Information misreporting in the credit market: analysis of a credit bureau's disciplinary rote.

Nabi, Mahmoud Sami ; Souissi, Souraya Ben

INTRODUCTION

Many studies (eg Stiglitz and Weiss, 1981) showed that asymmetric information dampens the efficient functioning of credit markets. On one hand, the borrowers need to get information about interest rates offered by banks in order to make their deposit decision. On the other hand, banks collect information about their potential clients in order to offer them the volume of credit and the interest rate that corresponds to their risk profile. This information search is costly for banks and borrowers in the absence of a centralized institution in charge of collection and diffusion of the information. Pagano and Jappelli (1993) show that information sharing between lenders reduces the information costs and increases the volume of lending. This result is confirmed by Padilla and Pagano (1997) who show that the exchange of information not only reduces information asymmetry between lenders and borrowers, but also reduces moral hazard and adverse selection. It is also confirmed by Elsas (2005) who shows that the relationship lending is positively related to better access to information. However, banks might not have any incentive to exchange information if this strategy will erode their client niches. Thus, false information reporting could be the strategy chosen by banks. This opportunistic behavior was analyzed by Semenova (2008), who investigated the following question: 'has a bank any incentive to get benefit from information sharing without losing its competitive advantage?' In a two-period model, she shows that banks choose the dishonest strategy in the last period because it generates higher profit. Interpreting this result she suggested that a Credit Bureau could be a solution to this misreporting market distortion.

It is known that information sharing is facilitated by information brokers such as Public Registers or Credit Bureaus that collect files and distribute information among the members of the information sharing system. Djankov et al. (2006) show that these two substitute organizations increase lending and favor the development of the information sharing system. It was also shown empirically that the availability of information about payment performance through Credit Bureaus increases the availability of credit for SMEs (Ganbold, 2008). According to Malhorta et al. (2006) the lenders' perception of the risk involved in lending to SMEs is reinforced when there is lack of information on their clients' credit profiles and histories. They argue that the establishment and use of Credit Bureaus as third party information providers could overcome the high cost of directly screening and monitoring clients. A World Bank survey (World Bank, 2004) shows that the availability of credit history information is likely to reduce processing time, processing costs and default rates. Despite its importance, the Credit Bureaus' role was not modeled in the literature about banks' information sharing strategies. This paper tries to contribute to filling this gap by answering the following question: could a Credit Bureau design an effective mechanism to incite banks to report correct information about their borrowers?

To answer this question we depart from Semenova (2008) and propose two extensions. As a first extension, we develop the Credit Bureau's role first in a three-period model and second in an infinite time horizon model. Indeed, no such role could take place in a two-period model since the misreporting strategy will remain the banks' optimal strategy during the second period whatever the incentive mechanism designed by the Credit Bureau. This is because the latter intervenes once the dishonest bank is discovered at the end of the second period. In our model the Credit Bureau intervenes through the withdrawal of the deviating bank's license and imposing on it a financial penalty. We show that the minimum level of this penalty depends on the structure of the credit market and the banks' far-sightedness about their future profits. This leads us to the second extension of the paper, which is moving from the pure price competition considered by Semenova (2008) to a spatial competition in line with Grimaud and Rochet (1994) and Salop (1979) with n interacting banks in the credit market. Brevoort and Wolken (2008) show that this is a more realistic modeling of the banking structure since distance matters in banking. More specifically they apply an empirical model of the United States market and find that proximity between borrowers and institutions has an impact on the delivery of financial services. The transportation cost could be interpreted as the activity specialization of the bank or the entrepreneur's cost of preparing a credit application. It can also be interpreted as the proximity between the bank's specialization and the firm's activity. Brevoort and Wolken (2008) show that the transportation cost can be interpreted as: (i) money and time costs incurred by an entrepreneur when requesting a financial service, (ii) transportation costs incurred by banks when visiting a firm looking for credit, and (iii) the cost of information collected either by the bank or the borrower. The remainder of the paper is organized as follows. The next section develops the basic model and the different interest rates in the case of honest reporting. After that, the section analyzes the role of the Credit Bureau in the case of dishonest information sharing. The subsequent section extends the basic model to an infinite time horizon and derives more general results. Finally, the last section summarizes the main findings.

BASIC MODEL

Departing from Semenova (2008) we develop an extended three-period model with spatial competition a-la Salop (1979) and a Credit Bureau with a disciplinary mechanism to prevent incorrect information sharing.

Environment

We consider a three-period credit market model with three types of actors: n identical banks indexed by i = 1, ..., n; a continuum [0, 1] of entrepreneurs, and a Credit Bureau. Banks are located symmetrically around a circle of measure 1 and entrepreneurs are uniformly distributed around this circle. Each entrepreneur is located at a distance [x.sub.i] [member of] [0, 1/n] from bank i and [x.sub.i+1] =l/n - [x.sub.i] from bank i + 1. The entrepreneur needs a loan of size 1 to undertake an investment project and asks it from bank i or bank i + 1 taking into account the interest rate he is offered and the transportation cost t.[x.sub.i](t.[x.sub.i+1]) he bears to reach bank i (respectively bank i + 1). Entrepreneurs have no initial wealth and are engaged for three periods on the credit market. They are divided into two types: high-ability entrepreneurs who are present in proportion [gamma] and low-ability entrepreneurs who are in proportion 1 - [gamma]. High-ability entrepreneurs undertake risky projects yielding [R.sup.*] with probability p and zero with probability 1 - p. Low-ability entrepreneurs undertake bad projects that yield nothing in all states of nature. The parameters [gamma] and p are public information from the beginning of the first period. This means that banks know the exact proportion of each type of entrepreneurs and know that a proportion [gamma]p among the high-ability ones end the first period with successful projects. However, initially, they are unable to distinguish on an individual base a high-ability entrepreneur from a low-ability one. This information is partially discovered at the end of the first period. In fact, the success of a project signals the high-ability type of the entrepreneur. However, a failing project could be a bad project (undertaken by a low-ability entrepreneur) or a risky project (undertaken by a high-ability entrepreneur) that failed. Table 1 presents the composition of the entrepreneurs' population at the end of the first period and the beginning of the second period and at the end of the second period and beginning of the third period.

At the end of the first and second periods, the population of entrepreneurs is composed as follows: [gamma]p the high-ability entrepreneurs whose risky projects succeed and 1 - [gamma]p entrepreneurs with failed projects. In addition, we assume in line with Dell'Ariccia (2001) and Padilla and Pagano (1997) that banks maximize profits in each period. This is not unrealistic if we consider that each bank has a monopoly on its location.

Time events

In the first period: each bank i = 1, ..., n offers a gross interest rate [R.sub.i1] to all types of entrepreneurs since it cannot distinguish between high-ability and low-ability ones. At the end of the first period, only a proportion p of the high-ability entrepreneurs has successful projects and repays their loans. We assume that banks can observe the interest rates charged by other banks. However, we assume that banks cannot observe for whom the interest rate was charged. Each bank i chooses the gross interest rate [R.sub.i1] to maximize its profit in the first period:

Max [[PI].sub.il] = ([gamma]p[R.sub.i1] - [bar.R])[D.sub.i] - f (1)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the demand for bank i, which is negatively dependent on [R.sub.i1] and positively dependent on the two neighbors' interest rate R1. The parameter [bar.R] [less than or equal to] [gamma]p[R.sub.i1] represents the exogenous cost of capital. The parameter f represents a fixed cost paid by each bank to adhere to the information sharing system managed by the Credit Bureau.

In the second period banks share information about the past results of the entrepreneurs rather than more complete information about their types. Padilla and Pagano (1997) show that this type of information sharing has a greater disciplinary effect on entrepreneurs. Bank i could report wrong information about its solvent high-ability clients not only to discourage the competing banks from attracting them but also to offer them relatively higher interest rates. In doing so, bank i makes additional profits on the loans granted to these solvent entrepreneurs. However, it runs the risk of being penalized by the Credit Bureau during the third period. This trade-off is analyzed in the third section. We assume that entrepreneurs have no self-financing capital and spend all their profits at the end of the first period. An entrepreneur can choose to stay with his initial bank or to move to another bank if he is offered a lower interest rate. However, this could take place only during the third period if his initial banks' deviation is revealed by the Credit Bureau and banned from the market. This is because a solvent entrepreneur could not signal his type directly to a competing bank.

In order to undertake a project during the second period entrepreneurs have to apply for a new loan. At the beginning of the second period, banks have acquired new information about the successful entrepreneurs. This enables them to charge differentiated interest rates. Denote by [R.sub.i21] the gross interest rate charged by bank i to the proportion [gamma]p of the high-ability entrepreneurs whose projects succeeded and who repaid their first-period loan. We denote by [R.sub.i22], the gross interest rate charged by bank i to the first-period defaulting entrepreneurs (whether they are of high-ability or low-ability type). The latter are composed of the high-ability entrepreneurs whose risky projects failed (their proportion is [gamma](l - p)) and the low-ability entrepreneurs (their proportion is 1 - [gamma]). Thus, the total proportion of defaulting entrepreneurs is 1 - [gamma]p. Bank i fixes the gross interest rates ([R.sub.i21], [R.sub.i22]) to maximize its second-period profit:

Max [[PI].sub.i2] = (p[R.sub.i21] - [bar.R])[D.sub.i1] + ([[[gamma]p(1 - p)]/[1 - [gamma]p]] [R.sub.i22] - [bar.R]) [D.sub.i2] - f (2)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are, respectively, the demand for loans addressed to bank i by the successful and defaulting entrepreneurs. The parameter f represents a fixed cost paid by each bank periodically to adhere to the information sharing system. Only a proportion p of the first category of loans will be repaid, whereas the proportion of successful projects financed by the second category of loans is p[gamma](1 - p)/ (1 - [gamma]p). Indeed, as shown in Table 1, among the total proportion 1 - [gamma]p only a proportion p of the [gamma](1 - p) high-skilled entrepreneurs (whose projects failed in the first period) will have successful projects during the second period.

In the third period, again, each bank i charges differentiated interest rates depending on the borrowers' payment history. We denote [R.sub.i31] the interest rate charged by bank i to the high-ability entrepreneurs whose type is revealed during the two preceding periods. (1) Therefore, their total proportion is [[mu].sub.1] = 2p(1 - p)[gamma] + [gamma][p.sup.2] or equivalently [[mu].sub.1] = (2 - p)[gamma]p. Let [R.sub.i32] be the gross interest rate charged to the proportion [(1 - p).sup.2][gamma] of high-ability entrepreneurs who defaulted during the two first periods and the proportion (1 - [gamma]) of low-ability entrepreneurs. Their total proportion is denoted [[mu].sub.2] = [(1 - p).sup.2][gamma] + 1 - [gamma]. Bank i chooses the interest rates ([R.sub.i31], [R.sub.i32]) that maximize its third-period profit:

Max [[PI].sub.i3] = (p[R.sub.i31] - [bar.R])[D'.sub.i1] + ([[gamma]p[(1 - p).sup.2]]/[[mu].sub.2]] [R.sub.i32] - [bar.R]) [D'.sub.i2] - f (3)

Where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are respectively the demand for loans addressed to bank i by the proportions [[mu].sub.1] and [[mu.sub.2] of the entrepreneurs. Only a proportion p of the first category of loans will be repaid, but the proportion of successful projects financed by the second category of loans is p[gamma][(1 - p).sup.2]/[[mu].sub.2]. Indeed, among the total proportion u2 only a proportion p of the [gamma][(1 - p).sup.2] high-skilled entrepreneurs (whose projects failed in the two first periods) will succeed their projects during the third period.

Entrepreneurs' utility

After obtaining a loan of size one at the beginning of period j = 1, 2, 3, high-ability entrepreneurs have to repay the principal and interests on the loans. This gross amount is also the gross interest rate which we denote [R.sub.ij]. The decision of an entrepreneur to ask for a loan from bank i or bank i + 1 is the result of his utility [U.sub.j] maximization at the beginning of period j:

[U.sub.j] = p([R.sup.*] - [R.sub.ij]) (4)

Finally, we assume that a low-ability entrepreneur (who knows his type since the beginning of period 1) undertakes a bad project because it provides him a positive utility.

Interest rates in case of honest information reporting

In this subsection we determine the gross interest rates charged by banks in the three periods in case of honest information sharing.

Proposition 1:. In the case of honest information reporting: (1) The gross interest rates are given by

[R.sub.1] = [[bar.R]/[gamma]p] + [t/n] (5)

[R.sub.21] = [[bar.R]/p] + [t/n] (6)

[R.sub.22] = [[bar.R]/p] (1 + [[1 - [gamma]]/[gamma](1 - p)]]) + [t/n] (7)

[R.sub.31] = [[bar.R]/p] + [t/n] (8)

[R.sub.32] = [bar.R]/p ([1 + 1 - [gamma]/[gamma][(1 - p).sup.2]]) + t/n (9)

(2) The profits of an individual bank are given by:

[[PI].sub.1] = [gamma]pt/[n.sup.2] - f (10)

[[PI].sub.3] = [[PI].sub.2] = (1 - [gamma])[bar.R]/n + [gamma]pt/[n.sup.2] - f (11)

Proof. See the appendix.

This proposition shows that buying information about borrowers reduces banks' profits but has no effect on the interest rates. Besides, it is clear that the transportation cost (which could be interpreted as the unitary degree of differentiation between banks) increases the bank's profit. This is intuitive since the higher this cost, the larger is the bank's local monopoly niche. On the other hand, profits are decreasing with the number of banks.

DISHONEST STRATEGY AND THE INCENTIVE MECHANISM OF THE CREDIT BUREAU

In the previous section we assumed that banks share correct information about their clients and charge interest rates depending on the entrepreneur's type. However, in maximizing their profits in each period they might share incorrect information about the solvent high-ability entrepreneurs when there is no control by the Credit Bureau. Nevertheless, all the actors of the credit market know that, at the beginning of each period, there is a proportion [gamma]p of solvent high-ability entrepreneurs. In this case, following Semenova (2008), we assume that the dishonest strategy is based on two types of incorrect information reporting. The first one is reporting a proportion of the solvent entrepreneurs as defaulters. The second one is reporting a proportion of defaulting entrepreneurs as high-ability entrepreneurs with successful projects. Proceeding this way, a bank may get additional profits in the second or third period. In our model, we extend the model of Semenova (2008) by assuming that the Credit Bureau can discover this deviation at the beginning of the second or the third periods. The Credit Bureau can observe (2) the additional profit realized by the dishonest bank. This abnormal profit signals to the Credit Bureau the dishonest that which will bear a penalty in the case of second period deviation. In the remainder of this section we determine the (gross) interest rates applied in case of misreporting and the additional profits realized by the dishonest bank. We will also specify the penalty threshold that the Credit Bureau should apply to prevent information misreporting. Depending on the values of the parameters [gamma] and p we identify two cases. In the first case, the proportion [gamma]p of high-ability entrepreneurs whose risky projects succeed is lower than the proportion 1 - [gamma]p of all the defaulting entrepreneurs. This correspond to the case 1 - [gamma]p > [gamma]p or [gamma]p < 1/2 which we denote case 1. The second case denoted case 2 is the opposite and takes place when [gamma]p [greater than or equal to] 1/2.

Case 1: 1 - [gamma]p > [gamma]p or [gamma]p < 1/2

In this case, a bank choosing the dishonest strategy will report all the proportion [gamma]p of its high-ability solvent clients as low-ability or high-ability defaulting entrepreneurs since it cannot distinguish the type of the defaulting entrepreneurs. Instead, it will select a proportion [gamma]p among the proportion 1 - [gamma]p) of its defaulting clients to report as solvent. This dishonest strategy could take place at the beginning of the second or the third periods.

The dishonest strategy at the beginning of the third period

Table 2 presents the reporting of a bank that chooses the honest strategy at the beginning of the second period and the dishonest strategy at the beginning of the third period.

This bank realizes an additional profit through charging a higher interest rates [R.sub.3,2] (see equation 9) for the proportion [theta] of the new-revealed high ability entrepreneurs. This is different from the case of honest information reporting where the entire proportion [gamma]p + [gamma]p (1 - p) of the revealed high-ability entrepreneurs (whose risky projects succeed at least once during the two first periods), are charged a lower interest rate [R.sub.3,1]. We can easily show that the third period additional profit relatively to the honest strategy is given by:

[DELTA][[PI].sup.dh.sub.i3] = ([R.sub.32] - [R.sub.31]) [theta]/n = [bar.R](1 - [gamma])/np [[1 + 1 - [gamma]/[gamma](1 - p).sup.2]]] (12)

The third-period profit is given by [[PI].sub.i3] + [DELTA][[PI].sup.dh.sub.i3] where [[PI].sub.i3] is the profit in case of honest reporting given by equation 11. In this case, the Credit Bureau discovers the dishonest bank at the end of the third period after observing the additional profit. However, the exclusion from the credit market does not make a sense and the dishonest bank will not pay the penalty.

The dishonest strategy at the beginning of the second period

A bank that chooses this strategy will realize additional profits in the second period. However, it will be discovered by the Credit Bureau at the end of the second period. The latter will charge it a penalty and exclude it from the credit market during the third period. The additional profit of this bank relative to its rivals will be realized through charging higher interest rate [R.sub.22] on the loans granted to the proportion [gamma]p of solvent entrepreneurs instead of [R.sub.21]. Using (6) and (7), it is straightforward to show that the additional profit (relative to the second-period profit in case of honest strategy) is given by:

[DELTA][[PI].sup.dh.sub.i2] = ([R.sub.22] - [R.sub.21]) [gamma]p/n = [bar.R]/n 1 - [gamma]/1 - p (13) (13)

Equation 13 shows that this additional profit is decreasing with the number of banks. If the credit market is competitive and the number of banks is high, we showed that the interest rates decrease and so do the profits. Hence, in a less competitive credit market, banks have more incentive to choose the dishonest information report strategy. To prevent banks from deviating, the Credit Bureau charges the dishonest bank a penalty [C.sub.1] and withdraws its license and excludes it from the credit market. Thus, not only does the bank pay the penalty [C.sub.1] but also abandons its third-period profit. Hence, a bank has no incentive to choose the dishonest strategy if the additional profit [DELTA][[PI].sup.dh.sub.i2] realized during the second period minus the penalty [C.sub.1] is lower than the third period misreporting profit [[PI].sub.i3] + [DELTA][[PI].sup.dh.sub.i3], which it abandons. Therefore, the penalty that the Credit Bureau should charge to prevent the misreporting strategy verifies:

[DELTA][[PI].sup.dh.sub.i2] - [C.sub.1] [less than or equal to] [[PI].sub.i3] + [DELTA][[PI].sup.dh.sub.i3] (14)

Proposition 2:. When the proportion of successful project is inferior to 50 % (case 1) the dishonest strategy is not optimal in the second period.

Proof. Using the expressions (12) and (13) and the fact that under case 1 we have [gamma]p < 1/2 it is straightforward to show that [DELTA][[PI].sup.dh.sub.i2] < [DELTA][[PI].sup.dh.sub.i3]. This means that condition (14) holds and the bank has no incentive to deviate in the second period. On the contrary, it is more profitable to it to misreport information at the beginning of period 3. Hence, when the proportion of successful projects is less than 50% there is no incentive for banks to deviate at the second period (this is because the condition [gamma](1 - p) + (1 - [gamma]) > [gamma]p is equivalent to 1/2 > [gamma]p).

Case 2: 1 - [gamma]p [less than or equal to] [gamma]p or [gamma]p [greater than or equal to] 1/2

In this case, the proportion p of high-ability entrepreneurs whose risky projects succeed is higher than all the defaulting entrepreneurs: 1 - [gamma]p. A bank that chooses the dishonest strategy will report the proportion 1 - [gamma]p of defaulting entrepreneurs as high-ability ones. However, it should report an additional [gamma]p - (1 - [gamma]p) = 2[gamma]p - 1 as high-ability entrepreneurs. It has no choice but selecting this latter proportion from the high-ability entrepreneurs whose projects has effectively succeeded. The remaining proportion of them ([gamma]p - (2[gamma]p - 1) = 1 - [gamma]p are reported as low-ability entrepreneurs.

Misreporting at the beginning of the third period

This situation occurs when the bank reports honestly the information at the second period and decides to misreport at the beginning of the third period. Table 3 presents the misreporting strategy of the bank.

This situation is equivalent to the one described in the subsection 'The dishonest strategy at the beginning of the third period' and we obtain the same additional profit given by equation 12. In this case also, the Credit Bureau discovers the dishonest bank at the end of the third period after observing the additional profit. However, the exclusion from the credit market does not make a sense and the dishonest bank does not pay the penalty.

Misreporting at the beginning of the second period

Table 4 presents the misreporting strategy of a bank that chooses the dishonest strategy at the beginning of the second period.

By choosing this strategy, the bank realizes an additional profit through charging the proportion p(1 - [gamma]p) of high-ability entrepreneurs the higher interest rate [R.sub.22] instead of [R.sub.21]. Using the expressions (6) and (7) of the interest rates we can easily calculate the second period additional profit:

[DELTA][[PI].sup.dh2.sub.i2] = ([R.sub.22] - [R.sub.21]) p(1 - [gamma]p)/n = [bar.R]/n (1 - [gamma]) (1 - [gamma]p)/[gamma](1 - p) (15)

Using equations 13 and 15 and the condition 1 - [gamma]p [less than or equal to] [gamma]p [less than or equal to] [gamma] is easy to show that the additional profit in case 1 is larger to that in case 2: [DELTA][[PI].sup.dh.sub.i2] [greater than or equal to] [DELTA][[PI].sup.dh2.sub.i2]. Hence, the additional profit is higher when the proportion of high-ability entrepreneurs whose risky projects succeed is lower than those who default. This is because the scope of misreporting is larger for the bank.

Proposition 3:. When the proportion of successful project is superior to 50% (case 2: [gamma]p [greater than or equal to] 1/2) the dishonest strategy is not optimal in the second period if and only if:

* the Credit Bureau withdraws the license of the dishonest bank during the third period if t [greater than or equal to [bar.t].

* the Credit Bureau withdraws the license of the dishonest bank during the third period and imposes a penalty strictly superior to [[bar.C].sub.2] if t < [bar.t] with:

[[bar.C].sub.2] = [gamma]pt/[n.sup.2]([bar.t] - t) [bar.t] = n[bar.R](1 - [gamma])/[([gamma](1 - p)p).sup.2] (p(1 - p) + [gamma]p - 1) (16)

Proof. See the appendix.

This proposition shows that the exclusion of the dishonest bank from the credit market during the third period is sufficient when the degree of differentiation of banks (the transportation cost) exceeds a determined threshold (t [greater than or equal to] [bar.t]). The intuition behind this is related to the trade-off that the dishonest bank faces. When t [greater than or equal to] [bar.t] the third-period profit exceeds the additional profit it could make if it chooses the dishonest strategy during the second period. However, when t < [bar.t] its monopoly power and its third-period profit are lower. In this case, the additional profit it could make during the second period is higher than the third-period profit. Therefore, the Credit Bureau has to impose a complementary sanction, which is a penalty strictly superior to [[bar.C].sub.2]. It is also interesting to note that an increase in the number of banks (n) lowers the penalty threshold [[bar.C].sub.2].

EXTENDING THE BASIC MODEL

In the section 'Dishonest strategy and the incentive mechanism of the Credit Bureau' we showed that the Credit Bureau induces banks to avoid the dishonest strategy during the second period by imposing to the deviating bank a sufficiently high penalty and withdrawing its license. However, it was shown that this is not possible when deviation occurs in the last and third period of the game. In this section, we extend the model to an infinite time horizon k = 0, 1, ..., [infinity] and show that the disciplinary role of the Credit Bureau holds for all the periods.

Entrepreneurs

We consider an overlapping generation of three-period lived entrepreneurs. At each period k a new generation [g.sub.k] is born containing a continuum [0,1] of entrepreneurs having the same characteristics as those described in the section 'Environment'. (3) Hence, at k = 0 there is only one generation go whereas two generations [g.sub.0] and [g.sub.1] coexist at k = 1 and for k [greater than or equal to] 2 we have three generations of entrepreneurs [g.sub.k - 2], [g.sub.k - 1] and [g.sub.k].

Banks

The n identical banks are located symmetrically around a circle of measure 1. Each bank i = 1, ..., n maximizes its discounted profits [V.sub.i] = [[summation].sup.[infinity].sub.k = 1] [[delta].sup.k - 1] [[PHI].sub.ik] where [delta] [member of] 0,1] represents the discount factor and [[phi].sub.ik] represents its total profit at date k.

Profit maximization under the honest strategy

Given the overlapping generation structure of the entrepreneurs (described in the section 'Entrepreneurs') the total profit [[PHI].sub.ik] is given by:

[[PHI].sub.i1 = [[PHI].sub.i1] [[PHI].sub.i2] = [[PHI].sub.i1] + [[PHI].sub.i2] [[PHI].sub.ik] = [[PHI].sub.i1] + [[PHI].sub.i2] + [[PHI].sub.i3] for k [greater than or equal to] 3 (17)

where [[PHI].sub.i1], [[PHI].sub.i2] and [[PHI].sub.3] are the maximized profits under the honest strategy and are given by (10) and (11). Therefore, we can write the discounted profits as following:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

Profit maximization in case of dishonest strategy

Let us now consider that bank i chooses the honest strategy during the first [k.sub.d] periods (k = 0, ..., [k.sub.d - 1]) and chooses the dishonest strategy at date [k.sub.d]. Figure 1 illustrates the impact of this choice on the bank's profits. This bank realizes additional profits of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (whose entrepreneurs reached the beginning of their third and final period of life at date [k.sub.d]) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (whose entrepreneurs reached the beginning of their second period of life at date [k.sub.d]) where [DELTA][[PI].sup.dh.sub.i3] and [DELTA][[PI].sup.dh2.sub.i2] are given by equations 12 and 15. Since the bank will be discovered by the Credit Bureau at date [k.sub.d] + 1 it renounces to all the subsequent profits, pays a penalty [C.sup.[infinity]] and stops its financial relationships with the entrepreneurs of generations [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. which were initially planned to end at dates [k.sub.d] + 2 and [k.sub.d] + 3, respectively. It is clear that the dishonest bank will not choose to deviate before date 3 in order to realize additional profits by misreporting entrepreneurs belonging to two generations rather than only one generation. Hence, [k.sub.d] [greater than or equal to] 3 and the total profit of the dishonest bank is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

Proposition 4:. When the proportion of successful project is superior to 50 % the dishonest strategy is not optimal at each date k = 1, ..., [infinity] if

* the Credit Bureau withdraws the license of the dishonest bank one period after its deviation when the bank is sufficiently foresighted: [delta] > [[delta].bar].

* the Credit Bureau withdraws the license of the dishonest bank one period after its deviation and imposes a penalty strictly superior to [[C.sub.bar].sup.[infinity]]([delta]) when the bank is not sufficiently far-foresighted: [delta] [greater than or equal to] [[delta].bar] with

[[delta].bar] = [DELTA][[PI].sup.dh2.sub.i2] + [DELTA][[PI].sup.dh.sub.i3]/ [3.summation over (j=1)] [[PI].sub.ij] + [DELTA][[PI].sup.dh2.sub.i2] + [DELTA][[PI].sup.dh.sub.i3] (20)

[[bar.C].sup.[infinity]]([delta]) = ([DELTA][[PI].sup.dh2.sub.i2] + [DELTA][[PI].sup.dh2.sub.i2])([delta] - [delta])/ (1 - [delta]) (21)

Proof. See the appendix.

Hence, a sufficiently far-foresighted ([delta] > [[delta].bar]) bank does not choose the dishonest strategy; otherwise it loses all its future profits, which have higher discounted value than the additional profits provided by the misreporting. However, if the bank cares less ([delta] [less than or equal to] [[delta].bar]) about the future profits lost, the Credit Bureau should complement the withdrawal of the license by imposing a penalty that makes the misreporting enough costly to induce the bank to choose the honest strategy. Finally, it is simple to note from equations 10 and 11 that [[summation].sup.3.sub.j = 1] [[PI].sub.ij] is increasing in the transportation cost, which gives us the following interesting variation of the threshold [[delta].bar] with the transportation cost t:

[partial derivative][delta]/[partial derivative]t < 0 (22)

This means that an increase in the bank's monopoly power (higher transportation cost) reduces the incentive of the bank to choose the dishonest strategy and consequently lowers the penalty threshold [[bar.C].sup.[infinity]]([delta]) (see equation 21).

CONCLUSION

In this paper we tried to answer the following question: could a Credit Bureau design an effective mechanism to incentivize banks to report correct information about their borrowers? To answer this question, we extended the model of Semenova (2008) in two directions. The first extension consisted in adding an explicit disciplinary mechanism by a Credit Bureau in an infinite-horizon framework. The second extension consisted in considering a credit market composed of n banks interacting in a spatial competition model a-la Salop (1979). The original result of this model is the sub-optimality of the dishonest strategy if the proportion of successful projects in the economy is less than 50%. In the opposite case, it was shown that the Credit Bureau could prevent the information's misreporting by withdrawing the license of the deviating bank and imposing a sufficiently high penalty one period after the occurrence of the deviation. We also showed that this penalty depends on many variables: the transportation cost, the proportion of high-ability entrepreneurs, the success probability of the risky investment projects and the banks' far-sightedness.

APPENDIX

Proof of Proposition 1

Each bank i = 1, ..., n fixes the interest rate [R.sub.i1] that maximizes its first period profit:

[[PI].sub.i1] = ([gamma]p[R.sub.i1] - [bar.R])[D.sub.i] - f (A.1)

Where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the demand addressed to bank i; [gamma]: the proportion of high-ability entrepreneurs; p: the probability of choosing successful projects; and [bar.R] the cost of capital. First, let's determine the expression of the demand [D.sub.i] addressed to bank i. Hence, we should determine the location x of the entrepreneur, situated between bank i and bank i + 1, who is indifferent between the two banks when asking for a loan.

[FORMULA NOT REPRODUCIBLE IN ASCII]

Bank i offers the interest rate [R.sub.i1] and bank i + 1 offers the interest rate [R.sub.1]. The indifference condition consists in equalizing the total cost of the two loans that could be granted by the two banks:

[R.sub.i1] + tx = [R.sub.1] + t(1/n - x) (A.2)

Then, we can determine the expression of bank's i demand:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A.3)

Thus, we have the following expression:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A.4)

Using, the expression [[PI].sub.i1] = ([gamma]p[R.sub.i1] - [bar.R])[D.sub.i] - f it is clear that + the profit of bank i is a concave function of [R.sub.i1] as shown by Figure 2.

Writing the first order condition, under the hypothesis that bank i considers the interest rate [R.sub.1] fixed by its rivals as exogenous (Cournot competition), we find the gross interest rate [R.sub.i1] that maximizes the profit [[PI].sub.i1]:

[R.sub.i1] = 1/2 ([R.sub.1] + t/n + [bar.R]/[gamma]p) (A.5)

This value is strictly superior to [R.sub.1] + t(1-n)/n under the condition [bar.R] [is less than or equal to] [gamma]p[R.sub.1], which is necessary to guarantee a non-negative profit during the first period. Now, we use the symmetry of the problem and set [R.sub.i1] = [R.sub.1] in (A.5). Hence, we obtain

[R.sub.1] = [bar.R]/[gamma]p + t/n (A.6)

Using (A.1) we obtain the following expression of the first-period profit

[[PI].sub.1] = [gamma]pt/[n.sup.2] - f (A.7)

It is now time to find the interest rates and the profit of the second period. The second period profit is given by

[[PI].sub.i2] = (p[R.sub.i21] - [bar.R])[D.sub.i1] + ([gamma]p(1 - p)/1 - [gamma]p [R.sub.i22] - [bar.R])[D.sub.i2] (A.8)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are, respectively, the demand for loan addressed to bank i by the successful entrepreneurs (whose proportion [gamma]p is uniformly distributed around the circle) and by the defaulting entrepreneurs (whose proportion 1 - [gamma]p is uniformly distributed around the circle). Hence, the expressions of these two types of demand are analogous to (A.4):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A.9)

It is clear from (A.8) that the maximization of [[PI].sub.i2] could be realized by maximizing separately its two components. In addition, these two separate problems are analogous to the maximization of [[II.sub.i1]. Therefore, to obtain the solution we have just to choose [gamma] = 1 and f = 0 in (A.1) for the maximization of the first term and replace p by p(1 - p)/(1 - [gamma]p) and f = 0 in (A.1) for the maximization of the second term. We can find the remainder of the results following the same reasoning.

Proof of Proposition 3

Taking in account the penalty and the exclusion from the credit market during period 3, the total profit of the deviating bank is given by:

[summation][[PI].sup.dh2.sub.i] = [[PI].sub.i1] + [[PI].sup.dh2.sub.i2] - [C.sub.2] (A.10)

Whereas the total profit of a bank choosing to deviate at the third period is given by

[summation][[PI].sup.dh1.sub.1] = [[PI].sub.i1] + [[PI].sup.h.sub.i2] + [[PI].sup.h.sub.i3] + [DELTA][[PI].sup.dh.sub.i3] (A.11)

where the additional profit [DELTA][[II.sup.dh.i3] is given by equation 12. Hence, the threshold penalty that delays the misreporting to the final period should verify:

[DELTA][[PI].sup.dh2.sub.i2] - [C.sub.2] [is less than] [[PI].sup.h.sub.i3] + [DELTA][[PI].sup.dh.sub.i3] (A.12)

And the penalty threshold is given by

[bar.[C.sub.2]] = [DELTA][[PI].sup.dh2.sub.i2] - [[PI].sup.h.sub.i3] - [DELTA][[PI].sup.dh.sub.i3] (A.13)

Finally, using equations 11, 12, 15 and A.13 we obtain the expression (16).

Proof of Proposition 4

The bank will not choose to deviate if [V.sup.h.sub.i] [greater than or equal to] [V.sup.dh.sub.i], which gives us the following condition [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] or equivalently

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The above inequality is also equivalent to [C.sup.[infinity]] [greater than or equal to] [[C.bar].sup.[infinity]]([delta]) with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If [[C.bar].sup.[infinity]] ([delta]) is negative this means that there is no need for the Credit Bureau to impose a penalty and the withdrawal of the bank's license is sufficient. This is the case ([[C.bar].sup.[infinity]] ([delta]) < 0) if the discount factor is superior to a threshold [[delta].bar]] given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Otherwise, the Credit Bureau should additionally imposes a penalty at least equal to [[C.bar].sup.[infinity]] ([delta]).

Acknowledgements

The authors would like to thank two anonymous referees and the editor for helpful suggestions on an earlier version of the paper. Any remaining errors are our own.

REFERENCES

Brevoort, KP and Wolken, JD. (2008): Does distance matter in banking?. Board of Governors of the Federal Reserve System (U.S.), Finance and Economics Discussion Series, 2008-34.

Dell'Ariccia, G. (2001): Asymmetric information and the structure of the banking industry. European Economic Review 45(10): 1957-1980.

Djankov, S, Mcliesh, C and Schleifer, A. (2006): Private credit in 129 countries NBER Working Paper No. W11078.

Elsas, R. (2005): Empirical determinants of relationship lending. Journal of Financial Intermediation 14(1): 32-57.

Ganbold, B. (2008): Improving access to finance for SME: international good experiences and lessons for Mongolia Visiting Research Fellow (VRF) paper, No. 438. Institute of Developing Economies, Japan External Trade Organization (IDE-JETRO), Chiba, Japan.

Grimaud, A and Rochet, J-C. (1994): L'apport du mode1e de concurrence monopolistique l'economie bancaire. Revue economique 45(3): 715-726.

Malhorta, M, Chen, Y, Criscuolo, A, Fan, Q, Hamel, II and Savchencko, Y. (2006): Expanding access to finance: Good practices and policies for micro, small, and medium enterprises. World Bank: Washington, D.C.

Padilla, AJ and Pagano, M. (1997): Endogenous communication among lenders and entrepreneurial incentives. The Review of Financial Studies 10(1): 205-236.

Pagano, M and Jappelli, T. (1993): Information sharing in credit markets. The Journal of Finance 48(5): 1693-1718.

Salop, SC. (1979): Monopolistic competition with outside goods. Bell Journal of Economics 10(1): 141-156.

Semenova, M. (2008): Information sharing in credit markets: Incentives for incorrect information reporting. Comparative Economic Studies 50(3): 381-415(35).

Stiglitz, JE and Weiss, A. (1981): Credit rationing in markets with imperfect information. The American Economic Review 71(3): 393-410.

World Bank. (2004): Doing business in 2004: Understanding regulation. World Bank: Washington, DC.

(1) They are composed of the high-ability entrepreneurs whose projects succeeded in the two periods (their proportion is [gamma][p.sup.2] and those whose project succeed in period 1 or period 2 (their proportion is 2[gamma]p(1 - p)).

(2) Technically, there are multiple methods to control the quality of the reported information. We can mention for example the comparison of the data reported by banks with those provided by the insurance agencies databases, the offices of taxation or other credit institutions.

(3) In particular, the entrepreneurs of each generation are uniformly distributed around the circle of measure 1 and each of them is located at a distance [x.sub.i] [member of] [0, 1/n] from a bank i and [x.sub.i + 1] = 1/n - [x.sub.i] from bank i + 1.

MAHMOUD SAMI NABI [1] & SOURAYA BEN SOUISSI [2]

[1] LEGI-Tunisia Polytechnic School, University of Carthage, BP 743, 2078 La Marsa, Tunisia and Economic Research Forum, Cairo, Egypt.

[2] LEGI-Tunisia Polytechnic School, University of Carthage, BP 743, 2078 La Marsa, Tunisia. Table 1: The composition of the entrepreneurs' population At the beginning At the beginning of the 1st period of the 2nd period High-ability: Solvent Solvent: [gamma] high-ability: [gamma]p [gamma]p Defaulting high-ability: Defaulting: [gamma](1-p) 1-[gamma]p Low-ability: Defaulting low- 1-[gamma] ability: (1-[gamma]) At the beginning At the beginning of the 1st period of the 3rd period High-ability: Solvent high-ability: Solvent: [gamma] [gamma][p.sup.2] [gamma][p.sup.2]+ Defaulting: [gamma]p(1-p) [gamma]p(1-p)=[gamma]p Solvent: [gamma]p(1-p) Defaulting: Defaulting: [gamma] [gamma]p(1-p)+ [(1-p).sup.2] [gamma][(1-p).sup.2]+ 1-[gamma]=1-[gamma]p Low-ability: Default: 1-[gamma] 1-[gamma] Table 2: Misreporting strategy only at the beginning of the third period Entrepreneurs' True type 1st period Reported proportion of the outcome of outcome entrepreneur's the loan at the beginning of the 2nd period [gamma]p High-ability Solvent Solvent (1-p) High-ability Default Default 1-[gamma] Low-ability Default Default Entrepreneurs' 2nd period outcome Reported outcome proportion of the loan at the beginning of the 3rd period [gamma]p Solvent [gamma][p.sup.2] Solvent: [gamma][p.sup.2] Defaulting [gamma]p(1-p) Defaulting: [gamma]p(1-p) (1-p) Solvent [gamma]p(1-p) Defaulting: [theta]=[gamma] [(1-p).sup.2]+1-[gamma] Solvent: [gamma][1+(1-p)(2p-1)-1 Defaulting Solvent [gamma] [gamma][(1-p).sup.2] [(1-p).sup.2] 1-[gamma] Default 1-[gamma] Solvent: 1-[gamma] Table 3: Misreporting strategy at the beginning of the third period Entrepreneurs' 1st period Reported outcome proportion outcome at the beginning of the loan of the 2nd period High-ability: Solvent: [gamma]p Solvent [gamma] Default: Default [gamma](1-p) Low-ability Default: 1-[gamma] Default :1-[gamma] Entrepreneurs' 2nd period outcome Reported outcome at proportion of the loan the beginning of the 3rd period High-ability: Solvent: Solvent: [gamma][p.sup.2] [gamma] [gamma][p.sup.2] Defaulting: [gamma]p(1-p) Defaulting: Defaulting [gamma]p(1-p) [theta]=[gamma] [(1-p).sup.2]+1-[gamma] Solvent [gamma]p(1-p) Solvent [gamma][1+(1-p)(2p-1))-1 Defaulting Solvent: [gamma][(1-p).sup.2] [gamma][(1-p).sup.2] Low-ability Default 1-[gamma] Solvent: 1-[gamma] :1-[gamma] Table 4: Misreporting strategy at the beginning of the second period Entrepreneurs' 1st period 2nd period outcome proportion outcome of the loan of the loan High-ability: Solvent: [gamma]p Solvent : [gamma][p.sup.2] [gamma] Defaulting: [gamma]p(1-p) Default: [gamma](1-p) Solvent: [gamma]p(1-p) Defaulting: [gamma][(1-p).sup.2] Low-ability: Default: 1-[gamma] Defaulting: 1-[gamma] 1-[gamma] Entrepreneurs' Reported type at the proportion beginning of the 2nd period High-ability: Solvent: p(2[gamma]p-1) [gamma] Default: p(1-[gamma]p) Solvent: (1-p)(2[gamma]p-1) Defaulting: (1-p)(1-[gamma]p) Solvent: [gamma]p(1-p) Solvent: [gamma][(1-p).sup.2] Low-ability: Solvent: 1-[gamma] 1-[gamma]