Forthcoming in the Japanese Economic Review

Land Price, Collateral, and Economic Growth #Masaya Sakuragawa∗
Yukie Sakuragawa **Abstract
This paper extends Kiyotaki and Moore (1997, JPE) to an endogenous growth model andinvestigates dynamic properties of a growing economy with binding credit constraint whenland is used not only as an input of production but also as collateral. There exists abalanced-growth path in an economy with binding credit constraint. In response to aonce-and-for-all productivity shock, the developed model shows the propagation mechanism
among output, capital, bank credit, and the land price in terms of the growth rate. The model’s
tractability allows us to derive interesting qualitative and quantitative findings.
JEL Classification Numbers: G30, O40#

We are greatly thankful to an anonymous referee, Naohisa Goto, Kaoru Hosono, Hidehiko Ishihara,
Tokuo Iwaisako, Shin-ichi Kitasaka, Tsutomu Miyagawa, Kiyoshi Mitsui, Kiyohiko Nishimura, Etsuro
Shioji for their valuable comments. Especially, we appreciate the referee because the referee’s
suggestion drastically improved this paper.
∗
Keio University, Department of Economics, 2-15-45 Mita, Minato-ku, Tokyo, 108-8345, Japan,
Tel&Fax: +81-3-5427-1832, E-mail: masaya@econ.keio.ac.jp**
Atomi University, Faculty of Management, 1-9-6 Nakano Niiza-shi Saitama, 352-8501, Japan, Tel:
+81-48-478-4110, Fax: +81-48-478-4142, E-mail: sakuraga@atomi.ac.jp

11. Introduction
It is widely recognized that land prices have played an important role in influencing credit
market conditions, corporate investment, and output in Japan. Kiyotaki and Moore (1997,
hereafter K&M) formulate a theoretical framework in which the appreciation in the collateralvalue mitigates credit constraints and thus amplifies fluctuations in investment and output
when land serves as collateral for loans.1 A number of empirical researches provide evidence
to support the important role of the collateral channel, including Ogawa, Kitasaka, Yamaoka
and Iwata (1996), Ogawa and Kitasaka (1998), and Ogawa and Suzuki (1998).
Over the post-war period, annual growth rates of GDP and land prices look like showing a
positive correlation in Japan. Until the period of the asset price boom at the end of the 1980s,
the persistent appreciation in land prices has been associated with faster economic growth.
Meanwhile in the 1990s when one may call the “Lost Decade”, the persistent depreciation inland prices has been associated with the slowdown in economic growth. The average annual
growth rates of GDP are 4.9% in the 1970s, 3.8% in the 1980s, and 1.5% in the 1990s, and theaverage growth rates of nationwide land prices are 9.2%, 6.1%, and -1.7%, respectively.
Observing this time-series behavior motivates us to understand how the K&M transmission
mechanism works in the growing economy.
In this paper we extend the K&M model to an endogenous growth model and investigate
dynamic properties of a growing economy with binding credit constraint when land is used not
only as an input of production but also as collateral. A balanced-growth path (BGP) is proved
to exist when the credit constraint is binding when entrepreneurs who borrow money discountthe future more than investors who lend to them. The equilibrium interest rate is determined tobe smaller relative to an economy with no binding credit constraint. When borrowers discount1

Since the seminal work by K&M, ample studies investigate the interaction between asset prices, credit,
and output in the economy with binding credit constraint, including Edison et al. (2000) and Chen
(2001)．

2

the future more and thus lenders discount the future less, the greater credit supply has to be
adjusted by the smaller equilibrium interest rate than otherwise. The resulting discrepancy
between the equilibrium interest rate and the marginal product of capital is, however,consistent with the equilibrium in the credit market because the borrowers’ debt capacity islimited by the liquidation value of land. This aspect of the equilibrium is contrasted with othermodels of binding credit constraint, including K&M, Kiyotaki (1998), and Aghion et al. (1999),
all of which identify the economy with binding credit constraint as the one with a smaller
interest rate that is given exogenously.
In response to a once-and-for-all productivity shock, our model exhibits the propagation
mechanism among output, capital, bank credit, and the land price in terms of the growth rate.
This finding is contrasted with K&M that derive the propagation mechanism in terms of levels.
The model’s tractability allows us to characterize the whole process of business fluctuations.
On impact, capital, the land price, and the equilibrium real interest rate show the procyclicalbehavior. Additionally, the model’s tractability allows us to derive not only the qualitative but
also the quantitative finding. The growth-enhancing effect of capital and output is shown to begreater in magnitude and is more persistent in the economy with binding credit constraint forplausible parameters.
By the introduction of a foreign asset into the model, the equilibrium with binding credit
constraint is divided into two regimes, depending on parameters. In one regime, theequilibrium shows a BGP with the endogenous determination of the interest rate, while inanother regime the interest rate is tied with the world interest rate and the equilibrium does notshow a BGP. The former regime is more likely to emerge if either the productivity is high or
the world interest rate is low.
This paper is organized as follows. In Section 2 we present the basic model. In Section 3
we study a benchmark economy with no borrowing constraint. In Section 4 we investigate an3

economy with binding credit constraint. In Section 5 we study the impacts of a one-tine
productivity shock on the dynamic of the economy. In Section 6 we introduce the foreign asset
with a constant interest rate into the model. Finally we conclude.

2. The Model
Consider an economy consisting of two types of agents who live infinitely,“entrepreneurs” and “investors”. The measures of entrepreneurs and investors are both unity. A
significant difference between our model and K&M model is the production technology. We
describe the production function as

Yt = At −1 K tα−1 L1t−−α1 K t1−−1α ,(1)where Yt is the output of production at period t , K t −1 and Lt −1 are capital stock and land
which are used as inputs for production at period t − 1 , K t −1 represents the aggregate capitalstock at period t − 1 that captures technological externalities (e.g. Romer, 1986), and At −1 isthe total factor productivity. Capital does not depreciate.
∞

The preference of each of investors is described as∑ β log C , where C is
t

t =0

t

t

consumption at period t , β ∈ (0,1) is the discount factor for future utilities, and Et is the
expectation formed at period t . At each period, there is a competitive one-period credit
market, in which one unit of good at date t is exchanged for a claim to Rt +1 units of good atperiod t + 1 .
On the other hand, the preference of each of entrepreneurs is described as∞∑ (θβ ) log C , where C
t

t =0

E
t

E
t

is the consumption of the entrepreneur at period t , and θ ≤ 1

is assumed. The case for θ < 1 captures a situation that entrepreneurs are more “myopic”
than investors.
Each of entrepreneurs is initially endowed with K 0 units of capital and one unit of land,4and each of investors is initially endowed with W0 units of the final good. The total amountof land is normalized unity. Only entrepreneurs have access to the production technology.
Investors earn interest income by lending their wealth to entrepreneurs. We assume thatθβ (1 + αA) > 1 . As is obvious below, this assumption allows us to focus on a growing
economy.
We make a critical assumption about financial arrangement. We assume that there is noenforcement mechanism to fulfill financial contracts between debtors and creditors. In thissociety, lenders cannot enforce on borrowers to repay their debt unless the debts are secured. Inorder to secure their debt, creditors can collect land that the debtor holds. Creditors cannot
seize output or capital of their debtors. In this environment, anticipating the possibility of theborrower’s strategic default, the creditor limits the amount of credit so that the debt repayment
due at the next period will not exceed the value of land that the borrower possesses.

3. Benchmark Economy
We start with the analysis of a benchmark economy with no enforcement problem. In thisfrictionless economy BGP exists only when both entrepreneurs and investors discount futuresat the same rate. Hence we focus on the case for θ = 1 .
Throughout this section and Section 4, we investigate an economy in the absence of
uncertainty concerning the technology shock and assume that At = A for all t ’s.
In the frictionless economy, each of entrepreneurs chooses a vector {CtE , Bt , K t , Lt }t∞=0 to∞maximize∑ β log C
t

t =0

E
t

subject to the flow-of-fund constraint (FFC), given by

AK tα−1L1t −−α1 K t1−−1α + Bt − Rt Bt −1 ≥ K t − K t −1 + Qt ( Lt − Lt −1 ) + CtE ,(2)given K 0 > 0 and L0 = 1 . Note that Bt is the amount of borrowing, Qt is the land priceat period t and Rt is the interest rate that is determined at t − 1 and is promised to repay att . The sequence of events is described Figure 1.
5∞∑ β log C subject to the

Each of investors chooses a vector {Ct , Wt }t∞=0 to maximize

t

t

t =0

budget constraint, given by

Wt = RtWt −1 − Ct ,(3)given W0 > 0 , where Wt is the asset that the investor holds at the beginning of period t . We
obtain Rt = 1 + αA , using K t = K t and Lt = 1 on the ground that entrepreneurs arehomogeneous and that entrepreneurs only demand land. Investors earn interest income bylending to entrepreneurs at the interest rate αA . The first-order condition for land implies(1 − α ) AK t + Qt +1 = (1 + αA)Qt ,(4)

where we again use K t = K t and Lt = 1 .
We characterize a BGP in the frictionless economy by

K t +1 Wt +1== β (1 + αA) ,
Kt
Wt

(5)

given the two transversality conditions, lim β t (Wt Ct ) = 0 and lim β t ( K t Ct ) = 0 .
t →∞

t →∞

Solving (4) forward, using (5) and the no-bubble condition for the land price,lim(1 + αA) − t Q t = 0 , we finally obtain
t →∞

Qt =

(1 − α ) A
Kt .
(1 + αA)(1 − β )(6)Equations (5) and (6) jointly imply that there exists a BGP with a growth rate of β (1 + αA)
on the wealth of investors Wt , capital invested by entrepreneurs K t , and the land price.

4. Economy with binding Credit Constraint
Now we turn to the analysis of an economy in which there is potentially an enforcement
problem. As will be made clear, a BGP will exist also when θ < 1 . Each of entrepreneurs
chooses a vector {CtE , Bt , K t , Lt }t∞= 0 to maximize

6∞∑ (θβ ) log C
t

t =0

E
t

subject to the FFC given

by (2), and the borrowing constraint (BC) given by

Rt +1Bt ≤ Qte,t +1Lt ,(7)given K 0 > 0 , B0 > 0 , and L0 = 1 , where Qte,t +1 is the expected land price at period t + 1
in which the expectation is formed at period t . Equation (7) states that entrepreneurs canborrow the amount so that the debt repayment that is obliged to make at t + 1 will not exceed
the expected liquidation value of the land as of t .
Solving for the entrepreneur’s problem, we formulate the following Lagrangean functionas∞Λ (Cte , K t , Lt , Bt ) = ∑ (θβ ) t [log CtE
t =0+ λt { AK tα−1 L1t −−α1 K t1−−1α + Bt − Rt Bt −1 − ( K t − K t −1 ) − Qt ( Lt − Lt −1 ) − CtE }+ η t {Qte,t +1 Lt − Rt +1 Bt }] ,where λt and η t are the Lagrange multipliers. Optimal conditions are given by four
first-order conditions, ∂Λ∂CtE= 0 , ∂Λ

complementary slacknesses, λt ≥ 0 , ∂Λ

∂Λ

∂η t

≥ 0 or η t × ∂Λ

∂η t

∂K t
∂λt

= 0 , ∂Λ∂Lt

= 0 , and ∂Λ≥ 0 or λt × ∂Λ

∂λt

∂Bt

= 0 , two

= 0 , and η t ≥ 0 ,= 0 , and the transversality condition, lim β T λT K T = 0 .
T →∞_When K t = K t and Lt = 1 are used, first-order conditions are described as

λt = (CtE ) −1 ,(8)λt= θβ (1 + αA) ,
λt +1

(9)

λt
η
Qt = t Qte+1 + θβ {(1 − α ) AK t + Qt +1} ,
λt +1
λt +1

(10)

and

7

λt
η
= ( t + θβ ) Rt +1 .
λt +1 λt +1(11)

The market clearing in the credit market requires the aggregate demand for credit to be equal
to the aggregate supply of funds, and is given by

Bt = Wt .

(12)

Finally, the investor’s optimal behavior remains to satisfy

Wt +1= βRt +1 .
Wt

(13)

Now we are prepared to solve the equilibrium. It follows from (8) and (9) thatCtE+1
= θβ (1 + αA) .
CtE

(14)

It follows from (9) and (11) thatθβ (1 + αA) = (ηt+ θβ ) Rt +1 .
λt +1

(15)

Equation (15) characterizes the condition under which the economy is (or is not)credit-constrained. It is obvious from (8) that λt > 0 for all t ≥ 0 , and thus (15) implies that

Rt < 1 + αA holds if and only if η t > 0 . Two cases are to be distinguished: in one case
Rt < 1 + αA holds and the BC given by (7) is binding with equality, and in the other case
Rt = 1 + αA holds and the BC given by (7) is not binding with equality.
Consider first the case when the BC given by (7) is binding with equality. Under theassumption of perfect foresight, Qte,t +1 = Qt +1 holds. Together with this, it follows from (9),(10), and (11) that

(1 + αA)Qt = (1 − α ) AK t +1 + αAQt +1 .
Rt +1

(16)
_

Combining the FFC given by (2) with the BC given by (7), together with K t = K t and

Lt = 1 , leads to
CtE = (1 + A) K t −1 − K t + Rt−+11Qt +1 − Qt .

(17)

The entrepreneur’s consumption is rewritten, using (16), as

CtE = (1 + A) K t −1 −

1+ A
Kt .
1 + αAIt follows from (14) and (18) that
8

(18)

1+ A
K t +1
1 + αA= θβ (1 + αA) .
1+ A(1 + A) K t −1 −Kt
1 + αA
Letting gtK ≡ K t K t −1 be the growth rate of capital, rearrangement of (19) leads to(1 + A) K t −(19)

1+ A K
gt +1 .
1 + αA(20)

gtK+1 = (1 + αA)(1 + θβ ) − θβ (1 + αA) 2 ( g tK ) −1 ≡ F ( g tK ) .

(21)

θβ (1 + αA)(1 + A)( gtK ) −1 = (1 + θβ )(1 + A) −Further rearrangement of (20) leads to

The Function F ( g tK ) is increasing, bounded above from (1 + αA)(1 + θβ ) , and satisfieslim
F ( gtK ) = −∞ . Letting g K ≡ g tK = g tK+1 be the stationary growth rate of capital, thereKg t →0are two solutions, θβ (1 + αA) and 1 + αA . Figure 2 illustrates the upward sloping curveF ( g tK ) that intersects the 45-degree line at two points. Since the higher stationary growthrate 1 + αA violates the transversality condition, the possible equilibrium is only around thesmaller stationary growth rate θβ (1 + αA) . Since the stationary state is dynamically unstable,
all the exploding paths violate the transversality condition. Hence the stationary growth pathwith a growth rate ofg K = θβ (1 + αA)(22)

is the unique equilibrium. We now turn to the determination of the interest rate. It follows from(12), (13), and the BC given by (7) that β Rt +1 =βRt +1 =Bt +1 Qt + 2 / Rt + 2=, and thus
Bt
Qt +1 / Rt +1

Qt +1
.
Qt

(23)

It follows from (16) and (23) that

K t (1 + αA)(1 − β )=.
Qt(1 − α ) A(24)

Equation (24) implies that capital and the land price grow at the same rate. It follows from (22)and this aspect of the BGP thatK t +1 Qt +1== θβ (1 + αA) .
Kt
Qt

(25)

Combining (25) with (23) leads to the interest rate:

Rt = θ (1 + αA) .
9(26)

Notice finally that Rt < 1 + αA , which requires θ < 1 . There exists an economy withbinding credit constraint only when entrepreneurs are more myopic than investors. Therefore,
if θ < 1 , the equilibrium is credit-constrained, and exhibits a BGP, where capital, the land
price , and the wealth of investors grow at the rate θβ (1 + αA) .
Consider secondly the case when the BC given by (7) is not binding with equality. It
follows from (9) and (10) that

(1 + αA)Qt = (1 − α ) AK t + Qt +1 .

(27)

The relation Rt +1 = 1 + αAt follows from (15) with η t = 0 . Combining the FFC given by (2)with (27) and the above relation leads to(1 + αA)( K t −1 + Qt −1 − Bt −1 ) − ( K t + Qt − Bt ) = CtE .

(28)

Letting Et ≡ K t + Qt − Bt be the firm’s equity value, it follows from (14) and (26) that

(1 + αA) Et − Et +1= θβ (1 + αA) .
(1 + αA) Et −1 − Et

(29)

From the analogous analysis as the economy with binding credit constraint, the uniqueequilibrium should satisfy

Et +1= θβ (1 + αA) .
Et

(30)

Equation (30) implies that the firm’s equity value grows at the rate θβ (1 + αA) , while thewealth of investors grows at the rate β (1 + αA) , which is implied by (13) and Rt = 1 + αA .
Considering a BGP in which K t , Qt , and Wt grow at the same rate, the equilibrium isfeasible only when θ = 1 . Therefore, if θ = 1 , the equilibrium is not credit-constrained, andexhibits a BGP, where capital, the land price , and the wealth of investors grow at the rate

β (1 + αA) . The equilibrium is summarized in the following proposition.

Proposition 1:(Case A) Assume that θ = 1 . The economy is not credit-constrained. The equilibrium exhibitsa BGP with a growth rate β (1 + αA) and the interest rate 1 + αA .
(Case B) Assume that θ < 1 . The economy is credit-constrained. The equilibrium exhibits a
BGP with a growth rate θβ (1 + αA) and the interest rate θ (1 + αA) .

There exists a BGP in an economy with binding credit constraint when the equilibrium interest
rate is endogenously determined to clear the credit market. In other words, the endogenous
10

behavior of the interest rate allows the economy to grow on the BGP. A simple experimentmakes this point transparent. Assume that a permanent productivity shock arrives. A permanentrise in the productivity A leads to a rise in the land price, and allows borrowers to demand
more credit because the appreciation in land mitigates credit constraints faced by them. This, inturn, drives the interest rate up, and allows investors to supply credit at higher speed. The risein A makes the resource allocation with higher growth rates of capital, the land price, andwealth of entrepreneurs and investors sustainable. This theoretical finding is contrasted with
K&M, Kiyotaki [1998] and Aghion et al. [1999], all of which identify the economy withbinding credit constraint as the one with an exogenously given interest rate that is smaller than
the return of productive asset.
When borrowers discount the future more and thus lenders discount the future less, thegreater credit supply has to be adjusted by the smaller equilibrium interest rate than otherwise.
The resulting discrepancy between the equilibrium interest rate and the marginal product of
capital is, however, consistent with the equilibrium in the credit market because the borrowers’
debt capacity is limited by the liquidation value of land.

5. Once-And-For-All Productivity Shock and Business Fluctuations
Having characterized the equilibrium of the economy with binding credit constraint, we areprepared to explore a number of implications of business cycles. We consider the response ofan unanticipated once-and-for-all productivity shock on the dynamics of the economy. We
assume that at the beginning of period T people experience a once-and-for-all productivity
shock from A to A' with A' > A .
We first examine the effect of the shock in the economy with no binding credit constraint.
We denote time subscript for A as necessity. For reference, it will be useful to express thecounterpart of (21) as the one with time subscript for A , i.e.,

g tK+1 = (1 + θβ )Letting gtE ≡(1 + αAt +1 )(1 + At )(1 + αAt +1 )(1 + αAt )(1 + At −1 )− θβ.
1 + At +1(1 + At +1 ) g tK

(21’)

Et
be the growth rate of the firm’s equity value, (29) is rewritten, with
Et −1

11

θ = 1 , asβ (1 + αAt −1 )(1 + αAt )( gtE ) −1 − (1 + β )(1 + αAt ) + gtE+1 = 0 .

(31)Assuming that AT = A' > A = AT −1 = AT +1 = AT + 2 = L , it is straightforward to findgTE+ 2 = β (1 + αA) . Introducing gTE+ 2 = β (1 + αA) into (31) with t = T + 1 leads togTE+1 = β (1 + αA' ) . Introducing further gTE+1 = β (1 + αA' ) into (31) with t = T leads to
gTE = β (1 + αA) .
It follows from (12), (13) and Rt +1 = 1 + αAt thatBt +1 Wt +1== β (1 + αAt ) . Since
Bt
Wt

Et + Bt = K t + Qt holds by definition, it follows thatK t +1 + Qt +1= β (1 + αAt ) .
K t + Qt
Letting g tQ+1 ≡

(32)

Qt +1
be the growth rate of the land price, (27) is rewritten asQt

K t 1 + αAt − g tQ+1=.
Qt
(1 − α ) At

(33)

It follows from (32) and (33) thatg tQ+1 =β (1 + At )(1 + αAt ) At +1.
β (1 + αAt ) At +1 + (1 + At +1 − g tQ+ 2 ) At

(34)

Since it is straightforward to see gTQ+ 2 = β (1 + αA) , the growth rate of the land price at period

T + 1 is finally expressed asgTQ+1 =β (1 + A' )(1 + αA' ) A.
β ( A − A' ) + A' (1 + A)

We calculate gTQ by plugging the predetermined value of(35)

K T −1 (1 − β )(1 + αA)=, given by
QT −1(1 − α ) A(6), into (33). We finally obtain gTQ = β (1 + αA) .
We next turn to the determination of gtK . Equation (27) is rewritten as

g tK+1 + g tQ+1 ( K t / Qt ) −1
= β (1 + αAt ) .
1 + ( K t / Qt ) −1
It follows from (33) and (36) that gtK becomes a function of gtQ , such that

12

(36)

gtK =β (1 + αAt −1 )(1 + At −1 ) − {β (1 + αAt −1 ) + (1 − α ) At −1}gtQ.
1 + αAt −1 − gtQ(37)

Substituting (35) into (37) leads tog TK+1 = β (1 + αA)(1 + A' ).
(1 + A)(38)

We obtain the growth rate of capital at T as g TK = β (1 + αA) by plugging

gTQ = β (1 + αA) andK T −1 (1 − β )(1 + αA)=
into (36). Using the relation Rt = 1 + αA ,QT −1(1 − α ) A

we obtain interest rates

as RT + 2 = 1 + αA , RT +1 = 1 + αA' , and RT = 1 + αA . Note thatnone of variables at T, KT , QT , or RT , reacts to any contemporaneous change in AT .
Entrepreneurs who have the log-utility do not change their consumption/investment decision to
the change in the interest rate. The impact of the shock arises with a one-period lag.2
We turn to the analysis of the economy with binding credit constraint. In response to the
productivity shock, under the assumption of perfect foresight Qte,t +1 = Qt +1 holds for anyt ≥ T + 1 , and the consumption of entrepreneur is described as (17) for t ≥ T + 1 . However, atT when the shock arrives, QTe −1,T = QT will be in general violated. The entrepreneur’s
consumption at period T can be expressed as

CTE = (1 + AT −1 ) K T −1 − K T + RT−+11QT +1 − QT + QT − QTe −1,T

= (1 + AT −1 ) KT −1 −1 + AT
KT + (QT − QTe −1,T ) .
1 + αAT(39)

Notice that the last term (QT − QTe −1,T ) captures the gain (lose) from the unexpected change
in the land price. Since it is straightforward to find gTK+ 2 = θβ (1 + αA) , it follows from (21)thatgTK+1 = θβ (1 + αA)

1 + A'
,
1+ A

(40)

2

Both KT and QT will react to the shock to the extent that entrepreneurs have an incentive tochange their consumption/investment decision to the shock. If, instead, the preference of entrepreneurs∞

is described as1∑ (θβ ) 1 − σ (C ) σ (σ ≠ 1) , K and Q react to the shock.
t

t =0

E 1−
t

T

13

T

which is qualitatively the same as the economy with no binding credit constraint. It follows
from (14), (17), and (39) that

1 + AT +1 K
gT +1
1 + αAT +1= θβ (1 + αAT ) .
e
1 + AT QT − QT −1,TK −1
(1 + AT −1 )( gT ) −+1 + αATKT(1 + AT ) −(41)

Plugging (40) into (41) leads togTK =where ∆qT ≡QT − QTe −1,TKT

θβ (1 + αAT )(1 + AT −1 ),(1 + AT ) − θβ (1 + αAT )∆qT

(42)

. Notice that ∆qT > 0 corresponds to the case when the debt

repayment is contingent on the anticipated land price, while ∆qT = 0 will correspond to thecase when the debt repayment is contingent on the realized land price.3
It will be useful to examine first the case for ∆qT = 0 . Equation (42) then reduces to

gTK = θβ

(1 + αA' )(1 + A)
, which should be smaller than the growth rate of capital unless the(1 + A' )productivity shock would arrive, i.e., β (1 + αA) . In response to the increase in QT , the debt
repayment due at period T would rise, and hence entrepreneurs who find their smaller net
worth would have to shrink investment in capital.
Consider next the interesting case for ∆qT > 0 , which we should study in details. It
follows from (24) and (25) that

QTe −1,T
KT −1=

QTe −1,T QT −1
QT −1 KT −1=(1 − α ) A'QT=andKT (1 + αA' )(1 − β )QTe −1,Tθβ (1 − α ) A. Note that the latter comes from the fact that
is
1− β
QT −1

equal to the value unless the shock would arrive. Finally, ∆qT is given byeθβ (1 − α ) A K −1
(1 − α ) A'
QT QT −1,T KT −1
∆qT =
−
=
−
( gT ) .
KT
KT −1 KT
(1 + αA' )(1 − β )
1− β

(43)

Plugging (43) into (42), and rearranging terms, the growth rate of capital at period T
3This case might be viable if the timing of debt repayment is after the arrival of the productivity shock.

14

becomesθβ (1 − α ) A
1− β.
gTK = θβ (1 + αA' )θβ (1 − α ) A'
(1 + A' ) −1− β(1 + A) −(44)

The following is established.

Proposition 2: Assume that at the beginning of period T there is an unanticipatedonce-and-for-all productivity shock. The growth rate of capital at period T is higher in theeconomy with binding credit constraint than without it if θβ + β > 1 , and the growth rate ofcapital at period T + 1 is smaller in the economy with binding credit constraint than withoutit.

Proof: We prove the former part by comparing g TK in (44) and θβ (1 + αA) . Definingφ ( A) ≡1 + αA, φ (.) is increasing if and only if θβ + β > 1 .
1 + A − {θβ (1 − α ) A} (1 − β )Together with this fact, the comparison leads to the former part. The latter part isstraightforward from the comparison between (38) and (40). Q.E.D.

Note that θβ + β > 1 is likely to be met since both β and θ are deemed close to unity.
We now turn to the determination of the growth rates of the land price. It is useful to express
the counterpart of (24) as

Qt(1 − α ) AtQ
(1 − α ) A'=
. We obtain T =, and
K t (1 + αAt )(1 − β )KT (1 + αA' )(1 − β )Qt(1 − α ) A=
for any t except for t = T . It is straightforward to see
K t (1 + αA)(1 − β )gTQ+ 2 = β (1 + αA) . Further calculation leads togTQ+1 =

QT +1 K KT
(1 + A' ) A= θβ (1 + αA' ),gT +1(1 + A) A'
KT +1
QT

15

(45)

and

θβ (1 − α ) A
Q
K
A'
1− β
gTQ = T gTK T −1 = θβ (1 + αA).
θβ(1 − α ) A'
A
KT
QT −1
(1 + A' ) −1− β(1 + A) −(46)

The following is established.

Proposition 3: Assume that at the beginning of period T there is an unanticipatedonce-and-for-all productivity shock. The growth rate of the land price at period T is higher
in the economy with binding credit constraint than without it if θβ + β > 1 , and the growthrate of the land price at period T + 1 is smaller in the economy with binding credit constraint
than unless the shock comes.

Proof: The former part is straightforward from the analogous proof with the former part of
Proposition 2. We prove the latter part by comparing gTQ+1 in (45) and θβ (1 + αA) . We
easily see gTQ+1 in (45) is smaller than θβ (1 + αA) by calculation. Q.E.D.
We derive the interest rates, using (23) ( Rt = β −1 gtQ ) , as RT + 2 = θ (1 + αA) ,θβ (1 − α ) A
A'(1 + A' ) A
1− β
RT +1 = θ (1 + αA' ), and RT = θ (1 + αA).
(1 − α ) A'θβA
(1 + A) A'(1 + A' ) −1− β(1 + A) −Figure 3, 4, and 5 illustrate the dynamic behavior of capital, the land price, and the real interest
rate in the two economies in response to the productivity shock. The straight line illustrates the
behavior of the economy with no binding credit constraint, and the dotted one with binding
credit constraint. Any of three variables react only at period T + 1 in the economy with nobinding credit constraints, while all of them react at both period T and T + 1 in the economy
with binding credit constraint.

16

On impact, KT +1 increase in both economies. Additionally, the effect of credit mitigationis added in the economy with binding credit constraint. In response to the resulting rise in QT ,K T increases. The growth-enhancing effect of the productivity shock is more persistent in theeconomy with binding credit constraint. Importantly, the impact of the shock begins one periodahead when credit constraints are binding.
One advantage of endogenous growth model is its tractability. We compare the magnitudeof the cumulative effect of the shock between the two economies by calculating gTK gTK+1 ofeach economy.

Proposition 4: Assume that at the beginning of period T there is an unanticipatedonce-and-for-al productivity shock. The magnitude of the cumulative effect of the shock isgreater in the economy with binding credit constraint than without it if

θ2(1 + αA' )1 + αA>.
θβ (1 − α ) A'θβ (1 − α ) A(1 + A' ) −(1 + A) −
1− β
1− β

(47)

Equation (47) is rewritten as θ 2φ ( A' ) > φ ( A) . Since φ (.) is increasing if θβ + β > 1 , thisinequality is likely to be met. The growth-enhancing effect tends to be greater in magnitude in
the economy with binding credit constraint than without it. Carlstrom and Fuerst (1997),
following Bernanke and Gertler (1989), calibrate an infinitely-lived-agent model with binding
credit constraint, and find that their agency-cost model replicates a sluggish and hump-shaped
dynamics in investment in response to a productivity shock. Our finding shows that credit
constraints do not necessarily become a source of sluggish dynamic in investment, but rather
suggests that if the marketable asset is used as collateral, the economy with binding credit
constraint may lead to the greater business fluctuations followed by a spike in investment, as in
the standard RBC model.

176. Introduction of a Foreign Asset
We have thus far analyzed the closed economy. Now we introduce the foreign asset with aconstant interest rate r into the model to satisfy αA > r . This extension potentially allows
investors to have another investment opportunity. If θ (1 + αA) ≥ 1 + r , nothing changes
because investors find it more beneficial to keep investing their funds in the domestic credit
market than to invest abroad. If θ (1 + αA) < 1 + r , investors now find it more beneficial toinvest abroad, and if the economy opens the capital account, the domestic interest rate will be
driven up until it is equal to the world interest rate 1 + r . It follows from (13) that the wealthof investors grow at rate β (1 + r ) . The derivation of g K does not need the market clearing
in the credit market (12), and thus the growth rate of capital remains θβ (1 + αA) . Since it issmaller than the growth rate of the wealth of investors, the demand for credit is smaller than
the supply of credit that investors can afford to make and the interest rate is kept to be tied withthe world interest rate over time. The land price also grows at rate θβ (1 + αA) .4 The
equilibrium does not show a BGP. Finally, the difference from the closed economy is reflectedin the capital-land-price ratio, which is given by

K t (1 + αA){(1 + r ) − θβ (1 + αA)}.
=Qt
(1 + r )(1 − α ) A(48)

The simple calculation shows that the capital-land-price ratio in the open economy is greaterthan that in the closed economy (given by (24)). By liberalization, the behavior of capitalremains unchanged, and the rise in the interest rate leads to a decline in the land price level.
The equilibrium is summarized in the following proposition

Proposition 5:(Case A) Assume that θ < 1 and θ (1 + αA) ≥ 1 + r . The economy is credit-constrained. The
equilibrium exhibits a BGP with a growth rate θβ (1 + αA) and the interest rate θ (1 + αA) .

4

The derivation is left to the Appendix.

18

(Case B) Assume that θ < 1 and θ (1 + αA) < 1 + r . The economy is credit-constrained. The
equilibrium does not exhibit a BGP. The wealth of investors grows at the rate β (1 + r ) , while
capital grows at the rare θβ (1 + αA) . The interest rate is 1 + r .5

By the introduction of a foreign asset into the model, the equilibrium with binding credit
constraint is divided into two regimes. In one regime (Case A) the equilibrium shows a BGP
with the endogenous determination of the interest rate, while in the other (Case B) theequilibrium does not show a BGP with the interest rate exogenously given. Assume that apermanent adverse shock to A happens when the economy is in the regime of Case A. As the
productivity declines, the equilibrium leads to declines in the growth rate and the interest rate
while keeping a BGP. If the productivity declines further beyond some critical level, theeconomy switches to the regime of Case B in which the growth rate of capital show a decline
but the interest rate does not, and hence the equilibrium does not show a BGP.
In also Case B, we can investigate the impact of the once-and-for-all shock. As for the
behavior of capital, the growth rates at T + 1 and T + 2 remain the same as Case A, but thegrowth rate at T is given byθβ (1 − α ) A
1 − β θ (1 + αA) (1 + r )g TK = θβ (1 + αA' ),θβ (1 − α ) A'
(1 + A' ) −1 − β θ (1 + αA' ) (1 + r )(1 + A) −(49)

which may or may not be greater than (44), depending on parameter values. Assume thatA = 0.03 , A' = 0.04 , r = 0.01 , and β = θ = 0.98 . The L.H.S. in (44) is greater thanthat in (48) if and only if α < 0.58.

5

When the economy is not credit-constrained( θ = 1 ), the analysis is trivial. Since 1 + αA ≥ 1 + r isalways satisfied, nothing changes by the capital account liberalization.

19

7. Conclusion
In this paper we extend the K&M model to an endogenous growth model and investigate
dynamic properties of a growing economy with binding credit constraint when land is used not
only as an input of production but also as collateral. The developed model exhibits the
propagation mechanism among output, capital, bank credit, and the land price in terms of thegrowth rate. The model’s tractability allows us to derive interesting qualitative and quantitative
findings. This analysis will be developed in several directions. First, it is interesting to extend
the model to allow for investors to hold land, for example, for housing. The enriched model is
expected to reproduce the observed behavior of the economy more accurately. Iacoviello(2005) calibrates the U.S. economy based on a version of the K&M model in which land isallocated between entrepreneurs and investors.
Second, it is worthwhile examining the possibility of debt renegotiation when the economy
faces the adverse shock. When the adverse shock arrives, the liquidation value of land will
depreciate below the promised debt repayment. The borrower will have an incentive of
renegotiating debt repayment down until the depreciated liquidation value of land. Thissituation may be captured by the case of ∆qT = 0 in Section 5, where the debt repayment is
contingent not on the anticipated but on the realized land price. It will be valuable to pursue
the asymmetry in the effects of the shock according to whether people face either the positiveor the adverse shock. Sakuragawa and Sakuragawa (2007) investigate the VAR-based impulse
responses and observe different impacts of the land price shock to other variables between
before and after around 1990 in Japan.
Third, although the credit constraint is thought of one source of inertia of the economy in
the literature of business fluctuations [e.g. Carlstrom and Fuerst (1997)], our finding show acontradictory finding that credit constraints do not necessarily become a source of inertia ininvestment, but rather may lead to the greater business fluctuations followed by a spike in20

investment. This finding suggests that investment dynamics will be influenced by themarketability of assets used as collateral in the financial contract. This aspect of collateral may
have an important insight on the role of credit constraints in business fluctuations
Fourth, it is interesting to examine the effects of government interventions. Sakuragawa
and Sakuragawa (2006) study the impact of land taxation on the economy with binding
collateral constraint and find several interesting results that would not arise in the economy
with no constraint.
Fifth, it is interesting to extend the developed framework to a monetary model or
multi-country model with international lending and borrowing. Our model with theendogenous determination of the interest rate will provide a useful framework to study theeffect of the monetary policy or the integration of credit markets.
Finally, it is interesting to use the developed model to replicate the Japanese economy. To
what degree our model reproduces the boom in the 1980s with appreciations in land prices andthe slump in the 1990s with depreciations is an important topic.

Appendix: The Calculation of Section 6
It follows from (16) and Rt +1 = 1 + r that

(1 − α ) AK t = (1 + αA)Qt −1 + αA
Qt +1
1+ r

(A-1)

It follows from (A-1) and (22) that

1 + αAQt +1
1+ r= θβ (1 + αA)
1 + αA(1 + αA)Qt −1 −Qt
1+ r

(1 + αA)Qt −Using g tQ+1 ≡

(A-2)

Qt +1
, rearrangement leads to
Qt

gtQ+1 = (1 + r ) + θβ (1 + αA) − θβ (1 + αA)(1 + r )( gtQ ) −1 .
When g Q ≡ gtQ = gtQ+1 , g Q = θβ (1 + αA) , or = 1 + r . Since g Q = 1 + r violates thetransversality condition,

21

(A-3)

g Q = θβ (1 + αA)(A-4)

is the unique solution. Finally, it follows from (A-1) and (A-4) that

K t (1 + αA){(1 + r ) − θβ (1 + αA)}=.
Qt
(1 + r )(1 − α ) A(A-5)

References
Aghion, P., A. Banergee and T. Piketty (1999) “Dualism and Macroeconomic Volatility”,
Quarterly Journal of Economics, Vol.114, pp.1359-1397.
Carlstrom, C.T. and T.S. Fuerst (1997) “Agency costs, net worth, and business fluctuations: acomputable general equilibrium analysis,” American Economic Review, Vol.87,
pp.893-910.
Hayashi, F. and E.C. Prescott (2002) “The 1990s in Japan: A Lost Decade,” Review of
Economic Dynamics, Vol.5, pp.206-235.
Iacoviello, M. (2005) “House Prices, Borrowing Constraints, and Monetary Policy in theBusiness Cycle”, American Economic Review, Vol.95, pp.739-764.
Kiyotaki, N. (1998) “Credit and Business Cycles”, Japanese Economic Review, Vol.49,
pp.18-35.
Kiyotaki, N. and J. Moore (1997) “Credit Cycles,” Journal of Political Economy, Vol.105,
pp.211-248.
Ogawa, K. and S. Kitasaka (1998) Sisan Shijyou to Keiki Hendou (in Japanese), Nippon Keizai
Shinbun Sha.
Ogawa, K., S. Kitasaka, H. Yamaoka and Y. Iwata (1996) “Borrowing Constraints and the Roleof Land Asset in Japanese Corporate Investment Decision”, Journal of the Japanese andInternational Economics, Vol.10, pp.122-149.
Ogawa, K. and K. Suzuki (1998) “Land Value and Corporate Investment: Evidence fromJapanese Panel Data”, Journal of the Japanese and International Economics, Vol.12,

22

pp.232-249.
Romer, P. M. (1986) “Increasing Returns and Long-Run Growth,” Journal of Political
Economy, Vol.94, pp.1002-1037.
Sakuragawa, M. and Y. Sakuragawa (2006) “Land Tax and Economic Growth under Credit
Market Imperfection,” International Journal of Economic Theory, forthcoming.
Sakuragawa, M. and Y. Sakuragawa (2007) “Tika to Nihon no Keiki Jyunkan,(Land Prices andBusiness Fluctuations in Japan),” Mita Keizai Gakkai Ronsyu, forthcoming

23

Figure 1: The sequence of events

T-1

T

At −1

{K t −1 , Lt −1 , Bt −1 , C tE−1 }At{K t , Lt , Bt , C tE }Yt
Rt Bt −1

24

T+1

Yt +1

Figure 2: The Existence of the Growth Pathg tK+1F ( g tK )θβ (1 + αA)

1 + αA25

g tK

Figure 3：Dynamic behavior of Capital in response to productive change

g tK

β (1 + αA) ×

1 + A'1+ A

β (1 + αA)

T-1

T

T+1

T+2

ｔ

Note) The straight line illustrates the behavior of the economy with no binding credit
constraint, and the dotted one the behavior of the economy with binding credit
constraint.

26

Figure 4：Dynamic behavior of the land price in response to the productive change

g tQ

β (1 + αA' )β (1 + αA)

T-1

T

T+1

T+2

ｔ

Note) The straight line illustrates the behavior of the economy with no binding credit
constraint, and the dotted one the behavior of the economy with binding credit
constraint.

27

Figure 5：Dynamic behavior of the interest rate in response to the productive changeR(1 + αA' )(1 + αA)

T-1

T

T+1

T+2

ｔ

Note) The straight line illustrates the behavior of the economy with no binding credit
constraint, and the dotted one the behavior of the economy with binding credit
constraint.

28