Appendix. Each partner receives an equal share of the total efforts, N

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1 Appndx In ths ppndx w provd th thortcl rsults on whch our hypothss n th mn ppr r bsd on. Frst w dscrb th modl st-up. roposton nd r lrdy shown by Kndl nd Lzr 99. Th rsults o roposton 3 nd 4 r nw nd show how tm sz cts pr prssur nd th orts lvls chosn n strt-up tm. Our thortcl mplctons r bsd on n xtnson o th modl n Kndl nd Lzr 99. W xplctly modl th ct o tm sz on ort n prtnrshps. W show tht dpndng on th montorng tchnology n strt-up prtnrshp tm pr prssur dpnds on. In th bsnc o spcc ssumptons t s not clr whthr pr prssur domnts r-rdng nd t whch sz o th tm. Thror w hv to ntroduc svrl ssumptons tht hold or strt-ups nd smlr prtnrshps nd thrby nlyz th ont ct o tm sz on ort n ths tms. W ssum n ccordnc wth Brüdrl t l. 996 tht strtup tms r rst nd ormost chrctrzd by thr prtculr prsonl rltonshp. W rgu tht clos nd stdy prsonl ntrcton wth hgh-rquncy dcson mkng s thr most mportnt chrctrstc. Foundrs n th srvc sctor or xmpl typclly work n on oc or on on loor sd by sd; oundrs n mnucturng my strt n on bg hll or smll lb. Du to ths sptl proxmty nd th typcl strt-up condton o contnully mkng st nd otn undmntl dcsons strt-up prtnrs constntly ntrct normlly ormlly or rndomly nd sty n clos prsonl contct. In our thortcl modl w buld on ths spcl tur o strt-up tms trnsltng t nto montorng tchnology whch s n ssnc chrctrzd by cost uncton tht s nvrnt to. W modl th mpct o ths montorng tchnology on pr prssur nd on th ont ct o both pr prssur nd rrdng n rlton to th sz o strt-up tm nd smlr prtnrshps. Bsd on Kndl nd Lzr 99 w us th ollowng gnrl ssumptons. W ssum strt-up wth homognous prtnrs = who shr thr tm output ch rnng / o th tm output. Ech prtnr chooss n ort lvl wth s cost uncton whr. Indvdul ort s not obsrvbl. Tm output s uncton o ch prtnr s ndvdul ort gvn by. To provd rson or tms w ssum tht s not sprbl n. W spcy th producton uncton s ollows 3.. tm sz cn vry wthout chng n producton tchnology. Th prtl drvtv on s ort s postv nd thr r dmnshng rturns n s ort. 4 5 Ech prtnr rcvs n qul shr o th totl orts /. 6 Furthr w ssum tht ort lvls r not complmntry.. th output o totl ort s no mor thn th sum o th outputs o th ndvdul orts. Snc w r not lookng t sls or prot o th strt-up s n output vrbl but t ndvdul ort xrtd by th prtnrs t sms nturl to ssum tht totl ort s th sum o ndvdul orts. Ths s n ln wth Adms ssumpton bout mdcl nd lgl prctcs whch us smlr producton tchnology.

2 Ths ssumptons drctly ld to th wll-known rst proposton on r-rdng n prtnrshps: Ech prtnr wnts to mxmz hs utlty.. th shr o th output mnus th costs o ort 7 u mx Hnc th rst-ordr condton or prtnr s u '. 8 solvs ths condton. Snc th cost uncton s convx s lowr thn th cnt lvl. Th cnt soluton rqurs tht th totl surplus s mxmzd: '. 9 onsquntly th ort lvl lls short o th cnt lvl s rsult o prot shrng; ths rcton ncrss wth. roposton. Ths r-rdng ct ncrss wth tm sz. Ech prtnr s wr o ths r-rdr problm. Thus tm mmbrs know tht ll othr tm mmbrs work t lowr thn cnt lvl.. lss thn n compny ownd by sngl prtnr. Hnc ch prtnr wnts th othr prtnrs to choos n ort lvl hghr thn thr ndvdully optml r-rdr lvl. Howvr orts r not obsrvbl nd nonsprbl producton uncton s ssumd montry ncntvs ntroducd by n ltrntv compnston schm othr thn qul prot shrng cnnot b usd to solv th problm. 3 Thus pr prssur sms to b good ltrntv s rgud by Kndl nd Lzr. Kndl nd Lzr 99 urthr lst two condtons or pr prssur to b ctv s motvtonl dvc p. 85: Frst th ort choc o tm mmbr should ct th utlty o th othr tm mmbrs. Thror th rst o th tm hs n ncntv to xrt prssur on. Gvn prot shrng ths condton s ullld. Scond ch tm mmbr should hv th blty to ct th chocs o. W show tht prtculrly n strt-up tms th scond condton s lso ullld. Followng Kndl nd Lzr w rstly ssum tht pr prssur countrcts r-rdng s ollows. r prssur s uncton o th ndvdul s ort choc nd o n stblshd group norm ē: ē. Th prssur ls dcrss wth hs ort wth dmnshng ct:. Wth th xstnc o pr prssur th mxmzton problm o tm mmbr bcoms u mx wth th rst ordr condton bng u. 3 Ths proposton hs lrdy bn provd.g. by Bru L nd Whnston 995:5. 3 ontrct thory stblshs tht pr prssur cn only xst th ctons o th group mmbrs r obsrvbl wthn th tm but not vrbl. Ths mns tht th ort lvls cnnot b provd n court o lw nd thror n ncntv contrct cnnot b bsd on ths ort lvls.

3 Th lvl o ort tht solvs ths rst ordr condton xcds th lvl o ort n th cs wthout pr prssur. r prssur thror stblshs cost on xrtng too lttl ort; howvr ths cost cn b dptd by n ndvdul s own ort choc. Th mor works th lss pr prssur h or sh ls. roposton In comprson to stuton wth r-rdng only th ort lvl chosn undr r-rdng nd pr prssur s hghr. Wth pr prssur th ort lvl s hghr thn n th cs wthout. A proo o ths proposton s lrdy gvn n Kndl nd Lzr 99:85. Although t s wll known tht tm sz drvs th r-rdr ct nmly prtnrshp sz t s lss clr wht drvs pr prssur. W rgu tht th pr prssur ct s drvn by th montorng tchnology usd n tm nd w show tht undr wll spcd condtons whch r typcl or strt-up tms mutul montorng dpnds on. r rssur by utul ontorng n Strt-Up Tms In rmwork wth mutul montorng n ddton to th ndvdul choc o ort ch tm mmbr cn lso montor othr tm mmbrs by undrtkng n cton. In strt-ups ths montorng ctons r typclly rsult o prsonl ntrctons btwn th prtnrs. As rgud bov tm mmbrs usully work n clos proxmty. Frqunt prtnr mtngs r typcl o strt-ups not only du to sptl proxmty but lso to constnt nd or dcson mkng nd coordnton owng to n otn hghly voltl nvronmnt. Accordngly th oundrs hv mny possblts to montor ch othr long th wy bcus o ths ntnsv ntrctons nd th xtnsv norml xchng o normton. In ths spcl st up pr prssur thus rsults rom th probblty o bng dtctd by pr whl wthholdng ort. W ssum tht th montorng ctons thmslvs hv no drct ct on tm output n th spcl orgnztonl sttng o strt-ups s dscrbd bov. W nclud montorng ctons n our modl by mns o dtcton probblty.. th probblty tht prtnr dtcts prtnr shrkng. Snc montorng ctons rqur ddtonl ttnton nd concntrton w urthr ssum tht thr s cost nvolvd n tkng montorng cton. So w rdn th cost uncton s nd ssum ccordngly:. 45 Th pr prssur on prtnr ncrss - but t dcrsng rt - montorng ctons o prtnrs ncrs. I thr s lrdy lrg mount o normton on s work vlbl ddtonl normton only hs smll nlunc on th pr prssur thy lrdy l. Accordngly w dust th pr prssur uncton whr nd Thus th ncntv not to shrk s hghr th mor montorng ctons occur whch s rsonbl bcus th probblty o bng cught ncrss wth rsng montorng ctons. As consqunc th ncntv to rduc pr prssur by showng grtr ort ncrss. Th mxmzton problm s thn 9 u mx Th nxt quston s wht th consquncs or optml ort n strt-up tms r. r prssur xplctly dpnds on th othrs montorng ctons nd rprsnts th dsutlty tht 3

4 4 ls. Both ctons.. th ort nd th montorng cton r chosn mor or lss smultnously. In ordr or montorng ctons to hv clr mpct on th ort dcson n th modl w ssum tht montorng ctons r chosn rst nd tht th ort dcson s tkn trwrds. To solv ths squntl structur w pply bckwrd nducton. Stg : choc o u Stg : choc o u ondton dtrmns th mount o montorng cton. Th rcton o th othr prtnr to th chosn montorng lvl wth rgrd to thr own ort choc s ncludd or vry. Ech prtnr knows how th othr tm mmbrs rspond. In ths rmwork mny qulbr my xst but w concntrt on th symmtrc qulbrum. Lmm A symmtrc qulbrum wth = =... = n und = =... = n xsts. I such symmtrc qulbrum xsts thn = s th bst nswr to = =... = nd by nlogy = s th bst nswr to = =... =. Group mmbrs mxmz thr utlty gvn th ort nd montorng chocs o th othr prtnrs - und -. In Stg th rst ordr condton quton dos not chng n th qulbrum cs. Evry prtnr hs th sm cost uncton nd th sm pr prssur uncton nd thror thy ll choos th sm ort lvl. In Stg th rst ordr condton chngs to u... u 3 From 3 t ollows tht n qulbrum xsts n whch ll mmbrs choos th sm lvl o montorng cton. Rrrng to th spcl stuton n strt-up tm ths mns or xmpl tht ll prtnrs prtcpt n th sm mtng or work n th sm room. Thus prtnr solvs th ollowng problm u mx 4 By usng th mplct uncton thorm w cn gur out how tm mmbrs rct by thr ort choc to th montorng cton o mmbr n qulbrum. Thus w drntt condton wth rspct to bcus th problm s symmtrc cross ll prtnrs. 5 Th numrtor s postv nd th dnomntor s ngtv so tht quton 5 s postv. Hnc ddtonl montorng ncrss ort smlr to wht s shown wth th modl o Kndl nd Lzr 99: 8. W now tk th nlyss on stp urthr wth our modl nd

5 study whthr thr s systmtc ntrcton btwn tm sz nd th lvl o montorng. 4 W rgu tht th numbr o tm mmbrs.. th numbr o montors s crucl. W show tht n ncrsng numbr o tm mmbrs ntnss pr prssur du to n ncrs o. I or xmpl mor prtnrs prtcpt n oundrs mtng th probblty tht ny knd o shrkng or consumpton on th ob wll b dscovrd ncrss bcus ch prtnr ntnss thr montorng ctons. An ncrs n tm sz lds to strongr r-rdng ct. Ths ct s wll known. Howvr t s not s obvous how tm sz cts th montorng ctons. A r-rdr ct mght rs hr s wll or on th othr hnd hghr ntnsty o montorng mght b chosn. I th montorng ctons dcrs or rmn constnt n th cs o n xpndng tm th r-rdr ct wll domnt th pr prssur ct. onsquntly ch tm mmbr wnts to compnst or ths ngtv ct by ntnsyng thr montorng lvl. Ths wll b th cs th ddtonl gn rom n ncrs n pr prssur s hghr thn th ddtonl ndvdul montorng costs. Ths trnslts nto our modl s ollows: roposton 3 Th ntnsty o mutul montorng ncrss wth th numbr o prtnrs d d. 6 roo : A dtld drvton cn b ound n th ppndx. Th probblty o dtctng somon shrkng ncrss wth. To drv ths rsult w ssum tht tht th montorng cton hs postv but dcrsng ct on ort whch s plusbl ssumpton or strt-up tms s wll s or othr prtnrshps 7 Thus th ntnsty o th ndvdul montorng cton rss wth. Tht s th probblty o dtctng shrkng ncrss. Ths rsult s drvn by th ssumpton tht th ndvdul costs do not dpnd on tm sz whch s rsult o th spcl stuton n strt-up tms wth ts sptl proxmty nd rqunt ntrcton. onsquntly th montorng costs o ch prtnr do not rs wth tm sz. Ths ssumpton sms vry plusbl or strt-ups bcus th ndvdul montorng costs.g. or prtcpton n mtng or wtchng co-workr n smll grg or oc r mor or lss ndpndnt o tm sz. Howvr mor prtnrs r sttng n mtng th probblty tht prtnr dtcts shrkng prtnr ncrss. So th quston s how dpndng on tm sz r-rdng nd pr prssur ntrct n strtup tms. Th wll-known ct o r-rdng s tht ort lvl s rducd wth ncrsng tm sz. r prssur countrcts r-rdng wth ncrsng sz bcus t cuss ort lvls to rs gvn th montorng tchnology tht cn b ssumd or strt-ups whr th probblty o bng dtctd whl shrkng rss wth th numbr o th tm mmbrs. Lmm Th ort lvl dpnds on two countrctng cts nd both cts ntnsy wth. To dtrmn how th ort lvl dpnds on w drntt wth rspct to Kndl nd Lzr 99 lso rs th quston whch ct group sz could hv on mutul montorng p. 8. Thy rgu tht th ct o th montorng cton on th ort choc s clos to zro. In contrst w r nlyzng th stuton n strt-ups whch r oundd by smll numbr o prtnrs so s smll nd t ll clos to zro but not to. A dtld drvton s ound n th mthmtcl ppndx. 5

6 d 8 d From ths w cnnot drv unnmous prdcton. Both th r-rdng ct nd th pr prssur ct ncrs wth th numbr o prtnrs but so r w cnnot prdct whch ct domnts or whch tm sz. To obtn rsult concrnng th rltonshp btwn ort nd tm sz w drntt scond tm. 6 d d Th ort uncton s concv n.. th ct o n ncrs n tm sz thr postv or ngtv dcrss wth. I w ssum n ntror soluton t ollows tht thr xsts n optml tm sz or whch th ort lvl s mxmzd. onsquntly n xtrmum dpndng on xsts t s mxmum ort lvl. roposton 4 Assumng n ntror soluton th ort lvl ncrss up to n optml tm sz nd thn t dcrss. Ths rsult llows th ollowng ntrprtton concrnng th rlton btwn r-rdng nd pr prssur: Snc th uncton s concv w hv hgh montorng rturns n smll tms. Th pr prssur ct domnts r-rdng up to th mxmum. Atr ths crtcl tm sz s rchd th rlton bcoms ngtv so th ort lvl dcrss bcus o th thn domntng r-rdr ct. Thus w cn drv th ollowng hypothss or our mprcl nlyss.. Th ort lvl ncrss wth tm sz lvls o nd thn dcrss tm sz kps growng.. Thr s n optml tm sz or whch th ndvdul ort o th strt-up tm mmbr s hghst. 9 thmtcl Appndx In symmtrc sh qulbrum nd ; urthr th rcton unctons or ch prtnr r ccordng to nd 3.. F nd F : F F 3 3 Th totl drntls r: df d d d df d d d 3 33 wth nd J 39 6 Agn s mthmtcl ppndx or th drvton. 6

7 roo o proposton 3 J 4 d J 4 d J b roo o quton 8 d J 4 d J J 43 c roo o quton 9/proposton 4 W ssum tht th thrd-ordr prtl drvtvs qul zro. F F F F F F F F F d 44 d F F wth F F ; F Insrtng th vlus nd solvng lds to F nd ; F d. 5 d 7