M
ortgage Refinancing in
Search Equilibrium
A master’s thesis by
Ran Wang
This thesis is submitted for the degree of
Master in Economics (QEM)
Autonomous University of Barcelona, Department of Economics and Economic History (IDEA) July 23rd, 2010
Mortgage Refinancing in Search Equilibrium
Ran Wang, QEM-UAB
Thesis Supervisor: José Suárez-Lledó
July 23rd, 2010
Abstract
The paper is proposed to build a search-equilibrium model to character-ize an economy with money, consumption and property investment and to analyze the influence of refinancing activities to general welfare, asset pric-ing and monetary policy choices. The model is built on Largos and Wright (2005), Berentsen et al(2004), and Suárez-Lledó (2009) with some critical reference to Hurst and Stafford (2002). Through the paper, the refinancing activities are for the first time introduced into the search-equilibrium model and are analyzed in a tractable manner. The paper makes good prepara-tion for welfare analysis of housing mortgage and refinancing activities.
1
Introduction
After the worst post-war financial crisis in the world, nobody would easily neglect the mortgage market and their influence on the overall economy any more. Indeed, the market has grown big enough to affect everyone. In the US, the mortgage market once was about the same size of US GDP during the booming period1. The breakdown of US housing sector led to falling property prices, loan defaults and dry-up of the credit market, finally resulting in a global recession.
However, besides affecting investments and financial market, the housing fi-nance may also have indirect effect on economic activities by changing consumer spending. As pointed out by Bernanke (2007), thanks to the development of refi-nancing activities, a large amount of housing equity,which was once illiquid asset, is now set free and probably brings greater consumption smoothing2
Here I propose to use a search-equilibrium model frameworked by Largos and Wright (2005) to research on the effect of mortgage refinancing. The framework, as a recent development of its kind, enables us to analyze economy and equilib-rium in a tractable and analytical manner. The fundamental idea here is to first build a model with refinancing activities and analyze the equilibrium so that we can use it to understand the real world phenomenon. Later in further studies, we can compare the equilibrium result with that of a prototype model going without refinancing. Then we may see that how the introduction of refinancing can reshape the economic landscape.
Before running into technical details, we need to understand some fundamental knowledge about housing market and refinancing activities. There are generally two types of refinancing: term-and-rate refinancing or cash-out refinancing.
Rate-and-term refinancing Rate-and-term refinancing pays off one loan with the proceeds from the new loan, using the same property as collateral. This type of loan allows you to take advantage of lower interest rates or shorten the term of your mortgage to build equity faster.
Cash-out refinancing (equity extraction) Cash-out refinancing leaves you with cash above the amount needed to pay off your existing mortgage, closing costs, points and any mortgage liens. You may use the cash for any purpose.
However, this categorization does not show the rationale underlying the choices. From a household’s point of view, a refinancing choice can come out of many situations. Specifically, there are three major purposes that are essential to our model here:
Purpose 1. Taking advantage of a decreased mortgage rate. Although mortgage rate is usually fixed after a mortgage deal has been made, by refinancing agents are always able to take advantage of an interest rate cut, and end up in a loan with lower interest payment.
Purpose 2. Taking advantage of inflating housing prices. While the mortgage in a property is usually fixed in the short term, housing prices can fluctuate much. 1According to Federal Reserve Statistics Release, there was totally 10.2 trillion dollars of US
mort-gage debt outstanding by the end of 2006, and at the peak it even reached 14 trillion dollars.
2Bernake also claimed, “This smoothing in turn should reduce the dependence of their spending on
current income, which, by limiting the power of conventional multiplier effects, should tend to increase macroeconomic stability and reduce the effects of a given change in the short-term interest rate. These inferences are supported by some empirical evidence.” See Bernanke(2007)
An inflated property price can directly lead to a gain on housing equity while its mortgage remains the same. Actually many abused subprime mortgage programs before the crisis were reported to be designed so that the borrowers could use the property gain to pay for the mortgages themselves, which would otherwise never be afforded by the borrowers.
Purpose 3. Usually taken as an empirical anomaly3, utility maximizing
house-holds would still choose to refinance in a world with stable interest rate and property prices. In my opinion, this probably comes from an intertemporal choice of consumption smoothing.
To understand these characteristics, we have to understand how an economy goes with collateralized debt. A lot of literature have come out in recent years. Usually in these models borrowers and lenders have different discount rates (im-patience rates) and thus heterogeneity is introduced (Bernanke and Gertler (1999), Iacoviello (2005), Kiyotaki and Moore (1997)). Topics such as optimal monetary policy with collateralized borrowing receive plenty of attention. Monacelli (2008) gives a good review of the topic, and the whole book of Campbell (2008) provides abundant source of different perspectives. While Monacelli (2009) is a classical paper providing a New Keynsian viewpoint of collateral constraint, Calzaet al
(2005,2009) actually provides a very comprehensive analysis of housing finance, relevant institutional characteristics and their economic influence. However, the framework they use leads to computational results, and thus have relatively poor analytical property.
Meanwhile, a framework developed by Largos and Wright (2005) has pro-vided a considerable alternative. Its analytical nature allows us to understand the coexistence of multiple assets as media of exchange while keeping the whole sys-tem tractable. Several papers have been built on this framework. For example, Berentsenet al(2004) uses the framework to analyze the role of financial interme-diaries in welfare improving; Kocherlakota (2003) uses the framework to explain why nominally risk-free bonds are essential in monetary economies; Geromicha-loset al(2007) explores a monetary model in which a real asset is valued for its rate of return and for its liquidity. Later, influenced by Berentsen et al(2004), Suárez-Lledó (2009) extends the model to analyze heterogeneous collateralized borrowing, laying down good ground for further and more specific analysis of housing equity. However, as far as I can see, nobody has ever used the framework to further analyze mortgage refinancing and its role in welfare and market equilib-rium. At the same time, Hurst and Stafford (2002) use a RBC model with fixed housing sector to analyze refinancing. Inspired by that, I try to incorporate the same setting into the search-equilibrium framework.
Therefore, in this paper, I use the search-equilibrium model frameworked by Largos and Wright (2005), and take many of the model assumptions the same as Suárez-Lledó (2009). However, there are some crucial changes that makes this paper different, and prepare it for refinancing modelling: Idiosyncratic shock on collateral prices has been replaced by housing value appreciation and mortgage rate benefit (collateral, or housing equity in this model, works as a shield against the mortgage interest rate payment, given the housing sector value fixed); the uni-fied interest rate charged for loans within each period has been replaced by a more subtle system where the refinancing rate for each agent equals his mortgage rate plus a monetary shift rate, thus monetary authority cannot change interest rate charged to everybody directly anymore, but can only shift the overall interest rate 3Hurst and Stafford (2002)
level and each agent would have their own rate from the new prime rate. Then we try to analyze the equilibrium, including static analysis on housing appreciation and different monetary policies.
2
The Environment
As claimed in the introduction, the model here utilizes a typical search-equilibrium framework from Largos and Wright (2005). There are a [0,1] continuum of agents who live for indefinite number of periods and have a discount rate for future utility at rate β ∈ (0,1). Time is discrete and each round (period) consists of three subperiods, or alternatively, three markets.
For each period, there is an endowed property sector with total value (in terms of consumption goods) ofH, which is fixed. Before the first market, agents have been into a mortgage contract(H −a), where ais the housing equity paid by agents themselves. Generally, awould always be smaller than H, and the gap is filled with the housing mortgage(H −a), which is set in the third market of the previous round and fixed across the first and second markets of the current round. For this mortgage, a idiosyncratic mortgage rate r is charged for each round. Besides, to provide incentive for agents to take the loan (H −a) rather than makea=Hto avoid the mortgage payment, there is an additional benefit of
δfor each unit of loan taken in(H−a). In real world, typically nobody pays 100% of his housing in the first place, thus it is reasonable to assume that, whatever the distribution ofris,δis always larger than its highest possible value, i.e.δ >rˆ, so that the incentive for mortgage exists for everybody.
In the first market, agents each decide how much they want to refinance within the period, in terms of additional loan l 4. In the second market, agents trade
in a decentralized market, while also supplying labor and consuming. Then, in the third market, agents trade in a centralized (Walrasian) market, and at the same time decide the labor supplyh, consumptionx, and the real estate equitya+1(thus
the mortgage(H−a+1)) for next period, while redeeming the investment of the
current round back into consumption goods, termed bya.
At the beginning of every period, agents receive a preference shock and they then find out that they are divided into two groups:sof them want to produce and sell (producers) in the decentralized market, while others want to be consumers. However, out of (1−s) consumers, only n of them successfully match with a producer. So in the second market, there will be three kinds of agents: sellers, buyers, and others. For buyers, they would need money to consume in the second and third market, while producers do not need it. If the economy goes without refinancing, agents may need to borrow more (larger mortgage) before each period in case they turn out to be buyers in this period. However, when there is chance to refinance with additional loan, buyers can borrow more within the period, while sellers are able to go with low mortgage all the time. The borrowingl is called refinancing loan or equity extraction. The interest rate for this loan i is called refinancing rate, and basically should follow a similar distribution asr. We can write it as
i=r+ ∆i
4We will see below that, given that the value of housing sectorH is fixed, additional loan equals
to cash-out refinancing. Meanwhile, we actually can use the negative loan to represent the mortgage prepayment, wherel <0.
. Where∆idenotes the shift of mortgage rate (from the original mortgage rate to the current refinancing rate), due to monetary policy change or macroeconomic shock. If there is no change in mortgage rate, then refinancing rate would be equal to mortgage rate, and this would capture the fact (Purpose 3) that with stable mortgagerandH, some people would still choose to refinance.
Therefore, within each round, the mortgage is fixed, and households can only change it by refinancing . While they do it, there will be two types of loans within each round of the economy, one fixed and one flexible. The mortgage rate r is fixed within each round (followingF(r) across the rounds), whileiis subject to change, possibly influenced by overall interest rate and monetary policy.
Following the similar setting as Largos and Wright(2005),we assume that pref-erences of agents are given by the functionU(q, Q, x, h) =u(q)−c(Q)+U(x)−h
whereqdenotes the consumption and labor in the decentralized,Qthe production in the decentralized market, x the consumption in the centralized market, h the labor supply in the centralized market, respectively. We also assume5thatu,cand
U are twicely continuously differentiable withu(0) = 0, u0 > 0, u0(0) = ∞,
U0 >0,u00 <0,c0 >0,c00 ≥0,c0(0) = 0andU00≤0. To simplify the proof in the following, we also takec(Q) = Q, which is not essential here. Suppose that there existsx∗ ∈(0,∞)such thatU0(x) = 1withU0(x∗)> x∗.
Before directly going to the equilibrium analysis, we still have two issues to clarify. The first one is the collateral constraint for the refinancing loan within the period. Partially inspired by Calzaet al(2005, 2009), I assume a downpayment requirement on the housing mortgage: for fixed housing sectorH, there is a down payment limitχfor each mortgage, within and across each round. Thus
a≥χH l≤ a−χH
φ(1 +i)
However, one thing worth noting is that, χ is normally around 0.2 in the U.S. market6 and thus we can always make a reasonable preassumption that the first constraint is not binding with a low enough value ofχ. Here the only constraint that matters is the second one above.
Another issue is the role of money. Fiat money, perfectly divisible and stored in any amountmt ≥ 0, is modeled here as a medium of exchange to facilitate trade in the second and the third market, while credit, in terms of mortgages and refinancing loans, exist at the same time. Following Suárez-Lledó (2009), we assume that money supply grows at rate ofγin each period, and the inflation arises in the third market where the real price of moneyφchanges toφ+1. Suppose the
balance of money isMt, then in stationary equilibria, the real balance is: φM =φ+1M+1≡z
To focus on the role of refinancing activities here, we assume the money supply grows at constant rate, thus
φ/φ+1 =M+1/M =γ
5the assumed conditions here will turn out to be very handle in obtaining an equilibrium for the model 6See Calzaet al(2005,2009)
3
The Equilibrium with Refinancing
Now, we start to analyze the equilibrium with refinancing loans through backward deduction starting from the third market of each period.
Third Market
As describe in the previous section, in this market agents redeem their housing equity, pay back the mortgage(H−a)and refinancing loanl, and make decision for the mortgage of the next period (H −a+1), while maximizing their utility
U(x). So for the third market, the value function is
V3(m, l, a) = max
x,h,m+1,a+1
[U(x)−h+βV1(m+1, a+1)] (1)
s.t. x+φm+1+a+1=h+φm−φ(1 +r+ ∆i)l+a+ (δ−r)(H−a)
a+1 ≥χH (2)
Here,ris the mortgage rate charged against the housing mortgage(H−a). How-ever, if there is only cost r on mortgages (H −a), individuals would not have any incentive to borrow, thus probablya would always equalH. To avoid this situation, we consider that additional unit mortgage borrowing has a benefit ofδ7. Obviously, here the housing equity ais used against mortgage interest pay-ment, and thus putting moreawould actually produce a benefit,i.e.(1 +r−δ)a
8, and thus the actual budget constraint is
x+φm+1+a+1 =h+φm−φ(1 +r+ ∆i)l+ (1 +r−δ)a+ (δ−r)H
(3) While keepingrfollowing a continuous distributionF(r), we actually model
(1 +r−δ)in the same way asδin Suárez-Lledó (2009), but hererhas a concrete meaning of mortgage rate. Then why shall it follow a probabilistic distribution rather than have a deterministic value? In fact, in the real world, although there is always a unified prime rate for mortgages, individuals usually take different mortgage rates from banks and lenders depending on their credit records, incomes and other personal conditions. Thus it is quite reasonable to assume that they have their own mortgage rate and that follows a distribution, symmetric or asymmetric, which may be even seen in real world data.
7Actually in the equilibrium where there is onlyrwithoutδ, there may be valuea∗< H, however,
we do not know the value ofH, so the result would be ambiguous. To avoid the uncertain and probably meaningless result, it is reasonable to add itemδhere. One may argue that there is benefit related toa
as well if we consider the benefit of housing, so the overall benefit should beδa+δ(H −a) = δH
instead, and there would still be no incentive to borrow, even we write it into the utility function. In fact, the fundamental problem relies on the assumption that the housing valueH is fixed, rather than equals the sum of mortgage and housing equity, otherwise there is clearly benefit of borrowing additional mortgage due to larger housing (e.g. more living room). However, assuming the alternative would make H nonconstant and complicate the equilibrium problem, for simplicity we just addδhere as a compromise.
8The benefit ofais quite obvious as it allows agents to store wealth across periods and also to earn
an interestr. However, there is even more important benefit of it: as collateral, it allows the agents to borrow within the period and finance their consumption.
Substituting the budget constraint (3) into (1) gives V3(m, l, a) = max x,m+1,a+1 [U(x)−X−φm+1−a+1+φm−φ(1 +r+ ∆i)l+a + (δ−r)(H−a) +βV1(m+1, a+1)] =φ[m−(1 +r+ ∆i)l] + (r+ 1−δ)a+ (δ−r)H + max x,m+1,a+1 [U(x)−X−φm+1−a+1+βV1(m+1, a+1)] (4) However, as we have explained before,χis normally around0.2and thus we can always make a reasonable preassumption that the first constraint is not binding with a low enough value ofχand there exists an interior solution.
From the first order conditions, we can get
U0(x∗) = 1 (5)
φ−1≥βV1m (00=00, if m >0) (6)
1≥βV1a (00=00, if a >0) (7) Actually (6) comes from the deduction ofφ ≥ βV1m+1, and (7) comes from the deduction of1≥βV1a+1. Thus,V1mis the marginal value of money taken into the first market for the current period t, andVa
1 is the marginal value of investment
on real estate equity in the first market for the current periodt. We can imagine that individuals are making trade off between the two assets, and thus the marginal value of them should be equal, in terms of one unit of real good, if there is interior solution.i.e.
V1a=V1φm=V1m×φ
Furthermore, we can obtain the envelope conditions as following:
V3m=φ; V3l=−φ(1 +r+ ∆i); V3a= 1 +r−δ. (8) Note that obviously the value function V3 is linear in m, a, l. Thus Largos and
Wright (2005) proves that there are no wealth effects, and a degenerate distribution of money and the asset is achieved. We actually directly use this result in the setting of our model so that there is no concern about the distribution of the money at all.
Second Market
As described in the Environment section, there are three types of agents entering this stage, namely, buyers (b), sellers(s) and others(o). In this market, sellers and buyers bargain bilaterally in decentralized market. To simplify the model we as-sume the buyers have full bargaining power. For others (o), their value for the second market should equal their value in the third market since there is no trans-action happening for them.For buyers they pay out cash but obtain consumption goods; for sellers, they sell goods and obtain cash. Thus
V2o(mo, lo, a) =V3(mo, lo, a) (9) V2b(mb, lb, a) =u(q) +V3(mb−d, lb, a) (10) V2s(ms, ls, a) =−c(q) +V3(ms+d, ls, a) (11) wheremj =m+lj, j =b, s. mis the same for all agents since the distribution of money is degenerate. dmeans the money paid to buyqconsumption. Thus the
bargaining problem would be
max
q,d u(q) +V3(mb−d, lb, a)−V3(mb, lb, a) s.t.−q+V3(ms+d, ls, a)−V3(ms, ls, a)≥0
d≤mb (12)
By the linearity ofV3and further assumptionc(q) =qwe actually need to solve
max
q,d u(q)−φd s.t. −q+φd≥0
d≤mb (13)
By Lagrange method we can have
if q∗ ≤φmb, then ( q=q∗, d= qφ∗, if q∗ > φmb, then q= ˆq, d=mb, (14) whereq∗s.t.u0(q∗) =c0(q∗) = 1
Here, similar to the proof of Suárez-Lledó (2009), we can prove that, forγ > β, agents will carrym = m∗, and a = χH, and q(z, χH) = q∗. Thus nobody would need to refinance though they still keep mortgage (actually they would pay just the required downpayment in the first place). Ifγ > β, thenm < m∗, and agents would like to borrow so as to get closer toq∗. This meansa > χH and
q(z, a) = ˆq < q∗. This result is essential when we prove the proposition of the existence and the uniqueness of the stationary monetary equilibrium later in this paper.
Meanwhile the F.O.C of the problem reads:
φ[u0(q)−1] =λd (15) As Suárez-Lledó (2009) have proved, in equilibrium, ifi > o, we always have
d=mbandq=φmb. Thusu0(q)>1and all buyers are constraint. Besides, from envelope theorem, we obtain that:
V2lj =−φ(1 +r+ ∆i), V2aj = 1 +r−δ, j=b, s.
V2ms =V3m=φ (16)
V2mb =V3m+λq =φu0(ˆq) =φu0(φmb)
First Market
As described in the environment section, before the starting point of this mar-ket, agents receive their types: either consumers or producers. Producers will become sellers, but consumers may find their matches to become buyers, or not. So in the beginning of the market, there are already three types of people, buyers (j = b), sellers (j = s) and others (j = o). The agent who enters the first mar-ket withmunits of money andaunits of the real assets invested in the property market equity. That also means he comes with a mortgage debt of value(H−a). In the first market, buyers may need to borrow more to finance their consumption, at a refinancing ratei = r+ ∆i, where∆is fixed so that the refinancing ratei
follows the similar distribution as mortgage rater, in the sense that, for the same agent, the issues that makes his mortgage rate deviate from the prime rate in cer-tain direction will still matter in the same way in his refinancing activities. This additional borrowing is equivalent to equity-extraction refinancing. On the other hand, for sellers and others, money are not necessary while lending it out means interest ati. When they lend their money out, the overall debt they have decrease, and this actually equals to prepayment refinancing.
Thus the value function is:
V1(m, a) = Z
[nV2b(mb, lb, a) +sV2s(ms, ls, a)
+ (1−n−s)V2o(mo, lo, a)]f(δ)dδ (17) wheremj =m+lj, j=b, s, o.
As pointed out by Suárez-Lledó (2009), the problems for sellers and others (consumers who do not match) are the same, and different from that of buyers. For lenders and borrowers who do not have match in the second market (j=s, o), they solve the following problem:
max lj
V2j(mj, lj, a) (18) s.t. 0≤m+lj
Then from the F.O.C., we obtain that
V2mj +λj +V2lj =o
where λ is the Lagrangian multiplier corresponding to the (lending) constraint. It is quite obvious that these people would lend out all the money they have to earn an interest at the refinancing rate. This activity actually equals to mortgage prepayment by refinancing. Thus in equilibrium, we have λj > 0. Using the results (16) from the second market, we obtain
φ+λj−φ(1 +r+ ∆i) = 0, j=s, o.
Borrows have an additional collateral constraint for the equity-extraction refinanc-ing max lb V2b(mb, lb, a) (19) s.t. 0≤m+lb l≤ a−χH φ(1 +r+ ∆i) (20)
Then from the F.O.C., we have
V2lb−λlb+V2mb +λb= 0
Using the results (16) from the second market and the fact that λb = 0 since m+lb >0for borrowers, we obtain
φ[u0(mbφ)−(1 +r+ ∆i)] =λlb (21) Identically,
So
φ[u0(z+ a−χH
(1 +r+ ∆i))−(1 +r+ ∆i)] =λlb
Now we want to show that, under certain assumptions, some borrowers are collateral-constraint and others are not.
We define
g(r) =u0(z+ a−χH
(1 +r+ ∆i))−(1 +r+ ∆i)
Since z, a are chosen to optimize the borrowers’ problem, we can always make sureλlb≥0, so we would always chosez, aso that
g(r)≥o
Furthermore, for given(z, a)∈R2+, sinceu00< o, so
u0(z+ a−χH
(1 +r+ ∆i))≤u
0(z)
asrincreases. That is to say,u0(z+(1+a−rχH+∆i))is bounded from above. However,
(1 +r+ ∆i)is obviously not bounded from above. Thus there would always ber
big enough such that
g(r)<0
So there must∃rc, s.t.
g(r) = 0
We assume thatg(r)is monotonic within(0,rˆ), otherwise no certain result can be obtained. Actually this is not such a strong assumption as it may appear to be. It can be proved that for some simple concave utility functions such asu(x) =x1/2, this assumption is not very hard to be met. This assumption also ensures that a possible critical value would be unique, if there is any. Multiple critical values would make the model indeterminant.
Now we show that ifg0(r)<0, there will be contradiction in the result. Ifg0(r)<0and we haverc∈ (0,rˆ), we would have two types of borrowers, constraint and unconstraint. Forr < rc, agents would haveg(r) = 0and thus from (21) we haveλlb = 0and are always constraint by the collateral-constraint; for r > rc, agents would haveλlb > 0 and thus are unconstraint. Then their refinancing loans would belbcandlub respectively,
(
lbc= φ(1+a−rχH+∆i), if r < rc lbu= φ(1+a−rχH
c+∆i), if r ≥rc
However, this is contradiction since forr ≥ rc, borrowers can borrow less than a−χH
φ(1+rc+∆i).
Thus, we should haveg0(r) <0, and thus we have uncontraint and constraint borrowers with their refinancing loanslbuandlcbrespectively,
(
lbc= φ(1+a−rχH+∆i), if r ≥rc lbu= φ(1+a−rχH
c+∆i), if r < rc
Thus, under the assumption thatg(r)0 >0, which should be met if there exists a feasible critical value, some borrowers with low mortgage rate will be uncon-straint since they have larger discounted borrowing budget. For them the first order condition reduces to
other borrowers with high mortgage rate will be constraint since they really want to borrow as much as possible given their low discounted borrowing budget. For them the first order condition reduces to
u0(q)>1 +r+ ∆i
On the other hand, we have the refinancing market equilibrium equation:
Z rc(z,a) 0 a−χH φ(1 +rc+ ∆i) f(r)dr+ Z rˆ rc(z,a) a−χH φ(1 +r+ ∆i)f(r)dr= 1−n n z (22) Moreover, by differentiating (17) we can have
V1m =φ Z [nu0(ˆq) + (1−n)(1 +r+ ∆i)]f(r)dr (23) V1a= Z {n[ u 0(ˆq) 1 +r+ ∆i−1] + (r+ 1−δ)}f(r)dr (24)
Combine the results (6) and (7) from the third market, we can obtain the liq-uidity premium equation:
1−β(1 + ¯r−δ) =β
Z
n[ u
0(ˆq)
1 +r+ ∆i−1]f(r)dr (25)
and equilibrium conditions:
γ−β(1 + ¯r+ ∆i) β(1 + ¯r+ ∆i) = Z ˆr rc(z,a) n[ u 0(ˆq) 1 +r+ ∆i−1]f(r)dr (26) 1−β(1 + ¯r−δ) β = Z ˆr rc(z,a) n[ u 0(ˆq) 1 +r+ ∆i−1]f(r)dr (27)
Following the same logic as Suárez-Lledó (2009), now we define a stationary equilibrium for this economy, and then prove the equilibrium exists and is unique.
Definition of the Equilibrium Given a distribution functionF(r)and an interest rate shift∆i, a symmetric stationary monetary equilibrium is a choice of real balances,z, and housing equitya, that satisfies (22) and (27), and a growth rate of moneyγthat comes in consistency with the monetary policy of interest rate shift∆iand also meets (26).
Proposition (Existence and Uniqueness of the Monetary Equilib-rium) There exists a unique symmetric stationary monetary equilibrium with credit. Equilibrium consumption is decreasing in∆iandγ, and also decreasing in the value of mortgage rater. Generally the equilibrium consumption is smaller than the efficient value, i.e.,qˆc<qˆu< q∗, withqˆ→q∗asγ →β.
Now we prove the Proposition.
Observing (26) and (27), we can find that, once zandahave been fixed, the value ofγis determined as well. To prove the existence and the uniqueness of the solution of (22), (26) and (27), we try to show that in the plane(z, a)they represent two monotonic curves that intersects at one single point. Let us write (27) as
F(z, a) = Z ˆr rc(z,a) n[ u 0(ˆq) 1 +r+ ∆i−1]f(r)dr− 1−β(1 + ¯r−δ) β = 0
From implicit function theorem we have dz da =− ∂F(z,a) ∂a ∂F(z,a) ∂z =− Rˆr rc(z,a)n[ u00(ˆq) (1+r+∆i)2]f(r)dr Rrˆ rc(z,a)n[ u00(ˆq) (1+r+∆i)]f(r)dr <0
Besides, by applying total differentiating to (22) we obtain that
dz da = n 1−n[ Z rc(z,a) 0 1 1 +rc+ ∆i f(r)dr+ Z rˆ rc(z,a) 1 1 +r+ ∆if(r)dr]>0
Thus, indeed, the two curves have slopes of different signs. However, to prove that they actually intersect at the first quarter of the plane(z, a)we have to show that the curve with positive slope actually starts below where the curve with negative slope starts in the first quarter of the plane. From (22), by makingz= 0, we have
Z rc(z,a) 0 a−χH φ(1 +rc+ ∆i) f(r)dr+ Z rˆ rc(z,a) a−χH φ(1 +r+ ∆i)f(r)dr= 0
Thusa−χH = 0. However, from (27), givenz= 0, we have
1−β(1 + ¯r−δ) β +n= Z rˆ rc(z,a) u0(1+a−rχH+∆i) 1 +r+ ∆if(r)dr
Because the left hand side is definitely positive, andu0 is decreasing, so we have
(a−χH) can always factor out, and basically this means here a−χH > 0. Thus the two curves described above intersects at one single point and there exists unique equilibrium.
The convergence of the equilibrium actually has been proved in the analysis of second market.
4
Implication and Conclusion
Having obtained liquidity premium equation and proven the existence and the uniqueness of the stationary monetary equilibrium for the model, we have suc-cessfully built a search-equilibrium model with housing sector and mortgage re-financing, and thus have laid good ground for analysis on the role of refinancing and mortgage in a model of money and its interaction with the monetary policy. Although we may not be able to go so far as to discuss the welfare effect and the optimal monetary policy in this paper, we do have some direct implications to understand refinancing.
The first implication is about the monetary policy. Suppose originally the over-all interest rate keeps constant, and each agent gets their own mortgage raterwith distributionF(r)falling around the prime rate of housing loans. When the mon-etary authority tends to decrease interest rate, or the public have such expectation that the overall interest rate may decrease in the short term,∆i <0and from
g(rc) =u0(z+
a−χH (1 +rc+ ∆i)
)−(1 +rc+ ∆i) = 0 (28) we haverc+ ∆iwould be a fixed value that solves the equation. When∆i <0, rc will change to a higher valuer0c. This means that more agents will be uncon-straint since the critical value increases and for the people withr∈(rc, r0c)would
not be constraint anymore on their refinancing borrowing. This reasoning is actu-ally consistent with common sense and empirical results9because the lowering of mortgage rate would normally lead to the booming of refinancing activities. This implication corresponds to the Purpose 1 that we described earlier in this paper.
The second implication is about the rising housing prices. Suppose within the period, the house prices increase. As the mortgage amount for each houseH is fixed before the beginning of the period, the rising house prices would actually lead to an increase in the real housing equitya. That is what lots of house buy-ers speculate on: borrow money to buy the house and wait for the price to rise. Similarly, from (28), we can find, if we keeprcfixed,
u0(z+ a−χH (1 +rc+ ∆i)
)−(1 +rc+ ∆i)<0
since u00 < 0 and a increases. Because we already have g0(r) > 0, the new
rc0 would be higher than rc. So the rising of house prices would lead to more refinancing activities as well. This goes with the intuition we have as well, and corresponds to the Purpose 2.
The last but not the latest, even there is no interest rate and no inflating housing sector, according to our model,there is still refinancing from borrowers to finance their consumption ifγ > β. This explains the Purpose 1.
Based on the framework of this paper, we can consider some further studies. The simple implications we have above can help us understand the effects of refi-nancing to some what extent. However, to make a complete evaluation for the role of refinancing activities, we may need to compare the results from the three cases with a more prototype model without refinancing market: Suppose there are totally two markets. Buyers are not allowed to borrow before entering the decentralized market, and the seller will always keep their money into next market rather than lend them out. This case is more similar to Largos and Wright (2005). Another important issue is to analyze and propose the optimal monetary policy to deal with refinancing activities based on a thorough welfare analysis.
To sum up, the paper builds up a search equilibrium model to analyze a mone-tary economy with housing investment and introduces the mortgage and refinanc-ing activities into the model, which extends the application of this framework to new area and help us understand the role of refinancing in the economy. Although we are not able to further analyze the optimal monetary policy and social welfare, the model does lay good ground for analysis of that kind.
5
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