Government intervention in credit allocation: a collective decision making model. Ruth Ben-Yashar and Miriam Krausz* Bar-Ilan University, Israel

Transcription

1 Govermet itervetio i credit allocatio: a collective decisio makig model Ruth Be-Yashar ad Miriam Krausz* Bar-Ila Uiversity, Israel Abstract The urose of this study is to address the imortat issue of govermet itervetio i credit markets through govermet loa rograms. This toic has sigificace for the ecoomic develomet of certai weaker sectors i society ultimately affectig overall ecoomic ad social coditios. We aly a collective decisio makig model to a bak's credit decisio rocess to show the effect of govermet loa rograms o credit allocatio. This allows us to aalyze the effect of the decisio makig structure of a bak (the collective decisio rule, e.g., cetralized ad decetralized, as a crucial factor i determiig the effectiveess of govermet itervetio. Key words: govermet loa rograms, collective decisio makig. JEL classificatio: G, G8, D7 *Corresodig author, kerausm@mail.biu.ac.il

2 Govermet itervetio i credit allocatio: a collective decisio makig model. Itroductio I this research we focus o the govermet's efforts to icrease the amout of credit available to certai tyes of borrowers such as miorities, wome ad develoig regios or idustries. We aly a collective decisio makig model to a bak's credit decisio rocess to show the effect of govermet itervetio o credit allocatio. The combiatio of these two strads of research allows us to aalyze the decisio makig structure of a bak as a crucial factor i determiig the effectiveess of govermet itervetio. Fiacial itermediaries have a imortat role i the ecoomy of trasferrig fuds from ivestors to roductive eterrises. But, due to asymmetry i iformatio, baks are ot always able to erform their task fully ad heomea such as credit ratioig (Stiglitz ad Weiss, 98 are observed. Govermets itervee i the credit market i order to rovide loas i cases where rivate markets will ot. They may be actig i resose to geeral credit ratioig i the market or to difficulties faced by certai sectors such as small busiesses. Govermet loa rograms have macro-ecoomic imlicatios (Esiosa-Vega, Smith ad Yi, as well as affectig certai targeted sectors. Ideed weak sectors such as miorities, studets ad small busiesses costatly face credit ratioig imlyig that o itermediary will suly them with credit eve whe they are willig to ay high iterest rates. Cosequetly, may govermets have loa rograms (e.g SBA loa rogram, Faie Mae, Sally Mae ad Freddie Mac i the U.S., studet's loa rogram i Caada ad SBS loa guaratees i the U.K. desiged to icrease the suly of loas to these targeted sectors. Such loa rograms effectively costitute govermet itervetio i the credit market. The two most commo methods of itervetio are loa guaratees ad loa subsidies ivolvig either existig itermediaries or secial ledig istitutios. However, oce these tools are i lace, baks are ot obliged to use them. The usual loa screeig

3 methods ad credit scorig techiques are alied to loa requests ad the bak decides whether or ot to grat the loas. If a govermet i a free ecoomy does ot wish to imose credit o itermediaries, ad leaves the bak to make its ow decisio withi the framework of a credit rogram, the govermet caot be esured that the suly of credit will match its exectatios eve whe govermets establish secial itermediaries for credit rograms. By aalyzig the bak's credit decisio rocess we wish to rovide a exlaatio as to why govermets sometimes oly artially succeed i icreasig the suly of credit ad how govermets should act i order to icrease ledig to target oulatios. I the field of bakig a large body of literature deals with credit scorig methods (Mester, 997, Altma ad Sauders, 997 ad Alle, Delog ad Sauders, 4, while a much smaller amout of research has bee devoted to the desig of the decisio makig rocess i baks. Most otably Stei ( discusses the effect of cetralized vs. decetralized decisio makig o the share of small busiess ledig. His focus is o the effect of the bak's decisio makig desig o the ability of soft iformatio, cocerig small busiess loas, to comete with loa requests made by large firms. Stei (5 discusses the desig of cut-off methods for credit decisios whe a credit score is rovided ad Adersso (4 carries out a exerimetal study that determies the role of a decisio maker's exeriece i the decisio to led. I this aer the bak's credit decisio whether to accet or reect a secific loa request is aalyzed withi the framework of a collective decisio makig model, which focuses o the aggregatio of the decisios of a grou of exerts, i.e, the (collective decisio rule. The subect of otimal grou decisio-makig i a committee of fixed size that is subect to huma fallibility has attracted a great deal of attetio i the ast coule of decades or so. Nitza ad Paroush (98, 985, Grofma et al. (983 ad Shaely ad Grofma(984 are the first works which lay the theoretical foudatios of the biary choice model. Be-Yashar ad Nitza (997 defied the otimal decisio rule i a more geeral framework, which allows asymmetric choice. Other studies aalyzed the otimal decisio rule uder costraits (Be-Yashar, Kraus ad Khuller, ; Be-Yashar ad Kraus,, the otimal decisio rule i olychotomous choice (Be-Yashar ad Paroush, ad other asects of collective decisio makig. The results obtaied by several studies are

4 alicable to a variety of ecoomic fields such as labor ecoomics (e.g., Nitza ad Paroush 98, Maagemet (e.g., Be-Yashar ad Nitza, 998 ad Ivestmet ad Reliability Theory (e.g., Sah ad Stiglitz 988 ad Sah 99, 99. The collective decisio makig model assumes a team of decisio makers whose task is to arove or reect a roect so as to maximize the exected utility of the orgaizatio. The decisios of the team members are aggregated by a cetral laer uder a collective decisio rule, to rovide a fial decisio whether to accet or reect the roect. Withi our alicatio of the decisio makig model, each team member has exertise i determiig whether or ot a loa should be grated. The grou of team members ca also be iterreted as a grou of criteria used i a credit decisio model. Both loa guaratees ad subsidies ca directly affect the secific decisio but ot ecessarily the decisio rocess. This deeds o whether the bak always adots the otimal decisio rule or abides to a fixed decisio rule that may ot ecessarily be otimal. I the former case the decisio rocess will chage whe govermets itervee ad this must be take ito cosideratio by the govermet. I the latter case the decisio rocess does ot chage whe the govermet itervees. The distictio betwee the case where the rocess is affected ad the case where it is ot, is imortat because differet istitutioal frameworks are used by govermets for their loa rograms. I some coutries govermet loa rograms are oerated through rivate ledig istitutios that rovide fiacial services to the geeral oulatio icludig loas. I other coutries secial ledig istitutios are established to deal oly with govermet loas. Differet istitutios may differ i their willigess or ability to adust their decisio rocess. However, i both cases the ledig istitutio emloys screeig methods ad credit decisio makig rocedures i order to exted credit to high quality borrowers that are most likely to reay the loa. We therefore aalyze govermet loa rograms both whe the decisio ad the decisio rocess are affected as well as whe oly the secific decisio is affected. We cosider govermet itervetio whe baks follow the otimal decisio rule/structure ad whe baks kee to a rigid decisio structure which may ot ecessarily be otimal. 3

5 . The model We assume that there are decisio makers i a bak whose task is to arove or reect a loa alicatio so as to maximize the bak's exected rofit. There are two kids of loa requests, good ( or bad (-. A good loa rovides the bak with a rofit while a bad loa creates a loss. We deote by x i decisio maker i s decisio, where x i = (accet or x i =- (reect, ad a decisio rofile x={x,,x }. Let us deote by + x the umber of decisio makers who decide i favor of accetig the loa. The idividual's decisio regardig the tye of loa is based o his iformatio such as ast exeriece i ledig to the alicat, the loa alicat's leverage ad other attributes of the borrower ad the loa alicatio. Decisio maker i s decisioal skill is rereseted by i (the robability that idividual i accets a good loa ad reects a bad oe. We assume that the decisioal skill is /< i < ad decisioal skills are statistically ideedet across decisio makers. A collective decisio is reached by the bak maager who alies a decisive decisio rule that is a fuctio f that assigs (aroval or - (reectio to ay x i Ω={,-}, f: Ω {,- }. Each loa of size L at a iterest rate r fiaces a roect with a retur which is kow to the alicat but ot to the bak. The bak faces a cost r of fiacig the loa, such that r > r. All alicats require the same loa size such that size of the loa is set by the alicat. Prices are set cometitively i the loa market. Facig asymmetric iformatio, the bak sigs a stadard debt cotract with the borrower i order to miimize moitorig costs. The state of a loa ca be good with a riori robability > α >, ad bad with a riori robability α. The state is determied as a fuctio of the radom retur o the roect beig fiaced, R, which is draw from a kow distributio of returs i the oulatio of borrowers, rereseted by the desity ( h such that the state is (good if R > Lr ad - otherwise. The distributio of returs i the oulatio of borrowers is kow to the bak maager while a idividual decisio maker gais iformatio oly about the articular loa she is facig. The decisio makers may be iterreted as comoets of a comuterized credit decisio model. 4

6 The govermet ca itervee i two ways. It ca give a loa guaratee which is activated i the case of bakrutcy (whe R < Lr such that the bak is comesated by a roortio g of missig icome, Lr R. The govermet ca also subsidize the loa by reducig the bak's cost of fuds, r, which is activated whether or ot there is bakrutcy. Whe there is govermet itervetio, the state is good if r gr r gr R > L. Note that Lr > L. Therefore, the rior robability that the g g loa is good icreases with g ad/or with the reductio i r. The rofit associated with the aroval ( (reect (- of a good ( loa is deoted by B(, (B(-,, where B(,>B(-,. Similarly, the rofits associated with aroval ad reectio of a bad (- loa are deoted by B(,- ad B(-,-, resectively, where B(-,->B(,-. B(=B(,-B(-, is the ositive et rofit from a good loa ad B(-=B(-,--B(,- is the ositive et rofit from a bad loa. Without loss of geerality, we assume that reectio of a roect (good or bad is associated with zero rofit. That is, B(-,= ad B(-,-=. Hece, B(=B(, ad B(- = -B(,-. Note that, B ad r gr L g ( = ( R + g( Lr R Lr h( R dr ( Lr r gr L g ( R + g( Lr R Lr h( R dr + L( r r h( R B( = dr. ( Lr The et rofits are also affected by the loa guaratee ad by the subsidy as described later o i the aer. The exected rofit is give by α[b(,t(f:+b(-,(-t(f:]+(-α[b(-,-t(f:-+b(,-(-t(f:-] where T(f: ad T(f:- are resectively the robabilities of reachig a correct collective decisio whe the state is (good ad - (bad, give the decisio rule f. The above exected rofit ca be reduced to the followig form E=αB(T(f:+ (-αb(-t(f:-- (-αb(-. (3 5

7 The otimal rule (Be-Yashar ad Nitza, 997 fˆ is a qualified weighted maority rule ad give by: where f ˆ = sig[ w i x i + λ + δ ]. (4 sig{ M} = i= M >, wi = M l i i α B(, λ = l, δ = l. α B( The otimal rule is defied by the otimal weight w i that is assiged to idividual i s decisio ad by some bias comoets determiig the extet of the otimal bias toward oe of the alteratives: λ ad δ. λ reflects the asymmetry i the riors of the two states, δ reflect the asymmetry of the rofits associated with the two states. Notice that λ+δ rereset the combied bias due to a riori robabilities ad the rofits, which are a fuctio of g ad r so that govermet itervetio affects the bias too. i We assume that idividuals have homogeeous skills, that is = = i. I this case the otimal rule is a qualified maority rule. That is [ w + λ ] fˆ = sig x i + δ, where w = wi = w = l. Note that the qualified maority rule ca be rereseted by k (Be Yashar ad Nitza 997, where more tha k decisio makers are required to decide i favor of the loa i order to accet a loa. More recisely, f k = x + > k otherwise We ca oit to several iterestig rules. k = is the simle maority rule, that is if the umber of decisio makers is odd the miimum umber of decisio Uder the assumtio that >, the weight is o-egative. i 6

8 + makers i favor of the loa must be i order to arove the loa. Two extreme decisio makig systems ca be idetified, hierarchy ad olyarchy. I a olyarchy (decetralized, the accetace of the loa by oe decisio maker imlies that the loa is aroved (i.e., k <. I a hierarchy (cetralized the bak aroves a loa oly if it is acceted by all decisio makers ( k <. The otimal k, kˆ, is give by: + kˆ δ λ =. (5 l I the symmetric case where B(=B(- ad α =, the bias elemets vaish ad the otimal aggregatio rule is the simle maority rule, k ˆ =. If δ + λ >, the bias is i favor of accetig the loa request ad therefore k ˆ <, i.e., less tha half of the decisio makers are required to decide i favor of the loa i order for a accet decisio. I the extreme case, whe the bias is very large, oly oe decisio maker is required to make a ositive decisio. This is a olyarchy. The oosite is true whe δ + λ <. It is ossible that the bak will ot ecessarily use the otimal decisio rule. This is because there may be costraits o the bak that determie the weights ad/or the bias. I this case we assume that the decisio rule is a qualified maority rule rereseted by the miimum umber of decisio makers, q, required to decide i favor of a loa i order for the loa to be aroved ( q = mi ( k, k +. For examle, if is a odd umber the q = +, where q is the simle maority rule. We aalyze both cases where the otimal rule is used ad where it is ot. 7

9 3. Results As show above, govermet itervetio is reflected i g ad r such that whe there is govermet itervetio, the state is good if r R > L gr g. That is α > ad < r. The et rofits from good ad bad loas are also affected by govermet itervetio such that B( icreases with g ad decreases with r while B(- decreases with g ad icreases with r. 3 B ( B( <, r That is B ( B( >, r < ad >. All the followig results are reseted for the case of a govermet guaratee where g is icreased but hold also for the case of a subsidy where r is decreased. Our first set of results assumes that the decisio rule is fixed ad is rereseted by q. Note that i this case, sice the decisio rule remais uchaged, the robabilities of accetig good roects, T ( q :, ad of reectig bad roects, T ( q :, do ot chage as a result of govermet itervetio. We assig the robability of arovig a loa uder the decisio rule q by Pr ob ( accet : q. Result : The robability of aroval icreases with the magitude of govermet rob itervetio, that is i the case of a guaratee: Proof: ( q : + ( α ( T ( : Pr ob( accet : q = α T q, ( accet : q >. 3 Usig Liebitz rule for differetiatig a itegral, i this case Lr R>. Also, ad sice L ( < Lr B ( r gr, R > g r gr L g = B ( Lr = r gr L g ( ( Lr R h( R dr + < Lr. The same method leads us to the results for ( ( Lr R h( R dr + > (Note that r gr. (Note that i this case R < L( r. g 8

10 :, = q where T ( q = (. = q ad ( T ( q : = ( rob ( accet : q = T ( q : ( T ( q : = q = ( T ( q : ( T ( : = q where ( = ( Sice (a >. { }. If (b >, >. 4 (c = a, =. 5 a a q >, the from (a ad (b rob ( accet : q >. If q <, the from (c we kow that = q rob with (a Q.E.D. = = q+ ( accet : q. Sice -q+>/, this last term is ositive from (b, ad >. Result is a straightforward result that imlies that give a fixed decisio makig structure i the bak, the robability of accetig loas icreases with govermet itervetio. This is achieved simly by the fact that the share of good roects has, from the bak's oit of view, icreased due to govermet itervetio. 4 ( ( > > > >. >. (Note that uder the model's assumtios >/ ad hece 5 a ( [ ( ( ] a a a a = ad a ( [ ( ( ] a a a a = a a =.. Hece, a a 9

11 The imlicatio is that some loas that would have bee reected before the guaratee was itroduced will ow be aroved ad the govermet succeeds i icreasig the umber of loas allocated. Furthermore, the larger the guaratee, the greater the icrease i ledig. Also, the bak will always wish to articiate i the loa rogram because the guaratee icreases the bak's utility from ledig. This ca be see by derivig E=αB(T(f:+ (-αb(-t(f:-- (-αb(- (equatio (3 with resect to g: E = T ( f B( ( ( B B : + α ( T ( f : B( + ( α > We have determied that the govermet ca use the guaratee as a tool to icrease ledig. We ow show that the structure of decisio makig i the bak has crucial imortace i determiig the magitude of the effect of govermet itervetio o the robability of loa accetace. Result : The effect of govermet itervetio o the robability of accetig a loa icreases as a symmetric fuctio of the decisio rule, as the simle maority rule is aroached from either side. I other words, it is determied as a symmetric fuctio of q that eaks at the simle maority rule. That is: ad rob rob ( accet : q rob( accet : q + i rob( accet : q i q ( accet : q + i q i > q < where i is a iteger. = q Proof: From (c above, if q < the = = q = q+ ( accet : q rob( accet : q + rob =. q q, therefore: Secifically this is true whe q is q + i + = q i. + q = q i, the by substitutig q = +,

12 Hece, Recall that, ( accet : q i rob( accet q + i rob : = q q rob ( accet : q = = q ad from (b above, Furthermore, the term o the RHS of this iequality decreases with i. Q.E.D.. = q > = q + i. Result imlies that structure of decisio makig i the bak affects the success of the govermet i imlemetig a loa rogram. The govermet ca exect the highest degree of success whe the bak uses the simle maority rule to arove a loa request. Thus if the govermet decides to establish a secial ledig istitutio that grats loas withi the govermet's loa rogram, these istitutios should embrace the olicy of the simle maority rule. Also, the govermet ca achieve the same level of success whether the bak imlemets a oliarchy rule whereby oly oe decisio maker is required to arove a loa or a hierrachy rule, whereby all decisio makers must arove a loa. However, i both these cases the govermet achieves the lowest level of success. Furthermore, there is symmetry i the level of success that ca be achieved whe movig away from the simle maority rule towards hierarchy ad oliarchy. The imlicatio of this result is that as a bak is either more cetralized i its decisio makig rocess or more decetralized its decisio makig rocess, the level of success of a loa rogram is reduced. The worst case from the govermet's oit of view is to face either extreme cetralizatio or extreme decetralizatio. Whe the govermet decides to imlemet its loa rogram through the commercial bakig system, the govermet will robably ot be able to imose a decisio makig rocess o the ledig istitutio. At the same time, a utility maximizig istitutio is likely to adot the decisio rule that maximizes its utility. However, the otimal decisio rule is a fuctio of the a riori robabilities ad the exected returs, both of which are affected by the guaratee. Hece whe a guaratee is itroduced the otimal rule chages. I other words, govermet itervetio affects the way i which bak make their decisios cocerig loa aroval. I the followig set of results we show how the otimal rule is affected by

13 govermet itervetio ad we aalyze the level of success of such itervetio uder the otimal rule. From equatio (5 above it follows that the otimal decisio rule, kˆ is egatively related toδ + λ. Sice B( icreases with g ad B(- decreases with g, ( ( = l B α δ icreases with g. Also, because α icreases with g, λ = l B α icreases with g. That is if g icreases the the otimal structure of the bak becomes more leiet toward aroval of the roect, i.e., a smaller roortio of decisio makers is ecessary for collectively choosig to arove the roect. That is kˆ = l <. Note that this derivative is ot a fuctio of the decisio rule. We deote by qˆ the miimum umber of decisio makers required to arove the loa so that the loa is acceted uder the otimal decisio rule. Result 3: Whe the bak chooses the otimal decisio rule: (a The robability of accetig a roect icreases with g. (b Give a decisio rule qˆ, the effect of g o the robability of aroval is greater whe qˆ is a otimal decisio rule that chages otimally with g, rather tha whe it is a fixed rule. Proof: (a Pr ob( accet : qˆ = α T ( qˆ : + ( α ( T ( qˆ : ( accet : qˆ ( qˆ : ( T ( qˆ : rob T = T ( qˆ : ( T ( qˆ : + α + ( α Note that the sum of the first two terms o the RHS is ositive (from Result. Also, ( qˆ : T ( qˆ : T > sice qˆ = (, lˆ < qˆ, = lˆ where lˆ is the ew otimal rule (i.e., the miimum umber of decisio makers required to be i favor of a loa uder the ew otimal rule, i order for the loa

14 to be acceted. As we have show the rule is more leiet i accetig loas as a result of the icrease i g therefore lˆ < qˆ. ( ( T qˆ : Also, ( T ( qˆ : Hece, rob (b Note that >, sice qˆ = (, lˆ < qˆ. = lˆ ( accet : qˆ rob >. ( accet : qˆ is comosed of two elemets: ( T ( qˆ : ( T ( qˆ :, which is the effect of g o the robability of accetace whe the rule is fixed, qˆ, ad ( ( qˆ : ( T ( qˆ : T α + ( α, which is the effect of g o the robability that arises from chagig the otimal rule. Thus, give a decisio rule qˆ, the effect of g o the robability of aroval is greater whe qˆ is a otimal decisio rule rather tha whe qˆ is alied as a fixed decisio rule. Q.E.D. Result 3 imlies that it is i the iterest of the govermet that the ledig istitutio adusts its decisio makig rocess to the otimal rule rather tha remaiig with the same rule which erforms as a fixed rule, because the the govermet ca achieve a greater icrease i ledig. Thus, whe facig a ledig istitutio with ay tye of decisio makig rocess, the govermet will always achieve more whe the istitutio adusts its decisio makig rocess as a result of the govermet loa rogram istead of remaiig with its rule fixed. It is also imortat to ote that the utility of the bak icreases whe it articiates i the govermet loa rogram uder the otimal decisio rule. This ca be see by the fact metioed above that the bak's utility icreases uder a fixed rule. Sice otimizig the decisio rule always imroves utility it must be that baks that otimize their decisio rule certaily icrease utility i the resece of 3

15 govermet itervetio. Furthermore, the utility icreases more whe the bak otimizes the decisio tha whe a fixed rule is used. 4. Coclusio I this aer we have show that the structure of decisio makig i baks is a crucial factor i determiig the effect of govermet loa rograms o the amout of ledig. This has imortat olicy imlicatios for govermets that wish to icrease ledig to certai sectors. I essece our results oit to the coclusio that govermets ca achieve better results from their rograms whe facig baks that have either cetralized or decetralized decisio makig systems. I fact, the closer the decisio makig rocess is to the simle maority rule, the greater the effect of the govermet rogram. Furthermore, the govermet ca achieve suerior results whe ledig istitutios are versatile i their decisio makig rocess, choosig the otimal decisio rule i accordace with the level of govermet itervetio. I our aalysis the govermet always succeeds i icreasig ledig ad baks always fid it worthwhile to articiate i the govermet rogram. There may however be cases where baks will ot be willig to exted loas through a govermet rogram, or alteratively exted far fewer loas tha the govermet iteded. I our model this ca be exlaied by adverse selectio that affects the a riori robabilities. A larger umber of bad loas will eter the market, kowig that there is a icrease i the robability of loa aroval. This will chage the distributio of oulatio of borrowers faced by the bak ad accordigly will affect the rior robabilities of facig good ad bad loas. 4

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