Max Max { { { S S S,,, } } } hr S i h ady a ock lvl givn a [5] c c S p 4 cpr Givn hi oluion o h cond-ag problm, in h fir-ag h ol onr choo h lvl of unk invmn in h dph of h ll uch ha h marginal co of invmn qual h dicound marginal bnfi from xracion. L u aum ha h iniial ock lvl i grar han S. L S b h lvl of invmn ha corrpond o S in [1]. From 4 i i clar ha along h opimal pah, h agn do no xrac any ar hnvr h ock lvl fall blo S. Thu h marginal bnfi from inving byond S ar zro. Hovr, for any lvl of invmn < S, h oal bnfi from inving i givn by c ä ä [6] B p d pr d 0 hr dno h im a hich h ock aain h lor bound. To a noaion, l S. Diffrniaing [7] ih rpc o, giv h marginal bnfi from inving a any lvl < S a 5 Th diffrnc bn h ar lvl in h ll and h lvl o hich i ha o b lifd 8

11 [7] B S r p c ' ä S S n h appndix, ho ha B i ricly poiiv and donard loping for < S. Figur ho h marginal bnfi and co curv of invmn. Th agn choo h lvl of invmn ha qua h marginal co of invmn φ ih h marginal bnfi of invmn. f φ S φ, hn h agn inv S and driv h ock o h ady a lvl, S. On h ohr hand, if φ > φ S, hn h opimal choic of invmn i l han S and givn by h inrcion of h marginal bnfi and co curv in figur. n hi ca, h ady a ock lvl i l han S. Givn ha h ll onr i a pric akr in h mark for ar, hi oluion alo dfin h ocial opimum in hi ing.. Homognou Agn n hi cion conidr h ca of o agn i 1, ho xrac from a common groundar aquifr and ar homognou in all rpc. A oppod o h ca of ol-onrhip, in a o-pron ca, h o-ag modl i much mor complx bcau of ragic bhavior. For a in xpoiion, hav dividd hi cion in o par. n h fir par, prn h Nah quilibrium oluion for h ca uually modld in h groundar liraur hr only h xracion dciion and no h capaciy choic dciion i akn ino accoun for inanc, Provnchr and Bur, 1993; Gir, Such a ing i uful in iuaion hr ihr capaciy can b quickly adjud o any chang in xracion nd and/or co of ing up capaciy ar ngligibl and o capaciy do no rprn a rigid conrain. n cond par of hi cion rlax hi aumpion and prn h o-ag modl ih capaciy and xracion choic..1 xracion choic ih no capaciy conrain Th opimizaion problm for agn 1 hr i givn a h ca of agn i ymmric 9

12 [8] Max c1 p1 d 1 0. [A] & r 1 [B] 0 1 Solving h o b rpon funcion for a Nah ymmric oluion g a rul analogou o h ol-onrhip ca. Th groundar ock i drivn o N by h mo rapid approach pah MRAP givn a [9] N r / 0 if if if > < N N N hr [10] c c N p p On comparing quaion [5] and [10] i i clar ha N < S. Thi i h andard rul of ovrxploiaion hn agn do no fully inrnaliz h xrnalii gnrad in h u of h common. Th gro payoff from xracion in hi ca ar givn a [11] π N N 0 p c N pr N hr N i h im a hich h ock rach h ady a lvl givn a N in 10.. To ag gam ih capaciy and xracion choic No l u conidr h ca hr agn hav o choo h lvl of invmn in capaciy prior o xracion. A ho blo, agn may no choo invmn lvl ragically in 10

13 ordr o forc xi or dr nry of h ohr agn. 6 n ordr o kp h analyi fairly gnral hr, aum ha agn choo invmn lvl qunially, ih h choic of mov bing ndognou o h gam. n a gam ih ymmric agn hi impli ha agn mov imulanouly. 7 ould b hlpful o cagoriz h J of opion availabl o ach agn a: J {D, A, } hr D and for driv ou, A for accommoda and for xi. For agn 1 o b abl driv ou agn, o condiion mu b aifid. Fir, in ag 1 of h gam, agn 1 mu inv mor han h xpc agn o inv, i.. 1 >. Scond, in ag, agn 1 mu driv don h ock o a lvl byond in fini im. By dfiniion, adopion of h driv ou opion by any agn impli forcd xi for h ohr agn and oghr h imply ha hr xi a im priod d, uch ha for all > d, hr i only on agn in h gam. A oppod o hi, h ragy o accommoda impli ha hr ar o agn in h gam for all. For accommodaion o ork, boh playr mu inv a h am lvl. Finally, no ha ach agn alay ha h opion of xiing ou of h gam in fini im, irrpciv of ha h ohr agn do. Givn hi ing, olv for h Courno-Nah quilibrium dfind a DFNTON 1: A ragy 1,, 1, i a Courno-Nah CN quilibrium if h n dicound payoff o agn i i 1,, from chooing i, i i maximizd givn h quilibrium ragy of h ohr agn. To olv for h CN quilibrium rcall ha N a found o b h Nah ady ock lvl in h abnc of capaciy conrain. L N dno h lvl of invmn in capaciy ha corrpond o N from [1]. No ha N, N canno b h quilibrium invmn ragy in h prnc of capaciy conrain. Thi i bcau ach agn by inving a mall amoun,, abov N can driv ou h ohr agn and g largr profi. To xamin hn uch a driv-ou ragy ould b 6 A numbr of papr in h indurial organizaion liraur hav xamind h choic of capaciy a an nry drrn ragy Tirol, 1988 provid a urvy. Th rul in h papr hav bn found o b qui niiv o h aumpion mad rgarding h iming of h mov, i.. hhr hr i imulaniy in choic of capacii or an xognouly givn qunialiy ih on playr incumbn having a fir movr advanag, poibly du o chnological lad. 11

14 1 chon, lay ou h xracion pah and h aociad gro payoff hn agn 1 puru h driv ou opion undr h xpcaion ha agn ould inv. No ha if agn 1 xpc o b l han S h opimal ady a invmn lvl undr ol-onrhip hn h ould choo o inv S and driv don h ock o S. Thi ca i imilar o h ol onrhip ca. Th mor inring ca ari hn agn 1 xpc o b grar han S. Thi i h ca conidr in h r of hi cion. Undr h driv ou ragy, agn 1 xracion pah in h cond ag i givn a [1] < > if 0 if if 1 r D Th gro payoff o agn 1 from xracing along hi pah can b rin a [13] d pr d c p S S D 0 1 π A on ould xpc, hi driv ou payoff rcivd by agn 1 i a daing funcion of proof in h appndix. By dfiniion, undr h abov driv ou ragy, agn i forcd o xi and hi xracion pah i givn a [14] < if 0 if Th gro payoff o agn from xracing along hi pah can b rin a [15] d c p S 0 π 7 Thi i bcau hn agn ar ymmric hy ould hav h am prfrnc ovr h choic of mov.

15 No ha h abov payoff i a funcion of. Undr compl informaion, agn kno ha h ould b drivn ou in h cond ag. Thrfor, givn φ, h choo invmn opimally in h fir ag. For φ >0, h opimal n xi payoff i givn a [16] Π φ S c p d φ hr 0 S c arg max p 0 d φ Th abov n xi payoff ha a pcial ignificanc in hi ing. ach agn can alay guaran for himlf hi minimum payoff by xiing ou of h xracion gam in fini im, irrpciv of h acion of h ohr agn. Thu Π φ rprn h rrvaion payoff in hi ing. Dfiniion : A ragy i, i for agn i i1, i individually raional if h n payoff o agn i from hi ragy i a la a larg a hi n xi payoff. A oppod o driving ou agn, agn 1 can alo accommoda him. f boh agn inv N and accommoda ach ohr hn h gro payoff ar givn by π N in [11]. A argud arlir, hi i an unabl quilibrium inc π N < lar i xpcd o inv N. Hovr, no ha inc xi an NC > N, uch ha for NC D π 1 N and o agn 1 prfr o driv ou agn if h D π 1 i a daing funcion of hr [17] D NC π 1 π N Propoiion 1: n h homognou ca ih capaciy conrain, if i i individually raional for boh playr o accommoda ach ohr and inv NC hn a NC, NC ar h Courno-Nah quilibrium invmn lvl. b N i h ady a ock lvl. 13

16 Proof: To chck if NC, NC ar h quilibrium invmn lvl conidr ha happn if hr i a on p unilaral dviaion by agn 1 o NC hr > 0, in ordr o driv ou agn. Givn h hap of π D, i follo ha π D NC NC N < π D π. Hnc hi dviaion i no profiabl. No conidr a unilaral dviaion by agn 1 o NC -. Sinc NC -. < NC, agn 1 canno driv ou agn. Agn, hovr, ould no find i opimal o driv ou agn 1. Sinc h n xi payoff i lor han h payoff from accommodaion a NC, hi dviaion i alo no profiabl for agn 1. Whn boh agn accommoda ach ohr, h ady a ock lvl i givn by N in 17. Rcall ha N a hon o b h ady a ock lvl in h ca ihou capaciy conrain alo. Hovr, h diffrnc in h ca ih capaciy conrain i ha ragic bhavior lad boh agn o inv mor, inc NC > N. Corollary1: n h homognou ca ih capaciy conrain, boh agn inv in xc capaciy. V Hrognou Agn n hi cion aum ha agn ar homognou in all rpc xcp in rm of h co of di hy fac. L h co of di b dnod by φ 1 and φ, rpcivly, for h o agn. To bgin ih, l u xamin ho h n payoff undr h diffrn ragi dfind in h prviou cion, vary ih φ i i 1,. Undr h ragy of accommodaing by inving NC, h n payoff ar givn a A N NC [18] Π i φ i π i φ i A Π i φi NC No ha. φ i On h ohr hand, h n xi payoff ar givn a Π i φ i in [16], and i follo from h nvlop horm ha Π i φ i φ i i. 14

17 No ha NC i no a funcion of φ i [17], bu i i a daing funcion of φ from A [16]. Thi impli ha aring from lo lvl of φ hr Π φ > Π φ a φ i ina i i i i A Π φ fall a a conan ra hil Π φ fall a a daing ra figur 3. Thi lad o i i h folloing lmma i i Lmma 1: Thr xi a φ 0 > 0, uch ha hn φ i > φ 0, h n xi payoff for agn i xcd hi n payoff from accommodaion. No l u xamin ho h ady a ock and invmn lvl vary, a inqualiy among agn ina. W dfin inqualiy a h diffrnc bn φ 1 and φ, and modl ina in inqualiy by a man prrving prad givn by daing φ 1 and inaing φ uch ha φ 1 φ / φ. Furhr aum ha φ < φ 0, o ha hn agn ar homognou, h n payoff from accommodaion ar highr han h n xi payoff. Lmma 1 hn impli ha for high nough lvl of inqualiy, uch ha φ > φ 0, agn may find i opimal o xi ou of h gam. n h folloing o propoiion xamin ho chang in inqualiy affc invmn and ady a ock lvl. Propoiion : Saring from h lvl of qualiy, for mall man prrving dviaion in marginal co, uch ha φ 1 < φ < φ 0, h o agn coninu o accommoda ach ohr. Th Courno-Nah quilibrium invmn lvl ar NC, NC and h ady a ock lvl i N. Proof: Sinc accommodaion i individually raional for boh playr, h proof follo dircly from propoiion 1. Propoiion 3: For larg man prrving dviaion in marginal co, uch ha φ 1 < φ 0 bu φ > φ 0 S a Th Courno-Nah quilibrium invmn lvl ar givn a [max,, ]. 15

18 S b Th ady a ock lvl i givn by min [, } ]. Proof: n hi ca hr agn ar vry hrognou, h qunialiy of mov bcom imporan in h folloing ay. Fir, conidr ha happn if agn mov fir. Sinc h opion of accommodaion by inving a NC i no longr individually raional for agn, h can ihr driv ou agn 1 or xi ou of h gam himlf. Sinc agn 1 fac a lor marginal co of invmn and inv afr obrving agn, driv ou by agn i no a faibl opion hr. Agn only opion i o xi and hu h choo o inv choo o inv [max S, 1 inv o maximiz hi n xi payoff. Givn hi, agn 1 ]. Nx conidr ha happn if agn 1 ha o mov fir. f agn, hn agn having obrvd agn 1 invmn, ill inv a lil mor han him and driv him ou. To avoid bing drivn ou, agn 1 ill hav o inv a a lvl hr agn n payoff from driving ou agn 1 qual hi n payoff from xi. Dno hi lvl of invmn by. Thu agn 1 ill inv hra agn ill inv. No ha > and o h payoff for agn 1 hn h mov fir ar akly lor han hi payoff from h gam hr agn mov fir. Hovr, agn payoff ar h am undr boh pcificaion of h gam. Thu, agn 1 akly prfr o mov cond hil agn i indiffrn. Thrfor, i follo ha if h qunialiy of mov i ndognou hn agn ill mov fir. Th quilibrium invmn S lvl ill b givn a [max,, ] and h ady a ock lvl ill b givn by S min [, ]. V. nqualiy, nvmn and Sady Sa Sock Lvl n hi cion u h rul from h prviou cion o map a rlaionhip bn inqualiy on h on hand, and invmn and ady a ock lvl on h ohr hand. n figur 4a and 4b, h marginal co of invmn of agn, dnod a φ, i hon along h horizonal axi. Th origin rprn h poin of prfc qualiy a hich φ φ. A on mov o h righ along hi axi, φ ina hil φ 1 da, prrving h man a φ. Thu a movmn o h righ along hi 16

19 axi rprn inaing lvl of inqualiy. Th aggrga invmn lvl and h ady a ock lvl ar hon along h vrical axi in figur 4a and 4b, rpcivly. n a mall nighborhood around h origin hr φ φ 0, i i individually raional for ach playr o inv NC and accommoda h ohr playr propoiion. So h aggrga invmn lvl and h ady a ock lvl ar h am a in h ca of prfc qualiy. W labl hi a h rang of lo inqualiy in figur 4. A φ ina furhr uch ha φ > φ 0, i i no longr individually raional for playr o inv NC and ay in h gam indfinily lmma 1. follo from propoiion 3 ha in a mall nighborhood o h righ of 0 φ φ, agn inv and i drivn ou by agn 1 ho inv and driv don h ock o. Thu h ady a ock lvl a ll a h invmn lvl ar lor in hi nighborhood han undr prfc qualiy. A inqualiy ina furhr, aggrga invmn fall monoonically inc i a daing funcion of φ. Th rlaionhip bn ady a ock lvl and inqualiy i omha mor complx. No ha h ady a ock lvl fall harply a φ φ 0 and hrafr ina a inqualiy ina. For h ca hr φ > φ 0, can diinguih bn h folloing rang for inaing valu of φ figur 4. 1 Modra inqualiy: hr aggrga invmn lvl i ock lvl i, ih < N < S. High nqualiy: hr aggrga invmn lvl i lvl i, ih N < < S. < NC and h ady a < NC and h ady a ock 3 Vry High nqualiy: hr aggrga invmn lvl i ock lvl i S. S < NC and h ady a Conrary o h concluion rachd by Baland and Plaau 1997, find ha h rlaionhip bn inqualiy and ady a ock lvl i non-monoonic. n paricular, aring 17

20 from prfc qualiy, a inqualiy ina, hr i a rang hich rfr o a h rang of modra inqualiy hr ady a ock lvl i lor han h lvl undr prfc qualiy. On h ohr hand, in h lo and high rang of inqualiy, h ady ock lvl i a la a larg a ha undr prfc qualiy. n h vry high inqualiy rang, ady a ock lvl i a h fir b lvl. V. Summary and Concluion Prviou ork on common pool rourc ha gnrally aumd agn o b homognou. n hi papr hav focud on on apc of hrogniy, namly ha ariing from diffrnial acc o an imporan inpu di ud in xracion from h CPR. Acc o di bcom paricularly imporan in CPR conx hr conidrabl unk invmn i ndd prior o xracion, and hr hi invmn influnc h naur and xn of xracion. Common xampl ar: groundar pumping and dp-a fihing. n modling uch ca, h folloing conidraion bcom imporan. Fir, h choic of boh invmn lvl and h xracion pah ar iical. Scond, inc agn impo xrnalii on ach ohr in xracion, and prior invmn lvl influnc h xn of h xrnalii, agn ar likly o ragically choo invmn lvl. Third, h iming of mov i imporan and o a on-ag gam i inadqua o capur h complxiy of h ragic iuaion hr. Kping h conidraion in mind, dvlopd a o-ag modl hr agn choo h lvl of unk invmn in capaciy and ubqunly, h xracion pah ovr h infini horizon. Sunk invmn rvd a a commimn dvic in hi modl o dr nry or forc xi. Sinc h co of di influnc h invmn choic, hrogniy among agn in rm of hir acc o di affc boh capaciy and xracion choic. Uing hi modl find ha conrary o rul drivd in prviou udi bad on a aic ing, h rlaion bn inqualiy and fficincy in rourc xracion i non-monoonic. Th ady a rourc ock i clo o h ocially opimal lvl hn ihr inqualiy i vry high or vry lo. For modra lvl of 18

21 inqualiy, ho ha h rourc ock may in fac b lor han ha undr prfc qualiy. Furhr, ho ha bcau of h ragic rol of invmn in hi ing, agn inv in xc capaciy in gnral, xcp hn inqualiy i high. n many CPR conx, uch a ha of groundar in mi-arid ndia, dirc rgulaion of xracion ra i gnrally rgardd a infaibl in h hor run. An imporan indirc policy ool hr i h adminiraiv diribuion of imporan inpu, uch a di ud in groundar xracion. Baland and Plaau hav argud ha in uch ca unqual diribuion of di ould lad o highr fficincy in u of common. Hovr, polici favoring a highly unqual diribuion of di may no b poliically faibl. Th conribuion of our papr li in hoing ha modra lvl of inqualiy in diribuion may, in fac, lor rourc ock vn blo ha undr qual diribuion. Our baic modl can b xndd in vral dircion. An imporan xnion ould b o xamin hhr our rul chang qualiaivly hn hr ar mor han o agn. Prviou papr ha hav modld only xracion choic and no capaciy choic uch a Baland and Plaau, 1997; Dayon-Johnon and Bardhan, 1996 found h rul o b qualiaivly imilar in a mulipl agn ing. n h CPR liraur hr ha alo bn a lo of inr in xamining ha happn hn h produc from xracion of h common i old in an imprfcly compiiv mark Corn al, n our modl, inc aumd agn o b pric akr, i follod ha xracion i a i opimal lvl hn hr i a ingl agn xracing from h aquifr. Hovr, hn hi agn alo hold mark por hn hr i likly o b ovr-conrvaion of h rourc and opimaliy ould rquir mor han on agn. Th ragic ffc of invmn ar likly o b much mor complx in uch a ing. 19

22 0 Appndix 1 Proof: Undr ol onrhip, oal bnfi from invmn ar a ricly poiiv and concav funcion of invmn for < S. For < S, h oal bnfi ar givn by TB d pr d c p ä ä 0 TB MB c p ä ä ä ä pr ä ä ä c p r ä c p r ä For < S, for all, hrfor r r 1 Thu, MB c p To chck h ign of MB no ha 0 < from [1], hrfor MB >0 if c p > S p p c c > 4 For < S, kno ha S > and o h abov condiion i aifid.

23 1 No conidr h ign of MB c p MB c 3 c p c 3 Givn h aumpion rgarding in 1 i follo ha MB < 0 for < S. QD. Proof: D 1 π i a daing funcion of for > S. D 1 π d pr d c p ä ä 0 1 D π ä c p r ä n h o-pron ca, r r 1 Thrfor, 1 D π c p r r

24 No ha < 0 from [1], hrfor 1 D π <0 if h folloing condiion hold > c p r r No ha r r < 1, hrfor h abov condiion ould hold if > c p S S > < No ha for h ca bing conidrd, > S and o h abov condiion i aifid. QD

25 Rfrnc 1. Aggaral R. M., Poibilii and Limiaion o Coopraion in Small Group: Th Ca of Group Ond Wll in Souhrn ndia, World Dvlopmn. Forhcoming, Baland, J. M. and J. P. Plaau, Do Hrogniy Hindr Collciv Acion? Working Papr, Dparmn of conomic, Univriy of Namur, Baland, J. M. and J. P. Plaau, Walh nqualiy and fficincy in h Common Par : Th Unrgulad Ca, Oxford conomic Papr. 49, Baland, J.M. and J.P. Plaau, Walh nqualiy and fficincy in h Common Par : Th Rgulad Ca, Oxford conomic Papr. 50, Bardhan, P., Raional Fool and Coopraion in a Poor Hydraulic conomy, Working Papr No. C93-015, Cnr for nrnaional and Dvlopmn conomic Rarch, Univriy of California, Brkly, CA Bhaia, B., Luh Fild and Parchd Throa- A Poliical conomy of Groundar in Gujara, conomic and Poliical Wkly. 19, A Clark, C. W., Mahmaical Bioconomic, Wily, N York, Rricd Acc o Common Propry Fihry Rourc: A Gam-Thoric Analyi, in Dynamic Opimizaion and Mahmaical conomic P. T. Liu, d. Plnum, N York Copland, B. R., Sragic nhancmn and Drucion of Fihri and h nvironmn in h Prnc of nrnaional xrnalii, Journal of nvironmnal conomic and Managmn. 19, Corn, R., C.F. Maon and T. Sandlr, Th Common and h Opimal Numbr of Firm, Quarrly Journal of conomic. 101, Dagupa, P. and G. Hal, conomic Thory of xhauibl Rourc, Cambridg Univriy Pr, Cambridg, U.K Dayon-Johnon, J. and P. Bardhan, "nqualiy and Conrvaion on h Local Common: A Thorical xrci, Working Papr No. C96-071, Cnr for nrnaional and Dvlopmn conomic Rarch, Univriy of California, Brkly, CA Dixi A., Th Rol of nvmn in nry Drrnc, conomic Journal. 90, Fudnbrg, D. and J. Tirol, Capial a a Commimn: Sragic nvmn o Dr Mobiliy, Journal of conomic Thory. 31, Gir, M., Groundar: Focuing on h Ral u, Journal of Poliical conomy, 91, Gordon, H. S., conomic Thory of a Common Propry Rourc: Th Fihry, Journal of Poliical conomy, 6,

26 17. Johnon, R. N. and G. D. Libcap, Conracing Problm and Rgulaion: Th Ca of Fihry, Amrican conomic Rvi, 7, Kanbur, R., Hrogniy, Diribuion and Coopraion in Common Propry Rourc Managmn, Working Papr No. 844, World Bank, Wahingon, DC Kurin, J., Ruining h Common and Rpon of h Commonr: Coaal Ovrfihing and Fihorkr Acion in Krala Sa, ndia, in Graroo nvironmnal Acion: Popl Paricipaion in Suainabl Dvlopmn D. Ghai and J.M. Vivian, d. Rouldg, London and N York Lvhari, D. and L.J. Mirman, Th Gra Fih War: an xampl uing a Dynamic Courno-Nah Soluion, Bll Journal of conomic. 11, Provnchr, B. and O. Bur, Th xrnalii Aociad ih Common Propry xploiaion of Groundar, Journal of nvironmnal conomic and Managmn. 4, Olon M., Th Logic of Collciv Acion, Harvard Univriy Pr, Cambridg, MA Orom,., R. Gardnr and J. Walkr, Rul Gam and Common Pool Rourc, Univriy of Michigan Pr, Ann Arbor, M Sandlr, T., Collciv Acion: Thory and Applicaion, Univriy of Michigan Pr, Ann Arbor, M Shah, T., Groundar Mark and rrigaion Dvlopmn, Oxford Univriy Pr, Bombay Spnc, M., nry, Capaciy, nvmn and Oligopoliic Pricing, Bll Journal of conomic. 8, Tirol, J., Th Thory of ndurial Organizaion, MT Pr, Cambridg, MA

27 A Lif B War Lvl Figur 1: Schmaic diagram of groundar aquifr dpicing h lif AB

28 Marginal Bnfi A Marginal Co B φ S O S nvmn Figur : Marginal bnfi and co of invmn undr ol-onrhip

29 N Payoff xi payoff Payoff from Accommodaion O 0 φ φ Figur3: N payoff from xi and accommodaion ragi.

30 nvmn NC N S Lo nqualiy Modra nqualiy High nqualiy Vry High nqualiy φ φ a φ Sady a Sock S N Lo nqualiy Modra nqualiy High nqualiy Vry High nqualiy φ φ b φ Figur 4: ffc of nqualiy on nvmn and Sady Sa Sock