13. External Capital Market

Bernanke and Gertler (1989) discuss financing frictions in a macroeconomic model. They introduce the financial accelerator (balance sheet channel), a positive feedback effect:

• a positive shock makes the borrower wealthier;
• more wealth makes it easier to borrow (better balance sheet) and incentivizes the borrower to invest more efficiently;
• so the borrower will be wealthier next period, and so on repeatedly.

Before Bernanke-Gertler (1989), we take a look at a simplified version of Holmstrom and Tirole (1997), which will be used later.

13.1.1 Simplified Holmstrom and Tirole (1997).

• Two periods \(t=0,1\). Entrepreneur \(E\) has net wealth \(N\) and a scalable efficient investment opportunity (scalable = constant returns to scale in the investment payoff).
• \(E\) needs to borrow \(I-N\) from an outside creditor \(C\) at \(t=0\);
• for each dollar invested, the payoff at \(t=1\) is \(Y\) w.p. \(p\in(0,1)\) and 0 w.p. \(1-p\); discount rate \(\frac{1}{1+r}\); investment efficient, \(pY\equiv R>1+r\);
• \(E\) can shirk to enjoy private benefit \(B\) but reduces \(p\) to \(p-\Delta\) (\(\Delta\in(0,p)\)); assume shirking cannot make \(C\) break even;
• \(E\) and \(C\) contract on: investment \(I\); the profit share per unit of investment to \(C\), denoted \(X\), and to \(E\), denoted \(Y-X\).
• Entrepreneur's problem: solve for the highest investment \(I_{\max}\) such that \(C\) breaks even:

$$\max_{I,X}\ p(Y-X)\times I$$

$$[\text{IC for }E]\quad \underbrace{p(Y-X)}_{\text{not shirking}}\ge\underbrace{(p-\Delta)(Y-X)+B}_{\text{shirking}} \tag{13.1}$$

$$[C\text{ breaks even}]\quad \frac{1}{1+r}pXI\ge I-N \tag{13.2}$$

To get the highest investment we want \(X\) as high as possible, so let IC (13.1) bind:

$$p(Y-X)=(p-\Delta)(Y-X)+B\ \Rightarrow\ \Delta(Y-X)=B\ \Rightarrow\ X=Y-\frac{B}{\Delta} \tag{13.3}$$

Plug into the binding break-even (13.2):

$$\frac{1}{1+r}pXI=I-N\ \Rightarrow\ \left(1-\frac{1}{1+r}pX\right)I=N\ \Rightarrow\ I_{\max}=\frac{N}{1-\frac{1}{1+r}pX}=\frac{N}{1-\dfrac{R-\frac{pB}{\Delta}}{1+r}} \tag{13.4}$$

(using \(pY=R\); implicitly assume \(pX<11\) investors would invest infinitely). Key: the maximum investment \(I_{\max}\) is proportional to net worth \(N\) — a leverage multiplier and the micro-foundation of the financial accelerator.

Bernanke-Gertler (1989) study technology-shock propagation in an OLG economy with net wealth \(N\) endogenized.

• A single consumption good is produced by two factors: labor \(L\) and capital \(K\), with production function \(A_t F(K_t,L_t)\). TFP \(A_t\) is i.i.d. with mean \(\bar A\) (the TFP shocks are idiosyncratic, which is crucial). Capital depreciates fully in one period; labor is inelastically supplied \(L_t=1\ \forall t\).
• A riskless saving technology with net return \(r\) is available, discount factor \(\frac{1}{1+r}\).
• Each period \(t\), generation \(t\) is born with unit measure and lives two periods; each generation has two types: entrepreneur \(E\) (measure \(\eta\)), household \(H\) (measure \(1-\eta\)). Both have the same preference:

$$C_{t,1}+\frac{1}{1+r}C_{t,2}$$

• Competitive factor markets: wage \(w_t=A_t F_L(K_t,1)\) (13.5); capital payoff \(R_t=A_t F_K(K_t,1)\) (13.6).
• Friction source: \(E\) at the end of his first period can invest \(I_t^i\) to produce next-period capital \(K_{t+1}^i\) (\(i\) = individual variable):

$$\text{no shirk: }\begin{cases}K_{t+1}^i=\frac1p I_t^i & \text{w.p. }p\\[2pt] 0 & \text{w.p. }1-p\end{cases}\qquad\text{shirk (perk }B\text{): }\begin{cases}K_{t+1}^i=\frac{1}{p-\Delta}I_t^i & \text{w.p. }p-\Delta\\[2pt] 0 & \text{w.p. }1-(p-\Delta)\end{cases}$$

\(E\) wants to borrow from \(H\) to produce capital; \(H\) knows \(E\) could shirk → friction.

• Frictionless benchmark (no shirking option): \(C\) aggregates investment into all \(E\)'s. With no aggregate shock, by the law of large numbers, \(K_{t+1}=\frac1p I_t\cdot p+0\cdot(1-p)=I_t\). The riskless return \(r\) pins down the equilibrium capital by the LLN over \(A_t\):

$$1+r=\mathbb{E}[A_t F_K(K_t,1)]=\bar A F_K(K_t,1) \tag{13.7}$$

so \(K_t=K^\star\) is the same every period, regardless of the idiosyncratic TFP shocks.

• With friction: \(E\) has the shirking option. Since investment is efficient, \(E\) always maximizes investment, so by (13.4):

$$I_t^i=\frac{N_t^i}{1-\dfrac{\mathbb{E}[R_{t+1}]-\frac{pB}{\Delta}}{1+r}} \tag{13.8}$$

Aggregating over \(E\) (\(N_t=\int_{i\in E}N_t^i\,di\), with \(N_t=\eta w_t\) (13.9)), substituting (13.5), (13.6), \(\mathbb{E}[A_t]=\bar A\), and using \(K_{t+1}=I_t\) by the LLN, we get

$$K_{t+1}=\frac{\eta A_t F_L(K_t,1)}{1-\dfrac{\bar A F_K(K_t,1)-\frac{pB}{\Delta}}{1+r}} \tag{13.10}$$

where \(A_t\) in the numerator is the aggregate TFP in period \(t\). Financial accelerator and shock propagation: from (13.10), next-period aggregate capital is an increasing function of \(A_t\). If period \(t\) suffers a negative technological (MIT) shock, then \(A_t\downarrow\Rightarrow K_{t+1}\downarrow\Rightarrow N_{t+1}\downarrow\Rightarrow K_{t+2}\downarrow\Rightarrow\cdots\), the financial accelerator: any shock has an infinite sequence of impacts in all future periods (propagation). This propagation is through a labor-income channel, which is not very realistic; so we should consider other channels such as asset prices (see below).

Price could play an important role in determining debt capacity. A firm would liquidate its assets at a huge discount if it needs liquidity facing financial distress (e.g. a margin constraint or interest payment) but the price of its asset is low. The price could be much lower than the value the firm evaluates, for many reasons:

• one reason is noise trader risk (see the example below);
• another is that only the wrong hands want to take over the asset. Shleifer-Vishny (1992) suggest: the seller is financially distressed; industry outsiders have much lower value for the assets than industry insiders; and industry insiders simultaneously face financial distress, so the seller has to sell to outsiders at a much lower valuation.

Fire-sale numerical example (noise trader risk + margin constraint): three periods \(t=0,1,2\). The asset has \(t=0\) price \(p_0=11.25\), \(t=2\) price \(p_2=12\) (certain); but at \(t=1\), noise trader risk pushes the price to \(p_1=10\). (A rational investor would set \(p_1=\frac{12}{1+r_f}\) since \(p_2\) has no uncertainty; with \(p_0=11.25\) we'd have \(p_1\ge11.25\), so \(p_1=10\) implies investors are noise traders.) A fund borrows \(D\) from households to finance a total purchase \(A\). The margin constraint (leverage = asset/equity ≤ 5):

$$\frac{A}{A-D}\le 5$$

If at \(t=0\) the fund borrows at most \(D_0=9\) to buy \(A=11.25\), then \(\frac{11.25}{2.25}=5\) exactly binds. At \(t=1\) the price drops \(11.25\to10\), \(A\) drops \(11.25\to10\), and the max debt \(D_1\) satisfies \(\frac{10}{10-D_1}=5\Rightarrow D_1=8\). Since \(D_0=9>D_1=8\), the fund must repay 1, but has no cash, so it liquidates \(\frac{1}{10}\) of the asset for 1 (assuming price doesn't change as it sells); the asset becomes 9, the new debt ceiling \(\frac{9}{9-D_1'}=5\Rightarrow D_1'=7.2\), so it must sell again to repay the 0.8 debt... this repeats until it sells fraction \(\alpha\) such that

$$\frac{(1-\alpha)\times10}{(1-\alpha)\times10-(D_0-\alpha\times10)}=5\ \Rightarrow\ \frac{10-10\alpha}{10-9}=5\ \Rightarrow\ 10\alpha=5\ \Rightarrow\ \alpha=\frac12$$

i.e. the fund has to sell half its asset at \(t=1\) at an unreasonably low price, even though it knows tomorrow's (\(t=2\)) price is very high for sure. This is a fire sale. It could be even worse if the price drops further as the fund sells.

Kiyotaki and Moore (1997) consider a macroeconomic model with the fire-sale idea embedded, to study the financial-accelerator effect of a shock on the future sequence of net worth.

13.3.1 Setup.

• The economy has infinite periods \(t=0,1,2,\dots\) and only one consumption good. Two types of agents:
• Measure 1 of entrepreneurs \(E\): zero endowment every period; with capital \(k_t^E\), produces the consumption good with the CRS technology \(F(k_t^E)=\lambda k_t^E\).
• Measure 1 of farmers \(F\): large endowment \(e\) every period; with capital \(k_t^F\), produces with the concave technology \(G(k_t^F)\) (\(G'>0\), \(G''<0\), with \(G'(0)=\infty\) and \(G'(\bar k)<\lambda\) so both \(E\) and \(F\) should produce at the first best).
• Both have the same preference: \(\sum_{t=0}^\infty\beta^t c_t\).
• The economy has a fixed capital supply \(\bar k\) every period, i.e. \(k_t^E+k_t^F=\bar k\ \forall t\), and capital never depreciates.
• Denote the price of capital in period \(t\) by \(q_t\) (in consumption good); the amount \(F\) lends to \(E\) in period \(t\) by \(b_t\) (repaid in one period); a risk-free saving return \(r_f\), discount factor \(\frac{1}{1+r_f}\).
• Limited pledgeability constraint: \(E\)'s asset value at \(t+1\) is \(\lambda k_{t+1}+q_{t+1}k_{t+1}\), but the pledgeable asset is only \(q_{t+1}k_{t+1}\) (pledgeable = the part households could take away). This is reasonable: \(F\) (creditor) cannot force \(E\) (debtor) to work to produce \(\lambda k_{t+1}\); the only part the creditor could collect is the market value of the asset.

13.3.2 Collateral constraint. Since the pledgeable asset is only part of \(E\)'s total asset, \(F\) will lend no more than the pledgeable part (as collateral) to \(E\):

$$b_t(1+r_f)\le k_{t+1}^E q_{t+1}\quad\forall t \tag{13.11}$$

which is the collateral constraint.

\(F\) solves \(\sum_{t=0}^\infty\beta^t c_t\) subject to

$$c_t+q_t k_{t+1}^F+b_t=e+G(k_t^F)+q_t k_t^F+(1+r_f)b_{t-1}$$

(since there's a risk-free saving alternative, the return of lending to \(E\) should be \(r_f\)). The Lagrangian:

$$\mathcal L=\sum_{t=0}^\infty\beta^t c_t+\mu_t\big[e+G(k_t^F)+q_t k_t^F+(1+r_f)b_{t-1}-(c_t+q_t k_{t+1}^F+b_t)\big]$$

The f.o.c.s:

$$k_{t+1}^F:\quad \mu_t q_t=\mu_{t+1}\big(G'(k_{t+1}^F)+q_{t+1}\big) \tag{13.12}$$

$$c_t:\quad \beta^t=\mu_t\ \Rightarrow\ \frac{\mu_{t+1}}{\mu_t}=\beta;\qquad b_t:\quad \mu_t=\mu_{t+1}(1+r_f)\ \Rightarrow\ \frac{\mu_{t+1}}{\mu_t}=\frac{1}{1+r_f}$$

(so \(\beta(1+r_f)=1\)). Plug into (13.12):

$$q_t=\frac{1}{1+r_f}\big(G'(k_{t+1}^F)+q_{t+1}\big) \tag{13.13}$$

iterating gives

$$q_t=\sum_{j=1}^\infty\frac{G'(k_{t+j}^F)}{(1+r_f)^j} \tag{13.14}$$

Since \(G''<0\), (13.14) tells us \(F\) has a downward-sloping demand curve for capital \(k_t^F\): if \(E\) wants to sell more capital to \(F\) in period \(t+j\) (raising \(k_{t+j}^F\)), and the other \(k_{t+k}^F\) (\(k\neq j\)) don't change, then \(q_t\) drops.

First the frictionless benchmark where \(E\) has no collateral constraint. \(E\) solves \(\sum_{t=0}^\infty\beta^t c_t\) subject to

$$c_t+q_t k_{t+1}^E+(1+r_f)b_{t-1}=F(k_t^E)+q_t k_t^E+b_t$$

Define \(E\)'s net worth \(n_t=F(k_t^E)+q_t k_t^E-(1+r_f)b_{t-1}\) (13.15); note the price \(q_t\) enters net worth \(n_t\) (crucial later). With multiplier \(\xi_t\), the f.o.c.s:

$$k_{t+1}^E:\quad \xi_t q_t=\xi_{t+1}\big(F'(k_{t+1}^E)+q_{t+1}\big)=\xi_{t+1}(\lambda+q_{t+1}) \tag{13.16}$$

$$c_t:\ \beta^t=\xi_t\Rightarrow\frac{\xi_{t+1}}{\xi_t}=\beta\ (13.17);\qquad b_t:\ \xi_t=\xi_{t+1}(1+r_f)\ (13.18)\Rightarrow\frac{\xi_{t+1}}{\xi_t}=\frac{1}{1+r_f}$$

Plug into (13.16):

$$q_t=\frac{1}{1+r_f}(\lambda+q_{t+1}) \tag{13.19}$$

Combine \(F\)'s (13.13) and \(E\)'s (13.19):

$$\frac{1}{1+r_f}(\lambda+q_{t+1})=\frac{1}{1+r_f}\big(G'(k_{t+1}^F)+q_{t+1}\big)\ \Rightarrow\ \lambda=G'(k_{t+1}^F) \tag{13.20}$$

which uniquely pins down \(k_{t+1}^{F\star}\) and thus \(k_{t+1}^{E\star}=\bar k-k_{t+1}^{F\star}\). Plug into (13.14):

$$q_t=\sum_{j=1}^\infty\frac{G'(k_{t+j}^{F\star})}{(1+r_f)^j}=\sum_{j=1}^\infty\frac{\lambda}{(1+r_f)^j}=\frac{\lambda}{r_f}$$

Now the friction case where \(E\) faces the collateral constraint \(b_t(1+r_f)\le k_{t+1}^E q_{t+1}\). Suppose \(E\) starts at capital \(k_t=\hat k\) too low, so \(\hat k+\frac{b_t}{q_t}

$$\xi_t q_t<\xi_{t+1}(\lambda+q_{t+1}) \tag{13.21}$$

so \(E\) borrows to the maximum \(b_t=\frac{k_{t+1}^E q_{t+1}}{1+r_f}\). The consumption f.o.c. (13.17) still holds, \(\frac{\xi_{t+1}}{\xi_t}=\beta\), which with (13.21) implies \(q_t<\beta(\lambda+q_{t+1})\), so \(E\) would give up \(q_t\) units of consumption today to buy one unit of capital (consumption today \(c_t=0\)). Plugging into the budget constraint (\(c_t=0\), \(b_t\) at the max):

$$c_t+q_t k_{t+1}^E+(1+r_f)b_{t-1}=F(k_t^E)+q_t k_t^E+b_t\ \Rightarrow\ \left(q_t-\frac{q_{t+1}}{1+r_f}\right)k_{t+1}^E=n_t\ \Rightarrow\ k_{t+1}^E=\frac{n_t}{q_t-\frac{q_{t+1}}{1+r_f}} \tag{13.22}$$

Combine \(E\)'s (13.22) and \(F\)'s (13.13):

$$k_{t+1}^E\left[\frac{1}{1+r_f}\big(G'(k_{t+1}^F)+q_{t+1}\big)-\frac{q_{t+1}}{1+r_f}\right]=n_t\ \Rightarrow\ \frac{G'(\bar k-k_{t+1}^E)\,k_{t+1}^E}{1+r_f}=n_t \tag{13.23}$$

By (13.23), both \(G'(\bar k-k_{t+1}^E)\) and \(k_{t+1}^E\) are increasing in \(k_{t+1}^E\), so \(n_t\) is increasing in \(k_{t+1}^E\). If the starting \(\hat k\) is too low so the constraint binds, then \(n_t\) is too low, and by (13.23) \(k_{t+1}^E\) is too low so the constraint still binds at \(t+1\) — the collateral constraint always binds for \(E\).

The financial accelerator system of equations: net worth (13.24), capital choice (13.25), price (13.26):

$$n_t=(\lambda+q_t)k_t^E-(1+r_f)b_{t-1} \tag{13.24}$$

$$k_{t+1}^E=\frac{n_t}{q_t-\frac{q_{t+1}}{1+r_f}} \tag{13.25}$$

$$q_t=\sum_{j=1}^\infty\frac{G'(\bar k-k_{t+j}^E)}{(1+r_f)^j} \tag{13.26}$$

The accelerator mechanism: suppose \(E\)'s productivity drops permanently, \(\lambda\downarrow\). By (13.24) \(n_t\downarrow\); by (13.25) \(k_{t+1}^E\downarrow\); by (13.24) \(n_{t+1}\downarrow\), similarly \(n_{t+s}\downarrow\ \forall s\ge1\); by (13.25) then \(k_{t+s}^E\downarrow\ \forall s\ge1\); by (13.26) \(q_t\downarrow\); and by (13.24) \(q_t\downarrow\) makes \(n_t\) drop even further. This logic repeats, so net worth \(n_t\) and investment \(k_{t+1}^E\) drop significantly. The price \(q_t\) entering net worth is the key amplification channel.