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Strategic competition in retail banking under expense preference behavior
Journal of Banking & Finance 24 (2000) 809±824 www.elsevier.com/locate/econbase
Strategic competition in retail banking under expense preference behavior Pedro Purroy a, Vicente Salas
b,*
a
b
Escuela de Empresariales, Diagonal 696, 08034 Barcelona, Spain Facultad de Econ omicas, Economics Department, University of Zaragoza, Gran Võa, no. 4, 50008 Zaragoza, Spain Received 8 October 1998; accepted 24 May 1999
Abstract This paper presents a theoretical analysis of strategic competition in retail banking when some of the ®rms show expense preference behavior. The literature on expense preference behavior by banking ®rms is fairly large, but to our knowledge, so far, the strategic interaction between pro®t maximizing banks and banks with expense preference behavior has not been investigated. The paper has also an empirical interest since it is motivated by the current situation in Spanish retail banking where commercial banks, pro®t maximizers, compete with savings banks, non pro®t maximizers. Ó 2000 Elsevier Science B.V. All rights reserved. JEL classi®cation: G21 Keywords: Expense preferences; Strategic competition; Spanish banking
1. Introduction The expense preference behavior of banking ®rms has been considered a consequence of the ownership and governance structure of the ®nancial insti-
*
Corresponding author. Tel.: 34-976-761803; fax: 34-976-761767. E-mail address: [email protected] (V. Salas).
0378-4266/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 4 2 6 6 ( 9 9 ) 0 0 0 6 6 - 7
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tutions, and in particular of the peculiar assigment of property rights in banking ®rms such as savings and loans in the USA. The ownership of these ®rms is unclear and this leaves room for their control by managers and workers with preferences dierent from shareholders' value maximization. Several papers have looked at the implications for demand and interest rates of having banking ®rms maximizing an utility function which depends on pro®ts and labor expenditures. 1 These papers, however, look at the utility maximizing ®rm in isolation, i.e., as a market monopolist. The situation where pro®t maximizers and utility maximizers compete in an oligopolistic market has not yet been explored, and it is the main concern of the present paper. Our theoretical model ®ts well in the present situation of the Spanish retail banking market, where commercial banks compete with savings banks (`Cajas de Ahorros'). Since the mid seventies savings banks have increased their share of bank deposits from 30% to 50%. The erosion of commercial banks' market share in favor of savings banks has occurred at the same time that the latter outperformed the former in pro®tability and solvency. 2 Part of the growth of the savings banks has taken place through their purchase of regional banks far away from the geographic region where the savings institution was originally established. In other words, some saving banks are competing among themselves and with commercial banks nationwide. Bankers have raised their voice against certain growth practices of savings banks arguing that competition is highly assymmetric since savings banks do not face the restriction of dividends payment and that while they can purchase banks, commercial banks cannot purchase savings institutions since their ownership status does not allow them to do so. 3 There may be several reasons why saving banks outperform commercial banks in the deposits market: savings banks have concentrated their activities in market segments (low and middle income groups) who value the social programs carried out by these institutions through their `Obra Social'; they have provided customers with higher level of services due to their larger branching network and faster technological innovation in, for example, automatic teller machines; workers of saving banks are more ecient since they feel in part as `owners' of the ®rm and this increases their motivation and eort, etc. But the recent debate has focused mainly on the eects of dierent ownership and governance structures between banks and savings institutions, and the paper will focus on this line of explanation.
1
Akella and Greenbaum (1988), Edwards (1977), Hannan (1979), Krinsky and Thomas (1995), Rees (1974), Verbrugge and Jahera (1981). 2 Lagares (1995), Maroto (1993), Rodrõguez L opez (1995) . 3 Coello (1994) investigates the symmetry of competition between banks and savings institutions under a dierent perspective.
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Savings banks are often viewed in Spain as closer to workers' cooperatives and not for pro®t organizations. The answer to the question, `who owns the savings banks?' is di®cult to ®nd and the ®nal result may be that workers and managers take eective control of the organization. Besides, accounting pro®ts of the savings banks have to be either retained or allocated to social welfare programs. Orthodox theoretical thinking should lead us to conclude that organizations with such loose ownership structure and diuse allocation of property rights, should clearly be outperformed when competing with ecient, pro®t maximizing ®rms. But this has not been the case in Spain and many people view the current situation as a puzzle. The paper is an attempt to explain the puzzle. Our theoretical analysis shows that an expense preference behavior ®rm may outperform in market share and pro®ts to a pro®t maximizing ®rm when they compete in an oligopolistic market. The basic explanation of this result is that expense preference lowers the eective marginal (labor) costs of the ®rm, compared with the pro®t maximizing case, and as a result of this, its reaction function moves in a direction which implies an increase in market share and pro®ts. The ®nal results, however, depends on whether ®rms compete on quantities with homogeneous products, or the ®rms compete on prices with dierentiated products. 4 Although the paper focuses on the issue of expense preference versus pro®t maximizing behavior highlighted in the literature about the economics of the banking ®rm, its conceptual framework can be integrated into the broader topic of what is the real objective function of the ®rm when ownership is separated from control (Berle and Means, 1932). For example, writers from the managerial theory of the ®rm, Baumol, Marris, Williamson and others, point out to size, growth and discretional expenditures as variables which enter into the objective function of the manager who sets the policy of the ®rm (see Marris and Woods (1971) for a review). Later on, agency theory has shown that even if shareholders write contracts which attempt to align the interests of their managers with value maximization, information assymmetries imply that only second best contracts are feasible and managerial discretion is still present (Jensen and Meckling, 1976). Therefore, it has been argued, Chamberlain and Gordon (1991), that shareholders-owned ®rms may also follow non value maximizing strategies. The present paper assumes that the banks' shareholders are able to enforce ®rst best contracts to their managers and therefore the objective function of banks is pro®ts. The interesting thing is that shareholders
4
The underlying problem and results are similar to those obtained in the literature on the strategic choice of managerial incentives (Vickers, 1985; Fershtman and Judd, 1987; Sklivas, 1987; Salas, 1992). But, at least for saving banks, the objective function of the ®rm can be justi®ed in terms of its ownership structure and not in terms of the contract between shareholders and managers.
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prefer non pro®t maximizing managers to compete with savings banks since, as the model shows, pro®ts end up being higher. In any case, the full integration of the results of the paper with the literature on the separation between ownership and control of the ®rm may be a promising line of research. Section 2 presents a description of the technology and preferences used to build the theoretical model. Section 3 shows the basic results from the strategic competition process. Section 4 focuses on the likely response of banks to the expense preference behavior of savings institutions. The conclusion summarize the main results of the paper. 2. Technology and preference The basic model considers a bank, represented by subindex 1 and a savings institution, represented by subindex 2. Both ®rms compete for deposits in a market with supply function, under the assumption of homogeneous product, given by rD a b D;
a; b > 0;
1
where rD is the interest rate and D is the total deposits in the market. Each ®rm has a production function with a single input, labor, and constant returns to scale, Di ki Li ;
i 1; 2;
2
where ki is the average and marginal productivity and Li is the number of per worker. workers. There is a perfectly elastic supply of labor at a cost of x Each dollar of deposits is distributed between k dollars to cash reserves and 1 ÿ k dollars which can be placed in the money market at an interest rate of rb . Therefore, each dollar of deposits generates an income of R 1 ÿ k rb k r0 ; where r0 is the interest rate of cash reserves. The economic pro®t of ®rm i is given by Li Pi R ÿ rD Di ÿ x and substituting rD and Li from (1) and (2) above, Pi R ÿ a ÿ b D Di ÿ x
Di r ÿ b D ÿ ci Di ; ki
i is the marginal cost of one dollar of deposits. where r R ÿ a and ci x=k As we remarked in the introduction, banks are assumed to maximize pro®ts, P1 , but the objective function of savings institutions is unclear. The literature on expense preference behavior assumes that the loose assignment of property
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rights in institutions such as savings and loans, allows the managers of such institutions to choose their own preference function in place of pro®t maximization, subject to the constraint of not having operating losses. For example, it is assumed that the manager's utility function will depend on pro®ts and L2 and oU2 =oP2 > 0; labor expenses, U2 U2 P2 ; E2 , where E2 x oU2 =oE2 > 0: Another way to justify the objective function of the savings institutions is to assume that they behave as cooperatives and consequently, their objective function will be to maximize the total income payed to the workers, equal to R ÿ rD D2 x L2 ; where x R ÿ rD D2 =L2 . The two situations can be reconciled as follows. Assume that U2 is linear in L2 ; i.e., P2 and E2 x L2 ; U 2 P2 h x where h is a positive parameter. have
5
L2 we Substituting P2 R ÿ rD D2 ÿ x
L2 : U2 R ÿ rD D2 ÿ 1 ÿ h x Notice that if h 0, U2 P2 and if h 1, U2 x L2 . Therefore, the ob L2 implies that the savings bank is an instijective function U2 P2 h x tution in between a capitalist ®rm h 0 and a worker's cooperative h 1: With these considerations in mind we shall assume that 6 U2 R ÿ rD ÿ c2 D2 h c2 D2 2: after substituting L2 1=k2 D2 and c2 x=k 3. Strategic competition 3.1. Quantity competition and homogeneous product We look for the Nash equilibrium solution to the optimization problems max R ÿ rD ÿ c1 D1 ; D1
max R ÿ rD ÿ c2 D2 h c2 D2 ; D2
s:t
5
rD a b D1 D2 :
h has to be positive because utility increases with labor expenses. The linear function for U2 is chosen to simplify the exposition. The results would also hold under more general functions as proposed in the literature. 6
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Assuming that both ®rms are equally ecient, k1 k2 ; the equilibrium solution implies DEP 1 DEP 1 PEP 1 PEP 2 rDEP
1 R ÿ a ÿ c 1 h; 3b 1 R ÿ a ÿ c 1 ÿ 2h; 3b 1 R ÿ a ÿ c 1 h2 ; 9b 1 R ÿ a ÿ c 1 h R ÿ a ÿ c 1 ÿ 2h; 9b 2 a hc R ÿ c : 3 3 3
EP EP EP It is easy to show that DEP 1 < D2 and P1 < P2 , where EP stands for `expense preference' solution. In other words, in equilibrium savings banks capture a larger share of the market and earn higher pro®ts than commercial banks. So the theory is consistent with the empirical evidence that savings banks gain market share and obtain more pro®ts than commercial banks. To explain why a non pro®t maximizer outperforms a pro®t maximizer under the assumptions above, notice that the objective function of the savings bank may be written as
U2 R ÿ rD ÿ c 1 ÿ h D2 : So, as long as h 6 1; expense preference implies that the saving bank maximizes a `pseudo pro®t' function with lower marginal costs of the deposits, than the costs considered by the pro®t maximizing ®rms. Therefore expense preference acts as a `strategic competitive advantage' in terms of production costs. To clarify further the results we draw Figs. 1 and 2, with reaction functions and isopro®ts under pro®t maximization, PM, and expense preference, EP. When maximize pro®ts the equlibrium solution is point ÿ both PM®rms and both ®rms obtain the same level of pro®ts. Under expense ; D A DPM 1 2 preference, the reaction function of the savings banks ÿ EPmoves to the right, from EP EP to / , to the new equilibrium solution B D ; D /PM 2 2 1 2 . The isopro®ts of the two ®rms also change under the new situation, and in the new equilibrium the pro®ts of the commercial bank are always lower than in the old one. Pro®ts of the savings bank may be higher or lower compared with pro®ts in equilibrium solution A depending upon the value of the parameter h: In Fig. 1 where h is between 1=2h1 and h1 , 7 savings banks obtain 7 Where h1 R ÿ a ÿ c=c is derived from the condition of positive market share for commercial banks, i.e., DEP 1 > 0 () h < h1 .
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Fig. 1.
Fig. 2.
lower pro®ts in equilibrium solution B than in equilibrium solution A, while in Fig. 2 where h < 1=2h1 ; their pro®ts even increase under expense preference compared with the pro®ts in the pro®t maximizing equilibrium solution.
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3.2. Price competition with product dierentiation We now consider the case where depositors dierentiate between savings and commercial banks and these choose interest rates as competitive variables. The new demand functions are D 1 l f r1 ÿ g r2 ;
3
D 2 l f r2 ÿ g r1 ;
4
where l; f ; g are positive parameters with f > g. Keeping the same situation as above, the Nash equilibrium solution will be obtained by solving max R ÿ r1 ÿ c l f r1 ÿ g r2 ; r1
max R ÿ r2 ÿ c l f r2 ÿ g r1 hc l f r2 ÿ g r1 : r2
The reaction functions and the equilibrium solutions are given by R ÿ cf ÿ l g r2 ; 2f 2f R ÿ cf ÿ l g h r1 c; r2 2f 2f 2 R ÿ c f ÿ g lf f 2 gh ÿ 2 c; 2f ÿ g 4f ÿ g2 R c f ÿ g lf f 2f 2 ÿ g2 h c; 2f ÿ g 4f 2 ÿ g2 1 gf h R ÿ cf ÿ l c ; 2f ÿ g 2f g 1 2f 2 h R ÿ cf ÿ l c : 2f ÿ g 2f g
/EP 1 r1 /EP 2 DEP 1 DEP 2 r1EP r2EP
Once again, savings banks outperform commercial banks in terms of market EP share, DEP 2 > D1 and they will oer a higher interest rate for the deposits, EP EP r2 > r1 : In terms of pro®ts, savings banks will obtain higher pro®ts than commercial banks if h is suciently low. 8 4. The response of commercial banks We are interested now in the response that commercial banks may give to the situation where the preferences of the saving banks introduce asymmetries 8
When h < h2 R ÿ c f ÿ g lg=f 2 c:
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into the market and lower their pro®t prospects. As above a distinction will be made between the case where ®rms compete in quantities and the case where ®rms compete in interest rates. 4.1. Quantity competition If saving banks were privatized and became pro®t maximizers, the banks would earn a pro®t 2 1 Rÿaÿc : PPM 1 b 3 However, with savings banks and expense preferences the banks' pro®t is 2 1 R ÿ a ÿ c hc ; ÿ PEP 1 b 3 3 when h > 0. Banks will thus prefer a symmetric which is lower than PPM 1 market where competition is only among pro®t maximizers. This result may help to understand why Spanish commercial banks have requested a `privatization' of the saving banks which would overcome current asymmetries (such as the fact that saving banks can purchase commercial banks, but not the other way round). A new issue is what the commercial bank may do while saving banks are there with the present ownership structure. The answer comes from the literature on the strategic choice of managerial incentives. 9 The shareholders of commercial banks delegate the operating decisions to managers whose performance will be evaluated in terms of pro®ts and market share (recall our assumption that ®rst best contracts are feasible). Since managerial salaries will be indexed to this performance measure the manager will consider that the function to be maximized depends on pro®ts and market share. Therefore, the new objective function of the commercial bank will be U 1 P1 q D 1 ; where q is a parameter to be determined by the shareholders. 9 This literature, see Fershtman and Judd (1987), Sklivas (1987), assumes that shareholders choose the incentive function of their managers taking into account that the choice will aect the competition in the market place. The choice is open to all ®rms so that the shareholders' decision corresponds to a new equilibrium solution. In this paper it is assumed that savings banks cannot manipulate the choice of h and therefore it is taken as exogenous. Whether h may be dierent among saving banks depending upon the regulation that aects them in each autonomous Region of Spain, would be an empirical question.
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The new `game' will be solved by backwards induction. First, for a given q, the Nash equilibrium solution will be obtained. Since this solution will depend on q; then shareholders will choose q which maximizes pro®ts. The ®rst problem is formulated as follows: max P1 qD1 ; D1
max P2 hcD2 ; D2
and the Nash equilibrium solution is R ÿ a ÿ c hc 2 ÿ q; 3b 3b 3b R ÿ a ÿ c 2hc 1 ÿ q; DN 2 3b 3b 3b 2 R ÿ c a hc 1 N q: rD 3 3 3
DN 1
Shareholders will choose q maximizing PN 1 q; i.e., max PN1 q
1 R ÿ c ÿ a ÿ hc ÿ q R ÿ c ÿ a ÿ hc 2q: 9b
The optimal solution is q 1=4 R ÿ a ÿ c 1 h > 0, since R ÿ a ÿ c 1 h has to be positive in order to have DEP 1 > 0: An optimal q > 0 implies that the commercial bank shareholders ®nd it pro®table to act strategically in their choice of managerial incentives. In order to interpret this choice notice that the utility function of the bank's manager is U1 R ÿ rD ÿ c ÿ q D1 ; i.e., the objective function U1 is a pseudo pro®t function with lower marginal costs, and symmetry with respect to saving banks is now restored. Substituting q in the Nash equilibrium solution for the ®rst stage of the game, we get the new market shares, interest rate and pro®ts for what is identi®ed as the `sales preference' (SP) solution: DSP 1 DSP 2 rDSP PSP 1 PSP 2
1 R ÿ a ÿ c ÿ hc; 2b 1 R ÿ a ÿ c 3hc; 4b 1 3 R ÿ c a hc; 4 1 2 R ÿ a ÿ c ÿ hc ; 8b 1 R ÿ a ÿ c ÿ hc R ÿ a ÿ c 3hc: 16b
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The commercial bank increases market share and pro®ts with respect to SP EP SP EP those of the EP solution, DSP 1 > D1 and P1 > P1 : The relation between P1 PM and the pro®t in the pro®t maximizing case, P1 depends on the value of h; for PM values of h suciently high, 10 PSP 1 will be lower than P1 : So banks improve their pro®ts by modifying the incentives of their managers, but if h is high this improvement will still keep them below the symmetric pro®t maximizing solution. Finally, notice that the strategic choice of incentives by commercial banks bene®ts depositors since equilibrium interest rates will be higher. In fact we have rSP > rEP > rPM : Fig. 3 show graphically the results of the analytical solution. The reaction SP function of commercial banks moves to the right from /EP 1 to /1 and the new SP SP Nash equilibrium is C D1 ; D2 : So banks would recover market share, with respect to equilibrium B; at the expense of saving banks. 4.2. Price competition The situation is identical to the one considered above, but with demand functions (3) and (4). The equilibrium interest rates and deposits for given q and h are R ÿ cf ÿ l 2f 2 fg 2 q 2 hc; 2 4f ÿ g 2f ÿ g 4f ÿ g2 R ÿ cf ÿ l fg 2f 2 2 q hc; r2 2f ÿ g 4f ÿ g2 4f 2 ÿ g2 f R ÿ c f ÿ g l f 2f 2 ÿ g2 gf 2 q ÿ hc; D1 4f 2 ÿ g2 2f ÿ g 4f 2 ÿ g2 f R ÿ c f ÿ g l gf 2 f 2f 2 ÿ g2 ÿ 2 q hc: D2 2f ÿ g 4f 2 ÿ 3g2 4f ÿ g2 r1
The choice of q which maximizes the bank's pro®t is obtained by solving the problem R ÿ c f ÿ g l 2f 2 q fghc R ÿ c f ÿ g l ÿ max f q 2f ÿ g 4f 2 ÿ g2 2f ÿ g 2 2 2f ÿ g q ÿ fghc ; 4f 2 ÿ g2 which implies 10
p In particular if h > 1 ÿ 32=6h1 ; where h1 is de®ned in footnote 7.
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Fig. 3.
q
g3 R ÿ c f ÿ g l2f gg2 hc ÿ 8f 3 ÿ 4fg2 8f 4 ÿ 4f 2 g2
and the ®nal values of interest rates, deposits and pro®ts are r1SP r2SP DSP 1 DSP 2 PSP 1 PSP 2
R ÿ c 2f 2 fg ÿ g2 ÿ l 2f g fg 2 hc; 2 2 4f ÿ 2g 4f ÿ 2g2 R ÿ c 4f 3 2f 2 g ÿ fg2 ÿ g3 ÿ l 4f 2 2fg ÿ g2 4f 2 ÿ g2 2 hc; 8f ÿ 4g2 4f 2f 2 ÿ g2 R ÿ c f ÿ g l g 2f g ÿ hc; 4f 4 4f 2 ÿ 3g2 f R ÿ c f ÿ g l ÿ 2 2 4f hc; 2fg ÿ g 4 2f 2 ÿ g2 4 2f 2 ÿ g2 1 2 R ÿ c f ÿ g l 2f g ÿ fghc ; 8f 2f 2 ÿ g2 1 A1 ÿ A2 hc A1 A2 hc ÿ 2g2 f hc ; 2 2 4f 2f ÿ g
where A1 R ÿ c f ÿ g l 4f 2 2fg ÿ g2 and A2 4f 2 ÿ g2 f : In Appendix A we show that the value of q will be negative for feasible values of h: Therefore under price competition shareholders choose strategically to increase the marginal costs in the utility function of their managers,
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Fig. 4.
U1 R ÿ r1 c ÿ q D1 with q < 0: The choice has opposite sign and eects to when the competition is in quantities. To further understand the results we present the graphical solution. The intersection of reaction functions under the initial expense preference EP EP and /EP solution, /EP 1 2 , gives the equilibrium A r1 ; r2 in Fig. 4. This implies higher interest rates than when both ®rms are pro®t maximizers; point C r1PM ; r2PM . The decision variables, interest rates, are now strategic complements and therefore a positive value of h implies an increase in interest rates, at equilibrium, of savings banks and commercial banks. When commercial banks strategically choose the incentives for their managers, pro®t maximization implies moving the reaction function to the left q < 0, and the interest rates of commercial banks and savings banks are both reduced with respect to the initial EP solution; point B r1SP ; r2SP : In this EP new equilibrium banks increase their pro®ts with respect to A; PSP 1 > P1 (and 11 even with respect to C if h is suciently low ), but their interest rate and their market share are lower than in the initial solution A and lower than the respective interest and market share of the savings bank.
11
In particular if h < R ÿ c f ÿ g l 4f 2 ÿ g2 ÿ 2f
p 2 2f 2 ÿ g2 = 2f ÿ gfgc:
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5. Conclusion This paper has an empirical motivation in trying to answer the question of whether the peculiar ownership structure of Spanish savings banks may give them strategic advantages over commercial banks, and in this way explain their gains in market share at the same time that they maintain high levels of pro®tability. At ®rst glance, the particular ownership structure of savings banks would point towards the opposite direction. The results of the paper show that in an oligopolistic market with a given number of ®rms, a ®rm with expense preference behavior can outperform a pro®t maximizing ®rm in terms of market share and pro®ts, even if both have the same costs and identical demand functions. Commercial banks can try to restore symmetry in the competitive process by modifying the incentives of their managers. Our analysis shows that such symmetry is in fact achieved in markets where competition is in quantities and products are homogeneous. In this situation banks choose incentives for their managers which induce more `aggressive' behavior in the market place, so imitating in this way the `aggressive' behavior of saving banks. However, when competition is in price and products are dierentiated, the strategic choice of incentives by bank's shareholders implies higher asymmetry between the two banking ®rms since the best response to expense preference behavior by bank's shareholders is now to induce less `aggressive' behavior in their managers than in the pro®t maximizing case. The paper may also provide some insights on what may happen as more savings banks expand beyond their original regional areas, to compete nationwide. In this scenario, savings banks are likely to compete among themselves and therefore we would have ®rms with expense preference behavior competing against each other. Our results show that competition will be more intense both, in homogeneous and in dierentiated product markets. So the question is if such behavior may endanger the solvency of savings banks as higher competition implies lower pro®ts. The empirical evidence shows that in recent years interest rate spread has been steadly decreasing for saving and commercial banks, presumably in response to such intense competition. Of course, a validation of the model needs a more profound and elaborated empirical analysis. In conclusion, the request of Spanish commercial banks for more `symmetry' in the retail market's competition, is understandable under their particular interest, according to the results of the paper. More dicult is to establish the public policy responses to such request since the present situation is more favourable to higher competition and higher interest rates on deposits, and social welfare increases. But, as saving banks compete more often among themselves, the intensity of competition may be too high to preserve the solvency of ®nancial institutions, especially if we would translate our results to the loans market.
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Acknowledgements We acknowledge the editorial help and comments of a referree, which have contributed to integrate the paper into the broader issue ownership, control and business behavior.
Appendix A The value of q depends on h, but h is implicitly restricted to have interior solutions: DSP 1 > 0 () h < h1 ;
where h1
R ÿ c f ÿ g l 2f g > 0; fgc
DSP 2 > 0 () h2 < h; where h2 ÿ
R ÿ c f ÿ g l 4f 2 2fg ÿ g2 < 0; f 4f 2 ÿ 3g2 c
r1SP > 0 () h3 < h; R ÿ c 2f 2 fg ÿ g2 ÿ l 2f g < 0; fgc > 0 () h4 < h;
where h3 ÿ r2SP
R ÿ c 4f 3 2f 2 g ÿ fg ÿ g3 ÿ l 4f 2 2fg ÿ g2 < 0; f 4f 2 ÿ g2 c p A1 2 4f 2 f 2 ÿ g2 g4 ÿ g2 > 0 () h < h5 ; where h5 > 0: cA2 3g2 ÿ 4f 2
where h4 ÿ PSP 2
SP SP SP It is easy to show that DSP 1 > 0; r1 > 0; r2 > 0 and P2 > 0 are always satis®ed as long as h > 0. So the only restrictions on h come from the condiSP tions DSP 1 > 0 and P2 > 0, i.e., h < h1 and h < h5 : But since h5 < h1 ; the restriction on h are 0 < h < h5 : From the equation which determines q in the main text we have q > 0 i h > h1 ; which contradicts the condition DSP 1 > 0: Therefore q has to be negative since h < h5 < h1 :
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Strategic competition in retail banking under expense preference behavior Pedro Purroy a, Vicente Salas
b,*
a
b
Escuela de Empresariales, Diagonal 696, 08034 Barcelona, Spain Facultad de Econ omicas, Economics Department, University of Zaragoza, Gran Võa, no. 4, 50008 Zaragoza, Spain Received 8 October 1998; accepted 24 May 1999
Abstract This paper presents a theoretical analysis of strategic competition in retail banking when some of the ®rms show expense preference behavior. The literature on expense preference behavior by banking ®rms is fairly large, but to our knowledge, so far, the strategic interaction between pro®t maximizing banks and banks with expense preference behavior has not been investigated. The paper has also an empirical interest since it is motivated by the current situation in Spanish retail banking where commercial banks, pro®t maximizers, compete with savings banks, non pro®t maximizers. Ó 2000 Elsevier Science B.V. All rights reserved. JEL classi®cation: G21 Keywords: Expense preferences; Strategic competition; Spanish banking
1. Introduction The expense preference behavior of banking ®rms has been considered a consequence of the ownership and governance structure of the ®nancial insti-
*
Corresponding author. Tel.: 34-976-761803; fax: 34-976-761767. E-mail address: [email protected] (V. Salas).
0378-4266/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 4 2 6 6 ( 9 9 ) 0 0 0 6 6 - 7
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tutions, and in particular of the peculiar assigment of property rights in banking ®rms such as savings and loans in the USA. The ownership of these ®rms is unclear and this leaves room for their control by managers and workers with preferences dierent from shareholders' value maximization. Several papers have looked at the implications for demand and interest rates of having banking ®rms maximizing an utility function which depends on pro®ts and labor expenditures. 1 These papers, however, look at the utility maximizing ®rm in isolation, i.e., as a market monopolist. The situation where pro®t maximizers and utility maximizers compete in an oligopolistic market has not yet been explored, and it is the main concern of the present paper. Our theoretical model ®ts well in the present situation of the Spanish retail banking market, where commercial banks compete with savings banks (`Cajas de Ahorros'). Since the mid seventies savings banks have increased their share of bank deposits from 30% to 50%. The erosion of commercial banks' market share in favor of savings banks has occurred at the same time that the latter outperformed the former in pro®tability and solvency. 2 Part of the growth of the savings banks has taken place through their purchase of regional banks far away from the geographic region where the savings institution was originally established. In other words, some saving banks are competing among themselves and with commercial banks nationwide. Bankers have raised their voice against certain growth practices of savings banks arguing that competition is highly assymmetric since savings banks do not face the restriction of dividends payment and that while they can purchase banks, commercial banks cannot purchase savings institutions since their ownership status does not allow them to do so. 3 There may be several reasons why saving banks outperform commercial banks in the deposits market: savings banks have concentrated their activities in market segments (low and middle income groups) who value the social programs carried out by these institutions through their `Obra Social'; they have provided customers with higher level of services due to their larger branching network and faster technological innovation in, for example, automatic teller machines; workers of saving banks are more ecient since they feel in part as `owners' of the ®rm and this increases their motivation and eort, etc. But the recent debate has focused mainly on the eects of dierent ownership and governance structures between banks and savings institutions, and the paper will focus on this line of explanation.
1
Akella and Greenbaum (1988), Edwards (1977), Hannan (1979), Krinsky and Thomas (1995), Rees (1974), Verbrugge and Jahera (1981). 2 Lagares (1995), Maroto (1993), Rodrõguez L opez (1995) . 3 Coello (1994) investigates the symmetry of competition between banks and savings institutions under a dierent perspective.
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Savings banks are often viewed in Spain as closer to workers' cooperatives and not for pro®t organizations. The answer to the question, `who owns the savings banks?' is di®cult to ®nd and the ®nal result may be that workers and managers take eective control of the organization. Besides, accounting pro®ts of the savings banks have to be either retained or allocated to social welfare programs. Orthodox theoretical thinking should lead us to conclude that organizations with such loose ownership structure and diuse allocation of property rights, should clearly be outperformed when competing with ecient, pro®t maximizing ®rms. But this has not been the case in Spain and many people view the current situation as a puzzle. The paper is an attempt to explain the puzzle. Our theoretical analysis shows that an expense preference behavior ®rm may outperform in market share and pro®ts to a pro®t maximizing ®rm when they compete in an oligopolistic market. The basic explanation of this result is that expense preference lowers the eective marginal (labor) costs of the ®rm, compared with the pro®t maximizing case, and as a result of this, its reaction function moves in a direction which implies an increase in market share and pro®ts. The ®nal results, however, depends on whether ®rms compete on quantities with homogeneous products, or the ®rms compete on prices with dierentiated products. 4 Although the paper focuses on the issue of expense preference versus pro®t maximizing behavior highlighted in the literature about the economics of the banking ®rm, its conceptual framework can be integrated into the broader topic of what is the real objective function of the ®rm when ownership is separated from control (Berle and Means, 1932). For example, writers from the managerial theory of the ®rm, Baumol, Marris, Williamson and others, point out to size, growth and discretional expenditures as variables which enter into the objective function of the manager who sets the policy of the ®rm (see Marris and Woods (1971) for a review). Later on, agency theory has shown that even if shareholders write contracts which attempt to align the interests of their managers with value maximization, information assymmetries imply that only second best contracts are feasible and managerial discretion is still present (Jensen and Meckling, 1976). Therefore, it has been argued, Chamberlain and Gordon (1991), that shareholders-owned ®rms may also follow non value maximizing strategies. The present paper assumes that the banks' shareholders are able to enforce ®rst best contracts to their managers and therefore the objective function of banks is pro®ts. The interesting thing is that shareholders
4
The underlying problem and results are similar to those obtained in the literature on the strategic choice of managerial incentives (Vickers, 1985; Fershtman and Judd, 1987; Sklivas, 1987; Salas, 1992). But, at least for saving banks, the objective function of the ®rm can be justi®ed in terms of its ownership structure and not in terms of the contract between shareholders and managers.
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prefer non pro®t maximizing managers to compete with savings banks since, as the model shows, pro®ts end up being higher. In any case, the full integration of the results of the paper with the literature on the separation between ownership and control of the ®rm may be a promising line of research. Section 2 presents a description of the technology and preferences used to build the theoretical model. Section 3 shows the basic results from the strategic competition process. Section 4 focuses on the likely response of banks to the expense preference behavior of savings institutions. The conclusion summarize the main results of the paper. 2. Technology and preference The basic model considers a bank, represented by subindex 1 and a savings institution, represented by subindex 2. Both ®rms compete for deposits in a market with supply function, under the assumption of homogeneous product, given by rD a b D;
a; b > 0;
1
where rD is the interest rate and D is the total deposits in the market. Each ®rm has a production function with a single input, labor, and constant returns to scale, Di ki Li ;
i 1; 2;
2
where ki is the average and marginal productivity and Li is the number of per worker. workers. There is a perfectly elastic supply of labor at a cost of x Each dollar of deposits is distributed between k dollars to cash reserves and 1 ÿ k dollars which can be placed in the money market at an interest rate of rb . Therefore, each dollar of deposits generates an income of R 1 ÿ k rb k r0 ; where r0 is the interest rate of cash reserves. The economic pro®t of ®rm i is given by Li Pi R ÿ rD Di ÿ x and substituting rD and Li from (1) and (2) above, Pi R ÿ a ÿ b D Di ÿ x
Di r ÿ b D ÿ ci Di ; ki
i is the marginal cost of one dollar of deposits. where r R ÿ a and ci x=k As we remarked in the introduction, banks are assumed to maximize pro®ts, P1 , but the objective function of savings institutions is unclear. The literature on expense preference behavior assumes that the loose assignment of property
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rights in institutions such as savings and loans, allows the managers of such institutions to choose their own preference function in place of pro®t maximization, subject to the constraint of not having operating losses. For example, it is assumed that the manager's utility function will depend on pro®ts and L2 and oU2 =oP2 > 0; labor expenses, U2 U2 P2 ; E2 , where E2 x oU2 =oE2 > 0: Another way to justify the objective function of the savings institutions is to assume that they behave as cooperatives and consequently, their objective function will be to maximize the total income payed to the workers, equal to R ÿ rD D2 x L2 ; where x R ÿ rD D2 =L2 . The two situations can be reconciled as follows. Assume that U2 is linear in L2 ; i.e., P2 and E2 x L2 ; U 2 P2 h x where h is a positive parameter. have
5
L2 we Substituting P2 R ÿ rD D2 ÿ x
L2 : U2 R ÿ rD D2 ÿ 1 ÿ h x Notice that if h 0, U2 P2 and if h 1, U2 x L2 . Therefore, the ob L2 implies that the savings bank is an instijective function U2 P2 h x tution in between a capitalist ®rm h 0 and a worker's cooperative h 1: With these considerations in mind we shall assume that 6 U2 R ÿ rD ÿ c2 D2 h c2 D2 2: after substituting L2 1=k2 D2 and c2 x=k 3. Strategic competition 3.1. Quantity competition and homogeneous product We look for the Nash equilibrium solution to the optimization problems max R ÿ rD ÿ c1 D1 ; D1
max R ÿ rD ÿ c2 D2 h c2 D2 ; D2
s:t
5
rD a b D1 D2 :
h has to be positive because utility increases with labor expenses. The linear function for U2 is chosen to simplify the exposition. The results would also hold under more general functions as proposed in the literature. 6
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Assuming that both ®rms are equally ecient, k1 k2 ; the equilibrium solution implies DEP 1 DEP 1 PEP 1 PEP 2 rDEP
1 R ÿ a ÿ c 1 h; 3b 1 R ÿ a ÿ c 1 ÿ 2h; 3b 1 R ÿ a ÿ c 1 h2 ; 9b 1 R ÿ a ÿ c 1 h R ÿ a ÿ c 1 ÿ 2h; 9b 2 a hc R ÿ c : 3 3 3
EP EP EP It is easy to show that DEP 1 < D2 and P1 < P2 , where EP stands for `expense preference' solution. In other words, in equilibrium savings banks capture a larger share of the market and earn higher pro®ts than commercial banks. So the theory is consistent with the empirical evidence that savings banks gain market share and obtain more pro®ts than commercial banks. To explain why a non pro®t maximizer outperforms a pro®t maximizer under the assumptions above, notice that the objective function of the savings bank may be written as
U2 R ÿ rD ÿ c 1 ÿ h D2 : So, as long as h 6 1; expense preference implies that the saving bank maximizes a `pseudo pro®t' function with lower marginal costs of the deposits, than the costs considered by the pro®t maximizing ®rms. Therefore expense preference acts as a `strategic competitive advantage' in terms of production costs. To clarify further the results we draw Figs. 1 and 2, with reaction functions and isopro®ts under pro®t maximization, PM, and expense preference, EP. When maximize pro®ts the equlibrium solution is point ÿ both PM®rms and both ®rms obtain the same level of pro®ts. Under expense ; D A DPM 1 2 preference, the reaction function of the savings banks ÿ EPmoves to the right, from EP EP to / , to the new equilibrium solution B D ; D /PM 2 2 1 2 . The isopro®ts of the two ®rms also change under the new situation, and in the new equilibrium the pro®ts of the commercial bank are always lower than in the old one. Pro®ts of the savings bank may be higher or lower compared with pro®ts in equilibrium solution A depending upon the value of the parameter h: In Fig. 1 where h is between 1=2h1 and h1 , 7 savings banks obtain 7 Where h1 R ÿ a ÿ c=c is derived from the condition of positive market share for commercial banks, i.e., DEP 1 > 0 () h < h1 .
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Fig. 1.
Fig. 2.
lower pro®ts in equilibrium solution B than in equilibrium solution A, while in Fig. 2 where h < 1=2h1 ; their pro®ts even increase under expense preference compared with the pro®ts in the pro®t maximizing equilibrium solution.
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3.2. Price competition with product dierentiation We now consider the case where depositors dierentiate between savings and commercial banks and these choose interest rates as competitive variables. The new demand functions are D 1 l f r1 ÿ g r2 ;
3
D 2 l f r2 ÿ g r1 ;
4
where l; f ; g are positive parameters with f > g. Keeping the same situation as above, the Nash equilibrium solution will be obtained by solving max R ÿ r1 ÿ c l f r1 ÿ g r2 ; r1
max R ÿ r2 ÿ c l f r2 ÿ g r1 hc l f r2 ÿ g r1 : r2
The reaction functions and the equilibrium solutions are given by R ÿ cf ÿ l g r2 ; 2f 2f R ÿ cf ÿ l g h r1 c; r2 2f 2f 2 R ÿ c f ÿ g lf f 2 gh ÿ 2 c; 2f ÿ g 4f ÿ g2 R c f ÿ g lf f 2f 2 ÿ g2 h c; 2f ÿ g 4f 2 ÿ g2 1 gf h R ÿ cf ÿ l c ; 2f ÿ g 2f g 1 2f 2 h R ÿ cf ÿ l c : 2f ÿ g 2f g
/EP 1 r1 /EP 2 DEP 1 DEP 2 r1EP r2EP
Once again, savings banks outperform commercial banks in terms of market EP share, DEP 2 > D1 and they will oer a higher interest rate for the deposits, EP EP r2 > r1 : In terms of pro®ts, savings banks will obtain higher pro®ts than commercial banks if h is suciently low. 8 4. The response of commercial banks We are interested now in the response that commercial banks may give to the situation where the preferences of the saving banks introduce asymmetries 8
When h < h2 R ÿ c f ÿ g lg=f 2 c:
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into the market and lower their pro®t prospects. As above a distinction will be made between the case where ®rms compete in quantities and the case where ®rms compete in interest rates. 4.1. Quantity competition If saving banks were privatized and became pro®t maximizers, the banks would earn a pro®t 2 1 Rÿaÿc : PPM 1 b 3 However, with savings banks and expense preferences the banks' pro®t is 2 1 R ÿ a ÿ c hc ; ÿ PEP 1 b 3 3 when h > 0. Banks will thus prefer a symmetric which is lower than PPM 1 market where competition is only among pro®t maximizers. This result may help to understand why Spanish commercial banks have requested a `privatization' of the saving banks which would overcome current asymmetries (such as the fact that saving banks can purchase commercial banks, but not the other way round). A new issue is what the commercial bank may do while saving banks are there with the present ownership structure. The answer comes from the literature on the strategic choice of managerial incentives. 9 The shareholders of commercial banks delegate the operating decisions to managers whose performance will be evaluated in terms of pro®ts and market share (recall our assumption that ®rst best contracts are feasible). Since managerial salaries will be indexed to this performance measure the manager will consider that the function to be maximized depends on pro®ts and market share. Therefore, the new objective function of the commercial bank will be U 1 P1 q D 1 ; where q is a parameter to be determined by the shareholders. 9 This literature, see Fershtman and Judd (1987), Sklivas (1987), assumes that shareholders choose the incentive function of their managers taking into account that the choice will aect the competition in the market place. The choice is open to all ®rms so that the shareholders' decision corresponds to a new equilibrium solution. In this paper it is assumed that savings banks cannot manipulate the choice of h and therefore it is taken as exogenous. Whether h may be dierent among saving banks depending upon the regulation that aects them in each autonomous Region of Spain, would be an empirical question.
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The new `game' will be solved by backwards induction. First, for a given q, the Nash equilibrium solution will be obtained. Since this solution will depend on q; then shareholders will choose q which maximizes pro®ts. The ®rst problem is formulated as follows: max P1 qD1 ; D1
max P2 hcD2 ; D2
and the Nash equilibrium solution is R ÿ a ÿ c hc 2 ÿ q; 3b 3b 3b R ÿ a ÿ c 2hc 1 ÿ q; DN 2 3b 3b 3b 2 R ÿ c a hc 1 N q: rD 3 3 3
DN 1
Shareholders will choose q maximizing PN 1 q; i.e., max PN1 q
1 R ÿ c ÿ a ÿ hc ÿ q R ÿ c ÿ a ÿ hc 2q: 9b
The optimal solution is q 1=4 R ÿ a ÿ c 1 h > 0, since R ÿ a ÿ c 1 h has to be positive in order to have DEP 1 > 0: An optimal q > 0 implies that the commercial bank shareholders ®nd it pro®table to act strategically in their choice of managerial incentives. In order to interpret this choice notice that the utility function of the bank's manager is U1 R ÿ rD ÿ c ÿ q D1 ; i.e., the objective function U1 is a pseudo pro®t function with lower marginal costs, and symmetry with respect to saving banks is now restored. Substituting q in the Nash equilibrium solution for the ®rst stage of the game, we get the new market shares, interest rate and pro®ts for what is identi®ed as the `sales preference' (SP) solution: DSP 1 DSP 2 rDSP PSP 1 PSP 2
1 R ÿ a ÿ c ÿ hc; 2b 1 R ÿ a ÿ c 3hc; 4b 1 3 R ÿ c a hc; 4 1 2 R ÿ a ÿ c ÿ hc ; 8b 1 R ÿ a ÿ c ÿ hc R ÿ a ÿ c 3hc: 16b
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The commercial bank increases market share and pro®ts with respect to SP EP SP EP those of the EP solution, DSP 1 > D1 and P1 > P1 : The relation between P1 PM and the pro®t in the pro®t maximizing case, P1 depends on the value of h; for PM values of h suciently high, 10 PSP 1 will be lower than P1 : So banks improve their pro®ts by modifying the incentives of their managers, but if h is high this improvement will still keep them below the symmetric pro®t maximizing solution. Finally, notice that the strategic choice of incentives by commercial banks bene®ts depositors since equilibrium interest rates will be higher. In fact we have rSP > rEP > rPM : Fig. 3 show graphically the results of the analytical solution. The reaction SP function of commercial banks moves to the right from /EP 1 to /1 and the new SP SP Nash equilibrium is C D1 ; D2 : So banks would recover market share, with respect to equilibrium B; at the expense of saving banks. 4.2. Price competition The situation is identical to the one considered above, but with demand functions (3) and (4). The equilibrium interest rates and deposits for given q and h are R ÿ cf ÿ l 2f 2 fg 2 q 2 hc; 2 4f ÿ g 2f ÿ g 4f ÿ g2 R ÿ cf ÿ l fg 2f 2 2 q hc; r2 2f ÿ g 4f ÿ g2 4f 2 ÿ g2 f R ÿ c f ÿ g l f 2f 2 ÿ g2 gf 2 q ÿ hc; D1 4f 2 ÿ g2 2f ÿ g 4f 2 ÿ g2 f R ÿ c f ÿ g l gf 2 f 2f 2 ÿ g2 ÿ 2 q hc: D2 2f ÿ g 4f 2 ÿ 3g2 4f ÿ g2 r1
The choice of q which maximizes the bank's pro®t is obtained by solving the problem R ÿ c f ÿ g l 2f 2 q fghc R ÿ c f ÿ g l ÿ max f q 2f ÿ g 4f 2 ÿ g2 2f ÿ g 2 2 2f ÿ g q ÿ fghc ; 4f 2 ÿ g2 which implies 10
p In particular if h > 1 ÿ 32=6h1 ; where h1 is de®ned in footnote 7.
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Fig. 3.
q
g3 R ÿ c f ÿ g l2f gg2 hc ÿ 8f 3 ÿ 4fg2 8f 4 ÿ 4f 2 g2
and the ®nal values of interest rates, deposits and pro®ts are r1SP r2SP DSP 1 DSP 2 PSP 1 PSP 2
R ÿ c 2f 2 fg ÿ g2 ÿ l 2f g fg 2 hc; 2 2 4f ÿ 2g 4f ÿ 2g2 R ÿ c 4f 3 2f 2 g ÿ fg2 ÿ g3 ÿ l 4f 2 2fg ÿ g2 4f 2 ÿ g2 2 hc; 8f ÿ 4g2 4f 2f 2 ÿ g2 R ÿ c f ÿ g l g 2f g ÿ hc; 4f 4 4f 2 ÿ 3g2 f R ÿ c f ÿ g l ÿ 2 2 4f hc; 2fg ÿ g 4 2f 2 ÿ g2 4 2f 2 ÿ g2 1 2 R ÿ c f ÿ g l 2f g ÿ fghc ; 8f 2f 2 ÿ g2 1 A1 ÿ A2 hc A1 A2 hc ÿ 2g2 f hc ; 2 2 4f 2f ÿ g
where A1 R ÿ c f ÿ g l 4f 2 2fg ÿ g2 and A2 4f 2 ÿ g2 f : In Appendix A we show that the value of q will be negative for feasible values of h: Therefore under price competition shareholders choose strategically to increase the marginal costs in the utility function of their managers,
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Fig. 4.
U1 R ÿ r1 c ÿ q D1 with q < 0: The choice has opposite sign and eects to when the competition is in quantities. To further understand the results we present the graphical solution. The intersection of reaction functions under the initial expense preference EP EP and /EP solution, /EP 1 2 , gives the equilibrium A r1 ; r2 in Fig. 4. This implies higher interest rates than when both ®rms are pro®t maximizers; point C r1PM ; r2PM . The decision variables, interest rates, are now strategic complements and therefore a positive value of h implies an increase in interest rates, at equilibrium, of savings banks and commercial banks. When commercial banks strategically choose the incentives for their managers, pro®t maximization implies moving the reaction function to the left q < 0, and the interest rates of commercial banks and savings banks are both reduced with respect to the initial EP solution; point B r1SP ; r2SP : In this EP new equilibrium banks increase their pro®ts with respect to A; PSP 1 > P1 (and 11 even with respect to C if h is suciently low ), but their interest rate and their market share are lower than in the initial solution A and lower than the respective interest and market share of the savings bank.
11
In particular if h < R ÿ c f ÿ g l 4f 2 ÿ g2 ÿ 2f
p 2 2f 2 ÿ g2 = 2f ÿ gfgc:
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5. Conclusion This paper has an empirical motivation in trying to answer the question of whether the peculiar ownership structure of Spanish savings banks may give them strategic advantages over commercial banks, and in this way explain their gains in market share at the same time that they maintain high levels of pro®tability. At ®rst glance, the particular ownership structure of savings banks would point towards the opposite direction. The results of the paper show that in an oligopolistic market with a given number of ®rms, a ®rm with expense preference behavior can outperform a pro®t maximizing ®rm in terms of market share and pro®ts, even if both have the same costs and identical demand functions. Commercial banks can try to restore symmetry in the competitive process by modifying the incentives of their managers. Our analysis shows that such symmetry is in fact achieved in markets where competition is in quantities and products are homogeneous. In this situation banks choose incentives for their managers which induce more `aggressive' behavior in the market place, so imitating in this way the `aggressive' behavior of saving banks. However, when competition is in price and products are dierentiated, the strategic choice of incentives by bank's shareholders implies higher asymmetry between the two banking ®rms since the best response to expense preference behavior by bank's shareholders is now to induce less `aggressive' behavior in their managers than in the pro®t maximizing case. The paper may also provide some insights on what may happen as more savings banks expand beyond their original regional areas, to compete nationwide. In this scenario, savings banks are likely to compete among themselves and therefore we would have ®rms with expense preference behavior competing against each other. Our results show that competition will be more intense both, in homogeneous and in dierentiated product markets. So the question is if such behavior may endanger the solvency of savings banks as higher competition implies lower pro®ts. The empirical evidence shows that in recent years interest rate spread has been steadly decreasing for saving and commercial banks, presumably in response to such intense competition. Of course, a validation of the model needs a more profound and elaborated empirical analysis. In conclusion, the request of Spanish commercial banks for more `symmetry' in the retail market's competition, is understandable under their particular interest, according to the results of the paper. More dicult is to establish the public policy responses to such request since the present situation is more favourable to higher competition and higher interest rates on deposits, and social welfare increases. But, as saving banks compete more often among themselves, the intensity of competition may be too high to preserve the solvency of ®nancial institutions, especially if we would translate our results to the loans market.
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Acknowledgements We acknowledge the editorial help and comments of a referree, which have contributed to integrate the paper into the broader issue ownership, control and business behavior.
Appendix A The value of q depends on h, but h is implicitly restricted to have interior solutions: DSP 1 > 0 () h < h1 ;
where h1
R ÿ c f ÿ g l 2f g > 0; fgc
DSP 2 > 0 () h2 < h; where h2 ÿ
R ÿ c f ÿ g l 4f 2 2fg ÿ g2 < 0; f 4f 2 ÿ 3g2 c
r1SP > 0 () h3 < h; R ÿ c 2f 2 fg ÿ g2 ÿ l 2f g < 0; fgc > 0 () h4 < h;
where h3 ÿ r2SP
R ÿ c 4f 3 2f 2 g ÿ fg ÿ g3 ÿ l 4f 2 2fg ÿ g2 < 0; f 4f 2 ÿ g2 c p A1 2 4f 2 f 2 ÿ g2 g4 ÿ g2 > 0 () h < h5 ; where h5 > 0: cA2 3g2 ÿ 4f 2
where h4 ÿ PSP 2
SP SP SP It is easy to show that DSP 1 > 0; r1 > 0; r2 > 0 and P2 > 0 are always satis®ed as long as h > 0. So the only restrictions on h come from the condiSP tions DSP 1 > 0 and P2 > 0, i.e., h < h1 and h < h5 : But since h5 < h1 ; the restriction on h are 0 < h < h5 : From the equation which determines q in the main text we have q > 0 i h > h1 ; which contradicts the condition DSP 1 > 0: Therefore q has to be negative since h < h5 < h1 :
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