Exam 1 on:

Credit expansion and credit misallocation

Alexander Bleck & Xuewen Liu
Journal of Monetary Economics
Graded 24 September 2026 · 8 / 10 · 3 of 25 parts
Reading note · PDF
Fable 5.1 (adjudication), GPT 6-Astra (verification), Opus 5.5 (setting the exam and grading), GPT 6-Sol (setting the exam and grading)

Table of Contents

Scorecard
1. Where the paper sits: collateral, credit traps and misallocation
2. Collateral priced by cash in the market
3. When more credit raises the interest rate
4. The crowding-out spiral
5. Misallocation and the optimal size of the injection
6. Carrying the numerical example
References

Scorecard

Question	Parts graded	Score
1. Where the paper sits: collateral, credit traps and misallocation	3 of 3	8 / 10
2. Collateral priced by cash in the market	0 of 5	—
3. When more credit raises the interest rate	0 of 4	—
4. The crowding-out spiral	0 of 4	—
5. Misallocation and the optimal size of the injection	0 of 5	—
6. Carrying the numerical example	0 of 4	—
Total	3 of 25	8 / 10

Scores are out of the points attempted so far; the exam carries 97 in all. Parts not yet answered are marked as such, not as wrong.

He has the institutional reading of the model right: 1b reproduces the whole Hart-Moore chain — unverifiable $C$ , liquidation threat, full borrower bargaining power, $B^*=P/(1+r)$ with the cutoff the right way round — and 1c does the algebra of (4) cleanly, including the cancellation of $B$ . The weak spot is the step from a correct derivation to what it implies: in 1c he turns the $B$ -independence of (4) into a claim that $B^*$ screens on project feasibility, when the point is the opposite ( $C$ and $I$ are identical across entrepreneurs, so $B^*$ rations on collateral alone), and he calls $I$ idiosyncratic when it is the aggregate shock. In 1a he names heterogeneous asset specificity but not the shared credit market with a single $r$ that converts Sector 2’s boom into Sector 1’s loss, and he assigns Benmelech-Bergman a cost on banks where the paper says harmless. Two things to revisit: what is heterogeneous across entrepreneurs versus across sectors (1a, 1c), and footnote 7’s reason for rejecting $T_2$ debt — capacity, $P>X$ , not timing (1b).

Margin notes on your answers: ✗ an error, ✓ done well, + extra context.

1. Where the paper sits: collateral, credit traps and misallocation

The paper builds on Kiyotaki–Moore (1997) collateral constraints, contrasts itself with the credit traps of Benmelech and Bergman (2012), and judges outcomes by the Hsieh–Klenow (2009) / Restuccia–Rogerson (2008) notion of misallocation. Dates $T_0,T_1,T_2$ , no discounting. Each entrepreneur owns a project that pays a non-pledgeable $C$ at $T_1$ only if an extra $I<C$ is invested at $T_0$ , and $\tilde x\in\{u,d\}$ at $T_2$ , of which an outsider who seizes the asset recovers only $X\le d$ (asset specificity). An entrepreneur with internal capital $A$ needs $B=I-A$ from a bank at net rate $r$ , repaid at $T_1$ ; the asset can be sold in a $T_1$ secondary market at price $P$ . Two sectors $i=1,2$ share one bank credit market; in Section 4.2, Sector 1 has $C_1>C_2$ and $X_1<X_2$ .

a) Credit traps versus crowding out

a) · 3 points

In Benmelech and Bergman, additional liquidity injections can fail to raise collateral values. In two or three sentences, state what that paper implies about the cost of excessive injections, and what Bleck and Liu add that overturns it. Name the one modelling ingredient responsible.

Hint

Look at how the introduction describes injections in Benmelech–Bergman: ‘ineffective but ___’.

Your answer · 2 / 3 (partial) · submission 1, 2026-09-24

Benmelech-Bergman describe excessive liquidity as mostly harmless to the (production) economy, as it doesn’t raise collateral values and thus form credit traps. The cost of liquidity, in this case, would be on the banks that finance such liquidity-trapped firms. ✗ Not in Benmelech and Bergman (2012): in a credit trap the injection simply fails to expand credit, so no one bears a cost — the paper’s words are “ineffective but harmless to the economy” (p. 28). Inventing a cost on banks blurs the very contrast the question asks for.

Instead Bleck-Liu implement a 2-sector model, so there is a crowding-out effect of credit acquirement, and it’s the high-friction sector that bear the brunt. More specifically, he microfounded the mechanism by introducing the borrowing loan $X$ ✗ $X$ is not a loan: it is the pledgeable part of the $T_2$ cash flow $\tilde x$ — what an outside investor can recover by seizing and operating the asset, i.e. the measure of asset specificity (Section 2.1). Buyers can borrow $X$ per asset at $T_1$ , but that is a consequence of the pledgeability, not its definition. which measures asset specificity and a threshold $\underline{X}$ that separates these two sectors ✗ Heterogeneous $X$ is only half the ingredient. Without the integrated bank credit market and the single rate $r=P_i/B^*_i-1$ for both sectors (eq. (5a), Section 4.1), Sector 2’s price boom could not raise Sector 1’s cost of credit and the crowding-out would vanish. . The case where $X \le \underline{X}$ and price $P(Q) = dd\mathbb{E}^L(\tilde{x})$ resembles the credit traps ✓ Right, and worth keeping: Proposition 3’s $X\le\underline{X}$ region, where $P(Q)=\mathbb{E}^L(\tilde x)$ for every $Q$ , is Benmelech and Bergman (2012)’s credit trap nested inside this model — which is why Sector 1 is the one that gets drained. ( $Q \in [0, \overline{Q}]$ ).

Model answer

In Benmelech–Bergman a credit trap makes extra injections ineffective but harmless: if collateral values do not respond, the injection does not expand credit, and nothing is lost. Bleck–Liu show that excessive injections can be harmful (and ‘beyond inflation’, since everything is in real terms): they misallocate credit across sectors and so hurt the aggregate economy. The ingredient is a second sector with lower asset specificity (higher $X$ ) that shares the same credit market, so that its cost of credit moves one-for-one with Sector 1’s (a single rate in the baseline). Its collateral price responds strongly to liquidity, pushes up the common $r$ , and drains credit out of the sector whose collateral value is stuck. With one sector, or with segmented credit markets whose rates do not co-move, the harm disappears. Exact equality of the two rates is not what matters: Section 5.3 shows that a constant spread $r_1-r_2=\Delta r$ leaves the results qualitatively unchanged.

In the paper: Section 1, p. 28; Footnote 2, p. 28; Section 5.3, p. 37.

b) Why debt capacity equals P

b) · 4 points

Using the Hart–Moore (1994) logic the paper invokes, explain why the most an entrepreneur can credibly promise to repay at

T_1

is the secondary-market price

P

, and derive from it the marginal borrower

B^*

of eq. (1). Say in one sentence why the bank does not instead lend against

X

Hint

What is the bank’s outside option in a renegotiation at

T_1

, and who has the bargaining power?

Your answer · 3.5 / 4 (partial) · submission 1, 2026-09-24

The project’s cash flow $C$ is unverifiable at lending time $T_0$ and thus uncontractible. The only thing that enables a contract is a collateral asset (the project itself). If the entrepreneur does not repay, the bank exercise its outside option, which is to liquidate the project to sell in the secondary market at $T_1$ . The entrepreneur has full bargaining power ✓ Clean statement of the Hart and Moore (1994) step the paper invokes: the borrower concedes exactly the bank’s outside option, so the pledge is capped at the resale value, not at $\mathbb{E}^H(\tilde x)$ or the borrower’s own valuation. , so he would not pay more than $P$ at $T_1$ . If banks rationally anticipate the collateral value of $P$ at $T_1$ , they would grant a loan to an entrepreneur with maximum amount $P/(1+r) := B^*$ + Writing the constraint first, $B(1+r)\le P$ , makes the discounting explicit and is the form the paper inverts into eq. (1); it is also the form you reuse in Proposition 4’s $1+r=P(Q)/B^*(Q)$ . . $B^*$ is a threshold at which entrepreneur whose borrowing needs $B < B^*$ proceed to undertake liquidity investment.

The bank does not lend against $X$ because $X$ materialized only at $T_2$ which is too late for a bank to commit to ✗ The obstacle is size, not commitment. Footnote 7 (p. 29) says long-term debt maturing at $T_2$ is available but dominated because $P>X$ — it raises strictly less external finance, since $P\ge\mathbb{E}^L(\tilde x)=d+\theta_L(u-d)\ge d\ge X$ ; it becomes infeasible for all entrepreneurs only if the support of $B$ were $[X,I]$ . , instead of lending against $P$ where it can collateralize at $T_1$ .

Model answer

$C$ is unverifiable, so it cannot be contracted on. The only enforcement device is the threat to seize the project at $T_1$ and sell it in the secondary market, which yields $P$ . In renegotiation the entrepreneur has full bargaining power, so he offers the bank exactly its outside option and never repays more than $P$ . A loan $B$ at rate $r$ requires $B(1+r)\le P$ , so banks lend at most $P/(1+r)$ and the marginal financed entrepreneur is

B^*=\frac{P}{1+r}.

Entrepreneurs with $B\le B^*$ can be financed; those with $B>B^*$ cannot, whatever their project’s npv.

The bank does not lend against $X$ because $X$ is only what an outsider recovers at $T_2$ ; since $P\ge\mathbb{E}^L(\tilde x)=d+\theta_L(u-d)\ge d\ge X$ (eq. (3)), short-term debt secured by the $T_1$ resale value raises weakly more than long-term debt due at $T_2$ (footnote 7). If the support of $B$ were $[X,I]$ , long-term debt alone would be infeasible for everyone.

In the paper: Section 2.3, p. 29; Footnote 7, p. 29; eq. (1), p. 31; eq. (3), p. 31.

c) Participation cancels internal wealth

c) · 3 points

Compare an entrepreneur’s

T_1

payoff from borrowing

B=I-A

and investing with the payoff from depositing

A

in a bank. Derive the participation condition (4) and say whether it depends on

B

; what does that imply about what

B^*

sorts on?

Your answer · 2.5 / 3 (partial) · submission 1, 2026-09-24

Payoff if the entrepreneur borrow: $C - B(1+r)$
Payoff if he deposit the capital in banks: $A(1+r)$

The entrepreneur only invest if $C - B(1+r) \ge A(1+r) \Rightarrow C \ge I(1+r)$ . This doesn’t depends on $B$ ✓ Correct, and the reason is worth naming: the deposit interest forgone on own funds, $A(1+r)$ , exactly offsets the interest saved on the smaller loan, so the wealth distribution drops out of participation entirely. .

Thus, $B^*$ only sorts on entrepreneur with feasible projects ✗ Backwards. In the baseline every entrepreneur has the same $C$ and the same $I$ , so (4) either holds for all of them or for none and sorts nobody; $B^*$ is a pure financing cutoff, and those with $B>B^*$ are rationed by collateral value despite an identical, positive npv. (their cash flow is enough to offset their idiosyncratic shock $I$ ✗ $I$ is the aggregate liquidity shock — the same amount for every entrepreneur (Section 2.2). What is idiosyncratic is internal capital $A$ , hence $B=I-A$ with density $f(B)$ on $[0,I]$ . )

Model answer

Investing yields $C-B(1+r)$ at $T_1$ ; depositing yields $A(1+r)=(I-B)(1+r)$ . The difference is

C-B(1+r)-(I-B)(1+r)=C-I(1+r),

so the entrepreneur invests iff $C-I(1+r)\ge0$ , i.e. the project’s npv at the market rate is non-negative. The condition is independent of $B$ : the forgone deposit income on own funds exactly offsets the interest saved on a smaller loan. Hence every entrepreneur wants to invest when (4) holds, and $B^*$ is a pure financing cutoff, not a profitability ranking: those with $B>B^*$ are rationed by collateral, not by return. Footnote 9 adds that (4) also guarantees every borrower can repay at $T_1$ , so there is no default.

In the paper: eq. (4), p. 31; Footnote 9, p. 31; Proposition 1, p. 31.

2. Collateral priced by cash in the market

Notation (one sector). A unit mass of entrepreneurs; entrepreneur $B$ needs external funds $B=I-A$ , with $B\sim f$ on $[0,I]$ and cdf $F$ . Banks lend at net rate $r$ (repaid at $T_1$ ) against the $T_1$ resale price $P$ ; the government injects loanable funds $Q\in[0,\overline{Q}]$ . At $T_1$ a fraction $\pi$ of entrepreneurs (drawn independently of $B$ ) turn optimistic and value the asset at $\mathbb{E}^H(\tilde x)=u\theta_H+d(1-\theta_H)$ ; the rest value it at $\mathbb{E}^L(\tilde x)=u\theta_L+d(1-\theta_L)<\mathbb{E}^H(\tilde x)$ . No short sales; only insiders can operate the asset, so only optimists buy; a buyer can borrow $X$ against each asset he holds. Write $K\equiv\int_0^I (I-B)f(B)\,dB$ .

The one-sector equilibrium $\{B^*,P,r\}$ solves

B^*=\frac{P}{1+r}\quad(1),\qquad \int_{B^*}^I (I-B)f(B)\,dB+Q=\int_0^{B^*}Bf(B)\,dB\quad(2),

P=\begin{cases}\mathbb{E}^H(\tilde x) & \Gamma>\mathbb{E}^H(\tilde x)\\ \Gamma(B^*,r) & \Gamma\in[\mathbb{E}^L(\tilde x),\mathbb{E}^H(\tilde x)]\\ \mathbb{E}^L(\tilde x) & \Gamma<\mathbb{E}^L(\tilde x)\end{cases}\quad(3),

\Gamma(B^*,r)=\frac{\pi\left\{\int_0^{B^*}[C-B(1+r)]f(B)\,dB+\int_{B^*}^I(1+r)(I-B)f(B)\,dB\right\}+X}{1-\pi}=\frac{\pi\,[C\,F(B^*)-Q(1+r)]+X}{1-\pi},

where the second form uses (2), together with the participation condition $C-I(1+r)\ge 0$ (4).

a) Bank clearing pins the cutoff

a) · 3 points

Rewrite the credit-market clearing condition (2) as a single equation in

B^*

and derive

dB^*/dQ

. Sign it, and say in one sentence why

B^*

is pinned down by

Q

alone, without reference to

P

r

Hint

Add

\int_0^{B^*}(I-B)f\,dB

to both sides.

Not yet answered.

b) Reading the price formula

b) · 4 points

Explain the economics of

\Gamma

: (i) why the buyers’ own cash enters multiplied by

\pi

; (ii) why the borrowing term is

X

and not

\pi X

; (iii) why the denominator is

1-\pi

; and (iv) why the price is truncated at

\mathbb{E}^L(\tilde x)

and

\mathbb{E}^H(\tilde x)

rather than always equal to

\Gamma

Hint

After trade, who holds all the assets, and how much can be borrowed against each one?

Not yet answered.

c) The price-trap cutoff

c) · 5 points

Eliminate

r

from

\Gamma

in the interior region to write

\Gamma

as a function of

Q

alone. Then derive an expression for the cutoff

\underline{X}

of Proposition 3, below which

P=\mathbb{E}^L(\tilde x)

for every

Q\in[0,\overline{Q}]

, and say why the cutoff is evaluated at

Q=\overline{Q}

Hint 1

Use

1+r=\Gamma/B^*

from (1) when

P=\Gamma

Hint 2

Which

Q

gives the highest

\Gamma

Not yet answered.

d) Lemma 2 by fixed point

d) · 6 points

Prove Lemma 2. In the interior region (

P=\Gamma

), derive

d\Gamma/dQ

and show that

C-I(1+r)>0

is sufficient for

d\Gamma/dQ>0

. Work from the total derivative

\frac{d\Gamma}{dB^*}=\frac{\partial\Gamma}{\partial B^*}+\frac{dr}{dB^*}\frac{\partial\Gamma}{\partial r}

with

r=\Gamma/B^*-1

Hint 1

Compute

\partial\Gamma/\partial B^*

from the integral form and

\partial\Gamma/\partial r

from the integral form too, then use (2) to simplify the latter.

Hint 2

d\Gamma/dB^*

appears on both sides; solve for it.

Not yet answered.

e) What moves collateral value

e) · 3 points

Interpret the two terms in the numerator of (A5),

[C-I(1+r)]f(B^*)

and

(1+r)Q/B^*

: what economic channel does each capture when one more unit of liquidity enters the sector?

Hint

Think about one more entrepreneur switching from depositing to investing, and then about what a fall in

r

(at a given price) does to what the private sector owes the banks.

Not yet answered.

3. When more credit raises the interest rate

One-sector economy as in Question 2. The equilibrium rate satisfies $1+r=P(Q)/B^*(Q)$ , where $B^*(Q)$ solves $I\,F(B^*)=K+Q$ and $P(Q)$ is the truncated cash-in-the-market price of (3). In the interior region $P=\Gamma$ and, from the proof of Lemma 2,

\frac{d\Gamma}{dB^*}=\frac{[C-I(1+r)]f(B^*)+(1+r)\frac{Q}{B^*}}{\frac{1-\pi}{\pi}+\frac{Q}{B^*}}.\qquad\text{(A5)}

Proposition 4: for $X\le\underline{X}$ , $r(Q)$ is strictly decreasing; for $X>\underline{X}$ , under some $f$ and parameters, $r(Q)$ is $\cup$ -shaped with a minimum $r_{\min}$ at an interior $Q$ . Assume $C>I(1+r)$ wherever a strict sign is needed.

a) Rate with a trapped price

a) · 3 points

Suppose

X\le\underline{X}

so that

P\equiv\mathbb{E}^L(\tilde x)

[0,\overline Q]

. Derive

dr/dQ

and sign it. Why does the sign need no restriction on the density beyond

f(B^*)>0

Not yet answered.

b) Condition for a rising rate

b) · 6 points

In the interior region, derive $dr/dB^*$ from (A5) and show that

\frac{dr}{dQ}>0\iff f(B^*)\left[\frac{C}{1+r}-I\right]>\frac{1-\pi}{\pi}\qquad(\text{given }C>I(1+r),\ Q\ge0).

Hint 1

Start from

r=\Gamma/B^*-1

and put everything over the denominator of (A5).

Hint 2

Use

\Gamma=(1+r)B^*

at the end.

Not yet answered.

c) Why a thick density matters

c) · 4 points

Explain in words why

r

can rise with

Q

only when

f(B^*)

is high. Use the decomposition

d\ln(1+r)=d\ln P-d\ln B^*

and say how many new projects a unit of

Q

funds, independent of

f

Hint

Differentiate (2): what is

dF(B^*)/dQ

Not yet answered.

d) The two roles of pi

d) · 3 points

(i) Take the limit

\pi\to0

(other parameters fixed,

X\le d

\theta_L>0

): what happens to the equilibrium price and to the shape of

r(Q)

? (ii) Using the condition in 3b, how does a higher

\pi

affect the possibility of a rising branch? Then explain why the paper nonetheless keeps requiring ‘

\pi

sufficiently small’.

Hint

Look at

\Gamma(Q)=\frac{\pi CF+X}{(1-\pi)+\pi Q/B^*}

\pi\to0

, and recall the truncation in (3).

Not yet answered.

4. The crowding-out spiral

Two sectors share one integrated, competitive bank credit market at $T_0$ : loans never default (by (4)), so there is a single risk-free rate $r$ for both sectors, and a non-investing entrepreneur in one sector can end up funding an investor in the other. The $T_1$ asset markets are segmented (only a sector’s own insiders can buy its assets), so $P_1\ne P_2$ in general. Let $Q_i$ be the net inflow of outside liquidity into sector $i$ (possibly negative); the government sets $Q$ but not the split. Solving each sector’s one-sector equilibrium given $Q_i$ yields a curve $r_i(Q_i)$ ; the two-sector equilibrium (5a)–(5e) requires

r_1(Q_1)=r_2(Q_2)=r,\qquad Q_1+Q_2=Q.

In the main configuration Sector 1 has $X_1\le\underline{X}$ , so $r_1$ is strictly decreasing, and Sector 2 has $X_2>\underline{X}$ with $r_2$ $\cup$ -shaped, minimum $r_{\min}$ at $Q_2^{\min}$ . Near a given equilibrium write $a\equiv -r_1'(Q_1)$ and $b\equiv r_2'(Q_2)$ ; in the main configuration $a>0$ and $b$ can take either sign.

a) Why Q1 peaks at rmin

a) · 3 points

Explain why the injection

Q^*

that maximises

Q_1

(Proposition 6) is the one that minimises the equilibrium interest rate, and write

Q^*

in terms of

r_{\min}

and the two inverse functions.

Hint

Sector 1’s inflow is a decreasing function of the common rate alone.

Not yet answered.

b) Local comparative statics

b) · 5 points

Differentiate the two equilibrium restrictions to obtain

dQ_1/dQ

dQ_2/dQ

and

dr/dQ

in terms of

a

and

b

, on a branch where

b<a

. Use them to characterise the allocation effect (

b<0

) and the crowding-out effect (

0<b<a

) of Proposition 7.

Hint

Totally differentiate each of the two restrictions in turn; remember

r_1'=-a

and that the split

dQ_1,dQ_2

must add up to

dQ

Not yet answered.

c) The spiral as geometric series

c) · 4 points

The paper describes the move to a new equilibrium after an extra

\Delta Q

beyond

Q^*

as a spiral: the new liquidity enters Sector 2, raises the rate, squeezes liquidity out of Sector 1, which flows to Sector 2, and so on. Formalise this spiral to first order as a sequence of rounds, sum it, and show that it reproduces the 4b multiplier

\Delta Q_2/\Delta Q=a/(a-b)

. State the condition for the spiral to converge.

Hint

Round 0:

\Delta Q

lifts

r_2

b\Delta Q

. How much must leave Sector 1 for

r_1

to match?

Not yet answered.

d) Crowding out with falling rates

d) · 4 points

The online appendix (Section 4.1, Fig. B1) constructs a configuration with

r_1'>0

r_2'<0

and

r_1'+r_2'>0

. Using the 4b formulas in raw slopes, sign

dQ_1/dQ

dQ_2/dQ

and

dr/dQ

there, say how the appendix engineers the two slopes, and state what this establishes about the interest-rate signature of crowding out.

Hint

Which price bound makes a sector’s rate fall for sure? Which condition on

f

makes it rise?

Not yet answered.

5. Misallocation and the optimal size of the injection

Section 4.2: the sectors now differ in both cash flow and asset specificity, $C_1>C_2>I$ and $X_1<X_2$ ( $X_1\le\underline X<X_2$ ), with the same $I$ , $f$ , $\tilde x$ and $\pi$ . The equilibrium is (6a)–(6e): as in Question 4, with $P_i=p(\tilde x,B_i^*,C_i,X_i,r)$ and $C_i\ge I(1+r)$ . Let $K\equiv\int_0^I(I-B)f(B)\,dB$ , so credit clearing in each sector is $I\,F(B_i^*)=K+Q_i$ , and write $F_i\equiv F(B_i^*)$ . Program 1 maximises $Q_1$ (equivalently $B_1^*$ ) over $Q$ ; its solution is $Q^*$ . Program 2 maximises the aggregate surplus of liquidity investments

W=(C_1-I)F_1+(C_2-I)F_2.

The aggregate return on capital is $\bar R=\dfrac{C_1F_1+C_2F_2}{I\,F_1+I\,F_2}$ and $R_i\equiv C_i/I$ .

a) Higher returns lose the race

a) · 4 points

The paper appeals to Hsieh–Klenow and Restuccia–Rogerson: efficiency requires equalising marginal returns across uses, and here the return in sector

i

is the constant

R_i=C_i/I

with

R_1>R_2

. Say what the efficient allocation of a given pool of loanable funds looks like, and explain why the competitive bank allocation need not deliver it, even before any crowding out.

Hint 1

With constant, unequal returns, equalisation is impossible; think about a corner.

Hint 2

What does a bank observe and contract on:

C_i

P_i

Not yet answered.

b) Surplus as scaled average return

b) · 4 points

Show that

W=C_1F_1+C_2F_2-2K-Q

, that

\bar R=\omega R_1+(1-\omega)R_2

with

\omega=F_1/(F_1+F_2)

, and hence that

W=(\bar R-1)(Q+2K)

. Then derive the first-order condition of Program 2 in terms of

\bar R(Q)

(the appendix’s (A10)).

Hint

Sum the two credit-clearing conditions.

Not yet answered.

c) The welfare trade-off at the margin

c) · 4 points

Starting from

W=(C_1-I)F_1+(C_2-I)F_2

, derive

dW/dQ

in terms of

dQ_1/dQ

. Give the exact local condition for an extra unit of injection to lower

W

, and relate it to the two-force decomposition the paper gives after Proposition 10.

Hint

Use

IF_i=K+Q_i

for each sector before differentiating, and

dF_1+dF_2=dQ/I

Not yet answered.

d) Program 2 versus Program 1

d) · 4 points

Let

Q_W

denote the maximiser of

W

over the range of

Q

on which neither sector is fully funded. Using 5c and the local derivatives of 4b, show that

Q_W\ge Q^*

, characterise when

Q_W>Q^*

strictly and when

Q_W=Q^*

is possible, and use this to explain the appendix’s ‘coincidence’ that both programs give

Q=1.1787

in the Proposition 8 example.

Hint

What is

dQ_1/dQ

Q^*

r_2

is differentiable at its minimum? What if

r_2

has a kink there?

Not yet answered.

e) Capping speculative leverage

e) · 3 points

The conclusion suggests regulating leverage in the more speculative industry. In the model, suppose a regulator caps the margin a Sector 2 buyer can raise per asset, effectively lowering

X_2

to at most Sector 2’s trap cutoff

\underline X

while

C_2

is unchanged. What happens to

r_2(Q_2)

, to the relation between

Q_1

and

Q

, and to the optimal injection? Is there any surplus cost within the model?

Hint

Proposition 3 and 3a apply to Sector 2 once

X_2\le\underline X

Not yet answered.

6. Carrying the numerical example

Appendix A’s example (Propositions 3–7): $I=1$ , $C=2.2$ in both sectors, $\pi=0.4$ , $\mathbb{E}^H(\tilde x)=1.3668$ , $\mathbb{E}^L(\tilde x)=1.0334$ , $f(B)=\log\frac{1}{1-B}$ on $[0,1)$ , $X_1=0.05$ , $X_2=0.35$ , $\overline Q=1.5$ . The appendix reports $\underline X=0.0501$ , $r_{\min}=0.3270$ for Sector 2, and $Q^*=0.6113$ with $Q_1=0.1950$ , $Q_2=0.4163$ . A calculator (or a few lines of code) is allowed.

a) Distribution and inflow bounds

a) · 2 points

Derive the cdf

F(B)

, the quantity

K=\int_0^1(1-B)f(B)\,dB

, and the range of net inflows

Q_i

a sector can absorb or release. Write the credit-clearing condition that pins

B_i^*

for a given

Q_i

Hint

Substitute

u=1-B

, then integrate by parts.

Not yet answered.

b) Reproducing the trap cutoff

b) · 3 points

Using

\Gamma(Q)=\dfrac{\pi C F(B^*)+X}{(1-\pi)+\pi Q/B^*}

, reproduce the appendix’s cutoff

\underline X=0.0501

and Sector 2’s maximum price

P_2=1.3667

. What does the second number tell you about how

\mathbb{E}^H(\tilde x)

was chosen?

Hint

Evaluate at the per-sector maximum inflow, where

B^*=1

Not yet answered.

c) Checking the reported optimum

c) · 5 points

(i) Compute

B_1^*

and the common rate at

Q_1=0.1950

and confirm that it equals

r_{\min}

. (ii) Compute

B_2^*

\Gamma_2

and

r_2

Q_2=0.4163

. (iii) Evaluate

f(B_2^*)[C/(1+r)-I]

there, compare it with

(1-\pi)/\pi

, and explain why the comparison confirms that

Q_2=0.4163

is (approximately) where

r_2

bottoms out.

Hint

Solve

B+(1-B)\ln(1-B)=0.445

by bisection; the root is between 0.75 and 0.80.

Not yet answered.

d) When Sector 2 is saturated

d) · 5 points

Sector 2 cannot absorb more than

Q_2=0.75

. (i) Find the aggregate injection

Q_{\text{sat}}

at which Sector 2 becomes fully funded in this example. (ii) Describe the equilibrium for

Q>Q_{\text{sat}}

: which condition in (5a)–(5e) must be relaxed, and how do

Q_1

and

r

move? (iii) What does this imply for Proposition 6’s claim that

Q^*=0.6113

uniquely maximises

Q_1

[0,\overline Q]

with

\overline Q=1.5

Hint

Q_2=0.75

r_2=P_2/1-1

. Which

Q_1

gives Sector 1 the same rate?

Not yet answered.

References

Benmelech, Efraim, and Nittai K. Bergman. 2012. “Credit Traps.” American Economic Review 102 (6): 3004–3032. doi:10.1257/aer.102.6.3004.

Hart, Oliver, and John Moore. 1994. “A Theory of Debt Based on the Inalienability of Human Capital.” Quarterly Journal of Economics 109 (4): 841–879. doi:10.2307/2118350.

Hsieh, Chang-Tai, and Peter J. Klenow. 2009. “Misallocation and Manufacturing TFP in China and India.” Quarterly Journal of Economics 124 (4): 1403–1448. doi:10.1162/qjec.2009.124.4.1403.

Kiyotaki, Nobuhiro, and John Moore. 1997. “Credit Cycles.” Journal of Political Economy 105 (2): 211–248. doi:10.1086/262072.

Restuccia, Diego, and Richard Rogerson. 2008. “Policy Distortions and Aggregate Productivity with Heterogeneous Establishments.” Review of Economic Dynamics 11 (4): 707–720. doi:10.1016/j.red.2008.05.002.