Bulletin of Monetary Economics and Banking, Vol. 21, No. 2 (2018), pp. 191  216
SUBPRIME MORTGAGES AND LENDING BUBBLES
Ali Yavuz Polat1
1Department of Economics, Abdullah Gul University, Kayseri, Turkey. Email: aliyavuzpolat@gmail.com
ABSTRACT
We consider a model with two types of households: the poor with no initial endowment
and the rich with positive endowment, and two types of assets: properties in a poor area and properties in a rich area. In the model, poor agents need credit to buy an asset, whereas the rich can draw from their endowment. We show that
Keywords: Subprime mortgage; Bubbles; Credit; Equilibria; Household asset.
JEL Classifications: D14; G10; G21.
Article history: 

Received 
: July 7, 2018 
Revised 
: September 16, 2018 
Accepted 
: October 17, 2018 
Available online 
: October 31, 2018 
https://doi.org/10.21098/bemp.v21i2.955
192Bulletin of Monetary Economics and Banking, Volume 21, Number 2, October 2018
I. INTRODUCTION
The years preceding The Global Financial Crisis (GFC) witnessed unprecedented access to credit for low income individuals with little or no credit history. This was partly facilitated by
Another channel is mobility. With the possibility of selling their houses at a high price, poor households can enjoy higher
We consider a model with two types of households: the poor (with no initial endowment) and the rich (with some endowment), and two types of assets: a house in a poor area (a poor asset) and a house in a rich area (a rich asset). In the model, poor agents need credit to buy an asset, whereas the rich can draw from their endowment. In other words, the poor need a 100%
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when old, but also enjoy a positive surplus. Under the
To analyze welfare, we consider an extension where both poor and rich assets depreciate through time. We consider a supplier who can build new houses, whenever the price of the house is higher than the cost. In that case, when the poor asset is underpriced (zero price), the supplier will not find it profitable to build new houses in the poor neighborhood. Thus, over time, the total stock of poor assets will decrease due to depreciation. In contrast, if there is a bubble in the poor market, the supplier will build new houses, keeping the total stock of assets stable. Comparing these two scenarios, the bubble brings a welfare improvement by stabilizing the total stock of poor assets.
One important empirical implication of this model is that bubbles grow faster in the poor market than in the rich market. Moreover, if at any point in time the 100% LTV policy is abandoned, there will be a mass default among the poor. The intuition of both results is due to poor agents having zero initial wealth and their dependence on credit.3 Figures 1 and 2 show the
This paper proceeds as follows. Section II reviews the literature. Section III outlines the model, and Section IV characterizes the equilibria. Section V sets forth extensions of the model. Section VI draws conclusions and implications for policy.
2
3
4
5
However, there is a risk that if the bubble bursts, there may be widespread defaults among the poor. We comment on this issue in the extension section below.
As discussed in the conclusion, the
See Figures 7, 8, and 9 in the Appendix for Los Angeles, New York, and Tampa.
For San Francisco and Miami, the difference between the low and high classes seems to be much more significant.
194Bulletin of Monetary Economics and Banking, Volume 21, Number 2, October 2018
II. RELATED LITERATURE
Bubbles are studied extensively in the literature and have attracted the attention of economists, academics, and policymakers due to their consequences on the allocation of resources. This section presents an overview of the vast literature on this topic without presenting a detailed discussion. Although there is no consensus among economists on the definition of the term “bubble,” one can define a bubble as sustained mispricing of an asset. Not every mispricing can be considered a bubble however. The term bubble refers to a period in which investors believe that price growth will continue, so they hold the asset at the ongoing
The literature describes various theoretical explanations for bubbles.7 One strand of models is the rational expectations models, where agents have identical information (Martin and Ventura, 2012; Galí, 2014). Considering the possibility of speculation when traders are assumed to have rational expectations, Tirole (1982) derives the conditions under which bubbles can be ruled out. This author shows that at least one of the following four conditions must be violated to sustain a bubble (Barlevy, 2015): (i) the number of potential traders is finite; (ii) all traders are assumed to be rational, which is common knowledge; (iii) traders should hold common prior beliefs about the environment; and (iv) resources are allocated efficiently
Another strand of literature considers asymmetric information bubbles (or heterogeneous belief models) where agents have different information, but based on a common prior distribution (Harrison and Kreps, 1978; Allen, Morris, and Postlewaite, 1993; Scheinkman and Xiong, 2003; Conlon, 2004; Brunnermeier, 2008; Conlon, 2015). In these models, prices reflect information but, contrary to the symmetric information case, the existence of a bubble may not be common knowledge.8 Other literature considers the interaction between
6This type of explanation for the bubbles is termed the
7See Brunnermeier (2008) and Brunnermeier and Oehmke (2013) for an overview.
8As Brunnermeier (2008) mentions, it can be the case that all agents are aware of the
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not be able to trade against the bubble. This “limits to arbitrage” argument implies that bubbles can exist, since arbitrageurs cannot drive prices to the fundamental.
The present study is more related to the class of “credit bubble” models, such as Allen and Gale (2000). These authors present a model built on a risk shifting argument. Investors, having limited liability, borrow from banks and bid up asset prices. When the value of their investment turns out to be low, they simply default and walk away. Barlevy (2014) develop a
III. THE MODEL
We consider an OLG model where each generation lives for two periods. Agents are heterogeneous in their initial endowment. There are two types of agents: the rich with an initial endowment A > 0, and the poor with no initial endowment. In the economy, there are also two types of assets9 whose consumption generate positive utility. More precisely, agent i ∈ {R,P} (R = Rich, P = Poor) derives utility ui (aj) from the consumption of asset aj ∈{ aR,aP} when young. There is a continuous mass m of poor agents and n of rich agents. Type aR asset represents a residential property in a rich area, whereas aP represents a residential property in a poor area. Consider a segregated city where the rich live in one neighborhood and the poor in the other. We assume that the rich have no intrinsic utility from consumption of a property in the poor area. This is just for simplification and, as long as the rich value the poor asset less than the poor do, our results follow. Formally, we have the following assumption.
Assumption [A1]
[A1] basically states that rich agents value the rich asset more than poor agents and rich agents assign no value for the poor asset.10 In contrast, poor agents derive some utility from the poor asset, albeit lower than the utility the rich derive from the rich asset.
The assets are fixed in supply. There is a continuum SP of poor assets and SR of rich assets. In any period t, the old generation retires and those who hold an asset can sell it to a member of the young generation at some price ptj, j= R,P.
9A durable consumption good like a residential property.
10The assumption, uR (aP)=0, is just for simplification. It does not affect our results qualitatively as long as the above utility ranking holds.
196Bulletin of Monetary Economics and Banking, Volume 21, Number 2, October 2018
Each agent, either poor or rich, can purchase no more than one house, either as consumers or investors. Moreover, the agents who cannot buy a house obtain zero utility.
Assumption [A2] m > SP > n > SR
[A2] helps us to identify equilibrium prices. Specifically, there are more poor (rich) agents than the supply of poor (rich) assets such that agents compete to buy the assets. The assumption that the supply of poor assets is larger than the mass of rich agents ensures that rich agents will not drive prices in the poor asset market, even if they may be interested in buying a poor asset for speculative reasons. This is discussed in the characterization of the equilibrium in Section IV below.
To simplify the illustration, assume all agents have a discount factor 

per 


period.11 


IV. CHARACTERIZATION OF EQUILIBRIA
We define the fundamental value of asset aj, j ∈ {R,P} as:
(1)
As will become clear, in equilibrium, the value of asset aj is entirely determined by the demand of type j individuals. We thus simplify the notation by setting uP (aP) ≡ uP and uR (aR) ≡ uR.
Consider an equilibrium where banks offer contracts with no initial down payment (100% LTV ratio), type i young agents borrow Btj from the bank to buy type j asset from the old agents, and the banks demand repayment rBtj, r≥1. Type i=R,P agent will buy a property when the following holds:
(2)
Consider the following environment:
•Banks are competitive, but assume that for each borrower there is an upper
bound for the credit available . This constraint is relaxed at a rate r2>1,
i.e., . This borrowing constraint may reflect the total amount of
11Assume that banks borrow from depositors whose discount rate is β and there are enough depositors so that the funding cost of banks will be determined by the depositor’s discount factor. Considering that the banking sector is competitive, banks will charge . This will imply that the opportunity cost of borrowers in the economy is .
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credit available in the economy12 and will be crucial for the determination of equilibrium. Whenever the borrower is expected to default, we also assume
that banks do not lend, so that 
. This implies that if a bank expects that 
, such that the anticipated price in the next period is not enough to 

cover the repayment, then 
. 
• In each period t, since there are more young than old who own an asset,13 the young compete to buy the asset.
Given this environment, consider first the poor asset aP. The poor have no 

initial wealth, so they pay a price such that 
. Then (2) becomes 

(3) 
Also, the young poor cannot pay an amount greater than the maximum borrowing:
(4) Lemma 1 For the poor asset market, if there exists a borrowing equilibrium
with Bt>0, then price grows at rate r or higher to avoid default, i.e., 
. This 
implies that agents obtain a positive surplus, i.e., (2) does not bind. 

Proof. In the model, there are two constraints for a young poor agent: the
borrowing constraint (4) and the
(2). Due to competition among the young, at any period at least one of these two constraints should bind (otherwise, if both are slack, one can outbid others
by paying where (2) still holds.). The agents do not default in an equilibrium since they cannot access credit if they are not expected to pay back
their debt, i.e., 
if banks expect that 
. Thus, 

. This implies that agents can borrow only if the price of the poor 
asset grows at rate r or higher. In turn, this implies that agents obtain a positive surplus since WTBP does not bind. To see this point, note that a binding WTBP would imply that agents cannot pay back their debt:
12We think that this assumption is related to the
13One can also think of the basic assumption of the classical OLG model, where population grows each period at a fixed rate. However, this assumption will be redundant in our model, since, in every period, only some young will get the asset and in the next period, when they become old, the new generation will keep competing for the asset considering that the mass of assets is always less than the young population in each period. Thus, we simply assume that the population is fixed each period.
198Bulletin of Monetary Economics and Banking, Volume 21, Number 2, October 2018
(5)
(5)then implies that in any equilibrium with positive borrowing, WTBP does not bind.
Lemma 2 Suppose [A1] and [A2] hold. Then, in the poor asset market:
i. If r2 ≥ r, there exists an equilibrium with prices and in any consecutive periods. Poor agents who hold an asset obtain a positive surplus, (2) holds with strict inequality, and there are no defaults in equilibrium.
ii.If r2 < r, there is a unique equilibrium with price ptP = 0 in every period.
Comparing equilibria (i) and (ii), the poor are better off under (i).
Proof. Lemma 1 shows that in equilibrium, (4) must bind due to competition among the young (since WTBP is slack). Thus, in case (i), the young must borrow
up to the maximum in equilibrium. Then, equilibrium prices are and
. Since r2 ≥ r, we have , implying that agents can pay back their debt and there are no defaults in equilibrium.
A slack in (2) implies that the young who hold an asset obtain a positive surplus:
(6)
Note that there may also exist no bubble equilibria with prices ptP = 0,∀t, if 

agents expect 
at any point in time. The proof is similar to 
the one provided below.
We now check whether the rich have an incentive to buy the poor asset when
r2 ≥ r. Even though they derive no utility, if r2 < r, the rich may also buy the poor asset to speculate on price growth. But Assumption [A2] indicates that they cannot
drive prices, since the mass of the poor asset is bigger than the mass of rich agents. Thus, the marginal buyer is poor and the equilibrium is as established above.
For case (ii), again, as proved above, if there exists a borrowing equilibrium
BtP>0, it has to be 
. However, considering that r2 < r, 
, 

this implies that the old cannot pay back their debt and default. But, then expecting 

a default, banks will never give credit, i.e., 
. Thus, the only equilibrium 

involves p P = 0, ∀t. 



t 


> n, the marginal 
Now suppose the rich buy the poor asset. Again since SP 
buyer is a poor agent and thus the previous argument applies.14
Comparing the two cases (i) and (ii), in (i) the surplus from purchasing an asset is bigger than that of case (ii). The reason for this is that, when there is a bubble, agents enjoy an excess surplus, on top of the utility they get from consumption, considering that price growth is bigger than the repayment to the bank. Formally,
14Note that, when r2<r, the rich do not have speculative incentives to buy the poor asset, since uR (aP)=0, whereas the poor derive a positive utility even though the price is zero. Thus, even without the assumption SP>n, the marginal buyer is a poor agent.
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the surplus under case (i) is , where the gain from the price growth
is positive for r2 > r. Whereas the surplus under case (ii) is since
. Then, comparing the surpluses we get:
Note that here not all young agents obtain an asset. Thus it should be the case that there will be some credit rationing.
Figure 3. Poor asset market equilibrium for r2 ≥ r where pt = Bt and pt+1 = Bt+1
The figure illustrates constraints for the poor when r2 ≥ r.
Pt+1 
r2Pt 




E* 
rPt 

Bt+1 = r2Bt 



rpt 
 u 
(0,0) 


Bt 

Pt 


Figures 3 and 4 provide a graphical illustration of the arguments. Figure 3
represents the constraints for the poor when r2 ≥ r. The red lines represent the two main constraints: WTBP (2) and the borrowing constraint (4). Rearranging (2), we get pt+1 ≥
. Thus, the equilibrium is at point E*.
15Note that for the poor, the price is equal to the borrowing pt=Bt and pt+1=Bt+1. Thus, (2) can also be represented in the graph by
200Bulletin of Monetary Economics and Banking, Volume 21, Number 2, October 2018
Note that the (0,0) point in Figure 3 can be also sustained as a
Figure 4. Poor asset market equilibrium for r2 < r
The figure shows poor asset market equilibrium when r2 < r. 








rPt 
rp 
t 
 u 













Pt+1 






r2Pt 

















t+1 = r2Bt 







B 









C 














(0,0) 











t 


Pt 






B 























Figure 4 shows the case where r2 < r. The
However, any is outside the shaded region, so that default would have
occurred. But then, expecting a default, banks will supply no credit. As a result,
and the only equilibrium involves pt = pt+1 = 0, ∀t. 

As stated above in Lemma 2, there are two main cases in the poor asset market: 

i. For r2≥r, the young bid the price up to the borrowing limit 
and 
, thus implying that price grows at rate r2 every period. Under this equilibrium, young poor individuals can pay back their debt, since price growth is fast enough. Thus, there is a bubble equilibrium and for individuals who bought a house ((2)) holds with strict inequality. The price fetched by the asset in the second period of an agent’s life is used to repay the principal plus the interest of the loan borrowed . Banks break even and there are no defaults. Under this parametric case, rich agents may also have an incentive to buy the poor
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prices, since the mass of rich agents is smaller than the mass of poor assets available in the economy. As a result, the marginal buyer is a poor agent.
ii.For r2 < r, the borrowing constraint is relaxed at a rate smaller than the cost of borrowing r, thus implying that even if the poor agent buys the asset by borrowing up to the maximum, price growth is still not large enough to cover the repayment in the next period; i.e., the price fetched by the asset in the
second period of life is lower than the repayment, . This implies that if an agent borrows and purchases a house, the agent will default when old. Thus,
the poor asset is not traded at a positive price and there will be an equilibrium where the asset price is below the fundamental, namely ptP = 0, in every period. In this scenario, no rich agent will have an incentive to buy the asset.
The above discussion implies that there are two types of equilibria for the poor housing market depending on the credit available in the economy. If there is strong credit growth in the economy (case i.), there will be a bubble equilibrium where agents who buy a house obtain positive surplus (WTBP(2) holds with strict inequality). If there is not enough credit growth in the economy (case ii.), then, in equilibrium, the poor asset is significantly underpriced (the equilibrium price is below the fundamental).
Now, consider the rich asset, aR:
Different from the poor, rich agents have an endowment A, which can be used to buy a house.
The rich again have two constraints: the
(7)
and the resource constraint
(8)
The rich can pay a down payment Dt and borrow Bt to buy the house, so that the price of the rich asset is ptR = Bt + Dt. Below, we focus on the case where the down payment is the same in each period, Dt = Dt+1 = D,16 so that the price is:
(9)
Thus, the
16Note that there can be other equilibria where Dt≠D and it grows each period. We restrict attention to a case where, under a bubble, the rich compete for the asset and, after some point, due to an increase in price, they have to pay their entire endowment. Thus, even though at the initial periods Dt<D, after some period, the young rich need to pay D to ensure buying the asset.
202Bulletin of Monetary Economics and Banking, Volume 21, Number 2, October 2018
(10)
and when the maximum is borrowed we obtain17
(11)
First, we need to check whether there can be an equilibrium where rich agents do not borrow, Bt = 0, and simply pay from their initial endowment so that there is no bubble, ptR = pt+1 = D.
Lemma 3 If , then there exists an equilibrium without borrowing,
and Bt* = 0. If A < FR; thus, in equilibrium, it must be Bt* > 0
(whenever ).
Proof. In any equilibrium, at least one of the constraints (7) or (8) must bind due to the competition among the young (as buyers). Now, consider an equilibrium where the young do not borrow and pay down payment D so that in each period
the price equals the down payment, . However, this means that the resource constraint (8) is slack (since agents do not borrow). Thus, (7) must bind, which implies:
The second part of the lemma follows from the fact that if A<FR, then WTBR does not bind, so the resource constraint must bind.
Lemma (3) basically states that, if the endowment A is big enough, there exists an equilibrium where the rich asset is traded at the fundamental value without any bubble (and the price is stable). If the initial endowment is small, then, similar to the poor market, the young borrow to buy the asset, i.e., Bt* > 0 provided that
.18
Now consider an equilibrium with a positive amount of borrowing. The rich have resources A, which can be used to buy the asset. They already paid D ≤ A as a downpayment when they were young. Then, when they are old, the
condition is 


. Since 
, rearranging the no 


default condition we get: 


Rearranging [WTB_Rich] (WTBR) we get:
17Note that we denote uR=uR (aR) for the rest of the paper.
18Note that still there may be a
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Below, Figures 5 and 6 show these two constraints (12) and (13) for the rich when r2 ≥ r and r2 < r, respectively. In both graphs, the dotted blue lines represent the
Figure 5. Constraints for the rich when r2 ≥ r
The figure illustrates constraints for the rich when r2 ≥ r.
Bt+1 
r2Bt 
WTBR (D > FR) 
Bt+1 = r2Bt 
E* 
WTBR (D = FR) 


rBt  A 
(0,0) 
Bt 
Bt 




In Figure 5, the maximum borrowing (point E*) is an equilibrium, though not unique, where the resource constraint binds and the agents obtain a positive surplus. Note that there can be other equilibria (discussed below), such as a point on the WTBR where D = FR and Bt+1 = rBt so that agents obtain zero surplus (since WTBR binds).
When wealth is smaller than a specific value , then the downpayment has to be small as well, since D ≤ A, so that the redline WTBR will lie below the default line (the blue dotted line). This case is similar to the poor asset market, where agents are wealth constrained. Thus, to have a meaningful difference between the poor and the rich, we assume that A is large enough. Lemma 6 in the Appendix considers the case where this assumption does not hold.19
19Basically, as shown in the Appendix, if the initial wealth of the rich is smaller than this
204Bulletin of Monetary Economics and Banking, Volume 21, Number 2, October 2018
Assumption [A3] A>A where .
Figure 6. Constraints for the rich when r2 < r
The figure illustrates constraints for the poor when r2 < r.
WTBR (D > FR) 
WTB 
R 
(D = FR) 


rB 
 A 







t 

Bt+1 



r2Bt 





Bt+1 = r2Bt 




(0,0) 
Bt 

Bt 








Figure 6 represents the constraints that the rich face when the credit growth rate is smaller than the borrowing rate, i.e., r2 < r. Considering a candidate bubble equilibrium, there are two possible cases, one where WTBR binds and the other where the
Note that there may be other equilibria where Bt and/or Dt grow at rates that are not constant. But we restrict attention to balanced growth paths.
Let be the initial borrowing limit in period 0.
Lemma 4
Suppose [A1], [A2], and [A3] hold. Then, in the rich asset market,
i. For the case r2 ≥ r; 





a) 
if 


or 



b) 
if A≥FR, r2>r and 
is sufficiently large, 




then there exist balanced bubble equilibria such that the maximum amount 

is borrowed 
, and rich agents who hold an asset obtain a positive 

surplus (WTBR (13) is slack). The price is 

, implying that the 

resource constraint (8) binds. 




c) 
if A ≥ FR, there also exist bubble equilibria where the rich agents obtain no 

d) 
surplus in all periods, i.e., WTBR (13) binds. 


, 

also in all cases, there exist equilibria with no bubble where 
∀t.
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205 


ii.If r2 < r, there is no balanced growth path involving a bubble; i.e., there exists a continuum of equilibria without a bubble (a steady state) where the asset is traded at the fundamental price ptR* = FR, ∀t such that:
a)If A≥FR, then (11) (WTBR) binds and ptR*=p*=B*+D*=FR for D*∈[0,FR] and B*∈[0,FR]
b)If A<A<FR, then (11) (WTBR) binds and ptR*=p*=B*+D*=FR for
and where .
Proof. We can rewrite the constraints for the rich.
If WTBR holds with inequality, then competition implies and Dt=A so that , i.e., the resource constraint binds.
Then we can rewrite WTBR as follows:









(14) 





















However, we know that 



. Taking the limit for t→∞ shows that (14) 




can hold for all t only if r2 ≥ r. If also 

, then the resource constraint 
always binds in every bubble equilibrium (and the rich who hold an asset experience positive surplus).
Consider then the case where. We need to look at the initial conditions.
If r2 > r andis large enough such that WTBR is slack (i.e., (14) holds at time zero), then we are back to the case above.
If is small or r2 = r so that
then either 
or D0 < A, or both. 


Whenever 




, bubble equilibria where WTBR binds are also possible. We 

now construct a bubble equilibrium where WTBR binds. 


Suppose that Dt=D is constant and Bt grows as follows 







, 
(15) 




then the rich make zero surplus in all periods. Consider a special case where
. Then Bt grows at a rate r < r2 in every period so that the borrowing
constraint (and the resource constraint as well) never binds.
206Bulletin of Monetary Economics and Banking, Volume 21, Number 2, October 2018
Note however that there may be other equilibria where Bt and/or Dt grow at rates that are not constant and satisfy (15). But we restrict attention to balanced growth paths.
For the case (ii), when r2 < r:
Note that due to the resource constraint,20 the price ratio 
is bounded above 

by r2 in the limit for →∞ . Since r2 < r, 






WTBR (7) implies so that even though we have binding WTBR,
Since we obtain a contradiction, this means that for r2 < r, there cannot exist a balanced bubble equilibrium.
Thus, consider an equilibrium where the price is constant, ptR = pR*. Then, from
WTBR, it must be that . However, due to competition, cannot be an equilibrium, since otherwise a young agent can pay pR*+ϵ and obtain the asset for sure. Thus, the equilibrium must be such that WTBR binds (agents obtain no surplus). The price is thus equal to the fundamental value in all periods.
The only difference between (ii)(a) and (ii)(b) is the maximum borrowing considering the default
(16)
Note that agents have a continuum of choices between borrowing and down payment. This is due to rich agents having two choice variables and the cost of borrowing being the same as the opportunity cost of consumption in the model. In that sense, agents are indifferent between borrowing less and consuming less today
20 
Recall that the resource constraint is 

and 



. 







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207 


but repaying a lower amount tomorrow vs. borrowing more and consuming more today and repaying a higher amount. Since we are interested in the equilibrium price, this continuum of choices does not affect our argument.21
Considering the different possible cases for r2, the intuition of Lemma (4) is as follows. Whenever the credit growth rate is larger than the cost of borrowing, there exists a bubble equilibrium where prices grow steadily and agents who hold an asset enjoy a positive surplus. This is due to the fact that the price fetched in the second period of life (when agents are old) is higher than the repayment to the bank.
Whenever the credit growth rate is smaller than the borrowing rate, r2 < r, borrowing is not sustainable. Since price growth would be smaller than the borrowing rate,22 a bubble could not be sustained, since agents would not be able to pay back their debt. Thus, the only equilibrium is a steady state where the rich asset is traded at the fundamental price.
Also note that when the initial credit limit is small and the endowment A is large, there exist bubble equilibria where the rich experience zero surplus.
Lemma 5 As a special case, when r2=r, in the rich asset market any borrowing 

level 
with prices p R*=B 
+D* and 
is an equilibrium 

where Bt+1=rBt 
and 
t t 





binds, i.e., the young who hold an asset get no 

• D* = FR for A ≥ FR and WTB 
R 

excess surplus 










• D* = A for A < FR and WTBR is slack, i.e., the young who hold an asset get excess surplus.
Proof. The poof is similar to case (i) in Lemma 4; the only difference is that
WTBR binds for A ≥ FR. 


Proposition 1 Suppose [A1], [A2], and [A3] hold. An equilibrium of this 

economy consists of price pairs 
and borrowing levels B j* for both asset 

markets j ∈ {R,P} and a downpayment D* for the rich such that, 
t 


i.if credit growth is fast, i.e., r2 ≥ r:
a)in the poor asset market, there exists an equilibrium where agents borrow
up to the maximum 
and 
. Poor agents who hold 
an asset obtain a positive surplus 

b)in the poor asset market, there also exits a steady state with price ptP=0 in all periods
21To understand this point clearly and formally. Consider an equilibrium for case (ii)(a). For the
given equilibrium price 




consider two strategies where a young pays 
at 

time t and borrows , 

vs. a young pays D=FR from the endowment and borrows nothing 



B 
=0 so that still pays same price p R=FR. Both of these strategies are equivalent in terms of the utility 

t 



t 










is paid, (7) 

generated. Since (8) does not bind and 

can be written as 





whereas when A is paid and 





is borrowed (7) 






















can be written as 







. Thus as long as the resource constraint, (8) does not 









bind, there exist a continuum of equilibria, 



where 
. 














22Actually, in the rich asset market, the price growth rate should be smaller than the growth rate of credit.
208Bulletin of Monetary Economics and Banking, Volume 21, Number 2, October 2018
c)in the rich asset market, if or if A ≥ FR, r2 > r and is

sufficiently large, then there exist balanced 
bubble 
equilibria such 


that the maximum amount is borrowed 
, 
and rich agents 


who hold an asset obtain a positive surplus (WTBR(13) is slack). The 


price is 
, implying that the resource constraint (8) binds. 


If A ≥ FR, then there also exist bubble equilibria where the rich agents obtain 

d) 
no surplus in all periods, i.e., WTBR(13) binds 
steady 
state with price 

in the rich 
asset market, there also exists a 



, ∀t. 


ii. if credit growth is smaller than the borrowing rate r2 
< r, then 


a) 
in the poor asset market there exists a unique steady state with price ptP*=0, ∀t. 

b) 
in the rich asset market, there exists a steady state (no bubble) where 


the asset is traded at the fundamental: i.e., 

, ∀t and (11) 
(WTBR) binds.
Proof. See proof of Lemmas 2 and 4 above.
When r2 > r, in the poor asset market, the only alternative to a bubble equilibrium is an equilibrium where the asset is severely underpriced, i.e., ptP=0 in all periods. In contrast, in the rich asset market, there exist equilibria where the price is equal to the fundamental. If the young rich expect no price growth, they will pay pt=FR. Thus, expectations are crucial in the equilibrium selection.
An interesting observation following from Proposition 1 is that, while the poor always obtain positive surplus in a bubble equilibrium (even though in the steady state), this is not generally true for the rich.
Note also that when credit growth is fast, rich agents also have an incentive to purchase the poor asset to speculate on price growth (even if they derive no intrinsic utility from the poor asset). However, considering [A2], the rich cannot drive the prices in the poor asset market. As for the rich asset market, since the rich can rely on their endowment and value the rich asset more than the poor, they can always price out poor agents since the poor cannot provide a down payment.23
Comparing the two types of equilibria (I).(a) and (I).(b), the poor are better off when there is a bubble in the poor asset market.
Proposition 2 For a given credit growth rate r2>r, in any bubble equilibrium such that the resource constraint binds, the bubble grows faster in the poor market compared to the rich market.
Proof. Compare the two bubbles in the poor and rich asset markets. For the
poor asset, 
and 

. This implies that the bubble grows 

at the rate r2. For the rich market, 

and 
. 
However, this means the growth rate of the bubble is less than r2 in the rich market, since D > 0.
23Note that this depends on the nature of the contract for the rich. If the contract requires a down payment, then the poor can never buy a rich house. If the contract requires no down payment, then we need to consider a scenario where the price of the poor asset grows faster than that of the rich asset. For example, if the price of the rich asset grows at r2 (for the case r2>r) and the price of the rich asset grows at rate r, at some point the poor asset will become more expensive. Thus, we need to assume that the price of the poor asset at any period is bounded by the price of the rich asset.
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The intuition of this result is simple. Since the rich can pay from their endowment, when there is a bubble in the rich market the price grows slower compared to the poor market. This result matches the empirical observations given in Figures 1 and 2.
Several assumptions of the model must be discussed. First, the
Second, when r2 > r, i.e., the credit growth rate is larger than the cost of borrowing, there exist ever growing bubbles in both markets. Considering the limited resources, these bubbles cannot be sustained forever, since at some point the bubble will become so big that all the credit of the economy will be allocated to the bubble.
V. EXTENSIONS
A. Depreciation
To illustrate some possible consequences of underpricing, consider an environment where the stock of houses depreciates over time at rate λ∈(0,1). More precisely,
if no new houses are built in period t, Suppose also that there is a producer of houses who can replenish the stock provided that the market price is above the marginal cost. We assume that the producer is a price taker (i.e., can only sell at market price) and has a constant marginal cost of producing a type j house, j∈{P,R}. Assume
Assumption [A4]. 0 < cP < cR ≤ FR
For simplicity, we restrict attention to the case where the total supply of houses cannot exceed an upper bound . For instance, this might be the case if there are building restrictions that limit the amount of land available for building. Similar to the previous sections, we assume
Clearly enough, whatever the price at time t, an old agent selling his property will receive only λptj. We compare two types of equilibrium: a bubble where the borrowing limit is binding and an equilibrium with no bubble. In general, the surplus generated by the transactions in market j∈{P,R} at time t+1 is:
(17)
210Bulletin of Monetary Economics and Banking, Volume 21, Number 2, October 2018
where ≥ 0 denotes production at time t+1. (Note that the total surplus (17) does not include the banks’ surplus, since credit is modeled in a reduced form. This is however consistent with the competitive case where banks make zero profits).
We restrict our attention to the poor asset market and consider a bubble
equilibrium where the producer supplies in every period, so that for all t. Assume also that λ is not too small, so that λr2 > r. This implies that price growth allows old agents to repay their debt with the proceeds of the house sale. Then, (17) becomes
(18)
At the other extreme, consider now an equilibrium with no bubble (so that ptP=0, ∀t and no new houses enter the market). In this case, (17) becomes
(19)
Clearly enough, (18) is always larger than (19). Moreover, in the second case limt ∞ Wt=0, so that the total surplus would converge to zero in the long run. It is also→clear that, while a bubble equilibrium may also generate a higher surplus in the rich market, so long as cR ≤ FR, the supply of rich assets will never go to zero in the long run (so that limt→∞ Wt > 0). This is because the price of the rich asset is bounded below by the fundamental FR.24
Proposition 3 Suppose Assumption [A4] holds. Then, comparing an equilibrium without a bubble (r2 < r) to an equilibrium with a bubble (λr2 > r), welfare is higher under the bubble scenario.
B. Change in LTV Policy
Now, consider a scenario where the banks no longer supply 100% LTV loans. This would mean that the poor cannot access credit, given that they have no endowment. When this change is announced, there is widespread default among the poor, since the young generation cannot receive any credit, implying that the old can find no buyers. Thus, the old at the time of the announcement are unable to pay back their debt. It is possible to enrich the model by endogenizing lending to generate endogenous credit freezes.
24Note that part of our welfare result is due to the fact that the interest rate r is fixed and does not respond to changed conditions.
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VI. POLICY IMPLICATIONS AND CONCLUSION
In terms of policy, our results suggest that there may be scope for market interventions aimed at sustaining the value of the assets held by
REFERENCES
Abreu, D., and Brunnermeier,M.K. (2002). Synchronization risk and delayed arbitrage. Journal of Financial Economics, 66,
Abreu, D., and Brunnermeier,M.K. (2003). Bubbles and crashes. Econometrica, 71,
Allen, F., and Gale, D. (2000). Bubbles and crises. The Economic Journal, 110, 236– 255.
Allen, F., Morris, S., and Postlewaite, A. (1993). Finite bubbles with short sale constraints and asymmetric information. Journal of Economic Theory, 61, 206– 229.
Barlevy, G. (2014). A
Barlevy, G. (2015) Bubbles and fools. Economic Perspectives, 39,
Blanchard., and Jean, O. (1979). Speculative bubbles, crashes and rational expectations. Economics Letters, 3,
Brunnermeier., and Markus, K. (2008). Bubbles. New Palgrave Dictionary of Economics, 8. 1.
Brunnermeier., Markus, K., and Oehmke, M. (2013). Bubbles, financial crises, and systemic risk. Handbook of the Economics of Finance, chapter 18. w18398 (Elsevier).
Conlon., and John, R. (2004). Simple finite horizon bubbles robust to higher order knowledge. Econometrica, 72,
Conlon., and John, R. (2015). Should central banks burst bubbles? Some microeconomic issues. The Economic Journal, 125,
De Long., Bradford,J., Shleifer,A., Summers,L.H., and Waldmann,R.J. (1990). Noise trader risk in financial markets. The Economic Journal, 98,
Diba, B.T., and Grossman, H.I. (1988). The theory of rational bubbles in stock prices. The Economic Journal, 98,
Ferreira, F., Gyourko,J., and Tracy,J. (2010). Housing busts and household mobility. Journal of Urban Economics, 68,
212Bulletin of Monetary Economics and Banking, Volume 21, Number 2, October 2018
Galí, J. (2014). Monetary policy and rational asset price bubbles. The American Economic Review, 104,
Harrison, J.M., and Kreps,D.M. (1978). Speculative investor behavior in a stock market with heterogeneous expectations. The Quarterly Journal of Economics, 92,
Martin, A., and Ventura,J. (2012). Economic growth with bubbles. The American Economic Review, 102,
Scheinkman, J.A., and Xiong, W. (2003). Overconfidence and speculative bubbles. Journal of Political Economy, 111,
Shleifer, A., and Vishny,R.W. (1997). The limits of arbitrage. The Journal of Finance, 52,
Tirole, J. (1982). On the possibility of speculation under rational expectations. Econometrica: Journal of the Econometric Society,
Tirole, J. (1985). Asset bubbles and overlapping generations. Econometrica: Journal of the Econometric Society,
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APPENDIX
Figure 1.
California
The figure shows the trend in the
Index Jan 2000=100
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1994 
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Home Price Index (High Tier) for San Francisco, California©
Home Price Index (Low Tier) for San Francisco, California©
Source: S&P Dow Jones Indices LLC
Fred.stlouisfed.org
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Figure 2.
The figure shows the trend in the
Index Jan 2000=100
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Home Price Index (High Tier) for Miami, California©
Home Price Index (Low Tier) for Miami, California©
Source: S&P Dow Jones Indices LLC
Fred.stlouisfed.org
Figure 7.
California
The figure shows the trend in the
Index Jan 2000=100
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Home Price Index (High Tier) for Los Angeles, California©
Home Price Index (Low Tier) for Los Angeles, California©
Source: S&P Dow Jones Indices LLC Fred.stlouisfed.org
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Figure 8.
The figure shows the trend in the
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Home Price Index (High Tier) for New York, New York©
Home Price Index (Low Tier) for New York, New York©
Source: S&P Dow Jones Indices LLC
Fred.stlouisfed.org
Figure 9.
The figure shows the trend in the
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Home Price Index (High Tier) for Tampa, Florida©
Home Price Index (Low Tier) for Tampa, Florida©
Source: S&P Dow Jones Indices LLC Fred.stlouisfed.org
216Bulletin of Monetary Economics and Banking, Volume 21, Number 2, October 2018
Figure 10.
The figure shows the trend in the
Index Jan 2000=100
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Source: S&P Dow Jones Indices LLC
Fred.stlouisfed.org
Lemma 6 Suppose [A3] is violated , then the equilibrium in the rich market is similar to that of the poor market.
i.If r2 ≥ r, then there exists an equilibriumwhere D*=A<FR. ICR does not bind and (8) binds, i.e the rich obtain a positive surplus from purchasing the asset.
ii.If r2 < r, then the rich borrow in equilibrium , and equilibrium
price is .
Proof. The proof is similar to Lemma 2. For part (ii) the only difference from the poor is any price smaller than the endowment can be satisfied as equilibrium, and no agent borrows. The intuition is similar to Lemma 2, when the credit growth is smaller than the cost of borrowing, rich agents have an upper limit for the borrowing which is determined by the binding no default condition. Thus, the
asset is traded below fundamental value since since .