15. Liability Side of Banking

During the Great Depression (U.S. 1929–1933), real GDP dropped 34%, prices 24%, and bank deposits 48.2%. The Fed had already been established (in the 1910s), but there was no deposit insurance at the time. We consider the bank-run problem under this background. Diamond and Dybvig (1983) provide a model with household liquidity shocks to understand how banks create liquidity for welfare improvement, and to explain a bank run as a self-fulfilling prophecy caused by no fundamental problem in banks at all. In a nutshell: a bank improves welfare by creating liquidity for consumption flexibility, but suffers from a self-fulfilling run.

15.1.1 Setup. Three periods \(T=0,1,2\). Two technologies (available to both households and banks):

• Technology A (short-term riskless saving): 1 unit deposited at \(T=0\) pays 1 unit at \(T=1\); 1 unit at \(T=1\) pays 1 unit at \(T=2\).
• Technology B (long-term risky investment): 1 unit at \(T=0\) pays \(\pi\le1\) if taken out at \(T=1\), and \(R>1\) if taken out at \(T=2\). We focus on \(\pi=1\) (then early withdrawal of the long technology equals the short one, the short technology is redundant, and all invest in the long technology).

The long technology only has liquidity risk (early \(T=1\) withdrawal destroys value from \(R\) to 1), but no uncertainty over \(R\) at all.

• The bank: takes deposits at \(T=0\), invests in the long technology, providing maturity transformation — investing short-term-liquidity deposits into long-term production.
• Household: a continuum on $[0,1]\(, each subject to an i.i.d. liquidity shock \)\theta$ (revealed at the start of \(T=1\)): with probability \(\alpha\), \(\theta=1\) (early type, enjoys only \(T=1\) consumption); with probability \(1-\alpha\), \(\theta=2\) (late type, enjoys \(T=1\) and \(T=2\)). Preferences:

$$u(c_1,c_2)=\begin{cases}u(c_1) & \text{if }\theta=1\\[2pt] \rho\,u(c_1+c_2) & \text{if }\theta=2\end{cases}$$

with \(\rho<1\), \(R\rho>1\), \(u'>0\), \(u''<0\). Each household has endowment 1 at \(T=0\). The bank announces at \(T=0\): pay \(y_1\ge1\) if withdrawn at \(T=1\), \(y_2>y_1\) at \(T=2\) if not; if \(\alpha\) proportion withdraw at \(T=1\), the bank liquidates \(\alpha y_1\) proportion of the long asset, with \((1-\alpha y_1)R\) paid at \(T=2\) to the remaining \(1-\alpha\) proportion.

15.1.2 Autarky. A household invests long itself, getting 1 if early (\(T=1\)) and \(R\) if late (\(T=2\)), with expected utility

$$\mathbb{E}[u(c_1,c_2)]=\alpha\,u(1)+(1-\alpha)\rho\,u(R)$$

This is the payoff a household takes as given; as an individual it cannot insure against the shocks.

15.1.3 With bank: liquidity creation and risk sharing. The bank has a continuum of customers, fully insuring by the LLN; with competition it offers the best feasible policy, the planner's problem:

$$\max_{c_1,c_2}\ \alpha\,u(c_1)+(1-\alpha)\rho\,u(c_2)\quad\text{s.t.}\quad \alpha c_1+(1-\alpha)\frac{c_2}{R}=1 \tag{15.1}$$

((15.1) is zero-profit break-even; no expectation since the social outcome is deterministic by the LLN.) Combining the Lagrangian f.o.c.s:

$$\frac{(1-\alpha)\rho\,u'(c_2^\star)}{\alpha\,u'(c_1^\star)}=\frac{(1-\alpha)\frac1R}{\alpha}\ \Rightarrow\ u'(c_1^\star)=\rho R\,u'(c_2^\star) \tag{15.2}$$

Since \(R\rho>1\), \(u''<0\):

$$c_1^\star

Focus on CRRA \(u(c)=\frac{c^{1-\sigma}}{1-\sigma}\) (\(\sigma>0\)). Then (15.2) gives

$$\left(\frac{c_2^\star}{c_1^\star}\right)^\sigma=\rho R\ \Rightarrow\ c_2^\star=c_1^\star(\rho R)^{\frac1\sigma} \tag{15.4}$$

Solving \(c_1^\star=\dfrac{1}{\alpha+(1-\alpha)\rho^{1/\sigma}R^{1/\sigma-1}}\) from (15.4) and (15.1). The condition \(c_1^\star>1\) is

$$c_1^\star>1\ \Leftrightarrow\ \alpha+(1-\alpha)\rho^{\frac1\sigma}R^{\frac1\sigma-1}<1\ \Leftrightarrow\ \rho^{\frac1\sigma}R^{\frac1\sigma-1}<1\ \Leftrightarrow\ \rho

Imposing the reasonable additional condition (15.5),

$$1

which is crucial for the run analysis.

Liquidity creation: the long technology's liquidity at \(T=1\) = being sold at \(T=1\) at a price close to the \(T=2\) fundamental \(R\) (with zero discount, \(R\) is the \(T=1\) fundamental price). Without a bank, the \(T=1\) price is only 1; with a bank, it is \(c_1^\star>1\) — the bank creates liquidity. Hidden individual type = the source of runs: the probability \(\alpha\) of \(\theta\) is known, but the individual shock is hidden; the bank cannot stop a \(\theta=2\) household from withdrawing \(c_1^\star>1\) at \(T=1\).

There are two possible equilibria:

• Good equilibrium: households don't run, simply follow shocks (only \(\theta=1\) withdraws at \(T=1\), \(\theta=2\) waits to \(T=2\)).
• Bad equilibrium: everyone, worried about getting their money, withdraws at \(T=1\) regardless of \(\theta\); the bank goes bankrupt, paying only a fraction of households in full while the rest suffer total loss.

The run mechanism. By (15.6) \(11\), which the bank cannot pay to all — only the first \(f=\frac{1}{c_1^\star}<1\) fraction in line get the full \(c_1^\star\), others get zero (the bank's sequential service constraint). Observing this, others' optimal action is to queue as fast as possible regardless of \(\theta\) (something beats nothing), so the run becomes a self-fulfilling prophecy = the bad equilibrium.

Two ways to rule out the bad equilibrium:

• External deposit insurance: once money is guaranteed safe as promised, a \(\theta=2\) household has no incentive to withdraw at \(T=1\), so only \(\theta=1\) withdraws at \(T=1\).
• Suspension of convertibility: the bank announces at \(T=0\) — if too many show up at \(T=1\), it pays only the first \(\alpha\) fraction, and those after get nothing until \(T=2\). Then a \(\theta=2\) household has no incentive to withdraw at \(T=1\) (knowing there's enough to pay \(c_2^\star\) at \(T=2\), unwilling to suffer the \(R\to1\) consumption loss).

15.1.6 Discussion (Jacklin 1987). Suppose there's a competitive market at \(T=1\) (after the shock) trading the demandable bank deposit. Then this market and the banking solution \((c_1^\star,c_2^\star)\) cannot co-exist — the banking solution is dominated by: investor A invests long itself at \(T=0\); if \(\theta=2\), A is happy (gets \(R>c_2^\star\)); if \(\theta=1\), A sells its \(T=2\) claim of \(R\) to \(\frac{R}{c_2^\star}>1\) "late" households and gets \(\frac{R}{c_2^\star}>1\) at \(T=1\) (better than the bank's \(c_1^\star\)). So everyone follows this, and no one deposits in the bank in equilibrium.

Other critiques of Diamond-Dybvig (1983) (discussed by Calomiris-Kahn 1991): (1) it ignores that bank failure is often not just from a liquidity shock but also fraud (e.g. banker stealing); (2) it doesn't recognize that runs are not randomly self-fulfilling but are often initiated by withdrawals of informed depositors.

Main idea: a runnable demandable deposit gives bankers the right incentive not to steal and depositors to monitor.

15.2.1 Setup. Three periods \(t=0,1,2\). The bank has one project, needs investment \(Y\) at \(t=0\), with payoff \(R^H Y\) or \(R^L Y\) only at \(t=2\); \(\tilde R\in\{R^H,R^L\}\) unobservable to the market; the banker can steal \((1-\alpha)\tilde R Y\) from the bank. \(Z\) investors: each deposits 1 at \(t=0\); at \(t=1\) each can acquire information at cost \(I\), a signal \(\sigma\in\{g,b\}\) (privately revealed), with prior \(\lambda\) for \(g\) and \(1-\lambda\) for \(b\); the probability of \(R^H\) under signal \(\sigma\) is \(\rho_\sigma\), with \(\rho_g>\gamma>\rho_b\) (\(\gamma\) the prior). An informed investor can report \(\hat\sigma=\hat g\) or \(\hat b\) (regardless of the truth), with utility \(U(\hat\sigma,\sigma)\) of reporting \(\hat\sigma\) while truly knowing \(\sigma\). In equilibrium \(K\) investors are active (acquire info), \(Z-K\) sleepy (don't), with sleepy investors treated as always reporting \(\hat\sigma=\hat g\).

15.2.2 Banker's absconding. The banker must repay \(P\) to investors at \(t=2\). If absconding with \((1-\alpha)\tilde R Y\) uncaught is profitable, he steals:

$$(1-\alpha)\tilde R Y>\tilde R Y-P\ \Rightarrow\ \tilde R<\frac{P}{\alpha Y}$$

i.e. he tends to abscond when the return \(\tilde R\) is low; all investors know this rule.

15.2.3 Constraints. Sleepy IR: \(\lambda U(\hat g,g)+(1-\lambda)U(\hat g,b)=1\); active truth-telling IR: \(\lambda U(\hat g,g)+(1-\lambda)U(\hat b,b)=1+r\); active truth-telling IC: \(\lambda U(\hat g,g)+(1-\lambda)U(\hat b,b)\ge\lambda U(\hat g,b)+(1-\lambda)U(\hat b,g)\).

15.2.4 Optimal contract. By the revelation principle and simple algebra, Calomiris-Kahn (1991) show the optimal contract satisfies: some investors choose to acquire signals while others don't; all who report \(\hat\sigma=\hat b\) get a constant payoff \(R\) at \(t=1\); all who report \(\hat\sigma=\hat g\) split the remaining payoff at \(t=2\); the bank is inefficiently liquidated if total withdrawal exceeds \(\underline N R\). This optimal contract is exactly a demandable deposit: the bank's sequential service constraint is key, providing monitoring incentives to investors; the fragility (run threat) of the demand deposit also provides anti-absconding incentives to managers.

15.3.1 Role of bank equity. Bank equity is generally a buffer absorbing loss (in Calomiris-Kahn the sleepy investors are the equity holders absorbing loss). Without project uncertainty, the potential run of demand depositors disciplines the banker, so a run is an off-equilibrium event that never happens; with project uncertainty, the bank's value fluctuates, runs can happen in down states, and bank equity absorbs loss to make runs less frequent. However, bank equity is more important than a loss absorber: the systematic risk effect takes place before bank equity is totally wiped out. Empirically, banks with low equity capital have extremely high risk premia and seldom offer loans to the real economy. Holmstrom-Tirole (1997) offers a framework to think about bank equity's effect on project investment and loan rates (real activity), more than just loss absorbing.

15.3.2 Setup. Two periods \(t=0,1\). Three risk-neutral types: households (\(t=0\) deep pocket); bankers (limited \(t=0\) endowment); entrepreneurs (unit continuum \(i\in[0,1]\), limited endowment \(A_i\)). Entrepreneurs' aggregate endowment \(K_f=\int_0^1 A_i\,dG(A_i)\) (\(G\) the asset CDF). An exogenous saving technology with gross return \(r\). Three projects all need investment \(I\), paying \(R\) w.p. \(p\) and 0 w.p. \(1-p\): good \(G\) (\(p=p_H\), zero private benefit); OK \(b\) (\(p=p_L\), private benefit \(b\)); bad \(B\) (\(p=p_L\), private benefit \(B\)). \(p_H>p_L\), \(B>b>0\). Parameter restrictions \(p_H R-rI>0\), \(p_L R+B-rI<0\): \(G\) strictly positive NPV, \(B\) strictly negative NPV (implying \(p_L R+b-rI<0\), so the OK project is also negative NPV).

Direct financing: entrepreneurs raise money directly from households; socially only positive-NPV projects are invested, so only the good project \(G\) in equilibrium. Contract \((T_0,R_f)\): \(T_0\) the household contribution, \(R_f\) the entrepreneur's payoff if success (household gets \(R-R_f\)). Constraints:

$$T_0+A\ge I \quad[\text{investment}] \tag{15.7}$$

$$p_H(R-R_f)-rT_0\ge0 \quad[\text{IR}] \tag{15.8}$$

$$p_H R_f\ge p_L R_f+B \quad[\text{IC}] \tag{15.9}$$

With \(\Delta p\equiv p_H-p_L\), (15.9) rearranges to

$$\Delta p\,R_f\ge B\ \Rightarrow\ R_f\ge\frac{B}{\Delta p} \tag{15.10}$$

(intuition: the entrepreneur must receive a not-too-small payoff from the good outcome \(R\) to have incentive not to pursue private benefit \(B\).) Let (15.7) bind, \(T_0=I-A\), into (15.8):

$$p_H(R-R_f)-r(I-A)\ge0\ \Rightarrow\ R-\frac{r}{p_H}I+\frac{r}{p_H}A\ge R_f \tag{15.11}$$

combine with IC (15.10):

$$R-\frac{r}{p_H}I+\frac{r}{p_H}A\ge\frac{B}{\Delta p}\ \Rightarrow\ A\ge I-\frac{p_H}{r}\left(R-\frac{B}{\Delta p}\right)$$

i.e. the minimum entrepreneur net worth under direct finance

$$\bar A=\underbrace{I-\frac{p_H}{r}\left(R-\frac{B}{\Delta p}\right)}_{\text{firm's agency cost}} \tag{15.12}$$

Summary: \(A_i<\bar A\) → project cannot be financed; \(A_i\ge\bar A\) → financed, entrepreneur devotes \(\bar A\) and borrows \(I-\bar A\) from the household. Intuition: \(\frac{B}{\Delta p}\) is the minimum payoff \(R_f\) to keep the entrepreneur choosing the right project, so the max pledgeable payoff to the household is \(R-\frac{B}{\Delta p}\), its discounted expected value \(\frac{p_H}{r}(R-\frac{B}{\Delta p})\) is the max the household would invest, and the entrepreneur fills the gap between \(I\) and that.

Introduce a bank as intermediary. Socially still only \(G\). Additional assumptions: if the banker monitors, the bad project \(B\) is eliminated; if not, the banker enjoys private benefit \(C>0\) and no project is eliminated (monitoring is unobservable to households, so the banker needs a monitoring incentive). Assume the monitoring cost is not too high, \(C<\frac{p_H}{\Delta p}(B-b)\) (15.13). The banker has endowment \(K_m\) (aggregate intermediary capital, treated as owned by the banker); the gross loan return \(\beta\ge r\); the bank capital market is competitive → the same rate \(\beta\), focus on one bank w.l.o.g. The bank gets \(R_m\), the household gets \(R-R_f-R_m\). Constraints:

$$I=I_f+I_m+I_h \quad[\text{investment}] \tag{15.14}$$

$$p_H(R-R_f-R_m)-rI_h\ge0\ \Rightarrow\ I_h=\frac{p_H(R-R_f-R_m)}{r} \quad[\text{household IR/break-even}] \tag{15.16}$$

$$p_H R_m\ge p_L R_m+C\ \Rightarrow\ R_m\ge\frac{C}{\Delta p} \quad[\text{bank monitoring IC}] \tag{15.17}$$

Let (15.17) bind, \(R_m=\frac{C}{\Delta p}\) (15.18). With monitoring, the entrepreneur's IC to choose \(G\):

$$p_H R_f\ge p_L R_f+b\ \Rightarrow\ R_f\ge\frac{b}{\Delta p} \tag{15.19}$$

Let it bind, \(R_f=\frac{b}{\Delta p}\) (15.20). Bank-loan demand: \(\beta I_m=p_H R_m\), by (15.18):

$$I_m(\beta)=\frac{p_H C}{\beta\Delta p} \tag{15.21}$$

Starting from (15.14) and substituting (15.21), (15.16), (15.18), (15.20) gives the minimum entrepreneur net worth

$$\underline A=\underbrace{I-\frac{p_H}{r}\left(R-\frac{b}{\Delta p}\right)}_{\text{firm's agency cost}}+\underbrace{\frac{p_H}{r}\frac{C}{\Delta p}\left(1-\frac{r}{\beta}\right)}_{\text{bank's agency cost}} \tag{15.22}$$

Summary: \(A_i<\underline A\) → cannot be financed; \(\underline A\le A_i<\bar A\) → can only borrow from the bank; \(A_i\ge\bar A\) → always borrows directly from the household (since \(\frac{B}{\Delta p}>\frac{b}{\Delta p}\), the entrepreneur keeps more under direct finance). The third term (bank agency cost) $>0$ (\(\beta\ge r\)), a necessary return for the bank to invest; higher \(C\) → larger agency cost.

Aggregate demand for bank loans \(D(\beta)\):

$$D(\beta)=\int_{\underline A(\beta)}^{\bar A}\frac{p_H C}{\beta\Delta p}\,dG(A)$$

with \(\underline A(\beta)\) from (15.22) and \(\bar A\) from (15.12). Aggregate supply: the opportunity cost of bank capital is the exogenous saving gross return \(r\), so supply is 0 when \(\betar\) only when \(D(\beta)=K_m\).

Before a shock: aggregate bank capital \(K_m\) is sufficient, \(\beta^\star=r\) is low, and \(K_0\) capital is invested by the bank. After a shock: aggregate bank capital drops from \(K_m\) to \(\hat K_m\), \(\beta^\star>r\) rises, and aggregate bank investment drops from \(K_0\) to \(\hat K_m\). Intensive margin: investment to each entrepreneur \(\frac{p_H C}{\beta\Delta p}\) drops; extensive margin: the threshold \(\underline A(\beta)\) rises, so fewer entrepreneurs are eligible for loans. So this supply-demand analysis explains the soaring loan rate and reduced loan provision in a crisis where bank capital drops significantly. Hence exogenous fluctuations in bank capital make real investment fluctuate together. This also embeds the idea of intermediary asset pricing: in a crisis, financial intermediaries are the marginal investors who determine market prices.