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Working Paper Series
The Effects of Open Market Operations
in a Model of Intermediation and Growth
WP 94-10
Stacey L. Schreft
Federal Reserve Bank of Richmond
Bruce D. Smith
Cornell University
This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/
Working Paper 94-10
THE EFFECTS OF OPEN MARKET OPERATIONS
IN A MODEL OF INTERMEDIATION AND GROWTH*
Stacey L. Schreft
And
Bruce D. Smith
Research Department
Federal Reserve Bank of Richmond
August 1994
l
Stacey L. SchrefI is an Associate Research Officer at the Federal Reserve Bank of
Richmond, and Bruce D. Smith is a Professor at Cornell University and a visiting
scholar in the Research Department of the Federal Reserve Bank of Minneapolis.
The
views expressed herein are those of the authors and do not necessarily reflect the views
of the Federal Reserve Banks of Minneapolis or Richmond or the Federal Reserve
System.
Abstract
We examine a standard model of capital accumulation in which spatial separation and
limited communication create a role for money and shocks to portfolio needs create a role for
banks. In this context we examine the existence, multiplicity, and dynamical properties of
monetary equilibria with positive nominal interest rates. Moderate levels of risk aversion can
lead to the existence of multiple monetary steady states, all of which can be approached from
a given set of inital conditions.
In addition, even if there is a unique monetary steady state,
monetary equilibria can be indeterminate, and oscillatory equilibrium paths can be observed.
Thus financial market frictions are a potential source of both indeterminacies and enhanced
economic volatility.
We also consider the consequences of monetary policy actions that rearrange the
composition of government liabilities.
Contractionary monetary policy activities can have
complicated consequences, depending especially on the nature of the steady state equilibrium
that obtains when there are multiple steady states. Under plausible conditions, however,
contractionary monetary policy activity raises both the nominal rate of interest ~IKJthe rate of
inflation.
Output can either rise or fall.
1
That financial market conditions “matter,” both for the level of real activity and its rate
of growth, is now a well-established proposition.’
A large number of theoretical studies detail
how the various functions of financial market institutions affect production and capital
accumulation.
Yet for many economies the most prominent financial market institution is
their central bank, and the most common exogenous source of day-to-day changes in financial
market conditions is monetary policy.
How do common monetary policy actions affect capital accumulation, real and
nominal rates of interest, and the rate of inflation ? Answering this question requires
analyzing an economy with at least three kinds of assets: money, government bonds, and
capital. Thus, an economy with a fairly rich set of asset markets is needed to examine the
consequences of monetary policy actions--such as open market operations--for real activity
and asset returns.
While a large theoretical literature exists on the integration of financial markets into
standard growth models, little of this literature can accommodate government liabilities in an
interesting way.2 This paper presents a model of capital accumulation with outside liabilities
and with banks that operate to insure agents against random liquidity needs.3 In this context
‘For theoreticaltreatmentsof this topic see Greenwood and Jovanovic (1990), Bencivenga and Smith (199 l),
Levine (199 l), Cooley and Smith (1992), Bencivenga, Smith, and Starr (1993 a,b,c), Khan (1993), Parente
(1993), and Greenwood and Smith (1993). Anecdotal evidence supporting the proposition in the text appears in
Patrick (1966), Cameron (1967), McKinnon (1973), and Shaw (1973). More formal empirical evidence is
presented by Goldsmith (1969), Antje and Jovanovic (1993) and King and Levine (1993 a,b).
‘Two exceptions to this statement are Azariadis and Smith (1993) and Schrefi and Smith (1994). Monetary
growth models, like those of Diamond (1965), Tirole (1985), Sidrauski (1967 a,b), and Brock (1974, 1975) do
allow for the presence of government liabilities and capital. However, those models do not give an explicit
allocative role to financial market institutions, and they deliver a variety of implications about the consequences
of monetary policy activity that are refuted by the data. For a discussion of the latter issue see Azariadis and
Smith (1993) or Schreft and Smith (1994).
31n doing so, it generalizes the model presented in Schreft and Smith (1994).
3
approaching the low-capital-stock steady state or follow any of the continuum of paths that
approach the locally stable high-capital-stock steady state. Economies suffering the former
fate will have permanently high nominal interest rates, which is a signal of their financial
systems’ inefficiency.
Economies approaching the high-capital-stock steady state will have
relatively low nominal and real rates of interest. In any event, two economies with similar, or
even identical, initial capital stocks can end up with different long-run output levels.
The possibility that a locally stable monetary steady state exists, even if the steady
state is unique, also indicates that dynamical equilibria of our economy may be indeterminate,
even when steady state equilibria are not. Moreover, it can easily transpire that paths
approaching the steady state display damped oscillations as they do so. Both of these are new
phenomena because virtually all existing monetary growth models deliver unique
nonstationary equilibria that converge monotonically to a steady state.5 Thus the integration
of more interesting financial market institutions into models of money and growth is a
potential source of both indeterminacy and enhanced economic volatility.
The effects of permanent changes in open market activity depend, of course, on the
number of steady state equilibria.
When there is a unique monetary steady state,
“contractionary” monetary policy actions (that is, increases in the ratio of bonds to money)
necessarily raise the nominal rate of interest. If bank portfolio allocations are not too
sensitive to changes in the nominal interest rate, contractionary monetary policies also lead to
a reduction in steady state output. This occurs for an obvious reason: an increase in
government bonds outstanding simply crowds out capital in private portfolios.
However, if
the demand for reserves by banks is sufficiently sensitive to changes in the nominal interest
‘The exceptionsto this statement are Azariadis and Smith (1993) and Schrefi and Smith (1994).
5
government liabilities.
The scheme of the paper is as follows. Section I describes the environment, while
section II considers the nature of trade and the role for banks. Steady state equilibria are
analyzed in section III, while section IV addresses dynamical equilibria.
Section V comments
on the possibility of development trap phenomena; section VI concludes.
I. Environment
We consider an economy consisting of an infinite sequence of two-period-lived
overlapping generations, plus an initial old generation. In addition, in each period agents are
assigned to one of two locations; we assume that at each date the locations are completely
symmetric and that, at the beginning of each period, each location contains a continuum of ex
ante identical young agents with unit mass.
We let
t
= O,I,... index time. At each date
t
there is a single final commodity that is
produced using a constant returns to scale technology with capital and labor as inputs. This
technology is commonly available to all agents. Any agent using K, units of capital and L,
units of labor can produce F(K, ,L, ) units of the final good at
We let f(k, ) = F(7c,1) denote
t.
the intensive production function, where k, E K, / L, is the time
t
capital-labor ratio employed.
We assume that f satisfies the following assumptions: (a) f(0) = 0, (b) j’(k) > 0 > j”(k), Vk
1 0, and (c) the standard Inada conditions.
At each date the final good can be either consumed or set aside as an investment to be
converted into capital. One unit of the final good set aside at
t
t
becomes one unit of capital at
+ I with probability qE(O,l] and becomes worthless with probability I - q. Investment
returns are assumed to be iid on each unit invested. Finally capital, once produced, is used in
7
fraction x E (0,l) of all young agents in each location is randomly selected to be transferred
to the other location. Although x is constant and known at the beginning of each period, the
identities of the specific agents who are to be relocated are discovered only after savings
decisions have been made.
We assume that neither capital investments, nor the consumption good, nor
government bonds can be transported between locations.
Thus money is the only asset that
can be carried between locations, which is the source of its liquidity advantage.6 In addition
we assume that spatial separation and limited communication preclude agents from
exchanging privately issued claims originated in “the other” location; hence only currency is
useful in interlocation exchange.7
For this reason, agents who discover that they are to be relocated will wish to convert
their other asset holdings into currency.
Thus random relocations play the same role here that
“liquidity preference shocks” play in the Diamond-Dybvig (1983) model. As in that context,
agents will wish to be insured against the event of premature asset liquidation.
The efficient
way for this insurance to be provided [see Greenwood and Smith (1993)J is through a bank
that takes deposits, holds the primary assets in the model directly, and structures the returns
paid to agents in a way that depends on whether they are relocated (in effect, on their date of
withdrawal).
As in Diamond-Dybvig, all savings will be intermediated through banks of this
type. Thus agents who are to be relocated simply make withdrawals from their banks and are
‘%r effect relocated agents face a cash-in-advance constraint on consumption purchases, while agents who are
not relocated do not. The inability to use bonds in interlocation exchange can be motivated by the realistic
assumption that bonds are issued only in large denominations.
‘This formulation follows Townsend (1987), Mitsui and Watanabe (1989), Homstein and Krusell (l993), and
Champ, Smith, and Williamson (1992). The last of these papers contains a detailed discussion of these
assumptions and some defense of their realism for the United States and Canada around the turn of the century.
9
A. Factor Markets
We assume that any agent can run the production process. Producers hire capital and
labor in competitive factor markets that operate in each location. Hence capital and labor are
each paid their marginal products, so the standard factor pricing relationships obtain:
(3)
r, = f’(k,);
(4)
w,= w(k,)=f(k,)-k,f’(k,);
where r, is the time
t20,
t
t20,
rental rate on capital and w, is the real wage rate at
t.
Notice that
w ‘> 0 holds. In addition, it will be convenient to make one additional technological
assumption. Define sZ@) = Wqw(k). Then assume that, for all k,
(a. 1)
R’(k) > 0.
Assumption (a.1) is equivalent to
(a.1’)
kw’(k) / w(k) E (0,l);
k2 0.
Assumption (a.1) [or (a.l’)] holds if, for instance, f is any CES production function with an
elasticity of substitution no less than one.
B. Banks
Banks take deposits, hold the model’s primary assets directly, and announce deposit
return schedules that depend on the depositor’s relocation status (or date of withdrawal).
addition, there is free entry into banking. Thus competition ensures that, in equilibrium,
In
11
The gross real return on money, pc /Pi+,, appears in (6) because agents who are relocated at
are given real balances at that date, which they then carry into
t
t
+ 1. The real return on these
money holdings is, of course, p, / p,+, . The promised real return to depositors, ~27,
incorporates this consideration.
For the fraction I- x of its depositors who are not relocated at
t,
the bank has
promised a gross return of d,” per unit deposited. Thus the bank owes (I- x)d,“w, to
nonmovers, which it pays at date
t
+ I. If I, >I, which we assume throughout, money is
dominated in rate of return. Hence the bank will not carry real balances between periods; or,
in other words, it holds money only as a reserve to pay agents who are relocated at
t.
Thus
payments to nonmovers will be financed solely out of the bank’s holdings of bonds and
capital, so that
(7)
(1 -n)d,“w,I
R,b, +qr,+,i,; t> 0
must hold.
It will be convenient to represent the bank’s choices in terms of the weights attached
to different assets in its portfolio.
To this end, let y, = m, / w, denote the bank’s ratio of
reserves to deposits, and let A, = i, / w, denote the ratio of its capital investments to its
deposits. Then equation (6) can be rewritten as
(6’)
d,“<r,(p, b,,,) / x; t20,
while (7) takes the form
(7’)
d,“s[qr,+,k,+R,(l-y,-h,)]/(k);
t20.
13
Thus y’(l) has the same sign as (p-l).
C. Eauilibrium
In equilibrium the factor pricing relationships (3) and (4) must be satisfied, as must the
no-arbitrage condition (8). In view of (3), (8) can be rewritten as
(11)
R, = qf’(k,+,);
t 2 0.
In addition, money supply must equal money demand at each date. Since all demand for
money here derives from banks’ demands for reserves, in equilibrium
(12)
m, =y (qv(k,);
t2 0
must hold. Finally, k,+, = qi, must hold. From (5), this condition is equivalent to
(13)
k,+, =q[w(k,)-b,-m,];
t20.
These conditions, plus the government budget constraint (l), constitute the full set of
equilibrium conditions of the model. For future reference, we note that (1) can be rewritten
as
(14)
forall
R,-,b,-,=m,- m,-,(P,-,b,) + b,
t2
1.
15
(21)
qf’(k) 2 1.
In addition (20) delivers nonnegative values for I iff
(22)
(1 +P)wdv)
holds. Finally, equation (20) defines a locus in figure 2 with a slope given by
(23)
dlldkl,,=(l
+P>qf”(k)l(l
-P[qf’(k)-1112<0.
Thus the locus defined by (20) has the configuration depicted in figure 2.
To obtain the remaining steady state equilibrium condition, substitute (2) and (15) into
(17) to obtain
(24)
k=qW[l
-y (Ml +P)].
Recalling the definition R(k) = k / qw@, we may rewrite (24) as
(25)
r(I)=[l-W)]l(l
+P).
This gives the second steady state relationship that must obtain between I and k.
Equation (25) defines a locus in figure 2 with slope
(26)
dl I dk I25=-CY(k)/(l
+P)y’(l).
Thus the slope of the locus defined by (25) is opposite in sign to y’(I). There
are now three cases to consider.
17
the vertical axis at the point [l -a
(0)] / (I -i- p). There are therefore three possibilities
regarding the existence of steady state equilibria with I > 1.
(a) Suppose, as in figure 2.c.i., that 1~ qj’(@ < (1 + p) / p. Then, if any steady
state equilibrium with I > 1 exists at all, there are necessarily at least two such equilibria.
These are represented by points A and B in the figure.
(b) It is possible that 1 < qf(& < (1 + @ / p holds, but that there are no steady state
equilibria with I > 1. This situation is depicted in figure 2.c.ii.
(c) Figure 2.c.iii depicts an economy with 4f(&.) < 1. This economy necessarily has
at least one steady state equilibrium with I > 1, and it is possible that this equilibrium is
unique.
B. Comnarative Statics
We now consider the comparative static consequences (that is, the consequences for
steady state equilibria) of open market operations.
Specifically, an increase in p constitutes an
increase in the bond-money ratio and hence corresponds to an open market sale, while a decrease
in p corresponds to an open market purchase. Thus an increase in p represents a contractionary
change in monetary policy, as conventionally conceived.
As before, it is necessary to distinguish between three cases in examining the
consequences of a change in p.
1. Case 1: o= 1
In this case--which is one where agents have logarithmic utility--y(l) E x holds.
Thus (25) reduces to
19
nominal interest rate. The effect on the steady state equilibrium value of the capital stock,
however, is ambiguous.
Whether the capital stock and output rise or fall in this case depends
on the elasticity of banks’ demands for reserves with respect to the nominal rate of interest
and possibly on the magnitude of p. We now state
Prouosition 1. Suppose that y’(l) I 0. (a) Then dk/dp <, 0 if p/(1 -p) 2 I holds.
(b) Suppose that rr I 0.5, p I l/3, and p I 1 hold. Then dk/d/3 > 0.
Proposition 1 is proved in the appendix.
The intuition of this case is obviously more complex than for case 1. As in case 1, an
increase in p--ceteris paribus--introduces more bonds, which tend to crowd out capital.
However, the increase in the nominal interest rate also reduces the demand for reserves, which
tends to reduce the ratio of total government liabilities to deposits--(1 + P)y(l)--held in the
portfolios of banks. If the latter effect is not too large, the steady state capital stock will fall;
if it is large enough, the steady state capital stock will rise. In the second case, a
“contractionary” change in monetary policy will not have a contractionary effect on steady
state output.
3. Case 3: D > 1
In this case y’(l) > 0 holds, and the determination of a steady state equilibrium is
depicted in figures 2.~. For simplicity, we focus here only on the case where flF,,
>I and
in which steady state equilibria exist. Hence there are at least two such equilibria.
As before, an increase in p shifts the locus defined by (20) up and to the right
21
unpleasant monetarist arithmetic result.
In case 1, dwdp < 0 necessarily holds; in cases 2 and 3, it might hold. Moreover, in
case 2 an increase in l3 raises the nominal interest rate. If it also raises Ic R = qf’(k)
falls, so that the inflation rate must increase in the steady state. Thus unpleasant monetarist
arithmetic must be observed in cases 1 and 2, and it can always be observed in case 3. In
other words, the possibility that contractionary monetary policy ultimately leads to higher
rates of inflation is ubiquitous here.
IV. Dynamics
To study dynamical equilibria, it will be convenient to study the special case of CobbDouglas production.
Therefore, we henceforth assume that j@,, = AP, with 01 E (0,l).
Since b, = pm, holds for all
t
2 0, equations (12) and (13) imply that the evolution of
the capital stock is described by
(30)
k,+, =qw(k,)[l-(1 +Pb(I,)];
Similarly, the date
t
t>O.
+ 1 version of the government budget constraint [equation (191 can be
written as
(31)
m,+,(l +P) =m,(P,~P,+,)[l+PR,(P,+,~P,)]=m,(P,b,+,)(l
Using R, = I, cb, /P,+) ) = qlV~+~ 1 to obtain
(32)
P,/P,+, =4fl(4+,)4
ts
+PI,);
t20.
23
holds, then equation (35) determines a monotone sequence (k, ). This sequence converges to
a unique steady state equilibrium capital stock, as depicted in figure 4. In addition, (34)
reduces to
(36)
I,=al{(l-a)(1
+p)[l -x(1 +p)] -ap>;
t20.
Thus the nominal interest rate is unchanging over time.”
Moreover, equations (35) and (36)
imply that there is a unique dynamical equilibrium, given the initial capital stock k,. This
dynamical equilibrium converges monotonically to the steady state.
B. Case 2: ~~(0.1)
To study this case (and the case p ~1 as well), we define the function Y(I) by
(37)
Y(I)=a(l
+pl)y(I)/(l-a)(1
+p)I[l -(l +P)y(I)].
Then we can rewrite equation (34) compactly as
(38)
r(I,+J=Y(I,);
t20.
Moreover, whenever p # 1, y-’ exists, so that (38) can be expressed in the form
(38’)
I,+1=y -@(I,)];
t2 0.
..
Equations (30) and (38’) are the equilibrium laws of motion for k, and I, whenever p #l.
“‘The value of I, given by (36) exceeds unity iff a/(1-a) > 1-x(1 + p) > [a /(I -a)][P/(l
holds. For the case of Cobb-Douglas production, this condition is identical to equation (27).
+ p)]
25
I,+, = I, holds whenever (41) is satisfied as an equality--that is, whenever I, =I*,
where I* is implicitly defined by
(42)
rj(I*) E (1-a)(l+P)
/ ~1.
A value I* > I satisfying (42) exists iff
(a.3)
n(l)=(l
+P)/[l
-x(1 +p)]>(l
-a)(1 +~)/cr>lim,+m~(I)=~
holds.”
Since p~(O,l) implies that rl’(l) < 0, if I* exists it is unique.
When I, > I* holds, the fact that q’(l) < 0 implies that (41) is satisfied and hence that
I,+, > I, . Similarily, when I, < I*, I,+I < I, holds.
Thus the behavior of I, over time is as
.
depicted in figure 5.
Equation (30) implies that
(43)
k,+, -k,=qw(k,)[l-(1
+P)y(I,)]-k,;
t20.
It follows that k,+, 2 k, obtains whenever
(44) [‘-Q(k,)]W+PCy(I,).
Since p~(O,l) implies that y’(I) < 0, (44) at equality defines an upward-sloping locus along
which the per-capita capital stock is constant. Combinations (I, ,kJ to the right of this locus
satisfy (44) as a strict inequality and hence have the capital-labor ratio increasing over time.
Thus figure 5 depicts the full dynamical behavior of this economy.
“For the case of Cobb-Douglas production, this condition is identical to (1 + p ) / p > qJ’ ($ &d
@(’ > 1, as described in section III.A.2.
27
Since pa (0,l) implies that y’(l) < 0 and rl’(l) < 0 both hold, it follows that aI, +, /a& > 1 .
Let 3Liand h, denote the eigenvalues of J.
Clearly h, = a~(O,l) and
x2 = aI, +, /dI, > I both hold, confirming that the unique steady state depicted in figure 5 is
a saddle.
C. Case 3: u > 1
When p > 1 holds, equations (30) and (38’) continue to describe the equilibrium laws
of motion for k, and I, . However, p > 1 implies that y’(l) > 0; hence, I, +, 2 I, holds iff’
(50
yf)'Y
(4)
or, equivalently, iff
(50’)
q(I,)‘(l
-cc)(l +p>/cL
As before, I,+, = I, will be satisfied iff
(51)
rl(I,)=(l
-a>(1 +PVa
holds. However, here an analysis of equation (51) is more complicated than previously.
It is useful to begin by stating some properties of the function rl.
Lemma 2. Suppose that (a.2) and p > 1 hold. Then
(a)
7j(l) =(l +p)/[l
(b)
q(l) > 0 holds for all I I y-‘[ll(l+P)].
(c)
For all Ics(1 ,y-‘[l/( l+p)]), q’(l) I 0 holds iff
-x(1 +p)]>O.
As I? y-‘[l /(l+p)], q(l) +oo.
29
and I, is constant, as in cases 1 and 2.
When (54) is satisfied, the steady state equilibrium is a sink, implying the
indeterminacy of monetary equilibria.
In particular, we can choose any initial value I, in a
neighborhood of I*, and the economy will asymptotically approach its steady state level. The
nominal interest rate will necessarily fluctuate as it does so. Thus (54) is a sufficient
condition for financial markets to result in the indeterminacy of an equilibrium and the
existence of economic fluctuations.
Examnle 2. Suppose that l3 > 0 and
?j(l)=(l
+P)/[l
-x(1 +p)]>(1 -a)(1 +P)lcx
hold. As I ? y -‘[l/(l+p)],
r~(l) + a~ and, in addition, (52) must be violated. ‘(In other
words rl’(l) > 0 holds for sufficiently large I.) Moreover, suppose that
(55)
[(p-l)/p]x(l
-x)(1 +p)2< 1 -rt(1 +P).
Equation (55) implies that rl’(l) < 0 . Then equation (51) is as depicted in figure 7, and if
there are any steady state equilibria with I > 1, then there are at least two of them. These
steady states are labeled A and B in figure 7.
When I, = IA, I,+, = I, holds. Moreover, for values of I, near I,, q’(I, ) c 0 holds.
Thus I, < (>) IA locally implies that I,+, > (<) I, . Similarly, I,+,=I, when I, =I, . However,
in a neighborhood of I, , q’(I, ) > 0 . Hence, I, >(< ) IB locally implies that I,+, >(<) I,.
Equation (44) continues to imply k,+, I k, ; (44) is now is a downward-sloping locus.
Thus figure 8 depicts the dynamic behavior of this economy.
31
D. Numerical Examules
We now provide some numerical examples illustrating many of the possibilities just
discussed.
Table 1 displays steady state equilibrium values for an economy with p > 1 and with
p = 0.13 The latter feature, of course, guarantees uniqueness of the steady state. As shown in
table 1, increases in p reduce
1&/al,
1 and, for p 2 2.5, the steady state is a sink. Paths
approaching it display damped oscillations as they do so.
Table 2 examines the effects of varying p in an economy that has two steady state
equilibria with I > 1.l4 As is apparen t from the table, increases in p (contractionary monetary
policy activities) raise I and reduce k in the high-capital-stock steady state, and hence they
necessarily raise the inflation rate (reduce p, /p,+, ). All of these effects are reversed with
reference to the low-capital-stock steady state.
Table 2 also illustrates the possibility that the high-capital-stock steady state is a sink.
For p < 1.0075, the parameter values of this example violate both equations (56) and (57).
Thus, as the example indicates, the conditions under which the high-capital-stock steady state
is a sink are much more general than just those given by the sufficient conditions of
proposition 2.
V. Multble
Steadv States and “DeveloDment TraW
Models of money and capital accumulation typically deliver a unique monetary steady
‘?he other parameter values employed in table 1, which are held constant, are A = 1.l, a = 0.5, 1c= 0.1,
and q = 0.5.
14The relevant parameter values other than p are A = 1.1, a = 0.335,
II = 0.25,
and q = 0.5.
33
development trap phenomena here are associated with the possibility that an economy’s
financial system might operate either relatively efficiently or relatively inefficiently, with the
efficiency of the financial system being determined endogenously.
VI. Conclusions
We have examined a model of capital accumulation in which spatial separation and
limited communication create a role for money and random shocks to agents’ portfolio needs
create a role for banks. We have seen that in such a model there is considerable scope for the
existence of multiple monetary steady states with positive nominal interest rates, for the
indeterminacy of monetary equilibria, and for endogenously arising economic fluctuations.
These possibilities arise because the severity of the financial market frictions that
agents face are--at least partly--endogenous in this environment.
When nominal interest rates
are low, agents perceive the costs of being forced to use currency in interlocation exchange as
being correspondingly low, and hence they perceive financial markets as providing, relatively
speaking, ample liquidity and as functioning “smoothly.” When nominal interest rates are
high, on the other hand, agents perceive high costs to being forced to use currency in
interlocation exchange, there is a premium on liquidity, and the functioning of financial
markets seems less smooth.
When y’ > 0 holds, low nominal interest rates (well-functioning financial markets) lead
banks to invest heavily in capital. Hence the financial system operates with relative
efficiency, holds low levels of government liabilities, and finances capital formation.
High
nominal interest rates, in contrast, are associated with banks holding comparatively large
quantities of government liabilities, with the result that there is less financing of capital
35
Atmendix
A. Proof of Proposition 1.
Differentiating equation (20) with respect to p gives
(A.l)
dIldf3=lf”(k)llf’(k)]dk/dp-[12/qf’(k)][1
-qf’(k)&f”(k)dkldp].
Differentiating equation (25) with respect to p gives
y(I) +(1 +fi)y ‘( I)dIldj3 = -Q’( k)dkldp.
(A-2)
Substituting (A.l) into (A.2) and rearranging terms yields the expression
Y(0
(A.31
-(R’(k)
31 ‘WY ‘(IMl -dwl
+(l + P)y’(I)r(k)
ww
=
/IS’(k)] + [I’pf”(k)/f’(k)]]dk/df3.
Thus, since Q’ > 0 by (a.1) and y’ < 0 by hypothesis, dk/dp is opposite in sign to the
expression
(*)
YU)U -Cl +PWY ‘WY(OI{[~
-qf’(Wqf’WH.
1. Proof of (a).
Since 1 < qf(‘k,, < (l+p)/p holds (see figure 3.b),
In addition, equations (9) and (10) imply that
37
holds for all I 2 1. Then (a.2) implies that r~(l) > 0 whenever I> 1.
(b)
Immediate from y(1) = x .
(c)
Straightforward differentiation of (40) yields
rl’(wl(~)=--~41 +PU-‘+V +P>Y’vv[l
(A.lO)
-Cl +P)Ym
The statement in the text is obtained by rearranging terms in (A. 10). q’(l)<0 follows from
the fact that p~(O,l) implies y’(l) < 0. 0
C. Proof of Lemma 2.
(a) is immediate from y( 1) = rt .
(b) From equation (40) it is apparent that q(I) > 0 iff l/( 1 + p) 2 y(l).
(c) Equation (A. 10) gives an expression for q’(l)/t~(l). Rewrite this as
wuvrlu)=-(l
(A.1 1)
+Po-‘+rIY ‘u)~YvMl
+P)Ywr1-(1
+P>rUN>.
Now use
IYW/Y (I)= (P-W
-YWP
in (A.1 1) to obtain
(A.12)
Iq '(0 /W)
= 31 + PO-' +KP - WPl[l
-Y (r>lkl + WYmm-41
+ WYml.
For all I as stated, q(l) > 0 holds. Hence r-1’(1)I 0 holds iff the right-hand side of
(A. 12) is nonpositive.
But this is equation (52) in the text. •I
39
But then (56) implies that aI,+,/iYI, 2 0 when evaluated at I, = I,. This establishes (a).
(b) aI,+,/aI, < 1 holds at I, = IA , as just argued. Thus it remains to show that
aI,+,/aI, > -1 at that same point.
The proof-is by contradiction.
Suppose to the contrary that--at the point I, = IA--
(A.19)
aI,+,/aI, I
-1.
(A.15) and (A.19) then imply that
(A.20)
12
-Cl +P)Yu,N /[1 31 +P)YvJl~w(P
- l)l/ [1 -YUJl(l
Moreover, y(IA )< l/( l+p) holds, so that [ 1 -y(IJ]-’ < (l+p)/p.
(A.2 1)
[Pep-w
+WP(l
+P(p[2-(l
Since IA > 1 holds, (A.21) implies that p/(p-1)p
+P)YqJm
+w,>.
But then (A.20) implies that
-31 +P)Yu‘JI’l.
> 1, contradicting (57). This establishes (b)
q
41
Goldsmith, Raymond W., 1969, Financial structure and development,
New Haven.
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manuscript, Federal Reserve Bank of Richmond.
Shaw, Edward S., 1973, Financial deepening in economic development,
Press: New York.
Oxford University
P
Steady State I
Steady State k
1.4
1.1146
.0609
-2.7865
1.7
1.1163
.0607
-1.5947
2.5
1.1189
.0604
-0.7459
3.0
1.1198
.0603
-0.5599
4.0
1.1211
.0601
-0.3737
10.0
1.1234
.0599
-0.1248
Parameter values are:
f3 = 0, A =
1.1, cc = 0.5, 71:=O.l, q =0.5
a+,/cv,
Figure 1
Timing of Events
agents work
when young
factors
art paid
banks make portfolio
allocation decisions
relocated agents
withdraw from banks
consumption occurs
1
I
I
t
production occurs
t
young agents
make bank deposits
t
rclocalions
are realized
I
t
relocation occurs
I
t+1
1
I
I
I
I
I
I
i
I
I
I
I
I
I
i
I
(201
I
I i
I
I
I
I
\
\
-A---
1
\
a-
I
i
I
l
I
I
1
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
i-----ly--
I
I
I
I
I
I
I
i!
I
I
k
qf(k)=(li
Figure 2b
Steady State Equilibrium
P N4~)
I
i
I
I
I
I
I
(20)
(25)
I
\
I
I
I
I
k
i
T
qf(k) = I
qf (k) = ( I+ PYP
Figure 2.c.ii
Steady State Equilibria
P>l
k
I
1’ (25’)
I
1 (20)
I
I T
I
I \t
i
I
i
I
I
(20’)
I
;
1
\ I
I
\I
\
(25)
I
I
I
f-
I
I
I
I
B
r
_I’
\
1
I
I
I
I
l
I
I
I
k
qf(k) = 1
qff(k) = U+ PYP
Figure 3.a
Contractionary Monetary Policy
p=l
I
I
I
I
I
(25)
7
--(257
--
i
-L A
I
--.
-i
I
I
I
I
I
I
I
I
I
I
-----
I
t
-------
k
k
qT(k) = I
qr(k) = (I+ l-94
Figure
3.c
Contractionmy Monetary Policy
P>l
k,
I
I
I
I
I
I
-
I
k*
w----e
I
I
I
I
I
c
Figure
5
Dynamical Equilibria
P E (&I)
I
I
I
I
I
I
I
I
I
I
I
I
I
LL
(1 - a)(1 + PI
I
I
-I --I
a
I
I
1
1
1
I
I
I
I
i
1
IA
*rl
I
I
I
Figure 7
I
I
y-’ [W + i-91
----
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