Supplement to Holmström & Tirole: Market equilibrium. The model outlined in Holmström and Tirole (1997) illustrates the role of capital,

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1 1 Jon Vsle; Septemer 2014 and out ECON 4335 Economcs of Bankng Supplement to olmström & Trole: Market equlrum The model outlned n olmström and Trole (1997) llustrates the role of captal, oth among entrepreneurs, unnformed nvestors and anks (or montorng nsttutons), for the undertakng of nvestment projects under market falure caused y moral hazard, oth among entrepreneurs and anks. (The frst group the entreprenuers should, rrespectve of how the project s fnanced, e nduced to choose the good lottery the only one eng socally enefcal wth a hgh proalty of return and wth no prvate eneft, B, that s reaped only f the ad lottery s chosen), whereas the anks should, under ndrect fnance e nduced to montor (at some cost c) and also get the entreprenuer to choose the good lottery wth no prvate eneft as gven n the montor case y < B, and also + c < B. In the model, some crtcal values are derved (as done n the artcle), from whch we can dvde the orrowers nto three groups: The captal rch entrepreneurs wth a good record who solely use drect fnance, entrepreneurs wth medum sze captal get ndrect fnance (from anks actng also on ehalf of unnformed nvestors), and captal poor entrepreneurs young frms wthout any record wll not undertake the nvestment (ut wll make ther funds avalale n the market). The memer of ths last group has too lttle captal to put nto the project so as to convnce external actors that they wll ehave dlgently or eng suffcently responsle for choosng the good lottery. (A orrower has to have enough skn or stake n the game n order to have any ncentve to ncrease the value of the project; ths ncentve s related to own captal put nto the project.) From the relevant ncentve and ndvdual ratonalty constrants we get to the upper crtcal value, denoted Ar (), wth r as the requred rate of return to unnformed nvestors, as the lowest value of captal that wll enttle an entrepreneur

2 2 wth drect fnance. Ths crtcal value s shown n the paper (not necessary to defne the remanng symols) to e gven y: The project s externally fnanced f and only f: A+ I ³ I or p B (1) A ³ I - I ³ I - [ R ] : A( r) 1 + r - Dp = where Ar () > 0y assumpton, wth I eng the amount nvested y outsde nvestors, ncreasng n r and ncreasng n the agency cost B = B D p - p p p p L as well. (Ths agency cost s hgher the hgher s the opportunty cost to the entreprenuer to work hard, B, as well the lower s the lkelhood rato Dp p whch sgnals the nformatveness of oservng hgh return. The hgher s ths lkelhhood rato, the more lkely s t that the entreprenuer has worked or not shrked (chosen the good lottery), when oservng hgh return, and the smaller s the agency cost. (An entrepreneur wth captal elow ths crtcal value wll not e granted credt wthout anks. A frm wth equty or wealth aove ths crtcal value, wll put her entre captal nto the project, whereas the unnformed nvestor wll at least put n I An underlyng assumpton s that p R -(1 + r ) I > 0 > p R -(1 + r ) I + B ; only L - A.) projects wth no shrkng wll e socally desrale to undertake. Entreprenuers that are ht y the constrant (1), wll e constraned even though the good project should e undertaken. There s credt ratonng n equlrum; a market falure. The queston then s: Can ths market falure e allevated y ntroducng anks? By ntroducng anks, that can montor and get the entreprenurs to work (wth a lower opportunty cost or smaller prvate eneft from choosng the ad lottery, ), the captal constraned entrepreneurs can now e fnanced and therefore undertake ther projects. For such ndrect fnance to take place, ncentve as well as ndvdual

3 3 ratonalty constrants have to e satsfed, when the anks requred rate of return s l (whch must exceed r ecause of montorng cost c, and that anks can act lke unnformed nvestors wthout undertakng the montorng task, y requrng a rate of return r ). We then fnd that a ank wll go n wth an nvestment I pc I ( l): =, decreasng n l, showng the smallest amount an (1 + l) D p entrepreneur wll orrow from a ank. (Indrect fnance s more expensve than drect fnance.) We then have that f A + I + I ( l) ³ I, a project wll e fnanced, whch can e translated to a lower crtcal value on own captal, elow whch no entreprenuer wll get her project fnanced: pc p + c (2) Ar (, l) = I-I ( l) - I = I- - ( R- ) (1 + l) D p 1 + r Dp Ths lower crtcal value s ncreasng n oth rates of return, r and l. (It s assumed that Ar (, l ) < Ar ().) We then have: Frms wth captal AÎ éar (, l), A() r ù ê ë ú û, n numer, GAr ( ( ))- GArl ( (, )), each eng fnanced y anks wth orrowng I ( l ), total demand for ank credt s: (3) D ( r, l) : = ég( A( r)) G( Ar (, l) ù I ( l) êë - úû Note that for a gven r, the lower crtcal value wll e ncreasng n l ; hence the numer of frms eng granted ank credt wll go down. Also, ecause I ( l ) s decreasng n l, a hgher requred rate of return from anks, for a gven requred rate of return from unnformed nvestors, wll reduce total demand for ank credt. D ence we have < 0. Because oth crtcal values are ncreasng n r, we cannot l state precsely the sgn of D r ; t wll depend on propertes of the dstruton GA. ( )

4 4 In equlrum, wth an exogenous ank captal, that s put nto the projects (showng the anks lendng capacty), as gven y s: (4) K = ég( A( r)) G( Ar (, l) ù I ( l) êë - úû For any r and K, the equlrum condton for ank credt K, ths market clearng condton wll determne a unque rate of nterest for ank credt (or nformed captal, the term used y &T), as gven y l (, rk ). Note that for a smaller value of K, l must e hgher. Why? Suppose the opposte, clamng that l wll go down. Then I ( l ) wll ncrease, and ecause the lower crtcal value wll e lower, we get that more frms eng granted a hgher loan from the anks. Ths s not compatle wth a lower ank captal. ence; the rate of nterest for ank credt must ncrease as K ecomes smaller;.e.; l(, rk) < 0. K To go a step further, we can look at the full market equlrum, where we consder the par of nterest rates ( r, l ) that wll smultaneously clear the market for ank credt as well as the market for unnformed captal. Let household savng e gven y an ncreasng savng functon Sr (), and let ths e part of the supply of unnformed credt, along wth supply of funds or captal y entrepreneurs wth captal/cash elow the lower crtcal value Ar (, l ). ence the supply of unnformed captal s Ar (, l) (5) Sr ( ) + AgAdA ( ) ò 0 whereas demand s gven y Ar ( ) ò ò ò (6) D( r, l) = éi A I ( l) ùg( AdA ) éi Ag ù êë - - úû + êë - úû ( AdA ) = [ I -Ag ] ( AdA ) -K Ar (, l) Ar ( ) Ar (, l)

5 5 where we have used (4). ence we have the fnal equlrum condton, requrng that total supply of external captal; LS of (7) elow s equal to aggregate demand for captal; the RS of (7), where the dstruton of A s defned on é0, ù êë úû : Ar (, l) ò ò 0 Ar (, l) (7) Sr ( ) + AgAdA ( ) + K = [ I-AgAdA ] ( ) What s now the mpact on equlrum of changes n exogenous varales, related ether to (negatve) shft n demand for captal, or shft n supply of captal. A negatve shft n demand can e traced ack to a collateral squeeze as frm values are reduced n a recesson (the alance sheet channel, as the dstruton G s shfted to left; accordng to frst order stochastc domnance. The supply can go down, ether ecause K can e reduced (through the lendng channel; a credt chrunch), so that the anks lendng capacty s reduced (due to hgh losses on loans), or savngs among households are suject to a negatve shft ( hoardng lqudty ). What s the mpact on the equlrum rates of nterest due to these changes? Comparatve statcs Suppose the supply of captal wll go down; the LS of (7) s reduced. To restore equalty, the RS must of course go down. One outcome s that the lower crtcal value must ncrease. Then prevously ank fnanced frms close to ths crtcal value wll now e dened fnance. If the reduced supply s the outcome of a reducton n K, l wll ncrease. ence, Ar (, l ) wll go up; entrepreneurs wth captal close to ths crtcal value wll now e moved from undertakng the project to ecome supplers of captal n the unnformed segment, whereas entrepreneurs close to the upper crtcal value are unaffected. owever, ecause the supply of unnformed captal now s ncreased, r mght go down, gvng a eneft to the captal rch entrepreneurs. (owever, ths lower r wll reduce savng from households.) At least one of the

6 6 nterest rates must ncrease under a captal squeeze. Now anks demand a hgher rate of nterest whereas ond owners may requre a lower rate of nterest. If households reduce ther savng, through a negatve shft n the supply functon Sr (), whle ank captal s kept fxed, the frst effect s, I guess, that Ar (, l ) must ncrease, as r s ncreased n the frst step. Wthout further nformaton aout GA, ( ) we cannot predct the mpact on l. (What we can say, ecause of (4), l and r must change so that (4) s oeyed, wth K kept unchanged.) At last, f a shock s explaned y a collateral squeeze, when the dstruton G moves to the left accordng to frst order stochastc domnance, then from (7), and wth Ke = Ag( AdA ), whch s reduced due to the squeeze (the avergae ò 0 entrepreneur has less captal), we have: ò (7) Sr ( ) + K + K = I gada ( ) = Ié1 -GAr ( (, l)) ù e êë úû Ar (, l) If the LS of (7) s reduced, when average frm value declnes durng a recesson, the RS must declne as well, through an ncrease n the lower crtcal value. (ere we have a fxed scale of nvestment.) At least one of the nterest rates must ncrease so as to ncrease the lower crtcal value. The weakest entrepreneurs, those entrepreneurs wth captal close to Ar (, l ) are hurt. Then we have the followng: Lower values of captal among entrepreneurs (as wll happen durng a recesson) wll reduce the fnancng optons for undertakng proftale nvestment projects, and hence amplfyng the ntal recesson. (Ths s sometmes called an accelerator effect.)