A Simple DSGE Model of Banking Industry Dynamics

Transcription

1 A Simple DSGE Model of Banking Industry Dynamics Akio Ino University of Wisconsin - Madison December 12, 217 Abstract In this paper I introduce imperfect competition and entry and exit in the banking sector into an otherwise standard DSGE model. The model generates an amplification mechanism through bank entry and exit. In particular, if there is a positive productivity shock, borrowing to finance investment increases leading to higher bank profits and an increase in entry. Increased competition leads to a reduction in markups and encourages further investment. Compared to the model with perfect competition, imperfect competition in the banking sector increases the standard deviation of output by 7.5%. I would like to thank Oliver Levine, Engin Atalay, and especially Dean Corbae for helpful comments and suggestions. I also thank seminar participants at the macroeconomics seminar in the University of Wisconsin-Madison. 1

2 1 Introduction As we saw in the recent financial crisis, financial factors play a big role in amplification and propagation of shocks. There are many papers, such as Bernanke, et. al. (1999), which include financial factors into dynamic stochastic general equilibrium (DSGE) models. Later, papers such as Gertler and Karadi (21), added banks to intermediate between households and entrepreneurs into DSGE models. While there has been much progress incorporating financial factors into business cycle models, some important data features of the banking sector is missing from these models. The banking industry is characterized by countercyclical markups, incomplete passthrough of costs to prices, and procyclical entry and countercyclical exit. According to Corbae and D Erasmo (213), the correlation between output and markups is.27, the correlation between output and entry and exit rate is.25 and.47, respectively. In addition, the Rosse-Panzar H statistic is 52% which implies incomplete passthrough. These characteristics are absent in most DSGE models with a competitive banking sector. In this paper I introduce imperfect competition and entry and exit in the banking sector into an otherwise standard DSGE model. The model with imperfect competition generates countercyclical markups and amplification of business cycles which is not present in the competitive case. I use the model to answer the following question. How much amplification of productivity shocks arise due to imperfect competition in the banking sector? I also examine how changes in market structure affect real variables. I need entry and exit in the banking sector to replicate the countercyclical loan markup in the data. When the positive productivity hits the economy, a higher demand for investment increases the profit of banks. Therefore, new banks enter into the market, making the market more competitive. As a result of tighter competition, loan markup decreases. This is not only consistent with the data, but also works as an amplification mechanism of productivity shock. This is not the first paper to include imperfect competition in the banking sector. Corbae and D Erasmo (213/217) consider dynamic, stochastic, industry equilibrium models with bank level heterogeneity and aggregate uncertainty. To solve those models, they need to use a variant of Krusell-Smith algorithm. Here I consider simpler symmetric frameworks with imperfect loan market (Cournot) competition that can be solved using Dynare, for ex- 2

3 ample. A benefit of the simple model is that I can incorporate much richer environemnt to capture aggregate dynamics, for example, physical capital, which does not appear in Corbae and D Erasmo (213/217). Gerali, et. al. (21) builds a DSGE model where banking sector is monopolistically compettitive and shows that the shock to the banking sector is an important source of fluctuation in the euro area. In their paper, except for the sticky loan rate, the loan markup is constant and moved by exogenous shock. In my model, the loan markup fluctuates endogenously throught the entry and exit into the banking sector. In this sense, this paper can be thought as a microfoundation of a shock to the banking sector. The framework in this paper builds on Jaimovich and Floetotto (28). In their paper, there is a finite number of firms imperfectly competing at each of a continuum of locations. With this assumption, while each firm has regional market power, it does not affect the aggregate state of the economy. I am applying this framework to the banking sector and obtain a new amplfication mechanism through imperfect competition in the banking sector. The remainder of the paper is organized as follows. Section 2 describes the environment of the model. Section 3 defines the equilibrium I am going to solve. Section 4 describes the calibration of the model. Section 5 discusses how the preference and technology parameters are chosen and section 5 provides results for the model. Section 6 concludes and lists a set of extensions to the model. 2 Environment My model is based upon the standard DSGE models as in Iacoviniello (25), augumented with imperfect competition in the banking sector. There is a unit measure of locations i, 1]. At each location there are three types of agents: Households, firms, and regional banks. There is a unit measure of households in each location who work for the final good producer, deposit in a regional bank, and consume. Final goods production technology hire labor and capital. A finite number of retail or regional banks take deposits in their region and provide funds to wholesale or national banks. National banks syndicate loans to entrepreneurs who purchase capital to be rented to regional final goods production firms. The only shock in this model is the aggregate productivity shock. There is no idiosyncratic uncertainty, and as a result, in equilibrium, all regions are symmetric. 3

4 From now on, I will describe the objectives and constraints faced by each agents. 2.1 Households Households in region i enter into period t with d t 1 units of deposit they made to regional banks. In period t, households receive interest on deposits Rt 1d d t 1 (i), wage income w t (i)l t (i), and dividends paid by retail and national banks, Dt r (i) and Dt n, respectively. Households can use these income for consumption c t (i) and deposit for next period d t. The utility maximization problem of the household in region i is given by ] max E β t c H t (i),lt(i),dt(i) H log(c H t (i)) (l t(i)) 1+φ (1) 1 + φ subject to t= c H t (i) + d t (i) w t (i)l t (i) + R d t 1(i)d t (i) + D r t (i) + D n t 1 (2) In the symmetric equilibrium, the stochastic discount factor of households is given by Λ t,t+1 β H c H t c H t+1 which will be used to discount the future income of regional banks. 2.2 Final good producer A final good producer in region i borrow capital from entrepreneurs, hire labor, and produce a homogeneous goods. The profit maximization problem of producers in region i is given by The first order conditions are max A t(kt d (i)) α (lt d (i)) 1 α r t kt d (i) w t lt d (i) (3) k t(i),l t(i) r t (i) = α y t(i) k d t (i), w t (i) = (1 α) y t(i) l d t (i), y t (i) = A t (k d t (i)) α (l d t (i)) 1 α. 4

5 2.3 Entrepreneur Entrepreneurs borrow b t from the wholesale banks and invest in physical capital k t to rent it to the final good producers and receive the rental rate r t. The entrepreneurs problem is given by max E β c E E t log(c E t ) t,kt,bt subject to t= c E t + R b t 1b t 1 + q t k t r t + q t (1 δ)]k t 1 + b t (4) and a borrowing constraint b t E t m E q t+1 k t (1 δ) ] where m E is a loan-to-value ratio. Condition (5) comes from Kiyotaki and Moore (1997). This borrowing constraint states that an entrepreneur cannot borrow more a certain fraction of the expected value of their capital. I will choose the parameters so that the constraint is binding around the steady state. Under this assumption, the first order condition of entrepreneurs is given by c E t = (1 β E )nw t, q k t k t = β E nw t + b t, R b t R b tb t = E t m E q k t+1k t (1 δ)], nw t = q k t (1 δ) + r t ]k t 1 R b t 1b t 1. where nw t is the networth of an entrepreneur. 2.4 Capital producers Capital producers use I t units of final goods to produce t= 1 κ i 2 ( I t I t 1 1 units of capital, and sell it to the entrepreneurs at the price q t. The profit maximization problem of the capital producer is then { ( E Λ,t q t 1 κ ( ) ] )} 2 i It 1 I t I t 2 I t 1 5 ) 2 ] (5) I t

6 The first order condition is 1 = q t 1 κ ( ) 2 ( i It It 1 κ i 1 2 I t 1 I t 1 ( ) ( ) ] c E 2 t It+1 It+1 + β E E t κ i 1 I t 2.5 Banks c E t+1 I t ) It In this paper, I assume that there are two types of banks, national banks and regional banks. The national banks borrow from regional banks in each region and make final loans, while the regional banks borrow from from households and make loans to national banks. There are infinitely many national banks in the model, so they will take the final loan rate and deposit rate as given. On the other hand, the loan market for regional banks is not competitive: in each region, there is only a finite number of regional banks. Each period, incumbent regional banks will exit with an exogenous probability η, and new banks can enter into the market by paying the fixed cost of entry, φ National bank National banks borrow from a finite number of regional banks and produce final loans which are supplied to entrepreneurs. They produce the final loans from regional loans using the following lending technology: I t 1 ] 1 b t = ] 1 b t (i) ω ω di where ω (, 1] is the substitutability of loans across regions and b t (i) is the amount of loans from region i: b t (i) = N t(i) j=1 b t (i, j) (6) If ω = 1, then the final loan is a simple sum of regional loans, and regional loans are perfect substitute. On the other hand, if ω, b t min i {b t (i)}, and regional loans are not substitutable. 6

7 The assumption that loans from different regions are not perfect substitute looks unrealistic. This assumption is a reduced form way to allow regional banks to have market power. This approach is also used in Gerali et al (21). This this lending production technology, national banks maximize their expected discounted present value of dividends by choosing the final loan supply b t and the loan demand for each region, {b t (i)} i : max b t,{b t(i)} i,1] E Λ,t {Rtb b t t= 1 s.t. b t = ] 1 b t (i) ω ω di 1 } ] Rt(i)b b t (i)di After solving the national banks problem, the demand for loans from region i by a national bank is given by This implies the following inverse demand function Regional banks ( ) 1 R b ω 1 b t (i) = t (i) bt (7) R t R b t(i) = R b tb 1 ω t b t (i) ω 1 (8) In each region i, there is a finite number of regional banks which borrow from households and make loans to natinal banks. They exit with probability η. Each period, regional banks compete in a Cournot fashion: taking the supply of loans from other banks as given, they choose the amount of loan to make. Even though there is a finite number of regional banks in each region, since there is a continuum of regions in this model, each regional bank are small in the aggregate economy. As a result, they take aggregate demand for loans, and aggregate interest {b t (i, j), b t, Rt} b t= as given. Therefore a regional bank j in region i solves ] W t (i, j) = s.t. max {b s(i,j)} s=t E t (1 η) s t Λ t,s (Rs 1(i) b Rs 1(i))b d s 1 (i, j) s=t R b s(i) = R b sb 1 ω s b s (i, j) + b s (i, j)] ω 1 (Inv. demand) 7

8 where Λ,t+1 denote the stochastic discount factor between period and t + 1 and W t (i, j) is the present value of the profits of incumbent banks. The supply of loans by a regional bank (b t (i, j)) doesn t change the future loan rate because each bank is small in the aggregate economy. So when the regional bank tries to choose b t (i, j), it only needs to think about the effect on the profit in period i. After the inverse demand curve is plugged in, the maximization problem above can be simplified as max b t(i,j) The first order condition is R b t b 1 ω t (b t (i, j) + b t (i, j)) ω 1 R d t (i) ] b t (i, j) (R b t(i) R d t (i)) + (ω 1) Rb t(i) b t (i) b t(i, j) = The first term represents the benefit of supplying additional unit of loans: when a regional bank increases its loan supply, then the profit will increase by R b t(i) R d t (i), if it does not change the loan rate. Here the competition in the banking sector is not perfect, so this is not the only factor which determine the loan rate. The second term represents the cost of supplying additional unit of loans: when a regional bank increases its loan supply, the loan rate drops, which will decrease the profit. Regional banks decide the optimal loan rate taking into account these two factors. Note that the supply of loan from region i is defined in (6) as b t (i) = Nt(i) j=1 b t(i, j). So in a symmetric equilibrium, each bank supplys 1/N t (i) times the total supply of loans from region i, b t (i, j) = b t (i)/n t (i). Under this situation, the first order condition becomes (R b t(i) R d t (i)) + (ω 1) Rb t(i) N t (i) = After some algebra, I obtain the loan rate in region i as a function of the number of bank in region i and the deposit rate: So the loan markup is R b t(i) = M t (i) = N t (i) N t (i) + ω 1 Rd t (i). N t (i) N t (i) + ω 1 > 1 (9) where ω (, 1) measures the substitutability of loans across regions. 8

9 2.6 Entrant Entrants pay the fixed cost of entry φ in order to begin operations in the market. There is no revenue in the period of entry because the loan is repaid in the next period, and equity issuance is used to cover entry cost. In the next period, the bank becomes an incumbent and receives W t+1. Therefore, the present value of the entrant s profits in the symmetric equilibrium is V t = E t Λ t,t+1 W t+1 ] Free entry condition implies that the entry profit should be equal to the entry cost: V t = φ (1) Let Nt E denote the number of entrant in period t. Since the fraction η of incumbents banks exit, the law of motion for N t is given by N t = N E t + (1 η)n t 1 3 Symmetric Equilibrium In this section I will define a symmetric equilibrium and describe relevant conditions for it. A symmetric equilibrium in this model is {c H t, l t, d t, c E t, k t, b t, y t, nw t, I t, q t, R b t, R d t, r t, w t, M t, N t } t such that Households solves their utility maximization problem ] 1 R d = β c H H E t t t c H t+1 (11) (l t ) φ = w t c H t (12) c H t + d t = w t l t + Rt 1d d t 1 + (Rt 1 b Rt 1)b b t 1 φnt E (13) Λ t,t+1 = β ch t c H t+1 (14) 9

10 Entrepreneurs solve their utility maximization problem c E t = (1 β E )nw t (15) q k t k t = β E nw t + b t (16) R b tb t = E t m E q k t+1k t (1 δ)] (17) nw t = q k t (1 δ) + r t ]k t 1 R b t 1b t 1 (18) Regional producers maximize their profit r t = α y t k t 1 (19) w t = (1 α) y t l t (2) y t = A t (k t 1 ) α (l t ) 1 α (21) Capital producers solve their profit maximization problem 1 = q t 1 κ ( ) 2 ( ) ] i It It It 1 κ i 1 2 I t 1 I t 1 I t 1 ( ) ( ) ] c E 2 t It+1 It+1 + βe t q t+1 κ i 1 I t c E t+1 k t = (1 δ)k t κ i 2 Regional banks maximize their profit I t ( ) ] 2 It 1 I t 1 (22) I t (23) R b t = M t R d t (24) M t = N t N t + ω 1 (25) Free entry condition for entrants is satisfied V t = φ (26) V t = E t Λ t,t+1 W t+1 ] (27) b t 1 W t = (M t 1 1)Rt 1 d + E t (1 η)λ t,t+1 W t+1 ] (28) N t 1 1

11 Market clearing conditions: deposit, final goods b t = d t (29) c H t + c E t + I t + φnt E = y t (3) The stochastic process for the productivity shock is given by log(a t ) = ρ log(a t 1 ) + ɛ A t (31) ɛ A t i.i.d(, (ζ A ) 2 ) (32) 4 Calibration There are two types of parameters, the parameters for real sector (β H, β E, φ f, α, m E, κ i, δ) and the parameters for the banking sector, (ω, φ, η). The time period of the model is quarterly. The parameters (β H φ f, α, m E, κ i, δ) are fairly standard in the literature, so I take the value of these parameters from the literature. I choose the value of β E so that the borrowing constraint for the entrepreneur is binding around the steady state. The important parameters in this model are the elasticity of substitution of regional loans ω, fixed entry cost φ, and exogenous exit rate η. For the exit rate, I choose η =.1 so that in annual the average exit rate is 4%. Parameter Value Target HH disc. factor β H.995 Policy rate Ent. disc. factor β E.95 Binding Const. Disutility of labor φ f 1 Gali(28) Cobb-Douglas parameter α.3 rk/y Ent. LTV ratio m E.88 LTV Inv. Adj. cost κ i.4 GNNS(21) Depreciation rate δ.25 K/Y Elast. of subst. across region ω.978 Loan markup=4.5% (annual) Fixed entry cost φ 4.1 # of banks in SS = 2 Exit rate η.1 Ave. exit rate = 4% (annual) Table 1: Parameters in this model 11

12 5 Results 5.1 Amplification of shock In order to understand how the productivity shock is amplified through the imperfect competition in the banking sector, I computed the impulse response functions to a 1% positive productivity shock. The result is reported in the Figure 1. We can see that the response of output is larger under the imperfect competition than under the perfect competition. This is due to the amplification though imperfect competition in the banking sector. When there is a positive productivity shock, entrepreneurs try to borrow more as the return on investment increases. This leads to a higher demand for loans, which increases the profits of banks. Knowing this, banks enter into the market and the number of banks increases. As a result, competition gets tight and markup decreases, which allow entrepreneurs to borrow at cheaper rate and encourage investment. In the first several periods of the impulse response, the number of banks decreases. This is because in the first several period, the effect on the stochastic discount factor is larger than the effect of profit so the present value of regional banks decreases. Eventually it increases and will be above the steady state level for a long time. The table 2 reports the effect of imperfect competition on volatilities of outputs, investments, and loan rate. From this table, we can see that the standard deviation of output and investments increases while the standard deviation of loan rate decreases. The increase in the standard deviation of output and investment is due to the amplification through the imperfect competition we have discussed so far. The decrease of loan rate comes from the countercyclical markup. When the productivity is high, the loan rate tends to be high as well because there are higher demands on loans. On the other hand, when the banking market is imperfect, bank enters into the market in good times and loan markup decreases. These two factors cancel out each other and decrease the standard deviation of loan rate. The table 3 reports the standard deviation and correlations of bank related variables with output. Under the perfect competition, loan markup is constant, so the volatility and correlation is zero. On the other hand, when I introduce the imperfect competition in the banking sector, the correlation of markup and output becomes negative (.4456), which is consistent with the data (.31). This also allows us to get the right sign on the correlation of 12

13 Productivity shock (A) Output Investment Cournot Comp.6 Number of local banks Loan markups.8 Debt Loan rate.4 Bank profit 1 Entreprener net worth Figure 1: The impulse response to 1% positive productivity shock Comp Cournot std(y t ) (+7.5%) std(i t ) (+2.6%) std(rt) b.17.8 (-52.9%) Table 2: The effect of imperfect competition on volatilities 13

14 Comp Cournot Data std(m t ).3.1 Corr(Y t, M t ) Corr(Y t, N t ).4456 Corr(Y t,profit t ).35 Corr(Y t, Rt) b Table 3: Correlation of bank-related variables outputs and loan rates. Under the perfect competition, it is positive (.545) while under the imperfect competition it becomes negative (.185) which is again consistent with the data (.29). 5.2 Sensitibity analysis In this section I change the fixed entry cost (φ) and substitutability of regional loans (ω) to see how economy react to changes in the market structure of the banking sector. The table 4 reports the effect of a change in the fixed entry cost. Here I double the entry cost from.41 in the bench mark case to.82. As we can see from the table, the higher fixed cost leads to more volatilities in general: the standard deviation of output and investment increases by 2.9% and 9.5%, respectively. This is because a higher entry cost makes the degree of imperfect competition in the banking sector high. Under this high fixed cost, the loan markup in the steady state increases by.475%, or 1.9% in annual. The table 5 reports the effect of a change in the substitutability of regional loans. Here I change ω from.971 in the benchmark case to.5. As we can see from the table, the lower substititability leads to more volatility as well. 14

15 Comp Cournot Cournot φ = 4.1, ω =.978 φ = 8.2, ω =.978 std(y t ) (+7.5%).383 (+1.4%) std(i t ) (+2.6%).1231 (+3.1%) std(rt) b.17.8 ( 52.9%).9 ( 47.1%) std(m t ).3.5 Corr(Y t, M t ) Corr(Y t, N t ) M SS 1.125% 1.6% Table 4: The effect of a change in fixed entry cost φ Comp Cournot Cournot φ = 4.1, ω =.978 φ = 4.1, ω =.5 std(y t ) (+7.5%).468 (+34.9%) std(i t ) (+2.6%).222 (+134.6%) std(rt) b.17.8 ( 52.9%).5 (+194.1%) std(m t ).3.47 Corr(Y t, M t ) Corr(Y t, N t ) M SS 1.125% 11.5% Table 5: The effect of a change in substitutability of regional loans (ω) 6 Conclusion In this paper I introduce imperfect competition and entry and exit in the banking sector into an otherwise standard DSGE model. The model generates an amplification mechanism through the entry and exit in the banking sector: if there is a positive productivity shock, investor tries to borrow more as the return on investment increases. This leads to higher demand on banks loans and higher bank profits. Knowing this, there is an increase in the number of entrant, which reduces the loan markup. As a result, loan rate gets cheaper and this encourage investments furthermore. Compared to the model with perfect competition, imperfect competition in the banking sector increases the standard deviation of output by 7.5%. 15

16 For the future research, I will introduce a sticky price flamework into this model so that the model can be used to evaluate monetary/fiscal policies. In addition, I will add endogenous exit which is costly. In that case, imperfect competition with higher profits may lead to more stability and less costs consistent with the concentration stability view of banking (Allen and Gale(24), Beck et al.(26)). This allows us to evaluate if the imperfect competiton in the banking sector is actually a bad idea or not. 16

17 References 1] Allen, F. and D. Gale (24) Competition and Financial Stability, Journal of Money, Credit, and Banking, 36, p ] Bernanke, B., M. Gertler, and S. Gilchrist (1999) The Financial Accelerator in a Quantitative Business Cycle Framework, in John Taylor and Michael Woodford (eds.), The Handbook of Macroeconomics, Amsterdam: North-Holland. 3] Corbae, D. and P. D Erasmo (213) A Quantitative Model of Banking Industry Dynamics, mimeo. 4] Beck, T., A. Demirguc-Kunt, and R. Levine (26) Bank concentration and crises, Journal of Banking and Finance. 5] Corbae, D., and P. D Erasmo (213) A Quantitative Model of Banking Industry Dynamics, mimeo. 6] Corbae, D., and P. D Erasmo (217) Capital requirements in a quantitative model of banking industry dynamics, mimeo. 7] Cuciniello, V. and Federico M. Signoretti (215) Large Banks, Loan Rate Markup, and Monetary Policy, International Journal of Central Banking. 8] Gerali, A, S. Neri, L. Sessa, and F. M. Signoretti (21). Credit and Banking in a DSGE Model of the Euro Area., Journal of Money, Credit and Banking, 42: p ] Gertler, M. and P. Karadi (21). A model of unconventional monetary policy., Journal of Monetary Economics, 58: p ] Iacoviello, M. (25) House prices, borrowing constraints, and monetary policy in the business cycle, American economic review, p ] Jaimovich, N., and M. Floetotto (28) Firm dynamics, markup variations, and the business cycle, Journal of Monetary Economics, 55(7),

18 12] Kiyotaki, N., and J. Moore Credit Cycles., Journal of Political Economy 15 (2): p

19 Appendix J-F: Price Elasticity of Demand with N(j) = 1 (i.e. Monopolistic Competition Consider the case where the final good Y is produced by a continuum of differentiated sectoral goods Q(j), each supplied monopolistically: 1 Y t = ] 1/ω Q t (j) ω dj where ω (, 1) is the substitutability of sectoral goods. Cost minimization of the final good producer implies the following demand function ] 1 pt (j) ω 1 Q t (j) = Yt, The price elasticity of demand is constant: P t Q t (j) p t (j) 1 P t p t (j) Q t (j) = 1 ω 1 ] ω 1 p t (j) ω ω ω 1 dj (A..1) Extend the previous model by assuming that each sectoral good Q(j) is produced by a finite number N(j) of differentiated intermediate goods x(j, i): N t(j) Q t (j) = N t (j) 1 1 τ x t (j, i) τ where τ (, 1) is the substitutability of intermediate goods. Note that if N t = 1, then it is equivalent to the prior case. The demand function for the intermediate good (i, j) is i=1 1 τ where x t (j, i) = ] 1 pt (j, i) τ 1 p t (i) Qt (j) N t (j) = ] 1 pt (j, i) p t (i) N t(j) p t (j) = N t (j) 1 τ 1 p t (j, i) τ i=1 τ 1 p t (j) τ 1 P t τ 1 τ ] 1 ω 1 Y t N t (j) 19

20 where The price elasticity of demand is x t (j, i) p t (j, i) p t (j, i) x t (j, i) = 1 τ 1 + p t (j) p t (j, i) = 1 N t (j) ( 1 ω 1 1 τ 1 ( ) 1 pt (j, i) τ 1 p t (j) In the symmetric equilibrium where p t (j, i) = p t (j), ) pt (j, i) p t (j) p t (j) p t (j, i) x t (j, i) p t (j, i) p t (j, i) x t (j, i) = 1 τ 1 + ( 1 ω 1 1 τ 1 ) 1 N t (j) (A..2) The price elasticity of demand depends on N t (j). If N t (j) = 1, then (A..2) is equivalent to standard monopolistic competition case (A..1) with constant markup which depends on substitutability ω in sectoral goods. If N t (j), then (A..2) also yields a constant markup but at a level that reflects substitutability τ in intermediate goods, not sectoral goods as in (A..1). If τ > ω, then (A..2) is consistent with countercyclical markups (i.e. if sectoral goods are less substitutable than intermediate goods). The model with default In this section I will discuss how the default choice can be incorporated in my model following Bernanke, Gertler, and Gilchrist(1999). An entrepreneur consists of many members. Each family member is given k t 1 unit of capital and sent to different locations to rent capital to production technology. There is an idiosyncratic shock to the depreciation (1 δ)ω t. The optimal contract is a standard debt contract which specifies 1. The threshold value for ω t, ω t, below which lenders audit the borrower, 2. The amount of capital k t and loans b t, and 3. The cost of borrowing, R b t. 2

21 Before each member return to the family, they have to pay back the debt by using the resource s/he has. Each member is left with the following amount of net worth n t (ω t ) = r t + q t (1 δ)ω t ]k t 1 R b tb t 1 Members default if n t (ω t ). The threshold value for ω t is determined by r t + q t (1 δ) ω t ]k t 1 = R b tb t 1 (A..3) Let G denote the expected value of ω conditional on the audit. G( ω t ) ωt ω t df (ω t ) Monitoring cost is proportional (µ) to the resource available. National banks borrows at R l t. Then the participation constraint is 1 F ( ω t )]R b tb t 1 + (1 µ)r t F ( ω t ) + (1 δ)g( ω t )q k t ]k t 1 R l t 1b t 1 Note that we assume that PC holds for any state in period t + 1. Using (A..3), we can rewrite the PC as {r t 1 µf ( ω t )] + (1 δ)γ( ω t ) µg( ω t )]qt k }k t 1 Rt 1b l t 1 where Γ( ω t+1 ) = 1 F ( ω t+1 )] ω t+1 + G( ω t+1 ) The net worth of the family of entrepreneurs is n t = {r t 1 F ( ω t )] + (1 δ)1 G( ω t )]q k t }k t 1 1 F ( ω t )]R b tb t 1 From (PC), the amount of repayment is 1 F ( ω t )]R b tb t 1 = R l t 1b t 1 (1 µ)r t F ( ω t ) + (1 δ)g( ω t )q k t ]k t 1 Using the equation above, we can eliminate R b t: n t = {r t 1 µf ( ω t )] + (1 δ)1 µg( ω t )]q k t }k t 1 R l t 1b t 1 21

22 Entrepreneurs solves ] max E β c E E t log(c E t ), subject to t,kt,bt, ωt t= c E t + q k t k t = b t + n t n t = {r t 1 µf ( ω t )] + (1 δ)1 µg( ω t )]q k t }k t 1 R l t 1b t 1 {r t 1 µf ( ω t )] + (1 δ)γ( ω t ) µg( ω t )]q k t }k t 1 R l t 1b t 1 where PC holds for each state. Under the assumption ln(ω t ) N( σ 2 /2, σ 2 ), Γ( ω) µg( ω) = (1 µ)φ( z σ) + ω1 Φ( z)] G( ω) = Φ( z σ) where z = (ln( ω)+.5σ 2 )/σ and Φ is the cdf of standard normal distribution. The lagrangian of entrepreneurs problem is L = ] E βe t log(c E t ) t= { + E λ t bt + {r t 1 µf ( ω t )] + (1 δ)1 µg( ω t )]qt k }k t 1 t= R l t 1b t 1 c E t q k t k t }] { } ] + E µ t {rt 1 µf ( ω t )] + (1 δ)γ( ω t ) µg( ω t )]qt k }k t 1 Rt 1b l t 1 t= The first order condition is c E t : β t E 1 c E t = λ t k t : λ t q k t = E t λt+1 {r t+1 1 µf ( ω t+1 )] + (1 δ)1 µg( ω t+1 )]q k t+1} ] + E t µt+1 {r t+1 1 µf ( ω t+1 )] + (1 δ)γ( ω t+1 ) µg( ω t+1 )]q k t+1} ] b t : λ t = E t R l t (λ t+1 + µ t+1 ) ] ω t : λ t { rt µf ( ω t ) + (1 δ)µg ( ω t )q k t } = µ t { rt µf ( ω t ) + (1 δ)γ ( ω t ) µg ( ω t )]q k t } 22

23 We can simplify this first order condition as c E t : β t E 1 c E t = λ t k t : q k t λ t = E t (λt+1 + µ t+1 ){r t+1 1 µf ( ω t+1 )] (1 δ)µg( ω t+1 )q k t+1} ] + E t (1 δ)q k t+1 {λ t+1 + µ t+1 Γ( ω t+1 )} ] b t : λ t = E t R l t (λ t+1 + µ t+1 ) ] ω t : µ t = where S t (1 δ)γ ( ω t ) S t λ t or S t = r t µf ( ω t ) + (1 δ)µg ( ω t )q k t 1 c E t qt k c E t = ] (1 δ)γ ( ω t+1 ){r t+1 1 µf ( ω t+1 )] (1 δ)µg( ω t+1 )qt+1} k (1 δ)γ ( ω t+1 ) S t+1 ] + E t (1 δ)qt+1 k 1 (1 δ)γ ( ω t+1 ) 1 (1 δ)γ ( ω t+1 )]S t+1 c E t+1 (1 δ)γ ( ω t+1 ) S t+1 ] = β E E t Rt l 1 (1 δ)γ ( ω t+1 ) (1 δ)γ ( ω t+1 ) S t+1 β E E t 1 c E t+1 c E t+1 where S t = r t µf ( ω t ) + (1 δ)µg ( ω t )q k t The following figures show the impulse response to 1% positive productivity shock when we include the default of entrepreneurs. 23

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