Appendix A: Sample contract – original and English translation (SAA 10,602, F. 1309)

FOR ONLINE PUBLICATION

Appendix A: Sample contract – original and English translation (SAA 10,602, F. 1309)

Xxxxx den 2e November 1772 compareerde voor mij Xxxxxx xxx xxx Xxxxx Openbaar Notaris binnen Amsterdam de xxxx Xxxxxxx xx Xxxxxxx Xxxxxx xx Xxxxx, voor en in de naam van zijn Compagnie luidende Xxxxxxx xx Xxxxxxx Xxxxxx xx Xxxxx & Co, Kooplieden binnen deeze stadt

Today, November 2, 1772, appeared before me, Xxxxxx xxx xxx Xxxxx, Public Notary in the City of Amsterdam, Xx. Xxxxxxx xx Xxxxxxx Xxxxxx xx Xxxxx, for and in the name of his company called Xxxxxxx xx Xxxxxxx Xxxxxx xx Xxxxx & Co, merchants in this town (hereafter: “the party present”).

en bekende bij deeze wel en deugdelijk schuldig te wezen aan de Xxxx Xxxxxxx Xxxxxxx, meede Xxxxxxx alhier de somma van 24.000 guldens bankgeld spruytende uyt hoofden en ter saake van sodanige somma als de selve den 22e Oktober laatstleden aan syn comp[arants] voorn[oemde] Compagnie heeft afgeschreven, […] en welke somma van f. 24.000 Bankgeld hij Comparant in de naam van zijn voorn[oemde] compagnie aanneemt

And declared to be indebted to Xx. Xxxxxxx Xxxxxxx, also merchant in this city for the sum of 24,000 guilders banco, originating from and relating to a withdrawal of such sum on October 22 last in favor of the present party’s said company, and the present party accepting that sum of 24,000 guilders banco in the name of said company.

en belooft aan voorn[oemde] Xxxx Xxxxxxx Xxxxxxx of zijn Co[mpagnies] rechthebbende kosten schadeloos alhier weeder te zullen restitueren en voldoen binnen de tijdt van ses maanden te reekenen van den 6 Oktober deeses jaars met den Interest van dien tegens vier percent ’t jaar en bij prolongatie gelijke interest

And promises to said Xx. Xxxxxxx Xxxxxxx, or his company’s legal representative, to return this sum (including any costs incurred), within the time of six months, counting from October 6 this year, with the interest of 4% annual, and in case of prolongation the same interest.

en zulks tot de volle en effectueele betaalinge toe tog de interessen te betaalen ieder 6 maanden des zo zal bij opeischinge of aflossinge den een den ander ses weeken voor de vervaltijd waarschouwent

And [promises] to pay the full and effective payment of the interest every six months

In case that the contract is not prolonged he will be notified 6 weeks in advance.

tot nakominge deezes verbind hij comparant zijn en zijn gemelde Compagnons persoon en goederen als na rechten en specialijk sodanige vijftienhonderd ponden sterling capitaal actien in de d’Oost Indische Compagnie van Engeland als tot London voor reekening van zijn comparants gemelde compagnie als pand ter minnen op de naam en reekening van gemelde H[eer] Xxxxxxx Xxxxxxx zijn getransporteerd […]

To honor this agreement, the present party pledges his own body and goods and especially 1500 Pounds Sterling capital in the stocks of the English East India Company, which have been transferred in London from the account of the present party’s company to the account of said Xx. Xxxxxxx Xxxxxxx as collateral.

[…]

en zulks meede een somma van f. 1500 indien deselve actien mogten komen te daalen op 180% en zo vervolgens van 10 tot 10 % om bij aflossinge en voldoeninge van gemelde capitaale somma gerescontreerd en geluiqideerd te werden, zullende de interessen van zodaanige restitutie kon te resteeren van dien dag af dat dezelve restitutie geschied is

And he also [promises] to transfer an amount of 1500 guilders banco if the price of said stock were to fall below 180% and similarly with every additional fall of 10%. Interest payments associated with these sums of money will be calculated until the moment the money is effectively transferred.

en hy comparant belooft meede in de naam van zyn gemelde Compagnie te zullen goed doen de provisie en onkosten die by ’t transporteren van dezelve Actien aan zijn compagnie zullen komen te vallen welk transport by aflossinge zal met ten geschieden door de correspondenten van zijn comparants gemelde Compagnie.

And he, the party present, promises in the name of his said Company to pay for the fees and other costs associated with transferring the stock to his Company the moment the loan is repaid, which will be arranged by the correspondents of the present party’s said company

Voorts verklaarde hy Comparant dezelve Heer Xxxxxxx Xxxxxxx specialijk te authoriseeren en consititueeren ommeindien zijn comparants gemelde compagnie in gebreken mogt komen te blijven de voorsz[egde] capitaale somma van f. 24000 bankgeld en interessen promptelijk te betaalen en voldoen ofte […] en meede zo wanneer bij vermindering der waarde van voornoemde Actien zijn comparants gemelde Compagnie op de eerste aanzegginge ’t surplus niet kwam te voldoen dezelve actien door een makelaar alhier ofte tot London te mogen verkopen omme daar uit te vinden ’t geene syn Xx[eles] uit kragte deezes zal zijn Competeerende ’t geene hy Comparant in de naam van zyn voornoemde Compagnie belooft voor goed vast en van waarde te houden en zoo wanneer dezelve minder mogten renderen zoo belooft hij comparant ’t mindere aan zijn Ed[elste] zullen opleggen en voldoen waar tegens gemelde Xxxx Xxxxxxx Xxxxxxx als meerdere aan zijn comparants gemelde Compagnie zal goed doen en hij Comparant bekende van syn Xx[ele] wegens voorsz[egde] actien een renvers[aal] te hebben ontvangen

Furthermore, the present party declares that, in case the present party’s company defaults on the obligation to repay said sum of 24,000 guilders banco and associated interest payments in a timely fashion, or when he fails (due to the fall in value of said stocks) to provide additional surplus after a first instigation, he authorizes Xx. Xxxxxxx Xxxxxxx especially to have the said stock sold through an official broker, either here or in London, and to retrieve from the proceeds the amount of money he is entitled according to this agreement with the present party’s company.

In case the sale yields less than the full amount, the present party promises to make up the difference. In case it yields more, Xx. Xxxxxxx Xxxxxxx will remit the resulting surplus.

The party present declares that he has received a counter-deed in reference to said stock.

Actum Amsterdam, 2 November 1772 Signed in Amsterdam, November 2, 1772

Appendix B: Additional figures and tables

Figure B. 1: Kernel densities haircuts before Christmas 1772

Raw Corrected for year and borr. FE

Density

.1 .2 .3 .4 .5

Haircuts

Raw vs corrected for year dummies and borrower fixed effects

May‐70 Oct‐70 May‐71 Nov‐71 May‐72 Nov‐72

Blaauw

Xxxxxxxxxxx

Xxxxxxxx and Xxxxxxxxx

Initial maturity First renewal Second renewal

Figure B. 2: The timing of collateralized loans extended by Xxxxx Xxxxxx Xxxxx, 1770-1772

Dec‐69 Jun‐70 Dec‐70 Jun‐71 Dec‐71 Jun‐72 Dec‐72 Xxxxxxx

Ximenes

Xxxxx Xxxxxx

Xxxxxx xx Xxxxx

Pie Rich & Xxxxxxxxx

Initial maturity First renewal Second renewal Third renew

Panel A: November (May) cycle Panel B: June (December) cycle

This figure illustrates the importance of timing in determining matches between lenders and borrowers with the example of lender Xxxxx Xxxxxx Xxxxx. Loan contracts were signed for 6 (or 12) months and were often silently renewed with another 6 (or 12) months. Xxxxx extended his loans either in the beginning of May/November or June/December. When loans were repaid after a multiple of 6 months, funds became available for new borrowers. The vertical axis indicates borrowing by different borrowers; the width of the bars indicates the size of the collateral behind a loan (in face value). The horizontal axis plots time and indicates when loans were originally extended and renewed.

Figure B. 3: Xxxxxx and borrower network – 1770-75

Figure B. 4: Surplus on loans to consortium

density

0 .05 .1 .15 .2

surplus (fraction of collateral value)

Margin Call (paper) Execution (realized)

Distributions of the ‘surplus’ on secured loans (the difference between the value of the collateral and the loan), right after the issuing of margin calls on December 29th and after the actual execution of the underlying collateral.

Figure B. 5: Distribution of EIC returns

1723 - 1772 (first half) 1770 - 1773 (first week)

.04

Density

.02

-30 -20 -10 0 10 20 30

6 month returns (%)

Returns calculated over 6 month periods (overlapping). Vertical line indicates the 6 month return over the second half of 1772.

1.5

Figure B.6: Debt and cash positions Consortium before Christmas 1772

log(debt/transactions) 2 3 4 5

log(balance/transactions) 0 .5 1

-.5

1770h1 1770h2 1771h1 1771h2 1772h1 1772h2

Consortium

95th perc. (excl. Cons.)

Rest of sample

date_yh

1770h1 1770h2 1771h1 1771h2 1772h1 1

Consortium

5th perc. (excl. Cons.)

Rest of sample

date_yh

Panel A: log(debti ,t / transactioni ,t ) Panel B: log(balancei ,t / transactionsi ,t )

These two figures calculate half-yearly averages of log debt and cash positions for the consortium compared to the mean and 95th/5th percentile of the sample.

Figure B. 7: Haircuts before and after Christmas 1772

Not exposed Exposed

before 1772 after 1772

Xxxxxxx

.0 .0 .0 .0

Xxxxxxxx

before 1772 after 1772

Density

.1 .2 .3 .4

Haircuts

Haircuts before and after Christmas 1772, differentiated by exposed and non-exposed lenders

Figure B. 8: Interest rates before and after Christmas 1772

Not exposed Exposed

before 1772 after 1772

Density

2.5 3 3.5 4

Interest rates (%)

before 1772 after 1772

Density

2.5 3 3.5

Interest rates (%)

Interest rates before and after Christmas 1772, differentiated by exposed and non-exposed lenders

Table B. 1: Descriptive statistics, EIC stock returns over 6 month periods (overlapping)

Sample	Prior to distress	Distress period	Full
	1723-1772*	1770-73**	1723-1794
Mean	0.0051	-0.034	0.0028
Median	0.0068	-0.019	0.0053
σ	0.089	0.108	0.089
Skewness	0.248	-0.49	-0.07
Maximum loss	-0.256	-0.358	-0.358
% of observations with loss>0.2	0.011	0.075	0.022

* first half ** first week of 1773

Table B.2: Haircuts and time varying borrower risks (dependent variable: haircuts)

	(1)	(2)	(3)	(4)	(5)	(6)
		Full sample			Pre-1773
log(debti ,t /	0.011***		0.009**	0.011**		0.011***
xxxxxxxxxxxxx,x )	(0.004)		(0.004)	(0.004)		(0.004)
log(balancei ,t /		-0.022***	-0.018**		-0.009	-0.016
xxxxxxxxxxxxx,x )		(0.007)	(0.007)		(0.013)	(0.012)
Non-EIC	-0.048***	-0.051***	-0.049***	-0.050***	-0.053***	-0.051***
	(0.008)	(0.008)	(0.008)	(0.008)	(0.008)	(0.008)
Lender type dummies	Y	Y	Y	Y	Y	Y
Borrower fixed effects	Y	Y	Y	Y	Y	Y
Adj. R2	0.565	0.560	0.573	0.541	0.522	0.542
N	317	317	317	272	272	272

This table presents estimates of the impact of two borower risk measures on haircuts, for the sample as a whole (cols 1-3) and for the period before xxx Xxxxxxxxxxx default (cols 4-6). debti,t: total margin loan position borrower i at time t. balancei,t(xxxxxxxxxxxxx,t): average daily balance (transaction volume) of borrower i in the Amsterdam Bank of Exchange during the 52 weeks prior to time i. All estimates include borrower fixed effects and lender type dummies. Robust standard errors (clustered at the lender level) are presented in parentheses, * p < 0.10, ** p < 0.05, *** p < 0.01.

Table B. 3: Lender attrition
	(1)	(2)	(3)	(4)
		Probit, 1 =	remains in sample
Exposed	0.075	0.079	0.145	0.128
	(0.054)	(0.052)	(0.088)*	(0.097)
Total lending before		6.230	5.438	5.806
1773 (£000)		(3.612)*	(3.618)	(3.635)
Fraction total lending			-0.090	-0.071
to consortium			(0.100)	(0.117)
Lender type dummies	N	N	N	Y
N	177	174	174	149
Adj. R2	0.012	0.041	0.045	0.062

Estimates of a probit model predicting whether lenders will remain in the sample. The table presents marginal effects, e.g. in Col (1) a lender is 7.5% more likely to stay in the sample if it was exposed to the consortium. Robust standard errors are presented in parentheses. * p < 0.10, ** p < 0.05, *** p < 0.01.

Table B. 4: Total lending

Including Van Seppenwolde Excluding Van Seppenwolde

	(1)	(2)	(3)	(4)	(5)	(6)
	Pooled OLS	Pooled OLS	FE	Pooled OLS	Pooled OLS	FE
Exposed	1.911	2.144		1.433	1.783
	(0.935)**	(1.166)*		(0.923)	(1.212)
Post 1772	0.326	0.175	-0.104	0.742	0.764	0.129
	(0.487)	(0.610)	(1.713)	(0.402)*	(0.517)	(2.275)
Exposed * Post	-1.186	-1.465	-3.648	-0.749	-1.232	-1.640
1772	(0.913)	(1.163)	(3.341)	(0.988)	(1.081)	(2.447)
non-EIC	3.337	3.499	2.782	2.243	1.832	1.720
	(1.346)**	(1.700)**	(3.612)	(0.895)**	(0.954)*	(3.345)
Constant	2.190	3.050	2.287	1.884	4.147	2.864
	(0.462)***	(1.405)**	(1.492)	(0.335)***	(1.518)***	(1.272)**
Year dummies	Y	Y	Y	Y	Y	Y
Lender type dummies Lender FE	N N	Y N	Y	N N	Y N	Y
N	202	175	202	128	113	128
N (if balanced panel)			50			30

R2 0.040 0.080 0.880 0.050 0.150 0.955

# lenders 177 152 177 113 99 113

Regression estimates for total lending at the lender level on the collateral of all English securities. Total lending is calculated before and after Christmas 1772; in £000s of face value of collateral. Exposed lenders are those who were forced to liquidate collateral after the events of Christmas 1772. The interaction between the post-1772 and the exposed dummies captures the diff-in-diff effect. Lender type dummies are as in Table

3. Lender fixed effects refer to fixed effects on the family level. Robust standard errors (clustered at the lender level) are reported in parentheses. * p < 0.10, ** p < 0.05, *** p < 0.01.

Table B. 5: Haircuts and time since event (dependent variable: haircuts)

	(1)	(2)	(3)	(4)	(5)
	OLS	OLS	FE	FE	FE
Exposed	-0.005	-0.003		-0.000
	(0.005)	(0.005)		(0.006)
Exposed * Post 1772	0.097 (0.030)* **	0.086 (0.033)***	0.086 (0.046)*	0.054 (0.030)*	0.101 (0.045)* *
Time since event	-0.001	0.023	0.008	-0.065	0.010
	(0.058)	(0.059)	(0.066)	(0.058)	(0.073)
Exposed * time since	-0.051	-0.041	-0.051	-0.031	-0.058
event	(0.044)	(0.044)	(0.045)	(0.042)	(0.048)
non-EIC	-0.058 (0.006)* **	-0.055 (0.007)***	-0.047 (0.012)* **	-0.053 (0.008)* **	-0.046 (0.014)* **
Constant	0.218 (0.007)* **	0.244 (0.018)***	0.243 (0.026)* **	0.212 (0.012)* **	0.460 (0.032)* **
Year dummies	Y	Y	Y	Y	Y
Lender FE	N	N	Y	N	Y
Borrower FE	N	N	N	Y	Y
Lender type dummies	N	Y		Y
Borrower type dummies	X	X	X
X	000	000	000	000	000
X (xx balanced panel)			166	77	33

R2 0.334 0.440 0.632 0.659 0.802

# groups (lenders)	177	152	177	152	177
# groups (borrowers)	72	70	72	70	72

Regression estimates for all English securities. Observations refer to new contracts and are weighted by the face value of the collateral. Haircuts are calculated as the fraction of the collateral value that is not financed with a loan. Exposed lenders are those who were forced to liquidate collateral after the events of Christmas 1772. The interaction between the Exposed and the Post 1772 dummies captures the diff-in-diff effect. Time since event is measured in years. The interaction between the Exposed and Time since event dummies captures the reversion of the treatment effect. For example, in Column 3 the immediate treatment effect on haircuts is .08 and decreases by .04 every year. Lender and borrower type dummies are as in Table 3. Xxxxxx and borrower fixed effects are at the family/firm level. Robust standard errors (clustered at the lender level) are reported in parentheses. * p < 0.10, ** p < 0.05, *** p < 0.01.

Table B. 6: Details of borrower-time fixed effects

Non-exposed: Exposed:

Lender Haircut Nom. value coll. (£)

Type Lender Haircut Nom. value coll. (£)

Type Diff.

Panel A: Borrower: Xxxxxxxxx

Xxxxxx	0.00	0000	X,0	Xxxxx	0.22	1000	M,9
Xxxxxxx	0.00	0000	X,0	Xxxxxxxxx	0.17	1000	P,7
Straalman	0.18	2500	M,N,7	Pereira	0.21	5000	M,S,10
				Steenis	0.25	4000	P,F,8
Wtd. av.:	0.18			Wtd. av.:	0.22			0.04
Panel B: Borrower: Penha
Poorten	0.15	2000	M,9	Xxxxxxxx	0.23	1000	M,P,10
Rutgers	0.30	1000	M,6	Boddens	0.31	1000	M,8
Straalman	0.30	1000	M,N,8	Bors	0.31	2000	N,F,8
				Hagen	0.31	2000	?,6
Wtd. av.:	0.23			Wtd. av.:	0.30			0.07
Panel C: Borrower: Tile
Winter	0.26	1000	M,7	Boddens	0.26	1000	M,10	0.00

Weighted difference: 0.05

This table documents the different transactions that drive the borrower-time fixed effects results. Information is restricted to the three borrowers who, after Christmas 1772, borrowed from both exposed and non-exposed lenders (Xxx Xxxxx xxx Xxxxxxxxx, Xxxxxxx Xxxxxxxxx & Xxxxx, and Xxxxxxx Xxxxxxxxxx Xxxx). The left hand side of the table shows lending transactions with exposed lenders, the right hand side presents transactions with non- exposed lenders. We aggregate information by borrower-lender pair. All averages are weighted by the total face value of the collateral underlying a set of transactions. All collateral refers to EIC stock. Lender type: P – Xxxxxxxxx, M – Merchant, N – Noble, F – Female, S – Specialist. Numbers refer to the decile of total lending each lender belongs to. E.g. P, 7 means that a lender is a Patrician whose total lending is in the 7th decile of the distribution of total lending activity.

Table B. 7: Extensive and intensive margin (dependent variable: haircuts)

	(1)	(2)	(3)
	Pooled OLS	Pooled OLS	Pooled OLS
Exposed	-0.003	-0.003	-0.003
	(0.005)	(0.004)	(0.006)
Exposed * Post 1772	0.066	0.052	0.077
	(0.023)***	(0.028)*	(0.039)**
non-EIC	-0.056	-0.056	-0.056
	(0.006)***	(0.007)***	(0.006)***
Absolute position with		-0.000
consortium (£ 000s)		(0.000)
* Post 1772		0.002
		(0.003)
Relative position with			-0.001
consortium (fraction)			(0.011)
* Post 1772			-0.026
			(0.038)
Constant	0.245	0.247	0.246
	(0.017)***	(0.016)***	(0.017)***
Year dummies	Y	Y	Y
Lender type dummies	Y	Y	Y
Borrower type dummies	Y	Y	Y
N	387	387	384
R2	0.440	0.443	0.442

Regression estimates for all English securities. Observations refer to new contracts and are weighted by the face value of the collateral. Haircuts are calculated as the fraction of the collateral value that is not financed with a loan. Exposed lenders are those who were forced to liquidate collateral after the events of Christmas 1772. The interaction between the Exposed and the Post 1772 dummies captures the extensive margin of adjustment. The absolute position with the consortium measures the total amount of the collateral the consortium had pledged with a specific lender around Christmas 1772 (in (£ 000s face value). The relative position with the consortium divides this measure by the total amount of collateral that was pledged with a specific lender before Christmas 1772. The interactions with the post-1772 dummy capture the intensive margin of adjustment. We do not measure this with a triple interaction because the position with the consortium for non-exposed lenders is always

0. Standard errors for the absolute and relative position measures are 5.26 and 0.39 respectively. Robust standard errors (clustered at the lender level) are reported in parentheses. * p < 0.10, ** p < 0.05, *** p < 0.01.

Table B. 8: Haircuts, excluding January 1773

	(1)	(2)	(3)	(4)	(5)
	Pooled	Pooled	FE	FE	FE
	OLS	OLS
Exposed	-0.005	-0.002		-0.001
	(0.005)	(0.005)		(0.006)
Exposed * Post 1772	0.068	0.058	0.050	0.039	0.060
	(0.022)***	(0.023)**	(0.035)	(0.024)	(0.037)
non-EIC	-0.059	-0.055	-0.047	-0.053	-0.046
	(0.006)***	(0.007)***	(0.012)***	(0.008)***	(0.014)***
Constant	0.218	0.246	0.245	0.210	0.458
	(0.006)***	(0.016)***	(0.024)***	(0.012)***	(0.034)***
Year dummies	Yes	Yes	Yes	Yes	Yes
Lender FE	No	No	Yes	No	Yes
Borrower FE	No	No	No	Yes	Yes
Xxxxxx observables	No	Yes		Yes
Borrower	No	Yes	Yes
observables
N	412	381	412	381	412
N (if balanced panel)			160	73	42
R2	0.299	0.296	0.229	0.706
# lenders	177	152	177	152	177
# borrowers	69	67	69	67	69

Regression estimates for all English securities. Observations refer to new contracts and are weighted by the face value of the collateral. Observations for January 1773 are excluded. Haircuts are calculated as the fraction of the collateral value that is not financed with a loan. Exposed lenders are those who were forced to liquidate collateral after the events of Christmas 1772. The interaction between the Exposed and the Post 1772 dummies captures the diff-in-diff effect. Lender and borrower observables are as in Table 3. Xxxxxx and borrower fixed effects are at the family/firm level. Robust standard errors (clustered at the lender level) are reported in parentheses. * p < 0.10, ** p < 0.05, *** p < 0.01

Table B. 9: Xxxxxxxx et al. analysis on haircuts – collapsing data into pre- and post-1772

	(1)	(2)	(3)	(4)
Exposed	-0.020		-0.005
	(0.010)*		(0.006)
Exposed * Post 1772	0.080 (0.025)***	0.052 (0.023)**	0.085 (0.016)***	0.080 (0.019)***
Post 1772	-0.027	0.007	-0.025	-0.012
	(0.020)	(0.019)	(0.014)*	(0.017)
Non-EIC	-0.042 (0.013)***	-0.037 (0.022)*	-0.057 (0.008)***	-0.039 (0.017)**
Lender type dummies	Y		Y
Lender fixed effects	N	Y	N	Y
Weighted	N	N	Y	Y
N	175	202	175	202
R2	0.157	0.848	0.439	0.845

In this table, we collapse our data into two periods only: pre- and post-1772, as suggested by Xxxxxxxx, Xxxxx, and Xxxxxxxxxxxx (2004). That means that we have at most two observations per lender. In cols 1 and 2 we assign each lender-period observation the same weight. In cols 3 and 4 we weight observations by the total lending activity of lender in that period (measured by the total face value of accepted collateral). Standard errors are reported in parentheses * p < 0.10, ** p < 0.05, *** p < 0.01.

Appendix C: Concentration of lending

To test if random matching of lenders and borrowers can explain the nature of lending in our sample, we calculate the Herfindahl index for every lender during the pre-crisis period:

where is the share of lending by lender to an individual borrower . If lenders repeatedly lent to the same borrower, to the exclusion of other investors, we would expect a high Herfindahl index. The left panel of Figure C. 1 presents the actual distribution of these Herfindahl indices for all lenders in ours sample. Many lenders only entered into a single transaction; these are highlighted for the observations where the Herfindahl index equals 1.43

The distribution is discontinuous, with zero weight between 0.68 and 1. This is the result of the way a Herfindahl index is constructed and the fact that most lenders only do a few transactions.

To compare the actual distribution with one arising by chance, we randomly pick a lender from our set of actual lenders. We determine how many new loan contracts he or she entered into before Christmas 1772, and then randomly draw a corresponding number of counterparties (taking into account that some borrowers are more active than others). Finally, we calculate the resulting Herfindahl index, and repeat the exercise 10,000 times. As the figure demonstrates, the two distributions are nearly identical. Both xxx Xxxxxxx Chi2 and the log likelihood test for the equality of distributions fail to reject.44

We use the Herfindahl indices to test whether the (possible) destruction of existing credit networks after xxx Xxxxxxxxxxx bankruptcy might explain our empirical findings. The idea is that lenders who lost their network would have been forced to lend to new borrowers. Since these individuals were relatively unknown, they would have initially charged higher haircuts. We start from the assumption that lenders that are heavily invested in a particular client relationship will have more concentrated portfolios. We then estimate the following equation

where includes time effects and both borrower and lender characteristics. is a random error. captures whether lenders increased haircuts more if they engaged in more

43 The y-axes are aligned to reflect equal fractions. Grey bars reflect lenders who entered into at least 2 transactions. The white bars indicated lenders who only lent out once.

44 P-values 0.43 and 0.505.

relationship lending before Christmas 1772 (a higher Herfindahl index). Table C.1 (Col 1) shows that this is not the case; if anything a higher degree of concentration before Christmas 1772 (more relationship lending) leads to lower haircuts. This effect is not statistically

significant though.

In Col 2 we include a triple interaction effect between the Herfindahl index, the post- 1772 dummy and the exposed dummy. This captures whether exposed lenders who had a more concentrated lending portfolio changed haircuts more aggressively after Christmas 1772. The idea is that exposed lenders with a relatively concentrated loan portfolio would have faced a larger disruption of their network. The triple interaction effect is insignificant and negative, suggesting that, if anything, exposed lenders with a more concentrated loan portfolio charged lower haircuts after xxx Xxxxxxxxxxx default.

Actual Herfindahl indices

Herfindahl indices from random draws

0 .2 .4 .6 .8 1

Xxxxxxxxxx Xxxxxxxxxx

Lenders with single transaction Lenders with multiple transactions

Frequency

100

150

Frequency

2000 4000 6000

8000

Figure C. 1: Herfindahl indices – actual vs simulated

For each lender we calculate the Herfindahl index of its lending before Christmas 1772. In addition, we construct a random distribution of Herfindahl indices. We randomly pick a lender from our set of lenders; we determine how many (x) new loan contracts it entered into before Christmas 1772; we randomly draw x counterparties (taking into account that some borrowers are more active than others); and we calculate the resulting Herfindahl index. We do this 10,000 times. The y-axes are aligned to reflect equal fractions. Grey bars reflect lenders who entered into at least 2 transactions. The white bars indicated lenders who only lent out once.

Tests on the equality of the distributions:

Test statistic

pvalue

Xxxxxxx'x Chi2

83.8

0.435

Log likelihood ratio

Obs. (Unique values)

37.9

178 (84)

0.505

Table C. 1: Haircuts and concentration lending before Christmas 1772

	(1)	(2)
Exposed	-0.002	0.022
	(0.005)	(0.012)*
Exposed * Post 1772	0.056	0.075
	(0.028)**	(0.056)
non-EIC	-0.056	-0.055
	(0.006)***	(0.006)***
Herfindahl (pre-event)	0.006	0.028
	(0.008)	(0.014)**
Xxxxxxxxxx (pre-event) * Post 1772	-0.030	-0.011
	(0.036)	(0.070)
Herfindahl (pre-event) * Exposed		-0.037
		(0.017)**
Herfindahl (pre-event) * Exposed * Post		-0.022
1772
		(0.088)
Constant	0.244	0.228
	(0.017)***	(0.018)***
Year dummies	Y	Y
Lender & borrower type dummies	Y	Y
N	384	384
R2	0.443	0.452
# lenders	149	149

Pooled OLS estimates for all English securities. Observations refer to new contracts and are weighted by the face value of the collateral. Haircuts are calculated as the fraction of the collateral value that is not financed with a loan. Exposed lenders are those who were forced to liquidate collateral after the events of Christmas 1772. The interaction between the Exposed and the Post 1772 dummies captures the diff-in-diff effect. The Herfindahl index (0-1) measures the concentration of a lender’s portfolio before Christmas 1772. The double interaction between Xxxxxxxxxx and Post 1772 captures whether all lenders with higher degrees of concentration charged higher haircuts after Christmas 1772. The triple interaction between Xxxxxxxxxx, the Exposed and Post 1772 captures whether exposed lenders with a higher degree of concentration adjusted haircuts more. Lender and borrower type dummies are as in Table 3. Robust standard errors (clustered at the lender level) are reported in parentheses.

* p < 0.10, ** p < 0.05, *** p < 0.01

Appendix D: Direct exposure to EIC price movements

It is possible that individuals who lent to the consortium overall had strong exposure to EIC stock through other portfolio holdings. Then, changes in haircuts could reflect managing this risk, rather than the shock of the default.

To investigate this issue we estimate the following equation

where includes time effects and both borrower and lender characteristics. is a random error. This equation tests whether exposed lenders in general charge higher haircuts when EIC prices are lower. Results are presented in Table 12.

Col 1 includes the interaction between the exposed dummy and the EIC stock price. The economic size of the coefficient is small and statistically insignificant. The average EIC price during 1770-1772 was 212%; in 1773-1775, it was 155%. The price decline corresponds an increase in haircuts by 1.9% (0.57*0.033). This is less than a third of the impact of the interaction effect with the post-1772 dummy (Table 6, Col 2). Col 2 includes

both interaction effects to perform a horserace: what has more explanatory power, the post- 1772 dummy or changes in the price of EIC stock? The estimates show that the interaction effect with the post-1772 dummy is much stronger; it increased haircuts by 6.8%. The coefficient on the interaction between exposed and the EIC price is now wrongly signed. Overall, these results show that EIC stock prices have no additional predicative power above and beyond the post-event dummy.

Table D.1: EIC factor (dependent variable: haircuts)

	(1)	(2)
	Pooled OLS	Pooled OLS
Exposed	0.003	-0.009
	(0.007)	(0.008)
Exposed * EIC price	-0.033	0.047
	(0.030)	(0.038)
EIC price	0.049	-0.015
	(0.029)*	(0.035)
Exposed * Post 1772		0.068
		(0.035)*
Constant	0.245	0.252
	(0.022)***	(0.023)***
Year dummies	Y	Y
Lender type dummies	Y	Y
N	288	288
R2	0.320	0.332
# lenders	127	127

Pooled OLS regression estimates for EIC stock only. Observations refer to new contracts and are weighted by the face value of the collateral. Haircuts are calculated as the fraction of the collateral value that is not financed with a loan. Exposed lenders are those who were forced to liquidate collateral after the events of Christmas 1772. EIC prices are in fractions of the face value. Average price before Christmas 1772 2.12, after Christmas 1772 1.55. The estimates in Col 1 indicate that such a price fall causes haircuts demanded by exposed to increase by 0.019 (0.57*0.033). The interaction between the Exposed and the Post 1772 dummies in Col 2 captures the benchmark diff-in-diff effect. Lender type dummies are as in Table 3. Robust standard errors (clustered at the lender level) are reported in parentheses. * p < 0.10, ** p < 0.05, *** p < 0.01

Appendix E: Further robustness checks

Disaggregation of haircut components

The change in the haircut we document can be disaggregated into two parts – the difference between the price at which a contract is signed and the pre-agreed level when a margin call is triggered, and the difference between the trigger level and the value of the loan. In Table E. 1, we analyse the shift in the haircut for its two components separately.

In Panel A, we examine the difference between market price and the trigger level for a margin call. The lenders who were exposed to the default increased the trigger level substantially, by 4-5 percent – very close to the change in the overall collateral requirements. In Panel B, we analyze the distance to loss, the difference between the margin trigger and the value of the loan. Here, there are only relatively small and mostly insignificant effects – lenders adjusted the risk profile of their lending by demanding margin earlier, and keeping the value of the loan overall lower relative to the market value at the time of signing.

Table E. 1: Disaggregation of haircut components

(1) (2) (3) (4) (5)

Pooled OLS

Panel (A): Distance to margin call

Pooled OLS

FE FE FE

Exposed	-0.009	-0.005		-0.006
	(0.007)	(0.006)		(0.006)
Exposed * Post 1772	0.063	0.042	0.039	0.028	0.046
	(0.023)***	(0.024)*	(0.036)	(0.028)	(0.043)
non-EIC	-0.036 (0.006)***	-0.029 (0.006)***	-0.029 (0.009)***	-0.027 (0.007)***	-0.031 (0.011)***
Constant	0.131 (0.006)***	0.167 (0.014)***	0.167 (0.020)***	0.116 (0.009)***	0.069 (0.026)***
R2 0.135		0.293	0.630	0.534	0.804
Panel (B): distance to loss Exposed 0.004		0.002		0.007
(0.006)		(0.005)		(0.006)
Exposed * Post 1772	0.012	0.022	0.025	0.010	0.027
	(0.012)	(0.012)*	(0.018)	(0.015)	(0.018)
non-EIC	-0.024 (0.007)***	-0.027 (0.006)***	-0.020 (0.008)**	-0.027 (0.007)***	-0.019 (0.010)*
Constant	0.087 (0.005)***	0.081 (0.017)***	0.088 (0.021)***	0.095 (0.008)***	0.098 (0.026)***
R2 0.306		0.390	0.672	0.614	0.825
Year dummies	Yes	Yes	Yes	Yes	Yes
Lender FE	No	No	Yes	No	Yes
Borrower FE	No	No	No	Yes	Yes
Lender type dummies Borrower type dummies N	No No 405	Yes Yes 374	Yes 405	Yes 374	405
N (if balanced panel)			166	77	33
# lenders	177	152	177	152	178
# borrowers	72	70	72	70	72

Regression estimates for all English securities. Haircut = distance to margin call + distance to loss. Observations refer to new contracts and are weighted by the face value of the collateral. Exposed lenders are those who were forced to liquidate collateral after the events of Christmas 1772. The interaction between the Exposed and the Post 1772 dummies captures the diff-in-diff effect. Lender and borrower type dummies are as in Table 3. Xxxxxx and borrower fixed effects are at the family/firm level. Robust standard errors (clustered at the lender level) are reported in parentheses. * p < 0.10, ** p < 0.05, *** p < 0.01.

East India Stock only

In the baseline results, we use lending against all assets in our database – East India stock, 3% annuities, and Bank of England stock. While we control for compositional change, it is interesting to examine how much of a shift we can find by focusing on EIC stock exclusively (the asset against which xxx Xxxxxxxxxxx syndicate predominantly borrowed).

In Table E. 2, Panel A, we show that lending requirements in EIC stock changes in very much the same fashion as in the universe of all assets. In the pooled estimation (Col 2), the coefficient suggests a rise in collateral requirements by 6.8 percent. The fixed effect estimates look very similar to the benchmark numbers in Table 6. However, estimates become (borderline) insignificant. This is because with fixed effects, the effective number of observations that can be used to identify the interaction effect is constrained to those that are in the sample before and after 1772. In addition, we lose observations by constraining the sample to EIC transactions.

In Panel B, we analyze lending against non-EIC assets only. Due to the limited number of observations, the fixed effect specifications cannot be estimated. The pooled OLS estimates are very similar to those for loan contracts collateralized with EIC stock. For example, the estimate of the interaction effect in Col 2 is 6.6% (versus 6.8% in Panel A). Overall, there is no reason to think that the estimated effects in our baseline specification only reflect changes in haircuts in one type of asset.

Table E. 2: Haircuts – different types of collateral

	(1) Pooled OLS	(2) Pooled OLS	(3) FE	(4) FE	(5) FE
Panel A: EIC only
Exposed	-0.002	0.000		-0.000
	(0.008)	(0.008)		(0.008)
Exposed * Post 1772	0.072	0.068	0.074	0.037	0.055
	(0.025)***	(0.028)**	(0.045)	(0.026)	(0.044)
Constant	0.222 (0.006)***	0.240 (0.016)***	0.225 (0.032)***	0.210 (0.013)***	0.169 (0.046)***
N	314	288	314	288	314
N (if balanced panel)			134	65	41
# lenders	147	127	147	127	147
# borrowers	60	57	60	57	60
R2 0.132		0.296	0.561	0.599	0.787
Panel B: BoE, SSC and 3% Annuities
Exposed	-0.007	-0.005
	(0.008)	(0.007)
Exposed * Post 1772	0.102 (0.016)***	0.066 (0.027)**
Constant	0.158 (0.007)***	0.226 (0.019)***
N	104	99
# lenders	70	64
# borrowers	27	26
R2	0.072	0.284
Year dummies	Yes	Yes	Yes	Yes	Yes
Lender FE	No	No	Yes	No	Yes
Borrower FE	No	No	No	Yes	Yes
Lender type dummies Borrower type dummies	No No	Yes Yes	Yes	Yes

Regression estimates for EIC and BoE, SSC and 3% Annuities separately. Observations refer to new contracts and are weighted by the face value of the collateral. Haircuts are calculated as the fraction of the collateral value that is not financed with a loan. Exposed lenders are those who were forced to liquidate collateral after the events of Christmas 1772. The interaction between the Exposed and the Post 1772 dummies captures the diff-in- diff effect. Lender and borrower type dummies are as in Table 3. Xxxxxx and borrower fixed effects are at the family/firm level. Due to a limited number of observations the fixed effects models cannot be estimated for the non-EIC securities. Robust standard errors (clustered at the lender level) are reported in parentheses.* p < 0.10,

** p < 0.05, *** p < 0.01

Outliers

It is possible that a few, extreme values for the haircuts influence our results. A standard way to deal with this issue is to winsorize the data. We winsorize the top and bottom 5 percent of observations, and re-estimate (see Table E.3). The results are largely unchanged. Coefficients

are significant throughout, and are statistically indistinguishable from those in the baseline specification. For completeness we do the same for interest rates and re-estimate our benchmark results. Again, results are virtually unchanged.

Table E. 3: Haircuts – Winsorized dependent variable

	(1) Pooled OLS	(2) Pooled OLS	(3) FE	(4) FE	(5) FE
Exposed	-0.005	-0.003		-0.001
	(0.005)	(0.004)		(0.006)
Exposed * Post 1772	0.072 (0.020)***	0.064 (0.022)***	0.060 (0.032)*	0.040 (0.022)*	0.059 (0.031)*
Non-EIC	-0.057 (0.006)***	-0.054 (0.006)***	-0.047 (0.011)***	-0.051 (0.008)***	-0.045 (0.014)***
Constant	0.219 (0.006)***	0.240 (0.014)***	0.236 (0.022)***	0.214 (0.011)***	0.199 (0.033)***
R2	0.369	0.470	0.677	0.655	0.842
Year dummies	Y	Y	Y	Y	Y
Lender FE	N	N	Y	N	Y
Borrower FE	N	N	N	Y	Y
Lender type dummies	N	Y		Y
Borrower type dummies	X	X	X
X	000	000	000	000	000
X (xx balanced panel)			166	77	33
# lenders	177	152	177	152	177
# borrowers	72	70	72	70	72

Regression estimates for all English securities. Observations refer to new contracts. Haircuts are calculated as the fraction of the collateral value that is not financed with a loan. Exposed lenders are those who were forced to liquidate collateral after the events of Christmas 1772. The interaction between the Exposed and the Post 1772 dummies captures the diff-in-diff effect. Observations are weighted by the face value of the collateral; the top and bottom 5% of the haircut distribution are Winsorized. Lender and borrower type dummies are as in Table 3. Xxxxxx and borrower fixed effects are at the family/firm level. Robust standard errors (clustered at the lender level) are reported in parentheses. * p < 0.10, ** p < 0.05, *** p < 0.01.

Table E. 4: Interest rates – Winsorized dependent variable

	(1) Pooled OLS	(2) Pooled OLS	(3) FE	(4) FE	(5) FE
Exposed	0.064	0.042		0.060
	(0.035)*	(0.033)		(0.041)
Exposed * Post 1772	-0.019	-0.008	-0.059	0.026	0.070
	(0.077)	(0.077)	(0.086)	(0.080)	(0.165)
non-EIC	-0.072	-0.087	-0.076	-0.104	-0.076
	(0.036)**	(0.033)**	(0.045)*	(0.049)**	(0.054)
Constant	3.534 (0.033)***	3.617 (0.092)***	3.690 (0.094)***	3.583 (0.070)***	3.620 (0.145)***
R2 0.464		0.515	0.736	0.659	0.840
Year dummies	Y	Y	Y	Y	Y
Lender FE	N	N	Y	N	Y
Borrower FE	N	N	N	Y	Y
Lender type dummies	N	Y		Y
Borrower type dummies	X	X	X
X	000	000	000	000	000
X (xx balanced panel)			166	77	33
# lenders	177	152	177	152	177
# borrowers	72	70	72	70	72

Regression estimates for all English securities. Exposed lenders are those who were forced to liquidate collateral after the events of Christmas 1772. The interaction between the Exposed and the Post 1772 dummies captures the diff-in- diff effect. Observations are weighted by face value of the collateral; the top and bottom 5% of the distribution are Winsorized. Robust standard errors (clustered at the lender level) are reported in parentheses. * p < 0.10, ** p < 0.05,

*** p < 0.01

Extreme observations

The final step is to examine the sensitivity of our results to the influence of a single lender or borrower. To this end, we re-estimate the baseline specification (Table 6, Col 2), dropping one lender or borrower at a time. Figure E. 1 Panels A-D shows the distribution of

coefficients (first row) and t-statistics (second row). The range of estimated coefficients is

small, with results ranging from 5.5 to 7.5 percent. The t-statistics never falls below 2. This shows that our results are not driven by a single lender or borrower.

Figure E. 1: Outlier analysis, dropping one lender (borrower) at a time

Dropping one lender at a time Dropping one borrower at a time

.05

.055

.06

.065

.07

.075

coef. estimate of interaction effect

.05

.055 .06

.065

.07

.075

coef. estimate of interaction effect

density 40

density

100 150

200

250

(A) Coefficient estimate (B) Coefficient estimate

2.2 2.4 2.6 2.8 3

3.2

t-stat of interaction effect

2.5

t-stat of interaction effect

3.5

density

1 1.5

2.5

density 2

Coefficients on the interaction effect and t-statistics are generated dropping one lender (or borrower) at a time. All estimates include lender and borrower observables.

Appendix F: Primary Sources

GAR: Gemeentearchief Rotterdam (City Archives Rotterdam); NA: Nationaal Archief (Dutch National Archives); OSA: Oud Archief van de Stad Rotterdam (Old Archives City of Rotterdam); SAA: Stadsarchief Amsterdam (City Archives Amsterdam)

SAA (library), ‘Stukken betreffende den boedel van Xxxxxxxx en Zoonen’, 1773-1779 SAA, Notariële protocollen Xxxxxx xxx xxx Xxxxx, 5075: 10,593 - 10,613 (various notary

protocols)

SAA, Tex den Bondt aanvulling 1 en 2, 30269: 347 (‘Staat en inventaris van de boedel van Xxxxxxxx xxx Xxxxxxxxxxx’)

SAA, Archief van de Stads Beleenkamer, 5043: 1 (‘Notulen van de vergaderingen van het ‘Fonds tot maintien van het publiek crediet’ (1773)’)

NA, Xxxxxxx van de familie Xxx xxx Xxxxx xxx Xxxxxxxx, 3.20.54: 381, 386, 396 (various correspondence)

OSA, 1.01: 3710 (‘Stukken betreffende de kasgeldlening groot fl. 300.000 door de stad aan J. en X. xxx Xxxxxxxxxxx, kooplieden te Amsterdam’)

GAR, Archief van de Maatschappij van Assurantie, Belening, etc., 199: 5, 40, 354 (various accounts and letters)

XXX, Xxxxxxx xxx Xxxxx Fundatie, 90: 52, 56 (various letters)

De Koopman, Vol. IV ( 1772-1773) (Dutch periodical)

Appendix G: Full model solution1

The appendix is structured as follows. In Section G.1 we first describe the setup of the model and the underlying assumptions, including the specific loan contract and matching frictions that we consider. Next, in Section G.2, we analyze xxx Xxxx bargaining problem that borrowers and lenders solve when they meet. To do so, we first derive each agent’s value function from obtaining a loan. We prove Proposition 3 in the main text that describes under what conditions borrowers will always accept a loan from a relatively pessimistic agent. In Section G.3 we then derive the equilibrium price as a function of outcomes in the loan market. We provide the conditions under which the general equilibrium exists and is unique. In Section G.4 we prove Proposition 1 in the main text that states that the optimal contract is always risk-free. We also show that Proposition 2 in the main text follows logically from the preceding results. In Section G.5, we analyse the local comparative statics of the model in closed-form and we prove Lemmas 4 – 7. Finally, in Section G.6 we provide global results through numerical analysis.

G.1 Setup and Assumptions

We model the market for collateralized lending as a matching market with frictions. Time is continuous and there is an infinite horizon. We focus on a steady state solution.

Apart from a risk free storage technology (with an instantaneous interest rate of zero), there is a single risky asset that is in unit supply. Following Geanakoplos (2003), the asset has a binomial payout where the good and the bad state occur with probability 1/2. The timing of the payout is uncertain and follows a Poisson process with intensity v > 0.

There are three type of agents in the market i ∈ {1, 2, 3} with masses Ni who are all

risk neutral and have a subjective discount rate of zero. Each agent is infinitesimally small. Crucially, the agents have different beliefs about the asset payout. They all agree that in the good state of the world the asset will pay r, they disagree about the payoff in the bad state of

the world: r1 < r2 < r3. Expected payouts are given by vi. All agents have a cash endowment of si. For simplicity, we assume that only type 3 agents will ever be cash constrained.

We focus on the case where v2 < p < v3. After the derivation of the equilibrium, we provide sufficient restrictions on the model’s primitives to arrive at this case. In this scenario, only type 3 agents are willing to buy the asset. They will try to borrow from type 1 and type 2 agents to 1We are indebted to Xxxxxx Xxxxx and Xxxxxx Xxxxxxxx for their assistance in the development of this appendix.

increase their asset holdings above and beyond what they can buy with their own capital s3. In

what follows, we will initially derive the general case where N1 N2. However, to generate

closed-form solutions, we will restrict the analysis to the special case where N1 = N2.

G.1.1 Contract Space and Loan Contracts

We consider the following (restricted) contract space. First, there are shorting restrictions and agents can only buy and hold the asset. Second, agents can contract loans from each other to increase their asset holdings. Loans are collateralized with the asset and are non-recourse. (In equilibrium this assumption turns out to be irrelevant since a borrower will invest all of her wealth in the asset.)

A loan contract stipulates the size of the loan per unit of the asset lj and an interest rate ρj

for j ∈ {1, 2}. In our stylized setting, loan contracts do not have a fixed maturity: a contract

ends if (1) the asset pays out, or if (2) the contract breaks down with an exogenous Poisson intensity λ. This greatly facilitates the computation of the steady state equilibrium. Interest payments are made lump sum at the end of the contract and are independent of the realized contract length. This may seem unrealistic, but notice that agents’ subjective discount rates are zero. Interest rates therefore only reflect the underlying risk of the collateral or the transfer of surplus from borrower to lender, both of which are independent of the length of the contract.

We assume that if the contract ends before the asset pays out, the interest payment is zero. The borrower simply uses the proceeds from selling the collateral to repay the principal of the loan. We need to make this assumption to ensure the existence of a steady state solution, but there is a clear economic intuition behind it. Again, interest rates only capture risk or the transfer of surplus. In steady state, a loan contract ending before the asset pays out is always risk free: the price at which the asset was bought and the price at which the collateral is sold to repay the loan has to be the same by the definition of a steady state. Furthermore, if a loan contract ends before the asset pays out, there is no surplus generated yet that can be shared.

If the loan contract ends upon the asset payout, there are two possible scenarios. In the good state of the world, the asset payout is always sufficient to settle both principal and interest. In the bad state of the world, this may not be the case: the borrower may be forced to default. In that case, a lender can seize the asset’s payout without any additional costs. It will charge a risk premium in the good state of the world to be compensated for this default risk. Crucially, borrowers and lenders can disagree about whether there will be default in the bad state of the

world or not. If a borrower is more optimistic, he might believe that the return in the bad state of world will be sufficient to repay principal and interest, while the lender believes this is not the case.

G.1.2 Population Dynamics

We closely follow Xxxxxx, Xxxxxxxx and Xxxxxxxx (2005) in setting up the population dynamics. Define Mj(t) as the total mass of matches between type 3 agents (borrowers) and type j ∈

{1, 2} agents (lenders) at time t. Let Uj(t) = Nj — Mj(t) be the unmatched mass of type j

lenders and U3(t) = N3 — M1(t) — M2(t) the unmatched mass of borrowers. N is the total

mass of agents in the economy. Lower case variables refer to proportions relative to N and are, by definition, bounded in between 0 and 1.

For simplicity, we assume that only borrowers actively search for lenders. They search with intensity µ and they cannot ex ante distinguish between type-1 and type-2 lenders or other type- 3 agents. Under the exact law of large numbers for random independent matches, a borrower is matched with a lender of type-j at a total rate of µu3(t)uj(t). At the same time, a fraction of existing contracts breaks down with intensity λ. The rates of change for u3(t) and mj(t) for

j ∈ {1, 2} are therefore given by

du3(t) = —µu3(t) [u1(t) + u2(t)] + λ [m1(t) + m2(t)] (G.1)

dmj(t) = µu3(t)uj(t) — λmj(t) (G.2) with the following restrictions

mj(t) + uj(t)	=	nj	(G.3)
u3(t)	=	n3 — m1(t) — m2(t)	(G.4)
n1 + n2 + n3	=	1	(G.5)

Proposition G.1 There exists a unique steady-state solution to equations (G.1)-(G.5).

Proof. Conjecture that a steady-state exists. Start by substituting (G.3)-(G.5) in (G.1) and setting the right hand side of (G.1) to 0. This gives the following quadratic equation:

G(u3) Ξ u2µ + u3(µ + λ — 2µn3) — λn3 = 0.

Now notice that G(0) < 0 and G(1) > 0, which implies that one of xxx xxxxx of G(.) must be negative while the other must lie in (0, 1). Let u3 be the positive root for the quadratic problem

above. To show that u3 < n3, all we need to prove is that G(n3) > 0. This follows directly from the fact that G(n3) = n3µ(1 — n3) > 0.

After setting the right hand side of (G.2) to zero, using (G.3), and realizing that mj =

u3µ+λ

u3µ nj and u3 is bounded in (0, 1), it is immediate that the equilibrium values for m1 and m2 are also bounded in (0, 1). When we analyze the special case where N1 = N2, it is straightfor- ward to see that u1 = u2 and m1 = m2.

G.2 The Loan Decision

G.2.1 Value Functions

We first define the borrowers’ steady state value functions. We define qj∗ as the number of assets a borrower can buy in equilibrium when he is matched to a type j lender, and N∗3,j as the expected profit per unit of the asset from a type-j loan for j ∈ {1, 2}. Vj is the value function from signing a loan contract with a type-j lender. qO∗ and N∗3,O are defined as quantities and expected profits when a borrower is not matched to a lender; VO is the corresponding value function.

We first conjecture that a steady state exists in which the price p, loan quantities lj∗ and interest rates ρj∗ are constant and in which Vj ≥ VO (full matching). In this case, the value functions are given by the following:

v + µ

V = v q∗N∗ + µ [u V

+ u V

+ (1 — u

— u )V ] (G.6)

V = v q∗N∗ + λ V

(G.7)

1 v + λ 1 1 v + λ O

V = v q∗N∗ + λ V

(G.8)

where

2 v + λ 2 2 v + λ O

and

N∗O = v3 — p (G.9)

qj∗

N∗ = 1 r — (1 + ρ )l∗ — s3 + 1 max r — (1 + ρ )l∗ — s3 , — s3 ,

lj + max

r3 — (1 + ρj )lj , 0

— p. (G.10)

which, realizing that —s3/qj∗ = lj∗ — p, can be rewritten as

Nj = 2 (r — ρjlj ) + 2

∗ 1 ∗

1 ∗

∗ }

The total amount of assets qj∗ that a borrower can purchase depends on the amount of capital

s3 he has and loan size lj∗:

Defining haircuts as

qj∗(p — lj∗) ≤ s3.

k∗ = p — lj∗ , (G.11)

j p

this constraint can be rewritten more intuitively as

qj∗ × p × k∗j ≤ s3:

the borrower’s capital should cover the total value of the assets purchased times the haircut. As long as Nj∗, the borrower’s expected profit per unit of the asset, is strictly positive, the borrower will want buy as much of the asset as possible and the constraint will be binding, in which case

q∗ = s3

p — lj∗

for j ∈ {1, 2}, and

q∗ = s3 .

O p

Solving equations (G.6)-(G.8) in terms of quantities and expected profits, we arrive at

VO =

(v + λ)2qO∗NO∗ + (v + λ)µu1q1∗N1∗ + (v + λ)µu2q2∗N∗2

(v + λ)[v + λ + µ(u1 + u2)]

(G.12)

V = (v + λ)λqO∗NO∗ + [(v + λ)(v + µu1) + vµu2]q1∗N∗1 + λµu2q2∗N∗2

(G.13)

1 (v + λ)[v + λ + µ(u + u )]

V = (v + λ)λqO∗N∗O + λµu1q1∗N1∗ + [(v + λ)(v + µu2) + vµu1]q2∗N∗2 . (G.14)

2 (v + λ)[v + λ + µ(u + u )]

Intuitively, each value function is a weighted average of all the possible payoffs. Under the exact law of large numbers, these weights can interpreted as the probabilities that a certain payoff N∗j will be realized.

We then consider the lenders’ value functions, where Lj is the value of not having a contract

for a type-j lender and Lj is the value of being in a loan contract. In steady state, the value functions are given by:

v + µ

v + λ

Lj = µ (u Lj + (1 — u )Lj ) (G.15)

v + λ

2 ρjlj + 2 [min

Lj = v q∗ 1 ∗ ∗ 1

{r , l∗(1 + ρ∗)} — l∗] + λ Lj , (G.16)

solving the system of equations in terms of interest rates and loan sizes, we arrive at the fol- lowing results:

v + λ + µuj

2 ρjlj + 2 [min

Lj = µuj q∗ 1 ∗ ∗ 1

{r , l∗(1 + ρ∗)} — l∗] (G.17)

v + λ + µuj

2 ρjlj + 2 [min

Lj = v + µuj q∗ 1 ∗ ∗ 1

{r , l∗(1 + ρ∗)} — l∗] . (G.18)

G.2.2 Nash Bargaining Solution For The Optimal Contract

When a borrower and lender meet they negotiate over the terms of the contract. We assume that they Nash bargain over the total surplus of the contract, where borrowers have bargaining power 8 ∈ [0, 1] that is determined outside the model. Before we solve the bargaining problem, note the following:

● We conjecture that the optimal debt contract is always risk free from both the perspective of the borrower and lender. We prove this after the derivation of the equilibrium. This means that lj(1 + ρj ) ≤ rj .

● As long as lj∗, ρ∗j ≥ 0, a lender will always be at least indifferent between signing a loan contract or not. We assume that a lender will always go ahead with the loan contract if the borrower strictly prefers to sign the contract.

● Borrowers do not always strictly prefer to go ahead with a loan. Specifically, they might decide to turn down a type-1 contract. In what follows, we focus on a “full matching” equilibrium where Vj ≥ VO for j = 1, 2 and a borrower will always accept a type-1 loan. After the derivation of the equilibrium, we will calculate the constraints on the model’s primitives that are sufficient to guarantee this.

● There are cases (specifically when lenders have all the bargaining power) that a borrower is indifferent between signing a loan contract or not. As long as the lender strictly prefers to go through with the loan, we assume that the borrower will go ahead and sign the contract.

Xxx Xxxx bargaining problem can be formulated as

lj,p

max(Vj — VO)8(Lj — Lj )1—8 (G.19)

such that

We define SB

lj ≥ 0

ρj ≥ 0

lj(1 + ρj ) ≤ rj

= Vj — VO as the borrower’s surplus associated with a type-j loan and SL =

Lj — Lj as the lender’s surplus. We can rewrite each agent’s surplus as follows:

j = (r + vu—j)qjNj — rqONO — vu—jq—jN—j

SL = e q 1 1 {r , l (1 + ρ )} — l ]

where

j j j

2 ρjlj + 2 [min j j j j

r =

v + λ + µ(u1 + u2)

vµ

6 =

, (G.20)

, and (G.21)

ej =

(v + λ)(v + λ + µ(u1 + u2))

v . (G.22)

v + λ + µuj

As long as the optimal contract is risk-free, we can use the expressions from (G.9)-(G.10) to rewrite the surpluses as:

SB = (r + 6u

) (v

— p — ρ l ) — r (v

— p) — 6u

—j

— p — ρ

(Gl .2)3)

j —j

p — lj

j j p 3

—j p — l 3

—j —j

SL = e 3 ρ l

(G.24)

j j p — l j j

The Lagrangian is given by:

—

L(lj, ρj ; η, l∗ j, ρ∗ j ) = (SB)8(SL)1—8 + η1lj + η2ρj + η3(rj — lj(1 + ρj ))

and xxx Xxxx-Xxxxxx necessary conditions for the constrained maximization problem are the following:

6SB

8(SB )8—1(SL)1—8 j

6SL

+ (1 — 8)(SB)8(SL)—8 j + x

— η (1 + ρ ) = 0 (G.25)

j j 6lj

6SB

j j 6lj 1 3 j

6SL

8(SB )8—1(SL)1—8 j

+ (1 — 8)(SB)8(SL)—8 j + x

— η l

= 0 (G.26)

j j 6ρ

2 3 j

We solve this problem in multiple steps:

(lj, ρj, rj — lj(1 + ρj )) ≥ 0 (G.27)

(η1lj, η2ρj, η3(rj — lj(1 + ρj )) = 0. (G.28)

1. Multiply (G.25) by lj and (G.26) by ρj , thus eliminating the restrictions related to η1 and

η2. We can thus rewrite (G.25) and (G.26) as the following:

(SB )8—1(SL)—8

"8SLl

6SB

+ (1 — 8)SBl

6SL #

= η l (1 + ρ ) (G.29)

j j j

j 6l

j j 6l

3 j j

(SB )8—1(SL)—8

"8SLρ

6SB

+ (1 — 8)SBρ

6SL #

= η ρ l

(G.30)

j j j

j 6ρ

j j 6ρ

3 j j

2. Suppose that the last constraint is binding such that rj = lj(1 + ρj ). Isolate η3 in (G.30) and substitute in (G.29) to obtain the following:

8SLl

6SB

+ (1 — 8)SBl

6SL

6SB

= 8SL(1 + ρ ) j

6SL

+ (1 — 8)SB(1 + ρ ) j .

j 6lj

j j 6l j

j 6ρ

j j 6ρ

This leads to the following solution for the optimal value of lj:

l (8, l

) = β(8, l

)r + (1 — β(8, l

)) "rj

— (v3 — p) + F—j p)#

where:

j —j

β(8, l—j) =

—j j

—j

8(v3 — rj )

n—j

1 + F—j

n—j

8(v3 — r ) + (1 — 8)(p — r )

1 +

—j

h F i ∈ [0, 1]

j j n—j

and

n—j = r + 6u—j

F—j

= rv3 — p + 6u p

v3 — p — ρ—jl—j p — l—j

—j

Note that the optimal loan contract lj(8, l—j) is the best response function, given the equi- librium contract with a type —j lender. We have therefore arrived at x Xxxx equilibrium that is characterized by the following system of equations:

2 F2 p — (p — r1)(v3 — p — r1)3

l1 = β(8, l2)r1 + (1 — β(8, l2)) 4

p 1 + F2

— r1

5 (G.31)

2 F1 p — (p — r2)(v3 — p — r2)3

l2 = β(8, l1)r2 + (1 — β(8, l1)) 4

1—8

p—rj

—j

+ 8

1—8

p—rj

a—j + (1 — 8)aj(1 — a—j) + (1 — a—j)8

which yields the following solution:

p 1 + F1

— r2

5 , (G.32)

r h$

+ 8 v3—rj i + (v

— p) ha

+ (1 — 8)a (1 — a

) — $

+ (1 — a

)8 p i

∗

—j

p—r—j

lj (8, p) =

h v3—r i

(v —p) h

p i

(G.33)

where:

aj =

r r

nj r + 6u—j

v + λ

v + λ + µuj

, and (G.34)

$8 = 1 — (1 — 8)(1 — a1)(1 — a2) (G.35)

µ2u1u2

= 1 — (1 — 8)(v + λ + µu ) (v + λ + µu ). (G.36)

1 2

Note that in the special case where N1 = N2 , we get that u1 = u2 and a1 = a2 = a.

If the constraint lj(1 + ρj ) ≤ rj is not binding (such that η3 = 0), it can be shown that no equilibrium exists. Borrowers and lenders can always find a better allocation by adjusting lj and ρj up to the point that the constraint binds. Intuitively, type-3 agents will borrow the maximum amount at which the loan contract is still risk-free.

3. Next, we explore the boundary cases where 8 = {0, 1}. For these two cases, we can solve the maximization problem in the same way as before. The solutions to lj∗(8, p) are identical to the limiting cases when we take 8 → 1 or 8 → 0 in equation (G.33), proving that the solution is continuous in 8. The solutions are given by lj → rj when 8 → 1 (full

bargaining power to the borrower) and lj → r

p j v3

when 8 → 0 (full bargaining power to

the lender). Note that in the case where 8 = 0 ,

k∗j (8, p) = k∗j (8)

= v3 — rj v3

and the equilibrium haircut does not depend on the price.

(G.37)

4. Finally, we calculate the equilibrium value for ρ∗j (8, p). Notice that the constraint lj(1 +

ρj ) ≤ rj is binding. This means that ρj∗(8, p) is given by:

ρ∗(8, p) = rj — lj∗(8, p). (G.38)

lj∗(8, p)

j v3

Is the solution l∗(8, p) feasible? The solution is bounded between the interval hr p , r i,

which is positive if rj > 0. Since lj(1 + ρj ) ≤ rj , this implies that the interest rate interval for ρj∗(8, p) is also bounded on the nonnegative side. We must now check if both SB ≥ 0 and

SL ≥ 0. Since ρjlj ≥ 0 in equilibrium for any value of 8, it follows that SL ≥ 0.

G.2.3 Proof of Proposition 3

As indicated before, it will not always be the case that SB

≥ 0. Specifically, it might not be

optimal for a borrower to accept a loan from a type-1 lender. We can calculate the restrictions on the model’s primitives that will guarantee that a borrower will always accept this loan.

Proof. of Proposition 3 (V1 ≥ VO).

The first step of the proof is provided by the following Lemma:

Lemma G.2 If SB l∗(1, p), ρ∗(1, p) > 0, then SB l∗(8, p), ρ∗(8, p) ≥ 0 68 ∈ [0, 1]

Proof. The case for 8 = 0 is trivial since the SB equals 0 if the lender has all the bargaining power. Define S(8) as short-hand for S(lj∗(8, p), ρj∗(8, p)). We prove the Lemma by contradic-

tion. Suppose Iˆ8 ∈ [0, 1] such that SB(ˆ8) < 0. Since SB(8) is continuous in 8, I˜8 ∈ (ˆ8, 1) such

that SB(˜8) = 0. This implies that SB(ˆ8)^8SL(ˆ8)1—^8 ≥ SB(˜8)^8SL(˜8)1—^8 = 0 ⇒ SL(ˆ8) = 0.

If the lender’s surplus is equal to 0, this is only possible if lj∗(ˆ8, p)ρj∗(ˆ8, p) = 0. Since the

loan contract is risk free, this implies that lj∗(ˆ8, p) = rj , resulting in SB(ˆ8) = SB(1) > 0, a

j j

contradiction.

The intuition for this Lemma is straightforward. The reason a borrower would not want to accept a type-1 loan would be that he is better off waiting for a type-2 lender. When 8 = 1, the borrower has full bargaining power and will capture all the surplus from the transaction. When 8 < 1, he will have to give some of that surplus to the lender. This means that if it is optimal to accept a type-1 loan when 8 = 1, it also has to be optimal when 8 < 1.

As a result, we only have to verify the case when 8 = 1, which is relatively simple since (lj∗(1, p), ρ∗j (1, p)) = (rj, 0). Setting 8 = 1 and substituting the expressions for N∗j from (G.9)- (G.10) into expression (G.23) for SB, the surplus from a type-1 loan, we find that the borrower’s surplus from this contract is strictly positive when 8 = 1 if and only if:

1 > a

p — r1 + (1 — a )p — r1

2 p 2

p — r2

z @¸,1 c z >¸,1 c

where a2 is given by (G.34). Isolating r2, and focusing on the special case where a1 = a2, we arrive at inequality (2) in the main text. We can also isolate r1 to obtain the following inequality:

r > p(1 — a2)r2

p — a2r

The RHS is strictly decreasing in p. Therefore, if we substitute v2 for p, we can derive the following general condition that holds for all v2 < p < v3.

r > v2 — a2v2 r

v2 — a2r2

z @¸,1 c

The inequality is intuitive. It states that the beliefs of type 1 and 2 lenders should not be too far apart. The maximum distance depends on a2. If a2 is close to 1, matching frictions are quite severe – either µ lies close to zero or u2, the fraction of free type-2 lenders in the population, is close to 0 – and the inequality holds for any r2. If a2 is close to 0 and matching frictions are negligible, the inequality will never hold.

G.3 General Equilibrium and Proof of Existence.

G.3.1 Market Clearing and Equilibrium Prices

The market clearing condition requires that total demand equals the (unit) supply of the asset:

1 = q∗(n

— m — m ) + q∗m + q∗m

= s3 u

s3 s3

+ m1 + m

(G.39)

p p — l1∗(8, p) p — l2∗(8, p)

It is not feasible to find a closed form solution for all values of 8, but we can easily calculate

the solution for the case when 8 = 0:

p∗(8 = 0) = Ns u

1 v

— r

+ m + m

v3 (G.40)

2 v

— r

G.3.2 Proof of existence

Starting with the closed form solution for p∗ when 8 = 0, we can prove the existence of a unique equilibrium for each possible value of 8, restricting the set of parameters such that v2 < p∗(0) < v3. The proof proceeds as follows.

1. We first calculate the restrictions on the parameter space such that 6p/68 ≥ 0 6(8, p). This is an intuitive condition that implies that if borrowers have more bargaining power, they will have more funding at their disposal, and this will lead to a higher price for the asset.

2. We then apply a version of Xxxxxx’x Existence and Uniqueness Theorem for Ordinary Differential Equations (ODEs) to prove existence and uniqueness.

Lemma G.3 As long as v3 > p, v2 > v3 — r1 will guarantee that 6p/68 ≥ 0 6(8, p).

Proof. Applying the implicit function theorem to (G.39), the derivative is equal to the following expression:

h1(8,p)2

h2(8,p)2

m1 6h1 + m2 6h2

68 = — 1 + m1 6h1 + m2 6h2

(G.41)

Nc3

h1(8,p)2 6p

h2(8,p)2 6p

8(v — r ) $ +3 8j

where kj(8, p) Ξ p — lj∗(8, p) /p is the equilibrium haircut, given by:

p$8 + (v3 — rj )1—8

+ (v3 — p)

a—j + (1 — a—j)

1—8 .

kj(8, p) =

8 p h

p—rj

h p ii

8 p—r—j + (1 — 8)aj

(G.42)

Expression (G.42) shows that the sign of 6p/68 directly depends on the signs of 6kj/68 and

6kj/6p.

1—8

(p—rj )2

—j

(p—r—j )2

p—rj

3 —

—j

—

—j

The first order derivative of kj with respect to p is equal to:

6kj

(v — r ) $

+ 8 h 8

(v3—rj )rj + (1 — a

)8 (v3—r—j )r—j i

3 —

1—8

6p np$

+ (v

r ) 8 p + (v

p) ha

+ (1

a ) h8 p + (1

8)a iio2

p—r—j

—

(G.43)

where aj and $8 are given by (G.34)-(G.35). This derivative is always positive for 6(8, p). We

proceed by listing the conditions under which the derivative of k with respect to 8 is positive. Given the complexity of the solution for 6k1/68, we list the conditions for the function to be strictly decreasing in 8 based on expression (G.42) for k(8, p) itself. If the denominator grows at a rate higher than the numerator with respect to 8, then it means that the function decreases.

1—8

Given that both $8 and 8 are strictly increasing in 8, p > v3 — r1 and v3 — p > 0 are necessary

and sufficient conditions for 6k1/68 > 0.

Having v3 —p is a necessary restriction for borrowers to be willing purchase the asset, which we assume is always the case. Condition p > v3 — r1 is less straightforward. Since there is no closed form solution for the equilibrium price p, we must define a lower bound that will hold for any feasible value of p. Given that p > v2, assuming that v2 > v3 — r1 ensures that the price is strictly increasing in 8, no matter the equilibrium p, as long as v2 < p < v3.

Proposition G.4 As long as v2 < p∗(0) < v3 and v2 > v3 — r1, there exists an unique solution to the problem defined by (G.19).

Proof. We use a variant of Xxxxxx’x Existence and Uniqueness Theorem for ODEs to prove this proposition.2 The initial condition for the ODE is provided by p∗(0) Ξ pO, defined by (G.40).

2See Xxxxxx and Davidson (2012) pp. 87 Theorem 5 for one of the many references in the literature.

We approximate the differential with the first order derivative of the conjectured equilibrium price function dp/d8 ≈ 6p/68. Let the RHS of (G.41) be defined as 5 (8, p), then we can rewrite (G.41) as

pj = 5 (8, p). (G.44)

If 5 (8, p) is continuous in both 8 and p and continuously differentiable with respect to p on the rectangle Y = {(8, p) : 0 ≤ 8 ≤ 1, a ≤ p ≤ b}, then if (0, pO) ∈ Y, there exists an unique solution to (G.44) with p(0) = pO.

First, we must define Y in our specific setup. Under the restriction listed in Lemma G.3, the equilibrium price is increasing in terms of 8. It is therefore natural to define {a, b} = {pO, p1} where p1 Ξ p∗(1). We already defined pO – clearly (0, pO) ∈ Y. Now, we must characterize p1.

Given lj∗(8 = 1, p) = rj , p1 is defined by the following equation:

p = Ns u

1 p

— r

+ m + m

p1 . (G.45)

2 p

— r

where p1 is a positive real root of the underlying cubic equation. By Xxxxxxxxx’ rule of signs, the solution can have either one or three positive real roots. Algebraic manipulation reveals that there is only one real root, which therefore has to be positive. Let b = p1 be this root.

Next we ensure that p1 < v3. We start from equation (G.45). Notice that the LHS is strictly

increasing in p1 while the RHS is strictly decreasing in p1. Therefore, if v3 is such that:

v > p

= Ns u

1 v

— r

+ m + m

2 v

— r

then it must be that p1 < v3. Note that this restriction is an upper bound and might be too restrictive.

Finally, we need to make sure that 5 (.) is continuous and continuously differentiable on p over Y. Analyzing equations (G.41)-(G.43), first notice that the image of kj(.) for (8, p) ∈ Y is bounded. The solution p(8) is an increasing function of 8, and therefore, if p1 < v3 and pO > v2, it follows that v2 < p < v3 for any p ∈ Y. This condition ensures that all the functions inside (G.41) are continuous except potentially when 8 = 1. Now we already know that kj(1, p) = p/ p — rj < ∞ for any p ∈ Y, which implies that 6kj/6p|8=1 < ∞ and 6kj/68|8=1 < ∞,

ensuring that 5 (.) is continuous in Y. In order to show that 5 (.) is continuously differentiable

with respect to p, one can easily define an open neighborhood U such that Y c U but none of the prior restrictions are violated, and show that partial derivatives of 5 (.) exist for (8, p) ∈ U ,

68p

682

6p2

thus proving that 5 ∈ C1. The partial derivatives will depend on 62hj , 62hj and 62hj which

again can be shown to be continuous in U , including the case when 8 = 1.

G.3.3 Aggregating Restrictions on Parameters

In the preceding analysis we have listed a number of restrictions to ensure that a unique, full matching equilibrium exists where v2 < p < v3. We now summarize these restrictions:

1. The proof of Proposition 3 states that r1 > v2—a2v2 r2 for V1 > VO (full matching equilib-

v2—a2r2

rium). Define

g Ξ v2 — a2v2 . (G.46)

v2 — a2r2

2. We know that for the price to be increasing in 8 (a necessary condition for uniqueness and existence of the equilibrium), v2 > v3 — x0, xx 0x0 + r2 > r3.

v3—r1

v3—r2

3. In addition, we need that v2 < pO < v3, where pO = Ns3 u3 + m1 v3 + m2 v3 .

Grouping these three restrictions, the model’s primitives must satisfy the following:

g2r2	<	r1 < r2	(G.47)
r2	<	r3 < (1 + 2g2)r2	(G.48)
r2	<	2pO — r < r3	(G.49)

Intuitively, the first two restrictions state that beliefs cannot lie too far apart. If beliefs diverge by too much, the equilibrium price will fall below v2, or borrowers will start to decline type-1 loan contracts. The third restriction effectively defines an upper and lower bound for what s3 can be. If s3 is too small, the price will dip below v2, if it is too large, the price will exceed v3.

G.4 Proofs of Propositions 1 and 2

Next, we prove Proposition 1 in the main text that states that the risk-free contract is the optimal contract, under the set of parameter restrictions defined before. Borrowers and lenders have different beliefs about r. This means that their expectations about whether the borrower will default in the bad state of the world might also differ.

Proof. of Proposition 1 (the equilibrium loan contract is always risk free: lj(1 + ρj ) = rj ).

We consider two deviations from the risk-free contract, one in which both borrowers and lenders belief that default in the bad state of the world will occur, one in which only the (more pessimistic) lenders think the borrower will default.

1. Default according to both borrowers and lenders: r3 < lj(1 + ρj ) ≤ r.

This case is ruled out because there are no gains from trade. We prove this by contradic- tion. Suppose there exists a contract with gains from trade. From (G.10) and (G.15) we can rewrite borrowers’ and type-j lenders’ profit functions (per unit of the asset) as

j j

1 r — p — ρ l + 1 [l — p] ≥ 0

j j

1 ρ l + 1 r — l ≥ 0

respectively. Combining the two inequalities implies that

rj ≥ lj(1 — ρj ) ≥ 2p — r ⇒ rj ≥ 2p — r ⇒ vj ≥ p.

This is a contradiction since p > vj.

2. Default according to lenders; full repayment according to borrowers: rj < lj(1 + ρj ) ≤

r3.

In this case, the lender’s surplus is given by

SL = e

s3 1

(ρ l — l

+ r ).

j j p — l 2 j j j j

This is the average of the contract’s payoff in the good and bad state of the world. Since the borrower does not expect to default, his surplus function is unchanged and given by (G.23).

Define

xB (8) = v3 — p — ρ∗j (8, p)lj∗(8, p) and (G.50)

p — lj∗(8, p)

xL(8) = ρj∗(8, p)lj∗(8, p)

(G.51)

p — lj∗(8, p)

where lj∗() and ρj∗() are the optimal risk-free contracts. xB(8) can be interpreted as a borrower’s profit from the risk-free contract with a type-j lender, per unit of his own capital, s3. xj is the lender’s profit. Any risky contract (ρ , lj) > (ρ∗, l∗) that attempts to

j j j j

maximize total surplus for any 8 ∈ [0, 1] must satisfy the following conditions:

— p — ρ l ) ≥ s x (8) (G.52)

p — lj

j j 3 j

s3 1

j j

(ρ l + r

— l ) ≥ s x (8) (G.53)

p — lj 2

j j

3 j

rj < lj(1 + ρj ) < r3. (G.54)

We concentrate on conditions (G.52) and (G.53), from which we can derive the following condition for ρjlj:

v3 — p — xB(8)(p — lj) ≥ ρjlj ≥ 2xL(8)(p — lj) + lj — rj

This interval is only non-empty if

v3 — p — xB(8)(p — lj) ≥ 2xL(8)(p — lj) + lj — rj .

After substituting for (G.50) and (G.51), we can rewrite the inequality as:

(v3 + rj — 2p) ≥ 0,

lj — lj∗

∗

p — lj

z >¸,O c

z ¸, c

which is a contradiction. The first LHS term is strictly negative since v3 +rj < 2v2 < 2p. To verify this, consider j = 2 and notice that 2v2 = r + r2 > v3 + r2. The second term

on the LHS is always strictly positive since any risky contract must feature a loan size that is strictly larger than the risk-free choice. This implies that a risky contract with lj > lj∗ (and ρj > ρj∗) makes borrowers worse off compared to the risk-free contract.

In sum, following the intuition from Geanakoplos (2003), a risky loan is never optimal. The proof of Proposition 2 in the main text follows logically from these results:

Proof. of Proposition 2 (as long as r1 < r2 ⇒ k1 > k2).

The inequality k1 > k2 follows directly from (G.11) and the fact that the optimal contract is risk free and l1∗ < l2∗.

G.5 Comparative Statics: proofs of Lemmas 4 – 7

Next, we analyse the comparative statics of the model and, in particular, provide proofs for Lemmas 4 – 7. Starting point is a situation where beliefs of type 1 and 2 lenders are identical,

i.e. r1 = r2. We then analyse the impact of a shock to r1, keeping r2 constant. In order to guarantee closed form solutions, we simplify the model in two dimensions. First, we calculate all derivatives at the point where r1 = r2 = r. The comparative static results are therefore local and only valid for small changes in r1. In the next section, we present global results through

numerical analysis. Second, we restrict the analysis to the special case where N1 = N2 and

a1 = a2 = a.

Proof. of Lemma 4: бh1 — бh2

< 0 for 6(8, p) ∈ [0, 1] × (v2, v3)

6r1

p 6r1

6r1

First, we calculate the differences in first order derivatives from expression (G.11):

6k1 6k2

6r1

—

= — 1 6l1∗

— 6l2∗ — 1 6p

6l1∗ — 6l2∗ + 6p

l1∗ — l2∗

r1=r

(G.55)

The third term on the RHS will be equal to zero as l1∗ = l2∗ = l∗. The same is true for the second

6r1 r1=r

term. To see this, start from expression (G.33) and notice that l1∗ and l2∗

depend on p in exactly

the same way. As a result, the derivatives of l1∗ and l2∗ with respect to p will be identical when

r1 = r2.

Q(l1∗)

1 — 8

p — r

(p — r

The first term on the RHS is different from zero. We first calculate the derivatives of lj∗ with respect to r1:

6l1∗

= 1 $ + 8 v3 — r1 + v3 — p

(r — l∗)

6l2∗

= 1 v3 — p

(1 — a) 8(p — l∗)

6r1

Q(l2∗) (p — r2)2

where Q(lj∗) is the denominator in expression (G.33) and $8 is given by (G.35). As long as

r1 = r2, Q(l1∗) = Q(l2∗) = Q(l∗). We then calculate the difference:

r1=r2

Q(l∗)

1 — 8

(p — r)2

6l1∗

6l2∗

—

(G.56)

= 1 $ + 8

(v3 — r) (p — r) + (v3 — p) [(r — l∗) — (1 — a)(1 — 8)(p — l∗)] .

If 8 = 0, the difference is always positive since, from expression (G.35), $8 > 0 for 68 ∈ [0, 1]. The second term on the RHS will be smallest when (1 — a)(1 — 8) is close to one, its maximum value. Since

(v3 — r) (p — r) + (v3 — p) (r — l∗) — (v3 — p) (p — l∗) = (p — r)2 > 0, (G.57) the RHS will always be positive.

бc3

r1=r2

This proof follows from the fact that

Proof. of Lemma 5: бh1 — бh2

= 0 for 6(8, p) ∈ [0, 1] × (v2, v3)

6k1 6k2

6s3

6s3 r1=r

—

1 6p

p 6s3

= —

6l1∗ — 6l2∗ — 6p

l1∗ — l2∗ . (G.58)

6s3

As long as r1 = r2, the second term on the RHS will be equal to zero. So will the first term:

since l1∗ and l2∗ depend on p in exactly the same way, the derivatives of l1∗ and l2∗ with respect to

p will be identical.

Proof. of Lemma 6: бp1 — бp2 7 0

In equilibrium, loans are risk free and the total loan payment equals the return in the bad

state of the world (from the point of the lender), that is lj(1 + ρj ) = rj . Therefore

1 6r

6ρ1

= 1 l∗ — r

6l1∗

(l1∗)2

6ρ2 6r1

= —(l∗)2 r

6l2∗ .

2 6r

r1=r

Taking the difference and imposing that r1 = r2 yields

6ρ1 6ρ2

6r1

—

= 1 "l∗ — r 6l1∗

— 6l2∗ # .

(xx)0

0x0

xx0

r1=r2

Using expressions (G.33) and (G.56) for l∗ and бl1 — бl2

to simplify, we arrive at

6r1 r1=r

6ρ1 6ρ2

—

1 1 8

v3 — p

[(1 — a) (1 — 8) [r(p — r) + p — l∗] — (r — l∗)] .

6r1

Q — —

6r1 r1=r2

(l∗)2 (l∗) 1 8 (p r)2

z >¸,O c

(G.59)

While the first term is always weakly positive, the sign of the second term is ambiguous and depends on the exact values of 8 and a. Note that if 8 = 0, the difference in derivatives is zero.

In this case, lenders extract all surplus, such that ρ∗(0, p) = v3—p , which does not depend on r .

j v3 1

If 8 = 1, interest rates will be zero, as borrowers have all bargaining power, and the difference in derivatives is not defined.

6r1

r1=r2 h

Proof. of Lemma 7: 6h1 — 6h2

Let

1 > 6p1 — 6p2 1

6r1

r1=r2 p

h,r1

r1=r2

∆o Ξ 6k1 — 6k2 1

p,r1

∆o Ξ 6ρ1 — 6ρ2 1 ,

r1=r2

where k and ρ are the initial haircut and interest rate given by p—l and r—l , respectively. ∆oh,r

p l 1

and ∆op,r are the differences in semi-elasticities of type 1 and 2 haircuts and interest rates with respect to a change in r1. The semi-elasticities measure the percentage change in haircuts or

interest rates in response to a unit change in r1. For example, if r1 falls by x, ∆oh,r1 captures the percentage difference in responses between k1 and k2.

1 — 8

Using expressions (G.55) and (G.56), we arrive at the following absolute value of ∆oh,r :

h,r1

p — l∗ Q(l∗)

∆o =

1 $ + 8

(v3 — r)(p — r) + (v3 — p)[(r — l∗) — (1 — a)(1 — 8)(p — l∗)]

(p — r)2

(G.60)

We use l∗’s functional form from (G.33) to simplify this expression. We first rewrite (G.33), noting that we are considering the case where a1 = a2 = a, and plugging in for $8 from (G.35):

l∗(p, 8)|

= 1 r $

+ 8 v3 — r + (v

— p)8(1 — a)r

r1=r2

Q(l∗)

1 — 8 p — r p — r

= r $

+ 8 (v3 — r)(p — r) + (v3 — p)(1 — a)(1 — 8)(p — r)

Q(l∗)

1 — 8

(p — r)2

(G.61)

Combining expressions (G.60) and (G.61), we arrive at

∆o

l∗ 1 1

h,r1 = ∗ + ∗ ∗

8 (v3 — p)[(r — l∗) — (1 — a)(1 — 8)(2p — l∗ — r)]

(p — l ) r p — l Q(l ) 1 — 8 (p — r) z ¸, c

From (G.57) we can see that ∆oh,r is always positive.

Using (G.59), it is straightforward to derive the following expression for ∆op,r :

(G.62)

∆o = 1 1 1 8 (v3 — p)[(1 — a)(1 — 8)[2p — r — l∗] — (r — l∗)]. (G.63)

p,r1

(r — l∗) l∗

Q(l∗) 1 — 8 (p — r)2 z

7¸,O c

Following the discussion below expression (G.59), the sign of ∆op,r1 is ambiguous and depends on the exact parameter values.

To calculate the difference between |∆oh,r1 | and |∆op,r1 |, we need to consider two cases:

1. [(1 — a)(1 — 8)[2p — r — l∗] — (r — l∗)] < 0 and |∆op,r1 | = —∆op,r1

In this case we have that:

|∆o

l∗

| — |∆o | =

+ 1 8 (v3 — p)[(r — l∗) — (1 — a)(1 — 8)(2p — l∗ — r)]

—

h,r1

p,r1

(p l∗) r

z >¸,O c

Q(l∗) 1 — 8 (p — r)2

z >¸,O c

× 1 — 1 r

p — l∗ r — l∗ l∗

z @¸,O c

= $8 + 1 8 (v3 — p) × Ф

with

Q(l∗)(p — l∗) Q(l∗)(p — l∗) 1 — 8 (p — r)2

Ф = (v — r)(p — r) 1 + (1 — a)(1 — 8)

—

v3 p

+[r — l∗ — (1 — a)(1 — 8)(2p — r — l∗)] 1

r — l∗ — (p — l∗) r

z >¸,O c h i

z >¸,O

r — l∗cz ¸, l∗

l∗

It can be shown that Ф > 0:

Ф > (v

— r)(p — r) 1

+ (1 — a)(1 — 8) + [1 — (1 — a)(1 — 8)] hr — l∗ — (p — l∗) r i

— p

≥ (v

— r) (p — r)

+ hr — l∗ — (p — l∗) r i

— p)

> (v

— r) (p — r)

— (v — r)

III

3 (v

— p)

= (v

— r) 2p — r — v3 > 0,

— p

where (>(I)) comes from replacing (2p — r — l∗) with (r — l∗). Since p > r and

r — l∗ — (1 — a)(1 — 8)(2p — r — l∗) > 0,

replacing p by r increases a positive coefficient multiplying a negative term, thus strictly decreasing the expression. (>(II)) comes from setting (1 — a)(1 — 8) = 0, its minimal value. (>(III)) comes from setting l∗ and v3 at their maximum attainable values ( l∗(8 = 1) = r and p = v3) in the negative term. Finally, (>(IV )) comes from p > v2 and 2v2 — r — v3 = 1(r — r3) > 0.

2. [(1 — a)(1 — 8)[2p — r — l∗] — (r — l∗)] > 0 and |∆op,r1 | = ∆op,r1

For this case, we were unable to find a closed-form proof. Extensive numerical analy- sis (available upon request) indicated, however, that |∆oh,r1 | > |∆op,r1 | for all feasible parameter values.

In sum, the Lemma holds for both cases.

G.6 Numerical simulations and global results

In the previous section, we derived closed form solutions for the relevant comparative statics when r1 = r2. These results indicate how haircuts and interest rates change in response to relatively small changes in r1. It is possible that responses look different when we consider larger shocks. In this section, we analyze this numerically. In general, results are consistent with the previous section.

Starting point for the numerical simulations is r1 = r2. We then reduce r1, keeping r2 constant, and trace out the impact on haircuts and interest rates. As discussed in Section G.3.3, to guarantee the existence of a (unique equilibrium), we can only let r1 decrease to g2r2 , where

g2 is given by expression (G.46). For this exercise, we normalize the model in two dimensions.

l∗

First, we impose that the average valuation of the asset by type 2 and 3 agents equals one, that

v2 + v3

= 1. (G.64)

This is a pure normalization. Second, we impose that in the scenario where lenders have all bargaining power (8 = 0) and r1 = r2, the price also equals unity: p(8 = 0, r1 = r2) = 1. A price lower than one would mean than a shock to r1 would sooner drive the price below

v2 , at which point the equilibrium breaks down. A price higher than one would imply that allocating more bargaining power to borrowers (8 → 1) would sooner drive prices above v3, also destroying the equilibrium. Imposing that p(8 = 0, r1 = r2) = 1 trades off these two

potential (numerical) problems. Keeping all else equal, equation (G.40) pins down the amount of optimist capital s3.

For our simulations, we fix the haircut a lender charges when 8 = 0 and r1 = r2 = r at k(8 = 0, r1 = r2) Ξ kO. Together with expression (G.64), this imposes a number of restrictions on r and r.

1. When 8 = 0, the haircut is given by expression (G.37):

k = v3 — r .

Combining this with expression (G.64), and noting that r2 = r, implies that

r = (1 — k ) 4 — r . (G.65)

3 — kO

2. The condition that v3 < r implies a lower bound on the values that r can take. Plugging in for (G.64) and (G.65), we arrive at:

r > .

4 — kO

3. Finally, we need that r3 > r. This implies an upper bound on the values that r can take. To see this, plug in for (G.64) and (G.65) to arrive at:

r < 1 + kO.

Points (2.) and (3.) define an interval for possible values of r. For simplicity, we use the midpoint of this interval in our simulations (results are generally robust to using other feasible values). Given kO, this choice for r pins down r through expression (G.65).

Apart from these normalizations, we need to fix the other parameters of the model. Most important is kO. Changing the other parameter values does not have important quantitative implications. We cannot use any value of kO in (0, 1): there is an upper bound. As discussed in Section G.3.3, differences of beliefs in the model cannot be too extreme, specifically,

r > r3

2 1 + g2

Using (G.64), and noting that r2 = r, implies that:

r < 2 — r .

1 + g2

Plugging in for (G.65) yields

< 2(r — 1) + g2(4 — r) Ξ k .

2 + g2(4 — r)

In our simulations we set kO equal to this upper bound. Qualitatively, results are similar for other kO ∈ (0, kO], but using kO is most conservative. Lemma 7 states that the difference (in absolute value) in semi-elasticities between type 1 and 2 loans with respect to changes in r1 is larger for haircuts than it is for interest rates. Unreported simulation results indicate that this difference is smallest when kO = kO.

Table G.1 gives an overview of the different normalizations and parameter choices and Figure G.1 presents the simulation results. On the x-axis we display r1 ∈ [g2r2, r2] Panels A and C show how haircuts charged by type 1 and type 2 lenders change as we move r1 away from r2. We consider three scenarios where 8 ∈ {0.1, 0.5, 0.9}. As expected, in response to a decrease in r1, type 1 haircuts increase. At the same time, market wide leverage falls and

the equilibrium price p drops. This causes type 2 haircuts to decrease. Panel E presents the difference between the two type of haircuts, normalized by their initial level where r1 = r2 =

r and k1 = k2 = k :

k1(r1) — k2(r1). (G.66)

k(r)

This expression shows how much the difference in haircuts goes up in response to a given decrease in r1 relative to the initial level of haircuts. In the main text, we calculate the difference in semi-elasticities at the point where r1 = r2. Expression (G.66) is the global equivalent. Consistent with Lemma 4, expression (G.66) is always positive.

Panels B and D repeat the exercise for interest rates. For both type 1 and 2 loans, interest rates increase as r1 falls. The interest rate is defined as surplus payments divided by the loan

amount. Loans extended by type 1 agents become smaller when r1 falls (“size effect”) and generate a smaller overall surplus which leads to a reduction in surplus payments to the lenders (“surplus effect”). The net impact on the interest rate is ambiguous. In the simulations, the size effect dominates and interest rates increase. Loans extended by type 2 agents tend to increase due to a fall in the equilibrium price and generate a higher overall surplus. Again, the net impact on the interest rate is unclear. In the simulations, the surplus effect always dominates and interest rates go up. Panel F shows the difference in interest rates, normalized by their initial level where ρ1 = ρ2 = ρ. In the simulations, type 2 interest rates always increase more than type 1 interest rates, but differences are small: Panels B and D are difficult to distinguish from each other. This discussion echoes the results from Lemma 6 that establishes that the sign of the difference is ambiguous.

Panel G looks at the differences between Panels E and F: are relative changes in haircuts larger than those for interest rates? Consistent with Xxxxx 7 in the main text, the impact on haircuts dominates for feasible changes in r1:

k1(r1) — k2(r1) > ρ1(r1) — ρ2(r1)

k(r) ρ(r)

Unreported simulation results indicate that when 8 approaches unity and the change in r1 is large, there are cases where the response in interest rates is larger than for haircuts. However, these are instances where interest rates are low to begin with, implying that we divide the difference between ρ1 and ρ2 by a number close to zero. We interpret this result as an artefact of the model, and our definition of semi-elasticties, rather than a substantive economic result. In any case, for moderate changes in r, Xxxxx 7 is always valid.

References

Xxxxxx, Xxxxxxx, and Xxxx X. Xxxxxxxx (2012). Ordinary Differential Equations. New York: Springer.

Xxxxxx, Xxxxxxx, Xxxxxxx Xxxxxxxx, and Xxxxx Xxxx Xxxxxxxx (2005). “Over-the-Counter Markets”. Econometrica, 73(6), 1815-1847.

Figure G.1: Simulation results

Table G.1: Parameter values numerical analysis

Parameter:	v	µ	λ	n1	n2	n3
Values:	1	1	1	1/3	1/3	1/3
Parameter	s3	r2	r3	r	kO
Values:	2.181	0.550	0.826	1.311	0.485

Normalization(s): p(8 = 0, r1

= r2

) = 1 v2 + v3 = 2 kO = 2(r—1)+g2(4—r)

2+g2(4—r)

v3—rj

O v3

r = 2 + 1+h0

4—h0 2

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Appendix A: Sample contract – original and English translation (SAA 10,602, F. 1309)

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