1. finance
2. personal finance
3. lending
4. student loans

Macroeconomics and Financial Markets
... or, better,...
Housing and Credit Markets: Bubbles and
Crashes
Veronica Guerrieri and Harald Uhlig∗
University of Chicago, NBER and CEPR
VERY PRELMINARY
COMMENTS WELCOME
First draft: July 7th, 2014
This revision: April 13, 2015
∗
We are grateful to Chiara Fratto, Ken Kikkawa and Chao Ying for their excellent
research assistance. Address: Veronica Guerrieri, University of Chicago, Booth School
of Business, 5807 South Woodlawn Avenue, Chicago, IL 60637, U.S.A., email: [email protected], Harald Uhlig, Department of Economics, University of Chicago,
1126 East 59th Street, Chicago, IL 60637, U.S.A, email: [email protected]. This
research has been supported by the NSF grant SES-1227280 and by the INET grant
#INO1100049, “Understanding Macroeconomic Fragility”. Uhlig has an ongoing consulting relationship with a Federal Reserve Bank, the Bundesbank and the ECB. Guerrieri
has an ongoing consulting relationship with the Chicago Federal Reserve Bank.
i
Contents
1 Introduction
1
2 A stark model
6
2.1 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . 13
2.1.1
An exogenous crash in leverage. . . . . . . . . . . . . . 17
2.2 An exogenous crash in house prices . . . . . . . . . . . . . . . 20
3 Related Literature: Households’ Leverage
23
4 Multiple Equilibria and Catastrophes
25
4.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.2 Multiple equilibria . . . . . . . . . . . . . . . . . . . . . . . . 32
4.3 Some numerical examples . . . . . . . . . . . . . . . . . . . . 33
5 Credit Cycles
34
6 Bubbles
34
7 Sentiments
38
7.1 The idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
7.2 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
7.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
7.4 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . 49
8 Evidence
49
9 Conclusions
59
ii
1
Introduction
In the recent years, the United States has experienced, at the same time, a
boom-bust episode in house price and a boom-bust episode in credit markets,
as reflected in figures 1 and 2.
Source: http://www.worldpropertyjournal.com/north-america-residential-news/spcase-shiller-home-price-indices-report-fordecember-2011-case-schiller-home-price-index-median-home-prices-the-national-composite-10-city-composite-index-20-citycomposite-home-price-indices-david-m-blitzer-5351.php
Figure 1: The S&P/Case-Shiller Home Price Indices
The purpose of this chapter is to explore the connection between financial
markets and the housing market and its effects on the macroeconomic activity. There is a large and growing literature that separately explores credit
cycles and house price bubbles. In this chapter we will try to connect these
two streams of literature and understand the potential feedbacks between
the two.
In particular, we will explore two different broad approaches to think
about this connection:
1
Source: The Financial Crisis Inquiry Report, National Commission, January 2011
Figure 2: Subprime Mortgage Originations
1. the house price boom-bust generates the credit boom-bust;
2. the credit boom-bust generates the house price boom-bust.
Moreover, we embrace the view that, in both cases, these connected boombust episodes generate a boom-bust episode in aggregate activity, which, in
turns, can feedback and amplify the boom and bust in the financial and
housing markets. Given that the relationship between the house price boombust episode and the credit boom-bust episode can be itself quite rich, in
this chapter we will mostly focus on that, and less on the connection with
the real economy.
We start the chapter by discussing a simple mechanical baseline model
in section 2, which is meant to describe the interaction highlighted between
the credit cycle and house prices, highlighted above. On purpose, it makes a
2
number of stark assumptions to avoid several thorny issues that arise in a fully
specified equilibrium model. In particular, we take as given the dynamics of
both leverage and house prices. We then perform two types of exercises: first,
we keep house prices constant and study the response to a boom and bust
in leverage, second, we keep the leverage constant and study the response to
a boom and bust in house prices. In the rest of the chapter, we try to go
deeper in understanding where these dynamics are coming from and connect
these two types of exercises to the existing literature.
We start exploring the idea that the boom and bust in the credit market was the fundamental shock that spilled over the housing market and
the real economy. In order to do so, in section 4, we completely abstract
from the housing price dynamics, and focus on the boom and crash in the
credit market. In particular, we propose a stylized static model of households
who borrow to become homeowners and intermediaries who lack information
about the quality of the borrowers. The main idea is that a simple increase
in the credit availability in the economy, that we interpret as “saving glut”,
can endogenously generate first a boom and then a crush in lending activity
because of adverse selection issues that can generate multiple equilibria. The
basic idea that a saving glut can generate an endogenous credit cycle because of multiplicity of equilibria arising from informational problems builds
on Boissay, Collard, and Smets (2014). In our model, the increase in credit
availability first increases the lending activity and hence increase the opens
“sub-prime market”. However, as the quality of the pool of borrowers decrease, good borrowers may decide to pay a cost to separate themselves and
get better credit terms. This in turns may make the sub-prime market to
collapse.
The next two sections discuss several related papers. In section 3,we
discuss a class of papers that started exploring the role of leverage in the
households’ sector and the real effects following changes in leverage, such as
3
Guerrieri and Lorenzoni (2011), Eggertsson and Krugman (2012), Justiniano, Primiceri, and Tamb
These papers tend to take the leverage dynamics as given and look at their
real effects. In section 5, we discuss another strand of literature that focuses
on credit cycles, although without connecting them to the household sector,
such as Kiyotaki and Moore (1997) or Myerson (2012).
Next, we move to the idea that the boom and crash in housing prices is
the key element of the interaction between housing and credit markets. In
particular, if house prices are expected to rise, banks are more willing to lend,
although this means to lend to worse creditors, e.g. sub-prime borrowers.
Moreover, speculating households are more willing to buy as house prices
appreciate. While appealing, formalizing this intuition tends to run into the
“conundrum of the single equilibrium”: if prices are expected to be high
tomorrow, then demand for credit and thus houses should be high today,
and that should drive up prices today, making it less likely that prices will
increase. Or, put differently, as prices rise, they eventually must get to a nearmaximum at some date: call it “today”. At that price, prices are expected
to decline in the future. But if so, banks and speculating households are less
likely to buy today: but then, the price should not be high today. These
types of bubbles are typically ruled out by thinking about the “last fool”
who is willing to buy at the highest price, when prices can only go down
from there: such fools should not exist in rational expectations equilibria.
To think about house price dynamics, we then discuss two types of bubble
models.
In section 6, we discuss a first class of bubble models, where the interest
rate demanded on assets is below or at most equal to the growth rate of the
economy. This can give rise to rational bubbles and stochastically bursting
bubble in, essentially, dynamically inefficient (or borderline efficient) overlapping generations models, as in Carvalho, Martin, and Ventura (2012) or
Martin and Ventura (2012). These authors employ various versions of OLG
4
models, in which, ideally, resources should be funneled from inefficient investors or savers to efficient investors or entrepreneurs. They assume that
there is a lending friction: entrepreneurs cannot promise repayment. They
can only issue securities, where the buyer hopes that someone else buys them:
call them “bubble”, “cash” or “worthless pieces of paper”. Equilibria then
exist, where newborn entrepreneurs create “bubble” paper. The existing
bubble paper in the hands of old agents as well as those created by newborn
entrepreneurs get sold to savers. Savers find investing in these bubbles more
attractive than investing in their own, inefficient technologies. This technology needs to be inefficient enough so that its return is on average below the
growth rate of the economy, creating the dynamic inefficiency for bubbles to
arise. In that case, the “fundamental” value of any asset paying even a tiny
amount per period is actually infinite. Or, put differently, the last fool to
buy the bubble at the highest price is happy to do so, since the value of the
bubble next period will not have gone down too much and since that fool is
desperate to save.
In section 7, we discuss a second class of bubble models, where the interest rate demanded on assets is above the growth rate of the economy.
Here, an aggregate bubble eventually must stop growing, being bounded
by the resources in the hands of “newborn” agents purchasing these assets.
Rationality considerations typically rule out such bubbles, see the “conundrum” above. We therefore investigate models with irrational optimism and
changing sentiments. We shall present our own sketch of a model, in which
an asset is bought in the hope of selling to an ever more foolish buyer. A
benchmark example in the literature, exploiting changing sentiments, is the
“disease” bubble model of Burnside, Eichenbaum, and Rebelo (2013). There
is some intrinsically worthless bubble component, which could be part of
the price of a house. An initially pessimistic population may gradually
become infected to be “optimistic” and believe the bubble component ac5
tually has some intrinsic value: once, everyone is optimistic (forever, let’s
say), there is some constant price that everyone is willing to pay. However,
“truth” may be revealed at some probability every period, and clarify that
the bubble component is worthless indeed. Then, during the pessimistically
dominated population epoch, prices rise during the non-revelation phase,
since the rise in prices there compensates the pessimistic investor for the
risk of ending up with a worthless bubble piece, in case the truth gets revealed. The price will rise until the marginal investor is optimistic: at that
point, the maximum price may be reached. Here, in essence, the optimistic
maximal price is not justified by fundamentals and the last fools bought
happily, thinking they got something of value (though: strictly speaking,
Burnside, Eichenbaum, and Rebelo (2013) stay silent on this issue).
Finally, in section 8, we juxtapose our findings to some lessons we have
drawn from the existing literature regarding the empirical evidence.
We wish this chapter to trigger further research and thinking on this
important connection. As shall become clear, the issues are far from resolved.
2
A stark model
In the financial crisis of 2008, the following interplay might be at work,
amplifying any initial shock: 1) as house prices fell, banks became more
reluctant to lend to new home buyers, and 2) as banks became more reluctant
to lend to new home buyers, demand for houses and thus prices for houses
fell as a consequence. In particular, banks became more reluctant to lend
because the drop in house prices negatively impacted their balance sheets,
hence generating a more general credit crunch, depressing real activity.
In this section, we introduce a very simple mechanical model, featuring
some of that interplay, but without describing the “deep reasons” for some
key elements: indeed, that is the point of the latter sections of the chapter.
6
Time is discrete and infinite, so t = ..., −1, 0, 1, . . .. There is a continuum
of households. Each period, a fraction λ of the households “die” or exit, and
they are replaced with a fraction λ of newborn households. Each period all
households who are alive earn some exogenously fixed income y and consume
a non-negative amount of goods. When a household is born, it buys a house.
Houses are in fixed supply and have all the same size.
We assume that a household born at time s is willing to buy a house for
any price ps ≤ p¯s , where the process for p¯t is exogenously given. In particular,
we assume that pt = p¯t as long as households can borrow enough to make
the payment. However, we will see that it may be that banks won’t have
enough liquidity to finance that price and ps may become endogenous.
This means that we make two very stark assumptions. First, households
do not optimize and, for exogenous reasons, are happy to buy for any price
ps ≤ p¯s . We will see in section 5 that this is an important simplification, as
it is not easy to get a model where rational households are happy to buy a
house at a high price when prices are going to drop in the future. Second, the
process for the price level that arises when there are no restrictions from the
banking sector is exogenous. In particular, we assume that there are three
phases of the price dynamics: 1) first prices are constant p∗ for any t ≤ 0;
2) then prices start rising exponentially for 0 < t ≤ T ; 3) prices crash and
go back to p∗ for t > T . It is key that we assume that when prices increase,
for any 0 < t ≤ T , market participants expect prices to increase further in
the future, that is, the crash is completely unexpected. Such an increase in
house price can arise from a number of sources, and we can think of two in
particular. One is that households are simply willing to pay more for houses,
due to a preference shift towards housing consumption or general household
optimism towards the future. The other is that lenders are willing to lend
more to households or allow them to borrow at higher rates of leverage. It is
the purpose of later sections to “look under the hood” here and to provide
7
better micro foundations.
To buy their house, households borrow from a banking sector. In particular, consider a household born at date s who buys the house at price
ps . Such a household has to make a down payment of θ to get a loan
ls = max{ps − θ, 0}. The mortgage contract is “interest only”, that is, the
household will be required to make interest payments r(ps − θ) in all future
periods, until the household exits. Thus, a household born at time s will
consume in her first period
cs = y − θ − r(ps − θ)
Given that ct ≥ 0, this implies that prices are bounded above by
ps ≤ pmax =
y − (1 − r)θ
r
In any subsequent period, the household will learn if she exits at the end of
that period. The non-exiting households will consume ct = y −r(ps −θ). The
exiting households will sell their house at current market price pt , seeking to
repay the principal. If pt + y ≥ (1 + r)(ps − θ), the household can repay the
interest and the principal, and before exiting can consume cft = pt + y − (1 +
r)(ps + θ), where f stays for final period. If pt + y < (1 + r)(ps − θ), the
household defaults, consumes nothing, and the bank only receives pt .
Given that banks discount future periods at rate 1 + r, if we assume that
households never default, the value v of this contract at the date of signing
satisfies the recursion
v=
1
(r(ps − θ) + (1 − λ)v + λps ) ,
1+r
or
v = ps − θ.
8
(1)
Below, we use v as the book value of the mortgages which have not yet been
repaid, even if the market value of these mortgages has been declining, due
to house price declines. Note further, that we obtain household heterogeneity,
when house prices are fluctuating, as the amount of outstanding mortgages
depend on the date of birth of the household.
The evolution of house prices and repayments in turn impact on the
balance sheet of the banks, as we describe below. So, total lending to new
households may be limited by resources of the banks, due to binding net
worth constraints. In that case, the market price pt may actually fall below
p¯t .
Banks only invest in households’ mortgages. They finance them with their
own net worth nt as well as deposits dt by a group of savers. Bankers need to
pay the market interest rate r on deposits. Note that we assume that interest
rates on deposits are the same that bankers use for internal discounting and
for the mortgages: we do so in order to keep the model simple.
Let us now examine the evolution of assets and net worth. We always
examine the balance sheet at the end of the period. The bank ends the period
with mortgages at book value at and net worth nt on the asset side, and
deposits dt on the liability side. Thus, equating assets to liabilities gives
at = nt + dt
(2)
The book value of the mortgages outstanding at time t at is equal to
at =
∞
X
j=0
λ(1 − λ)j (pt−j − θ)
= λ(pt − θ) + (1 − λ)at−1 ,
(3)
given that only young households, that is, a fraction λ of the population,
purchase a home in each period and the book value of their mortgage is
9
equal to the purchase price of the home net from the down payment, which
remains unchanged during the life time of the mortgage. When prices are
constant, pt = p∗ , then
at ≡ p∗ − θ.
(4)
Consider now the beginning of a new period. The bank receives interest
payments rat−1 on all “old” mortgages, increasing net worth by that amount.
Then, a fraction λ of “old” mortgages exit. The bank receives φt per exiting
mortgage or φt λat−1 in total, where 1 − φt is the default rate. More precisely,
φt at−1 =
∞
X
j=0
λ(1 − λ)j min{pt−1−j − θ, pt + y − r(pt−1−j − θ)}
(5)
The resulting net worth loss is (1 − φt )at−1 , as the book value at−1 of the
exiting mortgages is replaced by their payoff φt at−1 . In particular, if the
current market price is at least as high as all past market prices, then φt = 1
and there is no change in net worth. Moreover, the bank makes payments
rdt−1 to existing depositors, reducing net worth by that amount. The bank
also receives an inflow of new deposits dt − dt−1 and makes new mortgage
investments λ(pt − θ) which do not change net worth. At the end of the
period, the bank has a cash position on the asset side, which may be negative.
Finally the bank pays dividends cb,t to the bank shareholders, which we will
refer to as banker’s consumption, reducing the net worth by that amount.
In the baseline model we allow cb,t to be negative so to reduce to zero the
end-of-the period cash position of the bank. In this case, the price is always
equal to the exogenous process, that is, p¯t = pt .
Tracing through these various transactions allows us to calculate the endof-period net worth nt as well as cash balances. The end-of-period net worth
10
is equal to
nt = nt−1 + rat−1 − (1 − φt )λat−1 − rdt−1 − cb,t .
(6)
The banker’s consumption cb,t is such that the bank has zero end-of-the
period cash balances, that is,
cb,t = rat−1 + φt λat−1 − rdt−1 + (dt − dt−1 ) − λ(pt − θ).
(7)
Use this equation to replace cb,t in (6) to obtain
nt = nt−1 + dt−1 − λat−1 + λ(pt − θ) − dt
One can use (2) to examine the evolution of net worth. As one special
case, suppose that deposits are constant, dt−1 = dt = d. Then,
nt = nt−1 − λat−1 + λ(pt − θ)
(8)
i.e., the change in net worth is given by the book value difference between
newly created and exiting mortgages. At a superficial look, it would appear
that net worth is “magically” created by higher prices and that default on
exiting mortgages does not matter. However, it needs to be recognized that
these movements find their counterpart in the banker’s consumption cb,t , see
(9): to keep d unchanged, higher prices for new houses as well as larger
default on old mortgages reduce these shareholder pay-outs.
We then consider a version of the model with the restriction that cb,t cannot be negative, that is, shareholders cannot raise more equity. In this case,
depending on the price process, it may be that at some point bankers do
not have enough funds to lend p¯t − θ to the households, so that pt becomes
endogenous and such that cb,t = 0. As we will show in the next numerical
11
exercises, this is the interesting case because there is going to be an amplification in the house price decline due to the shortage of liquidity in the
banking sector.
Equation (8) also reveals that net worth stays constant, if deposits are
constant and prices are constant, as constant prices imply at−1 = pt−1 = p∗ ,
see (4). In this case equation (9) implies that bankers’ consumption is equal
to
c∗b = r(p∗ − d) + λθ = rn∗ + λθ,
(9)
where the last equality comes from equation (2), which simply says that
interest rp∗ on the asset side is split across the interest payments rd to
depositors and the interest payments rn∗ to bank shareholders.
We can now study the dynamics of our model, assuming that house prices
increase exponentially between time 0 and T and then drop back to the initial
level p∗ , that is,
pt = p∗ + αbt
for t = 0, ..., T − 1, and pt = p∗ for t < 0 and t > T . Given this process for
prices, we can first calculate what happens to assets. Using equation (3) we
obtain
at = p∗ − θ +
for t < T and
αλ
1 − (1 − λ)/b
bt
at = p − θ + α
1 − (1 − λ)/b
∗
for ≥ T . For simplicity assume p∗ = 0 and b = 1/γ. In the next subsection
we show some numerical simulations.
12
Ratio of current assets to current price
1
0.9
0.8
Ratio
0.7
0.6
0.5
0.4
0.3
0.2
1
1.05
1.1
Growth of prices
1.15
1.2
Figure 3: Ratio of assets to prices, for various values of γ, when λ = 0.05.
2.1
Numerical experiments
In this subsection, we evaluate some numerical experiments. We assume that
the interest rate paid on assets rA may be different (more specifically: larger)
than the interest rate rL paid on deposits.
It may be useful to derive a lower bound on the asset-to-price ratio. If
prices have been grown continuously at some rate γ forever, and there is no
base price or downpayment, and given the death/rebirth-rate λ, then
at = λ(pt − θ) + (1 − λ)at−1
λγ
=
(pt − θ)
γ+λ−1
13
(10)
(11)
The relationship between current price and the stock of outstanding assets
is plotted in figure 3.
Let κt be the end-of-period net worth-to-capital ratio, that is, κt ≡ nt /at .
Then, dt = (1 − κt )at and if there are no defaults, the bankers’ consumption
can be written as
cb,t = (rA + λ)at−1 − rL (1 − κt−1 )at−1 + (1 − κt )at − (1 − κt−1 )at−1 − λ(pt − θ)
Assuming κt = κ for all t, i.e. keeping the leverage ratio constant, one obtains
cb,t = (rA + κλ − rL (1 − κ))at−1 − κλ(pt − θ)
exploiting (10). Supposing that prices have been growing forever at rate γ
and that there are no defaults on repayments, and thus supposing (11), one
obtains
cb,t
= rA + κλ − rL (1 − κ) − κ(γ + λ − 1)
at−1
(12)
As an interesting benchmark, suppose that cb,t = 0. Then
1
γ = 1 + (rA − rL (1 − κ))
κ
(13)
which is intuitive. In particular, if rA = rL = r, then
γ = 1+r
(14)
so that the interest paid on net worth must exactly finance its growth. Put
differently, the values calculated for γ in (13) or (14) are the upper bounds for
the price growth rates γ to avoid that banker consumption falls into negative
territory during the price growth phase, when holding leverage constant.
We now conduct two numerical experiments to highlight the two approaches we discuss in the introduction:
14
1. We assume that house prices are constant and assume a boom and bust
in the leverage ratio κ;
2. We assume that the leverage ratio is constant, and assume a boom and
bust in house prices dynamics. In this case, we conduct two experiments:
A. First, we allow for equity infusion, that is, we allow for bankers’
consumption to become negative.
B. Second, we impose that bankers’ consumption cannot become negative and then house prices may endogenously fall to accommodate
the constraints on net worth.
The experiments without equity infusion raise a new issue, however. The
entire mortgage portfolio is held by banks: there is no outside market. But
if the banks make losses and there is no equity infusion, then there may be
some default on depositors and there would be no way to maintain the desired
leverage ratio. We address this issue by assuming that the government will
supply an emergency loan to the banks, making up the shortfall. In the
future, the banks will have to pay an interest rate rE on the outstanding
loans, and repay a fraction µE , i.e. we allow for the emergency loan to
potentially have a long maturity. We assume that the loan is “ignored” for
the purpose of calculating the leverage ratio, i.e. the leverage ratio determines
the amount of deposits relative to assets, not the amount of networth relative
to assets. Indeed, it is possible that the networth is negative: for sufficiently
favorable emergency loans, the banks may be able to “earn” their way out
of negative networth territory, per the income earned on their assets, which
they get to keep.
If banker consumption is imposed to be nonnegative, the price level and
default rate become endogenous. The price level will be at least θ, the
15
downpayment of new agents entering the housing market. If the price level is
pt = θ, one can immediately calculate the default rate of the exiting agents
from equation (29). For the price level to be higher than that, a nonnegative
banker consumption c˜b,t must result, using the equations below. With the
current price, one can then obtain the evolution of assets as in equation (3).
The evolution banker consumption, government emergency loans Lt , deposits
and networth is then summarized by the following equations
c˜b,t = (rA − (1 + rL )(1 − κt−1 )) at−1 − (rE + µE )Lt−1
+φt λat−1 − κt λ(pt − θ)
Lt = (1 − µE )Lt−1 − min{0, c˜b,t }
cb,t = max{0, c˜b,t }
dt = (1 − κt )at
nt = κt at − Lt
The equation for c˜b,t is used to solve for pt and the repayment rate φt (pt )
per calculating the highest θ < pt ≤ p¯t , so that c˜b,t ≥ 0 (though we have not
checked whether multiplicity could be an issue). Numerically, we shall solve
for it, using a grid of 1000 points for the range between θ and the maximal
price newly entering households would be prepared to pay in principle, if
they could obtain a mortgage of sufficient size. If no such value is available,
then pt = θ.
As parameters, we choose the following. The benchmark non-bubble
price p = 1 has been normalized as ten times (annual) income. We assume
a downpayment (relative to income) of θ = 2, which should be assumed
to be “saved up” from prior income before agents are born and enter the
housing market. Recall that θ is the initial downpayment, so that households
borrow pt − θ, if positive, or nothing, if the downpayment suffices to buy the
house. We allow the price to grow to 50% of the maximum, before it crashes
16
exogenously. The exit probability λ has been set equal to 0.1. We assume
that banks earn 4% on their assets and pay 3% on their liabilities, be they
depositors or the government (in case of emergency loans). Obviously, the
results are quite sensitive to these assumptions. We assume that government
loans have a maturity of 20 years, i.e., that 5% of the outstanding bonds
need to repaid each period.
p/y
10
θ
2
rA
0.04
rL
0.03
rE
0.03
µE
0.05
λ
0.1
γ
1.04
(max γ
1.04 )
p¯−1
0.5 ∗ α
¯
2.1.1
An exogenous crash in leverage.
To highlight the credit boom and crash as the key element of the relationship between housing and credit market, we consider an exogenous crash in
leverage. To that end, we assume that the allowed networth coverage ratio
is initially at κ = 0.05 until some date t = −1, implying a leverage ratio of
20, and then unexpectedly rises to κ = 0.2 at date t = 0, implying a leverage
ratio of 5. The price p¯ has been fixed to be constant at ten times annual
income.
In experiment A, bankers supply fresh equity, if needed. This is shown in
figure 4. There is a single period, when the leverage ratio suddenly changes,
necessitating an infusion of extra cash, modelled as negative banker consumption. Once the fresh equity is injected, everything continues as before, except
17
that banker consumption is now higher, given the new and lower leverage.
The price for houses remains at pt = p¯.
Banker Consumption (case A)
0.02
0
consumption
−0.02
−0.04
−0.06
−0.08
−0.1
−0.12
−20
−10
0
year
10
20
Figure 4: An exogenous crash in leverage: implication for banker consumption. Negative values should be interpreted as the injection of fresh bank
equity
Matters are more dramatic, in case there is no fresh infusion of bank
equity, as figure 5 shows. Prices crash endogenously, as can be seen in the
top left panel. There is a fairly brief period of default, as shown in the top
right panel. For the parameterization here, the banks gradually rebuild their
networth to the required new ratio, as indicated by the red-dashed line in
the left panel of the third row.
18
Prices p−bar and p rel. to inc. (case B)
Repayment Rates (case B)
1
8
0.8
Repayment Rates
10
price
6
4
2
0.4
0.2
0
−20
−10
0
year
10
0
−20
20
0
year
10
20
1
0.8
Repayment Rates
0.008
0.006
0.004
0.6
0.4
0.2
0.002
0
−20
−10
Repayment Rates (case B)
Banker Consumption (case B)
0.01
consumption
0.6
−10
0
year
10
0
−20
20
−10
0
year
10
20
Government Loan (case B)
Capital Ratio, n(t)/a(t) (case B)
0.16
0.2
0.14
0.15
Government Loan
0.12
ratio
0.1
0.05
0.1
0.08
0.06
0.04
0
0.02
−0.05
−20
−10
0
year
10
20
0
−20
−10
0
year
10
Figure 5: An exogenous crash in leverage without infusion of bank equity.
Prices crash endogenously, as can be seen in the top left panel. There is
a fairly brief period of default, as shown in the top right panel. For the
parameterization here, the banks gradually rebuild their networth to the
required new ratio, as indicated by the red-dashed line in the left panel of
19
the third row.
20
2.2
An exogenous crash in house prices
To highlight the housing price boom and crash as the key element of the
relationship between housing and credit market, we consider an exogenous
crash in house prices.
First, note that the maximal price-to-income ratio is
α
¯=
y
rA
We shall assume that prices rise until t = −1 and crash at t = 0. We
shall pick a value for the last pre-crash price p¯−1 in relation to its maximally
possible value α.
¯ We allow the price to grow to 50% of the maximum, before
it crashes exogenously. We set the required net worth ratio to κ = 0.1.
The results are in the four figures 6 and 7. Figure 6 shows what happens
when bank equity can be injected, i.e., when banker consumption can become
negative. House prices trade at the exogenously given levels p¯t . There is a
temporary dip in repayments, but they recover gradually, as the top right
panel shows. Of course, no government loans are necessary in this case.
Matters are more dramatic in 7, when no fresh injection of bank equity
is available. Now, when prices crash, they crash to the downpayment level
θ and stay there, for the chosen parameter configuration. Banker consumption never recovers. There is continued default on banker assets. These are
the legacy assets of pre-crash assets, which gradually disappear over time:
households with pre-crash loans continue to have difficulties repaying these
loans. Eventually, the government holds a bond position offset by a negative
amount of networth of bankers, without any corresponding assets.
20
Prices p−bar and p rel. to inc. (case A)
Repayment Rates (case A)
1
20
0.8
Repayment Rates
25
price
15
10
5
0.6
0.4
0.2
0
−20
−10
0
date
10
0
−20
20
−10
0
date
10
20
Assets and Networth (case A)
Banker Consumption (case A)
1.5
0.015
assets and networth
0.01
consumption
0.005
0
−0.005
−0.01
1
0.5
−0.015
−0.02
−20
−10
0
date
10
0
−20
20
Asset to current price ratio
−10
0
date
10
20
Government Loan (case A)
1.4
1
1.3
1.2
Government Loan
0.5
a(t)/p(t)
1.1
1
0.9
0.8
0
−0.5
0.7
−20
−10
0
date
10
20
−1
−20
−10
0
date
Figure 6: Results from experiment A, parameters 2.
21
10
20
Prices p−bar and p rel. to inc. (case B)
Assets and Networth (case B)
25
1.5
1
assets and networth
20
price
15
10
5
0
−0.5
−1
0
−20
−10
0
date
10
−1.5
−20
20
Repayment Rates
0.008
0.006
0.004
20
0.6
0.4
0.2
0.002
−10
0
date
10
0
−20
20
Asset to current price ratio
−10
0
date
10
20
Government Loan (case B)
7
1.4
6
1.2
Government Loan
5
a(t)/p(t)
10
0.8
0.01
4
3
2
1
0
−20
0
date
1
0.012
0
−20
−10
Repayment Rates (case B)
Banker Consumption (case B)
consumption
0.5
1
0.8
0.6
0.4
0.2
−10
0
date
10
20
0
−20
−10
0
date
Figure 7: Results from experiment B, parameters 2
22
10
20
3
Related Literature: Households’ Leverage
In describing the linkage between credit cycle and boom and bust in housing
prices, one key ingredient in our mechanical model is that households borrow
to become homeowners. One defining feature of the recent recession has been
a very high households’ gross debt to gdp ratio, which reached roughly 128%
by 2008. In the recent years, there has been a growing body of work that
explore the role of households’ leverage on the level of aggregate activity. As
we said in the introduction, we will not focus on the paper on the real side of
the model, but in this section we will describe some of the existing literature.
The classic representative agent model is not suited to think about the role
of gross households’ debt, as it focuses on the net asset position of a country.
This is why the first papers to address this issue had to go in the direction of
introducing agents’ heterogeneity into the picture. In particular, the first two
papers focusing on the importance of households’ leverage to understand the
recent recession are Guerrieri and Lorenzoni (2011) who use a Bewely type of
model and Eggertsson and Krugman (2012) who use a keynesian model with
two types of agents, a borrower and a saver. Both papers explore the idea
that when households are highly levered, the real effects of a credit crunch
can be particularly severe in the presence of some nominal rigidity. When
banks stop lending to the households, households that are in debt are forced
to to cut on their consumption to repay some of their debt. Moreover, when
there are more than two types of agents, even households that are not in
debt may cut on their consumption for precautionary reasons.
Let me describe more in detail the paper by Guerrieri and Lorenzoni (2011).
Households are heterogenous for their income process, they decide how much
to consume and how much labor to supply. Markets are incomplete and
households can smooth their consumption by trading a one-period risk-free
bond. However, they face an exogenous borrowing limit. The paper explores
23
the transitional dynamics in response to a one-time unexpected tightening
in the borrowing limit, that is interpreted as a credit crunch. The first main
result is that a credit crunch generates a large initial drop in the interest rate,
as households start deleveraging. The interest rate then gradually adjusts
back up towards the new steady state level, which is below the initial one. In
a model with nominal rigidities, this downward pressure on the interest rate
may easily drive the economy into a liquidity trap. Their second result is
that a credit crunch generates a recession, and so it might have contributed
to the Great Recession. Although, the more indebted households will work
more to help their deleveraging, the authors show that they tend to be the
less productive and, under reasonable parameters, the economy enters a recession even with no nominal rigidities. Morevoer, they show, that once one
introduces nominal rigidities into the picture, the recession is larger.
In a second part of the paper the authors enrich the model with durable
consumption. One shortcoming of the basic Bewly type of set up is that, for
reasonable parameters, it cannot account for a large level of households debt
to gdp, as houesholds borrow just to insure against temporary income shocks.
One interesting feature of the recent US outlook is that the households who
were more in debt before the Great Recession were not necessarily the poorer
households, but the households that indeed invested more in durable consumption. This makes this extension particularly relevant. Durable goods
can be used as collateral and are costly to be sold, so that they are imperfect substitute of bonds. The authors show that durable consumption drops
substantially in response to a credit crunch, while non-durable consumption
does not react much, which is consistent with the data.
More recently, there has been a growing body of papers working on related
models with heterogenous houseolds, such as Hall (2005), Kaplan and Violante (2014),
Midrigan and Philippon (2011), Huo and R´ıos-Rull (2013), . . .
The set of papers I just described focused on the real effects of a credit
24
crunch. Another important issue is to think more in detail about the source
and the mechanics of the credit expansion first and the following credit
crunch. Justiniano, Primiceri, and Tambalotti (2014) goes in the direction
of thinking more carefully about the modeling of the credit expansion that
preceded the Great Recession. They distinguish between borrowing constraints and lending constraints. The literature on credit constraints has so
far mainly focused on borrowing constraints, that are tightly linked to the
loan-to-value ratio by construction. They argue that in the years before the
recession this ratio did not move much and so it is more realistic to model
the expansion in credit as driven by a relaxation in “lending constraints”
instead. “Lending constraints” want to capture constraints to flows of funds
from savers to mortgage borrowers. They model them as constraints to saving, but they argue that they could as well model them as constraints to
financial intermediaries, such as capital requirements.
TO BE COMPLETED
4
Multiple Equilibria and Catastrophes
In this section, we focus on the boom and bust in the credit cycle, abstracting
from the dynamics of house prices. The main idea is that an increase in credit
availability, which can be interpreted as a “saving glut”, can endogenously
generate an increase in lending and then in a crash of the credit market in
the presence of private information about the quality of the borrowers. This
idea builds on Boissay, Collard, and Smets (2014), but the model that we
present here is quite different. In particular, to tailor this idea to the market
for households’ mortgage we have a richer baseline model at the expenses of
reducing the dynamics to 2 periods only.
25
4.1
Model
There are two periods t = 1, 2. The economy is populated by a continuum
of three types of agents: households, lenders, and banks. Lenders and banks
are homogenous, while households are heterogenous.
Households enjoy utility u(c, h) in period 2, where c is consumption of
a non-durable good and h is housing consumption. For simplicity, let us
assume that utility is linear, that is,
u(c, h) = c + γh.
¯ so that h ∈ {0, ¯h}, and their price is
Houses come in fixed size equal to h,
fixed to 1. Households receive an income draw y in period 2 and have to
decide whether to buy a house or not in period 1, so if they decide to buy a
house they have to borrow the full amount.
Households are heterogenous for their income process. Let ν ∈ [0, 1]
be the household’s type. Assume that ν is distributed according to some
distribution function G(ν) and affects the distribution of the household’s
income Fν (y). In particular, Fν2 (y) first order stochastic dominates Fν1 (y)
when ν1 < ν2 .
To buy a house in period 1, an household has to borrow 1 unit of funds
from the banks at some price p and promise to repay 1/p in period 2. At
the beginning of period 2, the household’s income y is realized and then the
household decides whether to repay her debt or not. If she does default on her
debt she suffers a penalty δ > 0. Let us denote by χ ∈ {0, 1} the repayment
decision. This implies that higher types are better potential borrowers, in
the sense that they have a lower probability of bad income realizations.
The household’s type ν is private information of the household. However,
at the beginning of period 1, households can choose to pay a fixed fee k to get
their type verified. Let v(ν) ∈ {0, 1} be the decision of paying the verification
26
cost (v = 1) or not (v = 0). If a household verifies her type, banks can make
the lending terms type-contingent, so that the price is going to be equal to
p˜(ν). If instead a household does not verify her type, banks do not know the
type of borrower and hence will offer a pooling price pP .
To sum up, households have three options: 1) do not verify their type
and borrow accepting a pooling contract, that is, h(ν) = 1 and v(ν) = 0 ; 2)
pay the cost to verify their type and access type-contingent contracts, that
is, h(ν) = v(ν) = 1; 3) do not borrow at all, that is, h(ν) = v(ν) = 0.
Let us proceed backward and consider the repayment decision, conditional
on borrowing in the first period, that is, on h(ν) = 1. A household with
realized income y would like to repay if
¯ ≥ y − δ.
y − 1/p + γ h
We will choose δ large enough so that in equilibrium any household would
like to pay back her debt if she can.1 However, she may not be able to
repay because her realized income is not high enough, and, in particular,
χ(y, p) = 1 iff y ≥ 1/p. Let π(ν, p) be the ex-ante repayment probability of
a household of type ν who borrow at price p, that is,
1
π(ν, p) = E(χ(y, p) = 1|ν, p) = 1 − Fν
p
Then we can show the following proposition.
Proposition 1. The repayment probability π(ν, p) is increasing in ν and
increasing in p.
Proof. First, π(ν, p) is increasing in ν, since Fν are ordered by first-order
stochastic dominance. Second, π(ν, p) is increasing in p, since 1 − Fν (y) is
1
We will show below that this is a necessary condition to have a non-empty set of
borrowers at the pooling price.
27
decreasing in y for any ν.
This simply means that that a household with higher type has a higher
repayment probability, for given price.
Let us now focus on the lending market. Let us assume that the banks can
borrow from the lenders at some rate R, which is exogenously given.2 Also,
they trade the loans, which we will refer to as assets from now on, on the
secondary market.3 Each asset is characterized by the type of the associated
borrower ν and has a different repayment probability π(ν, p) ∈ [0, 1]. This
implies that the pooling price and the type-contingent price are going to be
determined by the following no-arbitrage conditions:
pP =
h
i
1
E 1 − Fν pP |ν ∈ [ν, ν¯]
R
,
(15)
where ν and ν¯ are the worse and the best types of loans traded without
verification, and
p˜(ν) =
1 − Fν
R
1
p˜(ν)
.
(16)
Obviously, households who pay the verification fee will be able to borrow at
terms that are more convenient the better their type is, that is, we can prove
the following proposition.
Proposition 2. If it exists, the type contingent contract p˜(ν) is increasing
in ν and decreasing in R.
˜
Proof. First, let us make a change of variable and define R(ν)
≡ 1/˜
p(ν).
2
Think for example that they can borrow from the ”Chinese”.
One can potentially generalize to create MBS that pool different loans and a similar
mechanism would go through as long as there is a constraint on the measure of types that
can be pooled together.
3
28
Then we can rewrite equation 16 as
1−
R
˜
= Fν (R(ν)).
˜
R(ν)
˜
For any ν, let us solve this equation in R(ν).
First impose restrictions on Fν
such that the curves defined by the LHS and RHS of the above equation cross
at least one. Second, if they cross more the once, let us pick as a solution
the lowest one. This follows the idea that if there are two possible R(ν)
that banks would be willing to offer, and we are in an equilibrium where
households borrow at the highest one, it would always be profitable for them
˜
to deviate. Then, we pin down a unique R(ν)
and hence a unique p˜(ν) for
each ν. Moreover, if ν increases, the RHS shifts downward, given that this
˜
implies a first order stochastic dominant shift in Fν , and R(ν)
decreases. If
˜
instead R increases the LHS shifts downward and R(ν)
increases. Given the
˜
definition of R(ν),
this completes the proof.
Now, let us go back to the households’ problem. The utility of household
ν can be written as
U¯ (ν, pP ) = max U B (ν, pP ), U V (ν), U N (ν) ,
where U B (ν) is the expected utility of a household of type ν who decides to
buy a house and not verify her type (h(ν) = 1 and v(ν) = 1), U V (ν) is the
expected utility of a household who decides to borrow and verify her type
(h(ν) = v(ν) = 1), and U N (ν) is the expected utility of a household of type
ν who decides not to buy a house (h(ν) = v(ν) = 0). Then
B
P
U (ν, p ) =
Z
∞
1
pP
1
¯
(y − P + γ h)dF
ν (y) +
p
29
Z
1
pP
0
(y − δ)dFν (y),
(17)
U B (ν, p˜(ν)) − k,
(18)
Z
(19)
and
N
U (ν) =
ydFν (y).
The following proposition shows that the households’ optimal behavior can
be characterized by two cutoffs values, such that, households with low types
do not buy a house, households with high types buy a house and may verify
their type, and households in the middle range, buy a house but choose to
borrow at the pooling contract.
Proposition 3. Suppose d[fν (x)/(1 − Fν (x))]/dν is increasing in ν for all
x. Then, there exist two values p and p¯, so that for all pP ∈ [p, p¯] there are
two cutoffs ν(pP ) and ν¯(pP ) such that h(ν) = 0 iff ν < ν(pP ) and v(ν) = 1
iff ν ≥ ν¯(pP ). Moreover, ν(pP ) and ν¯(pP ) are respectively decreasing and
increasing in pP .
Proof. Let us rewrite
U B (ν, p) = Eν (y) + Z(ν, p) − δ,
U V (ν) = Eν (y) + Z(ν, p˜(ν)) − δ − k,
and
U N (ν) = Eν (y),
where recall that
1
1
¯
Z(ν, p) ≡ 1 − Fν
γh − + δ .
p
p
First, a type ν-consumer will choose to buy a house iff U B (ν, p) ≥ U N (ν),
30
that is,
Z(ν, p) ≥ δ.
Notice that ∂Z(ν, p)/∂ν > 0, given that Fν (x) first order stochastic dominates Fν ′ (x) for ν > ν ′ . If Z(0, p) < δ and Z(1, p) > δ, this implies that
there exists a unique cut-off ν(p) that is implicitly defined by
Z(ν(p), p) = δ.
(20)
such that U B (ν, p) ≥ U N (ν) iff ν ≥ ν(p). Moreover, it is straightforward to
see that ∂Z(ν, p)/∂p > 0, which implies that ν(p) is decreasing in p. Also, it
implies that to complete this step of the proof it is enough to pick p and p¯
such that Z(0, p¯) < δ and Z(1, p) > δ.
Next, notice that proposition 2 together with the fact that ∂Z(ν, p)/∂p >
0, ∂Z(ν, p)/∂ν > 0, and d[fν (x)/(1 − Fν (x))]/dν is increasing in ν for all x,
imply that dZ(ν, p˜(ν))/dν > dZ(ν, p)/dν. Hence, if Z(0, p) > Z(0, p˜(0)) − k,
Z(1, p) < Z(1, p˜(1)) − k, and Z(¯
ν (p), p) > δ, there exists a unique cutoff
ν¯(p) > ν(p), implicitly defined by
Z(¯
ν (p), p˜(¯
ν (p))) − k = Z(¯
ν (p), p),
which implies that U B (¯
ν (p), p) = U V (¯
ν (p)) and U V (¯
ν (p)) > U B (¯
ν (p), p) iff
ν > ν¯(p). Also, notice that ν¯(p) is increasing in p, given that ∂Z(ν, p)/∂ν >
0.
To complete the proof we must choose p and p¯ such that Z(0, p˜(0)) − k <
Z(0, p) < δ, Z(1, p˜(1)) − k > Z(1, p) > δ, and Z(¯
ν (p), p) > δ for all p ∈ [p, p¯].
(We can show that under mild conditions v(p) ≤ v¯(p))
An equilibrium can then be represented by a pooling price pP , a separating price schedule p˜(ν), and two cutoffs ν and ν¯ satisfying the following
31
conditions:
pP =
h
i
E 1 − Fν p1P |ν ∈ [ν, ν¯]
R
p˜(ν) =
4.2
1 − Fν¯
R
1
p˜(ν)
,
(21)
(22)
1
¯−
γh
+ δ = δ,
pP
(23)
1
¯
¯ + δ − R − k.
γ h − + δ = R˜
p(¯
ν) γh
p
(24)
1 − Fν
1
pP
1 − Fν
,
1
pP
Multiple equilibria
In our model, multiple equilibria can arise: we can have a good equilibrium
where good households do not verify their type, the price is high and hence it
is indeed optimal not to pay the verification fee; or a bad equilibrium, where
good households do verify their type, hence lowering the pooling price and
making it indeed optimal to pay the verification cost. We can then imagine
that a simple increase in credit availability, that is, a decrease in R, can
endogenously generate first a boom and then a crush in households’ loans.
Let us think about the following story.
1. Let us imagine that we start in an equilibrium where R is low and all
borrowers are pooled together, that is, ν¯(p) = 1.
2. Then, R declines due to a “saving glut” type of episode, pushing both
p and p˜(ν) up and hence both U B and U V shift up. This implies that
ν(p) decreases and more bad households become borrowers. However,
let us assume that ν¯(p) stays at 1. The change in the composition of
32
the loans tends to depress prices. However, prices have to go up on
net, so the interest rate effect has to dominate.4
3. If R decreases further, at some point ν¯(p) becomes smaller than 1
and some borrowers decide to verify that they are good types, hence
worsening the pool of households who borrow at p. There are two
possibilities:
(a) p increases, then both U B and U V shift further up and both ν and
ν¯ decline, dampening the increase in p.
(b) p declines, then U B has to shift down and ν increases. In this
case, it must be that the decline in ν¯ is strong enough to more
than compensate the pressure upward on p played by the increase
in R and in ν.
4. Then R increases again, but we are stuck in an equilibrium with some
separation!
The shift from an equilibrium of type (a) to an equilibrium of type (b) can
be interpreted as a market crash.
4.3
Some numerical examples
In this subsection, we show some numerical examples, where we used some
specific distributions H. In particular, we examined the following two cases:
1. A mixture of two exponential densities,
h(ν) = ω
λ2 e−λ2 ν
λ1 e−λ1 ν
+
(1
−
ω)
1 − e−λ1
1 − e−λ2
4
Imagine, by contradiction that p declines, then ν(p) has to increase, but then p has
to increase, generating a contradiction.
33
2. A mixture of an exponential density and a truncated normal (where,
in terms of the parameterization, we have not normalized the latter to
integrate to unity, just the density h(ν) as a whole),
e 2
λe−λν
e−(ν−ν ) /(2σ
√
h(ν) ∝ ω
+
(1
−
ω)
1 − e−λ
2πσ
2)
¯ = 2, δ = 0.1 for all. For the mixture of two exponentials, where
We chose γ h
κ = 0.15, λ1 = −20, λ2 = 5, ω = 0.8 (with κ or k the cost for verification),
we obtained a unique separating equilibrium for R = 1.4, multiple equilibria
for R = 1.58 and a unique pooling equilibrium for R = 1.65. A graphical
representation of the results is in figure 9. For the mixture of an exponential
density and a truncated normal, where κ = 0.25, λ = −20, ν e = 0.1, σ =
0.2, ω = 0.6, we obtained a unique separating equilibrium for R = 1.3, mul-
tiple equilibria for R = 1.4 and a unique pooling equilibrium for R = 1.5. A
graphical representation of the results is in figure 10.
5
Credit Cycles
To be completed. This section is a short review of the “credit cycles” literature. Examples are Kiyotaki and Moore (1997) or Myerson (2012).
6
Bubbles
This section is a short review of the “bubbles” literature, which, effectively,
assumes dynamically inefficiency. There, bubbles “die out” on their own,
rather than exploding.
For example Carvalho, Martin, and Ventura (2012), Martin and Ventura (2010),
Martin and Ventura (2012), Martin and Ventura (2014) employ various versions of OLG models, in which, ideally, resources should be funneled from
34
Density for ν
9
Equilibrium Manifold
0.9
8
0.85
7
0.8
6
Density
p
0.75
0.7
5
4
0.65
3
0.6
2
0.55
1
0.5
1
1.2
1.4
R
1.6
0
0
1.8
0.2
Fixed point calculation E[π]/R: Two Point Example
0.15
0.85
0.1
0.8
E[π] − p*R
E[π]/R
0.6
0.8
1
0.05
0.75
0.7
0.65
0
1.20
−0.05
−0.1
0.6
0.5
0.5
ν
Fixed point calculation: Two Point Example
0.9
0.55
0.4
1.20
0.6
−0.15
0.7
p
0.8
−0.2
0.5
0.9
Bounds for R=1.20,(blue:low,red:high)
0.6
0.7
p
0.8
0.9
Bounds for R=1.20,(blue:low,red:high)
0.8
0.8
0.6
0.6
ν
1
ν
1
0.4
0.4
0.2
0.2
0
0.5
0.6
0.7
p
0.8
0.9
0
0.5
0.6
0.7
p
0.8
¯ = 2, δ = 0.1, κ =
Figure 8: For the mixture of two exponentials, where γ h
0.2, λ1 = −10, λ2 = 10, ω = 0.85 (with κ or k the cost for verification), we
obtained a unique separating equilibrium for large enough, multiple equilibria
for small.
35
0.9
Density for ν
Equilibrium Manifold
16
0.9
14
0.85
12
0.8
10
Density
p
0.75
0.7
8
0.65
6
0.6
4
0.55
2
0.5
1
1.2
1.4
R
1.6
0
0
1.8
0.2
Fixed point calculation E[π]/R: Two Point Example
0.4
0.6
ν
0.8
1
Fixed point calculation: Two Point Example
0.62
0.1
0.6
0.05
1.65
0.56
1.58
0.54
0.52
E[π] − p*R
E[π]/R
0.58
1.40
0
1.65
1.58
1.40
−0.05
−0.1
0.5
0.48
0.5
0.52
0.54
0.56
p
0.58
0.6
−0.15
0.5
0.62
0.52
0.54
0.56
p
0.58
0.6
0.62
Volume lent at R and p: Two Point Example
Bounds for R=1.40,(blue:low,red:high)
0.9
1
1.65
0.8
0.8
0.7
Volume lent
0.6
ν
0.6
0.4
0.5
0.4
0.3
1.58
0.2
0.2
0.1
0
0.5
0.55
0.6
0.65
0.7
0.75
p
0
0.5
1.40
0.52
0.54
0.56
p
0.58
0.6
0.62
¯ = 2, δ = 0.1, κ =
Figure 9: For the mixture of two exponentials, where γ h
0.15, λ1 = −20, λ2 = 5, ω = 0.8 (with κ or k the cost for verification), we
obtained a unique separating equilibrium for R = 1.4, multiple equilibria for
R = 1.58 and a unique pooling equilibrium for R = 1.65.
36
Density for ν
Equilibrium Manifold
14
0.64
12
0.62
10
Density
p
0.6
0.58
0.56
8
6
4
0.54
2
0.52
1
1.2
1.4
R
1.6
0
0
1.8
0.2
Fixed point calculation E[π]/R: Two Point Example
0.4
0.6
ν
0.8
1
Fixed point calculation: Two Point Example
0.7
0.15
0.1
0.65
E[π] − p*R
E[π]/R
0.05
0.6
1.50
0.55
1.40
0
−0.05
1.30
1.40
1.50
−0.1
1.30
0.5
−0.15
0.45
0.5
0.55
0.6
p
0.65
−0.2
0.5
0.7
0.55
0.6
p
0.65
0.7
Volume lent at R and p: Two Point Example
Bounds for R=1.30,(blue:low,red:high)
0.9
1
0.8
0.8
1.50
0.7
Volume lent
0.6
ν
0.6
0.4
0.5
0.4
0.3
1.40
0.2
0.2
0.1
0
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
p
0
0.5
1.30
0.55
0.6
p
0.65
Figure 10: For the mixture of an exponential density and a truncated normal,
¯ = 2, δ = 0.1, κ = 0.25, λ = −20, ν e = 0.1, σ = 0.2, ω = 0.6, we
where γ h
obtained a unique separating equilibrium for R = 1.3, multiple equilibria for
R = 1.4 and a unique pooling equilibrium for R = 1.5.
37
0.7
inefficient investors or savers to efficient investors or entrepreneurs. They
assume that there is a lending friction: entrepreneurs cannot promise repayment. They can only issue securities, where the buyer hopes that someone
else buys them: call them “bubble”, “cash” or “worthless pieces of paper”.
Equilibria then exist, where newborn entrepreneurs create “bubble” paper.
The existing bubble paper in the hands of old agents as well as those created
by newborn entrepreneurs get sold to savers. Savers find investing in these
bubbles more attractive than investing in their own, inefficient technologies.
This technology needs to be inefficient enough so that its return is on average
below the growth rate of the economy, creating the dynamic inefficiency for
bubbles to arise. In that case, the “fundamental” value of any asset paying
even a tiny amount per period is actually infinite. Or, put differently, the
last fool to buy the bubble at the highest price is happy to do so, since the
value of the bubble next period will not have gone down too much and since
that fool is desperate to save.
Further papers utilizing related approaches are Edward L. Glaeser (2008)
7
Sentiments
7.1
The idea
The aim of this section is to formalize and examine the following and oftentold story about buyers and sellers in markets. We wish to use it in particular
for thinking about the housing market, but it might apply equally well to
the stock market or any other market in which assets get re-traded.
The story we have in mind is as follows. Prices for assets sometimes
bubble above their fundamental value and then come crashing down. They
do so due to buyers betting on greater fools. More precisely, when a buyer
38
buys an asset, she may realize that the price is above its fundamental value,
but is betting on being able to sell the asset at a future date at an even
higher price to a greater fool. What matters to the buyer is not, how foolish
it is to keep the asset itself, but how foolish other participants are.
There are variants of this story formalized in various way in the literature,
where ssets are trading above fundamental values due to diverse beliefs between optimists and pessimists, creating an “add-on” above the fundamental
value, possibly requiring short-sale constraints on the pessimists. Static versions can be found in Geanakoplos (2002) or as in Simsek (2013). Dynamic
versions are in Harrison-Kreps (1978) or in Scheinkman and Xiong (2003).
These papers incorporating diverse beliefs probably come closest to what
we wish to examine here. However, these models typically focus on the case
where agents are either pessimistic or optimistic: by construction then, the
optimists must be the “greatest fools”. By contrast or in extension, we wish
to incorporate the idea that the more optimistic buyers are typically not
yet the greatest fools themselves, but simply betting on even greater fools
out there. At the extreme end, the “greatest fool” must be someone willing
to pay for something that is intrinsically worthless (to all others), without
ever being able to sell to someone at an even higher price, and there may
be quite foolish people out there with overly strong optimism of being able
to sell to such “greatest fools”. So, at that end, the model may require
substantial irrationality. The key here is, however, how this suspected strong
irrationality at the upper end of the potential price distribution trickles down
to the price and sales dynamic among the less foolish or even rational part
of the population.
The model shares many elements with Mikhail Golosov and Tsyvinski (2014).
There, assets are traded in a sequence of bilateral meetings between agents
having different information regarding the fundamental value of the asset.
By contrast here, everyone understands the asset to be intrinsically value39
less: the differences arise in beliefs regarding the optimism of others. As
such, our chapter is more closely related to Abreu-Brunnermeier, Econometrica 2003. A benchmark example for a dynamic model exploiting heterogeneous beliefs and changing sentiments, is the “disease” bubble model
of Burnside, Eichenbaum, and Rebelo (2013). There is some intrinsically
worthless bubble component, which could be part of the price of a house. An
initially pessimistic population may gradually become infected to be “optimistic” and believe the bubble component actually has some intrinsic value:
once, everyone is optimistic (forever, let’s say), there is some constant price
that everyone is willing to pay. However, “truth” may be revealed at some
probability every period, and clarify that the bubble component is worthless indeed. Then, during the pessimistically dominated population epoch,
prices rise during the non-revelation phase, since the rise in prices there compensates the pessimistic investor for the risk of ending up with a worthless
bubble piece, in case the truth gets revealed. The price will rise until the
marginal investor is optimistic: at that point, the maximum price may be
reached.
For our showcase model below and in contrast to the models of section 6,
we do not assume that the economy is, effectively, dynamically inefficient. In
many models, bubbles can be traded forever, because the return to be earned
on these assets does not exceed the growth rate of the economy. It may be
important, however, to examine models in which the required rate of return
is higher than the growth rate. It is then clear from the start that the price
eventually must hit a ceiling: say, when the value of the asset exceeds all
resources in the hands of the buyers. Usual backward induction arguments
then rule out such bubbles in the first place, see Tirole. The purpose of this
section to tweak the rationality argument per introducing a “greatest fool”,
thwarting that backward induction. The “greatest fool” could alternatively
be interpreted as a rational “collector”, who just happens to value an asset
40
at high price, while nobody else does: the “bubble” then arises, when such a
collector is believed to be there and trade commences, until the collector has
purchased the item. While we like the “greatest fool” interpretation better,
we shall use the label “collector” to denote agents willing to pay a high price
for the asset (and not realizing that they cannot sell it at any higher price).
We will formulate a model where a single asset is traded in pairwise
random meetings of agents of different types. For a first-layer theory (which
we pursue below), agents differ in their belief regarding the fraction of agents
in the economy which are of the mythical “collector” type, willing to pay some
given high price for the asset. Agents who assign higher shares then value
the asset higher already for the simple reason that they believe it more likely
to meet such a mythical collector agent. I.e. if agents simply held the asset
until they meet the collector, that time is shorter for the more optimistic
agents, resulting in a higher price. This will be the main force below. It
additionally results in trading between less optimistic and more optimistic
agents, creating the bubble. Note that it does not actually matter whether
such collector agents are present: all that matters is the beliefs by the various
agents in the presence of such agents. It is truly here where we introduce
departures from the standard rationality paradigm. We allow for the belief
in such collectors to suddenly disappear: if that happens, the price crashes.
It should be clear, that one can construct higher-layer type theories too.
For a second-layer theory, all agents may agree that there are no collectors.
However, they may all believe that a certain fraction of agents out there
does believe such collectors to be there, and the more optimistic agents may
believe that fraction to be higher. Agents in such an economy will then not
per se wait to sell to a collector (they know they cannot), but wait to sell
to an agent who believes such collectors to be present. Furthermore, the
agents that are less optimistic regarding the existence of such believers will
sell to agents who are more optimistic regarding such believers. A third-layer
41
theory would be about agents differing in their beliefs of meeting agents who
believe that they can meet an original believer, etc.. We feel that it would
be fascinating to explore the ramifications ad variations of the simple model
below a lot further than we do. It is just meant as an inspiration and starting
point.
7.2
The model
Time is continuous, t ≥ 0. There is initially a continuum of agents of total
mass one. There is distribution of agent types θ ∈ [0, 1] in the economy,
characterized by the distribution function H(θ). We assume that H is differentiable, and shall denote its density with h(θ). We call agents of type
θ = 1 “collectors”. We shall assume that H(θmax ) = 1 for some θmax < 1,
and thus, the distribution H assigns no weight to collector types.
Agents may differ in their belief how many agents of type θ = 1 there
are, with θ parameterizing that belief. Specifically, we shall assume that,
initially, an agent of type θ uses the beliefs
Hθ (x) = (1 − θ)H(x) + θ1x=1
(25)
In other words, agent θ uses a weighted average between the true distribution and a point mass at the “collector” type. Agents are aware of their
differing beliefs, but they individually nonetheless insist on the beliefs they
hold. However, we allow for an aggregate revelation event, arriving at the
rate α ( or: instantaneous probability αdt), at which point all agents suddenly understand that there are no collector types and their beliefs switch to
the true distribution H. One might wish to assume that agents are unaware
that this revelation event could occur (“MIT shock”), but it turns out that
the mathematics is not much different if they do: so we shall assume that.
In the latter case, the better interpretation is that agents believe that, with
42
some probability αdt, the distribution of population types changes from Hθ
to H, interpreting this as a taste shift for other agents.
There is a single and indivisible asset (“coconut”), initially in the hand of
an agent of type θ = 0. There are random pairwise meetings between agents:
it suffices to describe the meetings between the agent that currently has the
asset and some other agents. These meetings happen at rate λ. If the agent
currently holding the asset is of type θ and if the revelation event has not
yet happened, then she will believe that she meets an agent drawn from the
distribution Hθ . The asset-holding agent (which we shall call the “seller”)
posts a take-it-or-leave-it-price qθ . The other agent (which we shall call the
“buyer”) decides to accept or reject the trade. If the trade is rejected, the
seller keeps the asset and keeps on waiting for the next pairwise meeting. If
the trade takes place, the buyer produces qθ units of a consumption good or
“cash”, at instantaneous disutility qθ , which the seller consumes, experiencing
instantaneous utility qθ . The future is discounted at rate ρ. The buyer
receives the asset and then in turn waits for the next pairwise meeting. If
the buyer turns out to be a collector, he will be willing to buy the asset at any
price at or below some exogenously fixed value v(1). The asset may provide
some intrinsic value to the collector or the collector may simply be the “last
fool”, failing to understand that he can sell the asset at an even higher value
in the future. In any case, the asset offers no intrinsic benefit to any agent
who is not a collector. In other words, we assume that non-collector agents
have preferences given by
Z
U = E[
∞
e−ρt ct dt]
(26)
0
where we allow ct to be negative, and where ct is the consumption flow
resulting from these trades. Crucially, we shall assume:
Assumption A. 1. ρ > 0
43
A few brief remarks may be in order. We have not used time subscripts
for qθ , though there may be equilibria, in which these prices do depend on
time. Here, we shall concentrate on time-invariant solutions, for simplicity.
More importantly, it may seem odd to consider only a single asset, given that
we have a whole continuum of agents at our disposal. This assumption is
helpful to simplify the analysis, however. Otherwise, one would have to distinguish between meetings, where the potential buyer already owns an asset
or not. Furthermore, over time, agents would need to keep track of the distribution of asset-owning types. It is plausible that these distributions shift
to the right over time, i.e., that it is the higher types holding assets, as time
progresses. In the decision problem to be analyzed, selling agents would then
need to forecast these evolutions, creating potentially intricate interactions
and complications that go beyond our purposes here. These evolutions are
likely to necessitate introducing a time-dependence of the posted price, for
example.
Note that, in essence, the bubbly economies described in section 6 can
be understood as featuring ρ ≤ 0 with α = 0 (and finite lives), so that
agents are willing to agree to a trade, in which they give up more today than
they receive later, or at least do not insist on getting more later on. We
rule out this channel. Note also that the search-theory models of money like
Kiyotaki and Wright (1993) and their successors assume that the sum total
of the consumptions is larger than zero, i.e., that the seller benefits more
from the sale than the buyer is hurt. If trades can only take place, using the
intrinsically worthless asset, the asset helps in achieving a better outcome
than autarky. Related modelling devices are used in Harrison-Kreps (1978)
or Scheinkman and Xiong (2003). Here, by contrast, we “shut down” any
benefits from the trade per se.
We shall formulate the strategies of buyers as threshold strategies. I.e., a
buyer of type θ picks some value vθ , and purchases the asset, if the take-it-or44
leave-it price is at or below that value, provided the revelation event has not
(a)
yet taken place. There is likewise a value vθ
when or after the revelation
event has taken place. For the collector type, v1 > 0 is a parameter, and
(a)
v1 = 0 (for completeness of the description). A seller of type θ picks a take(a)
it-or-leave-it price qθ before the revelation event and pθ after the revelation
(a)
(a)
event. A Nash equilibrium is then the four functions (vθ , vθ , qθ , qθ )θ∈[0,1] ,
so that the strategies of agent θ maximize the utility function (26), given the
strategies of all other agents.
Given these strategies, one now obtains the (random) paths for the prices,
starting at q0 , and then driven by the pairwise random meetings in the future.
One can use these random paths to describe their means and distributions,
etc..
7.3
Analysis
Proposition 4. After the revelation event, all agent types trade at price
(a)
qθ = 0.
Proof. (We have to show that this is true. Hopefully it is. It is reasonable:
after the revelation event, no agent would value the asset intrinsically, so
there is no hope of selling it at a positive price.)
Given this proposition, a seller before the revelation event can only hope
to sell the asset in the future before the revelation event takes place. Put
differently, we can assume that the seller discounts the future at rate α + ρ,
and that any before-revelation value of the asset to the seller in the future
is discounted at that rate too. We could introduce a new symbol for α + ρ.
In slight abuse of notation (or appealing to the “MIT shock logic”), we shall
continue to use ρ for that discount rate.
We proceed to analyze the before-revelation phase. Note that it does not
matter to an agent of type θ whether he exits time t holding the asset, because
45
a sale did not go through, or holding the asset, because he just purchased
it. Therefore, the maximal price vθ at which the agent is willing to acquire
the asset is equal to the continuation value, when exiting an unsuccessful
attempt at selling the asset.
Consider then a seller of type θ, contemplating a sales price q. The
seller assumes that the buyer type x is drawn from the distribution Hθ .
Furthermore, the seller assumes that buyers follow their equilibrium strategy
of accepting or rejecting a proposed price. A trade will then take place
is px ≥ q. From the sellers perspective, the trade then takes place with
probability
φθ (q) = Prob(“sale” | q) =
Z
1q≥px Hθ (dx)
Proposition 5. The probability of a sale φθ (q) is decreasing in q.
Proof. Obvious.
If the trade takes place, the seller receives q. If the trade does not take
place, the seller exits still holding the asset. Trading possibilities arrive at
rate λ. If the seller θ chooses q as its sales strategy, the value Vθ (q) of the asset
under that strategy can therefore be “heuristically” computed as follows. In
a very short time interval dt, the sale takes place with probability λφθ (q).
Else the seller will remain owner of the asset at time t + dt, still valuing the
asset at Vθ (q) then, if the aggregate revelation event has not taken place.
Therefore
Vθ (q) = λφθ (q)qdt + (1 − λφθ (q))(1 − ρdt)Vθ (q)
(27)
or, cancelling higher order terms,
Vθ (q) =
λφθ (q)q
=
ρ + λφθ (q)
46
q
ρ
λφθ (q)
+1
(28)
The optimal selling strategy q = qθ is the one maximizing Vθ (q), delivering
vθ = Vθ (qθ ))
Proposition 6. The value vθ is increasing in θ.
Proof. (We have to show that this is true. Hopefully it is. It is reasonable:
higher types are more optimistic about meeting collector-type agents, and are
therefore willing to obtain the asset at a higher price )
Proposition 7. For each sales price q, there is a least optimistic buyer type
or buyer threshold type x(q), so that all buyers of type x ≥ x(q) will buy the
asset and all buyers of type x < x(q) will not, i.e.
“x ≥ x(q)′′ ⇔ “vx ≥ q ′′
The function x(q) is increasing in q. Furthermore, for q ≤ v1 ,
φθ (q) = (1 − θ)(1 − H(x(q))) + θ
(29)
Proof. This follows immediately from the preceding proposition.
Proposition 8. For q > v1 , φθ (q) = 0.
Proof. (We have to show that this is true. Hopefully it is. It is reasonable:
collector types won’t buy at a price above v1 , so why should anyone else?)
To proceed using calculus, we make the following high-level assumption.
This assumption is not necessary for proceeding with the analysis, but it
simplifies the exposition.
Assumption A. 2. x(q) is differentiable in q.
47
Proposition 9. The optimal q = qθ satisfies the first-order condition
0=1+
λ
((1 − θ)(1 − H(x(q))) + θ) − ηθ (q)
ρ
(30)
where ηθ (q) is the elasticity of the sales probability,
ηθ (q) =
φθ (q)′ q
=−
φθ (q)
h(x(q))x(q)′ q
1
+ 1 − H(x(q))
(1/θ)−1
(31)
Proof. Differentiate Vθ (q) with respect to q.
One can rewrite that elasticity a bit further. Let
ψθ (x) = (1 − θ)(1 − H(x)) + θ
be the probability of meeting a buyer of type x or better (including the
collector), from the perspective of a type-θ seller. Define its elasticity
ηθ,ψ (x) =
ψθ (x)x
=−
ψθ (x)
1
(1/θ)−1
h(x)x
+ 1 − H(x)
Define the elasticity of the threshold buyer type,
x(q)′ q
ηx¯ (q) =
x(q)
Then
ηθ (q) = ηθ,ψ (x(q))ηx¯ (q).
This is the usual chain rule for elasticities, of course.
The results above suggest a strategy for obtaining an equilibrium. Suppose, one has some conjectured threshold buyer type function x(q), which
is increasing in q. With that, solve (30) for the optimal strategy qθ and
thereby for the value vθ = Vθ (qθ ). With the value, calculate the resulting
48
buyer threshold type function
x∗ (x) = argminx vx ≥ q
If x∗ (q) = x(q), one has obtained an equilibrium.
It may be possible to obtain analytical examples, for smart choices for
H. We shall pursue numerical examples instead.
7.4
Numerical examples
We use the strategy described above as a numerical recipe, using some initial
conjecture x(0) (q) at “j = 0” and then iterating on the procedure above to
obtain updates. We calculate everything on a grid. To solve for the optimal
choice of q, we numerically maximize (28). As parameters, we chose λ = 1,
ρ = 0.1 and H to be a uniform distribution on [0, 0.25]. The “collector
price” was normalized at v1 = 1. The results are in figure 11. As one can
see, agents with low θ pursue a strategy of seeking to sell to higher-θ agents,
but beyond θ = 0.1, agents now only wait for the collector to make the sale.
This can also be seen from the probability of sales. The jaggedness in the
sales probability graph is due to the numerical approximation with a grid,
and could be refined.
8
Evidence
What caused the subprime crisis and, by extension, the financial crisis of
2008? What moved “first”, what moved “later”? Answers to these questions
may be important for sorting through various explanations.
For example,
1.
• perhaps, house prices fell first for exogenous reasons, impairing
bank balance sheets and leading to a financial collapse,
49
Optimal q
1
0.7
0.95
0.6
0.9
0.5
0.85
0.4
0.8
q
v
Optimized v
0.8
0.3
0.75
0.2
0.7
0.1
0.65
0
0
0.05
0.1
θ
0.15
0.2
0.25
0
0.05
0.1
θ
0.15
0.2
0.25
Buyer thresholds
Optimized sales probability
1
0.45
0.4
0.8
0.6
x−bar
P("sale")
0.35
0.3
0.25
0.4
0.2
0.2
0.15
0.1
0
0.05
0.1
θ
0.15
0.2
0.25
0
0
0.2
0.4
0.6
0.8
q
Figure 11: Results from a numerical example. As parameters, we chose λ = 1,
ρ = 0.1 and H to be a uniform distribution on [0, 0.25]. The “collector price”
was normalized at v1 = 1. In the top left panel, we compare the optimal value
function to the value function obtained per only selling to the collector. As
one can see, agents with low θ pursue a strategy of seeking to sell to higher-θ
agents, but beyond θ = 0.1, agents now only wait for the collector to make
the sale. This can also be seen from the probability of sales. The jaggedness
in the sales probability graph is due to the numerical approximation with a
grid, and could be refined.
50
1
• or perhaps the banking system collapsed, leading to a reduction
in mortgage lending and a fall in house prices.
2.
• Perhaps, the fall in house prices triggered greater reluctance by
banks to issue subprime loans,
• or perhaps and conversely, mortgage lending and, in particular,
subprime borrowing, was reduced first, triggering a fall in house
prices.
3.
• Perhaps, subprime lending was reduced in the wake of higher
delinquency rates on subprime mortgages
• or perhaps subprime lending was reduced, and the subsequent fall
in house prices triggered delinquencies.
4.
• Perhaps delinquencies rose because the pool of borrowers worsened
• or perhaps short-term interest rates rose, leading to higher rates
for ARM mortgages and thereby higher delinquency rates.
The figures here are meant to shed light on the sequence of these events,
using “eyeball” reasoning.
1. Figures 12 and 13 show the S&P/Case-Shiller Home Price Indices.
There is a run-up in houseprices up to somewhere in the middle of 2006.
According to the 20-city index in 13, the peak is in July 2006. From
there, prices started to drop, falling from 206.52 in July 2006 to 192.98
in October 2007. House prices were falling, but the fall could perhaps
be viewed as modest. In October 2008, the date of the Lehmann crisis
and, in essence, the date of the financial collapse, house prices were
at 154.5, having fallen rather quickly, but continuously from October
2007. House prices fell a bit more subsequently, reaching their bottom
at an index value of 139.26 in April 2009.
51
Source: http://www.worldpropertyjournal.com/north-america-residential-news/spcase-shiller-home-price-indices-report-fordecember-2011-case-schiller-home-price-index-median-home-prices-the-national-composite-10-city-composite-index-20-citycomposite-home-price-indices-david-m-blitzer-5351.php
Figure 12: The S&P/Case-Shiller Home Price Indices
From the sequence of these events, it appears plausible to argue that
house prices fell first, and the financial system collapsed subsequently.
2. However, the share of mortgages in the form of subprime fell substantially, much earlier, as figure 14 reveals. Again, the peak subprime
lending share of all mortgage originations occured in 2006, at 23.5%,
with a dramatic fall to 9.2% in 2007 and a near-zero in 2008. The peak
occured roughly at the same time as the peak of the S&P/Case-Shiller
index in 12 and 13 and one might even wish to argue that the hump
near the peak looks rather similar here. However, the decline in the
peak subprime lending share from 2006 to 2007 was considerably more
dramatic than the fall in house prices.
From this comparison, it appears plausible that subprime lending rose
52
Case-Shiller 20-city Index
210
200
190
180
170
160
150
130
Jan-2005
Mar-2005
May-2005
Jul-2005
Sep-2005
Nov-2005
Jan-2006
Mar-2006
May-2006
Jul-2006
Sep-2006
Nov-2006
Jan-2007
Mar-2007
May-2007
Jul-2007
Sep-2007
Nov-2007
Jan-2008
Mar-2008
May-2008
Jul-2008
Sep-2008
Nov-2008
Jan-2009
Mar-2009
May-2009
Jul-2009
Sep-2009
Nov-2009
Jan-2010
140
Case-Shiller 20-city Index
Source: http://us.spindices.com/indices/real-estate/
sp-case-shiller-20-city-composite-home-price-index
Figure 13: The S&P/Case-Shiller Home Price 20 City Index
and fell together with house prices. If anything, subprime lending collapsed and decreased sharply, before house prices did. Thus, it may
have been the reduction in subprime lending, causing the fall in house
prices rather than vice versa.
3. One might wish to blame the decline in subprime lending on delinquency rates. Here, figure 15 is revealing. First, it shows that delinquency rates on fixed rate mortgages, be they prime or subprime, did
not noticeably increase in 2006 and 2007: if anything, subprime fixed
rate delinquency rates were near their all-time low of 2005.
The story is different for adjustable rate mortgages or ARMs. Here,
rates did go up somewhat in 2006 and then somewhat more from their
all-time low in 2004, but even there, the level in 2007 is rather compa-
53
Source: The Financial Crisis Inquiry Report, National Commission, January 2011
Figure 14: Subprime Mortgage Originations
rable to the levels before 2002, both for prime adjustable rates as well
as subprime adjustable rates, as figure 15 shows.
It is easy to miss that, when looking at the subsequent development
of delinquency rates, as shown in figure 16. Delinquencies later rose to
unprecedented levels (at least for this time interval), peaking at rates
somewhat above 40% for subprime adjustable rate mortgages around
the end of 2009. In particular, deliquencies on subprime mortgages and
prime adjustable rate mortgages had risen already considerably until
October 2008, the date of the financial collapse.
However, it does not seem plausible to argue that subprime lending
was reduced in 2007, because delinquency rates had increased already
at that point.
54
Source: Senator Schumer, Rep Maloney, Report of Joint Economic Committee, 2007
Figure 15: Subprime Deliquency Rates
4. If anything, perhaps the delinquency rates in 2007 and the overall movement on delinquency rates on adjustable rate mortages up to 2007 are
linked to short-term interest rates, as figure 17 show. The Federal Reserve increased the Federal Funds Rate in a sequence of small steps,
starting at 1 percent in June 2004 to 5.25 in July 2006, and then leveling
off, before dramatically reversing course at the end of 2007.
It is fairly plausible that the rise in short-term market interest rates
from mid-2004 to mid-2006 resulted in the rise of delinquencies on
adjustable rate mortgages from mid-2004 to mid-2006.
Finally, Alejandro Justiniano (fort) have argued that house prices rose
from 2000 to 2007 without an expansion of leverage, i.e. at rather constant
rates of mortgages to real estate, see the bottom-right panel of figure 18. The
subsequent fall in house prices then went along with an increase in leverage
55
Source: The Financial Crisis Inquiry Report, National Commission, January 2011
Figure 16: Subprime Deliquency Rates
and not necessarily a reduction in the volume of outstanding mortgages, see
the top left panel.
These authors argue, that, first, there was an increase in available funds
without an increase in leverage, leading to more mortgages at stable interest
rates and stable leverage ratios, leading to a run-up in house prices. In 2007
and beyond, they have argued that the collateralizability of houses relative
to available funds increased (or that available funds for lending decreased),
leading to a rise in mortgage rates and a collapse in house prices. There
certainly are some interesting comovements in 18 that deserve explanation,
though not all readers may buy into the hypothesis that the collapse in house
prices was caused by their relatively better collateralizability.
56
Source: http://www.zerohedge.com/article/comparing-fed-funds-rateprimary-credit-discount-rate-over-past-decade
Figure 17: Federal Funds Rate
57
Source: Justiniano et al Slides Household Leveraging and Deleveraging 2014
Figure 18: The Justiniano-Primiceri-Tambalotti Facts
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9
Conclusions
The purpose of this chapter was to examine a key connection between financial markets and macroeconomic activity. Specifically, we examined a
connection between two features
1. A boom-bust in house prices.
2. A boom-bust in credit markets.
To do so, we investigated several benchmark approaches and channels, and
relate it to the existing literature. Broadly, there are two possible channels:
1. The house price boom-bust generates the credit boom-bust. Both together generate the aggregate activity boom-bust.
2. The credit boom-bust generates the house price boom-bust. Both together generate the aggregate activity boom-bust.
It is already a challenge to understand the house price boom-bust together
with the credit boom-bust, without analyzing the aggregate activity repercussions. We therefore limited our focus on the interaction of the first two.
In this chapter, we examined the following appealing and popular intuition. If house prices are expected to rise, then
1. banks are more willing to lend. They are furthermore willing to lend to
worse creditors today, i.e. those with more risky income in the future
(“subprime mortgages”).
2. speculating households, that count on “flipping” the house are more
willing to buy.
While appealing, formalizing this intuition tends to run into the “conundrum of the single equilibrium”: if prices are expected to be high tomorrow, then demand for credit and thus houses should be high today (see
59
above), and that should drive up prices today, making it less likely that
prices will increase. Or, put differently, as prices rise, they eventually must
get to a near-maximum at some date: call it “today”. At that price, prices
are expected to decline in the future. But if so, banks and speculating households are less likely to buy today: but then, the price should not be high
today. Put differently, these types of bubbles are typically ruled out by thinking about the “last fool” who is willing to buy at the highest price, when
prices can only go down from there: such fools should not exist in rational
expectations equilibria.
The chapter first discussed a simple mechanical baseline model to represent the intuition above. On purpose, it is designed to avoid several thorny
issues that arise in a fully specified equilibrium model. The chapter then discussed various potential avenues to break the curse of the conundrum. We
furtermore juxtaposed our findings to some lessons we have drawn from the
existing literature regarding the empirical evidence.
We wish this chapter to trigger further research and thinking on this
important connection. As has become clear, the issues are far from resolved.
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