A STRUCTURAL ECONOMETRIC MODEL OF PRICE DISCRIMINATION
IN THE FRENCH MORTGAGE LENDING INDUSTRY
∗
Robert J. Gary-Bobo
∗∗
and Sophie Larribeau
∗∗∗
University of Cergy-Pontoise, THEMA
33, boulevard du Port, 95011 Cergy-Pontoise,Cedex, France
Email: robert.gary-bobo@eco.u-cergy.fr
January 2002; revised October 2002; revised July 2003, final revision
Abstract.
We propose a model of discrimination in the market for mortgages. The model explains
accepted loan applications and simultaneously determines loan sizes and in terest rates. A
competitive and a discriminating monopoly version of the model are proposed. Offered
interest rates and loan sizes are a function of observable borrower characteristics. The
competitive model rests on a marginal condition, reflecting contract optimality, to which a
zero-profit condition is added. In contrast, the discriminating monopoly maximizes profits
under a borro wer participation constraint, reflecting the availability of a rental marke t
as an outside option. Each version of the model is a bivariate, nonlinear model, and is
estimated by standard maximum lik elihood methods. The data used for estimation is a
sample of clients of a French network of mortgage lenders. We show the presence of ”social
discrimination” in the data, the loan conditions depending, not only on the borro wer’s
wage and down payment, but also on the borro wer’s occupational status. Abnormally
high risk premia in the competitive version of the model suggest the presence of market
power, justifying an attempt at estimating its monopolistic version. The discriminating
monopoly model estimates show that the borrowers’ price-elasticity of demand for housing
varies with occupational status, and is inversely related with the lender’s interest rate
markups. This confirms that the lender exploits structural differences in the preferences
to discriminate, as predicted b y standard theories.
Keywords: Mortgage Loans, Price Discrimination, Discriminating Monopoly.
∗
We thank Richard Arnott, Marc Ivaldi, and Daniel McFadden for their advice during a preliminary stage of
the present research, the participants and organizers of the 12th CREST-NBER Franco-American Seminar on
Economics and 1rst CEPR Workshop on Applied I.O., held in Toulouse, October 1999, the participants and
organizers of the CEPR Conference ”Discrimination and UnequalOutcomes”,heldinLeMans,France,January
2002, Florian Heider, Carmen Matutes, Peter Norman, Nathalie Picard, Jean-Charles Rochet and Nancy Wallace,
for comments and helpful discussions. The first author thanks John Quigley for his help and hospitality in June
2000. We finally thank Steve Martin and two anonym ous referees of IJIO for helpful remarks. We are responsible
for remaining errors.
∗∗
University of Cergy-Pon toise, THEMA, France, and CEPR.
∗∗∗
Univ ersity of Cergy-Pontoise, THEMA,
France.
1
1. Introduction
To the best of our knowledg e, only a handful of contributions ha ve attempted to estimate
a structural econometric model of price discrimination, and none have studied the related
phenomenon of interest rate discrimination, which can be viewed as a particular case, with
such methods
1
. In the present paper, we propose a structural econometric model of a
market for mortgages, and estimate it on micro-data.
The model determines interest rates and loan sizes simultaneously as a function of
observable borrower characteristics. It can thus be viewed as describing a case of first-
degree price discrimination in which the lender imposes differing price-quantity pairs on
different types of borro wers. In addition, our data set shows that the lender exerts a form
of mark et power. We have therefore specified and estimated two versions of the model, a
competitive and a discriminating monopoly version.
The competitive variant of our model rests on the idea of ”competition in con tracts”:
in equilibrium, any additional entry of a lender with loan offers cannot simultaneously
attract borrowers and mak e strictly positive profits. Two equations then describe the
equilibrium: the first is a zero-profit condition; the second is a necessary condition for
contract optimality, expressing the efficiency of trading between lenders and borro wers.
These two equations, once solved, give the amount lent and the interest rate charged as
functions of borrower characteristics. The discriminating monopoly variant of the model
rests on the idea of surplus-extraction by a monopolist: the lender maximizes profit subject
to the participation constraint of borrowers. Each borro wer has an outside option, which
is to ren t a house instead of buying one. The surplus-extraction or zero-surplus equation
says that in monopolistic equilibrium, borrowers are indifferent betwe en renting a house
or accepting a mortgage contract offered by the lender. A second equation is a necessary
condition f or contract optimality, as in the competitive ve rsion. This is consistent with
theclassicresultthatafirst-degree discriminating monopoly will propose efficient trades.
These two equations again determine the loan size and interest rate as functions of borrower
characteristics.
Theseassumptionsgiverisetononlinear,bivariate econometric models which are
estimated by standard maximum likelihood methods. The estimated risk-premia applied
to borrowers in the competitive version are mu ch too high to be reasonable. This indicates
thepresenceofmarketpower,becausetheestimatedrisk-premiaare,infact,interest-rate
markups imposed on borrowers. The discriminating monopoly version of the model, in
spite of its added complexity, giv es much more reasonable estimated values of the risk-
premium function.
We also findthattheamountlentandtheinterestratechargedvarysignificantly with
the borro wer’s income and down payment. There is a clear indication that differences in
treatment depend on the occupational status of borrowers: for instance, execu tives will
pay less to reimburse their loans than blue collar workers, everything else being equal.
More precisely, the mere fact of being identified as an executive by the lender would result
1
In con trast, there is an important strand of empirical literature on racial discrimination in mortgage lending
in the U.S., which is discussed belo w.
2
in better loan conditions, even if the executive’s income were in the range of a blue-
collar worker’s wage. Estimations also sho w that categories of borrowers have significantly
different preference parameters, and these differences are exploited by the monopolist,
as suggested by standard theory. For instance, workers have a smaller price-elasticity of
demand for housing than executives. The borrower’s price-elasticity of demand is in versely
related with the interest-rate markup c harged by the lender.
Tothebestofourknowledge,thistypeofnonlinear, structural approac h had not been
attempted before, and the estimations obtained show that the approach could be applied
to exploit richer data sets in other countries. We also think that our model could easily be
adapted to test for the presence of racial discriminationinmortgagelending,intheU.S.
(see the discussion in the next to last section of this paper).
Recent structural econometric approaches to nonlinear pricing or price discrimina-
tion include the pionneering work of Ivaldi and Martimort (1994), using data on energy
provision to French dairy producers, and Bousquet and Ivaldi (1997), on telephone pric-
ing. Third-degree price discrimination on the European car market has been studied by
Goldberg and Verboven (2001). Clerydes (2001) uses data on book sales to study discrim-
inatory pricing of paperbacks and hardcovers. Leslie (2001) studies price discrimination
in the sales of tickets for a Broadway show. Cohen (2000) shows that pac kage sizes are
used to price discriminate in the U.S paper to wel mark et. McManus (2001) tests for the
presence of second-degree discrimination, and therefore product design distortions, in the
price-quality menus o ffered by coffee shops surrounding the University of Virginia. Mi-
ravete (2001) studies nonlinear tariffs and consumer choice in a men u of optional calling
plans proposed by the Bell telephone company in Kentucky. Finally, Verboven (2002) uses
differing driver preferences for gasoline and diesel cars in Europe to estimate the extent
of price discrimination by manufacturers. Many of the above quoted studies use a form of
discrete c hoice model of product differentiation to represent the behaviour of heterogeneous
consumers
2
.
Relationships with the theoretical literature on price discrimination and credit rationing
The theoretical literature on credit and banking has emphasized screening under adverse se-
lection, proposing a theory which has the same formal structure as Rothschild and Stiglitz’s
(1976) model of competitive insurance markets. Among contributions to this topic, see
Milde and Riley (1988), in which loan size is used as a screening instrument, and Bester
(1985), establish ing that collateral is a screening instrument, under asymmetric informa-
tion. Calem and Stutzer (1995) have addressed the problem of racial discrimination with
a theoretical screening model in which the probability of rejection of mortgage loan ap-
plications is used as a screening device: clients choose from a menu of contracts in whic h
higher interest rates are traded o ff against higher probabilities of acceptance. This idea
cannot be applied in the following, since we use data on accepted loan files only. Our
theoretical model aims at modelling loans conditionally on acceptance. Brueckner (1994),
and Stanton and Wallace (1998) address the delicate problem of informational asymme-
tries about lender mobility, and the associated risks of premature repayment and mortgage
2
On the theory of discrete choice models and its application to oligopoly theory, see Anderson et al. (1992).
3
renegotiation. They construct a separating equilibrium in whic h borrowers with differing
mobility select fixed rate mortgages with different combinations of rates and points
3
.This
approach, which provides a good explanation for some of the observed mortgage ”men us”
in the United States, cannot be applied here, because the lender does not make use of
points in our data set. It is also quite certain that the Frenc h pro vincial housing markets
are characterized b y muc h less mobilit y than the U.S. markets (this is an important cultural
difference between the t wo countries which has not been studied, as far as we know). It fol-
lows that the interest-rate risk generated by prepayment on loans is apparently negligible
in France, at least as a first approximation.
In the theoretical model of Brueckner (2000), borrowers self-select by c hoosing differ-
ent initial loan-to-value ratios, high interest rates being associated with high LTV, and the
unobservable borrower characteristic driving self-selection is the level of personal default
costs. But Brueckner’s model is a refinement of the typically American strategic default
or default-option theory, which is not applicable to French data, because the French bor-
rower’s liability is not limited to the value of his (her) house. The initial LTV ratio (in
fact, the related ”down payment ratio”) pla ys a role in our empirical analysis; it has a
statistically significant effect as an argumen t of our estimated risk-premium function. We
find that the risk-premium is a decreasing function of the down pa ym e nt ratio. But this
could simply be the mechanical result of a better collateralization of the loan. The model
presented below maps the down payment, the income, and other exogenous characteristics
of borrowers into (interest-rate, loan size) pairs, and the LTV ratio is endogenous. Our
French lenders propose a very elementary kind of ”menu”, in which the interest rate varies
with the loan term (a twenty year loan will typically bear a higher rate than a ten year
one, everything else being equal). Thus, in principle at least, borrowers could self-select
according to some unobservable characteristic or preference parameter. The question is
then to discern which of these characteristics exactly is screened by the c hoice of the loan
term. In theory, it could be many things, such as the degree of risk aversion, the rate
of time preference, the probability of default, etc., and possibly a combination of sev eral
aspects simultaneously. The most common (and probably most reasonable) view is that
the lower the borrower’s income, the longer the chosen loan life, because, due to liquidity
constraints, poorer borrowers will simply try to spread reimbursement over a longer pe-
riod. The borrow er’s income and social status being observed by the lender, it is difficult to
iden tify another dimension of the space of consumer c haracteristics along which borrowers
would significantly self-select while choosing various loan durations, and which is not at
thesametimealreadyobservedbythebanker. Wehavecarefullytakentheloanterminto
account, but we treat it as exogenous, i.e.,asifitwereacharacteristicoftheborrower:
this is clearly a simplification, but a more sophisticated treatment seemed to be out of
reach.
To sum up, our French lenders, who are certainly somewhat old-fashioned, as com-
pared to US professionals, are also working in a different legal environment, which makes it
difficult to test some existing theories based on the idea of self-selection. This is an impor-
3
The idea of separation b y mobility, when borrowers are better informed than lenders about their probability
of moving, has been first modelled by Chari and Jagannathan (1989), but in a model in which the interest rate
is constant.
4
tant reason for modelling our lender as a first-degree discriminating monopolist: given our
data set, there are no banking practices, and no compelling dimensions of both the bor-
rower characteristic and the credit contract spaces along which to construct a reasonable
model of self-selection or second-degree discrimination
4
.
At this point, it should be noted that none of the structural econometric approaches
to price discrimination cited abo ve went as far as to use a condition that the ch oice of
product qualities maximizes profit to help identify their model’s parameters: product lines
are exogenously given. It is difficult to capture taste heterogeneity in a model of demand
for differentiated products and to derive a measure of the extent of price discrimination or
product-quality distortions, and the assumption that a producer’s product line is optimal
does not seem to have been tested or used as an identification restriction in the literature.
Given this difficulty, a contribution of the present paper consists in the use of assumptions
about the seller’s profit-maximizing beha viour (i.e., our surplus-extraction equation) to
help identify structural parameters in a price-discrimin ation problem.
Another difficu lty, pointed out above and by recent theoretical work, is that a real-
istic model of second-degree discrimination would be lik ely to entail several dimensions
of uncertainty about the borro wer’s characteristics. For instance, at least two dimensions
could be considered: the borrower’s marginal willingness to pay for a larger loan and a
parameter determining the utility level of his (her) outside option on the housing market.
A good model would also probably involve several screening instruments: loan size, loan
life, down payment, points, prepayment penalties, are possible instrumen t s. Th us, a mul-
tidimensional discrimination model would be required, as studied in the work of Rochet
and Chon ´e (1998). This is known to lead – apart from hard technical problems – to
much less separation of consumer types, and thus much less discrimination power than in
the classic, one-dimensional model. Roc het and Chon´e (1998) show that bunching is a ro-
bust feature of optimal solutions in the multidimensional screening problem. The optimal
solution cannot be explicitly computed in general, and with the exception of Armstrong
(1999), not much has been published on the approximation of the optimal discrimination
policy by simple pricing rules. A consensus on the form of the appropriate model has
not yet emerged. Our approach, which is to model data as if they reflected first-degree
discrimination based on observ ed client characteristics, is therefore justified, at least as a
first step.
In the following, section 2 is devoted to a description of the data and to the results of
a preliminary linear econometric investigation. Section 3 presen ts the t wo versions of the
model and section 4 their econometric estimation. Section 5 is devoted to a discussion of
the empirical literature on racial discrimination in mortgage lending, and section 6 contains
concluding remarks.
4
Why not a model of third-degree price discrimination then? This is essentially a question of terminology.
The received definition of third-degree price discrimination, is linear pricing (i.e., constant unit prices) combined
with market segmentation with respect to observable consumer characteristics. Our model is more general than
this, because it represents nonlinear pricing. On the other hand, first degree (i.e., perfect) discrimination exists
only in pure theory, because consumer characteristics are never perfectly observed in practice, implying that some
consumers with unobservable differences are equally treated by the seller. It follo ws that our approach could be
called ”nonlinear third-degree price discrimination”, instead of first-degree price discrimination.
5
2. The data
2.1. Description
We obtained a sample of observations on the clients of a French mortgage lender, the
Cr´edit Hypoth´ecaire de France (a nickname), hereafter CHF. The CHF is in fact a net-
work of building societies, scattered on the Frenc h territory, the BSs. These local BSs
have independent application screening and interest rate policies; they own in common a
financial institution, whic h borrows money on national and international bond markets,
and provides funds to the BSs. The CHF is a prudent and profitable institution, with a
long history and a solid reputation. The BSs do not securitize their loans. Rating agencies
have gran ted a very high note (AA+) to the CHF, so that the institution’s cost of funds is
well appro x imated by, and closely parallels, the long-term rate on French state bonds (the
”OAT” rate), with an almost constant difference of a few base points. In the absence of
sufficiently precise information on the cost of funds, we used the French OAT rate directly
in the estimations. Although the CHF has a special legal status, it is fair to describe the
behavior of the local BSs as profit maximization. Un til 1995, when the French go vernment
reformed its housing policy, the CHF had the privilege of distributing a particular kind
of state subsidized home loan. This privilege has disappeared today, since all commercial
banks can now initiate the same subsidized loans, but the CHF network has developed a
strong expertise in mortgage lending to the working class, and goodwill in accordance with
this specialization. Its clientele is composed of a vast majorit y of modest income employees
and workers. It is likely that many of the CHF clients would see their applications rejected
elsewhere.
On top of distributing state subsidized loans, the characteristics of which are tightly
regulated, the CHF also supplies the so-called ”free loans”, whic h are unregulated, ordinary
mortgages. Until recently, the vast majority of these mortgages have been classic, fixed
rate, fixed repayment mortgages. French mortgage law is in a sense simpler than that of
the U.S., since the borrowers’ liability is not limited to the value of the house (lenders
can pursue other borrower assets to mitigate default-related losses). In addition, house
prices ha ve not decreased very m uch in the provincialregions,whichare the geographical
origin of the sampled borrowers, during the observation period. It follows that strategic
default (or the exercise of the default option) is not empirically relevant in the sample. In
practice, m ortgage defaults seem to be triggered by consumer insolvency, mostly due to
loss of income. A form of unemployment insurance of mortgage loans does indeed exist,
but it is not compulsory, it is expensive, and limited in scope. Informational asymmetries
and moral hazard provide an explanation for the weakness of unemployment insurance of
mortgages in France (on this topic, see Chiappori and Pinquet (1999)). These loans can
in principle be renegotiated, the prepayment penalty being around 3% of the principal’s
remaining value in all BSs.
For the econometric investigations below, we have used a sample of 2610 observations
on accepted free loans, originated from various BSs across France between 1989 and 1994.
We ha ve eliminated the subsidized loans
5
. Thereisnoinformationonrejectedapplications,
and no observations of default or of repayment ”incidents”.
5
The interest rate, the amount lent, and the borrower characteristics of subsidized loans are strictly regulated,
6
Each observation corresponds to a file, including, 1
o
) the amoun t of the loan, 2
o
)the
loan interest rate (including insurance), 3
o
) the down payment (savings used to buy the
house by the borrower), 4
o
) the starting year, 5
o
)theloanterm,6
o
)theborrower’syearly
wage, 7
o
) the age of the borrower, 8
o
)thefamilysize,9
o
) the borrower’s occupational
status, falling in to 6 categories, and 10
o
) the geographical location of the BS granting the
loan. We k ept only four of the occupational status categories, blue-collar workers (1180
observations), white-collar employees (908 observations), so-called intermediate professions
(363 observations), and executives (159 observations). We also do not use the geographical
location variable in the estimations, and all regions have been pooled
6
.
The model presented below has been constructed to be estimated with this limited
set of information. It can of course easily be adapted to use more explanatory variables.
Table 1 provides descriptive statistics on the data. The amounts len t are not very
high. The down payment ratios, that is, (down payment / loan + down pa yment) are
small, around 17%. The loan terms are distributed between 1 year and 20 years (the
empirical distribution of loan terms is depicted on Figure 9). The nominal interest rates
are very high with a mean value of 11,7%, but the real in t erest rates were also very high at
the beginning of the nineties in France, the inflation rate being already quite low (around
2% per year). The interesting aspect of the data is the substantial variance of the loan
rates. This will allow the estimation of a risk-premium function, and of an interest-rate
elasticity of the demand for housing, in spite of the fact that the observation period is
short. Another striking fact is the markup on state bond rates, which is on average equal
to 3%. The lenders seem to exert a form of market power.
2.2. An exploratory linear model
To gain some understanding of our data set’s conten t, we have estimated a simple si-
multaneous equation model, explaining loan sizes and rates. More precisely, w e started
from the point of view that the relevant endogenous variable for this study is not the
interest rate itself, but the constan t repa y ment annuity, denoted p,anddefined as follows.
AhouseholdwhoborrowsforT y e ars at con tinuous-time rate r, will repay the amount,
p(r, T )=r/(1 − e
−rT
) at eac h instant of time to reimburse the loan completely, interest
and principal, per euro borrowed (see below, subsection 3.1). The discrete-time equivalent
of p, i.e., the amount repaid each year, whic h is denoted P ,is
P (¯r, T)=
1 −
1
1+¯r
1 − (
1
1+¯r
)
T
,
limiting the banker’s freedom considerably. They therefore constitute a very bad basis to test for a theory of
discrimination.
6
It would have been interesting to use geographical data but, (i), the available data is the location of the BS,
not that of the borrower’s house, and this regional information is coarse (we do not even know if the borrower
buys a house in the coun tryside, in the suburbs, or in the city center); (ii), it is hard to find good data on
local house prices in France; (iii) w e hav e introduced regional dummies in preliminary explorations using linear
econometric models: some of these dummies had significant coefficients, others did not; no interpretable pattern
seemed to emerge; the introduction of dummies did not change the other, important coefficients.
7
where ¯r = e
r
− 1 is the annu al in terest rate. If i denotes the lender’s cost of funds, define
C = P (i, T ). Thisisameasureofthelender’scost,whichiscommensuratewithP.If
borrowers are liquidity-constrained, P is a relevant variable, otherwise, the interest rate r
would be the only important one. (On liquidity constraints, see Deaton (1992)). Let then
m denote the amount lent and a the household’s down payment, so that H = m + a is
the value of the house. Let finally w denote the household’s yearly wage, and n the family
size. Using the subset of 1180 unregulated loans granted to blue-collar workers to estimate
a linear model explaining log(H)andlog(P ) sim ultaneously, we obtained the following
results
7
:
log(H)= 6.239
(23.15)
− 1.551
(−20.75)
log(P )
+0.102
(4.187)
log(w)+ 0.193
(19.60)
log(a)+ 0.028
(5.823)
n, (A)
and
log(P )= −0.4787
(−19.39)
+0.7239
(62.82)
log(C) − 0.01862
(−6.96)
log
³
a
H
´
. (B)
Note that a/H,thedown payment ratio,hasasignificant negativ e impact on P ,andthus
on the loan rate, as expected. But the striking fact is merely the significance of log(P )in
equation A, which determines house value. The coefficient of log(P ) can be interpreted
as a form of price-elasticity of housing demand. We estimated the same model separately
with the subsets of white collar employees, intermediate professions, and executives; the
3SLS estimated values of the coefficient on log(P )are−1.347 for white collars, −1.629
for intermediates, and −1.648 for executiv es, all highly significant. While white and blue
collars are not significantly different from each other, white collars are significantly different
from intermediate professions and executives
8
.Thesefindings suggest that the price-
elasticity of house size varies with occupational status and even seems to increase (in
absolute value) with the level in the social hierarchy. In the following, we try to provide
an explanation for this phenomenon, and to relate it with observ ed interest-rate markups.
7
Equations A and B hav e been estimated with the 3SLS method, to take care of simultaneity problems,
i.e., to estimate sim ultaneous equations systems with this classic method, the instruments used are the model’s
exogenous variables; in this case, log(C), log(a), n and log(w). The Student t-ratios are in parentheses. The
adjusted R
2
is .42 for equation A and .77 for equation B; the estimated correlation of A and B’s error terms is
.244.
8
We have estimated other variants of this system, all leading to the same conclusions. For a detailed account
of an econometric study of our data set with the help of standard linear econometric methods, see Gary-Bobo
and Larribeau (2003).
8
3. A model of mortgage lending
To describe the model, we first define the demand side, deve loping a continuous-time model
of an expected utilit y maximizing, infinitely-lived and liquidity-constrained consumer-
borrower. We then turn to the banking firm’s profit function. This preliminary mod-
elling work allo ws the presentation of two variants of the theory, one competitive, one
monopolistic, corresponding to two assumptions about competition on mortgage lending
markets.
3.1. The borrowers
Let α denote a household’s vector of charac teristics. Each household of type α is repre-
sented by an infinitely-lived, expected-utility maximizing consumer. The instantaneous
utility of consumer α,denotedu,isdefined on the set of bundles (c, h), where c is in-
stantaneous aggregate consumption and h is housing, measured in constant-quality square
meters. To sum up, u = u(c, h; α). Time t varies continuously, and it is assumed that
consumer α’s utility for a certain consumption path t → (c(t),h(t)) is
δ
Z
+∞
0
e
−δt
u(c(t),h(t); α)dt, (1)
where δ is a positive discount factor
9
. Now, each consumer is subject to a liquidity con-
straint of an extreme form: she cannot borro w against future income, and can only obtain
ahomeloanwiththeentirehouseasacollateral. For simplicit y, it is assumed that each
consumer is a worker with a constant wage w. Yet, the worker is subject to a risk of
default, which can be interpreted as loss of income, or as a layoff.Tothis,weaddthe
simplifying assumption (with a European flavor), that once laid off, the worker remains
unemployed forever. Default (or unemployment)randomlyoccursattimet
∗
.Theprob-
ability of defaulting between time 0 (the starting point of the loan here) and time t is
assumed to be exponential and its cumulative distribution function is denoted F ,thatis,
F (t
∗
≤ t)=1− e
−θt
, (2)
where θ is a nonnegative parameter. The density of F is f(t)=θe
−θt
. The only event
triggering mortgage defaults is unemployment, and it implies a complete loss of the house,
duetomortgageforeclosurebythebank,forsimplicity.
We consider classical, fixed in t erest, direct mortgage loans, with constant repayment
(or self-amortizing annuities). Mortgage loans are characterized by an amount lent m,a
fixed (cont inuous-time) interest rate r,atermT,andastartingdateT
∗
;theycanbe
described by the array (r, m, T, T
∗
). The borrower is endowed with a down payment a;
this represents money that has been accumulated in the past, and is assumed to be entirely
invested in the purch ase of the house. The consumer rents a house if he or she does not
borrow to buy one, and the accumulated sum of money a is then deposited in some bank
and yields an interest i
0
. To simplify the model, we assume that i
0
= 0 in the following.
9
The expected utility in equation (1) is multiplied by δ only for convenience and to simplify computations.
9
Now, if the house price per square meter is denoted π, then the house size (or housing
consumption level) of a borrower is simply
h =
m + a
π
. (3)
We define the down payment ratio of a consumer as a/(m + a).
The continuous-time constant amortizing repayment of a m euros loan of term T ,
originated at time T
∗
=0,isp(r, T )m,andsatisfies by definition,
me
rT
=
Z
T
0
p(r, T )me
rt
dt.
Straightforward int egration then yields,
p(r, T )=
r
1 − e
−rT
. (4)
For convenience, we call p(r, T )theprice of the loan, and denote it simply by p.Note
that all continuous-time variables can easily be transformed into their discrete-time coun-
terparts, with the appropriate formulas
10
.
Then, because of the assumed liquidity constraint, the borrower faces a budget con-
straintateachtimet. Taking consumption as a numeraire, the budget constraint has the
simple form,
c = w − pm. (5)
Definitions (3) and (5) show that, given w, a and π, there is a one-to-one relationship
between (p, m)and(c, h). Given T , the interest rate r can be retrieved from p by inverting
(4). It follows from this that a mortgage (r, m, T, T
∗
) is equivalently characterized by the
vector (c, h, T, T
∗
), given w, a and π. In the following, it will be more convenient to reason
in the (c, h) consumption-bundle plane, instead of the (r, m)plane.
The expected utility of a borrower
We define different levels of instantaneous utility as follows.
1
o
)Whenconsumerα repa ys a mortgage with price p and enjo y s a house of size h,
her utility level is u = u(w − pm, h; α).
2
o
) If the consumer defaults, she loses her income and her house. In this case, her
utility level is assumed to be independen t of p and h; it is denoted by u = u
0
(α).
3
o
) If the consumer never defaulted, and has completely repaid the loan, then u =
u(w, h; α).
4
o
) If the consumer loses her income after the loan is completely repaid, she does not
lose her house, and u = u(0,h; α).
5
o
) Finally, the consumer can rent a house. Let ρ denote the rent per square meter.
If the consumer rents, she will choose (c, h) ≥ 0soastomaximizeu(c, h; α) subject to the
10
For instance, if T isexpressedinyears,andˆr denotes the annual interest rate, then r=log(1+ˆr).
10
constraint c + ρh = w. The solution of this standard maximization problem is the pair of
demand functions (c
∗
(ρ, w; α),h
∗
(ρ, w; α)). The instantaneous indirect utility of a renter
is thus,
v
r
(ρ, w; α)=u[c
∗
(ρ, w; α),h
∗
(ρ, w; α); α]. (6)
Let now v(t, t
∗
,T
∗
,T; α) denote the instantaneous utility of a consumer of type α at
time t, given that default occurs at time t
∗
, under a mortgage contract (r, m, T, T
∗
). The
expression for v can be computed with the utility levels defined abov e, since by assumption,
the consumer rents her house from t =0untilT
∗
, loses h er income at t
∗
,andlosesher
house at t
∗
if T
∗
<t
∗
<T+ T
∗
.Wecannowdefine the discounted sum of utilities for the
mortgage (r, m, T, T
∗
)as
U(t
∗
,T
∗
,T; α)=δ
Z
+∞
0
v(t, t
∗
,T
∗
,T; α)e
−δt
dt. (7)
Using the assumption that the date of default is exponent ially distributed , we define the
expected utility of a loan of term T ,originatedatT
∗
, as follows,
V (T
∗
,T; α)=
Z
∞
0
θe
−θt
∗
U(t
∗
,T
∗
,T; α)dt
∗
. (8)
It is now possible to compute V . After some cumbersome, but straightforward computa-
tions, we obtain the following result
11
.
Result 1
V (T
∗
,T; α)=v
r
(ρ, w; α)[1 − e
−(θ+δ)T
∗
]
³
δ
θ + δ
´
+u
0
(α)[1 − e
−(θ+δ)(T +T
∗
)
]
³
θ
θ + δ
´
+u(w − pm, h; α)[e
−(θ+δ)T
∗
− e
−(θ+δ)(T +T
∗
)
]
³
δ
θ + δ
´
+u(w, h; α)
³
δ
θ + δ
´
e
−(θ+δ)(T +T
∗
)
+u(0,h; α)
³
θ
θ + δ
´
e
−(θ+δ)(T +T
∗
)
. (9)
From this result, one can derive the consumer’s tenure choice conditions, with static expec-
tations, that is, on the assumption that a mortgage characterized b y (r, m, T )willalways
be available in the future. These tenure choice conditions are in fact equivalent to the bor-
rower’s participation constraint, from the point of view of the banker. Define V
r
(ρ, w; α)
as the expected utility of a consumer who rents a house forever. Clearly, with the help of
(9), one easily checks that
V
r
(ρ, w; α)= lim
T
∗
→+∞
V (T
∗
,T; α)=v
r
(ρ, w; α)
³
δ
θ + δ
´
+ u
0
(α)
³
θ
θ + δ
´
. (10)
11
Detailed computations are relegated to the supplementary material web page.
11
Now, consumer α will accept a loan (r, m, T, 0) at time 0 if and only if, for all T
∗
≥ 0,
V (0,T; α) ≥ V (T
∗
,T; α). (11)
This condition has a remarkably simple equivalent formulation.
Result 2
Condition (11) is equivalent to
V (0,T; α) ≥ V
r
(ρ, w; α). (12)
To prove
12
this result, it is sufficient to show that V is decreasing with respect to T
∗
,if
and only if condition (12) holds.
From now on, and to clarify notation, we denote
¯
V (c, h, T ; α)=V (0,T; α), (13)
letting c and h explicitly appear as arguments of the expected utility. Parameter α must
be viewed as a vector of characteristics including θ, w, a, δ.
3.2. The lender
We now construct the bank’s expected profit function from a mortgage loan to a borro wer
of type α,startingattimeT
∗
= 0. From now on, the reader must understand that all
mortgages start at date T
∗
=0.
We assume that the bank’s liability structure can be modelled as if it sold bonds or
certificates of deposit in exchange for an interest rate denoted i to finance its loans. The
bank is also assumed to avo id any form of interest rate or refinancing risk. The timing
of payments on its debt, interest and principal, match es the timing of revenues from its
mortgage portfolio (amortization schedules are parallel). The loans are not securitized, and
kept on the asset side of the bank’s balance sheet (this assumption corresponds to common
French practice). The interest rate i is used to discount the bank’s future cash flows. If a
client defaults at date t
∗
, for the sake of simplicit y, w e assume that the liquidation value of
the house is zero. More realistic representations of mortgage foreclosure lead to complex
nonlinear expressions of expected profit, and are intractable with our data (see however
Appendix A, in which we sk etch a more sophisticated model).
We also simplify the model by assuming that administrative costs are zero. Given
that we do not have observations of the banker’s costs, and that this implies difficulties
to estimate cost parameters, our choice is a rea sonable approxim ation
13
. (This point is
12
For a proof of this result, see the supplementary material web page.
13
A realistic view is that there is a fixed administrative cost per loan file, because there are no obvious reasons
wh y a larger mortgage should be m ore costly to manage than a small one, at least as a first approximation. These
costs will be small relative to the size of loans, with a small impact on the interest rate charged in equilibrium.
Thepresenceoffixed costs per branch is an additional problem, albeit not a serious one. They affect the
competitive version of the model, but only the expression of the zero-profit condition in this competitive version,
since the necessary conditions for profit maximization do not depend on fixed costs. These costs are likely to
bias upwards slightly (i.e.,toaddasmallconstant)toparameterθ in equation (17) below. In contrast, fixed
costs do not intervene at all in the equations of the monopolistic version of the model: this is a clear advan tage
of this latter version.
12
discussed further in Appendix A).
With our simplifying assumptions, the date 0 net present value of a loan, conditional
on t
∗
,denotedb(t
∗
), is
b(t
∗
)=−m + m
Z
t
∗
0
p(r, T )e
−it
dt. (14)
The expected profit from a loan can therefore be computed as,
Π(c, h, T ; α)=
Z
T
0
θe
−θt
∗
b(t
∗
)dt
∗
+ e
−θT
b(T ). (15)
Straightforward computations yield the following formula
14
,
Π(c, h, T ; α)=−m +
p(r, T )m
p(θ + i, T )
. (16)
Given (16), the zero-profit condition Π(c, h, T ; α) = 0 leads to
p = p(θ + i, T ), (17)
meaning that interest rates charged can be expressed as i plus a risk-premium θ,incom-
petitive equilibrium.
With this formulation, a simple change of variables in (16), using (3) and (5), yields
the expression,
Π(c, h, T ; α)=
³
w
p(θ + i, T )
+ a
´
− πh −
c
p(θ + i, T )
, (18)
which is linear with respect to (c, h). The banker’s marginal rate of substitution between
c and h is therefore constant for each given type of borrower. More precisely,
∂Π
∂c
∂Π
∂h
=
1
πp(θ + i, T )
. (19)
3.3. Competitive equilibrium in mortgage contracts
We assume here, (i), that a large number of banks compete to supply loans to con-
sumers, and (ii), that a consumer’s t ype α is observable by every bank. Banks ”com-
pete in contracts”. For given T , the banker who offers the most advantageous contract
(c(T,α),h(T,α),T) will attract all the clients of type α. It follows that competition
will drive profits to zero for each pair (T,α). Formally, in equilibrium, if the contract
(c(T,α),h(T,α),T) is traded, Π[c(T,α),h(T,α),T; α] = 0. In addition, each contract
(c(T,α),h(T,α),T)inthemenuoffered in equilibrium, will satisfy the familiar necessary
condition for optimalit y,
∂Π
∂c
∂Π
∂h
=
∂
¯
V
∂c
∂
¯
V
∂h
, (20)
for otherwise, a competitor could offer a more advantageous contract to consumers of type
α,givenT .
14
For a proof of expression (16), see the supplementary material web page.
13
INSERT FIGURE 1 HERE
The competitive equilibrium is represen ted on figure 1(a). For any given T , the equilibrium
contract (c
∗
,h
∗
)=(c(T,α),h(T,α)) simply maximizes
¯
V (c, h, T ; α) under the zero-profit
constraint Π[c, h, T ; α] = 0, which is linear here.
3.4. The discriminating monopolist’s menu of contracts
If banks are endowed with a form of market power, due to concen tration in the mortgage
lending sector, or due to the fact that competitors consisten tly reject the applications of
some consumer types, then, a more appropriate model will be that of a first-degree dis-
criminating monopolist. Perfect discrimination requires that, for each type pair (T,α), for
which trade occurs, the contract (c(T,α),h(T,α),T) should maximize Π(c, h, T ; α)subject
to the borrower’s participation constraint (12), for given T . The necessary condition for
optimality (20) will then again be satisfied for all (T,α). But the zero-profitcondition
should be replaced by the zero-surplus condition,
¯
V (c(T,α),h(T,α),T; α)=V
r
(ρ, w; α), (21)
which says that the entire consumer surplus is extracted by the monopolist. This situation
is represented on figure 1(b): for each (T,α), the monopolist maximizes profit Π,with
respect to (c, h), subject to the constraint that
¯
V (c, h, T ; α) ≥ V
r
(ρ, w; α). Expected
profit is decreasing with respect to c and h: the bankers always p refer to reduce house
size h (lend less) for a fixed reimbursement flow pm, and they always prefer to increase
the rate, that is, reduce consumption c, for a given loan size m. The optimal contract is
therefore located on the lowest iso-profit line compatible with borrower participation.
4. Estimation of the models
We will first propose a wa y of estimating the competitive model described above, and
discuss the possibilities of testing for the presence of discrimination. We then present
some estimation results. The same will later be done with the discriminating monopolist
model.
4.1. Specification and estimation of the competitive model
In order to obtain a reasonably tractable formulation we assume that the instan taneous
utility u is quasi-linear, with the particular parametric form,
u(c, h)=u
0
+ c + γ
h
, (22)
where γ is a function of individual characteristics and , an individual preference parame-
ter
15
.Theparameterβ defined as
β =
1
1 −
,
15
The specification u=γ
1
c+γ
2
h
is of course equivalent to, and cannot be distinguished empirically from,
(22). The ratio γ=γ
1
/γ
2
only is identifiable, jointly with . The constant u
0
plays no role in the econometric
specification. These things become intuitively clear below.
14
with 0 <<1, can be interpreted as the ”price-elasticity of the demand for housing”, as
will be shown below. With the above specification of u,using(9)withT
∗
=0,onegets,
¯
V (c, h, T ; α)=u
0
+
wδ
δ + θ
e
−(δ+θ)T
+
cδ
δ + θ
(1 − e
−(δ+θ)T
)
+
γ
δ + θ
h
δ + θe
−(δ+θ)T
i
h
. (23)
From (23), it is easy to compute the marginal rate of substitution of a borro wer,
∂
¯
V/∂c
∂
¯
V/∂h
=
h
1
β
γK(θ, δ, T)
, (24)
where by definition,
K(θ, δ, T)=
1+(θ/δ)e
−(θ+δ)T
1 − e
−(θ+δ)T
. (25)
We remark that if the borrower becomes very impatient, then lim
δ→+∞
K(θ, δ, T)=1and
if the borrower becomes extremely patien t, lim
δ→0
K(θ, δ, T)=+∞.
The contract optimality condition (20) can then be rewritten,
h =
Ã
γK(θ, δ, T)
πp(θ + i, T )
!
β
. (26)
Using then the change of variable h =(m + a)/π and taking logs yields the expression,
log(m + a)=βlog(γ)+(1− β)log(π)+βlog[K] − βlog[p(θ + i, T )]. (27)
Expression (27) is very close to being a simple ”demand for housing” function, in the
expression of whic h p(θ + i, T )playstheroleofahousingprice.
Recall that the zero-profit con dition (17) writes p = p(θ + i, T ). If we assume that
both equations (27) and (17) are satisfied up to a random error term, (27) and (17) become
a bivariate nonlinear system.
Econometric specification
For the purpose of estimation, our bivariate model (17)-(27) should be fully specified. We
consider (a, w, T )asexogenousvariables. ThediscretevariableT couldinprinciplebe
treated as endogenous, but this would lead us to a muc h more complicated model with
three endogenous variables, and the equation determining T would be very difficult to
estimate. In addition the empirical distribution of T is very much concentrated on 10, 15
and 20 years (see figure9). ThisiswhywetreatT as exogenous. In addition, there are
other exogenous observations on the households, such as family size, age, and occupational
status, that can be used to explain differences in preferences for housing and default risk.
Let X denote the v ector of all exogenous variables, including, a, w, π, T and i.Thevector
15
(β,γ,δ,θ) fully characterizes a consumer. A possibility is then to assume that γ, β and θ
are themselves functions of X, and to specify functional forms γ(X), θ(X)andβ(X), with
parameters to be estimated (δ could also, in principle, be specified that way).
We have been able to estimate a somewhat simplified version of (17)-(27). To limit the
number of parameters, we ha ve set K = 1: this is tantamount to assuming that consumers
are very impatient
16
. We obtain the following model.
log(m + a)=β(X)log(γ(X)) + (1 − β(X))log(π) − β(X)log[p(θ(X)+i, T )] + e
1
, (28)
log(p)=log[p(θ(X)+i, T )] + e
2
. (29)
The perturbation vector e =(e
1
,e
2
) is assumed normally distributed with mean zero and
covariance matrix Ω. These error terms can be viewed as a mixture of several random
sources: unobserved borrower characteristics, measurement errors, and possibly optimiza-
tion errors on the part of economic agents. The random effect of unobserved agent char-
acteristics on p and m + a is likely to be correlated: the covariance matrix Ω is therefore
not constrained to be diagonal, and we’ll obtain an estimate of e
1
and e
2
’s covariance
17
.
Of course, in the above equation, it must be understood that p = p(r, T ). (Recall that
r and T are observed). The endogenous variables are simply log(m + a), the logarithm of
the total house price (in euros), and log(p), the logarithm of the contin uous-time constant
reimbursement annuity. The function log(γ(X)) is a linear function of the constant, log(w),
the age of the borrower (in logarithms), and family size (in logarithms). The parameter β
varies with occupational status dummies only (white collar, interm ediate professions and
executives), blue-collar workers being the reference group. Let Exec, Interm, Whitecol,
Bluecol respectively denote the executive, intermediate profession, white collar and blue
collar dummies. Finally, θ is specified as follows:
log(θ(X)) = θ
0
+ θ
1
log(w)+θ
2
log(w).(Exec)
+θ
3
log(w).(Interm)+θ
4
log(w).(Whitecol)+θ
5
log
³
m + a
a
´
, (30)
aspecification in which the wage (in logs) interacts with occupational status. The model
has been estimated b y the maxim um likelihood method. It is less simple than it seems,
since (m + a)appearsinθ(X) and is endogenous; this gives rise to complicated Jacobian
terms in the likelihood expression
18
. The price of houses π has been calibrated: we set its
value at 428 euros per square meter (this corresponds roughly to $45 per square foot), a
reasonable figure for the French prov inces.
Finally, information on the starting date of mortgage contracts is taken in to account
through the cost of funds i, which varies with time, but it is not included directly as an
16
An estimate of δ has been obtained in a variant of the monopoly model (see Appendix B), but at the cost
of simplifying the utility function (we then set β=1), and at the cost of estimating a simplified version of the θ
function. Parameter δ is difficult to estimate.
17
A model is structural when it is deriv ed from first principles (which is the case here), but the error terms
should ideally also have a clear interpretation. On the interpretation of error terms in structural econometric
models, see J. Rust (1994).
18
The likelihood function for this model is derived in the supplementary material web page.
16
explanatory variable
19
. Borrower age and family size variables do not work very well with
our data set. In principle, other specifications could be estimated, such as including the
occupational status variables in γ, or including them independently of the wage variable
in the θ function. Wehavetriedvariousspecifications of the β, γ and θ functions which
do not always work well
20
.
Estimation results
The results are presen ted in Table 2. All parameters are significant, except the white-
collar dummy coefficien t in β(X), the family size coefficient in γ(X), the intermediate-
professions×wage, and the executives×wage interaction coefficients in θ(X). Al l parameter
estimates also have the expected sign. We will concentrate our comments on β andonthe
estimated risk premium function.
First, the ”price-elasticit y of demand for housing” β varies significantly with occu-
pational status: for executives, β =2.1766; for intermediate professions β =2.1475; for
white collars β =2.1231; and for blue collars, we find β =2.1166. That is, the lower
in the professional hierarchy, the less price-elastic is the demand for housing
21
.Theβ of
a white collar is not significantly different from that of a blue collar, but executives and
intermediates do differ from blue collars in that respect.
Second, the estimated θ function shows that wages act as a discrimination device, as
expected: the higher a borrower’s wage, the lower the price cha rged. The coefficients of
wage×status interaction terms in the θ function are not all significant: thisisdisappointing,
but the estimates of these coefficients will become significant, with a nice pattern, in the
monopolistic version of the model below. At the same time, the inverse of the down
payment ratio has a positive coefficient, as expected.
Thepresenceofmarketpower
The mean value of θ in the blue-collar category is θ(Bluecol)=0.0398; it is θ(Whitecol)=
0.0425 for white-collars; θ(Interm)=0.0367 for intermediate professions, and θ(Exec)=
0.0330 for executiv e s. But the order of magnitude of these risk-premia is too high. Since
the probabilit y of default before term is F (T )=1− e
−θT
, we can estimate F (15) and the
figures are 0.464, 0.469, 0.380 and 0.350 for the blue collars, white collars, intermediates
and executives, respectively. These probabilities of default do not correspond to what is
known aprioriabout mortgage default rates in France. A reasonable figure would be
something like 3%, not 35% or 46%! The screening process in the local BSs is a kind
of ”old-style” job made b y some comit´es de cr´edit.Thelocalofficers are very likely to
19
A referee sugested that we could use this starting date information to capture shifts in the risk-premium
function due to changes in the unemployment lev el. This is a good idea in principle, but it yields poor results in
practice, presumably because changes in the unemployment rate are too slow during the estimation period
20
With the data and the estimation methods employed here, it is reasonable to use a somewhat parsimonious
specification: we should not demand too much from the data or try to estimate too many parameters.
21
The reader could have expected the reverse ranking: we provide an explanation belo w for this result.
17
use local information on markets, borro wers
22
, etc... The chief executives of the CHF
claim that their network does a very good screening job and that F(15) ≈ 0.01. This is a
sufficientindicationforthepresenceofmarketpower: thevaluesofθ should be interpreted
as markups, not as risk-premia. The fact that these markups are inversely related to the
corresponding price-elasticities of demand for housing, the βs, is an indication that price
discrimination is taking place. Workers are less price-elastic than executives, they therefore
pay more in (monopolistic) equilibrium. This is why we estimate a monopolistic version
ofthemodelinthenextsubsection.
Figures 2 to 4 represent n umerical simulations of the model with the estimated pa-
rameters. Figures 2 and 3 depict the interest rate charged to borrowers as a function of
thewageandofthedownpaymentratio,respectively. Todrawthesefigures, the values
of exogenous variables are set equal to their overall sample mean, with the exception of
the variable appearing on the x-axis. The sc hedules show differences of treatment bet ween
social categories, everything else being equal. The mere fact of being an executive leads to
a reduction of the interest rate of approximately half a percentage point, relative to white
collars. Figure 4 shows the house size schedule as a function of the wage. The same type
of social discrimination appears clearly.
4.2. The Monopoly Model
To establish the equations of the monopoly model, it is sufficient to replace the zero-profit
condition (17) with the zero-surplus condition (21). In order to obtain the analytical
expression of (21), we first compute V
r
, the indirect utilit y of a household on the ren tal
market. The rental demand for housing is obtained by maximizing u(c, h), as specified by
(22) above, with respect to (c, h), subject to the budget constraint c + ρh = w,whereρ is
the rent per square meter. This easily yields the instant aneous indirect utility of renting,
v
r
(ρ, w)=u
0
+ w +
1
β − 1
γ
β
ρ
1−β
, (31)
from which we deriv e the expected utility of renting forever, defined by (10) above:
V
r
(ρ, w)=v
r
(ρ, w)
δ
θ + δ
+ u
0
θ
δ + θ
. (32)
Now, we equate
¯
V ,givenby(23),withV
r
, to obtain the analytical form of (21), and
simple computations yield the following expression.
−pm +
βγ
β − 1
K(θ, δ, T )h
(β−1)/β
=
γ
β
ρ
1−β
(β − 1)(1 − e
−(θ+δ)T
)
. (33)
Note that u
0
has disappeared from (33). The model to be estimated is (27) and (33). As
before, we set δ =+∞, and thus K = 1 (see Appendix B, for an estimation of δ), and get
thefollowingbivariatemodel.
log(m + a)=β(X)log(γ(X)) + (1 − β(X))log(π) − β(X)log[p(θ(X)+i, T )] + e
1
, (34)
22
Our information on real defaults at the CHF is vague: we cannot provide a statistical analysis of defaulting
borrowers in the CHF network.
18
pm =
β(X)γ(X)
β(X) − 1
³
m + a
π
´
β(X)−1
β(X)
−
γ(X)
β(X)
ρ
1−β(X)
β(X) − 1
+ e
2
, (35)
where e
1
and e
2
are zero-mean random error terms, with a normal distribution, y
1
=
log(m + a)andy
2
= pm are the endogenous variables, and X is the vector of exogenous
variables. Of course, p = p(r, T ). The specifications of γ(X)andθ(X)areexactlythe
same as before (θ being defined by (30); γ being a loglinear function of the constant, w,
age and family size). For this estimation, we set β(X)=exp[b
0
+b
1
(Exec)+b
2
(Interm)+
b
3
(Whitecol)]. The price of a square meter and the yearly ren t per square meter are
calibrated at reasonable values, respectively π = 428 and ρ =46,ineuros.
The model is again estimated by standard, full information maximum likelihood meth-
ods
23
. Estimation results are summarized b y Table 3. All coefficients are significant, except
the coefficients of the intermediate professions dummy in β, of family size and age in γ,
and of the white-collar×wage interaction variable in θ.
Again, the values of β can be ranked according to occupational status, estimated
values being β =1.6117 for blue collars, β =1.6002 for white collars, and β =1.6308 for
executives.
The estimated mean values of θ for each occupational status reveal the expected
ranking of risk premia, from the lowest, the executives, to the highest, the white and
blue collars (see bottom of Table 3). ”Social discrimination” is present, since wage-status
interaction term s are significant for executiv es and intermediate professions in the risk-
premium function θ. Workers are discriminated against, just because they are workers,
not simply because their wage is low. A negativ e coefficient on the wage in the θ function,
which varies with occupational status,meansthatthewaythewageistakenintoaccount
to assess default risk depends on status. According to the banker, the richest borrowers
are the less risky. The extent of the interest rate reduction which is granted for a given
increase in the wage is higher, the higher the status.
The number of variables introduced in the risk-premium function is too small to guar-
anteethattheestimatesreflect prejudice against the working class (or a favorable prejudice
for executives). The observed discrimination could be a form of statistical discrimination,
that is, the occupational status variables are likely to act as proxy variables for unobserved
factors correlated with default risk (see the discussion in section 5 below).
Finally, these estimates of θ correspond to much more reasonable probabilities of
default than the competitive estimates obtain ed above. Computing F (15) = 1 − e
−15
¯
θ
with the estimated average values of the risk premium,
¯
θ, in eac h category, yields F = .169
for blue collars, F = .119 for white collars, F = .055 for intermediates and F = .018 for
executives: the expected ranking.
We conclude that housing demand elasticities (i.e.,theβs) are overestimated by the
competitive model, because risk-premia are also overestimated. Since equation (29) (i.e.,
the competitive pricing equation) is likely to be an incorrect representation, upward biased
risk-premia are transformed into upward biased housing demand elasticities while estimat-
ing equation (28) (i.e., the house size equation) sim ultaneously with (29). We probably
get better estimates of structural parameters with the monopoly version of the model, in
23
The lik elihood function for this model is derived in the present paper’s supplementary material section.
19
spite of the added complexit y of the zero-surplus equation (35).
We also conclude that workers seem to be discriminated against by banks, the origin of
the discrimination being mostly due to differing elasticities of demand, and secondarily to
differences in perceived default risks. These differences in elasticities β could simply capture
the fact that blue-collar workers are more likely to see their loan applications rejected by
other commercial banks, since the CHF is specialized in ”social loans.” Nev ertheless, the
results show that consumer heterogeneity is exploited by bankers to make more profit, as
illustrated by the simulations.
Figures 5 to 8 represen t numerical simulations of the model with estimated values
of the parameters. Figures 5 to 7 illustrate the same kind of phenomena as Figures 2 to
4, but under the assumption of monopolistic behavior. Figure 8 represents the estimated
reservation utility or participation lev els of the social categories, as defined by (21) above,
in the form of indifference curv es. It is easy to see that executive s consume more than the
workers for any given size of the house. To compute each of these indifference curves, we
used the sample mean values of the exogenous variables, except for the w age, the mean of
which is evaluated in each occupational status sub-sample.
In Appendix B, we propose a variant of the monopoly model, in whic h the impatience
parameter δ and function K are estimated. This has been done, as explained abo ve,
at the cost of drastic simplifications of the risk-premium and utility functions, to permit
identification. More work could be done in this direction, but we have chosen to emphasize
the links between interest-rate markups and the elasticit y of the demand for housing
24
.
5. Further remarks on racial discrimination in credit markets
Discrimination in credit markets has recently attracted considerable attention, and the
question of deciding whether or not – and why – lenders discriminate against minority
groups is a hotly debated topic among economists. The importance of the question is am-
plifiedbythefactthatracialdiscriminationinmortgage lending has been made illegal, in
the United States, by the Equal Credit Opportunity Act of 1974, and by the availability of
new data sources, allowing for new econometric tests. The recent literature on this ques-
tion is mostly empirical, and has concentrated on racial or sexual discrimination problems.
Empirical studies of discrimination in mortgage lending have developed with the debate
triggered by the con tributions of Shafer and Ladd (1981) and Munnell et al. (1996). Sev-
eral other important contributions to this literature are commen ted on in Ladd’s (1998)
survey article, and in the recen t book of Ross and Yinger (2002).
Adifficult problem in most empirical studies is to detect the presence of d iscrimination
in the sense of G. Becker (1957). More precisely, a lender, or seller, is said to discriminate
in the sense of Becker, if she is ready to forego profits just because of her prejudices. This
form of discrimination is based on a particular ”taste for discrimination” of the sellers, and
is not usually considered by standard I.O. theories of price discrimination.
In contrast, a lender might treat a minority group differently, because racial or ethnic
characteristics are correlated with some variables, important in the determination of credit
24
The estimation of time preference parameters is known to be very difficult in various fields of applied
econometrics, such as macroeconomics or finance.
20
worthiness and default risk, and which remain unobserved. This latter form of behavior
is called statistical discrimination in the sense of Arrow (1973) and Phelps (1972). For a
more recent treatment of this subject, see Loury (20 02). An important difficulty stems from
the fact that econometricians can never be sure of having introduced enough explanatory
variables to control for possible risk differences in their estimation of default probabili-
ties. It follows that a significant coefficient on race in regressions could only indicate that
statistical discrimination is taking place.
Another difficulty is the need to separate the effectofriskfromthatofmarketpower
in the formation of in terest rates
25
. T he risk premium charged on some borrowers can
also be interpreted as a standard monopolistic markup. To be more precise, it might be
that the borro wers’ preference parameters are correlated with their social, racial or ethnic
group, because individual preferences depend on a group’s particular economic conditions.
Then, if market power is present, standard price discrimination can in turn become the
explanation for differential treatment, without necessarily reflecting the presence of prej-
udice. (This type of approach, however, is not without its dangers, which would be to
attribute the bulk of observed differences in treatment to taste differences correlated with
race.) However, competition should tend to eliminate discrimination in the sense of Becker,
since prejudiced lenders would lose business in favor of the unprejudiced. It follo ws that
market power and discrimination in this sense must be closely interrelated.
Finally, the most difficult problem in detecting the presence of prejudice is that dis-
crimination in the market for mortgages might reflect the existence of discrimination in
other markets, suc h as the labour and housing markets, and th us be purely statistical in
nature. Some minorit y groups would pay higher mortgage rates because they have higher
probabilities of losing their jobs, and this, in turn, could simply be a consequence of their
employer’s behavior.
Much of the published work on discrimination in the mortgage market, to the best of
our knowledge, has been devoted to the study of default and of credit denial rates (again see
H. Ladd (1998), Ross and Yinger (2002)). In contrast, the model present ed above aims at
explaining the structure of accepted loan applications and determines loan sizes and interest
rates simultaneously. This model could be used to test for the presence of discrimination
in the sense of Becker, paying atten tion to the role of local market conditions, of preference
heterogeneity and of differing default risks. In theory, the model allows one to separate, in
theinterestrateandinloansizedifferences, what can be attributed to prejudice, from the
impact of differences in preferences and default risks. Variations in β, γ and δ can reflect
differing preferences while θ reflects default risk differences as perceived b y the banker.
An observable c haracteristic can significantly c hange (β,γ,δ),andthusleadtochangesin
treatment by the banker.
If X contains enough information (enough control variables) to estimate a risk pre-
mium reasonably, the function θ(X) should not significantly depend on race. If it indeed
does depend on race significantly, then, discrimination in the sense of Becker is taking
place. Of course, if the information contained in X is not sufficient to control for differ-
25
On the role of concentration in the explanation of interest rates, see Cavaluzzo and Ca valuzzo (1998). On
the empirical relationship between concentration and discrimination, see Berkovec et al. (1998). Graddy (1995)
finds a form of third-degree, race-based price discrimination on the New York fish market
21
ences in riskiness, then, statistical discrimination in the sense of Arrow-Phelps can be the
explanation for a significant coefficient on race in function θ.
The model presented above allow s for separation of these effects from plain discrim-
ination effects based on observable differences in preferences. There is a danger here,
however, which would be to attribute the bulk of differences in treatment to differences in
preferences: minority consumers would have relatively smaller or bad quality houses just
because they happen not to like nice housing! To avoid ambiguities of this type, it seems
to us that race should not be in troduced as a variable in the specifications of β, γ and δ.
6. Conclusion
The present contrib ution proposes a model of the mortgage lending market. The model
canbeusedtotestforthepresenceofdiscrimination, using only information on accepted
loan applications. It rests on the idea of discrimination by the lender, based on observable
attributes of the borrower. It explains the interest rate and the loan size of accepted loan
applications simultaneously. We study a competitive equilibrium variant and a discrim-
inating monopolist variant of the model, in order to take ph enomena related to market
power into account.
The model has been estimated with a sample of loan files originating from branches of
a French mortgage lender. We reject the competitive model because estimated interest rate
markups are too high to reflect default risks only. The monopolistic model gives a better
account of the discrimination phenomena at w ork in the data. We conclude that ”social
discrimination” is present, in the sense that ceteris paribus, a member of the working class
would pay higher interest rates than an (equally rich) executive, just because he or she is
identified as a blue collar. Part of the differences in interest rates must be attributed to
observable differences in preferences, since blue-collar workers have a significantly smaller
price-elasticity of demand for housing than executives. The model also shows how borro wer
characteristics and interest rates affect the size of granted loans.
22
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24
Appendix A. The Expected Value of Foreclosures
A more complex version of the expected profit function derived as expression (16) above
takes the expected value of foreclosures into account. Let Z(t
∗
,m,T; α)denotetheliq-
uidation value of borrower α’s house, given that default occurs at date t
∗
.FunctionZ
describes the expected net revenues of foreclosure. For convenience, w e assume that Z =0
for all t
∗
≥ T . We can then reformulate the banker’s profit function, conditional on t
∗
,as
b(t
∗
,c,h,T; α)=−m + m
Z
t
∗
0
p(r, T )e
−it
dt + e
−it
∗
Z(t
∗
,m,T; α).
Taking the expectation of b with respect to t
∗
,wegettheexpectedprofitfunction,
Π(c, h, T ; α)=−m +
pm
p(θ + i, T )
+
¯
Z(m, T ; α)
where,
¯
Z(m, T ; α)=
Z
t
∗
0
θe
(θ+i)t
∗
Z(t
∗
,m,T; α)dt
∗
,
is the expected present value of foreclosure. Now, Z itself should be specified more precisely.
Let ζ(m; α) denote the liquidation value of the house. Then, a reasonable specification of
Z can be written,
Z(t
∗
,m,T; α)=Min
h
mp
(1 − e
−i(T −t
∗
)
)
i
,ζ(m; α)
i
,
where (mp/i)(1 − e
−i(T −t
∗
)
)=pm
R
T
t
∗
e
−i(t−t
∗
)
dt is the value of the remaining debt (prin-
cipal and interest), when default occurs at date t
∗
.Now,toevaluateZ accurately, some
knowledge of future house prices, conditional on α and t
∗
is required. To be more precise,
ζ should depend on the banker’s house price expectations. This kind of information is not
present in our data set; it is always a problem to model expectations, and the expression
of
¯
Z is not simple. If we accept the simplifying assumption of constant house prices, and if
ζ is never high enough to ensure the banker completely against losses in case of foreclosure
(i.e., if the banker does not believ e that house prices will be high enough to recoup all
losses in case of default), the expression of
¯
Z can be studied numerically. According to our
simulations,
¯
Z happens to be appro x imately linear with respect to (m + a). Let us then
use a linear approximation and assume,
¯
Z = z
1
+ z
2
(m + a)
where z
1
and z
2
are parameters (to be estimated). We these assumptions, it easy to see
that the zero-profit condition becomes,
pm =(1− z
2
)p(θ + i, T )
h
(m + a) −
³
a + z
1
1 − z
2
´i
,
25
and that the contract optimality condition becomes,
h
1/β
γ(α)
=
1
π(1 − z
2
)p(θ + i, T )
.
There are problems to estimate these two equations. To get an int uition of the difficulties,
assume to simplify that a =0andz
1
=0,then,anidentification problem will arise,
because the respective magnitudes of (1 − z
2
) and of the constant in the θ function cannot
be determined.
Assume now that the value of foreclosuresiszero,butthatthereisafixed adminis-
trative cost k per loan. We easily obtain the zero-profit constrain t, in this case, by setting
z
2
=0andz
1
= −k. Equation (17) can then be rewritten as,
p = p(θ + i, T )
³
1+
k
m
´
,
and the contract optimality condition is still giv en by (27). Again, there is a problem to
estimate the value of k in this v ersion of the model, and it is easy to see that, k/m being
small in general, the above formulation is not very different from (17).
Assume now that the value of foreclosures is zero, that the cost per loan file is zero, but
that there are nonzero fixed costs in each branch, denoted A. This changes the zero-profit
condition (17) again. To get an in tuition of these changes, and to simplify the discussion,
assume that there is only one risk class of consumers (the discussion becomes more in volved
with several types of borrower, but leads essentially to the same conclusions). The identical
clients borrow m euros for T years and pay the price p. The profit of a BS can be expressed
as,
Π = −A − Nm +
Npm
p(θ + i, T )
,
where N is the n umber of identical borrowers. The zero-profit condition is then,
p = p(θ + i, T )
³
1+
A
Nm
´
.
In the absence of information on A and N, the multiplicative term (1 + A/Nm) cannot
be easily estimated. This term is also likely to be close to one, for Nm is a huge sum of
money. In the competitive version of the model, fixed costs will bias the estimation of the
constant in the θ function upwards.
To sum up, we have chosen a reasonable and tractable approach, which works well
numerically: the joint estimation of (17) and (27). This approach permits one to estimate
a risk-premium function θ jointly with preference parameters (in the β an d γ functions).
Thechosenapproachissuchthatestimatedparametersoftheθ function probably reflect
the expected liquidation value of the house, and the quality of the mortgage as a collateral,
as well as the personal cha racteristics of the borrower. In the competitive ve rsion of the
model, estimated parameters also reflect administrative costs. Ideally, we would have
preferred to separate completely, on the one hand, the impact of borrower characteristics
26
on default rates from that of expected property prices and of the down payment ratio
(both contributing to the quality of the mortg age), on the other hand.
Butthisisnottheonlydifficulty: in the data set, the effect of expected liquidation
values is entangled with substantial market power phenomena, i.e. with the markup
element in the θ function, and therefore difficult to iden tify. We think that the markup
element dominates in the data, and that the role played by expected liquidation values is
of secondary importance.
27
Appendix B. A Variant of the Monopoly Model
We also estimated a variant of the monopoly model, with a more parsimonious specification
of utility, that is,
u(c, h)=u
0
+ c + γlog(h), (36)
but we tried to estimate the impatience parameter δ.Withthisspecification, we find that
the instan taneous reservation utility of borrowers writes,
v
r
(ρ, w)=u
0
+ w − γ + γlog
³
γ
ρ
´
. (37)
The necessary condition for optimality (20) now writes,
h =
γK(θ, δ, T)
πp(θ + i, T )
, (38)
where K is still defined by (25). With the abo ve specification of u,using(9)withT
∗
=0,
one gets,
¯
V (c, h, T ; α)=
δc
δ + θ
(1 − e
−(δ+θ)T
)+
γlog(h)
δ + θ
h
δ + θe
−(δ+θ)T
i
+
wδ
δ + θ
e
−(δ+θ)T
+ u
0
. (39)
Then, equating (39) with V
r
(ρ, w; α), using (37), yields the specific form of the zero-surplus
condition (21), that is, after some computations,
(c − w)(1 − e
−(δ+θ)T
)+γlog(h)(1 + (θ/δ)e
−(δ+θ)T
)=γ[log(γ/ρ) − 1]. (40)
The system to be estimated is (38) and (40), with c − w = −pm and h =(m + a)/π.
Taking logs on both sides of (38), we find the system,
log(m + a)=log(γ(X)) + log[K(θ,δ, T)] − log[p(θ + i, T )] + e
1
, (41)
pm = −γ(X)
[1 + log(γ(X)/ρ)]
[1 − e
−(δ+θ)T
]
+ γ(X)log
³
m + a
π
´
K(θ, δ, T)+e
2
, (42)
where e
1
and e
2
are zero-mean random error terms, with a normal distribution, y
1
=
log(m + a)andy
2
= pm are the endogenous variables, and X are exogenous variables.
We assume that for each individual j, θ can be expressed as θ
j
= −(1/T
j
)log(1 − F)
where F is a common probability of default. The parameters to estimate are D = log(δ),
F , ρ and the parameters of function γ,whichisspecified as follows,
log(γ(X)) = γ
0
+ γ
1
log(w)+γ
2
log(age)+γ
3
log(T )
+γ
4
(Exec)+γ
5
(Interm)+γ
6
(Whitecol). (43)
28
This model was difficult to estimate. We estimated δ = e
D
and the γs by standard
maximum likelihood for fixed ρ and F , and then estimated ρ and F by a grid-search
procedure. The results are presen ted in Table 4. All the γ parameters are significantly
different from zero, except the coefficient on the white-collar dummy, meaning that white
collars can be merged with the blue collars. The estimated value of δ is 0.158, a reasonable
figure. Given that we assume liquidity constrained borrowers, they m ust be impatient
enough to be willing to borrow, so that it is reassuring to find δ above the maximal
interest rates in the sample. The estimated rent ρ, whic h is FFrancs 294 per square meter
and per year is also reasonable, given that our observations are outside the very expensive
Paris area (this makes a rent of $4.41 per square foot and per year, or an approximate
monthlyrentof$202foratwo-roomsflat of 550 square feet). Finally, the estimated value
of F is 0.011, very close to the expected 1% value. This, again, corresponds to a much more
realistic average estimated value,
¯
θ =0.00073, than those obtained with the competitive
model. In fact θ seems to be of the order of magnitude of 1 to 10 base points.
29
