Banking • In this handout we will address: – Why do banks exist? Or in other words, Are banks essential? – What determines the bank size distribution? – Why do some firms choose bank finance versus direct finance? – What are the implications of long term relationships on interest rates? – What are the macro consequences of frictions in the banking industry? • Some of this material draws on Freixas, X. and JC Rochet (2008) Microeconomics of Banking Cambridge: MIT Press. • Main functions of banks: 1. Offer liquidity and payment services 2. Transform assets 3. Manage risks 4. Screen and monitor borrowers

1

Maturity Transformation and Bank Runs • Reference: Diamond, D. and P. Dybvig (1983) “Bank Runs, Deposit Insurance, and Liquidity”, Journal of Political Economy, Vol. 91, p. 401-19. • This paper employs functions (1)-(3) and shows that banks can implement the first best allocation while direct bond finance cannot. • There is a temporal mismatch: Banks offer short maturity (liquid) demand deposits to savers and make long maturity (illiquid) loans to borrowers. This maturity mismatch is potentially risky (i.e. liquidity shocks are costly). • Three key elements of the model: 1. Individuals are uncertain about when they will want to make expenditures. This uncertainty produces a demand for liquid assets. This is what I mean by a liquidity shock. 2. Long term high yielding investment projects are costly to interrupt. 3. Expenditure decisions are made sequentially (think of the lines in Mary Poppins). If there is a fractional required reserve banking system, then beliefs matter a lot for whether there will be a bank run. 1

• The first two points highlight a role for an institution (we will call it a bank) to transform illiquid assets into liquid ones in order to provide insurance to people who receive private, random liquidity preference shocks. The third is critical for a belief-based run when there is a fractional required reserve banking system and policy responses like Deposit Insurance. • In particular, illiquidity of assets provides the rationale both for the existence of banks and for their vulnerability to runs. • A model of bank runs is important since the main motivation for the third central banking system in the U.S. came from the Panic of 1907, which caused renewed demands for banking and currency reform. During the last quarter of the 19th century and the beginning of the 20th century the United States economy went through a series of financial panics. According to many economists, the previous national banking system had two main weaknesses: an inelastic currency and a lack of liquidity. • As simply stated in wiki “The primary motivation for creating the Federal Reserve System (in 1913) was to address banking panics.”1 • The United States was the first country to establish an official deposit insurance scheme, the Federal Deposit Insurance Corporation, during the Great Depression banking crisis in 1933.

1.1

Environment

• Three periods t = 0, 1, 2. • Unit measure of ex-ante (i.e. t = 0) identical agents • All agents have 1 unit of the good at t = 0 and prefer to consume either at t = 1 or t = 2. • Storage technologies 1. Productive Storage technology: 1 unit of goods invested at t = 0 yields R > 1 units at t = 2. If the storage is interrupted at t = 1, the salvage value is the initial investment. t=0 t=1 t=2 −1 1 R 2. Pillow Storage technology: 1 unit of goods invested at t = 1 yields 1 unit at t = 2. t=1 t=2 −1 1 Storage in this technology is unobservable. 1 http://en.wikipedia.org/wiki/Federal

Reserve System

2

• Agents face a preference shock (θ) which is iid across agents and realized at t = 1. This determines their “type” (we will consider two different informational assumptions on type): 1. Early consumers: prob(θ = 1) = π with preferences u(c1 ).That is, they only want to consume at t = 1. 2. Late consumers prob(θ = 2) = (1 − π) with preferences u(c2 ).That is, they only want to consume at t = 2. • Given that these shocks are iid and there is a continuum of agents, π and (1 − π) also denote the population fractions of early and late consumers in the economy (subject to some technical details ). • Assume u0 (c) > 0, u0 (0) = ∞, u0 (∞) = 0,and CRRA≥ 1. Consumption is unobservable.

1.2

Autarkic Allocation

• Let W denote the amount withdrawn from the productive technology. • At t = 0 put endowment in productive technology. At t = 1, choose A W = 1 if θ = 1 and W = 0 if θ = 2. This generates cA 1 = 1 and c2 = R.

1.3

Planner’s problem when type is observable (First Best) max (c1 ,c2 )∈R+ ,(S,W )∈[0,1]

πu(c1 ) + (1 − π)u(c2 )

(1)

s.t.S + πc1

=

W

(2)

(1 − π)c2

=

R(1 − W ) + S

(3)

where the two constraints are resource feasibility at t = 1 and t = 2 respectively and the objective function should not be considered expected utility but the sum of utilities for each agent in the economy. • Since the short return between t = 1 and t = 2 (i.e. 1) is dominated by the return to the long asset (i.e. R), it is strictly better not to liquidate more than you need to cover expenditure by early consumers. Hence S = 0. In this case, problem (1)-(3) can be reduced to     R(1 − W ) W + (1 − π)u max πu W π 1−π yields u0 (c∗1 )

=

Ru0 (c∗2 ) 0

=⇒

u

=⇒

c∗1

since R > 1 and u0 (c) > 0. 3

(c∗1 ) <

(4) 0

>u

c∗2

(c∗2 )

• With concave preferences, autarky is ex-ante suboptimal relative to the first best (i.e. 1 ≤ c∗1 < c∗2 ≤ R). • For example, if u(c) = c1−α /(1 − α), then the 2 equations characterizing the first best are given by (4) and the consolidated resource constraint implied by (2)-(3) with S = 0 substituting for W : c∗2 (1 − π)c∗2 πc∗1 + R

= R1/α c∗1 =

1

which yields c∗1

=

c∗2

=

1 π + (1 − π)R

1−α α

,

1

Rα π + (1 − π)R

1−α α

.

But this implies that if α > 1, then c∗1 > 1 since 1 > π + (1 − π)R α−1 1 1 1 < R α and that c∗2 < R since πR + (1 − π)R α > R α .

1−α α

⇐⇒

• Hence we see the insurance role that a planner provides; she pools resources in order to insure agents against the unlucky event that they have to withdraw early. Ex-ante agents are willing to draw against the high return they receive in the event that they are a late consumer. • Note that the planner implements this by liquidating more of the long technology than π · 1.

1.4

Planner’s problem when type is unobservable (Second Best)

• While the planner doesn’t know who are early or late consumers, she can design an allocation mechanism such that households send truthful messages about their type (this is known as the revelation principle) • Truthful revelation requires that: 1. an early type doesn’t report that he is late. That is u(c1 ) ≥ u(0). The lhs is what the early type gets if he reports early and the rhs is what he gets if he reports late (doesn’t get anything at t = 1). Since utility is strictly increasing this requires c1 ≥ 0. 2. a late type doesn’t report that he is early. That is u(c2 ) ≥ u(c1 ). The lhs is what the late type gets if he reports late and the rhs is what he gets if he reports early (he stores the c1 in the short term technology till t = 2 then eats). Since utility is strictly increasing, this requires c2 ≥ c1 . 4

• These requirements are known as incentive compatibility or truth telling constraints2 , which must be added to the planner’s problem which is now: max (c1 ,c2 )∈R+ ,(S,W )∈[0,1]

s.t S + πc1

=

W

(1 − π)c2

=

R(1 − W ) + S

c2

≥

c1

πu(c1 ) + (1 − π)u(c2 )

But recall the solution to the problem without the incentive compatibility constraint actually satisfies the constraint. Hence the first best is actually implementable with private information. That’s not usually the case.

1.5

Decentralized solution 1: Competitive asset markets

• Competitive asset markets exist to transfer resources between agents over time. Asset payoffs cannot be conditioned on private information, but can be conditioned on publicly observable events like time. This assumption rules out standard insurance markets. Suppose there is a short term (one period) discount bond at price q1 and a long term (two period) discount bond at price q2 available at t = 0. • The household’s problem is max c1 ,c2 ,(L,W )∈[0,1],a1 ,a2

s.t.q1 a1 + q2 a2 + L =

πu(c1 ) + (1 − π)u(c2 )

1

c1

= W + a1

c2

= R(L − W ) + a2

• To construct an arbitrage argument, substitute for L from the first constraint and W from the second into the third to yield c2

= ⇐⇒

R(1 − q1 a1 − q2 a2 − [c1 − a1 ]) + a2   1 c2 = 1 + a1 (1 − q1 ) + a2 − q2 c1 + R R

• But this implies that in equilibrium, q1∗ = 1 and q2∗ = 1/R. 2 The revelation principle assures us that it is sufficient to restrict the allocation to respect simple messages which are incentive compatible. If ω ∈ {1, 2} are the two possible types of agents and m ∈ {1, 2} denotes the possible messages that an individual can report, then for a payoff that depends on true type and message v(ω, m), incentive compatibility requires that the type 1 agent not report he is type 2 and the type 2 agent not report he is type 1 or:

v(1, 1)

≥

v(1, 2),

v(2, 2)

≥

v(2, 1).

5

∗ ∗ ∗ ∗ • At  these prices, households choose a1 = 0, a2 = 0, L = 1, W (θ) = 1 if θ = 1 . That is, autarky is the equilibrium with private asset 0 if θ = 2 markets.

1.6

Decentralized Solution 2: Banks

• A deposit contract is specified as one in which agents deposit 1 unit at t = 0 in return for r1 units of the good if they withdraw at t = 1 and r2 units of the good if they withdraw at t = 2 as long as the bank is solvent at t = 1 : 3 t=0 t=1 t=2 −1 r1 r2 • Also have access to the productive and pillow technologies. • Withdrawals are served sequentially in random order until the bank runs out of assets. This is known as the sequential service constraint; a bank’s payoff to any agent depends on his place in line due to a solvency constraint and not on future info about agents later in line (effectively this rules out communication between agents). • To understand the solvency constraint, note that the most that a bank can withdraw from the long run technology at t = 1 is 1. Then if f fraction of agents go to the bank to withdraw at t = 1 and the bank offers r1 to each person, then the bank can only meet its obligations if f r1 ≤ 1. If f r1 > 1,then the bank is insolvent. • Specifically, the payoff associated with withdrawing at t = 1 or t = 2 per unit of deposit, which depends on one’s place in line, is denoted V1 (fj , r1 ) and V2 (f, r1 ) where fj is the fraction of withdrawers serviced before agent j and f is the total fraction of withdrawers serviced at t = 1. Then  r1 if fj r1 ≤ 1 V1 (fj , r1 ) = 0 fj r1 > 1 where the first line is the payoff to a type 1 individual if the bank is solvent and the second is if it is insolvent. For type 2 payoffs if solvent and insolvent respectively are ( r1 ) if f r1 ≤ 1 r2 = R(1−f 1−f . V2 (f, r1 ) = 0 if f r1 > 1 • Obviously, (r1 , r2 ) = (c∗1 , c∗2 ) with f = π is feasible and optimal. Competition between banks leads them to offer this deposit contract. 3 Solvency

means that the bank has resources remaining.

6

• If this is the deposit contract, is the banking system stable? That is, will banks be able to fulfill their contractual obligations in equilibrium? This depends on the behavior of late consumers, which in turn depends on their beliefs. – Suppose late consumers believe only early consumers will withdraw at t = 1. This implies f = π. ∗ Under this belief, a given late consumer believes his utility from withdrawing at t = 2 is u(c∗2 ). ∗ Now consider a deviation given these beliefs. If he chooses to withdraw at t = 1 and use the short term storage technology he will receive utility u(c∗1 ) at t = 2. ∗ Since c∗2 > c∗1 , this deviation is suboptimal given his beliefs. ∗ In this case we say that the first best banking allocation is a pure strategy Nash equilibrium (i.e. a “good” equilibrium).4 – Suppose that late consumers believe that other late consumers will withdraw at t = 1 (i.e. they believe there will be a run on the bank). This implies f = 1 (remember individual agents are of measure zero). ∗ Under this belief, f r1 > 1 since c∗1 > 1. ∗ In that case, if a given late consumer does what everyone else is doing and withdraws at t = 1, he receives c∗1 provided he is early enough in line (i.e. fj r1 ≤ 1) and zero otherwise. Since he can store that under his pillow, he receives either u(c∗1 ) or u(0). ∗ Now consider a deviation by the agent from what all other late consumers are doing. In particular, if he does not withdraw at t = 1, then under these beliefs he receives u(0). ∗ If f satisfies f r1 = 1, then since u(0) is dominated by an f chance of u(c∗1 ) or a (1 − f ) chance of u(0), it is a best response to withdraw early under these beliefs. 4 Informally, a set of actions or strategies is a Nash equilibrium if no player can do better by unilaterally changing his or her strategy. Formally, let

· · · · · ·

Si denote the action or strategy set for player i ∈ {1, ..., n}, S = S1 × S1 ... × Sn be the set of possible strategy profiles, x = (x1 , ..., xn ) ∈ S be a given strategy profile, x−i be a strategy profile of all players except for player i, fi (x) denote the payoff to agent i given a strategy profile x, and f = (f1 (x), ..., fn (x)) be the economywide payoff function.

A strategy profile x∗ ∈ S is a Nash equilibrium if no unilateral deviation in strategy by any single player is profitable for that player, that is ∀i, xi ∈ Si , xi 6= x∗i : fi (x∗i , x∗−i ) ≥ fi (xi , x∗−i ).

7

∗ Thus there is another pure strategy Nash equilibrium with bank runs which is pareto dominated by the above “good” equilibrium.5 • In the latter case, we see that a bank run can be the result of pessimistic beliefs despite the fact that there was no change in the tastes or technologies of the economy. The multiplicity of pareto ranked equilibria is rampant in coordination games like this. • Thus can construct sunspot equilibria where some of the time the banking system is stable and other times it is unstable. Under some parameterizations, this sunspot equilibrium might be ex-ante superior to autarky (i.e. a competitive asset market).

1.7

Optimal Responses to Runs

• If banks can commit to suspend convertibility when withdrawals are too numerous at t = 1 (i.e. the first j such that fj = π) and late consumers know this, they know the bank will be solvent and there is no need to run. • If the government (FDIC) commits to insure depositors such that even if the bank is not able to fulfill its obligations, the depositors receive the full value of their deposits which is funded by a tax system, then this implements the first best without any taxes actually ever being paid. • Deposit insurance is shown to be a better option if there is uncertainty about the true number of early consumers π. • One potential problem is the credibility or time consistency of the commitment to deposit insurance. For instance, when deposit insurance actually has to be paid out, might the choose not to honor that obligation? A paper which has considered this time consistency problem is by Ennis, H. and T. Keister (2009) “Bank Runs and Institutions: The Perils of Intervention”, American Economic Review, 99, pp. 1588-1607. • Another problem is that government insurance can mess up the incentives of the bank (if they know that the government will bail them out, then they may take excessive risk). A paper which considered how incentives are distorted by deposit insurance is by Keeley, M. (1990) “Deposit Insurance, Risk, and Market Power in Banking”, American Economic Review, 80, pp. 1183-1200. • Summary: While this does not prove that banks with deposit insurance are strictly essential (since we did not show there is no other way to implement the planner’s solution), it at least shows they provide an implementation result. 5 In this case, ALL agents receive a risky return that has mean 1 (i.e. f r + (1 − f )0 = 1). 1 This yields expected value lower than autarky.

8

2

A Model of Delegated Monitoring • Reference: Diamond, D. (1984) “Financial Intermediation and Delegated Monitoring”, Review of Economic Studies, Vol. 51, p. 393-414. • This paper employs functions (3)-(4) and shows why a bank may arise to economize on monitoring costs. • If there is asymmetric information, costly monitoring (be it screening exante or verifying ex-post) can alleviate the information asymmetry. In particular, if projects that need to be financed are large and private investors are small, then instead of having each investor pay a cost to monitor it may be efficient to have one bank do the monitoring. • However, in that case only the bank knows and not all the investors. Who’s going to monitor the monitor? • Diamond’s main proposition is that if the bank is sufficiently large (which means it is sufficiently diversified), then it is unlikely to be insolvent and incur bank bankruptcy expenses which could offset the gains from economizing on monitoring costs.

2.1

Environment

• One period (i.e. static model). • N identical firms want to borrow in order to finance productive projects. • Each firm requires 1 unit of capital to run the project. • Firm n0 s project yields a random amount yen which is unobservable to anyone but the firm. These returns are i.i.d. across firms. • By paying a monitoring cost K the lender can observe yen . • Assume that monitoring costs are not too big so that lending is profitable relative to investing in a riskless technology with net return r. That is, E[e y ] − K > 1 + r. • Each risk neutral private investor only has 1/m units of capital (so m of them are needed to finance a project) where m is a big number. There are at least m · N private investors. Thus there are lots of small investors. • There is also an auditing technology such that if an agent (be it a borrower or even a bank) declares it is insolvent (i.e. cannot pay what it promised), then it bears a cost C > K.

2.2

Direct Finance

• See Figure 2.2. • Total monitoring cost from direct finance is N · m · K 9

2.3

Bank Finance

• See also Figure 2.2. • Instead of having each private investor monitor, suppose there is a bank which accepts deposits from private lenders. • Total monitoring cost for the bank is N · K, lower than direct finance since m > 1. • The problem is that there is no incentive for the bank to report truthfully in this static model. It could claim that all the firms hP received i the lowest N en − N · y ≥ 0). amount y and abscond with the difference (i.e. E n=1 y If N is small, the probability that y is realized for all N borrowers is not low. The same goes for any value less than yen . • Given this moral hazard problem on the part of the bank, we have a question of monitoring the monitor. – Direct monitoring of the bank by each investor would clearly be inefficient (total cost is N · K + N · m · K). – The solution is that the bank offers a noncontingent debt or “deposit” contract under which each investor is promised (1 + rD )/m in exchange for a deposit of size 1/m and the bank is audited/liquidated P  N if its announced cash flow zeN = ei − K · N is less than i=1 y N (1 + rD ), the total sum pomised to depositors. Thus, the “deposit” contract is along the lines of Townsend (1979). • Given the outside option to depositors, the return on deposits rD must satisfy " (N )# X E min yei − K · N, (1 + rD ) · N = (1 + r) · N. (5) i=1

Notice that in the absence of deposit insurance, deposits are risky but that risk falls the more projects are funded by the bank.6 • The audit/liquidation costs which make this deposit contract incentive compatible (in the sense that the bank has an interest in sending a truthful declaration zeN ), are given by CN = E [max {(1 + rD ) · N − zeN , 0}] .

(6)

6 The expectation is on the outside since resource feasibility requires that bank outflows P (1 + rD ) · N must equal bank inflows N ei − K · N across all possible realizations. i=1 y

10

P  N So if the bank reports inflows which are low zb < ei − N · K , i=1 y the bank is insolvent and then the penalty is high. If the bank reports truthfully, there is zero penalty.7 • Thus delegated monitoring is more efficient than direct lending (i.e. the reason d’etre for a bank) provided K · N + CN < m · N · K.

(7)

Proposition 1 (D.0) If monitoring is efficient ( K < C1 ), investors are small (m > 1), and investment is profitable (E[e y ] − K > (1 + r)), financial intermediation (delegated monitoring) dominates direct lending if N is large enough (i.e. the bank is sufficiently diversified). Proof. Dividing (7) by N yields CN < (m − 1) · K N it is sufficient to show that " ( E min and

CN N

→ 0 as N → ∞.Dividing (5) and (6) by N yields )# N 1 X yei − K, (1 + rD ) = (1 + r) N i=1

" ( CN = E max (1 + rD ) − N

N 1 X yei − K N i=1

!

)# ,0

.

The strong law of large numbers implies that N 1 X yei = E[e y] N →→∞ N i=1

lim

Then since E[e y ] − K > (1 + r), (5) implies rD = r (i.e. deposits are riskless). Finally, (6) implies limN →∞ CN /N = 0. • Intuitively, as N → ∞, the bank is perfectly diversified and the chance that it is insolvent is zero, in which case there is no auditing cost at all. • Economizing on costs by large banks (ones that make a lot of loans), is one rationale for why people argue we should not break up banks. However, here there is no deposit insurance which can cause a different type of moral hazard problem. 7 Note

that ( min

N X

) yei − K · N, (1 + rD ) · N

i=1

( + max

(1 + rD ) · N −

N X

) yei − K · N, 0

i=1

is independent of reported cash flows, so the bank has no incentive to misreport.

11

3

Competition and Risk Shifting • This section is based on Allen-Gale (2004, JMCB) and Boyd-DeNicolo (2005, JF) • A simple static framework with an exogenous distribution of symmetric banks to understand how competition affects bank risk shifting in an industrial organization framework. • Different assumptions can lead to different implications for how competition affects economy-wide failure rates that are the subject of a vast empirical literature. – Beck, et. al. (2003) find banking system concentration (market share of top 1%) is negatively related to the probability of a banking crisis ( e.g. 2xhigher exit rate). – Berger et. al. (2008) find that concentration is positively related to default frequency.

3.1

Allen-Gale (2004, Section 2)

• The economy lasts two periods: 0 and 1. • There are two classes of agents, banks and depositors, and all agents are risk-neutral. 3.1.1

Banks:

• There are N banks that have no initial resources but have access to a set of constant return-to-scale risky technologies indexed by R. • The risky technology yields R with probability p(R) and 0 otherwise where p(0) = 1, p(R) = 0, p0 < 0, p00 ≤ 0. That is, choosing a riskier technology lowers the likelihood of success. • p(R) · R is a strictly concave function of R and reaches a maximum R∗ when p0 (R∗ )R∗ + p(R∗ ) = 0. Increasing R from the left of R∗ entails increases in both the probability of failure and expected output (similar to a Laffer curve). • The bank’s date 0 choice of R is unobservable to outsiders. • At date 1, outsiders can only observe and verify at no cost whether the investment’s outcome has been successful (positive output) or unsuccessful (zero output). • By assumption, contracts are simple debt contracts. • The bank has complete control over the choice of risk taking R.

12

3.1.2

Depositors

• Supply of deposits is represented by an upward sloping (convex) inverse supply curve, denoted by rD (·). PN • Deposits of bank i are denoted by Di and total deposits by D ≡ i=1 Di . • Deposits are insured, so that that the relevant supply does not depend on risk and banks pay a flat rate deposit insurance premium α > 0. • Banks compete for deposits in a Cournot Nash fashion. The rate of interest P N on deposits is a function of total deposits: rD = rD i=1 Di . 3.1.3

Cournot Nash Equilibrium

• In a Nash equilibrium, each bank i chooses (Ri , Di ) that is the best response to the strategies of other banks to solve   max 0, max p(Ri ) · (Ri − rD (D) − α) · Di (Ri ,Di )

• In a symmetric interior equilibrium where (Ri , Di ) = (R, D) > 0 and p(R) > 0, the FONC wrt (Ri , Di ) reduce to Ri : p0 (R) · (R − rD (N · D) − α) + p(R) = 0, Di : p(R) · (R − rD (N · D) − α) − p(R) ·

0 rD

(N · D) · D = 0.

(8) (9)

• Can show these 2 equations in 2 unknowns have a unique solution. • Prop. 1: In a symmetric equilibrium, the equilibrium level of risk shifting R is strictly increasing in N. As N → ∞, R → R. • Proof: Solve (8) for D, then plug into (9) and take derivative dR/dN. • Summary: More deposit market competition raises cost of attracting funds (lowering profits) and leads banks to take more risk given limited liability.

3.2

Boyd-DeNicolo (2005)

• Suppose now there are many entrepreneurs, who have access to projects of fixed size, normalized to 1, with the two-point random return structure previously described. • They borrow from banks, who cannot observe their risk shifting choice R, but take into account the best response of entrepreneurs to their choice of the loan rate rL . • In this case, as opposed to the previous case, the bank has no direct control over the riskiness of borrower’s projects. 13

• Given a loan rate rL entrepreneurs choose R ∈ [0, R] to maximize p(R) · (R − rL ) . • FONC wrt R : p(R) + p0 (R) · (R − rL )

0 ⇐⇒ p(R) h(R) ≡ R + 0 = rL p (R) =

• Taking the total differential 1 dR >0 = 0 drL h (R) since h0 (R) = 1 +

p0 (R)2 − p(R)p00 (R) >0 p0 (R)2

where p0 < 0, p00 ≤ 0. • An increase in loan rates causes entrepreneurs to take on more risk. • Assume the inverse demand for loans rL (L) is a decreasing function of the 00 0 ≤ 0 with rL (0) > rD (0). < 0, rL total supply of loans L where rL P  N • Banks have no equity so L = D = D i i=1 • Given the decision rules of entrepreneurs, in a Nash equilibrium each bank i chooses Di that is the best response to the strategies of other banks to solve   max 0, max p(R) · (rL (D) − rD (D) − α) · Di Di

subject to h(R) = rL (D). • Since R = h−1 (rL (D)) , we can rewrite the bank’s problem as max p(h−1 (rL (D))) · (rL (D) − rD (D) − α) · Di Di

subject to 0 ≤ h−1 (rL (D)) ≤ R. • Prop. 2: In a symmetric Cournot Nash Equilibrium, the equilibrium level of risk shifting R is strictly decreasing in N. As N → ∞, the Nash equilibrium converges to the competitive outcome rL = rD + α. • Intuition. – Increased competition lowers loan rates and entrepreneurs optimally respond by decreasing the risk of their investment projects. 14

3.3

Summary

• Probability of success is a decreasing function of technology choice p0 (R) < 0. • Market structure affects technology choice and hence economy-wide failure rates – Prop. 1: More deposit market competition leads to more failure (R is increasing in N ) as banks take more risk given lower profits. – Prop. 2: More deposit market competition leads to less failure (R is decreasing in N ) as entrepreneurs take on less risk if borrowing rates (via pass-through) are low. • Market structure has implications for financial crises. Reduced form empirical literature regressing economywide failure rates on banking concentration. • With entry and exit, failure rates have implications for banking concentration. Potential endogeneity problems with reduced form literature. • What’s missing from A-G and B-D? – Market structure, and degree of competition, is exogenously given by N. ∗ What determines N ? Free entry condition? – All banks are the same size (i.e. symmetry) is inconsistent with data.

4

Spatial Heterogeneity in Banking • In the prior sections, all loans and deposits were the same (i.e. a homogeneous good). But a loan in L.A. may be different from a loan in Madison (i.e. one dimension of differentiation is location). Furthermore, a home loan may be different from a corporate loan (i.e. another dimension of differentiation is type). In general, there is a lot of product differentiation. • Once there is product differentiation, it is possible for a bank (or firm in general) to be a monopolist in its perfectly differentiated product realizing that there are imperfect yet nearby substitutes that may still attract potential buyers of its specific product (i.e. it faces some competition). Combining the two yields a model of monopolistic competition. • In this section, the number of banks (N ) will be determined endogenously.

15

4.1

Environment

• There are equal numbers of identical households located at each point on a unit circle (A Salop (1979) model). • Banks are indexed by n = 1, 2, ..., N and are also located on the circle, but because there is a fixed cost F to set up a bank, there will only be a finite number of them. • Any deposits that bank n takes in at interest rate rnD it will loan out at an exogenously higher interest rate r > rnD (which could the world return on capital facing the small economy). • Households can only save in a bank (i.e. they don’t have access to the bank’s loan/storage technology). Hence this is a model “with” banks rather than “of” banks. • It is costly for a household who lives at a location distance x from a bank to travel to the bank. In particular, it costs γx.8 • Depending on the number of banks N, since households are uniformly distributed on the line, the optimal  (in terms of minimizing travel costs) 3 5 2N −1 1 , 2N , 2N , ..., 2n−1 . placement of banks are at locations 2N 2N , . . . , 2N • Figure 3.7 shows the total distance costs for different numbers of banks. The horizontal axis can be thought of as distance from the bank of the households on the line [0, 1]. The vertical axis can be thought of as cost to each person. The area of the triangles correspond to the total cost (sum of all households’ costs) of there being N banks in the economy.  −1 1 • For instance, if N = 1, the bank is located at 2N 2N = 2 . The household at point 12 on the horizontal axis has cost 0 on the vertical axis of going to the bank. But the households furthest away at points 0 and 1 have cost 12 γ of going to the bank. The sum of all costs is the sum of the two triangles. In this case, the base is 21 , the height is 12 γ, so the area of each triangle is 1 1 1 1 2 · 2 · 2 γ = 8 γ and since there are two triangles, the total distance cost is 1 2 · 8 γ = 0.25γ.9  1 −1 3 • If N = 2, banks are located at 2N = 41 , 2N 2N = 4 . The households at points 14 and 34 have costs 0. But the households furthest away at points 0, 21 , and 1 have cost 14 γ of going to the bank. The sum of all costs is the 8 An alternative to the location interpretation is that households have a very diverse set of savings needs but there are only a standardized set of savings options so they must bear the cost of finding a standardized savings instrument to fit their needs. 9 To see that if N = 1, it is optimal to locate at equal distances (i.e. at 1 ), consider what 2 the total cost would be if the bank located at say 43 . In that case, the total cost would be

1 3 3 1 1 1 · · γ + · · γ = 0.3125γ 2 4 4 2 4 4 as opposed to 0.25γ.

16

sum of the four triangles. Each triangle is distance cost is 81 γ = 0.125γ.

1 2

·

1 4

· 41 γ =

1 32 γ

so the total

• In general, if there are N banks the total distance cost is   γ 1  γ  1 · 2N = · 2 2N 2N 4·N which is the area of the triangle times the number of triangles. Notice that the total distance cost is decreasing in the number of banks N. • If each bank costs F to set up, then the total fixed cost of setting up banks is F · N. Notice that the total setup costs are increasing in the number of banks N .

4.2

Planner’s solution

• Total costs T C to society are given by TC =

γ + F · N. 4·N

These costs are graphed in Figure 3.8 as a function of the number of banks N. • If the objective is to minimize (social) costs, since total costs in Figure 3.8 are convex, there is a well defined minimum. The minimum occurs where dT C 10 This yields the (socially) optimal number of banks:11 dN = 0. −

γ +F 4N 2

=

N∗

=

0 ⇐⇒ r 1 γ 2 F

(10)

Thus, the higher are the fixed costs of setting up a bank, the less banks there should be. Also, the higher the distance costs, the more banks there should be.

4.3

Decentralized solution

• (10) gives the socially optimal number of banks. It is as if a central social planner decided what’s best taking into account all the costs. Now we ask what will happen in a decentralized monopolistically competitive economy where banks have to choose whether to pay the cost F to enter the economy or not. The answer will obviously be based on bank profitability. 10 Of

course, we are implicitly neglecting integer constraints (i.e. assuming N is continuous). a matter of interpretation, note that in the first line, the MB of adding another bank γ is that distance costs fall (i.e. 4N 2 ) while the MC is F, so setting MB=MC yields (10). 11 As

17

• Specifically, suppose N banks enter simultaneously, locate uniformly over n the unit circle [0, 1], and set deposit rates rD , n = 1, ..., N. Recall that for a monopolist, it is equivalent to choose a quantity as an interest rate since given the demand or supply for its product, a choice of a quantity is the same as the choice of a price. A monopolistically competitive firm is a monopolist in its “location”. • To determine the supply of deposits to bank n (i.e. the set of households in [0, 1] who want to deposit at bank n), consider a depositor who is considering bank n or bank n + 1. See Figure 3.9
where the depositor is bn from bank n + 1. distance x bn from bank n and distance N1 − x • The depositor who is distance x bn from bank n is indifferent between going to bank n or bank n + 1 if the return she gets net of distance costs from the two different banks are equal or:   1 D D rn − γb xn = rn+1 − γ −x bn . (11) N This can be re-written x bn =

D rD − rn+1 1 + n . 2N 2γ

(12)

• Since bank n draws depositors whose costs are less than going to banks n + 1 or n − 1, bank n’s potential supply of deposits is the sum of those 2 distances: DnS

= =

D D rD − rn+1 rD − rn−1 1 1 + n + + n 2N 2γ 2N 2γ D D D 2r − rn+1 − rn−1 1 + n . N 2γ

(13)

Note that the supply of deposits to bank n rises if it offers a higher interest rate rnD but falls if bank n’s distant competitors raise their interest rates (drawing customers away from bank n). This latter effect is where “competition” enters into monopolistic competition. • Profits of bank n are D D Πn = max(r − rnD ) · Dns (rnD , rn+1 , rn−1 ) D rn

18

• The optimal choice of interest rate (or first order condition) satisfies dΠn drnD

=

0 ⇐⇒

0

=

D D −Dns (rnD , rn+1 , rn−1 )  dD s

(14)



n

D drn

 s dDn  · D +(r − rnD ) ·  + drn+1  s dD + drDn · n−1

D drn+1 D drn D drn−1 D drn

   

D D , rn−1 ) corresponds to the loss the bank – The first term −Dns (rnD , rn+1 makes when it raises the rate it must pay households on all their existing deposits dD s

– The second term (r − rnD ) · drDn corresponds to the gain the bank n makes when it raises the rates since more households are attracted to dD deposit at the bank (i.e. drDn > 0). n

– The third and fourth terms correspond to the losses the bank makes D drn+1 dr D > 0 and drn−1 > 0) and n n drD D s s dDn dDn (i.e. drD < 0 and drD < 0). n+1 n−1

if other banks also raise their rates (i.e so the households are attracted away

• If all banks move simultaneously, then the other banks cannot react and dr D

change their interest rates when bank n changes its interest rate (e.g. drn+1 = n D 0), then (14) can be written (using (13) on the lhs and its derivative on the rhs): D D Dns (rnD , rn+1 , rn−1 )

=

D D − rn−1 2rD − rn+1 1 + n N 2γ

=

(r − rnD )

=

dDns ⇐⇒ drnD   1 (r − rnD ) · ⇐⇒ γ (r − rnD ) ·

D D 2rD − rn+1 − rn−1 γ + n N 2

(15)

• Since we assumed that all banks are identical in their technologies and all depositors are spread equally across locations, (15) has to hold for all n = 1, 2, ...N. A symmetric solution to all these equations is where D D rnD = rn+1 = rn−1 =r−

γ . N

(16)

This uses the fact that for the last bank (i.e. n = N ) it is the case that D D rN +1 = r1 by virtue of the fact that we are on a circle and for the first D . bank (i.e. n = 1), r0D = rN

19

• Equation (16) yields an ordering of bank loan and deposit rates that is consistent with a positive interest margin (i.e. the return on deposits is lower than the return on investments due to a cost that can be interpreted as the distance the depositor has to travel to the bank (or a cost associated with the bank not having exactly the type of deposit options the household wants), which depends on how close its competitors are (which itself depends on how many competitors there are). • In this case, equilibrium profits are given by  D D  2rD − rn+1 − rn−1 1 + n Π∗n = (r − rnD ) · N 2γ   h  γ i 1 γ = r− r− · = 2 N N N

(17)

Thus, with monopolistic competition banks earn positive profits in equilibrium (unlike the case of perfect competition). The size of the profits depends on the number of banks N that the bank is competing with. If it competes with lots of banks (i.e. N is big), then profits are low. • Since it costs F to set up a bank, from (17) a bank will only enter if profits net of set-up costs exceed 0 or γ − F ≥ 0. N2 • Since profits are decreasing in N in (17), if Nγ2 − F > 0 then another bank will enter (raising the number to N + 1) which lowers the benefit of γ entering to (N +1) 2 − F . Entry will continue while the benefit of entering exceeds the cost or until γ − F = 0. (18) M (N C )2 • (18) is known as the free entry condition. This implies r γ MC N = F

(19)

This condition implies that if F is arbitrarily small, there will be lots and lots of banks (i.e. virtually perfect competition). • Comparing the socially optimal number of banks in (10) with the decentralized number of banks (19), we see r r 1 γ γ ∗ N = < = N MC 2 F F so monopolistic competition generates “too many” banks from a social perspective. Each bank in the free market chases profits and doesn’t take into account the negative externality it imposes on other banks (stealing its competitor’s depositors) . 20

5

Bank Debt vs Market Debt • From Freixas and Rochet Section 2.5 and Diamond, D. (1991) “Monitoring and Reputation: The Choice between Bank Loans and Directly Placed Debt”, Journal of Political Economy, 99, 689-721. • Bring back the corporate finance decision; since in practice direct debt is less expensive than bank loans, it is usually considered that loan applicants are only those agents that cannot issue direct debt.

5.1

Static Analysis

• I = 1 (normalization) • Firms choose between two technologies: – Rg with probability pg and zero otherwise – Rb with probability pb and zero otherwise – pg Rg > 1 > pb Rb (so that only good projects have positive NPV, Rg > Rb and pg > pb ) • Success or Failure can be verified by investors but not choice of technology, except by a bank at cost K. • The cost of choosing the good technology is that you are more likely to have to pay the direct lender back and since choice of technology cannot be verified except by the bank, the return that you must pay back is independent of your choice. 5.1.1

Direct Market Debt

• First consider what if K = ∞ (so that only market debt is possible to finance investment). • Let R`M denote the debt repayment to lenders by the firm in the event of success. • Moral hazard problem: Since pg Rg > 1 > pb Rb , firm chooses good technology if pg (Rg − R`M ) ≥ pb (Rb − R`M ) M

so that there is a critical level of market debt repayment R` below which M the firm chooses the good technology (i.e. R`M ≤ R` ) where M

pg (Rg − R` ) M

R`

= =

21

M

pb (Rb − R` ) ⇐⇒ pg Rg − pb Rb . (pg − pb )

(20)

• Then the lender’s likelihood of repayment is given by ( M pg if R`M ≤ R` M π(R` ) = M pb if R`M > R` • The lender participation constraint is given by π(R`M )R`M ≥ 1 so that competition implies R`M =

1 . π(R`M )

(21)

• Since 1 > pb Rb , lender participation only arises if the borrower chooses the good technology (i.e. if the bad technology is chosen, then R`M = 1/pb and limited liability implies Rb ≥ R`M = 1/pb =⇒ pb Rb ≥ 1 (a contradiction)). M

• Since the borrower only chooses the good technology if R`M ≤ R` , then a necessary condition for equilibrium with direct market debt is M

pg R` ≥ 1.

(22)

• Thus, for funding to take place, the moral hazard problem cannot be too bad. Unlike the Holmstrom-Tirole moral hazard where there is an explicit opportunity cost B to taking the good action, here the cost is related to the likelihood ratio ((pg − pb )/pg ).12 5.1.2

Bank Debt

• Now suppose the monitoring technology is feasible, so if a bank monitors the borrower, they can enforce the choice of the good technology with positive NPV. • Competition among banks implies their participation constraint satisfies pg R`B − K = 1 ⇐⇒ R`B = M

1+K . pg

• If pg R` < 1 so that a direct market debt equilibrium is infeasible, in order for the borrower to participate in the absence of market debt, it must be case that  
1+K . ≥0 pg Rg − R`B ≥ 0 ⇐⇒ pg Rg − pg ⇐⇒ pg Rg − 1 ≥ K. 12 That

is, (20) into (22) yields the necessary condition:
  pg − pb pg R g − pb R b ≥ . pg

22

In other words the monitoring cost has to be less than the NPV of the good project.   1 . • In other words, bank lending exists if pg ∈ 1+K , M Rg R`

5.1.3

Equilibrium

The above results are summarized by the following proposition. Proposition 2 (D1) Assume monitoring costs are sufficiently small so that bank finance is feasible, then there are three possible debt market equilibria depending on the likelihood of success: (1) if pg ≥ 1M then firms issue direct debt; R`   1 (2) if pg ∈ 1+K , then firms borrow from banks; and (3) if pg < 1+K Rg , M Rg , R`

then no investments are made.

5.2

Dynamic Analysis

• In Diamond’s (1991) dynamic (here only a two date t = 0, 1) extension of the previous t = 0 model, successful firms can build a reputation that allows them to issue direct debt instead of using bank loans, which are more expensive. • Assume further that only a fraction f of firms have access to both technologies, while (1 − f ) have access to only the bad technology. • We will construct an equilibrium (under certain parametric assumptions) such that – at t = 0, all firms borrow from banks – at t = 1, firms that have been successful at t = 0 issue direct debt, while the unsucessful ones continue with bank loans – banks monitor all firms who borrow from them. • Working backwards, consider successful firms at t = 1. They can issue direct debt only if the probability of success at t = 1, conditional on success at t = 0, denoted PS is sufficiently high (consistent with (22) from the static problem since at t = 1, all that remains is the last period so the above results have to apply): PS ≥

1

(23)

M

R` where

2

f · (pg ) + (1 − f ) · pb Pr(success at t = 0 and t = 1) PS ≡ = Pr(success at t = 0) f · pg + (1 − f ) · pb 23


2

• If (23) is satisfied, then successful firms will be able to issue direct debt at rate RS = 1/PS . • On the other hand, the probability of success at t = 1 of firms which have been unsuccessful at t = 0 is given by
f · (1 − pg ) · pg + (1 − f ) · 1 − pb · pb PU = . f · (1 − pg ) + (1 − f ) · (1 − pb ) • Proposition D1 implies that if 1 1+K < PU < M Rg R`

(24)

then unsuccessful firms will borrow from banks at rate RU = (1+K)/PU . • Finally, it is necessary to establish that in t = 0,at parameter values which satify (23) and (24), all firms choose bank lending. – To do so, first note that the unconditional probablity of success at t = 0 is P0 = f · pg + (1 − f ) · pb – Reputation building comes from the fact that PU < P0 < PS . That is, the probability of repayment of debt is initially P0 but increases if the firm is successful and decreases if it is unsuccessful. M

– Because of that, the critical level of debt repayment R0 at t = 0 (above which moral hazard appears) is higher than in the static case. In particular, firms know that if they are successful at t = 0, they will obtain cheaper finance at t = 1 so they have an added incentive to “appear” good. – To see this, if the discount rate is β < 1, the critical level of debt at M t = 0, denoted R0 , at which the firm is indifferent between choosing the good and bad technologies at t = 0 is no longer given by (20) but by h
i 
 M U pg · Rg − R0 + βpg · Rg − RS + (1 − pg ) · 0 + βpg · Rg − R(25) h
i 
 M = pb · Rb − R0 + βpg · Rg − RS + (1 − pb ) · 0 + βpg · Rg − RU ∗ The rhs is the EPD profits of a firm that chooses the bad technology at t = 0 but the good technology at t = 1 (either because under the conjecture, an unsuccessful firm will borrow from a bank in t = 1 in which case since they are monitored they must choose the good technology or they were successful and can issue debt at the low rate RS ). 24

∗ The lhs is the EPD profits of a firm that chooses the good technology at t = 0 and t = 1 (we need to check that it is optimal to choose good after being successful, but that is given under the parameterization in Proposition D1). M

– Solving (25) for R0 yields M

R0

=

pg Rg − pb Rb + βpg (Rg − RS ) − βpg (Rg − RU ) (pg − pb )

=

R` + βpg (RU − RS )

M

(26) M

Proposition 3 (D.2) Under   the following assumptions: P0 ≤ 1/R0 ; PS > M 1+K 1 1/R` ; and PU ∈ Rg , M , the equilibrium of the two period Diamond model R`

is characterized as follows: (1) At t = 0, all firms borrow from banks at rate R0 = (1 + K)/P0 ; (2) At t = 1, successful firms issue direct debt at rate RS = 1/PS whereas the rest borrow from banks at rate RU = (1 + K)/PU > R0 . Proof. Apply Proposition D.1 repeatedly after adjusting parameters for the different cases. In particular, part (1) of Proposition D.2 comes from part (2) of Proposition D.1 since P0 > PU and by assumption 1+K 1 < PU < P0 ≤ M . g R R0 Further, part (2) of Proposition D.2 comes from parts (1) and (2) of Proposition M D.1 since by assumption PS > 1/R` holds (so succesful firms issue direct debt M M by part (1)) and since R` < R0 by (26), then unsuccesful firms use bank debt by part (2) since 1 1 1+K < PU < P0 ≤ M < M . Rg R R 0

`

QED. • Although the model is simple, the fact that (PU < P0 < PS ) induces several important features of credit markets: – Data: Succesful mature firms can issue direct debt at low interest rates – Data: Unsuccesful firms pay higher rates than new firms (RU > R0 ) M

– Theory: Moral hazard is partially alleviated by reputation R` M R0 .

25

<

6

Relationship Banking • From Freixas and Rochet (Section 3.6.1). Based on Sharpe, S. (1990, JF) “Asymmetric information, bank lending and implicit contracts: A stylized model of customer relationships” and Rajan, R. (1992, JF) “Insiders and Outsiders: The choice between informed and arm’s length debt”.

6.1

Environment

• Two periods t = 0, 1. • Risk Neutral borrowers and lenders discount future at rate β. • Zero return on risk free technology. • Firms require 1 unit of investment at the beginning of each period which yields R with probability p and 0 otherwise. • A bank loan to a firm requires an initial fixed cost K but once the bank incurs this cost, the bank is able to grant future loans to the same firm without any additional cost. • As a consequence of this last assumption, maintaining a continued relationship with the same bank avoids duplication of such screening costs. At the same time, it provides ex-post monopoly power to the incumbent bank.

6.2

Equilibrium

• At t = 1,if a firm switches from one bank to another (or a new entrant wants a loan), the IR constraint at t = 1 for a bank to lend to the “switching” firm is pR1S − K ≥ 1 in which case banking competition implies the firm will have to repay R1S = (1 + K)/p. • The t = 0 IR constraint for a bank in a long run relationship with a firm is given by pR0L − K + βpR1L ≥ 1 + β · 1. (27) • Indifference between staying in the long run relationship and switching means R1L = R1S . • In that case, competition implies (27) pins down R0L as pR0L

=

R0L

=

(1 + K + β · 1) − β(1 + K) ⇐⇒ (1 − β) (1 + K) + β . p 26

• Therefore R0L < R1L since (1 − β) (1 + K) + β p 1

(1 + K) ⇐⇒ p < (1 + K) .

<

• This example shows the bank uses its ex-post monopoly power during the second period, while competition among banks drives down rates at the initial stage of the relationship. • Thus we would observe a profit from the ex-post monopolistic state in the last period and a loss in the first period such that zero profits hold over the entire relationship. • Thus relationship banking leads to a holdup problem in the last period with an effect on prices that is similar to what happens in a switching cost model. • Although in general the existence of a holdup problem generates a cost for the firm, it also implies in this context that banks will finance more risky ventures because they will ask for a lower first period repayment. • Petersen and Rajan (1994, JF) argue this is important for small firms.

7

A DSGE Model of Banking • While there are now many DSGE models with a banking sector (e.g. Gertler and Karadi (2011, JME)), here we will illustrate how you can take a model like Diamond and Dybvig and turn it into a DSGE model. • This section presents Gertler and Kiyotaki (2015, AER) who add runs to a simplified version of Gertler and Karadi (2011).

7.1

Environment

• Two types of agents: household and bankers. • Two goods: consumption good and capital. • Supply of capital is fixed at 1: Kth + Ktb = 1 • Both HHs and banker can use capital to produce consumption goods, but there is an additional cost of managing capital for HHs:

27

– Technology of bankers: date t input

n Ktb capital

date t + 1 ( Zt+1 Ktb cons. good → output Ktb capital

– Technology of households: date t ( Kth capital input f (Kth ) cons. goods 7.1.1

date t + 1 ( Zt+1 Kth cons good → output Kth capital

Households

Receive 1. endowment Zt W h h 2. production from its capital holdings (Zt + Qt )Kt−1 ,

3. return on deposit from bankers Rt Dt−1 and use it for 1. consumption Cth , 2. production for next period, Qt Kth + f (Kth ), 3. deposits Dt . • The utility maximization problem of a household is given by "∞ # X i h max Et β log(Ct+i ) h ,K h ,D ∞ {Ct+i t+i }i=0 t+i

i=0

h Cth + Qt Kth + f (Kth ) + Dt = Zt W h + (Zt + Qt )Kt−1 + Rt Dt−1

7.1.2

Bankers

• Bankers accept loan from households and use it to purchase capital. • With probability 1 − σ a banker exits and is replaced by a new banker. • An exiting banker uses all their resource for consumption: cbt = nt . • A surviving or new banker will use net worth nt and deposit dt to purchase capital ktb at price Qt : Qt ktb = dt + nt . 28

• In the beginning of period t, the net worth of a surviving banker is given by b nt = (Zt + Qt )kt−1 − Rt dt−1 , and a new banker starts its life with endowment nt = w b . • Banker’s preferences: maximize consumption at the period of exit. "∞ # X i i−1 b Et β (1 − σ)σ ct+i i=1

• Bankers may divert and get θ fraction of its total asset. So the contract must satisfy the incentive compatible constraint: Vt ≥ θQt ktb . where Vt is the value of a banker, defined as "∞ # X Vt = max Et β i (1 − σ)σ i−1 cbt+i , {ktb ,dt }

subject to

i=1

Vt ≥ θQt ktb Qt ktb = dt + nt nt+1 = (Zt+1 + Qt+1 )ktb − Rt+1 dt cbt = nt .

7.2

Equilibrium without bank runs

• Assume that households do not anticipate the bank run (no uncertainty in Rt+1 ). • FOC of HHs Et [Λt,t+1 ]Rt+1 = 1

(28)

h Et [Λt,t+1 Rt+1 ]

(29)

=1

Ch

h ≡ where Λt,t+1 ≡ β C ht is the stochastic discount factor and Rt+1 t+1

is the marginal return on capital when household manage it. • Banker’s problem in recursive form:   Vt = max Et β(1 − σ)cbt+i + βσVt+1 , {ktb ,dt }

Vt ≥ θQt ktb Qt ktb = dt + nt nt+1 = (Zt+1 + Qt+1 )ktb − Rt+1 dt cbt = nt . 29

subject to

Zt+1 +Qt+1 Qt +f 0 (Kth )

• Solve banker’s problem with Guess and verify: Vt = ψt nt . • Growth rate of net worth is given by nt+1 Zt+1 + Qt+1 Qt ktb dt = − Rt+1 nt Qt nt nt b = (Rt+1 − Rt+1 )φt + Rt+1 , b where Rt+1 ≡

Zt+1 +Qt+1 Qt

and φt ≡

Qt ktb nt

is the leverage multiple.

• Since Vt = βEt [(1 − σ)nt+1 + σVt+1 ],   nt+1 nt+1 ψt = βEt (1 − σ) + σψt+1 nt nt  b = βEt (1 − σ + σψt )[(Rt+1 − Rt+1 )φt + Rt+1 ] = µt φt + νt b where µt ≡ βEt [Ωt+1 (Rt+1 − Rt+1 )], νt ≡ βEt [Ωt+1 ]Rt+1 , and Ωt+1 ≡ (1 − σ + σψt+1 ).

• In the same way, the IC constraint can be rewritten as θφt ≤ ψt = µt φt + νt ⇐⇒ (θ − µt )φt ≤ νt • Banker’s Bellman equation is then: ψt = max{µt φt + νt }, φt

subject to

(θ − µt )φt ≤ νt . • As long as µt ∈ (0, θ), the constraint is always binding, and the solution is given by ψt θ    ψt b ψt = βEt (1 − σ + σψt+1 ) (Rt+1 − Rt+1 ) + Rt+1 θ φt =

(30) (31)

Qt ktb = φt nt . • Since φt is same for all bankers, we can aggregate across bank to obtain Qt Ktb = φt Nt . Since Qt Ktb = Nt + Dt , Dt = (φt − 1)Nt .

30

(32)

• Aggregate law of motion for bank capital: the fraction σ will survive and 1 − σ enters with wb . b Nt = σ[(Zt + Qt )Kt−1 − Rt Dt−1 ] + (1 − σ)wb .

(33)

• Exiting bankers will consume their capital: b Ctb = (1 − σ)[(Zt + Qt )Kt−1 − Rt Dt−1 ]

(34)

• An eqm w/o bank run is (Cth , Ctb , Kth , Ktb , Dt , φt , ψt , Nt , Yt , Qt ) s.t. – HH’s optimization: (28), (29). – Banker’s optimization: (30), (31), (32), (33), (34). – Total output is the sum of output from capital, HHs endowment, and bank endowment: Yt = Zt + Zt W h + (1 − σ)wb . – Market clearing: Yt = f (Kth ) + Cth + Ctb 1 = Kth + Ktb .

7.3

Unanticipated Bank runs

• Let Q∗t denote the price of capital under the bank run . • In this paper, a bank run is characterized by Negative net worth: (Zt + b Q∗t )Kt−1 < Rt Dt−1 . • In a bank run, consumers receive some fraction of promised return, Rt = ¯ t , where xt = (Zt + Q∗t )K b /Rt Dt−1 . xt R t−1 • Assume that 1. New bankers cannot operate in the bank run period, 2. Right after the crisis, net worth is given by Nt∗ +1 = (1 − σ)wb + σ(1 − σ)wb . • The bank run is a self-fulfilling event: 1. Suddenly HHs think that a bank run will occur. 2. Since bankers are defaulting, HHs withdraw deposit from banks: Dt = 0. 3. Bankers sell all the capital to HHs (Kth = 1) to repay deposits. 4. The capital price Qt decreases because HH’s technology is less efficient. 31

5. A decrease in Qt reduces the net worth of bankers and eventually makes it negative, so bankers actually default. • If Dt−1 = 0, a bank run cannot happen → possibility of bank run depends on the strength of banker’s balance sheet. • A bank run may happen if xt =

b (Zt + Q∗t ) Q∗t Kt−1 Rb∗ φt−1 = t < 1. ∗ Qt Rt Dt−1 Rt φt−1 − 1

That is, a bank run may happen if 1. The return on bank assets, Rtb∗ , in a bank run is high relative to deposit rates, Rt , and/or 2. Deposits, Dt−1 , are small relative to total assets under bank runs, Q∗t Kt−1 .

7.4

Parameterization

• One period= one quarter. • β = 0.99 and ρz = 0.95. • Six parameters specific to this model: (θ, wb , σ, α, W h , Z). • Targets: φss = 10, b Rss

− R = 0.01, 1 = Expected horizon of bankers = 20 (5 years) 1−σ ZW h = 3Z Qt = 1

7.5

Algorithm

• Paper computes impulse responses to shocks in Zt as the nonlinear perfect foresight solution. • Assume that the economy is back to the steady state after period T . • In the period of bank run, we know Kth∗ = 1 and Dt∗ = 0. Given this, we can solve the equilibrium after period t∗ backwards. 1. Given Kth∗ = 1, Dt∗ = 0, and XT = Xss , solve the transition path to the steady state. 2. Given {Xt }Tt=t∗ +1 , compute Xt∗ (especially Q∗t∗ ). ∗

t −1 3. {Xt }t=0 is the same as in no bank run case.

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Figure 1: Impulse response: no bank run

7.6

Impulse Response Functions

• Even if depositors expect a bank run, as long as the balance sheet of bankers is healthy enough (φt−1 is low enough), then the bank run doesn’t happen. • On the other hand, capital requirements restrict loans by bankers and hence reduce output when a bank run doesn’t happen. • Suppose we implement a capital requirement: nt ≥ τ Qt ktb . As long as τ is high enough, this is always binding. So (32) will be replaced by 1 Qt Ktb = Nt . τ

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Figure 2: Impulse response: unanticipated bank run

Figure 3: Impulse response: capital requirement

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