An Equilibrium Model of Entrusted Loans

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1 An Equilibrium Model of Entrusted Loans Ying Liu Job Market Paper November 21, 2017 Click here for the latest version Abstract Entrusted loans are inter-corporate loans and a major component of shadow banking in China. In a model with entrepreneurial moral hazard and bank moral hazard, entrusted loans arise endogenously when the banking sector is competitive. Entrusted loans involve a lending chain in which high-capitalized firms channel bank loans into medium-capitalized firms. High-capitalized firms obtain cheap bank loans and overborrow to form shadow banks with the extra capital. Medium-capitalized firms simultaneously borrow from both banks and shadow banks, while low and semi-highly capitalized firms borrow only from banks. As a result of lower bank monitoring, entrusted loans improve entrepreneurs welfare. However, entrusted loans destroy firms value because firms earn reduced expected profits. Default risk is increased and real efficiency reduced. Keywords: Shadow Banking, Entrusted Loans, Moral Hazard, Corporate Structure JEL: G21, G28, G32, G38 I would like to thank my supervisor Norman Schürhoff for his helpful comments and advice. I am also grateful for comments received from Theodosios Dimopoulos, John Duca, Artem Neklyudov, Steven Ongena, Jean-Charles Rochet, and from seminar and conference participants at IFABS Ningbo conference, UNIL brown bag seminar and HEC Paris Finance PhD workshop. University of Lausanne and Swiss Finance Institute; ying.liu@unil.ch; Website:

2 1 Introduction Since the global financial crisis, shadow banking in China has undergone rapid growth. The size of this sector accounted for 39 percent of China s GDP at the end of 2012, and in 2016, it already amounted to about $8.5 trillion, equivalent to 80 percent of China s GDP, as reported by Moody s. A dominant form of China s shadow banking is entrusted loans. 1 Entrusted loans are inter-corporate loans in which a financial institution acts as a trustee, and the lender and borrower directly negotiate the terms of the loan. The trustee merely acts as a middleman to facilitate the transaction for legal reasons. Classical banking theories posit that banks differom other credit suppliers in that banks screen or monitor borrowers. 2 Entrusted loans are associated with direct inter-corporate lending, so the borrowing firms are subject to little or no monitoring by the lending firms. Despite being suboptimal to bank loans regarding monitoring, entrusted loans are becoming prevalent in China. 3 This phenomenon raises a puzzle. Why do firms rely on otheirms to obtain financing, rather than specialized financial intermediaries such as banks? In this paper, I provide an economic rationale for why entrusted loans endogenously emerge as a form of shadow banking. I show that when the banking sector is perfectly competitive, the interplay of entrepreneurial hazard and bank moral hazard results in a lending chain in which firms with access to cheap bank loans recycle the loans by granting entrusted loans to otheirms. In addition to characterizing the feature of firms engaged in entrusted loans, my model shows that entrusted loans improve entrepreneurs welfare but destroy firms value and reduce real efficiency. 1 Shadow banking has different forms in China, such as, entrusted loans, wealth management products and trust loans (Elliott, Kroeber, and Qiao (2015), Elliott and Qiao (2015)). 2 See, for example, Diamond (1984). 3 Entrusted loans were the largest component of China s shadow banking sector until Since 2015, entrusted loans are the second largest component and wealth management products become the largest one. As documented by Allen, Qian, Tu, and Yu (2017), entrusted loans constituted 32% of shadow banking in terms of RMB amount at the end of Outstanding amount of entrusted loans has jumped to 13.2 trillion RMB ($1.92 trillion) in

3 I develop a model based on Besanko and Kanatas (1993) with firms differing in their capitalization, as in Holmstrom and Tirole (1997). Each firm invests own capital and raises the remaining part through bank loans or entrusted loans to undertake risky investment project. Limited liability induces entrepreneur to choose effort that influences the success probability of project. Banks grant loans and monitoirms so that entrepreneurs exert more effort than they would without bank monitoring. The new element I introduce is that firms may form shadow banks and issue entrusted loans to otheirms. The difference between banking and shadow banking, apart from the source of financing, is that shadow banks exert no monitoring efforts compared to banks. Shadow banking is, thus, a different lending technology. When the banking sector is perfectly competitive, there exists an equilibrium in which high-capitalized firms over-borrow from banks and form shadow banks with their extra capital. The reason for which entrusted loans arise in the presence of a competitive banking sector is the following. In a competitive environment, the bank gets zero surplus, and borrower surplus is maximized. Given that bank monitoring is costly and unobservable, banks are subject to the moral hazard problem when choosing monitoring intensity. Monitoring costs must be incorporated into loan interest rates to incentivize banks to provide monitoring. As firm s capitalization reduces, bank loan demand increases, the interest rate on bank loan also increases so that bank has a greater incentive to monitor the firm to receive the higher payoff if the project succeeds. The bank monitoring cost is an increasing, convex function of monitoring intensity, so bank interest rate has increasing return to scale. Therefore, high-capitalized firms have the advantage of obtaining cheap bank loans and choose to over-borrow. The increasing return to scale of bank interest rate creates opportunities for high-capitalized firms to re-lend extra capital to otheirms that incur expensive bank loans. Medium-capitalized firms simultaneously borrow from both banks and shadow banks while low and semi-highly capitalized firms only borrow from banks. Banks have an ad- 2

4 vantage in monitoring firms to alleviate entrepreneurial moral hazard, but banks cannot commit to monitoring. The bank moral hazard induces banks to choose a monitoring level to maximize bank profits, rather than maximizing entrepreneur utility. In this case, bank monitoring forces entrepreneur to supply a level of effort that exceeds the one that would be supplied without bank monitoring. Shadow banks do not monitoirms, so entrepreneurs avoid effort costs by replacing bank loans with entrusted loans. A marginal increase of entrusted loans has two effects on entrepreneur s expected utility: (1) the success probability of investment project is reduced because of less bank monitoring; (2) entrepreneur obtains higher payoff because of reduced effort. Low-capitalized firms are highly leveraged and entail severe entrepreneurial moral hazard, a marginal increase of entrusted loans causes large reduction of success probability which dominates the gain of entrepreneurial payoff, so these firms are better off not taking entrusted loans. Semi-highly capitalized firms entail low entrepreneurial moral hazard, so the yield difference between bank loans and entrusted loans is small, the increase of entrepreneur payoff by replacing bank loans with entrusted loans is lower than the reduction of success probability, so entrepreneurs are better off not taking entrusted loans. Therefore, the demand for entrusted loans displays a hump-shaped pattern oveirm capitalizations. In the extension, I also consider the case that bank operates as a monopolist. Entrusted loans do not exist in this situation. When the bank has monopoly power, the bank loan contract is set to maximize bank s profit, and the bank appropriates the surplus. Bank internalizes the surplus when chooses monitoring intensity. The high surplus motivates the bank to provide a high level of monitoring. The bank moral hazard problem is alleviated so there is no role for entrusted loans. To study the impact of shadow banking on social surplus and real efficiency, I examine a benchmark model in which entrusted loans are prohibited. In the benchmark, firms only take bank loans which are just enough to finance their investment projects. In the presence 3

5 of entrusted loans, entrepreneurs welfare are improved, but firms value reduced. The reason is that entrepreneurs maximize their utilities rather than firms profits. Bank monitoring forces entrepreneurs to exert more effort than they would do without bank monitoring. By participating in shadow banking, entrepreneurs can shirk and exert less effort. As a result, the success probabilities of firm investment projects are reduced and thereby, real efficiency declines. This model can be extended in a number of directions. First, I include the restricted industries which are prohibited from bank lending. 4 The lenders of entrusted loans are high-capitalized unrestricted firms; the borrowers become medium-capitalized unrestricted firms and high-capitalized restricted firms. Interest rates of entrusted loans increase with the proportion of restricted firms. Shadow banks are the only external financing source for restricted firms, and thus, behave like a monopolist. Interest rates of entrusted loans and the default risk of restricted firms are both higher than those of the unrestricted firms. Second, I consider the case that lending firms can better monitor the borrowing firms than banks if lending firms and borrowing firms are affiliated. For example, the lending firms and borrowing firms belong to the same industry, or same business group. The main results remain valid, and the model generates several new features: (i) credit rationing is alleviated, (ii) the borrowing firms of entrusted loans earn higher expected profits and have lower default risk than the benchmark. The reason is that substituting bank loans with shadow bank loans results in the greater entrepreneurial effort and, thus, reduces default risk and improves firm s profit. This paper also provides interpretations for some stylized facts. First, the result is consistent with the findings from Allen, Qian, Tu, and Yu (2017) that the lenders of entrusted 4 In China, such industries include the real estate, coal mining and shipbuilding industries (Elliott, Kroeber, and Qiao (2015)). Allen, Qian, Tu, and Yu (2017) find that close to half of non-affiliated entrusted loans flew into the real estate and construction industry. A Wall Street Journal article A Partial Primer to China s Biggest Shadow: Entrusted Loans(May 2, 2014) reports that from 2004 to 2013, around 20% of entrusted loans went to property sector. 4

6 loans are high-capitalized firms that can obtain cheap bank loans, and the borrowers are medium-capitalized firms. 5 The model shows that it is due to the competition and moral hazard of banks. Second, Allen, Qian, Tu, and Yu (2017) and He, Lu, and Ongena (2016) find that lenders of entrusted loans usually charge very high interest rates to non-affiliated borrowing firms. The model assumes that the lenders do not monitor the borrowers, which can be interpreted as the lenders and borrowers not being affiliated, so the lenders do not have the knowledge or ability to monitor the borrowers. The lack of monitoring makes the borrowers riskier, and thus the lenders have to charge high interest rates. Third, the lenders simultaneously borrow from banks and re-lend to the borrowers in the model. This borrow in order to lend activity is consistent with findings in Shin and Zhao (2013). Forth, Allen, Qian, Tu, and Yu (2017) and He, Lu, and Ongena (2016) show that the average abnormal return for the lenders of non-affiliated entrusted loans is negative after the entrusted loans announcement. This paper shows that entrusted loans destroy firm value and severely increase firm default risk because entrepreneurs do not behave diligently. The rest of this paper is organized as follows. Section 2 relates this paper to previous research. In Section 3, I describe the model and solve the benchmark. In Section 4, I introduce the shadow banking sector and derive the equilibrium results. I discuss welfare implications, and list the theoretical predictions in Section 5. In Section 6, I give three possible extensions of the model. Section 7 concludes the paper. 2 Literature Review To the best of my knowledge, this is the first theoretical model on entrusted loans. The most related paper is that of Allen, Qian, Tu, and Yu (2017), in which the authors examine entrusted loans with transaction-level data. They divide entrusted loans into affiliated loans 5 Allen, Qian, Tu, and Yu (2017) find that lenders are well-capitalized firms with a median value of 4 billion RMB. The borrowers are medium-size companies with a median value of 0.4 billion RMB. 5

7 and non-affiliated loans on the basis of firm relationships and show that the loan interest rates incorporate both fundamental and informational risks. Another related paper is that of He, Lu, and Ongena (2016), in which the authors assess the valuation effects of entrusted loans by measuring the abnormal returns of stock prices after entrusted loan announcements. Chen, Ren, and Zha (2017) focus on the impact of monetary policy on entrusted loans and emphasize the different roles of state-owned banks and non-state-owned banks that work as trustees of entrusted loans. Yu, Lee, and Fok (2015) find a significant and positive correlation between high-interest entrusted loans and firms last period cash holdings. This paper is also related to the theoretical studies on credit rationing and constrained bank lending. On the one hand, some studies feature moral hazard. My paper belongs to this strand. Similar papers, include those of Besanko and Kanatas (1993), Petersen and Rajan (1995), Holmstrom and Tirole (1997), Repullo and Suarez (2000) and Allen, Carletti, and Marquez (2011). On the other hand, studies focus on asymmetric information, such as Stiglitz and Weiss (1981), Besanko and Thakor (1987a), Besanko and Thakor (1987b). Different from the above papers, I include firm heterogeneity, and I also introduce the channel of shadow banking. Therefore, I can model the endogenous emergence of shadow banking and identify the characteristics of the firms engaged in this activity. Moreover, by focusing on the moral hazard problem, I can study the impact of shadow banking on entrepreneurial behavior and firm performance. This paper also belongs to the branch of literature about China s shadow banking. Acharya, Qian, and Yang (2016) document another large component of shadow banking: wealth management products (WMP). They show that when small and medium-sized banks face strong competition from the Big Four banks to acquire deposits, they significantly increased the issuance of WMPs after the financial crisis. Dang, Wang, and Yao (2014) explain the growth of shadow banking with information sensitivity, which is a measure of tail risks. The rise of shadow banking is associated with the asymmetric perception of 6

8 information sensitivity from banks, shadow banks, and investors. Both Hachem and Song (2017) and Chen, He, and Liu (2017) link shadow banking activities to fiscal policy. Hachem and Song (2017) argue that the shadow banking system is related to the competition between small and medium-sized banks and the Big Four banks. Shadow banking is an intended consequence of the liquidity rules. Chen, He, and Liu (2017) show that the growth of shadow banking after 2012 is a hangover effect of the 4 trillion stimulus package. Other theoretical studies regard shadow banking as a problem of regulatory arbitrage. Buchak, Matvos, Piskorski, and Seru (2017) find that the increased regulatory requirements of banks account for 55% of shadow bank growth, and 35% of the growth comes from the development of financial technology. Farhi and Tirole (2017) show that financial intermediaries can migrate to shadow banking in response to regulatory requirements, and they also study optimal regulation in the presence of the shadow banking sector. In my paper, shadow banking is not caused by the regulatory burden of traditional banks, and the banks do not have capital constraints in my model. Shadow banking is a market reaction to the high competition of the banking sector. Moreover, entrusted loans are very similar to trade credit as documented by Petersen and Rajan (1997) and Biais and Gollier (1997) in the sense that both are inter-corporate financing. However, entrusted loans are not necessarily granted between suppliers and customers, and there are no goods or service purchases involved in entrusted loans. 3 Model The model is built on Besanko and Kanatas (1993) with two innovations. First, entrusted loans are introduced as an outside option, besides firm s own investment project. Firms have to decide whether to participate in entrusted loans. Second, firms have different capitalizations, as in Holmstrom and Tirole (1997). When taking risky investment projects, firms 7

9 invest all own capital and borrow the remaining from external financing. The introduction of firm heterogeneity allows me to study the firm characteristics that are engaging in entrusted loans. The model contains two types of agents: firms and banks. All agents are assumed to be risk neutral. There are two dates in the model: date 0 and 1. At date 0, each firm tries to locate funding for a risky investment project. At date 1, returns from projects are realized, firms are liquidated, and banks are paid off. There is a continuum of firms. Each firm is characterized by the capital w [0, 1] it holds. The set of firms is described by the probability density function g(w). All firms have access to the same technology and invest the same risky project in the model. The project requires investment 1 at date 0. If w < 1, the firm needs at least 1 w external financing to undertake the project. At date 1, the project generates return either Q > 1 with probability p (success) or 0 with probability 1 p (failure). Firms are run by entrepreneurs. The probability that the project succeeds depends on the effort the entrepreneur spends on the project. The cost of effort required to achieve a success probability p is p2, where β (0, 1 ] is the reciprocal of the marginal cost of effort. 2β 2 For simplicity, p is also the total effort from the entrepreneur. If the entrepreneur wants to increase the success probability of the project, she has to expend more effort, which will reduce her utility. Moral hazard arises since the effort of the entrepreneur is unobservable by the market. When the firm is not able to be self-financed, it has to take a loan from the bank. The banking sector is perfectly competitive, so profit of each bank is driven to zero. The function of banks is twofold: providing financial credit to the firm; offering monitoring services to improve entrepreneur performance. Banks have a comparative advantage at monitoring firms, which alleviates the entrepreneur moral hazard problem. Monitoring carries a cost to 8

10 the bank, which is assumed to be (p p 0 ) 2 /2m, if p p 0, M(p p 0 ) = 0, if p < p 0. p is the effort level desired by the bank, p 0 denotes the entrepreneur effort were there no bank monitoring and thus called entrepreneur non-monitoring effort, m (0, β) denotes the reciprocal of the marginal cost of monitoring. entrepreneur is smaller than the bank, so m < β. The marginal monitoring cost of the Bank monitoring raises the success probability from p 0 to p. In case the bank desired level p is lower than the entrepreneur non-monitoring effort level p 0, the bank does not monitor and the total effort p equals to non-monitoring effort p Entrepreneur effort and bank monitoring The timing of the model is as follows. Banks set up loan contracts which specify the loan size L B and corresponding repayment R B. Firms decide how much loans to take and invest in the project. Then entrepreneurs choose effort level p 0 and banks choose monitoring intensity p p 0. By back-ward induction, I first solve the optimal entrepreneur non-monitoring effort p 0 and bank monitoring p p 0 for given bank loan size L B and repayment R B. The entrepreneur chooses the effort to maximize her expected utility that is given by p 0 = arg max p p(q R B ) p2 2β + L B + w 1 = min{ β(q R B), 1}, where 1 is the gross risk-free interest rate. For ease of exposition, two assumptions are made: 9

11 Assumption 1. βq 1. Assumption 2. βq2 2r 2 f 1 > 0. Assumption 1 states that the first-best level of effort, equivalently the success probability is not bigger than 1. Under this assumption, the non-monitoring effort is thus p 0 = β(q R B). (3.1) Assumption 2 ensures that the maximized NPV of self-financed project is strictly positive, otherwise the entrepreneur will not undertake the investment project. Assumption 1 and 2 2 together require that r β f < Q. β Banks do not commit to monitoirms, and bank monitoring is also unobservable to the market, so banks also incur moral hazard problem. The bank monitoring intensity is chosen to maximize the expected profit of the bank, for given non-monitoring effort p 0, that is max p pr B (p p 0) 2 2m L B. The optimal bank monitoring is derived as p p 0 = mr B. (3.2) Equation (3.1) and (3.2) show that, both entrepreneur non-monitoring effort and bank monitoring expenditure are proportional to their payoffs. The payoff to entrepreneur is the total investment output deducted by repayment to the bank, so entrepreneurs of high-capitalized firms are willing to spend more effort. Instead, bank monitoring intensity is higheor medium-capitalized firms. Overall, the total optimal entrepreneur effort is given as p = βq + (m β)r B 10

12 3.2 Benchmark: bank lending with no shadow banking For comparison, I start with the benchmark in which shadow banking is prohibited. In the benchmark, banks are the only outside financing for all firms. Then the equilibrium bank lending problem is equivalent to the question in Besanko and Kanatas (1993), except that firms have different initial capital. The equilibrium loan size L B and bank repayment R B are chosen to maximize the entrepreneur s expected utility, subject to three constraints: the zero-profit condition of the bank, and two participation constraints of firms. The optimization problem of entrepreneur is given as max U = p (Q R B ) p 2 L B,R B 2β + L B + w 1, (3.3) subject to L B = p R B (p p 0) 2, 2m (bank zero-profit condition) L B + w 1, (firm participation constraint 1) R B Q. (firm participation constraint 2) The entrepreneur utility is composed by three parts: the expected discounted firm profit p (Q R B ) ; the cost of entrepreneur effort p 2 ; and retained capital L 2β B + w 1. The zeroprofit condition of bank comes from the fact that banking sector is perfectly competitive. Participation constraint 1 states that total capital of each firm is enough to invest in the project. If not, the firm will return the loan L B to the bank and give up the project. Participation constraint 2 ensures that the payoff to the entrepreneur should be nonnegative to incentive the expenditure of effort on the project. The maximization problem is simplified by substituting the zero-profit condition into the objective function and thereby eliminating L B. Moreover, the optimal entrepreneur non- 11

13 monitoring effort level (3.1) and bank monitoring (3.2) are impounded into the objective function, which yields max R B (m β m2 β )R2 B + βq2 (1 w) (3.4) 2rf 2 subject to (m 2β)R 2 B + 2βQR B 2r 2 f 1 w, (3.5) Q R B 0. (3.6) A third assumption is further required: Assumption 3. m < β 2 Assumption 3 ensures that the second order condition of the optimization problem is negative, so there exists an solution. Solving the optimization problem of entrepreneur, yields the following results. Lemma 1. In equilibrium, only firms with initial capital w w N 1 β2 Q 2 from banks, otheirms are not able to obtain bank loans. 2(2β m)r 2 f borrow Note that w N 0 when Q 2 2 (2 m β β )r2 f, that is, the profitability of investment project is high enough such that all firms are financed. While w N > 0 when Q 2 < 2 β (2 m β )r2 f, then firms with initial capital w < w N are credit rationed by banks. Lemma 2. In equilibrium, firms with capital w [w N, 1] take bank loan: L B = 1 w L OB B. (3.7) 12

14 The required repayment to banks is R B = βq β 2 Q 2 2(2β m)(1 w)r 2 f 2β m R OB B. (3.8) Equation (3.7) states that entrepreneur only issues the minimum amount of debt that is just enough to finance the project. Because of the marginal cost of debt, bank loan interest rate, is higher than the marginal return of retained capital, which is risk-free rate. So there is no need to hold capital. The interest rate of bank loan is computed as R OB B L OB B = βq β 2 Q 2 2(2β m)(1 w)r 2 f (2β m)(1 w). (3.9) Note that the interest rate decreases in firm capitalization w. The negative relationship between bank loan interest rate and firm capitalization is due to two reasons. First, banks price loans by incorporating risks of investment projects. When the firm is low capitalized and invest less own capital in the risky project, the entrepreneur s incentive to exert effort declines, which results in a high probability of investment failure. Banks thus charge a higher interest rate to compensate the risk of default. Secondly, banks also incorporate the monitoring cost into the interest rate. Bank s monitoring cost is a convex function of expenditure; the expenditure is proportional to the exposure R B as shown by equation (3.2). So the average monitoring cost increases in loan demand. Low capitalized firms take a large amount of bank debt and thus incur higher monitoring cost. Overall, banks charge a higher interest rate on low capitalized firms. 13

15 4 Shadow banking In this section, shadow banking is introduced. Since shadow banking in this paper refers to inter-corporate entrusted loans, there are still two types of agents in the model: firms and banks. However, among the firms, some endogenously form shadow banks and provide financing to otheirms in the form of entrusted loans. To differentiate, the formeirms are merely called lenders and the latter ones borrowers of shadow banking. The first question is to determine characteristics of firms participating in shadow banking. Specifically, what kind of firms become lenders and borrowers of shadow banking. Because all firms are in short of capital for investment project, the way lending firms can finance borrowing firms is through over-borrowing from banks 6. Therefore, firms decision of whether participating in shadow banking as lenders is equivalent to whether over-borrowing from banks or not. Lemma 2 shows that firms optimally choose to borrow the minimum amount of bank loan which is just enough to finance the risky project. That is because retained capital only generates a return of risk-free rate, which is smaller than the borrowing rate. However, once firms have access to another option which returns with a much higher rate, firms might be willing to over-borrow and invest extra capital in the new option. That is the way I model the emergence of shadow banking. Specifically, in the first step, shadow banking is simply formalized by an outside investment option which generates an expected discounted rate of return r s > 1. Firms can invest extra capital L B + w 1 in the outside option. The value of r s cannot be smaller than 1. Otherwise, firms are better off depositing the extra capital and earn the discounted rate of return 1. The lenders are firms that choose to over-borrow bank loans. The borrowers are merely firms choosing to take shadow bank loans. After lenders are determined, and shadow 6 Purely conducting lending activities without investing in own business is not allowed, otherwise, the firm becomes a bank and has to follow all the requirements of bank and function like a bank. 14

16 banks are formed. All firms except lenders thereby have both banks and shadow banks as outside financing sources. These firms will optimally choose from both sources. The existence of shadow banking depends on the rate r s. The equilibrium value of r s is determined by market clearing condition. That is, the total supply of entrusted loans should be equal to total demand. Total supply is the aggregate extra capital from lenders, and total demand is the aggregate demand from borrowers. The existence of shadow banking is therefore equivalent to having a solution r s > 1 which solves the market clearing condition. Although the emergence and lending of shadow banking are sequential, I take the two procedures as simultaneous. In essence, the two procedures are both external fundraising activities of firms. In a static model, I can study how banks and shadow banks interact in providing loans to firms. The timeline is given in figure 1. Figure 1: Time line of the model Firms decide whether to take bank loan, if yes, firms also choose loan volume L B. Firms choose the optimal bank loan L B and shadow bank loan L S. 0 1 Banks set the loan contract. Firms which take loan L B > 1 w form shadow banking, and set the entrusted loan contract Firms choose the effort p 0, banks choose the monitoring intensity p p 0 Projects mature, claims are settled 15

17 4.1 Endogenous emergence of shadow banking Similar to the benchmark, except that return from extra capital is r s instead of 1. The bank loan contract {L B, R B } is set so as to maximize the entrepreneur s utility, which is given as max U = p (Q R B ) p 2 L B,R B 2β + r s(l B + w 1), (4.1) subject to L B = p R B (p p 0) 2 2m, (4.2) L B + w 1, (4.3) R B Q. (4.4) Solving the optimization problem of entrepreneur, yields the following results. Lemma 3. For given r s > 1, firms with capital w over-borrow from banks when r s r s (w) = 1 + m 2 β + β m 2β m βq β 2 Q 2 2(2β m)(1 w)rf 2 1. (4.5) First, note that r s > 1, which is consistent with the assumption that r s > 1. Secondly, r s is firm specific, which is a decreasing function of w. It implies that firms that hold high capitalization tend to over-borrow and thus become lenders of shadow banking. Inverse solution of (4.5) shows that lenders of shadow banking have initial capital w 1 β2 Q 2 2r 2 f [ (r s 1) (2β m)r s+ 2m2 β m ] [ (2β m)r s+ m2 β β ] 2. This threshold value increases in r s. That is, high expected rate of return of shadow banking drive up firm participations. Proposition 1. For given r s > 1, firms with initial capital w 1 β2 Q 2 2r 2 f [ ] (r s 1) (2β m)r s+ 2m2 β m [ ] 2 (2β m)r s+ m2 β β over-borrow and form shadow banks with extra capital. Bank loan size and required repayment 16

18 are L B = β2 Q 2 2r 2 f R B = [ (r s 1) (2β m)r s + 2m2 β ] m [ (2β m)r s + m2 β β ] 2 L S B, βq(r s 1) (2β m)r s + m2 β RS B. β When entrepreneur chooses to over-borrow, the optimal loan size L S B does not depend on firm capitalization w anymore. Instead, all lendeirms take the same amount of bank loan. The firm that is indifferent from over-borrowing or not has capital w = [ (r s 1) (2β m)r s+ 2m2 β m ] 1 β2 Q 2 2rf 2 [ ] 2 = 1 L S (2β m)r s+ m2 β β B. Note that the expected profit from outside option r s (L B + w 1) is a lineaunction of loan size L B, so firms are willing to take as much bank loan as possible. However, on the other hand, taking more debt means entrepreneurs payoff from investment project will decline, and further discouraging entrepreneur behaving diligently, worsening the expected profit of firm s investment project. Therefore, the optimal amount of bank loan is determined such that the return from shadow banking compensate the reduction of entrepreneur utility. Also, the required repayment R B and bank loan interest rate R B L B are also independent of w. Each lendeirm has the same bank loan contract, each entrepreneur pays the same amount of effort and bank expends the same monitoring. Entrepreneurs of lenders firms obtain the same amount of utility, which equals to the utility of entrepreneur of the boundary firm w = 1 L B could get. 4.2 Shadow bank lending In the presence of shadow banks, all firms except lenders derived in section 4.1 optimally choose loans from banks and shadow banks. The difference between banks and shadow banks are twofolds: first, shadow banks do not provide monitoring service. Second, shadow banks 17

19 require a rate of return to be at least r s > 1, while the rate of return to bank is 1. Since shadow banks compete with formal banks in providing capital to firms, the rate of return from shadow banks is squeezed to r s. Similarly, the optimal bank loan contract {L B, R B } and shadow bank loan contract {L S, R S } are selected to maximize entrepreneur s utility, which is given as max U = p (Q R B R S ) p 2 L B,R B,L S,R S 2β + L B + L S + w 1 (4.6) subject to p R B (p p 0) 2 = L B, (4.7) 2m p R S = r s L S, (4.8) L B + L S + w 1 0, (4.9) R B + R S Q. (4.10) The inclusion of shadow banks bring several changes. First, entrepreneur s optimal nonmonitoring effort becomes p 0 = β(q R B R S ). The optimal bank monitoring effort p p 0 keeps but the total entrepreneur effort becomes p = βq βr S+(m β)r B. Second, if firms decide to take both bank loan and shadow bank loan, that is, L B > 0 and L S > 0, total loans should be enough to start the project, as indicated by (4.9). Meanwhile, total loan repayments R B + R S should not exceed the investment return Q. The rate of return from retained capital L B + L S + w 1 is only 1 because there is no second-tier shadow banking opportunity. In one case, suppose firm A over-borrows from both bank and shadow bank, and A wants to lend out extra capital to firm B. Within the capital A obtained, shadow bank requires a rate of return r s. When A lends to B, A expects a rate of return which is higher than r s. However, in this case, B could borrow directly 18

20 from shadow bank, instead of A. So A loses the borrower market because of competition from shadow bank. In another case, suppose firm A only borrows from banks and overborrows, then A lends to B. In this case, A must have initial capital w 1 L S B, which means A belongs to the lenders derived in section 4.1. Therefore, there is no second-tier shadow banking opportunity and the rate of return from retained capital is 1. Solving the entrepreneur s optimization problem gives the equilibrium bank and shadow bank lending. Proposition 2. Depending on the value of the expected rate of return of shadow banking r s, optimal financing strategies of firms are given as ) 1. If r s (1 1 + mβ m + m2 r 3 β β 2 s, all firms with capital w [w N, 1 L S B ] take only bank loans. Each firm borrows the minimum amount L B = 1 w, and bank requires return R B = βq β 2 Q 2 2(2β m)(1 w)r 2 f 2β m R OB B. (4.11) Firms with capital w < w N can not get any financing. 2. If r s < r s, firms with capital w [w 1, w 2 ] borrow from both bank and shadow bank, where w 1 = 1 βq 2(2 m β )r2 f [ 1 ( ) ] 2 (r s 1)(m 2 +2mβ 2β 2 )+m(2 mβ )(m β 2 (1+2r s 3rs)+2mβ(r 2 s 1)+m 2 ) 2(r s 1)(m, 2 mβ+β 2 )+2m 2 (2 m β ) w 2 = 1 βq 2(2 m β )r2 f [ 1 ( ) ] 2 (r s 1)(m 2 +2mβ 2β 2 )+m(2 mβ )(m+ β 2 (1+2r s 3rs 2)+2mβ(rs 1)+m2 ) 2(r s 1)(m. 2 mβ+β 2 )+2m 2 (2 m β ) The equilibrium bank loan size L B LM B, bank repayment R B RM B, the equilibrium shadow bank loan size L S LM S, shadow bank repayment R S RM S, are solutions to 19

21 the equation system U L B = U L S L B = β R rf 2 B (Q R S + ( m 1)R 2β B ) L S = β r sr R f 2 S (Q R S + ( m 1)R β B ) (4.12) L B + L S = 1 w L B 0, L S 0 Firm with capital w [w N, w 1 ) or w (w 2, 1 L S B ] borrow only from banks. The bank loan size is L B = 1 w and repayment to bank is R B = ROB B. 4.3 The existence of shadow banking The existence of shadow banking is equivalent to the existence of a solution r s (1, r s ) which satisfies the market clearing condition: total supply of shadow bank capital equals to total demand of shadow bank loan. The supply of shadow bank comes from the extra capital of lenders derived in section 4.1, which is T S = 1 1 L S B (L S B + w 1)g(w)dw. The demand of shadow bank loan is the total demand of the borrowers derived in section 4.2, that is, T D = w2 w 1 L S Sg(w)dw. The shadow banking exists if and only if there exists a solution r s (1, r s ) that solves T D = T S. Note that both total supply T S and total demand T D are functions of the expected shadow banking rate r s, one can derive some properties of T S and T D with respect 20

22 to r s. First, total supply T S is a monotonically increasing function of r s, since the bank loan demand L S B of lender is an increasing function of r s, so T S 1 L S B = g(w)dw + (L S B + w 1)g(1 L S r s 1 L S r B) LS B > 0. B s r s Intuitively, when the expected return from shadow bank lending is high, lenders prefer to borrow more from banks so that they can lend more through shadow banking, which drives up the total supply. Total demand T D is a monotonically decreasing function of r s, since w 1 is an increasing function of r s, w 2 and L M S are both decreasing function of r s, so one gets T D w2 L M S = g(w)dw + L M S g(w 2 ) w 2 L M S g(w 1 ) w 1 < 0. r s w 1 r s r s r s Second, one can compute the values of T S and T D at two boundaries, respectively: lim T S = 0 r s 1 lim T S > 0 r s r s 1 LS lim T D = B L M S g(w)dw > 0 r s 1 w N lim T D = 0. r s r s Since the total supply is a monotonically increasing function, the total demand T D is a monotonically decreasing function, and values of both functions are positive on r s (1, r s ), by the fixed point theorem, there should exist a solution r s (1, r s ), which solves T S = T D. Lemma 4. There always exist a shadow banking sector, with expected rate of return r s (1, r s ) that solves 1 (L S 1 L S B + w 1)g(w)dw = w 2 B w 1 L S S g(w)dw. 21

23 I solve the model numerically by setting Q = 2.2, β = 0.5, m = 0.2, = 1.1 and assuming w follows uniform distribution on [0, 1]. The bank and shadow bank lending are demonstrated in figure 8. Figure 8 shows that shadow bank loan demand L M S is a concave function of firm capitalization w. That is due to the entrepreneur effort. When firms simultaneously borrow from banks and shadow banks, banks provide monitoring which guarantees a high level of success probability. Shadow banks free ride banks monitoring and thus charge a lower interest rate because of no monitoring. So entrepreneurs are willing to take shadow bank loans and thereby reduce the demand for bank loans. However, the increase of shadow loan reduces the entrepreneur effort p because bank monitoring declines. So entrepreneurs have to balance the tradeoff between increased payoff and reduced success probability. Foirms with low capitalization w [w N, w 1 ), the reduction in p is much larger than the increase in entrepreneur payoff, so entrepreneurs do not take shadow bank loans. Foirms with high capitalization w (w 2, 1 L S B ], the increase in entrepreneur payoff is smaller because these firms enjoy low bank interest rate, shadow loan rate is not competitive as the bank loan rate, so entrepreneurs choose not to take shadow loans. Foirms which take both bank loans and shadow bank loans, the two loan demands are determined that the marginal profit of bank loans and shadow bank loans be equal, as shown by (4.12). I then compute and plot the interest rates of bank loan and shadow bank loan in figure 2. Figure (2a) shows the bank loan rates with and without shadow banking (benchmark), as well as shadow bank loan rate. The green line represents the equilibrium bank lending rate; the gray dotted line plots the bank lending rate in benchmark, the blue line plots the shadow bank lending rate. Figure (2b) gives a clear illustration of how the bank lending rate changes compared with the benchmark. Firms with the capital w [1 L S B, 1] are lender firms, firms with capital w [w 1, w 2 ] are borroweirms of shadow banking, otheirms do not participate in shadow banking. First, note that both banks and shadow banks charge lower interest rate to high- 22

24 Figure 2: The comparison with and without shadow banking (a) The lending rates (b) The change of bank lending rate from the benchmark capitalized firms and higher interest rate to medium-capitalized firms. They are mainly driven by the default risk caused by moral hazard problem. Next, shadow bank interest rates are smaller than bank interest rates. The difference comes from the bank monitoring service. Banks monitoirms by paying a convex cost, so banks incorporate the cost into the interest rate, which can be seen by deriving the interest rate from bank s zero profit equation (4.7): R B = L B p ( 1 + (p p 0) 2 2mL B ) Shadow banks do not monitoirms, so shadow banks can charge a lower rate than bank lending rate, as given by (4.8): R S L S = p r s. Since it is costly to monitoring medium-capitalized firms, the average monitoring cost increases in firm capitalization w, so is the difference between bank and shadow bank lending rates, as demonstrated by the green line and the blue line. 23

25 Moreover, in the presence of shadow banking, banks increase the lending rate to both lendeirms and borroweirms, which is illustrated in figure 2b. The adjustment is also positively correlated with the involvement of shadow banking activities. That is because participation of shadow banking discourages entrepreneur to perform diligently. The default risk (1 p) of both lenders and borrowers thus increase, so banks incorporate the high risk into the interest rates. 5 Welfare Analysis To study the impact of shadow banking on social surplus and real efficiency, I compare the equilibrium results with the benchmark where shadow banking is prohibited. Figure 9 illustrates the equilibrium results with and without entrusted loans (benchmark). Since banks always obtain zero profit under the two situations, and proposition 2 shows that the presence of shadow banking does not change the credit rationing, so I only focus on firms profits. Then, I study the impact of shadow banking on real efficiency by comparing the success probabilities p of the investment project. 5.1 Firm s profit Entrusted loans affects two types of firms: firms with capitalization w (1 L S B, 1] that become the lenders of entrusted loans, and firms with capitalization w (w 1, w 2 ) that become the borrowers. Note that the equilibrium bank loan contract {L B, R B } and entrusted loan contract {L S, R S } are chosen to maximize the entrepreneur s utility U, which is different from firm s profit π: U = π p2 2β. (5.1) 24

26 The entrepreneur s utility U is lower than the firm s profit π by p2, that is the entrepreneur 2β effort cost. Using results from proposition 1 and 2, firm s profit in the presence of shadow banking is computed as π OB F = (β m)[β2 Q 2 rf 2(2β m)(1 w)]+(β2 mβ+ m 2 )Q β 2 2 Q 2 2rf 2(2β m)(1 w) (2β m) 2 rf 2, w N w < w 1 π F = πf M = [β(q RM S )+(m β)rm B ](Q RB M RM S ), w rf 2 1 w < w 2 π OB F = (β m)[β2 Q 2 rf 2(2β m)(1 w)]+(β2 mβ+ m 2 )Q β 2 2 Q 2 2rf 2(2β m)(1 w) (2β m) 2 rf 2, w 2 w < 1 L S B π S F = βq2 [2m 4 +2β 4 r 3 s 2m 3 β(r s+1) mβ 3 r s(r s+1)+2β 2 m 2 r s(r s+2)] 2r 2 f(m 2 mβr s+(2r s 1)β 2 ) 2 r s (1 w), 1 L S B w 1 (5.2) And firm s profit in benchmark is π OB F = ] (β m) [β 2 Q 2 r 2f (2β m)(1 w) + (β 2 mβ + β m2 )Q 2 2 Q 2 2rf 2 (2β m)(1 w) (2β m) 2 r 2 f (5.3). Comparing lending firm s profit π S F and borrowing firm s profit πm F respectively, yields with the benchmark πob F, Lemma 5. In the presence of shadow banking, both the lending firms and borrowing firms get lower expected profits: π S F π OB F, π M F π OB F. The declines in firms profits are due to the agency problem. Entrepreneur chooses the debt level to maximize her utility, which equals to firm s profit minus entrepreneur effort cost, as shown by equation (5.1). Since bank also incurs moral hazard problem, bank chooses the monitoring effort to maximize its profit, not entrepreneur utility. So the bank s monitoring forces entrepreneur to exert more effort than her desired level. More specifically, banks force entrepreneurs to exert effort p, which is higher than entrepreneur s desired level 25

27 p 0. In the presence of entrusted loans, the lending firms replace bank loans with entrusted loans and avoid the corresponding bank monitoring. The lending firms of entrusted loans take excess loans which result in higher debt level, so the entrepreneurs also behave less diligently because of moral hazard. Overall, by engaging in entrusted loans, entrepreneurs obtain higher welfare, but firms earn lower expected profits as entrepreneurs exert less effort than the benchmark. I then compute the profits with the numerical results and plot the change of profit 100 (π F πf OB)/πOB F in Figure 3. The profits are reduced in the presence of entrusted loans. The reduction is very significant for high-capitalized lending firms because these firms invest a large amount in shadow banking. For the borrowers of entrusted loans, the reduction is a concave function of firm capitalization, which is proportional to the demand of entrusted loans of these firms. In other words, firms which participate more in shadow banking loose more profit. Figure 3: The change of firm profit with and without entrusted loans 26

28 5.2 Real efficiency The probabilities of success or the total efforts from entrepreneur are demonstrated in figure 4. Figure 4a compares the total entrepreneur effort with the benchmark, figure 4b demonstrates the change of entrepreneur effort compared with benchmark 100 (p p OB )/p OB. The total entrepreneur effort declines in the presence of entrusted loans. Similarly, the reduction is positively correlated with the involvement of shadow banking and more significant for the lending firms. However, the reductions are due to different factors for the lending firms and borrowing firms. Note that the total entrepreneur effort is the sum of entrepreneur non-monitoring effort and bank monitoring effort. The reduction of lenders comes from the decreased entrepreneur non-monitoring effort, while the reduction of borrowers is from the decreased bank monitoring effort. The lenders increase the demand for bank loan since part of the loan is invested in shadow banking, so the repayment to bank also increase, and the projected return to entrepreneur is reduced. Entrepreneur thus reduces the non-monitoring effort since it is proportional to the return: p 0 = β(q R B). For the borrowing firms, the demands of bank loans are reduced since part of bank loans are replaced by shadow bank loans. Banks cut the monitoring effort because banks receive a reduced payoff. While shadow banks do not provide monitoring service, the total entrepreneur effort thus is decreased. 27

29 Figure 4: The comparison with and without shadow banking (a) The entrepreneur total effort (b) The change of entrepreneur total effort Taken together, the reduction of entrepreneur effort of both lending firms and borrowing firms makes these firms riskier. Therefore, banks increase the lending rates of these firms, as shown in figure 2b. In conclusion, entrusted loans improve entrepreneurial welfare but reduce firms profit, because entrepreneurs exert less effort than desired by the shareholders of firms. The drop of entrepreneurial efforts also causes an increase in default risk. 5.3 Theoretical predictions The theoretical model has shown that the existence of entrusted loan is related to the competitive environment of the banking sector. Entrusted loans do not exist when the bank has monopoly power (described in section 6.1). When the banking sector is perfectly competitive, entrusted loans arise endogenously. Therefore, we should observe the banks are more competitive along with soaring of entrusted loans. 28

30 Essentially, the cause of entrusted loan is the heterogeneity in the interest rates of bank loan each firm has. Firms that can obtain cheap bank loan re-lend the loan to firms that could only borrow with the high rate from the bank. So I postulate the first hypothesis: Hypothesis 1. The bank lending rate is a decreasing function of firm capitalization, or an increasing function of loan volume. This hypothesis is based on Lemma 6. The theory shows that the high-capitalized firm has lower monitoring cost, the entrepreneur also pays more effort on the investment project. So the bank charges lower lending rate. The next two hypotheses are related to the welfare analysis Hypothesis 2. Foirms with the same capitalization, the bank lending rates are higher to firms that engage in shadow banking activities, than firms that do not. Hypothesis 3. Foirms with the same capitalization, the default risk(probability of success) is higher to firms that engage in shadow banking activities, than firms that do not. Hypothesis 2 could test whether the entrusted loans from the lending firms are naturally the over-borrowed bank loans. Fixing the firm capitalization, the lenders of entrusted loans have higher bank loan demands, according to hypothesis 1, the lending firms have to pay a higher interest rate to the bank. Hypothesis 3 reflects the negative impact of the entrusted loans to the involved firms. As an alternative investment project, entrusted loans offer a higher payoff, the entrepreneurs thus reduce the efforts on firms investment projects, resulting in a high default risk. 6 Extensions Three extensions are considered in this section. In the first extension, I assume the banking sector is non-competitive. In the second extension, I include restricted industries which are prohibited from bank lending. In the third extension, I consider a special case that shadow 29

31 banks can better monitor borrowing firms than banks if the lending firms and borrowing firms are affiliated. 6.1 Non-competitive banking sector In this section, it is assumed that the banking sector is non-competitive. The bank has monopoly power. I show that under this setting, entrusted loans do not exist. The reason is that with monopoly power, the bank obtains positive surplus. A high surplus induces the bank to provide a high level of monitoring, so bank moral hazard is alleviated. Note that entrusted loans result from bank moral hazard when the bank moral hazard is not severe, entrusted loans do not exist. To prove this, I first solve the optimal loan contract with a non-competitive banking sector. The optimization problem is given as p R B max (p p 0) 2 L B (6.1) L B,R B,p 2m subject to p (Q R B ) p 2 2β + r s(l B + w 1) 0 (6.2) L B + w 1 0 (6.3) 0 R B Q (6.4) When the bank has monopoly power, the loan contract {L B, R B } are set to maximize the bank s profit. Three constraints have to be satisfied. 6.2 establishes that each firm gets positive profit. 6.3 ensures that the firm gets enough financing for the investment project. Last, the required repayment to the bank cannot exceed the total return from the project. Note that the rate of return of retained capital L B + w 1 is r s, the following result will show that even if the firm has a profitable option (investing in shadow banking), the firm 30

32 chooses not to invest in this option. Proposition 3. Under non-competitive banking sector, only firms with initial capital w w N can obtain bank financing, the firms with w < w N are not financed. The optimal loan size, the bank repayment, and monitoring effort are given as L B = 1 w, (6.5) R B = p = βq 2β m Rnc B, (6.6) β β m p 0 = β 2 Q (2β m) p nc. (6.7) Each firm only borrows the amount of capital that is enough to undertake the investment project, thus, shadow banking does not exist. Each firm only borrows the minimum amount of capital 1 w from the bank. It implies that shadow banking does not exist in a non-competitive banking environment. Further, I compare the bank lending rates and total entrepreneur efforts (probability of success) under a non-competitive banking environment to the benchmark. The comparisons are given as follows. Lemma 6. The lending rates are R nc B L nc B R OB B L OB B = = βq (2β m)(1 w), βq β 2 Q 2 2(2β m)(1 w)rf 2, (2β m)(1 w) and R nc B L nc B ROB B. (6.8) L OB B 31

33 The entrepreneur total efforts are p nc = βq p OB = βq β 2β m, β 2β m + β m 2β m 1 2(2β m)(1 w)r2 f β 2 Q 2, and p OB p nc. (6.9) Equations (6.8) and (6.9) show that under non-competitive banking environment, the bank charges higher lending rate and the entrepreneur exerts less effort on the project, compared to those in a competitive banking environment. The comparisons are demonstrated in figure 5. The solid red line illustrates the non-competitive banking sector; the green line illustrates the competitive banking sector. Figure 5: The comparison between competitive and non-competitive banking environment (a) The bank lending rates (b) The total entrepreneur efforts The bank lending rate under a non-competitive banking environment is higher than that in a competitive environment. The lending rate increases with firm capitalization w. The bank has monopoly power and takes the most surplus from the lending. Equation (6.6) 32

34 shows that bank requires the same amount of repayment from each firm, and the repayment is more than half of the investment return. Since the bank s utility is negatively correlated to the loan size L B, the bank only lends the minimum amount of capital to each firm. As a result, high-capitalized firms borrow less but pay the same amount to the bank than other firms. Thus the interest rates are higheor the high-capitalized firms. Moreover, no firm has the incentive to over-borrow and forms shadow bank. Because replacing bank loan with shadow bank loan does not bring any benefit to the borrower. Since the bank always charges the same amount of repayment and provides the same amount of monitoring, the firm s profit does not improve by reducing bank loan. Therefore, shadow banking does not exist. The total entrepreneur effort is smaller under non-competitive environment and is independent of firm capitalization w. That is because the bank charges the same amount of repayment from each firm, which results in equal entrepreneur non-monitoring effort and bank monitoring. When the bank has monopoly power, the bank takes the most surplus by charging high lending rate. Worsening the entrepreneur s incentive to put effort into the project. 6.2 Restricted Industries Since 2005, the China Banking Regulatory Commission has put a more severe restriction on bank lending to certain industries, including real-estate, core mining, shipbuilders and local government financing platform, etc. The limited access to the bank loan, in turn, creates demand for entrusted loans. Unlike the formal banks, shadow banks are not under regulatory supervision. Firms that can issue entrusted loan fills the gap by lending to restricted industries. In this section, the model is extended by taking the restricted industries into account. More specifically, it is assumed that each firm is prohibited from bank loan with probability q [0, 0.5). q can be interpreted as the proportion of the restricted industries among all the 33

35 industries. Note that the firms in restricted industries are different from the credit rationed firms. Credit rationing in Lemma 1 is an endogenous decision of banks that not lending to firms with low capitalization. But the restricted firm is caused by an exogenous policy rule imposed on the bank, and it is irrelevant from capitalization. The equilibrium derived in section 3 and 4 is a special case when q = 0. When q > 0, all firms are divided into two types: restricted firms and unrestricted firms. Unrestricted firms behave same as described in section 3 and 4: high-capitalized firms over-borrow from banks, otheirms optimally choose to borrow from banks and shadow banks. The restricted firms can only borrow from shadow banks. In equilibrium: the lenders of the entrusted loan are high-capitalized unrestricted firms, and the borrowers are medium-capitalized unrestricted firms and high-capitalized restricted firms. I start from deriving the shadow loan contract to restricted firms. The bank loan and shadow loan contracts to unrestricted firms follow the results from section 4. The equilibrium expected rate of return r s from shadow banking is derived by market clearing. For the restricted firms, entrusted loans are the only external financing source. Therefore, the shadow bank has monopoly power. The shadow loan contract, including loan size L S and return R S, are set to maximize the profit of shadow bank. Since shadow bank does not monitor the firms, the optimal level of entrepreneur effort equals to the non-monitoring effort level, that is, p = p 0 = β(q R S). (6.10) For given effort level p, shadow bank chooses the loan size L S and payoff R S to maximize the expected profit: max L S,R S p R S L S. (6.11) 34

36 subject to p (Q R S ) p 2 2β + L S + w 1 0 (6.12) L S + w 1, (6.13) p R S r s L S. (6.14) Equation (6.12) and (6.13) are the participation constraints of the firm. Equation (6.12) indicates that the firm gets non negative profit, and equation (6.13) ensures that the firm obtains enough financing to undertake the project. The last constraint (6.14) is the participation constraint of the shadow bank, which requires that the shadow bank obtains rate of return at least r s. The optimal loan size is solved as L S = 1 w, (6.15) and the optimal return as R S = Q 2. (6.16) With monopoly power, the shadow bank sets a higher lending rate, so each firm only borrows the minimum amount 1 w which is just enough to undertake the investment project. Furthermore, the participation constraint of the shadow bank requires that w 1 βq2 4r s r 2 f w S. (6.17) Hence, only firms with an initial capital w w S can get entrusted loans. Note that the threshold w S > w N, which implies that the credit rationing with shadow bank is more 35

37 severe than the one with formal banks. That mainly results from the monitoring role banks play. Lemma 7. To firms from the restricted industries, shadow bank is the only outside financing source. Firms that have initial capital w [w S, 1] borrow from the shadow bank. The optimal loan size is L S = 1 w, and the required return to shadow bank is R S = Q. Firms that have 2 initial capital w < w S cannot get financing from shadow bank. The lending to restricted firms is part of the entrusted loans demand. Another part comes from the unrestricted firms, which is derived in section 4.3. So the total demand of entrusted loans is w2 1 T D = (1 q) L M S g(w)dw + q (1 w)g(w)dw. w 1 w S The supply of shadow loan comes from the unrestricted firms, which is 1 T S = (1 q) (L S B + w 1)g(w)dw. 1 L S B The expected rate of return of shadow banking, denoted as r s is derived by equalizing the total demand to total supply. Shadow banking exists if and only if there exists a solution r s (1, + ) which solves T D = T S. Based on Lemma 4, one has the following conclusion. Lemma 8. If there is a proposition q (0, 0.5) of firms being in restricted industries, the equilibrium expected rate of return r s is an increasing function of q, so one gets r s > rs. Shadow banking sector grows when q increases. When q > q( r s ( q) = r s ), the unrestricted firms are crowded out by the restricted firms, so only restricted firms get shadow loan. The inclusion of restricted industries increases the demand for entrusted loans and also reduces the total supply because the total amount of unrestricted firms decreases, further driving up the interest rates on entrusted loans. Therefore, the expected rate of return r s increases in q. Note that the lenders of entrusted loans are unrestricted firms which have 36

38 initial capital w (1 L s B, 1]. When q increases, r s and L S B also increase since they are both increasing function of q. So the range of lenders, which is (1 L s B, 1] expands, meaning that the shadow banking sector expands with q. However, the demand for entrusted loans from the unrestricted firms shrinks when r s increases. When r s is above r s, no firm from unrestricted industries takes entrusted loans, so the demand side is only composed of the restricted firms. q solves the equation r s ( q) = r s, so when q > q, one has r s > r s, and only firms from restricted industries borrow from shadow bank. Figure 10 plots the equilibrium lending with restricted industries. 6.3 Shadow banks also monitoirms In this section, I consider the case where shadow banks can monitoirms. More specifically, the lending firms can even monitor the borrowing firms better than banks. This could happen when lending firms and borrowing firms have a certain relationship, for example, suppliers and customers, parent firm and subsidiary firm, oirms belonging to the same business group. Empirical findings from Allen, Qian, Tu, and Yu (2017) and He, Lu, and Ongena (2016) show that around 75% entrusted loans are made between affiliated firms. 7 Compared with banks, lending firms may have an informational advantage or have efficiently monitor and enforce repayment when law enforcement is difficult (see Degryse, Lu, and Ongena (2016)). Moreover, two firms may have mutual interest, especially when one firm is controlling shareholder of another one. All could help the lending firms monitor the borrowing firms more efficiently. Suppose shadow bank has the same monitoring technology as the bank. That is, 7 In Allen, Qian, Tu, and Yu (2017), there are 800 affiliated entrusted loans vs. 289 nonaffiliated entrusted loans in their sample. In He, Lu, and Ongena (2016), there are 536 affiliated entrusted loans vs. 183 nonaffiliated loans in their sample. 37

39 shadow bank also incurs a convex monitoring cost, given as (p p 0 p b ) 2 /2m s, if p p 0 + p b, M s (p p 0 p b ) = 0, if p < p 0 + p b, where p b denotes the bank monitoring level. Similar to the bank, shadow bank monitors the entrepreneur so that entrepreneur exerts more efforts. The marginal monitoring cost of shadow bank is 1 m s. Since shadow bank can better monitor the firm, one requires m s > m. Moreover, similar to assumption 3, one also needs m s < β 2 to ensure that the second order condition of entrepreneur maximization problem is well defined. derived as Shadow bank chooses the monitoring intensity to maximize its own profit, which is p p 0 p b = m sr S. The entrepreneur non-monitoring level p 0 and bank monitoring level p b are the same as shown by equations (3.1) and (3.2). So the total entrepreneur effort is given as p = βq + (m β)r B + (m s β)r S. (6.18) Next, I solve the equilibrium bank loan contract {L B, R B }and shadow bank loan contract {L S, R S }. The shadow bank monitoring only affects the borrowing firms. The lending firms have the same borrowing strategy, that is, firms with capital above 1 L S B over-borrow from banks, and each firm borrows L S B and repays RS B to the bank. For otheirms, entrepreneurs optimally choose from both bank loans and shadow bank loans, the optimization problem is given as max U = p (Q R B R S ) p 2 L B,R B,L S,R S 2β + L B + L S + w 1 (6.19) 38

40 subject to p R B mr2 B 2r 2 f = L B, (6.20) p R S m SR 2 B 2r 2 f = r s L S, (6.21) L B + L S + w 1 0, (6.22) R B + R S Q. (6.23) Solving this optimization problem gives the shadow loan demand L S, which is a function of the expected rate of return r s. Given the lenders firms optimal bank loan size L S B, one can derive the total shadow loan supply. By equalizing the total supply with total demand, one can derive the equilibrium value of r s, and further get the equilibrium bank and shadow bank contracts. The equilibrium can be characterized as Lemma 9. If shadow banks also monitor the firms and shadow banks can better monitor the firms than banks, the equilibrium has the following features. 1. Credit rationing can be alleviated, that is, firms with initial capital w < w N can also get external financing. 2. The borrowers firms only take the minimum amount of external financing (i.e., L B + L S = 1 w). 3. The borrowers firms can get higher profit π and higher success probability p than the benchmark, the lenders firms get lower profit π and lower success probability p than the benchmark. 39

41 7 Conclusion A model is built to characterize the endogenous emergence of an important type of shadow banking in China: entrusted loans. The model shows that with entrepreneurial moral hazard and costly bank monitoring, entrusted loans arise when the banking sector is competitive. Lenders of entrusted loans are firms which have high capitalization. Borrowers of entrusted loans are medium-capitalized firms. Entrusted loans involve a lending chain in which high-capitalized firms over-borrow from the bank, then re-lend to medium-capitalized firms through shadow banking. Medium-capitalized firms mix the outside financing with both bank loans and entrusted loans. Entrusted loans improve entrepreneurs welfare but reduce firms profits. Moreover, firms default risk is increased and real efficiency reduced. The reduction is correlated with the involvement of shadow banking activity, and especially significant for lending firms. This result is consistent with the empirical observation that abnormal stock returns of lendeirms are negative after the entrusted loans are publicly announced. In the extension, I consider a particular case when shadow banks can better monitor the borrowing firms than banks when the lending firms and borrowing firms are affiliated, so the lending firm has better knowledge or stronger controlling of borrowing firm. In this case, entrusted loans can alleviate credit rationing. Low-capitalized firms that would not be financed without entrusted loans would be financed when entrusted loans are available. Entrusted loans also improve profits of borrowing firms and reduce their default risk, because of a higher level of monitoring from lending firms. Although the impact on lending firms is still negative: lending firms earn a lower expected profit, the total social welfare might be improved. It is more interesting to derive a general result with different bank and shadow bank monitoring. I have assumed that all firms have positive NPV investment projects and lack of 40

42 capital to undertake the projects. Entrusted loans are essentially bank loans that pass through a lending chain. It is useful to consider another case that entrusted loans are from retained earnings of firms that have run out of good investment projects. In this case, entrusted loans can be beneficial for lending firms, because of efficiently capital allocation. However, the impact to borrowing firms is unclear, and it is worth to have a study. 41

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46 Appendices A List of Variables Variable w g(w) w N Description Firm capitalization The probability density function of firm capitalization w Threshold of firm capitalization under which firms are credit rationed (w 1, w 2 ) The range of firms that borrow from both banks and shadow banks p 0 p 1 β 1 m Q r s r s L B R B L S R S L S B R S B L M B R M B L M S R M S L OB B R OB B T S T D Non-monitoring effort of entrepreneur Total effort of entrepreneur/ Probability of success of investment project Marginal cost of entrepreneur effort Marginal cost of bank monitoring Return of the investment project if succeed Risk free rate Expected return of shadow banking Upper bound of r s Firm s demand of bank loan Return of bank loan Firm s demand of shadow bank loan Return of shadow bank loan Equilibrium bank loan to the firms which become lenders of shadow bank Equilibrium bank return from the firms which become lenders of shadow banking Equilibrium bank loan to the firms which borrow from both bank and shadow bank Equilibrium bank return from the firms which borrow from both bank and shadow bank Equilibrium shadow banking loan to the firms which borrow from both bank and shadow bank Equilibrium shadow banking return from the firms which borrow from both bank and shadow bank Equilibrium bank loan to the firms which only borrow from the bank Equilibrium bank return from the firms which only borrow from the bank Total supply of shadow bank Total demand of shadow bank 45

47 B Proof of Lemma 1, 2, 3 and Proposition 1 The optimization problem of (3.3) is a special case of (4.1) when r s = 1, so the proofs are put together. Results of Lemma 1 and 2 are derived in the analysis for different values of r s. Replacing L B with the bank s zero-profit condition (4.2), and substituting the optimal value of p 0 and p with equations (3.1) and (3.2), respectively, the optimization problem (4.1) of the firm is given as max R B (mr s 2βr s + β m2 β )R2 B + 2β(r s 1)QR B + βq 2 2r 2 f r s (1 w) subject to (m 2β)RB 2 + 2βQR B 2rf 2 1 w, Q R B 0. The Lagrangian is L = (mr s 2βr s + β m2 β )R2 B + 2β(r s 1)QR B + βq 2 +λ 2r 2 f [ (m 2β)R 2 B + 2βQR B 2r 2 f 1 + w ], r s (1 w) where λ is the Lagrange multiplier associated with (4.3). Differentiating the Lagrangian with respect to R B gives L R B = 2(mr s 2βr s + β m2 β )R B + 2β(r s 1)Q + λ(2(m 2β)R B + 2βQ) = 0. (B.1) First, if we assume constraint (4.3) binds, that is, each firm only borrows the minimum amount that is just enough to cover the investment cost. Equation (4.2) and (4.3) give that 46

48 L B = (m 2β)R2 B +2βQR B 2r 2 f = 1 w. We can solve the optimal bank repayment as R B = βq β 2 Q 2 2(2β m)(1 w)r 2 f 2β m R OB B, which requires that β 2 Q 2 w 1 2(2β m)rf 2 w N. So if the firm has very low capitalization, the agency cost will be too high for bank to lend to the firm, hence, firm with initial capital w < w N are credit rationed by the bank, as shown in Lemma 1. The assumption that constraint (4.3) binds requires that λ 0. Substituting RB into equation (B.1), yields λ 0 implies that λ = βq(r s 1) + ( m2 β β + 2βr s mr s ) βq β 2 Q 2 2(2β m)(1 w)r 2 f 2β m βq + (m 2β) βq β 2 Q 2 2(2β m)(1 w)r 2 f 2β m. r s 1 + m 2 β + β m 2β m βq β 2 Q 2 2(2β m)(1 w)rf 2 1 r s. Note that r s > 1, so the benchmark applies since r s = 1, and the equilibrium results are given as in Lemma 2. On the other hand, if r s > r s, then λ < 0, which implies that constraint (4.3) does not bind. That is, the firm decides to over-borrow from the bank. If constraint (4.3) does not bind, then λ = 0, from equation (B.1) we can derive that R B = βq(r s 1) m 2 β + 2βr RB, S β s mr s 47

49 which is smaller than Q when r s > r s. And optimal bank loan size is derived from (4.2) as L B = (m 2β)R B + 2βQR B 2r 2 f = β2 Q 2 2r 2 f That are the results of Lemma 3 and Proposition 1. (r s 1)((2β m)r s + 2m2 m) β ( m2 β + 2βr L S B. β s mr s ) 2 C Proof of proposition 2 To solve the optimization problem (4.6), note that L B and L S can be replaced by equations (4.7) and (4.8), respectively. In addition, substituting p = p 0 + mr B, the optimization problem can be simplified as max R B,R S β 2r 2 f [ ( Q R S + ( m ) ( β 1)R B Q R S + 2 R S ( m ) ] r s β 1)R B m β R2 B + w 1, (C.1) subject to R B (Q R S + ( m 2β 1)R B ) + R S r s ( Q R S + ( m ) β 1)R B r2 f (1 w), (C.2) β R B + R S Q, (C.3) R B 0, (C.4) R S 0. (C.5) 48

50 Let λ, µ be the multiplier of constraint (C.2) and (C.5), respectively, one gets the Lagrangian problem as L = β 2r 2 f +λ [ ( Q R S + ( m β 1)R B ( [R B Q R S + ( m 2β 1)R B +µr S. ) ( Q R S + 2 R S ( m ) ] r s β 1)R B m β R2 B + w 1 ) + R S r s ( Q R S + ( m ) β 1)R B r2 f (1 w) β ] Taking derivative with respect to R B and R S respectively, yields L = ( m [ 2 R B β 1) R S 2( m ] [ r s β 1)R B 2m β R B+λ Q + ( m m β 2)R β B + ( 1 ] 1)R S = 0 r s [ (C.6) L = 2( 1 1)Q 2( 2 1)R S + 2 ( m R S r s r s r s β 1)R Q B +λ 2R m S + ( 1 ] β 1)R B +µ = 0. r s r s r s (C.7) First, assuming constraint (C.5) is binding, that is, R S = 0. And, one should have µ > 0. Furthermore, assuming constraint (C.2) is also binding. The optimal value of R B can be derived from (C.2) as R B = R OB B = βq β 2 Q 2 2(2β m)(1 w)r 2 f 2β m The binding assumption of constraint (C.2) requires that λ > 0, which can be proved by replacing R B into (C.6): λ = 2( m2 m + β 2 β 1)R B Q 2 4(1 m ) r2 f (1 w) 2β β > 0. In addition, µ is derived from (C.7), µ = 2(1 1 )Q + 2 (1 m r s r s β )R B 2 (Q + ( m ) ( m r s β 1 r s)r B β 1)2 RB + m β R B Q + ( m. (C.8) β 2)R B 49

51 µ > 0 is equivalent to [ (r s 1)Q + (1 m ] [ β )R B Q + ( m ] [ β 2)R B Q + ( m ] ( ) β 1 r s)rb m 2 β m 2 β + 1 RB > 0, which is a quadratic function of R B. Note that if r s 1 3 (1 + mβ m β + m2 β 2 ), the determinant of the quadratic equation is negative, so the inequality always holds. When r s < (1 1 + mβ ), m + m2 the inequality reduces to either 3 β β 2 or RB > (2β2 2βm)(r s 1) + m 2 + m β 2 (1 + 2r s 3rs) 2 + 2βm(r s 1) + m 2 Q (2β 2 2βm)(r s 1) + 2m 2 (r s + 1) 2 m3 β RB < (2β2 2βm)(r s 1) + m 2 m β 2 (1 + 2r s 3rs) 2 + 2βm(r s 1) + m 2 Q. (2β 2 2βm)(r s 1) + 2m 2 (r s + 1) 2 m3 β That is, w < βq 1 2(2 m β )r2 f w 1, 1 (r s 1)(m 2 + 2mβ 2β 2 ) + m(2 m β )(m β 2 (1 + 2r s 3r 2 s) + 2mβ(r s 1) + m 2 ) 2(r s 1)(m 2 mβ + β 2 ) + 2m 2 (2 m β ) 2 or w > βq 1 2(2 m β )r2 f w 2. 1 (r s 1)(m 2 + 2mβ 2β 2 ) + m(2 m β )(m + β 2 (1 + 2r s 3r 2 s) + 2mβ(r s 1) + m 2 ) 2(r s 1)(m 2 mβ + β 2 ) + 2m 2 (2 m β ) 2 Instead, if w 1 w w 2, µ > 0 is violated, so constraint (C.5) is not binding, which means that R S > 0 and µ = 0. Then the optimal value of R B, R S and value of λ are solved from 50

52 equations (C.2), (C.6) and (C.7): ) ) R B (Q R S + ( m 2β 1)R B + R S rs (Q R S + ( m 1)R β B = r2 f (1 w) β [ ] [ ] L R B = ( m 1) 2 β r s R S 2( m 1)R β B 2mR β B + λ Q + ( m 2)R β B + ( m β 1 r s 1)R S = 0 [ ] = 2( 1 r s 1)Q 2( 2 r s 1)R S + 2 r s ( m 1)R Q β B + λ r s 2R S r s + ( m β 1 r s 1)R B = 0. L R S which is equivalent to equation system (C.9) λ > 0 should always hold, since the constraint (C.2) is always binding. If not, one should have λ = 0. Replacing λ = 0, equation (C.6) reduces to ( m β 1)R S = r s [ m β + ( m β 1)2 ] R B, but the left hand side is non positive, since m < β, and the right hand side is non negative, 2 which is contradiction. Therefore, constraint (C.2) is binding. D Proof of Lemma 4 As have been proved that the total supple function is monotonically increasing with respect to r s and total demand function is monotonically decreasing w.r.t r s. The two functions are illustrated in figure (6). When r s = 1, total demand T D > 0, and when r s = r s, T D = 0. When r s = 1, total supply T S = 0 and T S increases in r s. Now, I prove that the intersection of the two curves is in between 1 and r s. Suppose rs r s, then one should have T D(rs) < T D( r s ) = 0 and T S(rs) > T S( r s ) > 0, but one function is negative and anotheunction is positive, there is no way to have intersection of the two curves. Thus, rs < r s. 51

53 Figure 6: Plots of total demand and total supple functions E Proof of proposition 3 Let λ and µ be the multipliers for constraint (6.2) and (6.3), respectively. The Kuhn-Tucker necessary conditions are given as R B : p β(p p 0) + λ( p ) = 0, (E.1) m L B : 1 + λr s + µ = 0, (E.2) p : R B p p 0 m + λ(q R B p ) β [ ] = 0, (E.3) λ p(q R B ) p2 2β + r s(l B + w 1) = 0, (E.4) λ 0, (E.5) µ [L B + w 1] = 0, (E.6) µ 0 (E.7) 52

54 The first step is to prove that constraint (6.3) is binding. If not, then µ = 0, and λ = 1 r s from (E.2). Replacing λ in equations (E.1) and (E.3) and solving for p and R B yields p = R B = β β + m( 1 r s 1) p 0, (1 1 r s )(β + m r s ) β(2 1 r s ) m(1 1 r s ) 2 Q. As λ > 0, the constraint (6.2) must bind, which derives the optimal loan volume: L B = p 2 p(q R B) 2β + 1 w. (E.8) r s Since L B > 1 w, one must have p > Q R B 2β, that is, β m < 2(1 1 r s ) < 2. But we require that m < β 2, contradiction. Therefore, constraint (6.3) is binding and L B = 1 w. The next step is to calculate the optimal loan return R B and monitoring effort p. Assuming constraint (6.2) is not binding, then λ = 0. Furthermore, R B and p are solved from (E.1) and (E.3) as R B = p = β 2β m Q, β β m p 0 = (E.9) β 2 (2β m) Q. (E.10) Moreover, constraint (6.3) is verified to be no binding by replacing R B and p into (6.3). At last, substituting the optimal value of L B, R B and p into equation (6.1), non negative profit of the bank requires that w 1 βq2 2(2 m β )r2 f = w N. 53

55 F Proof of Lemma 5 I first prove that the lendeirm s profit is lower than that of benchmark, that is, π S F < πob F. The lendeirm s profit π S F is a function of r s, and r s [ r s (w), r s ). One can get some properties of π S F : π S F ( r s ) = π OB F, πf S (m β) (r s = r s ) = r s [ ] [ ] β 2 Q 2 2(2β m)(1 w)rf 2 β 2 Q + (β m) β 2 Q 2 2(2β m)(1 w)rf 2 (2β m) ( β 2 + m 2 βm ) βqr 2 f πf S lim > 0, r s + r s 2 πf S rs 2 = β3 Q 2 ( β 2 + m 2 βm ) ( 4m 4 βm 3 (r s + 12) + β 2 m 2 (5r s + 14) 3β 3 m(3r s + 2) + 6β 4 ) r s ( m 2 βmr s + β 2 (2r s 1) ) 4 > 0 r 2 f < 0, So π S F is a convex function on r s [ r s, + ). Furthermore, one can compute the value of derivative of πs F r s at r s and get πs F r s (r s = r s ) < 0. So π S F r s [ r s, r s ], then one should have π S F πs F (r s = r s ) = π OB F. that is, π M F of π M F is a decreasing function of r s on Next, I prove that the borroweirm s profit is also lower than that of benchmark, < πob F < πob F. Note that the profit of benchmark πob is equivalent to the proof that πm F R S πf M = p(q R B R S ), taking the derivative w.r.t R S gives πm F R S F = π M F (RS = 0), so the proof < 0. As the profit of borroweirm is = p + Q R B R S p R S. In the equilibrium, one has frac p R S < 0, since shadow bank loan replaces bank loan, but bank provides monitoring which increases p but shadow bank does not monitoring, so the increase of shadow bank loan reduces bank monitoring and results in lower success probability p. Overall, πm F R S < 0, so π OB F = π M F (RS = 0) > πm F. 54

56 G Proof of Lemma 7 Replacing the optimal entrepreneur effort p, one gets the Lagrangian problem as L = β(q R S)R S r 2 f L S + λ [ β(q R S ) 2 2r 2 f + L S + w 1 ] + µ(l S + w 1), where λ is the multiplier of constraint (6.12) and µ is the multiplier of constraint (6.13). Taking derivative with respect to L S and R S respectively, yields L L S = 1 + λ + µ, (G.1) L = β(q 2R S) R S rf 2 λ β(q R S). (G.2) rf 2 Assuming constraint (6.13) binds, so L S = 1 w and one needs µ > 0. Further, assuming (6.12) also binds, then one should have R S = Q, which contradicts with the requirement that R S < Q. So (6.12) does not bind, then one gets λ = 0. Equation (G.2) gives the optimal shadow bank return RS = Q. Substituting λ = 0 into equation (G.1), one gets µ = 1 > 0, 2 which confirms the assumption that constraint (6.13) indeed binds. At last, replacing the optimal values of L S and R S into shadow bank constraint (6.14), one gets w ws. H Proof of Lemma 8 First proving that the equilibrium expected rate of return r s is an increasing function of q. Taking q 2 > q 1 > 0, suppose r s is not increasing function of q, then one should have either r s (q 2 ) > r s (q 1 ), or r s (q 2 ) = r s (q 1 ). In the first case, assuming r s (q 2 ) > r s (q 1 ). Since T D is a decreasing function of r s, one gets T D( r s (q 2 )) > T D( r s (q 1 )). One the other hand, T S is an increasing function of r s, so one gets T S( r s (q 2 )) < T S( r s (q 1 )). By the definition of r s, one should have T D( r s (q 1 )) = T S( r s (q 1 )), and T D( r s (q 2 )) = T S( r s (q 2 )). But we have 55

57 T S( r s (q 2 )) < T S( r s (q 1 )) = T D( r s (q 1 )) < T D( r s (q 2 )), which is a contradiction. In another case, assuming r s (q 2 ) = r s (q 1 ), one should have T S( r s (q 2 )) = T S( r s (q 1 )). But T S is an decreasing function of q, so T S(q 2 ) < T S(q 1 ), which is again a contradiction. Therefore, r s is an increasing function of q. So for any q > 0, r s(q) > r s(q = 0) = rs. When q = 0, there exists an upper bound r s such that for any r s r s, the total demand from unrestricted industries shrinks to zero, that is T D(r s ) = 0. But when q > 0, the total demand expands by the restricted industries. When r s r s, the demand from the unrestricted industries shrinks to zero, but the demand from the restricted industries is still positive. Hence, when q is large enough, the equilibrium r s could go above r s. In this case, only restricted industries require shadow bank loan, which is plotted in figure 7. Figure 7: Total demand and total supply function with restricted industries 56

58 I Proof of Lemma 9 First, when shadow bank does not monitor the borrowers firms, firms with low capitalization w [w N, w 1 ) only get bank financing. The bank loan contract is determined by the bank zero-profit constraint, that is p(q R B ) mr2 B 2r 2 f = (m 2β)R2 B + 2βQR B 2r 2 f = 1 w. Solving the quadratic function of R B gives the optimal bank repayment RB. But when w < w N, the determinant of the quadratic function is negative, which means there is no solution. Intuitively, it is because when firm has low net wealth, the agency cost is very high, so bank has to charge very high repayment, but then the entrepreneur will incur negative utility, which is not optimal. When bank can better monitor the borrowers firms, the entrepreneur still takes the minimum amount of loan, that is L B + L S = 1 w. To prove that credit rationing could be alleviated, one can prove that there exists positive solution R B and R S which solves the constraint L B + L S = 1 w. Replacing L B with equation (6.20) and L S with equation (6.21), substituting p, one can rewrite the constraint as a function of R B : [ (m 2β)RB+2 2 βq + (m s β)r S + m β R S r s The determinant is computed as ] R B + 2 r s R S (βq+(m s β)r S ) m2 sr 2 S r s 2r 2 f(1 w) = 0. = β 2 Q 2 2(2β m)rf 2 (1 w)+2βqr S ( β [ β+m s )+RS 2 (m s β + m β ) 2 + 2β m ] (m 2 s + 2m s 2β). r s r s r s Note that when w = w N, sum of the first two components results in zero. The sum of last two components is an increasing function of m s, and when m s β, 2βQR 2 S( β r s β ) + 2 [ ] RS 2 ( β + m β 2 r s ) 2 + 2β m r s ( β 2 2 β) > 0. So for large m s, the determinant is positive, 57

59 which means that there exists solution R B and R S which solves the constraint equation. In other words, the bank and shadow bank can extend credit to medium-capitalized firms. Second, the borrowers firm only takes the minimum amount of financing, that is L B + L S = 1 w. The entrepreneur optimization problem can be simplified as max R B,R S β 2r 2 f subject to [ ( Q + ( m s β 1)R S + ( m ) ( β 1)R B Q ( m s β + 1)R S + 2 R S ( m ) ] r s β 1)R B m β R2 B m srs 2 +w 1, βr s (I.1) ( R B Q + ( m s β 1)R S + ( m ) 2β 1)R B + R ( S Q + ( m s r s 2β 1)R S + ( m ) β 1)R B r2 f (1 w), (I.2) β R B + R S Q, (I.3) R B 0, (I.4) R S 0. (I.5) Let λ, µ be the multiplier of constraint (C.2) and (I.5), respectively, one gets the Lagrangian problem as L = β 2r 2 f +λ [ ( Q + ( m s β 1)R S + ( m β 1)R B ( [R B Q + ( m s β 1)R S + ( m 2β 1)R B ) ( Q ( m s ) + R S r s Taking derivative with respect to R B and R S respectively, yields L = ( m [ R B β 1) ( 2 2m s r s β )R S 2( m β 1)R B β + 1)R S + 2 R S ( m ) ] r s β 1)R B m β R2 B m srs 2 + w 1 βr s ( Q + ( m s 2β 1)R S + ( m ) ] β 1)R B r2 f (1 w). β ] [ 2m β R B+λ Q + ( m m β 2)R β B + ( 1 ] + m s r s β 1)R S = 0 (I.6) [ L = 2( 1 1)Q+2( m s R S r s r s β m2 s β 2 2 1)R S +( 2 m s r s r s β )(m β 1)R Q B+λ + ( m m s r s β 1)2R S β + ( 1 + m s r s r s β 1)R B (I.7) Assuming the constraint (I.2) does not bind, then λ = 0. Substituting λ = 0 into equations (I.6) and (I.7) and solve R S as ] +µ = 0. R S = β ( β 2 + m 2 βm ) Q(r s 1)r s β 2 (m((r s 2)r s + 2) + m s r s ) + βm ( m(r s 1) 2 + 3m s r s ) mms r s (m + m s r s ) + β 3 (r s 1) 2 < 0, 58

60 which violates the constraint that R S > 0. So constraint (I.2) binds, that is, L B +L S = 1 w. In terms of firm profit π F and success probability p. For borrowers firms, the success probability is increased by taking shadow bank loan. Since p = βq+(m β)r B+(m s β)r S, taking derivative of R S, yields p R S = ms β + p R B R B R S = ms β + β m = ms m > 0. Foirm profit π, since entrepreneur utility U = π p2, the utility should increase when entrepreneur takes 2β shadow bank loan. On the right hand side, the increase of p causes reduction of dis-utility p2. To have an equality, π must be increased. That is, the borrowers firms get higher 2β profits. For lenders firms, the success probability p is reduced, it s the same reasoning as the main model part. over-borrowing from the bank increases the debt level of firms, which reduces the entrepreneur non-monitoring effort and in the end reduces the success probability. 59

61 Figure 8: The lending of bank and shadow banking 60

62 Figure 9: The bank lending with and without shadow banking 61

63 (a) when q = 0.03 (b) when q = 0.1 Figure 10: The bank and shadow bank lending with restricted industries 62