Collateral Crises∗
Gary Gorton†

Guillermo Ordoñez‡
April, 2011PRELIMINARY AND INCOMPLETE

Abstract
Short-term, collateralized, debt is efficient if agents are willing to lend without
producing costly information about the value of the collateral. When the economy relies on this informationally-insensitive debt, information is not renewed
over time, generating a credit boom where firms with bad collateral start borrowing. We show a complex and opaque financial system is an optimal response toaccommodate increasingly worse collateral and sustain more credit.
The longer an economy remains in an information-insensitive regime, thelarger the fraction of collateral that looks similar. This creates fragility, since asmall aggregate shock to collateral values is more likely to generate a large systemic collapse in output and consumption. Furthermore, if a crisis triggers information production, it renders expansionary policies less effective in speeding up
recoveries.

∗

Preliminary and Incomplete. Please do not cite. We thank Mikhail Golosov, Guido Lorenzoni, Warren Weber and participants at Berkeley, Rutgers, Stanford, Yale and the ASU Conference on ”Financial
Intermediation and Payments” for their comments. We also thank Lei Xie for research assistance. The
usual waiver of liability applies.
†
Yale University and NBER (e-mail: gary.gorton@yale.edu)‡
Yale University (e-mail: guillermo.ordonez@yale.edu)

1Introduction

The modern financial structure has been characterized by rising levels of complexityand declining transparency. After the recent financial crisis these elements have beenat the forefront of the blame and the debate. Many regulatory changes have been
proposed to induce more information production and transparency, regarding boththe actions of financial intermediaries and the quality of financial instruments. Is thelack of transparency inherently pervasive? What are its costs and benefits? What
determines the information available in an economy? How do information dynamics
shape the size and likelihood of booms, crashes and recoveries?
In this paper we study the production and evolution of information in financial markets. We show that a complex and opaque financial system can be an endogenous
optimal response of financial intermediaries to efficiently sustain more credit in theeconomy. These potential benefits come at the cost of increasing system fragility toaggregate shocks and the consequent size of crises and credit crunches. Furthermore,
the evolution of information in the economy determines the speed of recoveries andthe effectiveness of policies to deal with crises.
Financial intermediaries perform two key functions. On the one hand, they reallocate resources in the economy, from less productive agents to more productive ones.
On the other hand, they provide efficient transactions services by designing their liabilities so that they can be used to trade without adverse selection. Examples are
pre-Civil War bank notes (in the U.S.), demand deposits, and, of late, sale and repurchase agreements (”repo”). These bank liabilities can be used directly for transactions
or can be transformed into a fixed amount of cash easily. In order to provide transaction services, bank liabilities are designed to be information-insensitive (i.e., it is
not profitable for any agent to produce information about the backing assets) so thatusers do not face adverse selection. To accomplish this, bank money is ”backed” bycollateral (designated state bonds in the case of pre-Civil War free banks, or specificbond collateral that the depositor takes possession of in repo), or by a diversified
portfolio of loans (in the case of demand deposits).
A crisis occurs when there is a shock causing bank liability holders to suspect thatthe backing collateral has deteriorated in value such that it has become informationsensitive. The bank debt holders then seek to withdraw their cash or they do not roll2

over the debt (e.g., in repo), causing a potential liquidation of the banking system.
While the U.S. had an extraordinary period of quiet with respect to banking crises
from 1934 to 2007, banking crises are, in fact, very common and very costly. Thereis a connection between recessions and financial crises. Financial crises tend to occur
near business cycle peaks, when economic activity is starting to weaken and banking
crises are often preceded by credit booms.1
In this paper we build on these micro foundations to investigate the role of suchinformation-insensitive debt in a macro economy. We do not explicitly model the
trading motive for short-term information-insensitive debt. We assume that households have a demand for such debt and, further, we assume that the short-term debt isissued directly by firms to households to obtain funds and finance efficient projects.
The debt that firms issue is backed by collateral. We show that, while the lack of
information acquisition improves trade and fuels credit booms, it also causes mostcollateral to look similar over time, increasing the potential losses in case of a crisisand slowing down a potential recovery.
The amount and nature of the borrowing depend on households’ beliefs about thevalue of the collateral, and on the households’ decisions about whether or not to produce information about the collateral value. Each firm’s collateral can be either goodor bad and its perceived quality is given by the probability that the collateral is good.
To determine the real quality of the collateral is costly. If households have incentives
to learn about the true quality of the collateral, firms may prefer to cut back on theamount borrowed to avoid costly information production, a credit constraint. However, firms whose collateral is of intermediate perceived quality may prefer to haveinformation produced, and offer an interest rate that covers the cost of information
production by households.
We show that if, the expected quality of the collateral available for a firm is low, thenfirms maximize borrowing by structuring the collateral so that is complex, increasing
the cost of information acquisition (e.g., a complex structured mortgage backed security). As more credit is needed in an economy, and lower quality collateral is needed,
more credit is obtained by making the collateral in the system more and more complex. When the increase in complexity is an endogenous response to the need to use
1

See Gorton (1988), Desmirguc-Kunt and Detragiache (1998), and Reinhart and Rogoff (2008 and2009), Caprio and Klingebiel (1996 and 1997) and Laeven and Valencia (2008).

3lower quality collateral, it makes the system more sensitive to negative aggregate
shocks to the expected value of collateral.
We focus on the dynamics of firms issuing information-insensitive debt and thoseissuing information-sensitive debt. If households know a firm has good collateral,they are willing to lend the optimal amount of capital that firms require. If they knowa firm has bad collateral, they are not willing to make any loan to the firm. We assume
all collateral changes quality with a certain probability, so if households do not renew
the information, they decide on the loan based on the perceived collateral quality.
When the perceived quality is high enough firms with good collateral can borrow, but
in addition some firms with bad collateral can borrow. In fact, consumption is highest
if there is no information production, because then all firms can borrow, regardlessof their true collateral quality. In our setting opacity dominates transparency and theeconomy enjoys a blissful ignorance.
If there has been information-insensitive lending for a long time, there can be a significant depreciation of information in the economy and only a small fraction of collateralwith known quality. In this setting we introduce aggregate shocks that may increaseor decrease the average value of collateral in the economy. We show that the longer
information about the collateral has not been renewed, the greater the credit boom
but also the greater the fragility characterized by a larger drop in consumption whena negative aggregate shock hits. In other words, a shock of a given size can have alarger impact on consumption the longer the preceding credit boom. The reason isthat a negative aggregate shock affects more collateral than the same aggregate shock
when the value of collateral is known. Hence, the size of the downturn depends on
how long debt has been information-insensitive in the past.
The crisis may trigger information production or not, having key implications for the
speed of recovery after a crisis. If the negative aggregate shock does not trigger information production (firms reduce borrowing to avoid information acquisition) therecovery takes more time to occur. However, expansionary policies that intend to
speed up the recovery are more effective if the crisis does not induce information
production. In other words, information production speeds up recoveries if governments are not expected to react to the crisis, but delay recoveries if agents expect the
government to take actions to improve the economic conditions.

4Related LiteratureThis paper is related to two strands of literature: Financial intermediation and its effects on macroeconomics and business cycles. As suggested above, our starting pointis that the raison d’être of financial intermediation is the production of short-termbank debt for transactions purposes. Gorton and Pennacchi (1990) argue that intermediaries exist to create trading securities that are immune to adverse selection whenused by agents in markets. In their setting, the securities used were riskless debt securities, backed by bank assets. The notion of ”liquidity” proposed was the idea that
trade was facilitated by securities that were immune to adverse selection. Diamondand Dybvig (1983), on the other hand, view ”liquidity” as consumption smoothing.
While both papers assumed that debt was optimal, Dang, Gorton, and Holmström(2011) show that debt is in fact the optimal trading security. The debt need not beriskless, but its defining characteristic is that it is ”information-insensitive,” whichmeans that it is common knowledge that it is not profitable to produce (costly) information about the payoff on the security.2 There is no adverse selection when trading. A shock, however, can cause previously information-insensitive debt to become
information-sensitive, possibly leading to a crisis.
As in Bernanke and Gertler (1989), Kiyotaki and Moore (1997) and Holmström andTirole (1998), our paper focuses on the magnification and persistence effects of financial markets on real activity. However, unlike those papers, we focus on the dynamics
of information acquisition and its effects on generating credit booms and magnifying
crises. Differently than Kiyotaki and Moore (1997), collateral is not directly neededin production but helps to relax financial frictions in obtaining credit to produce. Endogenous information production determines the fraction of collateral that can beused to sustain credit and the fraction of collateral that suffers in case of negative
aggregate shocks.
Our paper is also related to the literature that studies the role of uncertainty on booms,
crises and recoveries. Bloom (2009) shows that uncertainty shocks can create recessions with delayed recoveries, since firms may freeze hiring and investment. Bachmann and Moscarini (2011) show that recessions can generate an increase in uncertainty, since firms are more willing to experiment with prices to learn about their true
demand when economic conditions are poor. In this literature, uncertainty can be2

See also Andolfatto (2010) and Andolfatto, Berensten, and Waller (2011)5

captured by the cross sectional dispersion of firms in variables such as stock returns.
In our paper, a negative aggregate shock on the value of collateral also generates
an increase in uncertainty, since information about collateral is acquired and there ismore heterogeneity in the access to credit across firms.
Our paper is also related to the literature on leverage cycles developed by Geanakoplos (1997 and 2010). His work relies on low volatility and innovation for the buildup
of leverage, and a jump in uncertainty for the sudden decline. In our paper, crises are
generated by a negative aggregate shock in the expected value of collateral, which
generates a jump in uncertainty. Furthermore, we explicitly derive real effects andwelfare implications from endogenous regime change in terms of information production. Finally, the paper also adds to the literature on asymmetric cycles. Veldkamp (2005) and Ordonez (2009) study slow booms and sudden crashes generated
by learning about endogenous economic conditions. Here, endogenous dynamics ofinformation acquisition also generates asymmetric cycles, but from loosing information rather than processing it to learn.
In the next Section we present the model. In Section 3 we study debt decisions bya single firm. In Section 4 we study the aggregate and dynamic implications of information sensitiveness. In Section 5 we show a numerical simulation to illustrate
our results. We consider the choice of collateral in Section 6 and policy implicationsin Section 7. Empirical evidence is briefly presented in Section 8. In Section 9, weconclude.

2

Model

In the economy, two overlapping generations coexist per period. Each cohort is composed by a mass 1 of individuals who live for two periods: “young” and “old”. Eachgeneration is risk neutral and there is no discounting. We interpret the “young” generation as “households”, with income and no entrepreneurial ideas or managerial
labor, and the “old” generation as potential “firms”, with no income on their own but
with managerial labor.
Each household is born with an endowment of the numeraire good, K. This numeraire good is perishable and cannot be stored for consumption in the next period.
6

When getting “old”, each household develops a fixed amount of managerial labor L∗ .
We interpret L∗ as entrepreneurial ideas or managerial abilities, which does not generate any disutility. In the economy there is a mass 1 endowment of land X, which
is non-perishable. A firm is defined as a combination of a unit of land, labor L∗ andsome amount of the numeraire good K used as capital that can produce Y units ofnumeraire good K with the following stochastic Leontief production function:Y =A min{K, L}0with prob. q
with prob. (1 − q)

where A > 1. Since firms do not have any endowment of K, they need to borrow fromthe households to produce. We assume that the expected marginal product of capitalis higher than its marginal cost qA > 1, hence production is efficient. Naturally,
given the Leontief production function, K ∗ = L∗ is the level of capital that maximizes
profits and is the amount that each firm tries to borrow. We also assume K > K ∗ ,hence households have enough endowment to finance efficient production by firms.
The land has an alternative use besides production. It is possible to extract C units ofK from a good unit of land each period, while it is not possible to extract any K froma bad unit of land. When the land is used alternatively it cannot be used to sustain
production anymore.
In every period, with probability λ the true quality of each unit of land remains unchanged and with probability (1−λ) there is an idiosyncratic shock that changes land
quality. In this last case, land becomes good with a probability p̂, independent of its
current type. Even when the shock is observable, the realization of the new quality
is not, unless a certain amount of the numeraire good γ is used to learn about it. The
idiosyncratic shock and the realization are assumed independent of the idiosyncratic
production shock q. Conditional on the whole history of idiosyncratic shocks andmonitoring results, all agents in the economy share a belief p, which is the probability
the land is good.3
Having discussed the technology, preferences and objectives that agents face, we needto discuss the characteristics of the market for land and the market for loans. In thisenvironment, young households lend K to firms at the beginning of the period and3

We abstract from the possibility of asymmetric information, under which the firm may potentiallyknow more about the collateral than the lenders.

7buy land from firms at the end of the period to be used when becoming a firm in thenext period, when old. On the other hand, firms who acquired land when young,borrow K at the beginnings of the period to produce and sell the land at the end of
the period to the next cohort of firms.
With respect to the market for land, we assume that each buyer matches with one
seller and buyers have all the bargaining power (they make a take-it-or-leave-it offer),
which means sellers will be indifferent between selling or not, in which case they
just obtain the alternative product from the land.4 Hence the price of a unit of land
which is good with probability p is pC, the expected consumption of using the land in
the alternative activity. We also assume that at the time of the transaction, the buyer
knows whether the land suffers an idiosyncratic shock that affects its type next period
or not (this is whether the land remains with the same p or gets a new draw which isgood with probability p̂).
With respect to lending, we assume lenders cannot observe or seize the production of
the firms they finance, which means there is no lending unless there is some collateral
lenders can liquidate in case of no repayment. We assume the firm can use the unitof land X that is required to produce as collateral. We also assume that firms have
all the bargaining power when borrowing, which means lenders will be indifferent
between lending or not.5
The firm can decide to use a fraction x of land as collateral. In case the firm decides not
to pay back its debt, the lender can seize a fraction x of the proceeds the firm obtains
from selling the land at the end of the period, this is xpC. Recall, in the aggregate,
the mass of firms is independent of the number of firms that can pay back the debt
or not. This is because firms end up selling the collateral to households that become
firms in the next generation, regardless of their results. This implies there are no firesales in our environment.6
4

It is straightforward to modify the model to sustain this assumption, for example if a small fractionof households not only develops L∗ when becoming firms but also get an endowment of new land. In
this case there will be more firms selling land than households buying land. Since sellers who do notsell just deplete their unsold land, the mass of land sustaining production in the economy is invariant.
5
It is straightforward to modify the model to sustain this assumption, for example if only a fractionof households develops L∗ when becoming firms there will be more potential lenders than potential
borrowers every period.
6
This is a key difference with Kiyotaki and Moore (1997). In their paper credit cycles are generated
by cyclical fluctuations in the endogenous price of collateral. In this paper, they will be generated bycyclical fluctuations in the endogenous information available in the economy about the collateral.

8In every period, the expected consumption of a household (“young” generation) that
lends and buys land that is good with probability p is K −K(p)+E(repay|p)−pC. The
expected consumption of a firm (“old” generation) that borrows and sells land thatis good with probability p is E(Y |p) − E(repay|p) + pC. This implies that aggregate
consumption in each period is the sum of aggregate consumption of households andfirms. This isZ
1[E(Y |p) − K(p)]f (p)dp

Wt = K +0

where f (p) is the distribution of beliefs about collateral types in the economy. In theunconstrained first best all firms borrow and operate with K ∗ , regardless of beliefs
p about the collateral. This implies that the unconstrained first best aggregate consumption in every period is
W ∗ = K + K ∗ (qA − 1)since E(Y ) = qAK ∗ in the first best.
subsection
the end of Section
2 on Model: function of short-term debt, issued
The modelProposed
is intended
toat capture
the economic
by financial intermediaries, e.g., demand deposits, sale and repurchase agreements,
private banknotes, etc, in a completely self-contained macroeconomic model. The
The model is intended to capture the economic function of short-term debt, issued by financial
e.g., demand
deposits,
sale andagreements, private bank notes. The setting
setting weintermediaries,
have in mind
is shown
below
inrepurchase
Figure 1.
we have in mind is shown below in Figure #.

Figureof# Short-Term Debt
Figure 1: Economic Function

Lend $
Firm

Deposit $Bank

Household
CollateralCollateral

It is clear that our model has abstracted from the bank as we have the households lending directly to the

It is clear that
our model
abstracted
the
as we
have
the households
firms. Furthermore,
for hassimplicity,
we have notfrom
modeledthebanktransactionrole ofthe short-term
debt forthe households,
is theFurthermore,
focus of Dang, Gorton,
Holmström (2011).
Insteadwemodeledfocus on the thelending directly
to thewhichfirms.
for andsimplicity,
we havenotinformation dynamics. But, in so doing we do not want to be viewed as taking a stand on the role of thetransactiondebt
roleof the short-term debt for the households, which is the focus of Dang,
in lending, as opposed to its function as a transaction medium.
Gorton, and Holmström (2011). Instead we focus on the information dynamics. But,
When we look at some empirical tests, the model will be closer to the figure. But, for exposition in what
in so doingfollows
we we
doomitnotthewantbanks.to be viewed as taking a stand on the role of the debt inlending, as opposed to its function as a transaction medium. When we look at some
empirical tests, the model will be closer to the figure. But, for exposition in what
follows we omit the banks.
9

We will proceed in three steps. First we analyze the borrowing decision of a single firm that has collateral which is good with probability p. Then, we study theaggregate output when the distribution of beliefs about the collateral quality in theeconomy has a constant mean p̂ and an endogenously evolving variance. Finally, we
introduce aggregate shocks that affect all collateral in the economy, in addition to theidiosyncratic shocks, changing the mean of that distribution.

3

A single firm

In this section we study the short-term debt contract between a single lender and asingle firm and characterize the optimal debt the firm issues considering the possibility that the lender may want to produce information about the true quality of thecollateral.
First we solve the optimal borrowing decision of a single firm that lives a single period, and decides whether to issue debt that triggers information production or not.
Triggering information production (information-sensitive debt) is costly for the firm
because it raises the cost of borrowing. However, not triggering information production (information-insensitive debt) is also costly because it may reduce the possible
size of the loan. This trade-off determines the information sensitiveness of the debt
and then the volume of information in the economy.

3.1

A Single Period

3.1.1

Information-Sensitive Debt

Lenders can learn the true quality of the land by paying an amount γ of the numeraire
good. Since lenders are competitive and risk neutral, they break even based on theface value of the debt RIS (p) and the fraction xIS (p) of a land with expected price pC
posted by the firm as collateral. This is, lenders should be indifferent ex-ante between
producing information or not.7 For simplicity we just denote the face value of debt
7

We assume this interest rate R is not renegotiable after the lender finds out the collateral is good,otherwise there would never be incentives to produce information in the first place.

10and the fraction of collateral posted as R and x.
p(qRIS + (1 − q)xIS C − K) = γ

⇒

RIS =

pK + γ − (1 − q)pxIS C.
pq

This equality shows the relation between RIS and xIS . However there is an additional
condition that should be fulfilled, which is RIS = xIS C, this is debt is risk free because
the firm should pay the same in case of success or failure. It is not possible that
RIS > xIS C because the firm would always prefer to hand in the collateral, also incase of success. Similarly, it is not possible that RIS < xIS C because the firm would
always prefer to sell the collateral directly in the market at a price C and give lenders
RIS rather than xIS C in case of failure. This condition pins down the fraction ofcollateral a firm which land is good with a probability p will post in equilibrium.
RIS = xIS C

⇒

xIS =

pK + γpC

with the natural restriction that xIS ≤ 1. This implies that the firm will be able to bor∗
row the optimal K ∗ against a good collateral when pKpC+γ ≤ 1 (this is, for all collateralγ∗with p ≥ C−K∗ ). Otherwise the firm will be able to borrow less than K , even posting= 1 (this is K = pC−γ). Naturally, there is no
all the good collateral. In this case pK+γ
pC
p
p0+γ
lending, even if the firm post all the good collateral if pC > 1 (this is, for p < Cγ ).
The expected profits from borrowing that triggers information production areE(π|p, IS) = p(qAK − xIS C) + pC
Plugging the equilibrium xIS above, in case of borrowing K ∗ , expected profits areE(π|p, IS) = pK ∗ (qA − 1) − γ + pC
The firm will decide to borrow rather than just sell the land if pK ∗ (qA − 1) ≥ γ, orγγγ∗. If K ∗ (qA−1)> C−K
p ≥ K ∗ (qA−1)∗ , or qA < C/K , the firm stops borrowing before thelender decides to lend less than K ∗ when discovering the land is good. Based on thisassumption, expected profits are11 pK ∗ (qA − 1) − γ + pC
E(π|p, IS) =
 pC

3.1.2

γ
if p ≥ K ∗ (qA−1)γ
if p < K ∗ (qA−1)

Information-Insensitive Debt

Another possibility for firms is to borrow without triggering information acquisitionby lenders. Still it should be the case that lenders break even in equilibrium
qRII + (1 − q)pxII C = K

⇒

RII =

K − (1 − q)pxII C.
qAs in the previous case, the condition RII = pxII C holds in equilibrium. Then we canobtain the optimal posting of collateral
RII = xII pC

⇒

xII =

K
≤1
pC

Since xII ≤ 1, a restriction in borrowing without producing information is K ≤ pC.
The problem the firm faces when issuing information insensitive debt is that lenders
may decide to deviate, check the value of the collateral and ex-post decide whether
to lend or not at the specified fraction xII , before the firm gets to know such an information. Lenders want to deviate if the expected profits from acquiring information,
evaluated at xII , are greater than the cost of acquiring information γ. Hence, there are
no incentives to deviate and acquire information ifp(xII C − K) < γ⇒(1 − p)pxII C < γthat is, if the expected gains of producing information (learning the collateral is goodallows the lender to sell it a higher price later xII C while lending the collateral is badallows the lender not to lend) is smaller than the cost of producing information, γ.
It is clear from the previous condition that the firm can discourage information production about collateral, by reducing borrowing and output. If the condition is notbinding, then there are no strong incentives for lenders to produce information and12K = K ∗ . If the condition is binding, the firm will borrow as much as possible given
the restrictions of no triggering information acquisition,K=γ.
(1 − p)Hence, information-insensitive borrowing is characterized by the following debt size:γ
E(K|p, II) = min K ,, pC(1 − p)∗(1)

We focus on the case in which information-insensitive borrowing is characterized bythree regions.8

K(p) =











K∗

γ
if K ∗ ≤ (1−p)(no credit constraint)

γ
(1−p)

γ
if K ∗ > (1−p)(credit constraint)pC

γ
if pC < (1−p)(collateral selling)

The first kink is generated by the point at which the constraint to avoid information
production is binding when evaluated at the optimal loan size K ∗ , this is when financial constraints start binding more than technological constraints. The second kink is
generated by the constraint xII ≤ 1, below which the firm is able to borrow up to the
expected value of the collateral pC without triggering information production.
Expected profits under information-insensitive borrowing are:qAK − xII pC + pC,
i.e., with probability q production is successful, the firm always pays back xII pC andthe collateral has an expected value of pC. Then:
E(π|p, II) = E(K|p, II)(qA − 1) + pC

(2)∗

This is the more natural case when C > K ∗ and γ is not large (specifically γ < KC (C − K ∗ )). If∗
C > K ∗ and γ > KC (C − K ∗ ), then there are only two regions, where the middle region disappears.
If C < K ∗ , then the first region is given by pC and not K ∗ , since pC is always smaller than K ∗ . In allcases the main conclusions we derive are the same.
813

In the case we are focusing on, considering explicitly the kinks,K ∗ (qA − 1) + pC
γE(π|p, II) =(qA − 1) + pC
(1−p)



 pC(qA − 1) + pC

3.1.3

γ
if K ∗ ≤ (1−p)γ
if K ∗ > (1−p)γ
if pC < (1−p)

Optimal Debt

Figure 2 shows the profits under these two regimes for each possible p (after deducting the expected value of the collateral pC in all expressions). From the comparison ofwhich one is larger we can obtain the values of p for which the firm prefers to borrowwith an information-insensitive loan (II) or with an information-sensitive loan (IS).
Figure 2: A Single Period Expected Profits𝛾(𝑞𝐴 − 1)(1 − 𝑝)𝐾 ∗ (𝑞𝐴 − 1)

𝑝𝐾 ∗ (𝑞𝐴 − 1) − 𝛾

𝐿
𝑝𝐼𝐼

𝐿
𝑝𝐼𝑆

II

𝑝𝐶𝑙

𝑝𝐶ℎ
IS

𝑝𝐻
II

The cutoffs highlighted in Figure 2 are determined in the following way: pH is the pthat generates the first kink of the profit function of an information-insensitive loan.
This is the p below which firms have to reduce borrowing to prevent information14

production:
pH = 1 −

γ
.
K∗

(3)

The cutoff pLII is obtained from the second kink of the profit function of an informationinsensitive loan,9
r
γ
1
1
L
− .
(4)
pII = −
2
4 C
Similarly, the cutoff pLIS is obtained from the kink of the profit function of an informationsensitive loan:γ
pLIS = ∗.
(5)
K (qA − 1)with the natural restriction that pLIS = 1 if γ > K ∗ (qA − 1).
Finally, the cutoffs pCh and pCl are obtained from equalizing the profit functions of
information sensitive and insensitive loans and solving for the quadratic roots of:γγ= pK ∗ −.
(1 − p)(qA − 1)

(6)





γ
γ
1
1
+ ∗ 1+=0p −p 1+ ∗K (qA − 1)
K
(qA − 1)
2

There are only three regions of financing. Information-insensitive loans are chosen byfirms with collateral with high and low values of p, while information-sensitive loans
are chosen by firms with collateral with intermediate values of p.
To understand how these regions depend on the information cost γ, the four arrows
in the Figure show how the different cutoffs and functions move as we reduce γ. In
an extreme, when γ = 0, all collateral is information-sensitive (i.e., the IS region isp ∈ [0, 1]), which is intuitive since information is free.
Contrarily, as γ increases, the two cutoffs pCh and pCl get together and the IS region
shrinks until it disappears (i.e., the II region is p ∈ [0, 1]) when γ is large enough∗(specifically, when γ > KC (C − K ∗ )).
Having characterized how the information-sensitiveness of debt depends on collateral beliefs p, we can analyze expected consumption, which is our measure of welfare
9

The positive root for the solution of pC = γ/(1 − q)(1 − p) is irrelevant since it is greater than pH ,and then it is not binding given all firms with a collateral that is good with probability p > pH canborrow the optimal level of capital K ∗ without triggering information production.

15

in the aggregate at a given period. By construction, at the cutoffs pCl and pCh , where
the system changes from information-insensitive debt to information-sensitive debt,
the expected consumption between the two regimes is the same. This is because, atthe cutoffs
WtIS ≡ pK ∗ (qA − 1) − γ + K = K(qA − 1) + K ≡ WtII .
In Figure 3 we show how financial constraints reduce aggregate consumption when
the perceived quality of the collateral p declines. Efficient consumption is the blueline and the financially constrained aggregate consumption is the red line.
Figure 3: A Single Period Expected Consumption�𝐾 ∗ (𝑞𝐴 − 1) + 𝐾𝛾�(𝑞𝐴 − 1) + 𝐾(1 − 𝑝)�
𝑝𝐾 ∗ (𝑞𝐴 − 1) − 𝛾 + 𝐾

�
𝐾
𝐿
𝑝𝐼𝐼

3.2

𝐿
𝑝𝐼𝑆

II

𝑝𝐶𝑙

IS

𝑝𝐶ℎ

𝑝𝐻
II

Optimal Information Production Costs

The cost of information production, γ, is fixed. However, we can study the case inwhich a planner can choose the optimal monitoring costs γ ∗ that maximizes aggregate
consumption at each period t given financial constraints.

16

Proposition 1 Optimal monitoring costs are γ ∗ = ∞ if C > K ∗ , γ ∗ = [0, ∞) if C = K ∗and γ ∗ = 0 if C < K ∗ .
∗Proof If C > K ∗ , as γ → ∞, nE(π|p, II) = mino {K , pC} (qA − 1) + pC, which isγ, 0 (qA − 1) + pC for all p. As we regreater than E(π|p, IS) = max pK ∗ − (qA−1)γ
duce γ, E(π|p, II) declines for values of p where (1−p)< pC, reducing aggregate
consumption since those collateral can borrow less. This implies that making the system information-insensitive for all collateral achieves a higher aggregate consumption than making it information-sensitive

If C < K ∗ , when γ = 0, E(π|p, IS) = max {pK ∗ , 0} (qA − 1) + pC, which is larger
than E(π|p, II) = 0 for all p. As we increase γ, E(π|p, II) can never increase above
pCqA which is smaller than E(π|p, IS) for all p. This implies that making the system
information-sensitive for all collateral achieves a higher aggregate consumption thanmaking it information-insensitive
In the threshold, C = K ∗ , the firm chooses
the maximum
between E(π|p, IS) =noγ∗pC(qA − 1) + pC and E(π|p, II) = min K , (1−p) , pC (qA − 1) + pC, which cannot be greater than E(π|p, IS). Hence any value of γ gives an aggregate consumption
equal to the aggregate consumption of the information-sensitive regime.
Q.E.D.
The intuition for this result is the following. When C > K ∗ , without producing information firms can borrow up to pC for sure. However, when producing information
firms will borrow in expectation pK ∗ < pC and spend γ. Hence, it is optimal to
discourage information, even if information is free.
When C < K ∗ , without producing information firms can borrow up to pC for sure.
When producing information firms will borrow in expectation pK ∗ > pC and spend
γ. Hence, it is optimal to make information production as cheap as possible.

4

Aggregate Results

In this section we characterize the evolution of information about the collateral and its
impact on aggregate consumption. First, we study a case without aggregate shocks
to collateral and discuss the effects of endogenous information production on the
dynamics of credit booms. Then, we introduce aggregate shocks and study the effects
of endogenous information on the size of crises and the speed of recoveries.
17

4.1

No Aggregate Shocks

Assume that initially (at period 0) there is perfect information about which collateralis good and which is bad. The probability each firm’s collateral becomes good when
it is hit by an idiosyncratic shock is always p̂. In what follows we study the aggregate
consumption in this economy as time evolves, for different values of p̂, and we show
that, as long as p̂ is large enough consumption is growing with time since there is no
information production about the collateral.10
First, for notational simplicity, we define borrowing depending on p, based on theanalysis of Section 3:

K(p|γ) =













K∗

if pH < pγ
(1−p)

if pCh < p < pHγ
pK ∗ − (qA−1)

if pCl < p < pCh

γ
(1−p)

L
< p < pCl
if PII












pC

if

p < pLII

From the definition of the cutoffs, we know K(p) is monotonically increasing in p.
The aggregate consumption of the economy at period t is defined by:
Z 1
K(p)(qA − 1)f (p)dp + K̄,

Wt =0

where f (p) is the distribution of beliefs about all the collateral in the economy. In thesimple stochastic process for idiosyncratic shocks we assume, and in the absence of
aggregate shocks, this distribution has a three-point support: 0, p̂ and 1.
If p̂ > pCh or p̂ < pCl , information is not reacquired and at period t, f (1) = λt p̂,f (p̂) = (1 − λt ) and f (0) = λt (1 − p̂). Since K(0) = 0,
WtII = λt p̂K(1) + (1 − λt )K(p̂) (qA − 1) + K̄.

(7)

If p̂ ∈ [pCl , pCh ], information is reacquired in every period t for the fraction (1 − λ)10In this case, without aggregate shocks, the average quality of collateral in the market is given byp̄ = p̂ in all periods.

18

of collateral that gets the idiosyncratic shock. Then f (1) = λp̂, f (p̂) = (1 − λ) andf (0) = λ(1 − p̂). Considering K(0) = 0,
WtIS = [λp̂K(1) + (1 − λ)K(p̂)] (qA − 1) + K̄.

(8)Some interesting implications can be obtained from the previous equations. If theeconomy is in an information-insensitive regime (this is, p̂ > pCh or p̂ < pCl ), theevolution of aggregate consumption depends on p̂. As can be seen, W0II = p̂K(1) + K̄and limt→∞ WtII = K(p̂)(qA − 1) + K̄. If p̂K(1) = K(p̂), aggregate consumption isconstant over time, which occurs when:γ= p̂∗ K ∗ ,(1 − p̂∗ )which is fulfilled for pCh < p̂∗ < pH .
For all p̂ > p̂∗ , aggregate consumption grows over time. In particular the case inwhich the average collateral does not introduce financial restrictions (this is, p̂ > pH ),is characterized by an aggregate consumption increasing over time (since pH > p̂∗ ).
This is because more and more firms are borrowing, a credit boom.
Contrarily, if the average collateral implies information production (this is pCl < p̂ <pCh ), aggregate consumption WtIS does not depend on t, being constant at the level atwhich information about the collateral that suffers idiosyncratic shocks is reacquired
at every period .

4.2

Aggregate Shocks

In this section we introduce negative aggregate shocks that transform a fraction (1−η)
of good collateral into bad collateral. As with idiosyncratic shocks, the aggregate
shock is observable, but which good collateral changes quality is not. This implies
that when the shock hits, there is a downward revision of the perception about thequality of each unit of collateral. For example, collateral that has a p = 1, gets a newbelief p0 = η after the aggregate shock. Similarly, all collateral with p = p̂ get revised
downwards to p0 = η p̂.

19

We also consider positive aggregate shocks that transform a fraction α of bad collateral into good collateral. In this case beliefs are revised up for all collateral. Collateralwith p = 0 get revised to p0 = α and collateral with p = p̂ is revised to p0 = p̂ + α(1 − p̂).
In this section we focus on negative aggregate shocks and in the policy section wewill discuss how policies that look like positive aggregate shocks can be constructed
to deal with economic crises.
We will focus on the case where prior to the negative aggregate shock, the average
quality of the collateral is good enough such that there are no financial constraints(that is, p̂ > pH ). Later we will justify why, allowing for endogenous choice of collateral, this is in fact the situation that naturally arises.
The next Proposition shows that the longer the economy did not face a negative aggregate shock, the larger the consumption loss when such a shock does indeed occur.
Proposition 2 Assume p̂ > pH and a negative aggregate shock η in period t. The reduction
in consumption ∆(t|η) ≡ Wt − Wt|η is non-decreasing in η and the time t elapsed previously
in the absence of a shock.
Proof Assume a negative aggregate shock of size η. Since we assume p̂ > pH , the average collateral does not generate information production. The aggregate consumption
before the shock is given by equation 7 and after the shock aggregate consumption is:
Wt|η = λt p̂K(η) + (1 − λt )K(η p̂) (qA − 1) + K̄.
Then we can define the reduction in aggregate consumption as ∆(t|η) = Wt − Wt|η∆(t|η) = [λt p̂[K(1) − K(η)] + (1 − λt )[K(p̂) − K(η p̂)]](qA − 1).
Since K(p) is decreasing in p, ∆(t|η) is non-decreasing in η and t.

Q.E.D.

The intuition for this Proposition is straightforward. If there is little information aboutcollateral in the economy (that is, there is a small fraction of collateral with either
p = 0 or p = 1), a negative aggregate shock affects a high fraction of collateral in theeconomy, reducing borrowing and consumption a lot.

20

This result goes beyond the mechanical effect in which more collateral is affected.
After a negative shock to collateral, either a higher amount of the numeraire goodshould be used to produce information or borrowing is excessively restricted to avoid
such information production.
Since the fraction of collateral with information about their true quality decreases
over time in an information-insensitive regime, if we define ”fragility” as the probability aggregate consumption declines more than a certain value, then the following
corollary follows immediately from the previous Proposition.
Corollary 1 Given a structure of negative aggregate shocks, the fragility of an economy
increases with the number of periods the debt in the economy has been informationallyinsensitive, and then increases with the fraction of collateral of unknown quality.
The next Proposition shows that under a negative shock, information production
speed up the recovery.
Proposition 3 Assume p̂ > pH and a negative aggregate shock η in periodqt. The recovery

is faster when information is produced after the shock if η p̂ < η p̂ ≡ 12 +1
− Kγ∗ , where4IIIS
for all η p̂ < η p̂.
pCh < η p̂ < pH . This is, Wt+1
> Wt+1

Proof If the negative shock happens in period t, the distribution in period t is: f (η) =λt p̂, f (η p̂) = (1 − λt ) and f (0) = λt (1 − p̂).
In period t + 1, if information have been produced (IS case), after the idiosyncratic
shocks the distribution of beliefs is fIS (1) = λη p̂(1 − λt ), fIS (η) = λt+1 p̂, fIS (p̂) =(1 − λ), fIS (0) = λ[(1 − λt p̂) − η p̂(1 − λt )]. Hence, aggregate consumption at t + 1 if
information is acquired is,IS
Wt+1= [λη p̂(1 − λt )K ∗ + λt+1 p̂K(η) + (1 − λ)K(p̂)](qA − 1) + K(9)In period t + 1, if information have not been produced (II case), after the idiosyncratic
shocks the distribution of beliefs is fII (η) = λt+1 p̂, fIS (p̂) = (1−λ), fIS (η p̂) = λ(1−λt ),fIS (0) = λt+1 (1 − p̂). Hence, aggregate consumption at t + 1 if information is notacquired is,IIWt+1= [λt+1 p̂K(η) + λ(1 − λt )K(η p̂) + (1 − λ)K(p̂)](qA − 1) + K

21

(10)

Taking the difference between aggregate consumption at t+1 between the two regimesof information production.
IS
II
Wt+1
− Wt+1= λ(1 − λt )(qA − 1)[η p̂K ∗ − K(η p̂)]

(11)∗This expression
qis non-negative for all η p̂K ≥ K(η p̂), or which is the same, for allη p̂ < η p̂ ≡ 21 +1
− Kγ∗ . From equation (6), pCh < η p̂ < pH .
4

Q.E.D.

The intuition for this Proposition is the following. When information is produced
after a shock η that constrain credit a lot, information costs are spent at the time ofthe shock and then a fraction η p̂ of collateral can sustain the maximum borrowing K ∗ .
When information is not produced after a shock η that constrain credit, collateral thatremain with belief η p̂ should restrict credit in following periods, until beliefs moveback to p̂. This is equivalent to restrict credit proportional to monitoring costs allfollowing periods as well. The Proposition generates the following Corollary.
Corollary 2 There is a range of negative aggregate shocks big enough (η such that η p̂ ∈[pCh , η p̂]) in which agents do not acquire information, but recovery would be faster if they do.

5

Numerical Simulations

We illustrate our main results with the following numerical exercise. We assumeidiosyncratic shocks happen with probability (1−λ) = 0.1, in which case the collateralbecomes good with probability p̂ = 0.92. Other parameters are q = 0.6, A = 3 (these
two assumption imply that investing in the project generates a return of 80%), K̄ = 10,
L∗ = K ∗ = 7 (which means the endowment is enough to invest in the optimal projectsize), C = 15 and γ = 0.8.11
Given these parameters we can obtain the relevant cutoffs for our analysis. Specifically, pH = 0.89, pLII = 0.06 and the sensitive information region is in the valuesp ∈ [0.31, 0.83]. As discussed above, these cutoffs are obtained from comparing expected profits from taking a loan producing information with one without producing
information. Figure 4 plots these functions and the respective cutoffs.
11∗

Recall these parameters fulfill the assumptions qA < C/K ∗ and γ < KC (C − K ∗ ) in the text.

22

Figure 4: Expected Profits and Cutoffs
6

Expected Profits

54

E()IS

3

2
E()IS

1

0
0

0.2

0.4

0.6

0.8

1

Beliefs

Using these cutoffs we simulate the model for 100 periods. As in the main text, we
assume that at period 0 there is perfect information about the true quality of every
collateral in the economy. Over time there are idiosyncratic shocks that make this information vanish unless there is costly information acquisition about the realizations
after idiosyncratic shocks.
We introduce a negative aggregate shock that transforms a fraction (1 − η) of goodcollateral into bad collateral in periods 5 and 50. We also introduce a positive aggregate shock that transforms a fraction α = 0.2 of bad collateral into good collateralin period 30. We compute the dynamic reaction of consumption in the economy for
different sizes of negative aggregate shocks, η = 0.97, η = 0.91 and η = 0.90. We will
see that small differences in the size of a negative shock can have important dynamic
consequences in the economy.
Figure 5 shows the average probability that collateral is good in the economy for thethree possible negative aggregate shocks (this is the real collateral quality existing in
the economy). While aggregate shocks have a temporary effect on quality of collateral, after aggregate shocks occur the average quality converges back to p̂ = 0.92. As
can be seen, the negative aggregate shocks were constructed such that η p̄ is above
pH when η = 0.97, is between pCh and pH when η = 0.91 and is less than pCh when
η = 0.90.
23

Figure 5: Average Quality of Collateral
0.94
=0.91

=0.97

Average Quality of Collateral

0.92

0.9
pH
0.88

0.86
=0.90
0.84

0.82
0

pCh

20

40

60

80

100

Periods

Figure 6 shows the evolution of aggregate consumption for the three negative aggregate shocks. The first result to highlight is that when η = 0.97, aggregate shocks do
not affect the evolution of consumption at all. The reason is that shocks do not introduce financial constraints. The second result is that positive shocks do not affectthe evolution of consumption and the reason is that p̂ > pH , and hence improvements in the belief distribution do not relax the financial constraints even more. This
introduces an asymmetry on how shocks affect aggregate consumption.
The third result is that the reduction in consumption from the negative aggregate
shock in period 5, when not much information has vanished yet, is much lower than
the reduction in consumption from the same size negative aggregate shock in period
50. The reason is that the shock reduces financing for a larger fraction of collateral
which information has vanished over time but was good enough to finance projects
successfully. This is the result proved in Proposition 2.

24

Figure 6: Welfare
15.8
15.6

=0.97Aggregate Consumption

15.4
15.2
15
Always produce
information
about idiosyncratic
shocks14.8
=0.9014.6
14.4

=0.91

14.2
14
13.8
0

20

40

60

80

100

Periods

In Figure 7 we illustrate that small differences in the size of shocks have very different
consequences on the variance of beliefs about the collateral. A shock η = 0.91 does
not trigger information production but a shock η = 0.90 does it. Given that after many
periods without shock most collateral looks the same, these differences in information
production implies that these differences have large consequences on the variance ofbeliefs and the information available about most collateral.
This effect of information acquisition implies that, even when the real quality of collateral is the same under the two shocks, a slightly larger shock that induces information acquisition implies a faster recovery, which is the result from Proposition 3.

25

Figure 7: Standard Deviation of Distribution of Beliefs
0.4Standard Deviation of Beliefs

0.35
0.3
=0.90
0.25
0.2
0.15
0.1

=0.91
=0.97

0.05
0
0

20

40

60

80

100

Periods

6

The Choice of CollateralIn this Section we study an environment with collateral heterogenous in two dimensions, the expected value of the collateral p̂ and the cost of information acquisition γ.
Firms can choose freely the characteristics of the collateral to use in order to maximize
borrowing using such a collateral.
In previous sections we analyzed the effects of p on borrowing for a given cost of
information acquisition γ. In that environment we obtained that borrowing was increasing in p. The proof relies on the monotonically increasing function of the borrowing function K(p|γ).
Similarly, we can analyze borrowing for a given collateral with belief p, for different
levels of information costs γ. In this case the amount borrowed critically depends onwhether the collateral creates financial constraints or not and in case it does, whether
the financial constraint locates the collateral in an information-sensitive or insensitive
region. The next Proposition summarizes these results.
Proposition 4 Take collateral characterized by the pair (p, γ), these are the expected probability the collateral has value C and the cost of information acquisition when the collateral ishit by an idiosyncratic shock.
26

1. Fix γ. After an idiosyncratic shock:(a) No financial constraint: Borrowing is independent of p.
(b) Information-sensitive regime: Borrowing is increasing in p.
(c) Information-insensitive regime: Borrowing is increasing in p.
2. Fix p.
(a) No financial constraint: Borrowing is independent of γ.
(b) Information-sensitive regime: Borrowing is decreasing in γ.
(c) Information-insensitive regime: Borrowing is increasing in γ if higher than pC
and independent of γ if pC.
Proof Point 1 is a direct consequence of K(p|γ) being monotonically increasing in p
for p < pH and independent of p for p > pH .
e
To prove the second point 2 we derive the function K(γ|p),which is the inverse of
the K(p|γ) and analyze its properties. Take a situation where information acquisition
is not possible (or γ = ∞). In this case the limit to financial constraints is the point
at which K ∗ = pC. This is because lenders will not acquire information but will notelend more than the expected value of the collateral pC. Then, the function K(γ|p)
has
K∗
K∗
two parts. One for p ≥ C and the other for p < C .
∗1. p ≥ KC :eK(γ|p)
=







K∗

if γ1H ≤ γ

γ

if γ L ≤ γ < γ1H

(1−p)



pK ∗ −

γ(qA−1)

if γ < γ L

where γ1H comes from equation 3. Thenγ1H = K ∗ (1 − p)(12)and γ L comes from equation 6. Thenγ L = pK ∗(1 − p)(qA − 1)
(1 − p) + (qA − 1)
27

(13)

∗

2. p < KC :eK(γ|p)
=







pC

if γ2H ≤ γ

γ

if γ L ≤ γ < γ2H

(1−p)



pK ∗ −

γ(qA−1)

if γ < γ L

where γ2H in this region comes from equation 4. Thenγ2H = p(1 − p)C(14)

and γ L is the same as above.
e
It is clear from the function K(γ|p)
that, for a given p, borrowing is independent of γ inthe first region, it is increasing in the second region (information-insensitive regime)
and it is decreasing in the last region (information-sensitive regime).
Q.E.D.
Figure 8 shows the borrowing possibilities for all combinations (p, γ). As a further
illustration of how borrowing is affected by these two dimensions, Figure 9 showsdifferent levels of borrowing from high (dark red) to low (blue).12
Figure 8: Borrowing for different type of collateral

𝛾

𝒑𝑪

𝜸
(𝟏 − 𝒑)

𝛾2𝐻

12

0

𝑲∗

𝛾𝐿

𝒑𝑲∗ −

𝛾1𝐻
𝜸
(𝒒𝑨 − 𝟏)𝐾 ∗�𝐶The numbers that created this Figure are K ∗ = 2, C = 4, q = 0.7 and A = 3.

28

1

𝑝

Figure 9: Borrowing for different type of collateral - A numerical example

If we assume the price of collateral is the fair value pC, then firms would prefer toacquire collateral with the lowest possible price that maximize the borrowing size. In∗
this sense firms would start using collateral with p = KC and γ > γ1H evaluated at thatp. Then they will use collateral with a slightly higher p and γ > γ1H for that p.
This implies that endogenously the collateral used for borrowing will be biased towards relatively high p̂ and relatively high γ. This way the first collateral firms use
to borrow are those that most likely allow the firm not to be financially constrained,
either because the collateral is very likely to be of high quality, or because it is very
expensive to acquire information about its quality.

29

7

Policy Implications

In this section we discuss some policies the government may use to increase welfare,measured as∞
X
Ut = Etβ τ −t Wt .
τ =t

Unlike households and firms, the planner cares about the discounted utility of all
generations. Hence, a planner takes into account the possibility of a crisis in the
future. What can the government do to maximize welfare? This is, what can thegovernment do to reduce the size of crises without reducing growth? What can thegovernment do to speed up recoveries?
We explore three types of policies. Collateral policies, lending policies and information policies.

7.1

Collateral policies

This set of policies is intended to boost the expected quality of collateral (p̂). After anegative aggregate shock η, the natural and trivial reaction of the government should
be to eliminate such a shock by improving p̂. The effectiveness of collateral policies
depends on how fast the government is able to react to the negative shock, for example guaranteeing the quality of the collateral. This policy manifests itself as an α
positive aggregate shock, one period after the negative aggregate shock.
The next Proposition shows that, if there is a positive aggregate shock after a negative
aggregate shock that takes the average collateral η p̂ to a higher new level above pH ,the recovery from the negative shock is faster if there was no information production
as a response to the negative aggregate shocks.
Proposition 5 Assume a positive aggregate shock of size α, immediately following a negative
aggregate shock η, such that p0 = η p̂ + α(1 − η p̂) > pH . This policy is more effective in
speeding recovery after a shock η that induces information acquisition if that informationII
II
is not acquired. More specifically ∆II > ∆IS (where ∆II ≡ Wt+1|α− Wt+1and ∆IS ≡IS
IS
Wt+1|α− Wt+1) for all η p̂ ∈ [pCl , pCh ].

30

Proof As in Proposition 3, if the negative shock happens in period t, the distributionin period t is: f (η) = λt p̂, f (η p̂) = (1 − λt ) and f (0) = λt (1 − p̂).
Without information production, in period t + 1, after the idiosyncratic shocks, thedistribution of beliefs is fII (η) = λt+1 p̂, fII (η p̂) = λ(1 − λt ), fII (p̂) = (1 − λ) andfII (0) = λt+1 (1 − p̂).
In case the government introduces a policy that transforms a fraction α of bad collateral into good, for example by guaranteeing they pay C even when they are bad,in the period t + 1, following the negative shock that occurred in period t, beliefs
change from η to α + η(1 − α), from p̂ to α + p̂(1 − α), from η p̂ to α + η p̂(1 − α)and from 0 to α. 13 The distribution of beliefs become: fII (α + η(1 − α)) = λt+1 p̂,fII (α + η p̂(1 − α)) = λ(1 − λt ), fII (α + p̂(1 − α)) = (1 − λ) and fII (α) = λt+1 (1 − p̂).
Since we assume p̂ > pH and η > pH , the positive shock does not affect the borrowingof those beliefs. Since we assume α + η p̂(1 − α) > pH , borrowing for firms that previously had a collateral with beliefs η p̂ increase borrowing from K(η p̂) to K ∗ . Similarly,
borrowing for firms that previously had a collateral with beliefs 0 increase borrowing
from 0 to K(α).
Since the distribution does not change, just the beliefs assigned to collateral, we cancompute the aggregate consumption from the positive policy and compare it to theone without the positive policy, from equation 10. Then,IIII∆II ≡ Wt+1|α− Wt+1= λ(qA − 1)[(1 − λt )(K ∗ − K(η p̂)) + λt (1 − p̂)K(α)](15)

With information production, in period t + 1, after the idiosyncratic shocks, thedistribution of beliefs is fIS (1) = λη p̂(1 − λt ), fIS (η) = λt+1 p̂, fIS (p̂) = (1 − λ),fIS (0) = λ[(1 − λt p̂) − η p̂(1 − λt )].
With the positive policy beliefs change from η to α+η(1−α), from p̂ to α+ p̂(1−α), andfrom 0 to α. Also, beliefs 1 remain 1. Since we assume p̂ > pH and η > pH , the positive
shock does not affect the borrowing of those beliefs. Finally, borrowing for firms that
previously had a collateral with beliefs 0 increase borrowing from 0 to K(α). We cancompute the aggregate consumption from the positive policy and compare it to the13

The same results hold if the policy is introduced in subsequent periods.

31

one without the positive policy, from equation 9. Then,ISIS∆IS ≡ Wt+1|α− Wt+1= λ(qA − 1)[(1 − λt p̂) − η p̂(1 − λt )]K(α)(16)

Taking the difference between equations (15) and (16),∆II − ∆IS = λ (1 − λt )[K ∗ − K(η p̂)] + [λt (1 − p̂) − (1 − λt p̂) + η p̂(1 − λt )]K(α)= λ(1 − λt ) [K ∗ − K(η p̂) − (1 − η p̂)K(α)]
In the range of interest, at which η p̂ < pCh and there are incentives for informationγ.
production, avoiding information production would imply K(η p̂) ≤ η p̂K ∗ − (qA−1)
This implies the government imposing the restriction of no information production
borrowing should be lower than the borrowing level that induces information production. Using this upper bound to evaluate the expression above,
II

IS

∆ −∆


γ
− (1 − η p̂)K(α)(qA − 1)γ
≥ λ(1 − λt )(1 − η p̂) [K ∗ − K(α)] +>0(qA − 1)(1 − η p̂)≥ λ(1 − λ ) K ∗ − η p̂K ∗ +t

Q.E.D.

The intuition for this Proposition relies on the speed of recovery of information. If an
aggregate negative shock does not generate information production, when in average
the collateral quality recovers, borrowing can recover fast for the high fraction ofcollateral without information about their true quality. If an aggregate negative shock
generates information production, then, even when in average the collateral quality
recovers, the fact that bad collateral has been identified, restricts lending to such a
collateral until they face idiosyncratic shocks.
Figure 10 introduces a shocks α = to the numerical simulation in period 51, right after
the assumed negative shock. As can be seen the collateral policy that replenishes theconfidence on the average collateral is more effective when the negative aggregate
shock did not generate information than in the case in which the negative aggregate
shock did generate information. This implies that, if the planner can use a collateralpolicy to deal with a crises, it will be more effective doing if the shock is such that32

it just creates a credit crunch where no information is produced. This is the effectdiscussed in Proposition 5.
Figure 10: Effectiveness of Collateral Policy
15.8
15.6

=0.97Aggregate Consumption

15.4

=0.91

15.2
15
Always produce
information
about idiosyncratic
shocks14.8
=0.90
14.6
14.4
14.2
14
13.8
0

20

40

60

80

100

Periods

7.2

Lending policies

Information production after the negative aggregate shock renders less effective collateral policies that try to reintroduce confidence in the average value of collateral.
The planner may discourage information production by introducing lending policies.
Assume p̂ < pH such that the shock reduces efficient borrowing, or even worse, assume p̂ < pCh such that firms face a credit crunch where borrowing also triggers
information production. The government can tax households and transfer to firmsγγ
an amount K ∗ − (1−p)
directly and let the firms borrow the remaining (1−p)
such that∗firms can efficiently get the optimal K , and no information is produced.

7.3

Informational policies

Differently than the previous two options, informational policies are preventive innature. Conditional on suffering a negative aggregate shock and not being able to
make the economy recover without producing information, as was the objective of33

the previous two policies, informational policies may prevent huge crises by generating some information along the way to a credit boom. When there has been no
information acquisition for a long time, and a negative shock hits, it affects most of
the collateral in the economy, causing a large decline in production. Not only there
is a reduction in production, but a lot of information is acquired at the same time.
In fact, even when there is no information production (in the range between pCh andpH ), firms borrow as if there were information production.
The government can produce and release some information over time to reduce thepotential size of the recession that a negative aggregate shock generates. The costof such a policy is to reduce production in the period at which the policy is implemented. The benefit is to reduce the size of the potential shock and to speed up
the recovery after such a shock. However, given the risk neutrality assumption thispolicy is typically not beneficial ex-ante, unless the government correctly predicts anegative aggregate shock is about to happen.

8

Some Empirical Evidence(Very Preliminary and Incomplete)

In this section we examine some of the model’s predictions using U.S. historical data.
The model predicts that:
1. The higher or longer the boom period during which output grows, the deeper
the crash when a negative shock is realized
2. The deeper the crash, the faster the recovery conditional on acquiring information and no collateral policy is implemented.
3. The standard deviation of beliefs (about collateral quality) declines over theboom
4. The longer the decline in the standard deviation of beliefs the more sharply thestandard deviation increases with a crash, because information is produced.
The first hypothesis is associated with the increasing credit build-up as more firms
borrow based on p̂. Since the average quality of the collateral that is the basis for34

borrowing is declining, when this is discovered when information is produced, more
firms will be cut-off from credit. The second prediction states that if information is
produced, then it is faster for the economy to recover when compared to the casewhere information was not produced. The third hypothesis relates the convergenceof beliefs to p̂ with the length of the boom. And finally, the fourth prediction is essentially the converse of the first one, beliefs move increasingly towards p̂ over time
until a negative shock is realized, possibly causing information to be produced.
To make tests operational, some definitions of terms like ”higher” and ”deeper” areneeded. Also, the model distinguishes between negative shocks such that when they
arrive information is produced (roughly, a depression) and shocks that cause firms tobe credit-constrained (roughly, a recession), but where no information is produced.
But, the shocks are unobservable, and whether information is produced or not is unobservable. In order to examine the hypotheses, we need a long time series so that
there are a sufficient number of crises. We want to examine a period over which there
is no central bank, so agents’ beliefs and actions are not affected by expectations associated with possible central bank or government intervention, which could contaminate the empirical data. For this reason we first focus attention on the United States
before the existence of the Federal Reserve System, that is, prior to 1915. We thenexamine the whole history of crises in the U.S. from 1815-2010, and the subperiod of
1915-2010, when the Federal Reserve is in existence.

8.1

The Pre-Federal Reserve Era

8.1.1

Data Description

To examine the pre-Fed period we will use the annual index of American industrial production, 1790-1915, produced by Davis (2004) and New York Stock Exchange
stock price data for the period 1815-1925, collected by Goetzmann, Ibbotson, andPeng (2001). Davis (2004) uses annual physical-volume data on 43 manufacturingand mining industries to construct an index. In this subsection we use the industrial production and stock price series through the year 1914, after which the Fed isin existence. The Davis industrial production series can be used to examine the firsttwo hypotheses. The third and fourth hypotheses concern the standard deviation ofbeliefs about collateral value. We will proxy for these beliefs using the cross section35

of stock returns. This is a measure of uncertainty which has been previously used by,e.g., Lougani, Rush, and Tave (1990), Brainard and Cutler (1993), and Bloom, Floetotto, and Jaimovich (2009). The idea here is that the standard deviation of the crosssection of stock returns should decline during the credit boom, as more and more
firms are borrowing based on collateral with a perceived value of p̂. That is the firms
are increasingly viewed as being of the same quality. The previous work cited above
takes this measure of uncertainty as exogenous; we are trying to determine how it
arises endogenously.
The focus of our empirical analysis is on the period prior to and just after a crisis, thetime at which there is a negative shock. So, we examine business cycles. Davis’s annual data results in a different business cycle chronology than the National Bureau of
Economic Research (NBER); see Davis (2006). The NBER dating has not been revised
since first announced (see the discussion in Davis (2004, 2006)). Table 1 below shows
the NBER business cycle chronology and Davis’ chronology. There are some NBER
cycles that Davis does not include. Some of Davis’ cycles do not display much of adownturn.14 Where the NBER and Davis agree on the cycle existence, there is also
agreement on the date of the peaks. There is most disagreement about the troughdates. The differences concerning the trough dates make dating the start of the recovery somewhat problematic.
We focus on Davis’s chronology, as it is the most current. This gives us a sampleof nineteen cycles, omitting the wartime cycles, which were somewhat special as theshock was arguably the start of the war rather than anything else.15 Our basic strategy
is to examine the four hypotheses over the business cycle, taking the date of the cycle
peak as the date that the negative shock arrives. We also take the date of the cycle
trough as the start of the recovery. Below, we discuss the trickier problem of thetiming of information production after the negative shock.
The period from 1790-1915 includes the National Banking Era, 1863-1914, which was
followed by the Federal Reserve System, starting in 1914. It also includes the Civil
War period, and the period prior to the Civil War where banks were overseen by thestates and issued their own private money. Broadly speaking we think of recessions
14
With regard to the cycles with peaks at 1811, 1822, and 1836, Davis (2004, p. 1203) states that thesecycles had losses ”that do not exceed the minimum postbellum loss.”
15
Davis (2004, p. 1203) says of the wartime cycles: ”Two Civil War cycles (1861 and 1865 troughs)
are omitted. Although their inclusion would not meaningfully affect calculations.”

36

that included a widespread bank panic as corresponding to instances where information was produced. Table 1 lists the major panics and minor panics, although there aremany more events that some have classified as minor panics. There is no consensus as
to which events constitute a panic in all cases.16 Part of the problem is that some panics were regional rather than national. We will use all the cycles, whether there was apanic or not. But, it is notable that a panic involves households monitoring banks bydemanding their money back, in other words, trying to produce information about
their banks. Indeed, panics are associated with the deepest recessions.
At the onset of a banking panic, banks suspended convertibility, an event organizedby clearinghouses during most of this period. During the period of suspension following a panic, some banks and firms would be discovered to be insolvent and would
be closed. This raises two issues. First, the suspension of convertibility was organized by bank clearinghouses, which acted to stop production of information aboutbanks; they suspended the release of bank-specific accounting information. See Gorton (1985). This institutional arrangement may have altered the predicted relations
following the shock. Second, if there is information produced, about firms independent of the bank clearinghouses, then the question of the issue of the timing of information production arises. We assume that information is produced rather quickly,
but this may not be correct.
Our basic empirical strategy is to look at these business cycles and measure threevariables (measured in different ways) based on Davis’ industrial production index.
We will measure the boom by looking at the number of years from the last troughto the business cycle peak (BOOM1) and also by the cumulative output from the lasttrough to the peak (BOOM2). We will also look at measures of the decline in economic
activity measured as the cumulative loss from the trough to the next year when theprevious peak level was achieved (LOSS1), and also at the cumulative loss from peak
to trough (LOSS2). Finally, we will measure economic recovery as the cumulative
gain from the trough to the next peak (GAIN1). Another measure of the recovery
is the cumulative industrial production gain from trough to two years after trough(GAIN2). Table 2 below lists the different variables names and their definitions.
As mentioned, we proxy for agents’ beliefs using the standard deviation of the crosssection of stock returns, based on the monthly stock price data from Goetzmann, Ib16

Jalil (2009) compares the different panic dates produced by different authors.

37

botson, and Peng (2001). As they discuss, there are some missing values. Because
Davis’ data are annual, we convert the monthly standard deviations to annual bysimple averaging.17 There are two more substantive issues. The first issue concerns
whether the cross section returns should be price-weighted (the only available measure for weighting). Goetzmann, Ibbotson, and Peng (2001) produce a return index
that is price-weighted. Price-weighting dampens the variation in the cross section of
returns.18 We consider both price weighted and unweighted measures.19 The correlation between the annual series of the standard deviation of the cross section of
returns with and without price-weighting is 0.78. Secondly, although the stock pricedata starts in 1815, there are between eight and twenty stocks until 1833. The year
1837, following President Andrew Jackson’s veto of the re-charter of the Second Bankof the United States, marks the beginning of the Free Banking Era, during whichsome states allowed free entry into banking. We will look at two periods, 1815-1914
and 1839-1914.
The Panic of 1893 was a major panic. Figure 11 illustrates some of the hypotheses that
we will examine more closely below. The figure shows the period 1889-1897, which
includes the business cycle with peak in 1892, trough in 1897, and the widespreadbanking panic in 1893. The cumulative loss in industrial production was 29.97 (Davis(2004)).20 For purposes of the figure, both BELIEFS1 (standard deviation, not priceweighted) and BELIEFS2 (standard deviation, price weighted) have been multipliedby 10,000. The figure shows the rise in the industrial production index (the solid line)
and the drop starting in 1892. BELIEFS1 falls, as predicted, and then sharply rises,
as predicted. This is followed by a decline. BELIEFS2 does not show a sharp rise. In
17

When monthly values are missing, the annual average is the average over the remaining months.
The entire year 1867 is missing; its annual value was interpolated.
18
They argue that bid-ask bounce is a large potential problem; see Goetzmann, Ibbotson, and Peng
(2001, p. 9).
19
The price-weighted standard deviation of the cross section of returns is calculated as follows:n
ST Dweighted,t =
n−1

n
X

!1/2
¯ t)
P RCW GTi,t × (reti,t − ret

i=1

pricei,t−1¯ t is the price-weighted average return for all
where P RCW GTi,t = Pn priceis the weight, ret
j,t−1
j=1
n
stocks at time t. Note that the standard deviation is multiplied by n−1
because the sum of P RCW GTi,tis 1, but for a small sample the weights are adjusted to make estimation unbiased.
20
Davis (2004) measures the cumulative loss as the ”sum of percentage-point shortfalls in the logarithm of the index between peak and subsequent years below the peak” (p. 1203).

38

6

general, there is this large difference between the two measures of beliefs.
Figure 11: The Panic of 1893

We now turn to examining the hypotheses. Hypothesis (1) predicts that measures of the boom in output
8.1.2 priorHypothesis
Testing
to the negative
shock should be positively correlated with measures of the loss (measured as apositive number) following the shock. Table 2 lists the relevant measures. Over the whole sample
We nowturn
examiningare:
the hypotheses. Hypothesis (1) predicts that measures
period,thetocorrelationsof the boom in output prior to the negative shock should be positively correlated
BOOM2as a positive number) following the shock. The
with measures of theBOOM1
loss (measured
LOSS1
0.182
-0.074
correlations should all be positive, and Table 3 shows the evidence is mixed.
LOSS2-0.013
0.026
The second hypothesis states that the LOSS measures should be positively correlated
correlations
shouldall absence
be positive,the evidenceis mixed.
with The
recovery(GAIN)
in theof socollateralpolicies.
In the period pre Federal

Reserve Bank this is indeed the case. Table 3 shows they are. There are, however,
The second hypothesis states that the LOSS measures should be negatively correlated with recovery
severalpotential
problems(GAIN).
But, they
are not:with our recovery measures. The first, referred to above,is that the NBER and Davis seem to frequently disagree on the dating of the trough,which is the startingGAIN1
point forGAIN2
the GAIN measures. Another potential problem may
LOSS1
0.389
0.365 one where info is produced and one where nobe that we have two types of crashes:
LOSS2 is produced.
0.154This is0.215
information
not exactly captured by correlation.

The third hypothesis predicts that the cumulative change in the standard deviation
There
are several
potential
problems
with isournegatively
recovery measures.
first,of cross
sectionof stockreturns(BELIEFS)
correlatedThewiththereferredrise into above, is that
the NBER and Davis seem to frequently disagree on the dating of the trough, which is the starting point
economic activity as more firms borrow BOOM. As the boom grows, the standard
for the GAIN measures. Another potential problem may be that we have two types of crashes: one
where info is produced and one where no information is produced. This is not exactly captured by39
correlation.

The third hypothesis predicts that the cumulative change in the standard deviation of cross section ofstock returns (BELIEFS) is negatively correlated with the rise in economic activity as more firms borrowand Implications,” University of California, Berkeley, working paper.

Lougani, Prakash, Mark Rush, and William Tave (1990), “Stock Market Dispersion and Un
Journal of Monetary Economics 25, 367-388.
deviation of the cross section of stock returns (BELIEFS) should fall, as more firms areperceived to be of quality p̂. The correlations are as predicted.
Table 3: Hypotheses 1, 2 and 3

BOOM1
BOOM2
GAINS1GAINS2LOSS1LOSS2Hypothesis 1
0.182-0.013-0.074
0.026
Hypothesis 2
0.389
0.154
0.365
0.215

BELIEFS1 BELIEFS2Hypothesis 3-0.161-0.317-0.190-0.346

Hypothesis (4) says the longer/larger the decline in the standard deviation of thecross section of beliefs prior to the shock, the more sharply the standard deviationis revised upwards following a crash, because information is produced. Beliefs arerevised (R-BELIEFS), if information is produced. We first examine the related prediction that the standard deviation of the cross section of returns (R-BELIEFS) increases
when shock arrives. That is, the average standard deviation of the change in the crosssection of returns from the year just before the peak through the peak year should bepositive. In fact, it is:• BELIEFS1, NOT price weighted, average: 0.098.
• R-BELIEFS2, price weighted, average: 0.031.
Hypothesis (4) then predicts that the decline in BELIEFS prior to the shock, corresponding to the credit boom, should be negatively correlated with the increase in thestandard deviation of the cross section of returns after the shock hits (R-BELIEFS),
because of information production. BELIEFS goes down and R-BELIEFS goes up if
there is information production. Also, measures of the recovery, from the trough onwards should be positive correlated with R-BELIEFS; both go up. We at look at thesepredictions over the two sample periods in Table 4.

40

1839-1913
GAIN1
GAIN2
BELIEFS1
BELIEFS2

R-BELIEFS1-0.174
0.054
0.418
0.366

R-BELIEFS2-0.150-0.3950.038
0.201

Table 4: Hypothesis 41822-1913 (in parenthesis, 1839-1913)
GAIN1
GAIN2
BELIEFS1
BELIEFS2

R-BELIEFS1-0.127 (-0.174)-0.099 (0.054)
0.814 (0.418)

R-BELIEFS2
0.051 (-0.150)
0.374 (-0.395)
0.637 (0.201)

Measures of the recovery (GAIN) should go up, and so should the standard deviationof the cross section of returns after the shock if information has been produced. Overthe period 1822-1913, this is true with R-BELIEFS2 and both measures of GAIN, but
it is not true for R-BELIEFS1. Over the period 1839-1913, this is only found for the
GAIN2 and R-BELIEFS1 combination.
The standard deviation of the cross section of returns should go down prior to theshock (BELIEFS), corresponding to the credit boom. After the shock hits, beliefs arerevised; the cross section of stock returns goes up (R-BELIEFS). The negative correlation between BELIEFS and R-BELIEFS is sometimes present over 1822-1913, but notover the period 1839-1913.
The evidence for hypothesis (4) is mixed at best. The test of this hypothesis has two
problems. First, as mentioned above, there is disagreement about the dating of thetroughs. Second, according to our theory information is produced in some eventsand not others. But, these can’t be distinguished. One possibility is to only look at
downturns that had panics, assuming that these events correspond to information
being produced. But, then a decision must be made about which events were panics.
This is the subject of more research.
In general, the evidence is mixed. It is stronger for predictions that concern the precrisis period, that is, hypothesis (3) that the upswing in production is accompanied bya decline in the standard deviation of the cross section of beliefs. Evidence on what
happens at or after the peak is much more mixed. There are many measurement
problems, including the uncertainty about the timing of the trough dates, the factthat annual data is probably not the best given that the panic date may be the date at41

which information starts to be produced. Also, since information production is notobservable, we are potentially mixing the two events. Further research is needed.

9Conclusions

In this paper we discuss the positive and negative effects of information-insensitive
debt. At the one hand, information-insensitive debt generates a credit boom because
firms with bad collateral are able to borrow when information is not produced. Thiseffect has been also highlighted in different contexts by Hirshleifer (1971) and Dang,
Gorton, and Holmström (2011). On the other hand, the longer an economy is issuinginformation-insensitive debt and the longer the credit boom, the larger the systemfragility, such that a negative aggregate shock to collateral of a given size is more
likely to create a large systemic reduction in output and consumption. Furthermore, if
the crisis triggers information production, the economy recovers faster in the absenceof collateral policies, but slower in their presence.

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44

3This institutional arrangement may have altered the predicted relations following the shock. Second, if
there is information produced, about firms independent of the bank clearinghouses, then the questionof the issue of the timing of information production arises. We assume that information is produced
rather quickly, but this may not be correct.
Table
1: Pre-Federal
Reserve BusinessBusiness Cycles
and Panics
Table 1:Pre-Federal
Reserve
Cyclesand Panics
NBER Chronology
PeakTrough
1796
1802
1807
1811
1815
1822
1825
1828
1833
1836
1839
1845
1847
1853
1856

1799
1804
1810
1812
1821
1823
1826
1829
1834
1838
1843
1846
1848
1855
1858

1860
1864

1861
1867

1869
1873
1882
1887
1890
1892
1895
1899
1903
1907
1910
1913

1870
1878
1885
1888
1891
1894
1896
1900
1904
1908
1911
1914

Davis Chronology
PeakTrough
Pre-Civil War Business Cycles
1796
1798
1802
1803
1807
1808
1811
1812
1815
1816
1822
1823

Bank Panics
Major
Minor
1796

1819
Jan. 1825

1828
1833
1836
1839

1829
1834
1837
1840

Nov. 1833
Oct. 1839
Oct. 1857

Sept. 1854
1856
1858
Aug. 1857
Civil War Cycles
1860
1861
1864
1865
Post-Civil War Pre-Federal Reserve Cycles
1873
1883

1875
1885

Sept. 1873
May 1884

1892
1895

1894
1896

May 1893

1903
1907
1910
1913

1904
1908
1911
1914

Dec. 1861

1890

Oct. 1907
July 1914

Sources: NBER; Davis (2004); Gorton (1988); Jalil (2009).

Our basic empirical strategy is look at these business cycles and measure three variables (measured indifferent ways) based on Davis’ industrial production index. We will measure the boom by looking at
the number of years from the last trough to the business cycle peak (BOOM1) and also by the45

4cumulative output from the last trough to the peak (BOOM2). We will also look at measures of thedecline in economic activity measured as the cumulative loss from the trough to the next year when theprevious peak level was achieved (LOSS1), and also at the cumulative loss from peak to trough (LOSS2).
Finally, we will measure economic recovery as the cumulative gain from the trough to the next peak
(GAIN1). Another measure of the recovery is the cumulative industrial production gain from trough to
two years after trough (GAIN2). Table 2 below lists the different variables names and their definitions.
Table 2: Variables
Variable
LOSS1

LOSS2
BOOM1
BOOM2
GAIN1
GAIN2
BELIEFS1

BELIEFS2

R-BELIEFS1

R-BELIEFS2

Table 2: Variables

Definition
Cumulative industrial production loss frompeak to the date where the part peak isachieved.
Cumulative industrial production loss frompeak to trough.
Years from trough to peak.

Comments
Output loss as a result of the negative
shock: peak to year when previous levelis achieved again.
Output loss as a result of the negative
shock: peak to next trough.
Measure of upswing prior to thenegative shock, as credit is growing.
Cumulative industrial production change Measure of upswing prior to thefrom prior trough to peak.
negative shock, as credit is growing.
Cumulative industrial production gain from Measure of recovery growth after
trough to next peak.
negative shock.
Cumulative industrial production gain from Another measure of recovery.
trough to two years after trough.
Standard deviation of the cross section of Measure of the agents’ beliefs during
stock returns, NOT price weighted, from the upswing prior to the negative shock,as credit is growing.
prior trough to peak.
Standard deviation of the cross section of Another such measure.
stock returns, price weighted, from priortrough to peak.
Standard deviation of the cross section of Measure of the agents’ beliefs after thestock returns, NOT price weighted, from negative shock, during which timethe year before the peak through peak.
information may be being produced.
Standard deviation of the cross section of Another such measure.
stock returns, price weighted, from theyear before the peak through peak.

As mentioned, we proxy for agents’ beliefs using the standard deviation of the cross section of stock
returns, based on the monthly stock price data from Goetzmann, Ibbotson, and Peng (2001). As they
discuss, there are some missing values. Because Davis’ data are annual, we convert the monthly
standard deviations to annual buy simple averaging. 4 There are two more substantive issues. The firstissue concerns whether the cross section returns should be price-weighted (the only available measure
for weighting). Goetzmann, Ibbotson, and Peng (2001) produce a return index that is price-weighed.
4

When monthly values are missing, the annual average is the average over the remaining months. The entire year
1867 is missing; its annual value was interpolated.

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