THE EFFECT OF COST CHANGES IN AN OLIGOPOLY WITH APPLICATION TO BANK LENDING DISCRIMINATION. August 2002

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1 THE EFFECT OF COST CHAGES I A OLIGOPOLY WITH APPLICATIO TO BAK LEDIG DISCRIMIATIO August 00 Wllam D. Bradford School of Busness Unversty of Washngton Seattle, WA 9895 Prevous versons of ths paper were presented at the Econometrc Socety Meetngs (00), the Mdwest Economc Assocaton Meetngs n Chcago (00), and at a semnar at the Unversty of Washngton (00). Helpful comments from partcpants n those semnars and from Wayne Ferson are apprecated. bradford@u.washngton.edu Phone: Fax: /7/00

2 THE EFFECT OF COST CHAGES I A OLIGOPOLY WITH APPLICATIO TO BAK LEDIG DISCRIMIATIO Abstract Ths paper extends prevous analyses of the effect of exogenous cost changes on the performance of frms n an olgopoly. The analyss s framed n the context of dscrmnaton n bank lendng. Recent U.S. bankng studes have assumed that greater lendng dscrmnaton (bas) aganst ethnc mnortes results n mnortes payng an ncreasngly hgher nterest rate than nonmnortes as bank concentraton rses. Usng a framework of asymmetrc Cournot-ash competton, I show how the dscrmnaton tself changes concentraton, and that bas and concentraton may be postvely or negatvely assocated. Also, the profts and market shares of based banks may rse or fall as bas ncreases.

3 THE EFFECT OF COST CHAGES I A OLIGOPOLY WITH APPLICATIO TO BAK LEDIG DISCRIMIATIO. Introducton The effect of exogenous cost changes on the performance of frms n an olgopoly has been explored by Kmmel (99), Lahr & Ono (988) (997), and Zhao (00). Kmmel analyzed the outcomes when the costs of all of the competng frms exogenously change. Lahr & Ono and Zhao (the latter for the case of lnear demand) analyzed the case when the costs of one frm (or / frms) change. I extend ther analyses n three ways. Frst, the model developed heren allows for any subset (/ through ) of the frms costs to change, for both lnear and non-lnear demand. Second, I extend the framework to consder the case when competng frms products are lnearly heterogeneous as well as homogeneous (the papers above assume homogenous products). Thrd, I show the mplcatons of the exogenous cost changes for concentraton among the competng frms. Ths paper reflects my pursut of a model that reflects Becker s dscrmnaton coeffcent for banks competng n an olgopoly. In hs grounreakng work on dscrmnaton, Becker (97) focused prmarly on labor markets, and not the dscrmnaton of sellers aganst buyers n product markets. Hs wage dscrmnaton model concluded that non-economc dscrmnaton reduces the frm s profts. Thus hghly compettve markets would purge dscrmnatory behavor from the market place, whle non-economc bas could be sustaned n Other related papers nclude Levn (985), Okuguch (993) and Salant and Shaffer (999). These papers have more restrctve assumptons concernng the dstrbuton of the cost changes than does ths paper. one of these papers relate the nature of market demand to the effect of cost changes, nor do they nclude product dfferentaton n the context of cost changes for olgopolsts. Anderson, S., de Palma & Kreder (00), Delpalla & Keen (99) and Stern (987) examne the effect of costs n the form of taxes on profts and welfare, but they assume that the competng frms have dentcal costs; thus they do not consder volume for ndvdual frms, market shares or concentraton. In Becker s analyss the sellers laborers are homogenous and the labor market s compettve. Here the sellers are banks wth asymmetrc costs operatng n a Cournot-ash olgopoly.

4 3 less compettve markets. Berkovec et al (998), Cavalluzzo and Cavalluzzo (998) and Cavalluzzo, Cavalluzzo and Wolken (00) translated Becker s conclusons nto bankng markets, concludng that when lendng dscrmnaton exsts among banks, ) mnorty borrowers pay hgher nterest rates than non-mnorty borrowers, and ) the dfferences n the nterest rates pad by mnorty and non-mnorty borrowers expand as bankng markets become less compettve. These studes test these effects, but do not buld a model of bankng that provde a theoretcal framework for the tests. What does a formal model of bankng ndcate about these relatonshps? In ths paper, some banks have a bas aganst lendng n the subject market, where the market s defned as loans to a specfed group of borrowers (e.g., racal mnortes, women). Bank nterprets ts bas as a cost, b, n ts proft functon when t determnes how much t wll lend n ths market. The bas s exogenous to the loan producton costs. I examne the mpact of the bas on the structure and performance of the competng banks, when b s postve for n of the banks (0 < n ) and the banks compete n an asymmetrc Cournot-ash olgopoly. I fnd that frst, the dsfavored group pays a hgher rate of nterest and receves a lower loan volume when bas ncreases among the banks. Second, whle profts and market share always ncrease for unbased banks as bas ncreases, profts and market share may also ncrease for some of the based banks. Thrd, an ncrease n bas among the based banks can ncrease or reduce the concentraton of loans among banks n the market, dependng on the sze of the based banks, and the nature of the market demand for loans. Fourth, an ncrease n the proporton of based banks can magnfy or reduce the changes n profts and market concentraton, dependng on the dstrbuton of costs of the based banks among the total group of banks. Thus n the framework of a Cournot-ash olgopoly, the dsfavored group pays hgher nterest and receves a lower volume of loans, consstent wth the assumptons of the studes on bank lendng dscrmnaton. However, n contrast to those studes assumptons, bank

5 4 concentraton may ncrease or declne wth bas among banks. We can observe markets wth hgher bas and thus hgher nterest rates, but lower ndces of bank concentraton. Fnally, an ncrease n bas may not penalze the based banks n profts and market share. In fact, both may ncrease. More generally, these results are nconsstent wth both the Structure-Conduct- Performance paradgm that hgher concentraton s ted to hgher nterest rates on loans (see Hannan, 99) and the Relatve-Effcency paradgm that hgher concentraton would be assocated wth lower nterest rates on loans (Demsetz, 973). In a Cournot market, when costs (and thus market shares) dffer, addng a layer of costs reduces output, ncreases the prce, redstrbutes market shares and changes the concentraton n the market. The redstrbuton and change n concentraton depend on how relatve costs change and how demand changes as quantty suppled changes; thus concentraton may ncrease or decrease. The cost changes n ths paper relate to bas, but n other cases can reflect other charges, such as taxes and fees. The remander of ths paper s organzed as follows. Part ntroduces the settng and the basc model. Part 3 examnes the mpact of bas on the equlbrum nterest rate and total loan volume. Parts 4 and 5 dscuss the loan volume at ndvdual banks and the profts of banks as bas changes, respectvely. Part 6 analyzes the relatonshp between changes n bas and changes n market shares and concentraton among banks. Part 7 examnes the mpact of changes n bas when bank loans are lnearly heterogeneous. Part 8 contans an example usng a constant elastcty demand curve, and Part 9 concludes the paper.. Basc Approach Consder a bank that ssues deposts, D, and uses deposts and captal funds, K, to make M categores of loans. Varable costs are separable by actvty (admnsterng loans or deposts) and by category of loans. The markets for each type of loan and deposts dffer from each other. The profts of the bank can be expressed as

6 5 M (a) π = ( r a ) L ( r + a ) D F j= j j j d d where r and a represent the nterest yeld and admnstratve cost rate assocated wth loans and deposts, and F denotes fxed costs. The borrowers wthn each loan category have the same credt rsk. At ths pont I wll relate the analyses on endogenous costs to bank lendng dscrmnaton. In hs analyss of prejudcal dscrmnaton, Becker hypotheszed that those who have a taste for dscrmnaton behave as f they were wllng to pay somethng, ether ectly or n the form of a reduced ncome, to ndulge those tastes (Becker, 97, p. 4). 3 The money costs of a transacton do not always completely measure net costs, and a dscrmnaton coeffcent acts as a brdge between money and net costs. Assume that market s the dsfavored group. Thus defne b as the bank s dscrmnaton ( bas ) coeffcent for lendng to customers n market, where b 0. In ths market banks seek to maxmze proft, defned as revenues less expenses; and the expenses nclude a charge for the dsutlty of lendng to the borrowers n ths market. Thus the endogenous cost analyzed n Lahr-Ono, Kmmel and Zhou s emboded n the bas coeffcent, b. The bank s profts can be restated as M (c) π = ( r a b) L+ ( rj aj) Lj ( rd + ad) D F j= The pecunary costs of makng a loan may dffer among banks only due to dfferences n the admnstratve costs of lendng. When I place b, the bas coeffcent, nto the proft functon n calculatng π, the defnton of profts here shfts closer to a measure of utlty, measured n unts 4 of dollars. In other words, bank. b s a monetary equvalent and π s a monetary equvalent to the 3 Zzzo and Oswald (000) provde recent evdence on the wllngness of people to pay to reduce others ncomes. 4 See Frexas and Rochet (997), chapter 3 for a standard dervaton of these relatonshps. The typcal dervaton ncludes securtes as an alternate asset. I omt securtes snce ncludng securtes provdes no addtonal nsghts.

7 6 From ths pont, the analyss focuses only on the market for type loans, and the subscrpts refer to the banks that compete n ths market. Type loans are assumed to be homogeneous (I op ths assumpton later), and the banks n ths market compete n a Cournot- ash olgopoly. There are banks and no possblty of entry. 5 The total volume of bank loans n ths market s L, and the market nterest rate on loans s r, where r = r and r ( L ) < 0. For ash stablty t s assumed that (d) r ( L ) + Lr < 0 for all. Ths assumpton corresponds to the normal case n Seade (980a) and to strategc substtutes n Bulow et al. (985). The margnal costs, a, are constant for each bank over the relevant range, but vary across banks. Gven ths framework, bank s reacton curve s mplctly defned by the frst order condton: dπ (a) = r+ r L a b = 0, dl or (b) r+ r L = a + b A Cournot-ash equlbrum n the market for these loans s an output vector such that each bank s loan volume, L, s a best response to the vector of all of the other banks best response. Ths equlbrum s graphcally the ntersecton pont of all reacton curves and algebracally the soluton to (). Thus n equlbrum: (3) ( r a b) L = r The costs of lendng for bank have a techncal component and a bas component. Snce r (L) 5 Entry nto bankng s regulated and restrcted. However, non-bank lenders can compete for loans, dependng on the type of loan. Small busness lendng has hstorcally been very costly, because of the paucty of nformaton about small frms and the hgh cost of personnel requred to obtan and evaluate even that nformaton. Banks domnate that market. The emprcal studes on lendng dscrmnaton cted above assumed that banks operate n olgopolstc markets.

8 7 < 0, the sum (a + b ) and L are nversely related: relatvely hgh cost banks have lower loan volumes than low cost banks. But a based bank wth hgh qualty technology,.e. low a, may have a lower loan volume than an unbased bank wth low qualty technology. Assume that b s n postve and equal for n banks: b = b for n and b = 0 for n. Defneθ. The fracton θ of the banks are based, where 0 θ. Frst, the equlbrum nterest rate for loans, r, s derved. Sum L over all to obtan total loan volume, L: (4) L= ( r a ) r b ( L ) = = Then r can be expressed as: (5) r = a + θ b Lr ( L ) = Lr = a + θb where a = a. In ths equlbrum the nterest rate on loans s a functon of both the = technology of the banks, a, and the bas of the n banks,θb. The market premum for bas s the mean of the bas coeffcent over the total number of banks. I now adess comparatve statcs of the model, assumng an equal ncrease n the bas coeffcents of the n (= 3. The Interest Rate on Loans and Total Loan Volume θ ) based banks. Suppose = > 0 for θ banks and = 0 for (-θ) banks. The changes n the nterest rate and loan volume resultng from a common change n the bas coeffcents of the θ based banks are as follows: Proposton : (6) θ r = > 0 ( + ) r + Lr dl θ and = < 0. ( + ) r + Lr

9 8 All proofs are n the Appendx. An ncrease n bas ( > 0) ncreases the nterest rate and decreases the total loan volume. As the fracton of based banks rses, the hgher the ncrease n the nterest rate and the greater the declne n loan volume, as measured by expresson (6). If only one bank s based all banks are stll affected by the ncrease n bas, but the mpact s smaller. 6 Expresson (6) may be compared to expresson (A) n Kmmel (99), expresson (.) n Lahr and Ono (997), and proposton () n Zhao (00). Kmmel assumes that θ =, or that all banks have the same exogenous cost ncrease. Lahr and Ono (997) and Zhao (00) assume that θ = / or that only one bank has the cost ncrease. Ther analyses are specfc assumptons about θ. One can also express as (7) θ = + + E( L) where E(L) s the elastcty of r (L) wth respect to L: (8) L Lr EL ( ) dl r r The change n the nterest rate s a functon of the fracton of banks that are based, θ, and E(L). The latter s dscussed n Seade (980b, 985) and Kmmel (99). Two standard demand relatonshps are useful for understandng the relatonshp between E(L) and. Frst, a constant elastcty nverse demand curve such as r = AL /ε results n E(L) = ; where ε, the ε standard measure of demand elastcty, s negatve. Exhbt shows the demand curve, M, for the curve defned as r = 000L /(-0.35) ; for L n the 0 30 range (ths curve wll be used later n an 6 I assume that θ banks have the same ncrease n bas. It s well known that n Cournot models, ncreases n total costs of the same amount wll produce the same change n nterest rate and output, regardless of the composton of the change (for example, see Bergstrom and Varan (985)). Thus (6) can reflect the effect of any combnaton of bas ncreases among the banks, as long as the changes sum to the same amount as assumed n (6).

10 9 example). Lne C s the slope of M, r (L), whch changes over L. Thus E(L) s the elastcty of C wth respect to L. Snce ε = -0.35, E(L) s a constant over the range of L. Expresson (7) also translates that > (overshftng occurs) when E(L) < - ( θ). If demand s lnear, as assumed by Zhao (00), then E(L) = 0, snce the slope of a lnear θ demand curve s constant. Here = + <, whch approaches as θ and ncrease. Fnally, E(L) s negatve (postve) for a demand curve that s convex (concave) at the relevant level of L. 4. Loan Volume at Indvdual Banks. The mpact of bas on the loan volume of ndvdual banks depends on whether or not the bank s a based bank, as demonstrated n Proposton (Prop.). Defne the proporton of total loan volume held by bank as Proposton : S, where S L L ; (9a) (9b) dl = [ + SE r ] f b = b dl = [ + SE ] r f b = 0 dl and 0 > f b = 0. Recall that dl dl dl = < 0. Snce = > 0 for each unbased bank, the sum of changes n loan volume for based banks must be negatve and larger n absolute sze than the sum of changes for unbased banks. When demand s lnear, then E(L) = 0 and that dl <. Here (9a) shows < 0 n ths case. All based banks lose volume when bas ncreases. But f demand s

11 0 not lnear, loan volume may ncrease at some bas banks when bas coeffcents ncrease. 7 Substtute (7) nto (9a) and defne K 0 (a) dl > 0 f S < K 0 when E(L) < - (- θ), and (b) ( θ ) +. θ ( θ ) + + EL ( ). In the case of based banks, θ E dl > 0 f S > K 0 when E(L) > Suppose E(L) < - (- θ) < 0. 8 The ntuton (see Kmmel, p. 443 for a dscusson S L r b a when θ = ) s that from expresson (3), = =. If S L r b a j j j > (whch holds when E(L) < - (- θ) ), as b ncreases r b ncreases and S S j approaches. The market shares of banks wth S < K0 ncrease and offset the declne n the total loan volume, so that the loan volumes of those small based banks ncrease. 9 At the extremes, whenθ =, dl > 0 when S < + E( L) dl. However, when θ =, the requrement for E > 0 s + EL ( ) S <. The stablty EL ( ) requrement (d) translates nto + E > 0. Therefore, + EL ( ) EL ( ) s negatve, meanng that f only one bank s based, t cannot gan volume when ts bas coeffcent rses; regardless of how otherwse effcent t s. 7 Expressons (9a) and (9b) may be compared to (A) n Kmmel (99). Kmmel does not analyze the mpact of a change n cost on the volume at ndvdual frms. 8 Dvde (d) by r (L) and sum over all. The stablty requrement becomes + E(L) > 0, or that E(L) > -. Thus E(L) < - (- θ) - < 0 whle E(L) > - f - < E(L) < - (- θ) -. Ths set can be nonempty f θ > /. 9 In ther model of a homogeneous olgopoly, Salant and Shaffer (999) conclude that volume must declne at frms that have the exogenous cost ncrease. However, n ther model total ndustry costs are held constant, thus the nterest rate s constant; and the mpact of the change n the nterest rate dscussed here does not occur n ther model.

12 Alternatvely, suppose E(L) > ( θ ) + θ > 0. From (7) < f E(L) > 0. As b ncreases r b declnes, and the effect of dfferences n producton technology (a and a j ) on market shares ncreases. The market shares of banks wth S > K 0 ncrease enough to offset the declne n total market loan volume. At the extremes, f θ =, then dl > 0 requres that S > + E( L) dl. If θ = then E > 0 requres that S > + EL ( ), but EL ( ) + EL ( ) EL ( ) >, whch exceeds the doman of S. As wth EL< ( ) 0, when EL ( ) > 0 and only one based bank exsts ( θ = ), that bank wll lose volume when ts bas coeffcent ncreases, regardless of the bank s loan producton effcency. 5. Profts As the lendng bas ncreases, the nterest rate on loans ncreases and total loan volume declnes; but the loan volume at ndvdual banks may ncrease or declne dependng upon the loan producton costs and the bas of the bank. How do these relatonshps affect the profts of ndvdual banks? The effect of a common change n the bas of theθ based banks on bank profts s as follows: Proposton 3: (a) dπ L[ E( θs ) ( + ( θ))] = f b = b, θ dπ (b) = L [ + S E ( L )] > 0 f b = 0. The profts of unbased banks ncrease as the bas coeffcent rses. Ths follows because r ncreases for all banks and the loan volume of unbased banks ncreases. In the case of based dπ banks, < 0 when demand s lnear, snce E(L) = 0 n (a). When demand s not lnear,

13 (E(L) 0), consder the term nsde [ ] n the numerator of (a). Defne K + ( θ ) + EL ( ) θ E. In the case of based banks, (a) dπ > 0 f S < K when E(L) < - (- θ), and (b) d π > 0 f S > K when E(L) > [ θ + ] ( ) θ. Suppose E(L) < - (- θ) < 0. Here dπ > 0 f S < K. otce that K = K 0. The range of S wthn whch π > 0 s wder than the correspondng range for whch d dl > 0. Ths reflects that proft can ncrease for some banks wth lower loan volume when the ncrease n the nterest rate, r, s suffcently large. In the extremes, when θ =, dπ [ + E] > 0 f S <. E When θ =, the requrement s ncrease for one based bank s not possble. [ + EL ( )] S <. But EL ( ) [ + EL ( )] EL ( ) s negatve, and a proft Suppose E(L) > [ θ + ] ( ) θ dπ > 0. Here > 0 when S > K. Here K > K 0. The range of S wthn whch π > 0 s smaller than the correspondng range wthn whch d dl > 0. Snce r ncreases at a slower rate than b when E(L) > 0, t becomes more dffcult to acheve the volume ncreases necessary to ncrease profts. At the extremes, when θ =, dπ > 0 when [ + E] S >. When E θ =, dπ [ + EL ( )] > 0 when S >. Snce EL ( ) S, a proft ncrease for the one based bank s not possble. 6. Market Shares and Bank Concentraton Proposton 4: The dscusson of changes n market shares begns wth Prop. 4. [ + EL ( )] EL ( ) > and

14 3 (4) a a + b θb S = + Lr where b = b for a based bank and b = 0 for an unbased bank. Snce r (L) < 0, hgher loan producton cost and more bas wll reduce market share. But lower producton cost can offset lendng bas, and hgher producton cost may offset a lack of bas, so that hgh and low market share banks may nclude both based and unbased banks. Prop. 5 relates to the mpact of an ncrease n the bas coeffcent on the market shares of banks. Proposton 5: ( θ) Lr + ( θ )[ a a ( θ) b] ds (5a) = [ Lr ] f b = b θ Lr + ( θ )[ a a + θb] ds (5b) = [ Lr ] f b = 0 and ds > 0 f b = 0. Prop. 4 and 5 here are Kmmel s Prop. 3 reformulated to adjust for any fracton of frms havng an exogenous cost ncrease. Prop. shows that the total loan volume declnes wth ncreases n the bas coeffcent, and Prop. 3 shows that the loan volume at each unbased bank ncreases when the bas coeffcent ncreases. Thus one should expect that market shares of unbased banks ncrease as the bas ncreases (5b). In the case of based banks, the frst term n the numerator of (5a) s negatve, reflectng the reducton n the demand for loans assocated wth the hgher nterest rate. The change n S can be postve only f [ θ ][ a a b( θ )] s postve, or f [ θ ] and [ a a b ( θ )] have the same sgn. For lnear demand curves, θ > 0 always holds, whle

15 4 for nonlnear demand, θ 0 as E(L) -. 0 If ds θ > 0 then > 0 only f the bank s a large effcent bank,.e., a a > b( θ ). If θ 0 < then ds > 0 only f the bank s less effcent ( a a < ( θ ) b). A change n bas also affects the concentraton rato, snce market shares change when the bas coeffcents change. The measure of concentraton used here s the Herfndahl ndex, H, S = where H. H has been used as a measure of concentraton by the Federal Reserve n ts decsons on allowng mergers between banks. I wll analyze the change n H: H = Hˆ H, resultng from a change n the bas coeffcent, where Ĥ and H are the Herfndahl ndex after and before the change, respectvely. H can be expressed n component parts as ˆ = = (6) H = S S, where S S + and S s the change n the market share of bank. Then ˆ S (7) H = S S + ( ) = = S Wth ths relatonshp n mnd, Prop. 6 dscusses the effect of a change n bas on the Herfndahl ndex of concentraton. Proposton 6: (8a) dh ( b ) ( = θ S θ )( H ) f H >, Lr dh θ ( θ ) (8b) = [ Lr ] f H = 0 θ[( + E] From (7), θ =. For lnear demand curves, E(L) = 0. The stablty condtons requre + + E that + + E(L) > 0; thus for non-lnear demand curves, the sgn of θ comes from [( + E(L)].

16 5 where S b s the average fracton of the total volume of loans held by the θ based banks: S b S j = where S j n. θ Expresson (7) s the bass for the proof to Prop. 6. When H > the frst term n (7) s nonzero and domnates the second term, so that normal calculus technques (whch assume that the second term s zero) can be used, resultng n (8a). When H =, the frst term n (7) s zero, and (8b) shows the result. Prop. 6 shows that concentraton may rse or fall when bas ncreases, dependng on the total volume held by based banks and the sgn and magntude of θ. Recall that the mnmum H s and that r ( L ) s negatve. Suppose S b >, meanng the average based bank holds more loans than the average unbased bank n the market. If S b > and θ then dh < 0. If the based banks are relatvely large and f the nterest rate ncrease s larger than the fracton of based banks, then concentraton falls as the bas ncreases. Based banks, whch are the larger, more effcent banks, become less relatvely dh Snce H = S, then usng calculus, = = ds S. When H >, the second term n (7a) s = very small compared to the frst term. When H = (whch requres that S b = ), expresson (8a), whch reflects the frst term n (7), s zero. Thus expresson (8a) s zero when H =. For example, expresson (9) n Dxt and Stern (98) s ncorrect when H =. The second term n () can be used to show the change n H as bas changes. Salant and Shaffer (999) dscuss the mpact of cost changes on the Herfndahl Index, but they do not model the specfc change; and more mportant, they assume that the sum of the costs (the sum of producton costs and bas n ths paper) does not change. If total costs do not change, then the nterest rate and total loans do not change. I show that the change n the nterest rate produces a change n the dstrbuton of loans amongst the banks. Thus a change n the rate of nterest assocated wth a change n bas causes a change n concentraton.

17 6 effcent and lose market share. Market shares overall become more evenly dstrbuted, and concentraton declnes. Here t s unambguous that concentraton declnes as bas ncreases. Suppose S b <, or that the average based bank holds less loans than the average unbased bank. In ths case, when θ, then dh > 0. Snce based banks are smaller and ther costs ( a + b ) are larger, the dfference n costs between based and unbased banks wdens. The ncrease n bas causes the dstrbuton of loans to become less even, and H ncreases. In ths case concentraton ncreases as the bas of the based banks ncreases. These are the only unambguous cases. The other possbltes lead to ambguous outcomes. When S b < (based banks are small ) and θ < then dh s postve (negatve) f ( θ )( H )s greater (less) than θ ( Sb ). Or when b S > (based banks are large ) and θ > then dh s postve (negatve) f ( θ )( H ) s greater (less) than θ ( S b ). The outcome of more bas can be ether more concentraton or less concentraton among banks. I showed earler that the nterest rate on loans ncreases wth more bas: > 0. Thus more bas wll always ental hgher rates pad by borrowers n the market. On the other hand dh > 0 or dh < 0. More bas may produce hgher concentraton or lower concentraton, dependng upon the cost structure of the based and unbased banks, the proporton of banks that are based, and the amount of loans demanded by borrowers gven the rate of nterest on loans.

18 7 7. Lendng Bas n a Model of Dfferentated Bank Loans It may be more realstc to assume that banks have some degree of product dfferentaton n the market for loans to the dsfavored group. Ths mples a model that allows for heterogenety among banks. Suppose the loans of the banks are horzontally dfferentated such that r, the nverse demand functon for bank, s (9) r = v β L w Lj j where v, β, and w are all postve parameters, wth 0 < w < β. I wll normalze w and β so that β, whch mples that w (0,). The symmetrc degree of loan substtutablty between any two banks s measured by w. If w = 0, the demand for the loans of each bank s ndependent from that of the other banks, whereas f w =, the loans are perfect substtutes, and the market for these loans s a homogenous Cournot olgopoly wth lnear demand. I mantan the assumpton that b = b for a based bank and b = 0 for an unbased bank. The sngle perod soluton for L, L and π under Cournot competton are (0) () ( v a θb) L =, + w ( ) L = ( v w Lj a b ) j ( wv ) [ + w ( )]( a + b) + w( a+θb) =, and ( w)[ + w( )] () π = L. Gven these relatonshps, t follows that (3) [( ) w+ ][ a a + θb b] S = + ( v a θb)( w) Ths demand structure for dfferentated markets s used by Spence (976), Majerus (988), Yuan (999) and Hackner (000), among others.

19 8 Expressons (0) and () show that the effect of b s to reduce to reduce the overall supply of loans and the loans of bank. Gven ths framework, Prop. 7 dscusses the outcomes when bas changes. Proposton 7: When the market for bank loans s dfferentated by the functon: r = v β L w Lj, The followng relatonshps hold: a. Loan Volume j (4a) (4b) dl w [ ( θ ) ] = ( w)[ + w( )] dl wθ = ( w)[ + w( )] < 0 f b = b > 0 f b = 0 b. Profts dπ w[ ( θ )] (5a) = L ( w)[ + w( )] dπ wθ (5b) = L ( w)[ + w( )] < 0 f b = b, > 0 f b = 0 c. Market Shares ds G[ θ ( v a) p( v a)] (6) = ( v a θb) where p = f bank s based, p = 0 + w ( ) ds f bank s unbased, G > 0, and ( w ) > 0 f b = 0. d. Concentraton (7a) dh θ b = G( S ) ( H ) f v a θb H > (7b) dh θ( θ) G = ( v a θb) f H =

20 9 As mentoned earler, EL ( ) = 0 when demand s lnear; and for lnear demand, as bas ncreases, loan volume and profts always declne for based frms and ncrease for unbased frms. Wth regard to market shares, the market shares of unbased frms always ncrease as bas ncreases. The market shares of based frms wll ncrease or decrease dependng on sgn of the term nsde [ ] n (6). When θ = /, ths term can be rewrtten as follows: (8) θ(v a ) (v ā) = j a j ( ) v From (), v > a for all postve L s. There are ( ) a s and ( ) v s. Thus θ(v a ) (v ā) s always negatve f θ = /: when only one bank s based, t loses market share f ts bas ncreases. For θ > /, (8) can be rewrtten as (9) θ(v a ) (v ā) = (- θ)(ā u v) θ(a ā b ) where ā b and ā u are the average loan producton costs of the based and unbased banks, respectvely: θā b + (- θ)ā u = ā. Thus for a based bank, ds > 0, f (30) a - ā b < ( θ ) ( au v) θ Snce v > ā u, the RHS of 30 s negatve. Thus a based bank s producton cost must be lower than the average producton cost of based banks n order for the bank s market share to ncrease ds as bas ncreases. If θ =, the requrement for 0 > s a < a. Effcent (and large) banks gan market share wth more bas. In terms of concentraton, smlar to the case wth homogeneous banks, H may ncrease or declne wth more bas. The key term s S b. If ths term s suffcently postve, then concentraton falls as bas ncreases, and f t s negatve, concentraton rses as bas ncreases. If based banks hold a suffcently hgh market share (reflectng lower costs), then the dstrbuton of loans becomes more evenly dstrbuted when bas ncreases. But f based banks are small

21 0 (reflectng hgher costs), ther loss of loan volume as bas ncreases can lead to more concentraton of loans held by unbased banks. 8. Example Exhbt shows the equlbrum outcomes n nterest rate, loan volume, market shares, concentraton, and profts under the constant elastcty demand structure, r = GL /u ; where G =,000, u =0.35 and = 5 banks. Here EL ( ) = = The exhbt shows the.35 equlbrum outcomes of sx dfferent cost structures under ths demand functon. Case 0 s the base case. Here b = 0 for all banks. Bank has the hghest loan producton cost (a), banks, 3 and 4 have equal producton costs and bank 5 has the lowest producton cost. In cases through 5, an ncreasng number of the banks dscrmnate, and the bas coeffcent s the same for all based banks. In case only bank has a postve bas coeffcent. Banks and are based n case. Banks, and 3 are based n case 3. In case 4, banks,, 3 and 5 are based. All of the banks have a postve bas coeffcent n case 5. Comparson of the outcomes n r, L, L, S, H and π across the demand structures demonstrate several of the propostons above. They show that as the degree of bas n the market ncreases:. The nterest rate ncreases and total loan volume declnes. The equlbrum n case 0, the no bas case, has the lowest nterest rate and hghest total loan volume. Borrowers are worse off n all cases of lendng bas.. For ndvdual banks, market share may ncrease (compare market shares n case 0 wth those n case 5 for banks through 4) or declne (compare case 0 wth case 4 for the banks,, 3 and 5); market share always ncreases for unbased banks (compare the market share of bank 4 n case 0 to ts market share n cases through 4). 3. Concentraton (Herfndahl ndex) may ncrease (compare case 0 wth cases through 3) or decrease (compare case 0 wth cases 4 and 5).

22 4. At ndvdual banks, profts may ncrease or decrease. Consder bank, whch s based n cases through 5. Profts are hgher n cases 3, 4 and 5 than n case 0, and lower n cases and than n case 0. I should add that the emprcal studes have dffered n ther measurement of concentraton. Berkovc et al use the Herfndahl ndex of bank loans, whle Cavalluzzo et al use the Herfndahl ndex of bank deposts. The ssue behnd the use of concentraton measures s the extent to whch a few banks exhbt control over nterest rates on loans. 3 Thus the use of depost concentraton s a proxy for the concentraton of loans n the market. Depost concentraton mght not correlate wth control of prces and quanttes n specfc loan markets, and thus may be only a nosy measure of control n loan markets. But bank regulators use depost concentraton n ther analyss of control n bankng markets. 9. Concluson A Cournot model has been extended to consder the mpact of lendng bas on the performance of commercal banks. The loan volume, profts and market shares of based banks may ncrease as they ncrease ther degree of bas, even when there are competng unbased banks. In addton, hgher bas leads to hgher nterest rates on loans but loan concentraton may ncrease or declne. Ths latter fndng conflcts wth the assumpton used by several emprcal studes that f bas exsts, nterest rates should ncrease as concentraton ncreases. The result that hgher bas can lead to proft ncreases for some banks begs the queston of how prejudcal dscrmnaton s defned. Becker (97, p. 4) states If an ndvdual has a taste for dscrmnaton he must act as f he were wllng to pay somethng, ether ectly or n the form of a reduced ncome, to be assocated wth some persons nstead of others. When actual dscrmnaton occurs, he must, n fact ether pay or forfet ncome for ths prvlege. 3 See Gerosk (983).

23 I underlned the last sentence to emphasze that n Becker s framework, dscrmnaton occurs only when the dscrmnator suffers a loss. 4 I argue that dscrmnaton should nclude cases n whch the dscrmnator gans from bas, f the dscrmnator was actng as f he were wllng to pay somethng along the lnes consdered here. When banks add a non-economc charge to ther proft functons for lendng to dsfavored customers, some of those banks may ncrease profts whle others suffer proft reductons. But the cost of borrowng always ncreases and the amount of loans declnes the borrower always loses. Because of the negatve mpact of the bas, all who add the charge should be consdered as practcng prejudcal dscrmnaton. 4 Becker (993, p. 8) wrtes that dscrmnaton n the marketplace conssts of voluntarly relnqushng profts, wages, or ncome n order to cater to prejudce.

24 3 APPEDIX Proof of Proposton Proof: Totally dfferentate (b) to obtan: (A) r dl+ Lr dl+ r dl = da + In the case of an ncrease n the bas coeffcents of the n based banks, da = 0, = n and = 0 for n. Sum (A) over all and make use of the relatonshp that L = L. The result s = (A) dl θ = ( + ) r + Lr dl By the stablty assumpton, the denomnator of (A) s negatve so < 0. Wth regard to the change n the nterest rate on loans as bas changes, dl (A3) = r θ = r. ( + ) r ( L ) + Lr dl Snce r (L) < 0 and < 0 then 0. > Q.E.D. Proof of Proposton Proof: Restate Expresson (A) wth da = 0 as dl dl dl (A4) r + Lr + r = For based banks =. Use (A) to substtute for dl and (8) to derve (9a). For unbased banks dl = 0, and use (A) to substtute for and (8) to derve (9b). In (9b), r s

25 4 postve and from the stablty requrement (d), + SE s postve; therefore (9b) s postve. Q.E.D. Proof of Proposton 3: For based banks, from (c) d (A5) π dl = ( r a b) + L( ) As b changes, L responds so that dl remans 0. Thus use the frst order condton (b) to replace (r a b) and (9a) to replace dl. The result s (A6) dπ = L [ (θ ) EL ( ) + θ S EL ( )] θ θ From (7) the followng holds: (A7) θ = [ + + E( L)] Use (A7) to substtute for the last θ n the numerator of (A6) and (a) follows. For unbased banks, from (c) dπ dl (A) = ( r a) + L Use (b) to replace r a (set b = 0) and (9b) to replace dl, and (b) follows. From the stablty assumpton (c) t follows that + S E(L) > 0; thus Proof of Proposton 4 Proof: Dvde L n (3) by L to obtan: dπ > 0 for unbased banks. Q.E.D. (A8) L ( r a b ) S = L Lr

26 5 Use (5) to substtute for r, whch results n (4). 5 Q.E.D. Proof of Proposton 5. For based banks, from (4) the dervatve of S wth respect to b s dlr [ ] ( θ ) Lr + [ a a b( θ )] ds (A9) = [ Lr ] The proof s completed by showng that dlr [ ] = ( θ ): dlr [ ] dl Lr = r [ + ] = [ + E]. From (A7), r [ + EL ( )] = ( θ ). Ths completes the proof for based banks. For unbased banks, dlr [ ] θ Lr + [ a a + θb] ds (A0) =. [ Lr ] Use the relatonshp that dlr [ ] ds = ( θ ), and (5b) follows. To show that 0 > always holds for unbased banks, frst note from (A7) that θ [ + E] ( θ ) = + + E( L), and ds [Lr (L)] > 0. Thus gven (A3), 0 > f (A) θ [ + E][ a + θb a ] θ Lr [ + + E] > 0. From (5), Lr a+ θb= r+, and from (b), r a = r L, snce b = 0 for an unbased bank. Substtute these nto (A) and the requrement becomes Lr (A) θ[ + E][ Lr S ] θlr [ + + E] > 0. 5 See Kmmel (99) for an alternate proof of Prop. 4 for the case n whch θ =.

27 6 Dvde by θ Lr, whch s postve, collect terms and cancel out the. The requrement for a postve ds s (A3) + S [ + E] > 0. ds The stablty assumpton (c) translates nto + SE > 0. Thus 0 > always holds when b = 0, or when the bank s unbased. Q.E.D. Proof of Proposton 6. When H >, from (7), (A4) dh ds = S = ote that consstent wth calculus methodology, the second term n () s assumed to be zero. By usng (5a) and (5b), (A7) can be restated as ( θ )[ a a b + θb] θlr dh Sb (A5) S θ = + [ Lr ] Lr = where b = b for a based bank and b = 0 for an unbased bank; and S b s the mean market share of the θ based banks ( S / θ ). The second term n (A5) reflects that the θ based banks n have a ( - θ)lr (L) term n (5a) whle the unbased banks have a -θlr (L) term n (5b). Smplfy (A5) by separatng out the θ Lr term from the frst part of (A5): ( θ )[ a a b + θb] θ S dh Sb (A6) = θ = S + [ Lr ] Lr Lr = Usng S =, and further refnement brngs =

28 7 (A7) ( θ ) dh [ a a b + θb] θ( Sb) = S Lr = Lr Lr The term nsde the { } brackets s equal to S. Ths and the facts that S = and = H S = produce (A8) ( θ ) ( θ ) dh θ ( Sb ) = H Lr Lr Lr b [ θ( S ) ( θ )( H )] = Lr In order to obtan (8b), when s calculated as S = for all, then H = and (8a) s 0; thus the change n H (A9) dh ds = = = = ( θ ) b θ ( a a + b θ ) b + Lr Lr Lr where b = b for a based bank, and b = 0 for an unbased bank. But n (A30) ( a a + b θb) Lr = S. When dh θ ( θ ) S =, (A30) becomes = [ Lr ]. Q.E.D. Proof of Proposton 7 a. Loan Volume dl p[ + w( )] +θ w (A0) = where p = for a based bank, and p = 0 for an unbased ( w)[ + w( )] bank. Ths expresson results n (4a) and (4b) for based and unbased banks, respectvely. a. Profts

29 8 Snce π = L, then d (A) π dl = L. Plug (A0) nto (A), whch produces: (A) where d π w p[ w( )] = L θ + ( w)[ + w( )] p = for a based bank and p = 0 for an unbased bank. b. Market Shares Dfferentaton of (3) produces ds ( θ p)[ v a θb] + θ[ a a + θb b ] (A3) = G ( v a θb) where p = and b = bfor a based bank; and p = 0 and b = 0 for an unbased bank. Collect terms and (6) s obtaned separately for based and unbased banks. c. Concentraton (A4) dh ds = S = From substtute for ds from (A3), whch results n dh G( θ p) θg( a a + θb b) (A5) = S + S v a θb ( v a θb) Use the relatonshp that = = θga ( a + θb b) θ = S ( v a θb) v a θb to obtan expresson (7a). In order to obtan (7b), recall that when n H s calculated as S = for all, then H = and (7a) s 0; thus when S = the change (A6) dh ds = = G( θ p ) = = v a θb = θ G ( θ ) ( θ) G θ + ( v a θb) ( v a θb)

30 9 The frst term reflects the θ based banks and the second term reflects the ( θ ) unbased banks. Cancelng out the redundant terms results n (7b). Q.E.D.

31 30 REFERECES Anderson, S., A. De Palmaand B. Kreder (00) Tax Incdence n Dfferentated Product Olgopoly Journal of Publc Economcs, 8: Becker, Gary S. (97) The Economcs of Dscrmnaton. Chcago: Unversty of Chcago Press. Berkovec, J., G. Canner, S. Gabrel and T. Hannan (998) Dscrmnaton, Competton, and Loan Performance n FHA Mortgage Lendng Revew of Economcs and Statstcs, May, Bergstrom, T. and H. Varan (985) When Are ash Equlbra Independent of the Dstrbuton of Agents Characterstcs? Revew of Economc Studes 5: Bulow, J., J. Geanakoplos and P. Klemperer (985) Multmarket Olgopoly: Strategc Substtutes and Complements Journal of Poltcal Economy 93: Cavalluzzo, Ken S. and Lnda C. Cavalluzzo (998) Market Stucture and Dscrmnaton: The Case of Small Busnesses Journal of Money, Credt, and Bankng 30:4 ovember, Cavalluzzo, Ken S. and Lnda C. Cavalluzzo and John Wolken (00) Competton, Small Busness Fnancng, and Dscrmnaton: Evdence From a ew Survey Workng Paper, Board of Governors, Federal Reserve System. Forthcomng, Journal of Busness. Delpalla, S. and M. Keen (99) The Comparson Between Ad Valorem and Specfc Taxaton Under Imperfect Competton Journal of Publc Economcs, 49: Demsetz, H. (973) Industry Structure, Market Rvalry and Publc Polcy Journal of Law and Economcs, 6: -9. Frexas, X. and J. Rochet (997) Mcroeconomcs of Bankng, Cambrdge: MIT Press. Gerosk, P. (983) Some Reflectons on the Theory and Applcaton of Concentraton Indces Internatonal Journal of Industral Organzaton :, Hackner, J. (000) A ote on Prce and Quantty Competton n Dfferentated Olgopoles Journal of Economc Theory 93:, Hannan, T. (99) Foundatons of the Structure-Conduct-Performance Paradgm n Bankng Journal of Money, Credt and Bankng, 3: Kmmel, Sheldon (99) Effects of Cost Changes on Olgopolsts Profts The Journal of Industral Economcs 40:4 December, Lahr, S. and Y. Ono (988) Helpng Mnor Frms Reduces Welfare The Economc Journal 98:4 December, Lahr, S. and Y. Ono (997) Asymmetrc Olgopoly, Internatonal Trade, and Welfare: a Synthess Journal of Economcs (Zetschrft fur atonalokonome) 65:3, Levn, D. (985) Taxaton Wthn Cournot Olgopoly Journal of Publc Economcs 7:8-90.

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33 3 Exhbt The Demand for Loans 0.5 r = L r 0. M Slope of M C L

34 33 EXHIBIT : EXAMPLE USIG COSTAT ELASTICITY DEMAD Iseolastc Demand: r = GL -/u Example: G =,000 u = 0.35 = 5 banks Base: a Wth bas coeffcent: a + b Costs (a + b ): Structr 0 Structr Structr Structr 3 Structr 4 Structr 5 Bank Bank Bank Bank Bank Total Costs Interest Rate ( r ) Tot Loan Vol (L) Loan Volume (L ) Bank Bank Bank Bank Bank Total Market Shares (S ) Bank Bank Bank Bank Bank Total Herfndahl (H) Profts Bank Bank Bank Bank Bank Total Calculatons: r = Total Costs/( - /u) Bank Loan Volume = S L Total Loan Volume = (G/r) u Market Share (S ) = {[r - (a + b )]u}/r H = S + S + + S 5 Bank proft = [r - (a + b )]S L

35 34