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Chicago
o r K in g r a p e r s e r ie s
Liquidity Effects, Monetary Policy
and the Business Cycle
L a w re n c e J . C h ris tia n o a n d M a rtin E ic h e n b a u m
3
W o rk in g P a p e rs S e rie s
M a c ro E c o n o m ic s Is s u e s
R e s e a rc h D e p a rtm e n t
F e d e ra l R e s e rv e B a n k o f C h ic a g o
J u ly 1 9 9 2 (W P -9 2 -1 5 )
.
:i | liiiiiii iiiin iii |hi ..
J m . iiiiiiiiiii iii
FEDERAL RESERVE BA NK
O F CHICAGO
L iq u id ity Effects, M o n etary P olicy and the Business Cycle
by
Lawrence J. C hristiano
F ed eral R eserve B an k of M inneapolis
M a rtin Eichenbaum
N o rth w e ste rn U n iv ersity , N B E R an d th e F ed eral R eserve B an k of C hicago
J U L Y 1992
ABSTRA CT
T his p a p e r p resen ts new em p irical evidence to su p p o rt th e h y p o th esis th a t p o sitiv e m o ney supply
shocks d riv e s h o r t- te r m in te re s t ra te s dow n. W e th e n p resen t a q u a n tita tiv e , general
e q u ilib riu m m odel w hich is co n sisten t w ith th is h y p o th esis. T h e tw o key featu res of o u r m odel
are th a t (i) m o n ey shocks h av e a heterogeneous im p a c t on ag en ts a n d (ii) ex p o st inflexibilities
in p ro d u c tio n give rise to a v ery low s h o r t- r u n in te re s t e la stic ity o f m o n ey dem an d. T o g eth er,
th e se im p ly th a t, in o u r m odel, a p o sitiv e m oney su p p ly shock g en era tes a larg e d ro p in th e
in te re s t r a te co m p arab le in m a g n itu d e to w h at w e fin d in th e d a ta . In sh arp c o n tra st to sticky
n om inal w age m odels, o u r m odel im plies th a t p o sitiv e m o n ey su p p ly shocks lead to increases in
th e re a l w age. W e re p o rt evidence th a t th is is co n siste n t w ith th e U .S. d a ta . F in ally , w e show
th a t o u r m odel can ra tio n a liz e a version of th e R eal B ills D o ctrin e in w hich th e m o n e ta ry
a u th o rity acco m m o d ates technology shocks, th e re b y sm o o th in g in te re s t rate s.
W e w ould lik e to th a n k F ab io B ag lian o , M arv in G o odfriend, N o b u h iro K iy o tak i, G ian carlo
M arini, D an P eled , Ju lio R o tem b erg , S teve S tro n g in , a n d M ike W oodford for helpful
conversatio n s. In a d d itio n we w ould lik e to th a n k J o n a s F ish er for his research assistance, and
th e N a tio n a l Science F o u n d a tio n for fin an cial s u p p o rt. T h e view s expressed h ere do n o t
necessarily reflect th o se o f th e F ed eral B an k of C hicago, th e F ed eral R eserve B an k o f
M inneapolis, o r th e F ed eral R eserve S y stem .
1
.
Introduction.
Economists have long studied the mechanisms by which monetary policy affects
aggregate economic activity and interest rates.
Much of the recent literature has
emphasized the alleged role of "sticky" nominal wages and prices in explaining the
expansionary effects of monetary policy.
In contrast, this paper studies an alternative
channel, one which emphasizes the liquidity effects on interest rates of unanticipated
changes in the money supply.
W hy emphasize this particular monetary transmission mechanism?
The answer is
that, in our view, post-w ar U.S. data support the conclusion that exogenous increases in
the supply of money generate substantial, persistent declines in short-term interest rates.
This view contrasts sharply with that of the traditional literature on the subject, which has
tended to conclude that money supply shocks raise, rather than lower, short-term interest
rates (Reichenstein 1987). Section 2 of this paper presents new evidence which, when taken
in conjunction with a number of recent papers on the interest rate effects of monetary
policy (surveyed in section 2), casts considerable doubt on the basic conclusion reached in
the traditional literature.
Surprisingly, existing quantitative models of money are inconsistent with the view
that positive money supply shocks drive interest rates down.
For example, King (1992)
discusses the difficulty of reconciling sticky wage and sticky price models with this view.
Modified real business cycle (RBC) models where money is introduced simply via
cash— advance constraints (as in Greenwood and Huffman (1987), Cooley and Hansen
in—
(1989), or Christiano (1991)) or a transactions role for cash (as in Kydland (1989),
Marshall (1987) or den Haan (1990)) are also inconsistent with this view. This is because a
generic implication of these models is that, if money growth displays positive persistence,
then unanticipated shocks to the growth rate of money drive the nominal interest rate u p ,
but employment and output d o v m .
This reflects the fact that, in these models, money
1
shocks affect interest rates exclusively through an anticipated inflation effect.
The o n ly
way for an exogenous shock to the money supply to drive the interest rate down in these
models is for the shock to signal a subsequent decline in money growth. Not surprisingly,
this requires grossly counterfactual assumptions regarding the law of motion for the money
supply.
So, an important challenge is to identify the features of the real world which are
missing from existing models and which prevent them from replicating the negative
interest rate response to money shocks.
In section 3 of this paper, we present a model
which allows us to explore the quantitative importance of two features. The first is that
money injections have a heterogeneous impact on agents. In stressing this feature, we are
following a tradition of theoretical papers which argue that the key to understanding the
nonneutralities of money shocks is to understand that they impact differently on different
agents (Grossman and Weiss 1983;
King and Rouwenhorst 1990.)
Rotemberg 1984;
Woodford 1987;
Baxter, Fischer,
This paper follows Lucas (1990) and Fuerst (1992a) in
supposing that firms and financial intermediaries are the key subset of agents which
absorbs a disproportionately large share of money supply shocks. To generate this result,
we
suppose,
as
do
Lucas
and
Fuerst,
that
households
make
consumption—
saving decision before the realization of monetary policy.
their
nominal
This assumption
reflects the view that, in reality, firms and financial intermediaries respond virtually
instantaneously to movements in asset prices induced by central bank open market
operations, while households’ responses are more sluggish. It is well known that whenever
a subset of agents is forced to absorb a disproportionate share of a money injection, it is
possible that the equilibrium rate of interest will fa ll.1 Indeed, in our model, heterogeneity
per se guarantees this result.
'I t is im p o rtan t to em phasize th a t this is only a possibility. As long as there are anticipated inflation effects
associated w ith a m oney supply shock, then it is possible th a t these could sw am p, in equilibrium , th e liquidity
effects associated w ith heterogeneity. For example, C hristiano (1991) shows th a t this is th e case in a plausibly
param eterized version of F u erst’s (1992a) model.
2
But, we find that heterogeneity alone does not generate a large enough interest rate
response by comparison with the data. This motivates the second key feature of our model.
Specifically, we assume that money shocks occur' at a time when firms have already
precommited themselves to particular production plans, and that these are difficult to
adjust ex post.
This is important because we assume firms must finance their variable
inputs (i.e., labor) on a p a y -a s-y o u -g o basis with cash. Since revenues do not accrue until
the end of the production period, firms are forced to borrow working capital in advance.
The need for money to carry out production gives rise to a well-defined demand for money
on the part of firms.
The assumption that production plans are difficult to adjust once
initiated gives rise to a very small ex post, or short-run, interest elasticity of demand for
money on the part of firms. This characteristic of the model conforms well with the view,
widely held in the U.S. Federal Reserve System, that the short-run interest elasticity of
the demand for total reserves is very close to zero (see Strongin 1992.) This low elasticity
greatly amplifies the interest rate impact of a money supply shock in our model.
Section 4 reports the dynamic effects of a money supply shock in a fully
parameterized version of our model.
We find that the contemporaneous response to an
unexpected increase in the growth rate of money is a decline in the nominal interest rate,
and an increase in employment, the real wage, consumption, and output. W hile positive,
the contemporaneous rise in the rate of inflation is less than the percentage increase in the
growth rate of money. Thereafter, the nominal interest rate and the rate of inflation rise,
overshoot and then gradually return to their steady state values. During the overshooting
phase, nominal interest rates are higher than they were before the initial increase in the
growth rate of money. Consumption, employment and the real wage fall after their initial
increase and then also gradually return to their steady state values.
Finally, after some
delay, investm ent also increases and then slowly reverts to its steady state level. Taken
together, the qualitative response of the system to unanticipated changes in monetary
policy is very similar to that described by Friedman (1968) in his 1967 Presidential
3
Address.
In addition, section 4 presents evidence on a key implication of our model which
distinguishes it horn an important competing model of the monetary transmission
mechanism. In our model, a positive money supply shock leads to a rise in the real wage.
Sticky wage models of the sort analyzed by Fischer (1977), Cho and Cooley (1990), and
King (1992) and King and W atson (1992) imply the opposite.
W e show that various
empirical measures of the real wage rise in response to a money supply shock. We interpret
this evidence as supportive of our m odel.2
Section 5 briefly investigates a subset of our model’s policy implications. The same
features of our model that generate a liquidity effect also imply that the monetary
authority has greater flexibility than households to quickly direct cash to the financial
sector when it is needed. Because of this, the model can rationalize a version of the Real
Bills Doctrine.
According to this doctrine, it is welfare improving for the monetary
authority to increase the money supply in response to unanticipated changes in the real
production opportunities facing the private sector.
Unless the monetary authority stands
ready to supply needed working capital in times like this — say, by rediscounting
commercial paper — productive opportunities will go unexploited.
Interestingly, this
perspective on monetary policy is very close to the one that m otivated the United States
Federal Reserve Act, which begins by stating, among other things, that the central bank
should "... furnish an elastic currency, to afford means of rediscounting commercial paper,
..." (Federal Reserve Board, 1988.)
2.
Some New Evidence on the Interest Rate Effect of a Money Shock
2T his evidence does n o t distinguish betw een our m odel and sticky price models of the sort analyzed in Cho and
Cooley (1990), King (1992), and King and W atson (1992). These models im ply a rise in th e real wage after a
m oney supply shock.
4
This section presents new empirical evidence to support the hypothesis that positive
money supply shocks drive short-term interest rates down. In addition, we reconcile our
results with those in the traditional literature.
The results in the traditional literature are based on identifying money supply
shocks with unanticipated movements in broad monetary aggregates. When the analysis is
redone using the measure of money that is directly affected by open market operations,
namely nonborrowed reserves (NBR), the results in the literature are exactly reversed. In
particular, innovations to NBR are associated with sharp, persistent declines in short-term
interest rates.
In addition, innovations in NBR are followed by persistent increases in
broader monetary aggregates (Strongin 1992.)
A straightforward explanation of these
results is that liquidity effects are quantitatively important and that NBR innovations
primarily reflect exogenous shocks to the supply of money, while innovations to broader
monetary aggregates primarily reflect shocks to demand (say, disturbances to costs of
financial intermediation.) Goodfriend (1993), Meulendyke (1989) and Strongin (1992) have
sketched models of the Federal Reserve’s operating procedures which are consistent with
this view.
Using a very different style of analysis, Bernanke and Blinder (1990), Gali
(1992), King and Watson (1992) and Sims (1986, 1992) also interpret innovations to broad
monetary aggregates as primarily reflecting shocks to money demand.
To measure the interest rate response to an exogenous money supply shock, one
must first take a stand on an empirical measure of that shock.
The traditional literature
identifies the money supply shock with the disturbance term in a regression equation of the
form,
(2 .i)
Here,
log M t = « n t) + e,.
is a tim e t information set to be discussed momentarily,
is a linear function and M is the money stock.
^.
5
is orthogonal to flt , (
To rationalize interpreting
as the
exogenous shock to the money supply, (2.1) must be viewed as the monetary authority’s
decision rule for setting M^. The set
includes the set of variables (past, and possibly
some current) that the monetary authority looks at when setting the money supply. The
fitted residual in this regression, e^., is the empirical measure of the date t money supply
shock.3
The interest rate response to a money shock is measured by the regression
A
coefficients of the interest rate on current and lagged et ’s. These coefficients coincide in
population with the impulse response functions emerging from an appropriately specified
vector autoregression. We exploit this fact in the calculations reported below.
To proceed, one must specify flt , a measure of Mt and a measure of the short-term
interest rate, R^.. In practice, the choice of short-term interest (the three-m onth Treasury
bill rate, the short-term commercial paper rate, or the federal funds rate) does not impact
on inference. For simplicity, we work with the federal funds rate. Here we assume that
is composed of lagged values of the log of real gross national product (G N P), the log of the
GNP deflator, log
and log R^..
The solid line in Figure 1 depicts our point estim ate of the dynamic response of Rt
to an expansionary policy shock for three different measures of Mt .
The dashed lines
represent a tw o-standard deviation confidence band about our point estim ates. The three
measures of M underlying Figures la — lc are nonborrowed reserves (N B R ), the monetary
^.
base (MO) and M l. Seasonally adjusted, quarterly data for the period 1966:1 — 1991:2 were
used.4 Figure 1 reveals that when M^ is measured by either MO or M l, positive money
supply shocks give rise to persistent increases in R^. This finding reproduces the results
3O ther papers th a t adopt this general strategy for m easuring m oney supply shocks include B arro (1977, 1976),
Barro an d R ush (1980), King (1992), Leeper and G ordon (1992), and Mishkin (1983).
4T he im pulse response functions reported in Figure 1 were based on estim ating a four variable VAR, Zt =
A (L )Z t-i + vt, where v t is iid and E v tv t' = V. Here, Zt = [log M t, log G N P t, log P t, log R t], an d P t is the
G N P deflator. Also, A(L) = Ao + AjL + ...+ A n Ln , where L is the lag operator, and n = 5 when M is
m easured by N B R or MO, and n = 9 when M is m easured by M l. Lag lengths were selected based on the
Q—
statistics discussed in Doan (1990). T he m oney supply shock is identified as th e first elem ent of D vt, where
D is lower trian g u lar w ith ones on the diagonal, and D D ' = V. T he confidence intervals were com puted using
the m ethod described in Doan (1990) using 100 draw s from the estim ated asym ptotic d istributio n of th e VAR
coefficients. F or fu rth er discussion, see C hristiano and Eichenbaum (1992b).
6
underlying claims in the literature that positive money supply shocks drive interest rates
up, not down.
In sharp contrast, when
is measured by NBR, a positive money supply shock
produces a sharp, persistent, statistically significant decline in R^..
Christiano and
Eichenbaum (1992b) show that the qualitative features of Figure la — lc are robust to (i)
the use of monthly data with industrial production replacing GNP, (ii) splitting the sample
at the end of 1979, and (iii) alternative specifications of flj.
In particular, they consider
four alternatives which involve different specifications of which date t variables enter f2t .
Specifically, in one case, they include log Rt ; in the second, log GNPt ; in the third, log P t ;
and in the fourth, log P^ and log G N Pt .
The negative dynamic response of Rt to an
innovation in NBRt reflects in part the fact that N B R displays a strong negative
correlation with the federal funds rate (Christiano and Eichenbaum 1992b).
This is
apparent from Figure 2, which displays the detrended federal funds rate and detrended
NBR. s
The dramatic differences in the results based on N BR and MO are due to the
behavior of borrowed reserves (BR) (i.e., reserves borrowed by banks at the Federal
Reserve’s discount window.)
To see this, consider Figure Id which reproduces Figure lb
using MO minus BR as the measure of money.
resemble closely those based on NBR.
Notice that results based on MO — BR
This finding mirrors the result in Christiano and
Eichenbaum (1992b) that MO —BR and the federal funds rate display a strong, statistically
significant negative correlation, while MO and the federal funds rate display a positive
correlation.
These results might seem surprising given the small absolute magnitude of BR.5
5These correlations correspond to variables which have been logged an d then rendered stationary via the Hodrick
Prescott (1980) filter. C hristiano and Eichenbaum (1992b) show th a t the negative relation docum ented in
Figure 2 is robust to alternative detrending m ethods. We em phasize th a t this filter was used only for the
purpose of estim ating correlations. It was not used for com puting im pulse response functions, which are based
on the log levels of the d ata.
7
Indeed, as column (2) of Table 1 indicates, the average value of the ratio of BR to MO, is
only 0.66 percent.
But for second moments what matters is not that BR is small, but
rather that its changes are typically very large. Column (3) in Table 1 presents evidence
on this point. There we report statistics which measure the changes in BR relative to the
changes in Total Reserves (T R ), MO and M l.
These statistics are calculated as follows.
First, define the absolute change in a variable, y^, relative to T R as
(22>
Ty “ “
W e normalize by T R in order to ensure that the variable being averaged is stationary. The
change in BR relative to the change in another variable, y, is defined as V gp/V y.
Table 1: Magnitude of Level and Changes in Borrowed Reserves
1966Q1 - 1990Q4
(1)
(2)
Variable (Y )
mean. B R /Y
T otal Reserves
MO
Ml
Note:
(3)
.0250
.0066
.0023
-B R ^ Y
.64
.13
.05
Column 1 — variable analyzed, as indicated, in colum ns (2) and (3)
Column 2 — sam ple m ean of ratio, borrowed reserves to Y
Column 3 — see equation (2.2) in the text.
As column 3 of Table 1 indicates, changes in BR are on average 6 4 ,1 3 and 5 percent
of changes in TR , MO, and M l, respectively. In light of this, it is not surprising that BR
could have such a large impact on the estimated impulse response functions of T R and MO.
The sign switch in these functions reflects the w ell-know n fact (documented, for example,
8
in Christiano and Eichenbaum (1992b)) that BR displays a strong positive correlation with
the federal funds rate.
Goodfriend (1983) and others have argued that this correlation
reflects the propensity of banks to increase borrowing at the Federal Reserve’s window
when the spread between the federal funds rate and the discount rate increases.
In evaluating the model of section 3, it is useful to have a sense of the magnitude of
the interest rate response to a one percent, exogenous money supply shock. According to
Figure la , the response of Rt to a one standard deviation innovation in NBR is —
.001 to
—
.002, depending on whether one focuses on the first or second quarter response.
At the
same tim e, the standard deviation of an innovation to NBR is .015, or 1.5 percent.
The
average value of the ratio of NBR to M0 over the period 1965:1 — 1990:1 is about 1/4, so
that a 1.5 percent innovation in NBR corresponds to a 1.5/4 = .375 percent innovation in
the money supply (as measured by M0).
This implies that a one percent jump in the
supply of money leads to a 100*.001/.00375 = 26.7, or a 100*.002/.00375 = 53.3 basis
points change in the quarterly federal funds rate, depending on whether one uses the first
or second period interest rate response.
One source of bias in the previous calculations leads them to understate the interest
rate effect of a one percent unexpected increase in the money supply. Federal Reserve
discount window lending increases with higher interest rates. Consequently, an exogenous
jump in NBR would not show up dollar for dollar in total reserves.
Strongin (1992), for
example, takes the extreme position that the discount window is operated in such a way
that total reserves are completely insulated in the short run from exogenous shocks to
NBR.
In this case, the proper term to have used for the denominator in the above
calculations would have been zero, and we would have reported an infinite liquidity effect!
These considerations suggest interpreting our previous calculations as providing a lower
bound on the interest rate response to an exogenous shock in the money supply.
To summarize, movements in the federal funds rate are positively associated with
movements in broad monetary aggregates, and are negatively associated with NBR.
9
Algebraically — at least for MO and NBR — this reflects the role of BR.
An important
challenge for students of monetary policy is to develop an integrated explanation for these
facts. This would require a detailed model o f the links between N BR , BR, total reserves,
MO, the federal funds rate, other interest rates and the impulses which impact on these
variables. Clearly, this is beyond the scope o f this paper. Still, our evidence on NBR, in
conjunction w ith the institutional arguments in Goodfriend, Meulendyke, and Strongin, as
well as the evidence in Bernanke and Blinder, Gali, King and W atson, and Sims strongly
suggest that a first order property of monetary policy is this:
exogenous increases in the
money supply drive short-term interest rates down, not up.
3.
One W ay to Think About Liquidity Effects
This section presents a model which is capable of rationalizing the evidence that a
positive money supply shock leads to a sharp decline in short-term interest rates.
3.1
The Model
The model economy is populated by three types of perfectly com petitive agents:
households, goods producing firms and financial intermediaries. W e represent each type of
agent by a single, representative agent. In addition, there is a monetary authority. At the
beginning of tim e t, the representative household is in possession of the economy’s entire
beginning-of—
period money stock, M^. B y the end of the period, the entire money stock is
held by the representative firm.
The cash flow pattern from the household, the
representative financial intermediary and th e monetary authority to the firm is displayed
graphically in Chart 1.
At the beginning of time t, the household allocates its cash between two uses: loans
to the financial intermediary and purchases of the consumption good.
10
In particular, the
household lends
dollars, at the gross nominal interest rate Rt , to the financial
intermediary, and sets aside
goods.
dollars for the purpose of purchasing consumption
By assumption nominal consumption must be fully financed with cash.
This
cash— advance constraint can be satisfied using current wage earnings as well as M( —
in—
V
In addition to N^, another source of funds for the financial intermediary is lump sum
injections, X^, of cash by the monetary authority.
The financial intermediary lends its
cash, Nj. + Xj, to the firm which requires working capital to finance its production
activities.
To capture the notion that working capital is required for production we
suppose that, while investment is a credit good, labor must be paid in cash on a
pay— you— basis.*
as—
go
7
Absent other sources of cash, the firm must therefore borrow
enough working capital to cover its labor costs.
Chart 2 provides a graphical description of the way in which money flows back to
the household. As owner of the firm, the household receives dividends, F^., equal to all of
the cash which the firm has at the end of the period. Since investment is a credit good, Ft
simply equals the firm’s nominal revenues from selling consumption goods, net of its
interest plus principal payments to the financial intermediary. The financial intermediary
passes the cash it receives from the firm on to the household in two forms.
dollars are sent to the household in payment for the
intermediary at the beginning of the period.
First, Rt Nt
dollars lent to the financial
The remaining cash, which reflects profits
from lending the monetary injections to the firm, is sent to the household in the form of
dividends, D t . These payments reflect the fact that the financial intermediary is owned by
the household. Finally, the household also receives wage payments from the firm.
W e now present a formal description of our model by discussing the objectives and
flWe allow households to spend their current wage earnings in order to minimize the im pact of inflation on
average em ploym ent in the model. For a further discussion of this, see C hristiano (1991).
7We m ake investm ent a credit good in an effort to minimize the im pact of inflation on average e m p l o y m e n t in
the model. For a further discussion of this, see C hristiano (1991) and Stockm an (1981).
11
constraints facing the firm, the household, and the financial intermediary.
F ir m
The tim e t technology for producing new goods is given by
(3.1)
*(Kt ,zt Ht ) = K “ (ztHt ) 1-a( + (l-tf)K t ,
0 < a < l,0 < $ < l,
where
(3.2)
z = exp(/zt + 0j):
Here Kt is the stock of capital at the beginning of tim e t, Ht represents a weighted average
of hours worked over the period, 6 is the rate of depreciation on capital and the function
f( •, •) denotes new time t output plus the undepreciated part of capital. The variable zt
denotes the tim e t level of technology which has an unconditional growth rate of /z. As is
standard in the RBC literature, we assume that the technology shock 0^ evolves according
to
(3.3)
6t ~ p061-1 + e0t’
where 0 < P q < 1 and
Cfa
is an iid shock to 0^ with standard deviation a
The variable
is assumed to be orthogonal to all other variables in the model.
There are a variety of ways to capture the sort of ex post inflexibilities in
production alluded to in the introduction. A simple way to capture these is to consider a
technology in which date t output requires a sustained flow of labor input over the
production period. To this end, we suppose that
12
is a function of two discrete sequential
labor inputs,
(3.4)
and H2^, which are combined via the technology:
Ht =
Here
and
+ .5(H2 t) ( a - 1 ) / a ]£r/ ( a“ 1) >0 < a .
denote labor hours in the first and second parts of the production period,
respectively. The fact that we split the period into only two parts rather than allowing
to depend on a continuous flow of hours worked throughout the period is motivated by
considerations of tractability.
(One way to interpret (3.4) is that
represents time
spent producing a nonstorable intermediate input which is later combined with H2t and Kt
to generate final output.)
substitution between
In equation (3.4), the parameter a is the elasticity of
and
inputs become perfect substitutes.
in production.
In contrast, as a goes to zero, (3.4) corresponds to a
Leontieff technology in which
substitution is possible.
inflexibility.
As a goes to infinity, the two labor
and
are related by fixed coefficients and no
Equation (3.4) characterizes a production process with ex post
This is because at the tim e H2t is selected, the precommited value of H jt
imposes a restriction on the firm’s production technology.
Given our cash flow assumptions, the firm must borrow working capital from the
financial intermediary to cover its labor costs.
In particular, it must borrow
dollars to finance labor in the first part of the production period and
finance labor in the second part of the production period.8 Here,
dollars to
denotes the tim e t
dollar price for a unit of type i labor, i = 1,2. W e denote the gross nominal rate of interest
on these two types of loans by R^, i = 1,2.
intermediary at the end of period t.
All loans must be repaid to the financial
Consequently, the total tim e t costs, inclusive of
financing costs, associated with hiring labor equals
8We rule out the possibility th a t firms borrow more th an they need to finance production in the first p art of the
production period. T his is a nonbinding restriction on firms since com petitive behavior on the p art of financial
interm ediaries im plies th a t firms cannot increase their discounted expected profits by holding extra cash from
the first to the second p art of the production period.
13
(3.5)
R l t W » H» + R2 tW2 tH2 f
To capture the notion that open market operations may occur in the midst of ongoing
production operations we suppose that Hl t is chosen before, and
after, the tim e t
realization of monetary policy, X t .
Each period the firm also invests in capital. Because we assume capital goods are a
credit good, the end— period cash position of the firm is given by
of—
(3.6)
F , = P t {f(K t ,zt Ht ) - Kl+ 1 } - R U W U HU - R 2tW 2tH2 t,
where P^. denotes the time t dollar price of a unit of the consumption good.
W e assume
that Ft is distributed to the firm’s owner, the household, at the end of each period after the
consumption good market closes.
At this point it is convenient to define the information sets Q^q,
n tQ includes aggregate
and
where
and the values o f all model variables dated tim e t— and
1
earlier,
includes fl^g and <t ,
?
g includes
and X^..
Acting in the best interests of its owner, the firm maximizes the present discounted
value of the dividend flow to the household.
utility of consumption of the household.
E {^ =/
denote the tim e t+ 1 marginal
Then the problem of the firm at tim e 0 is to
choose contingency plans for { H ^ ,
(3.7)
Let U q
t > 0} in order to maximize:
+ 1 ^ t ± i F t | !!„ },
IT"1
14
subject to the technology for producing new goods, (3.1) — (3.4), and the definition of
dividends given by (3.6). The contingency plans for H^t and
are constrained to be
functions of the elements of f l^ while the plan for Hjj. is constrained to be a function of
the elements of
Pj, X j, U q
functions of
In solving its maximization problem, the firm takes {R^., W 2 t,
t0 be known functions of
an(I takes { W ^ , R ^ , 0^} to be known
It behaves competitively by taking these objects to be exogenous and
beyond its ability to control.
The firm’s criterion function, (3.7), reflects our timing assumptions regarding the
distribution of dividends to the household.
The term
t + i / P t + i *s
utility to a household of a dollar received at the end of time t.
marginal
The reason that the
subscript t+ 1 appears in this expression is that tim e t dividends cannot be spent on
consumption until tim e t+ 1 .
Household
At the beginning of tim e 0, the household ranks alternative streams of consumption
and leisure according to the criterion function:
(3.8)
E { E J =(/ u ( C t,Jt)|nt0}.
Here /3 is a subjective discount rate between 0 and 1, C denotes consumption at tim e t,
^.
and Jt denotes hours of leisure at tim e t,
(3.9)
J t = 1 - L1{. - L2 t,
where L^, i = 1,2 denotes the number of type i hours worked by the household at time t
and the h o i'^ W d ’s time t endowment of hours is normalized at unity.
15
Throughout we
assume that the function U(*,*) is given by:
(3.10)
U (C t , Jt ) = [C |“ 7J
for
= ( l - 7 )ln(Ct ) + Tin(Jt )
for
< if < 1 , i , * 0,
= 0,
where 7 is scalar between zero and one.
The household’s optimization problem consists of maximizing (3.8) subject to (3.9),
(3.10) ,
< Mj., its cash constraint,
(3.11)
Mt - N t + Wl t Llt + W2tL2 t > P t Ct ,
and its budget constraint,
(3.12)
M[+1 = Rt N, + Dt + F , + (M, - N, + WU LU + W2, I 2t - P,Ct ).
In ( 3 . 1 2 ) Dj. denotes time t dividends received from the financial intermediary and is
discussed below.
The maximization occurs by choice of contingency plans for setting
of the elements of Q^q,
function of the elements of
as a function
as a function of the elements of fl^ , i = 1 ,2 , and C as a
^.
solving its optimization problem, the household
behaves com petitively by taking { R ^ , W ^ , 0^} to be given functions of
and {P^, R2 t,
Rt , W 2 t, F t , D t , X t ) to be given functions of
Financial Intermediary
Recall that the financial intermediary has two sources of funds:
and cash
injections, X^, from the monetary authority. However, by assumption its supply of loans
16
for finan cin g ty p e
1
lab o r, N j^ , is d eterm in ed p rio r to th e realizatio n of th e tim e t cash
injection . C o n seq u en tly , th e fin an cial in te rm e d ia ry faces th e sequence of cash co n strain ts:
(3.13)
N j j < N^.,
and
(3.14)
N ^t + N gt < N t + X t -
T h e v ariab le N 2t denotes th e su p p ly of loans for financing ty p e
2
lab o r.
T h ro u g h o u t we
assum e an in te rio r so lu tio n for N jt , i = 1,2, for w hich (3.14) holds w ith eq u ality .
re stric tio n is n o n b in d in g as long as
>
1
T his
.
T o d isp lay th e fin an cial in te rm e d ia ry ’s p ro b lem , we begin b y n o tin g th a t its n et
cash p o sitio n a t th e en d o f th e p eriod, D^, is given by
(3.15)
D t = R l t N l t + & 2 t^ 2 t — ^ t N t '
T hese are d is trib u te d to th e household a t th e en d of tim e t a fte r th e co n su m p tio n good
m a rk e t h as closed.
A ctin g in th e b est in te re s ts of its ow ner, th e fin an cial in te rm e d ia ry
m axim izes:
(3.16)
E g { P C ' t +-l.D t | i l t l }
PH 1
by choice of co n tin g en cy p lans for {N l t , N 2 t:t > 0} su b ject to (3.13) a n d (3.14).
ad d itio n , th e co n tin g en cy p la n for
i =
In
is co n strain ed to be a fu n ctio n of th e elem en ts of f l^ ,
, . T h e fin an cial in te rm e d ia ry is p erfectly co m p etitiv e an d tak es { R ^ } to be a know n
1 2
17
fu n ctio n of
an d { R ^ R ^ } to be know n fu n ctio n s o f
T h e in te re s t r a te
is d eterm in e d by th e co n d itio n th a t th e in te rm e d ia ry earn s zero
p ro fits on funds received from th e household. T h is req u ires th a t
(3 .1 7 )
Rj. = [ N ^ R i t +
—
T h e m a rk e t s tru c tu re w hich we h av e im p o sed allow s th e firm an d financial
in te rm e d ia ry to in te ra c t only in seq u en tial sp o t m a rk e ts for lo an s. O th e r a rran g em en ts are
of course possible.
In considering th e se a lte rn a tiv e s , it is im p o rta n t to b ear in m in d th a t
th e re is no w ay for ag en ts to diversify aw ay fro m th e risk arisin g fro m ag g reg ate shocks to
th e m oney su pply.
F o r exam ple, th e fin an cial in te rm e d ia ry m ig h t pro m ise to d eliv er a
n o n c o n tin g e n t level of N 2t a t a n o n co n tin g en t r a te of in te re s t p rio r to seeing th e re a liz a tio n
of X t . In th is case, th e fin an cial in te rm e d ia ry w ould h av e to e n te r in to a sp o t m a rk e t for
fu n d s if th e cash in jectio n tu rn e d o u t to be low er th a n a n tic ip a te d .
I t w ould en d up
b o rro w in g funds from th e firm a t a p rem iu m . In effect th e fin an c ial in te rm e d ia ry w ould be
p ay in g a s ta te — n tin g en t can c ellatio n fee on th e p rev io u sly n e g o tia te d
co
N 2^ lo an s.
F e a sib ility w ould req u ire th is, since it w ould s till b e th e case th a t to ta l loans c a n n o t exceed
Nt
+
Xt -
A lth o u g h th e d is trib u tio n of d iv id en d s b etw een th e firm
a n d fin an c ial
in te rm e d ia ry w ould differ in th is m a rk e t s tr u c tu re from th e one u sed in th is p a p e r, we
su sp ect th a t th e asso ciated e q u ilib riu m allo catio n s an d th e liq u id ity effects of u n a n tic ip a te d
m oney shocks w ould n o t b e different.
M a r k e t C le a r in g a n d E q u ilib r iu m
In a d d itio n to o p tim izin g b eh av io r on th e p a r t of th e d ifferen t ag en ts in th e m odel
w e also re q u ire th a t, in eq u ilib riu m , m a rk e ts clear. F o r th e lo a n m a rk e t, th is c o n d itio n is
given b y
= W ^ H jt , i =
, . T h e co n d itio n t h a t la b o r m a rk e ts clear is g iv en b y
1 2
18
=
H jt , i =
1
, 2 , w hile th e co n d itio n th a t th e goods m a rk e t clears is given by C t + K t +
(1—5)Kt =
1
—
F in ally , we req u ire th a t th e ag g reg ate dem an d an d sup p ly of
m oney a re eq u ated .
T h is requires th a t th e v alu e o f M t _ ^ in ( 3 . 1 2 ) equals th e m oney
supply.
T o co m p lete o u r specification of th e m odel, we specify th e following law of m o tio n
for th e g ro w th ra te of m oney, x t = X t / M t = (M t +
1
— M t ) /M p
xt = (l-px)x + pxxt-1 + ext + vtft
(3.18)
T his law of m o tio n is a slig h tly m odified version o f th e specification used in m ost
m o netized R B C m odels.
See, for ex am p le, C ooley an d H an sen (1989), den H aan (1990),
K y d lan d (1989), C ho an d C ooley (1990), H odrick, K o ch erla k o ta and Lucas (1991),
M arshall (1987), K in g (1992), an d K in g an d W a tso n (1992). In (3.18),
is an iid m oney
supply shock th a t is o rth o g o n al to all v ariab les d a te d t — 1 an d earlier, as well as to
all s.
W e d en o te th e s ta n d a rd d e v ia tio n of ex t b y a
lite ra tu re is th a t u = 0.
f°r
. T h e s ta n d a rd assu m p tion in th e
In sectio n 5 we w ill also an aly ze policies in w hich th e m o n e ta ry
a u th o rity acco m m o d ates technology shocks, th a t is, u >
0
.
A ra tio n a l e x p ec tatio n s e q u ilib riu m co n sists of fu n ctio n s { C p N g p H g p L g p P p
W gt) P-2 t ’
^ t 2 ’ fu n c t*ons { ^ it> H l t , L l t , K t + 1 ,W l t , R ^ t ) of
an d a fu n ctio n N t
of f it Q such th a t ag en ts o p tim ize an d m a rk e ts clear. O b ta in in g th e se functions e x ac tly is
not possible. In s te a d we follow C h ristian o (1991) in c o n stru c tin g ap p ro x im atio n s. D etails
are p ro v id ed in a n ap p en d ix to th is p a p e r, C h ristia n o an d E ich en b au m (1992a), w hich is
av ailab le on req u est. In a d d itio n we discuss th e ex isten ce an d uniqueness of th e lin e a r
a p p ro x im a te eq u ilib riu m in th a t ap p en d ix .
W e conclude th e p re s e n ta tio n of o u r m odel b y su m m arizin g th e tim in g conventions
and th e ir in te rp re ta tio n . T h e first decision m a d e d u rin g a perio d is N p w hich is a fu n ctio n
of D 1 0 * T h en , K ^ p H p are decided b ased o n f l ^ a n d fin ally , C t , H g j are d e te rm in e d as
19
a fu n ctio n of
W e in te rp re t th ese tim in g assu m p tio n s as c a p tu rin g in an a n a ly tic a lly
co n v en ien t w ay th e n o tio n th a t, in re a lity , d ifferent decisions are m ad e b y d ifferent a g en ts
a t differen t frequencies in tim e re la tiv e to th e frequency w ith w hich op en m a rk e t
o p eratio n s a re c arried o u t an d w ith w hich shocks to tech n o lo g y o ccu r.
T h u s, in effect w e
assu m e
are
th a t
h o u seh o ld
p o rtfo lio
decisions,
as
c a p tu re d
by
N^,
revised
m o st
in freq u e n tly . F irm in v e stm e n t decisions an d in itia l p ro d u ctio n co m m itm en ts (i.e., H ^t ) are
revised m o re freq u en tly , b u t s till a t a low er frequency th a n th a t a t w hich o pen m a rk e t
o p e ra tio n s a re ca rrie d o u t.
F in ally , household co n su m p tio n an d ongoing p ro d u ctio n
decisions (i.e., H 2 t) are assu m ed to b e m ad e at th e sam e freq u en cy as open m a rk e t
o p eratio n s. T h e im p a c t of th e se assu m p tio n s o n o u r an aly sis is discussed below .
3.2
T h e R ole of H etero g en eity .
T h e tw o k ey d istin g u ish in g fe a tu re s o f o u r analysis are t h a t (i) m o n e ta ry shocks
h av e a h etero g en eo u s im p a c t o n ag en ts, an d (ii) p ro d u ctio n is in flex ib le ex p o st. In th is
su b sectio n w e discuss th e im p a c t of th e first featu re.
T o h ig h lig h t
th e
role th a t
h e te ro g e n e ity p lay s, we consider a special case of o u r m odel in w hich th e ex p o st
in flex ib ility fe a tu re is n o t p re se n t. W e refer to th is as th e sluggish saving model, w hich is
defined b y th e co n d itio n th a t th e m oney shock is know n a t th e tim e th a t L^^.,
an d
are d e term in e d . T h e m odel is id e n tic a l to o u r m odel in all o th e r resp ects. In p a rtic u la r, we
re ta in th e a s su m p tio n t h a t
m u s t be chosen p rio r to th e re a liz a tio n of th e m o ney shock.
As C h a rt I m akes clear, th is im plies t h a t m o n ey shocks h av e a h etero g en eo u s im p a c t on
ag en ts, since firm s m u st ab so rb a d is p ro p o rtio n a te sh are
(10 0
p e rc e n t) of m o n ey in jectio n s.
A cco rd in g to th e follow ing P ro p o sitio n , in th e sluggish savings m odel, in te re s t ra te s
drop, w h ile em p lo y m en t an d th e re a l w age rise in response to a p o sitiv e m o ney su p p ly
shock.
20
P ro p o sitio n
(i)
1
: Suppose th a t
th e household an d financial in te rm e d ia ry cash co n stra in ts, (3.11) an d (3.14),
are satisfied as a s tric t e q u a lity in d a te t,
th e household an d firm first o rd er co n d itio n s are satisfied as a s tric t e q u ality
a t d a te t,
> 0.
(ii)
(iii)
T h en , in th e sluggish saving m odel,
< °- Lx ,t > ° ' “ x ,t > °-
H ere, L
, = d lo g (L u + L 2 t ) /d x t , R
t = d R t / d x t , u y t = d (W j t / P t ) /d x t , i = 1,2, w here
d x t is an u n a n tic ip a te d shock to m o n ey . 9
*
In A p p en d ix A, we prove th e p o rtio n of P ro p o sitio n
1
p e rta in in g to L
..
x ,i H ere, we
sk etch th e pro o f o f th e rem a in d er of th e p ro p o sitio n .
W e do so in a w ay th a t em phasizes
*
th e crucial role th a t firm lab o r d em an d p lays in d eterm in in g R .. T h e b asic id e a is th a t
x ,i
th e in te re s t r a te m u st d rop by an am o u n t sufficient to in d u ce firm s to v o lu n ta rily h ire th e
increase in eq u ilib riu m em p lo y m en t.
In A p p en d ix A, we show th a t, in th e sluggish sav in g m odel,
A lso, L l t = I ^
= L an d R t = R ^t = R g ^
»t
eq u atio n for H jt , i =
(3.20)
Wt
-p ^ -
= W 2 1 . = W^..
U n d er th e se circu m stan ces, th e firm ’s E uler
, , is
1 2
l
5
fH t / R t -
9It is difficult to establish conditions under which the assum ptions of Proposition 1 hold w ith probability o n e.
However, one can establish th a t (i) holds in the nonstochastic steady state version of the m odel as long as Rt >
1. This is equivalent to the restriction (l+ x )ex p [—p^l— y)lp[/0 > 1. In addition, it is easy to determ ine
w hether (ii) and (iii) are nonbinding in nonstochastic steady—
state. W e assume th a t if model param eters are
such th a t (i) — (iii) hold in nonstochastic steady state, then they will hold with arbitrarily high probability m
the stochastic version of the model, for sufficiently sm all shocks.
21
(3.20) can b e ex p ressed as a s ta tic la b o r d em an d schedule in re a l w age, em p lo y m ent space.
T his sched u le is d ep icted b y th e d o w n w a rd -slo p e d solid lin e in F ig u re 3.
As in s ta n d a rd
R B C m odels, increases in th e c a p ita l sto ck o r p o sitiv e tech n o lo g y shocks sh ift la b o r
d em an d to th e rig h t, ex ertin g ex p an sio n a ry p ressu re on ag g reg ate em p lo y m en t a n d o u tp u t.
U nlike in s ta n d a rd R B C m odels, a fall in R t also sh ifts th e la b o r d em an d curve to th e
rig h t.
T h is is b ecau se th e firm eq u ates th e m a rg in al p ro d u c t of la b o r to th e re a l cost of
h irin g la b o r, ta k in g th e cost of w orking c a p ita l in to acco u n t.
Now consider la b o r su pply.
C o n d itio n al o n a given level of co n su m p tion, th e
household E u ler eq u atio n s for L-t , i = 1,2, define a s ta tic , u p w a rd —
sloped la b o r supply
schedule:
(3.21)
T his schedule is d ep icted b y th e u p w ard — p ed solid lin e in F ig u re 3 . 1 1
slo
Since a p o sitiv e m oney shock in creases em p lo y m en t, it also increases o u tp u t.
a ssu m p tio n , in v e stm e n t c an n o t resp o n d to a m o n ey shock.
By
I t follows th a t eq u ilib riu m C t
m u st rise in resp o n se to a p o sitiv e m o n ey shock, so th a t th e la b o r su p p ly curve sh ifts to th e
left. T h is is d ep icted b y th e u p w ard —
sloped d ash ed line in F ig u re 3. H ere, C ' d en o tes th e
new level o f co n su m p tio n .
as re q u ire d b y P ro p o sitio n
T h e only w ay, th e n , for eq u ilib riu m em p lo y m en t to in c re ase —
1
— is for th e la b o r d em an d cu rv e to sh ift to th e rig h t. B u t, th is
requires th a t R . d ro p , th u s e stab lish in g t h a t R . < 0.
t
v
x.x1
0
10In ( 3 .20), ( l / 2 )fjjt is f g
or f g
it
In F ig u re 3, w e d en o te th e new ,
, taking into account th a t in th e sluggish savings model
= E ^.
n
^ A p a rt from the case ^ = 0, our "consum ption co n stan t" concept of labor supply differs from the "A constant"
concept used in the em pirical labor literature (A i th e m arginal utility of w ealth.) For proving our results, the
s
consum ption constant concept turns out to be m ore convenient.
22
low er v alu e of R b y R '.
W ith th e la b o r d em an d cu rv e sh iftin g to th e rig h t, an d th e la b o r
*
supply cu rv e sh iftin g to th e left, th e real w age m u st rise, th u s estab lish in g th a t u . > 0 .
x ,t
3.3
T h e R ole of E x P o s t In flexibilities.
In th is su b sectio n we discuss th e im p act of ex p o st in flexibilities in p ro d u ctio n .
T hese arise because o f o u r assu m p tio n s th a t (i)
an d (ii)
is chosen p rio r to th e realizatio n of X t
are im p erfect s u b stitu te s tech n o lo g ically as long as a < a> T h e tw o
.
an d
p ro p o sitio n s discussed in th is su b sectio n esta b lish th a t th e role of ex p o st inflexibilities in
p ro d u ctio n is to m agnify th e q u a n tita tiv e response of th e sy stem to m oney su p p ly shocks.
To
convey
th is
in
th e
sim p lest w ay
possible,
*
n o n sto ch astic s te a d y s ta te . T o th is end, le t L an d
X
we e v a lu a te
th e
im p act effects in
*
*
R d en o te th e v alu e of L . an d R . in
X
XjX
XjX
*
n o n sto ch astic s te a d y s ta te . In A p p en d ix A we p ro v e th e follow ing p ro p o sitio n :
P ro p o sitio n
2
: Suppose th e co nditions of P ro p o sitio n
1
hold. T h en , in o u r m odel,
dL
Lx > Lx> wx > °» Z T ~ °-
H ere, Lx = dlog(L ^t + L 2 t ) /d x t , an d
stead y — ta te .
s
A ccording to P ro p o sitio n
= d (W 2 t / P t ) /d x t , e v a lu a te d in n o n sto ch astic
2
, th e em p lo y m en t response to a m o n ey shock in
our m odel exceeds th a t in th e sluggish saving m odel. As in th e sluggish saving m odel, th e
real w age rises in resp o n se to a m oney su p p ly shock. F in ally , th e p ro p o sitio n in d icate s Lx
is n o t a fu n ctio n of a.
In A p p en d ix A we prove th e follow ing pro p o sitio n :
P ro p o sitio n 3 : Suppose th e co n d itio n s o f P ro p o sitio n
23
1
hold. T h e n , in our m odel,
0
R
)
is d ifferen tiab le and m o n o to n e in a,
dR „
> 0
(ii)
“ 37
(iii)
th e re ex ists a a >
0
such th a t for all a < a, R x < R x .
H ere, R x d en o tes dR ^./dxt , e v a lu a te d in n o n sto c h a stic ste a d y s ta te .
So, w hen
an d Lgj.
are su fficien tly im p erfectly s u b stitu ta b le , th e in te re s t im p a c t of a m oney shock exceeds
th a t in th e sluggish saving m odel.
T o g ain in tu itio n in to th e fact th a t R x falls as a goes to zero, recall th a t th e firm is
th e m a rg in a l ag en t w ho m u st ab so rb u n a n tic ip a te d cash in jectio n s.
C o n seq uently, th e
in te re s t e la s tic ity of its d em an d for real b alan ces p lay s a p rim a ry ro le in d eterm in in g R x .
T h e in te r e s t e la stic ity th a t is re le v a n t is an ex p o st e la stic ity ,
77
, w hich ta k es in to acc o u n t
th e fact th a t firm s h ave alread y in itia te d p ro d u c tio n p lan s (b y s e ttin g L ^ ) a t th e tim e a
cash in je c tio n occurs. W h e n e v a lu a te d in th e n o n sto c h a stic ste a d y s ta te o f th e m o d e l , 12
77
(3.22)
=
1
a + 1/cr
N o te t h a t
77
is a s tric tly in creasin g fu n ctio n o f t . N o t su rp risin g ly , as
r
77
goes to zero, R x
becom es m o re n eg ativ e (see A p p en d ix A ). C o n seq u en tly , R x is in creasin g in a.
T h e in tu itio n b eh in d th e fact th a t
considerin g th e e x tre m e case w hen a =
0
77
is in creasin g in a can b e o b ta in e d b y
, w hen th e firm ’s ex p o st m o n ey d em an d e la stic ity
12T o o b tain (3.22), note th a t the firm ’s first order condition for H 2t implies W 2tR 2t = Ptf-rr
• D ifferentiating
x 2i t
x
this, holding m arket prices, the state of technology, H it and th e capital stock fixed, we get d H 2t/dR« 2t =
e x p (-/* t)(W 2t/P t) /[ e x p ( - /it) f H2H2J = e x p ( -/it) (fH 2t/ R 2tV lexp( - / i t )fH 2H 2l^ ' Here’ fH 2H 2lt “ the second
derivative of (3.1) with respect to H 2t- In nonstochastic steady state, H it = H 2t = H t = H, where H t is defined
in (3.4). T h en , it is easily verified th a t in steady—
state, exp(—/X t)fg ^ ^ ^
+ 1 / cr]/(4H), where f ^ is
the derivative of the product of ( 3 . 1 ) and exp(—/it) w ith respect to Ht, evaluated in steady—
state. E quation
(3.22) follows by substitution, by using the fact th a t in steady—
state, exp(—M t)fg ^ = *^H* ^ l t = ^ 2t = W ,
an d R 2t = R lt =
an(I from the definition, 7] = —dlog(W itH it+ W 2tH 2t)/^log(R 2t)-
24
is zero.
In th is case, dev iatio n s of Lgt from th e level p lan n ed w hen L l t was chosen,
g en era te n o e x tr a o u tp u t.
As a re su lt, th e re is no red u ctio n in
sufficiently g reat to
induce firm s to v o lu n tarily ab so rb e x tra w orking cap ital, since th e derived m arginal
p ro d u c t of t h a t w orking c a p ita l equals zero.
4.
E m p irica l R esu lts
In th is section we an aly ze th e q u a n tita tiv e p ro p erties of our m odel.
describe how we assigned values to th e m o d el p a ra m e te rs.
F irs t, we
Second, we co m pute th e
q u a n tita tiv e response of th e m odel variab les to a m oney su p p ly shock. A p art from th e fact
th a t in te re s t ra te s go dow n, an im p o rta n t d istin g u ish in g fe a tu re of o u r m odel is th a t real
w ages are p re d ic te d to rise a fte r a m oney su p p ly shock. In th e fin al subsection, we rep o rt
evidence on th e em p irical p la u sib ility of th is im p lic atio n .
4.1
P a ra m e te r V alues
O u r m odel has
12
T h ro u g h o u t, 0 w as set a p rio ri to (1.03) ‘
and
/?, i , 0, a ,
p
free p a ra m e te rs:
7
, 6 //, pg, a ^
,
an d ^ was set to zero. T h e p a ra m e te rs x, px
w ere set to th e values discussed in C h ristian o (1991).
<7
x , px ,
H e rep o rts, using d a ta on
MO covering th e p erio d 1959Q1 — 1984Q1, sam p le e stim a te s for th e se ob jects equal to
.0119, .80 a n d .004.
E stim a te s of px b ased on N B R an d M l are low er, an d so we also
consider a v alu e of px = .32.
F o r pg an d a^g we use th e p o in t estim ates o b ta in ed by
B urnside, E ich en b au m a n d R ebelo (1992):
pg = .9857 an d a (g = .01369.
T h e rem aining
p a ra m e te rs w ere e stim a te d using ag g reg ate U .S. tim e series d a ta .
T h e d a ta for Y t , C t , I>t , K t an d I t co rresp o n d to th e series discussed in C h ristian o
(1988), an d cover th e p erio d 1959Q1 — 1984Q1. T h e per c a p ita co n su m p tio n m easu re is th e
sum of p riv a te secto r co n su m p tio n of n o n d u rab le s a n d services, th e im p u te d re n ta l value of
25
th e
sto ck
of
consum er
d u rab les,
an d
g o v ern m en t
co n su m p tio n .
The
p er
c a p ita
h o u rs— orked d a ta consist of H an sen ’s (1984) ho u rs w orked d a ta . T h e per c a p ita sto ck of
w
c a p ita l w as m e asu re d as th e su m o f th e sto ck of co n su m er d u rab le s, pro d u cer s tru c tu re s
an d eq u ip m e n t, g o v ern m en t an d p riv a te re sid e n tia l c a p ita l, an d go v ern m en t n o n re sid e n tia l
c a p ita l. D a ta o n p er c a p ita in v e stm e n t,
1
^, a re th e flow d a ta t h a t m a tc h th e c a p ita l sto ck
concept.
T h e p a ra m e te r 6 was e q u a te d to th e sam p le av erag e ra te of d ep reciatio n on c a p ita l,
i.e., th e sam p le average of
1
— I^ )/K ^ . T h is yields a v alu e for 6 eq u al to .0212. In
— (K t ^
our m odel, th e av erag e g ro w th r a te of eq u ilib riu m o u tp u t equals f , th e gro w th ra te o f p er
i
c a p ita o u tp u t. In lig h t of th is, w e set f equal to .0041, th e sam p le average g ro w th r a te of
i
p er c a p ita G N P .
O u r p o in t e stim a te s of a a n d
th e m eans of
7
w ere designed to eq u a te th e m o d e l’s im p lic atio n s for
+ I ^ , an d K^/Y^. w ith th e sam p le averages of o u r em p irical m easu res o f
th e se v ariab les. W e a p p ro x im a te th e m o d el’s m e an im p lic atio n s fo r
by th e ir n o n sto c h a stic s te a d y - s ta te values.
a n d K ^/Y ^
U sing th e assu m p tio n th a t th e re p re s e n ta tiv e
household has a tim e en d o w m en t of 1460 h o u rs p er q u a rte r, w e o b ta in p o in t e stim a te s o f a
an d
4 .2
7
eq u al to .357 a n d .76, resp ectiv ely . 13
Q u a n tita tiv e P ro p e rtie s o f th e M odel
13Denote the steady state values of L it, L 2t and H t by H, and the steady sta te value of K t/Y t by K /Y . T h e
sam ple average of hours worked per person is 320, which translates into an estim ate of 320/1460 = .219 for th e
average fraction of available tim e worked. T he sam ple m ean of the capital— u tp u t ratio is 10.59. T hus, our
o
estim ation strategy involves choosing values of O and 7 to ensure: H = .219/2, K /Y = 10.59. T h e firm ’s Euler
f
equation for investm ent implies th a t, in nonstochastic steady state, exp (fl)/0 — a (Y /K ) + (1—<$), where we
have used the fact th a t, in nonstochastic steady—
state, C t + i/ C t = exp ( i . S etting Y /K , f and S to the v a lu e s
f)
l
specified in th e tex t and solving for O, we obtain O = .347. T he household and firm Euler equation for
c
c
im ply th a t in steady state 7 = {(C /Y )[ 2 H / ( 1 —2H)]R.2/(1—O ) + l} " 1,where R 2 = R = ( 1 + x )//?. In
c
steady— te , K t+ i/K t = exp (l. It follows th a t C /Y = 1—[exp(/z)—(1—<5)]K/Y. Using th e previously a ss ig n ed
sta
f)
values of 2H, x, O, f, 6 and K /Y and solving or 7 , we obtain 7 = .76. T he value of C /Y im plied by our p oin t
c l
estim ates is .73, after rounding. T he average of th e ratio of consum ption to o u tp u t is .7246. All sam ple
averages used in this footnote were taken from C hristiano (1988, T able 1 ).
26
T ab le 2 p resen ts th e co n tem p o ran eo u s p ercen t change in q u a rte rly hours w orked,
L i t + L g ^ an d th e p ercen tag e p o in t change in th e no m in al in te re s t ra te , R t , in response to
a one percen tag e p o in t shock to th e g ro w th r a te of m oney. W e d en o te th ese m ag n itu d es,
w hen e v alu ate d in n o n sto ch astic ste a d y — ta te , b y Lx an d R x>
s
It is useful to com p are th e p ro p erties of o u r m odel w ith a version in w hich all
decisions are m ad e a fte r th e re a liz a tio n o f 6t an d x^ (i.e., one in w hich th e in fo rm a tio n in
n ^ 2 is co n tain ed in
cash-in-advance model
w hen p
an d
W e call th is version o f our m odel th e
Its p ro p erties are also re p o rte d in T a b le
= 0, (i.e., th e m o n e ta ry shock is p u rely tra n s ito ry ), th e n L
cash— — v an ce m odel.
in ad
T his is n o t su rp risin g .
2
basic
. As row
1
= R
in th e basic
=
0
indicates,
A pu rely tra n s ito ry shock to th e grow th
ra te of m oney corresponds to a p e rm a n e n t in crease in th e level of th e m oney stock.
It is
well know n th a t th is k in d of d istu rb a n c e is n e u tra l in th e b asic cash— — v an ce m odel,
in ad
i.e., it has no effects o n e ith e r q u a n titie s o r re la tiv e prices.
T h e only effect is a
p ro p o rtio n al ju m p in th e p rice level w hich leaves b o th th e r a te o f in flatio n a n d th e nom inal
in te re s t r a te unaffected.
Row s
2
an d 4 in d icate th a t if p
>
0
, th e n R
> 0 an d L
< 0. N ote th a t th e larger
px is, th e la rg e r is th e rise in th e n o m in al in te re s t r a te an d th e la rg e r is th e fall in hours
w orked. T o u n d e rsta n d th is re s u lt, i t is useful to th in k of a p e rs is te n t in crease in
as a
co m b in atio n of a p u rely tra n s ito ry in crease in x t an d an a n tic ip a te d in crease in th e fu tu re
g ro w th ra te of m oney.
T h e la rg e r px is, th e la rg e r th e m a g n itu d e of th e a n tic ip a te d
increase in th e fu tu re g ro w th r a te of m oney.
T ra n s ito ry increases in x^ do n o t im p a c t on
th e n o m in al in te re s t ra te , th e in flatio n r a te o r h o u rs w orked.
H ow ever, th e a n tic ip a te d
increase in x^ ex erts u p w ard p ressu re on th e r a te o f in flatio n . T h is in tu r n induces a rise in
th e n om in al in te re s t ra te . W ith th e cost of w orking c a p ita l u p , th e n e t cost o f h irin g lab o r
increases, in d u cin g firm s to red uce th e ir d em an d for lab o r.
eq u ilib riu m hours w orked dow n.
C o n sisten t w ith row
3
N ot su rp risin g ly , th is drives
, th e only w ay for a positive
in n o v a tio n in x^ to g en era te a fall in th e n o m in al in te re s t ra te an d an in crease in hours
27
w orked is for th e in crease in x^ to signal a s u b s ta n tia l fall in th e fu tu re g ro w th r a te o f
m oney (a n d in fla tio n ), i.e., px < 0. T h is is grossly co u n te rfa c tu a l.
T a b le 2 in d ic a te s, th a t in th e sluggish savings m odel, a o ne p ercen tag e p o in t shock
in th e g ro w th r a te o f m oney drives hours w orked u p b y .11 o f a p e rc e n t an d drives R . dow n
by a b o u t 17 basis p o in ts .14 T h is ap p ears to be s u b sta n tia lly less th a n w h at is ob served in
th e d a ta (see sectio n 2.) So, w hile h e tero g en eity can in d u ce a fall in in te re s t ra te s follow ing
a m oney su p p ly shock, th e m a g n itu d e o f th a t fall seem s sm all.
C onsider now th e resu lts for o u r m odel.
(T h ese a p p e a r in th e colum ns lab elled
"sluggish saving an d inflexible p ro d u c tio n " .) W ith a = 10, a one p erce n tag e p o in t shock to
Xj. drives h o u rs w orked u p b y .56 of a p e rc e n t an d d riv es
(i.e., 45 b asis p o in ts).
dow n b y ab o u t .45 p erce n t
C o n sisten t w ith P ro p o sitio n 3, w h en a falls from 10 to .5, th e
in te re s t r a te effect becom es la rg e r so th a t now a one p erce n tag e shock to x t drives R t dow n
by n early one p erce n t.
F ig u re
4
displays
th e
d y n am ic
im p u lse
resp o n se
fu n ctio n s
of
th e
b asic
cash— — v an ce m odel (solid lin e) an d o u r m odel (a = .5, d ash ed lin e) to a one s ta n d a rd
in ad
d e v ia tio n shock in th e g ro w th r a te of m oney.
u n d er th e assu m p tio n th a t px equals .8.
T h ese resp o n se fu n ctio n s w ere g e n e ra te d
C o n sid er first th e b asic cash — — v an ce m odel.
in ad
N otice th a t in th e im p a c t p erio d of th e shock, th e in te re s t r a te R^ rises. A t th e sam e tim e ,
in v e stm e n t 1^ rises w hile co n su m p tio n C t falls.
T h is is b ecau se th e rise in R t acts like a
ta x on th e cash good (co n su m p tio n ) a n d a su b sid y o n th e c red it good (in v e stm e n t). N o tice
also th a t th e fra c tio n of tim e w orked (L t ) falls.
T h is effect can b e view ed as refle ctin g a
le ftw a rd sh ift in th e la b o r d em an d cu rv e a n d a rig h tw a rd sh ift in th e la b o r su p p ly curve.
T h e fo rm er is in d u c e d b y th e rise in R t , w h ile th e la tte r is in d u c ed b y th e fall in C t - B o th
shifts c o n trib u te to a fall in th e re a l w age r a te W ^ /P ^ . T h a t
falls reflects th a t th e shift
in th e la b o r d em an d cu rv e d o m in ates th e sh ift in th e su p p ly curve. G iven our assu m p tio n
14Note th a t for both models, the m agnitude of the em ploym ent and interest rate responses to a m oney shock are
independent of px. T his is consistent w ith the results in A ppendix A.
28
of d im inish in g m a rg in a l la b o r p ro d u c tiv ity , th e m arg in al cost of h irin g lab o r, R ^ W ^ /P t ,
m u st rise since
falls. F in ally , since
has fallen an d th e sto ck o f cap ital is unchanged,
c u rre n t o u tp u t m u s t also fall. W ith o u tp u t dow n an d th e sto ck of m oney u p , prices rise by
m ore th a n th e p erce n tag e ch an g e in th e m oney supply.
Since px > 0, m o n e ta ry g ro w th co n tin u es to be high re la tiv e to its s te a d y - s ta te
level a fte r th e shock. W ith th e g ro w th r a te o f m oney declining over tim e, th e in flatio n ra te
also declines to w a rd its s te a d y - s ta te value.
C o n seq u en tly , R t is also high re la tiv e to its
stead y — ta te v alu e, b u t declining over tim e.
s
Since R^ is declining, co n su m p tion slowly
rises to its ste a d y — ta te level, w hile in v e stm e n t declines to its s te a d y - s ta te level. Since a
s
high valu e of R t depresses la b o r dem an d , as long as R t is high, ho u rs w orked an d th e real
w age sta y low , an d th e m a rg in a l cost of h irin g la b o r stay s high.
In sh arp c o n tra s t to th e b asic cash— — v an ce m odel, o u r m odel im plies th a t th e
in ad
co n tem po ran eo u s v alues of R t falls w hile C t an d L^. rise in resp o n se to a po sitive m oney
shock. W ith
u p a n d w ith dim in ish in g m a rg in al la b o r p ro d u c tiv ity , th e m a rg in al cost of
hiring la b o r, R t W t / P t , falls. T h e co n tem p o ran eo u s in crease in th e p rice level is m u te d by
th e in crease in ag g reg ate o u tp u t.
As a re su lt, th e in itia l rise in th e in flatio n ra te is less
th a n p ro p o rtio n a l to th e in itia l p ercen tag e in crease in th e m oney su p p ly .
T h e in tu itio n reg ard in g th e d y n am ic response of th e sy stem th e re a fte r is sim ilar to
th a t for th e b asic cash — — v an ce m odel. W ith p
in ad
> 0, th e g ro w th r a te of M . continues
to be high, only slow ly re v e rtin g to its s te a d y s ta te level.
C o n seq u en tly in flatio n is also
high re la tiv e to its s te a d y s ta te , b u t declining over tim e. Follow ing th e im p a c t period, R t
a ctu ally rises above its in itia l v alu e, reflectin g a n tic ip a te d in flatio n effects. T h ere after, R t
declines to its s te a d y s ta te v alu e.
In v e stm e n t, co n su m p tio n an d ho u rs w orked respond in
th e ex p ected m a n n e r as th e n o m in al in te re s t r a te (th e re la tiv e p rice of cash goods) first
rises a n d th e n falls to its ste a d y s ta te level. T h e cash goods — hours w orked and
co n su m p tio n — first fall an d th e n rise to th e ir s te a d y s ta te levels, w hile th e cred it good —
in v e stm e n t — first rises an d th e n g ra d u a lly falls to its s te a d y s ta te level.
29
O n th e basis of
F ig u re 4, w e conclude th a t o u r m odel ratio n alizes, a t le a st a t a q u a lita tiv e level, th e
d e sc rip tio n o f th e effects o f ex p an sio n a ry m o n e ta ry policy given b y F rie d m a n (1968).
A lth o u g h F ig u re 4 in d icate s th a t o u r m odel can acco u n t fo r th e co n tem p o ran eo u s
co m p o n en t o f th e in te re s t r a te response to a m oney su p p ly shock, th e m odel clearly does
n o t a cc o u n t for th e p ersisten ce o f th a t effect. U sing th e sluggish sav in g m odel, C h ristia n o
an d E ic h e n b a u m (1992c) show th a t p ersisten ce can b e in tro d u c e d b y assum ing costs of
a d ju stin g Q t = M t ~N ^. T h e effect of th is is clear b y in sp ectin g C h a rt 1. W ith
slow to
a d ju st, w h en M ^+X ^ re tu rn s to th e h ousehold a t th e en d o f p erio d t, m u ch o f a p erio d t
m oney shock is a u to m a tic a lly p assed on to financial in term ed iaries an d from th e re on to
firm s in p erio d t+ 1 . T h u s, by m ak in g assu m p tio n s th a t cause Q^. to a d ju st slow ly, firm s
are, in effect, forced to ab so rb a d isp ro p o rtio n a te sh are of a m o n ey in jectio n for sev eral
periods.
B y sp read in g o u t, over tim e , th e h eterogeneous im p a c t of a m oney shock th e
liq u id ity effect also is sp read o u t over tim e .
P re su m a b ly , in c o rp o ra tin g costs of a d ju stin g
Q t in to o u r m odel w ould also cause liq u id ity effects to b e p e rs is te n t. W e h av e n o t done so
on th is p a p e r in o rd er to keep th e an aly sis focused as sh a rp ly as possible on th e
h e te ro g e n e ity an d ex p o st in flex ib ility featu res of o u r m odel.
4.3
W h y T a k e O u r M odel of L iq u id ity Effects Seriously?
A n im p o rta n t d istin g u ish in g fe a tu re o f our m odel is its im p lic a tio n th a t th e
eq u ilib riu m real w age rises a fte r a p o sitiv e m oney shock.
In th is sectio n w e discuss th e
em p irica l p la u sib ility o f th is im p lic atio n .
F ig u re 5 d isp lay s th e im p u lse resp o n se fu n ctio n of several m easu res of th e re a l w age
to a n in n o v a tio n in N B R , u sin g th e m eth o d o lo g y d escrib ed in sectio n 2.
T h e e s tim a tio n
p erio d is 1966:1 — 1991:2. E ach o f th e th re e colum ns o f g rap h s re p re se n ts re su lts b ased on
d ifferen t m easu res of th e real w age.
In th e first co lu m n th e re a l w age is m e asu re d by
av era g e h o u rly earn in g s in th e to ta l, p riv a te , n o n a g ric u ltu ra l secto r (C itib a s e
30
d a ta
m nem onic LE H 77).
In th e second an d th ird colum ns, th e real w age is m easured b y real
co m p en satio n in th e n o n ag ric u ltu ra l secto r (L B C P U 7 ) an d real av erag e hou rly earnings in
m a n u fa c tu rin g (L E H M , d eflated b y th e co n su m er p rice in dex, P U N E W ).
corresponds to a different w ay of co n stru ctin g
E ach row
in th e m o n e ta ry policy fu nction, (2.1). In
each case, lagged values of o u tp u t, th e p rice level, th e r a te of in te re s t, an d th e real w age
are in clu d ed in
W h a t differen tiates th e a lte rn a tiv e specifications of
is th e list of
v ariab les w hose co n tem p o ran eo u s v alu e is in clu d ed in f lt . T h e first row corresponds to a
specificatio n of
in w hich th e co n tem p o ran eo u s values of th e p rice level, th e real w age
and th e level of o u tp u t are included.
T h e second row corresponds to a specification in
w hich th e co n tem p o ran eo u s value of no v a ria b le is in clu d ed . T h e th ir d row corresponds to
th e case in w hich o u tp u t, price, th e in te re s t r a te an d th e real w age are included. F in ally ,
th e fo u rth row corresponds to a sp ecificatio n of
o u tp u t an d th e p rice level are included.
in w hich th e co n tem p o ran eo u s values of
T h e solid lin e in each g ra p h rep resen ts o u r p o in t
e s tim a te o f th e real w age response to a m o n ey su p p ly shock.
T h e dashed lines rep resen t
plus an d m in u s one s ta n d a rd d ev iatio n lin e s .15
T h e strik in g re su lt in F ig u re 5 is th a t for all specifications of
an d for all th re e
m easures of th e real w age, th e real w age responds p o sitiv ely to a p o sitiv e m oney supply
shock.
In several cases, th e po sitiv e sign of th e resp o n se is s ta tis tic a lly q u ite significant.
In one sense, th e se findings are clearly su p p o rtiv e of th e m o n e ta ry tran sm issio n em bedded
in o u r m odel.
A t th e sam e tim e , afte r som e lag , th e real w age response is so larg e th a t it
15The impulse functions and confidence intervals were com puted using the sam e methodology as the one u sed for
Figure 1, and described in an earlier footnote. In particular, the results in the first row of Figure 4 are b a sed
on estim ating a five variable VAR, Zt = A (L )Z t-i + vt, where v t is iid and E v tv t' = V and Zt = [log G N P t ,
log P t, log (w /p )t, log M t, log Rt], where M t is m easured by nonborrowed reserves, R t is m easured using the
federal funds rate, P t is the G NP deflator, and w /p is th e real wage, m easured as indicated in th e text. A lso,
A(L) = Ao + AjL + ...+ A nLn , where L is the lag operator, an d n = 5. T he m oney supply shock is identified
as the fourth elem ent of D vt, where D is lower triangular w ith ones on the diagonal, and D D ' = V. For the
results in the second row, Zt = [log M t, log G N P t, log P t, log R t, log(w /p)t], and the m oney supply shock is
the first elem ent of Dvt; in the th ird row, Z t = [log G N P t, log P t, log R t, log (w /p )t, log Mt], and the m o n e y
supply shock is the fifth elem ent of Dvt; in the fourth row, Z t = [log G N P t, log P t, log M t, log R t, log (w/p)t]
and the money supply shock is the th ird elem ent of Dvt.
31
d o m in ates th e in te re s t ra te response (co m p are F igures l a an d 5.)
cost o f h irin g la b o r first falls, b u t th e n rises.
m odel.
T h a t is, th e m a rg in al
T h e in itia l response is co n sisten t w ith o u r
H ow ever, th e lagged resp o n se w ould u n d o u b te d ly b e a p ro b lem from th e p o in t o f
view o f a m odified v ersion o f th e m odel w hich im p lies p ersisten ce in th e liq u id ity effect.
5.
P olicy Im p licatio n s.
In th is sectio n we briefly discuss th e fact th a t in o u r m odel, fixed k— ercen t m oney
p
g ro w th rules of th e ty p e ad v o ca ted by M ilto n F rie d m a n are n o t o p tim al.
A m ong o th e r
th in g s, th is discussion serves to h ig h lig h t th e b asic frictio n s in o u r m odel econom y. In o u r
se tu p , p riv a te ag en ts can n o t qu ick ly d irec t cash to th e fin an cial secto r in response to
u n a n tic ip a te d tech n o lo g y shocks.
B ecause o f th is, fav o rab le p ro d u c tio n o p p o rtu n itie s go
u n ex p lo ited , a t le a st in th e sh o rt ru n .
Specifically, according to th e follow ing p ro p o sitio n ,
th e co n tem p o ran eo u s em p lo y m en t response to a tech n o lo g y shock is zero.
Suppose th e co n d itio n s of P ro p o sitio n 1 h o ld a n d v — 0.
P ro p o sitio n 4:
T h en , in th e
sluggish sav in g m odel an d in o u r m odel:
*
Le~
L 0=
°-
P roof: See A p p en d ix A.
H ere, L q —
*
L q d en o te th e d e riv a tiv e of e q u ilib riu m em p lo y m en t w ith resp ect to an
u n a n tic ip a te d tech n o lo g y shock, e v a lu a te d in n o n sto c h a stic s te a d y s ta te , for o u r m odel a n d
th e sluggish savings m odel, resp ectiv ely .
T a k e n to g e th e r, P ro p o sitio n s 1, 2 a n d 4 suggest t h a t it m a y be w elfare im p ro v in g
for th e m o n e ta ry a u th o rity to in crease th e m o n ey su p p ly in response to u n a n tic ip a te d
32
technology shocks, i.e., to set v > 0 in (3.18). We interpret such a policy as embodying a
version of the Real Bills Doctrine. A simple way to see this is to focus on the sluggish
saving model. Figure 6 displays the response of the sluggish saving model economy to a
one—
standard deviation shock in technology, e ^ . The solid line corresponds to u = 0, the
case of nonaccommodative monetary policy. Consistent with Proposition 4, employment
does not respond during the impact period of the shock. At the same time, there is a
substantial rise in the interest rate, due in part to the surge in investment stimulated by
the technology shock.16
The dashed line in Figure 6 displays the response of our model economy to a one
standard deviation technology shock when v =
.3, so that monetary policy is
accommodative. Note that now equilibrium employment increases in response to the
technology shock.
Moreover, because of the liquidity effect in our model, this policy
response has the effect of smoothing the interest rate response to a technology shock. Not
surprisingly, we found that moving from v = 0 to v = .3 leads to a small increase in the
representative agent’s utility function, (3.10).
The previous results are consistent with
related findings reported in Fuerst (1992a,b).
In future research we plan to pursue, in
greater detail, the nature of optimal policy in models of the sort described in this paper.
6.
Conclusion
This paper presents a model in which heterogeneity and ex post production
inflexibilities are required to account quantitatively for the observed interest rate response
16To see the role of investment, substitute out for the real wage in the firm’s and household’s first order condition
for labor to get, Rt = .5[(1—
7)/7](l-2Lt)fH t/(Yt+(l—
6)Kt—
Kt+i), where Yt = Kt'(ztHt)^1 a\ fg t =
(Kt/Ht) “zf1
and zt is given in (3.2). Given that Lt does not respond to a shock in O the only way for
t,
Rt to change in response to a technology shock is via its impact on zt and Kt+1. Other things the same a
jump in Kt+i drives up the rate of interest. It follows that, because Kt+i is positively related to $t>
equilibrium jump in the interest rate in Figure 6 would have been smaller, had we specified that Kt+i is chosen
prior to the realization of 01-
33
to a money supply shock. We conclude by highlighting some of the model’s shortcomings
and important areas for future research.
First, our model cannot address the empirical
links between nonborrowed reserves, higher monetary aggregates and short-term interest
rates. For example, it cannot simultaneously account for the fact that short-term interest
rates comove negatively with nonborrowed reserves, but positively with broader monetary
aggregates like the monetary base and M l.
In our view, formally accounting for these
features of the data will require explicitly modeling the Federal Reserve discount window
and the money multiplier. The latter task will certainly involve a more interesting model
of financial intermediaries and a distinction between inside and outside money. We view
this as an important area of future research. Second, a key assumption of the monetary
transmission mechanism in this paper is that the household’s nominal consumption—
saving
decision is sluggish over a significant interval of time. In current work with Charles Evans,
we are investigating the empirical plausibility of this assumption using flow of funds data.
34
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37
Appendix A: Proof of Propositions 1 —4
1.
Proof of Proposition 1.
Given the discussion in section 3.2, all that remains to be proved is the result, L* .
x,t
> 0. We do that here. In the sluggish saving model, the household’s first order conditions
for Ll t and L2t imply W l t = W2t and the financial intermediary’s first order condition
implies R^t = R ^ . The firm’s first order conditions for type 1 and type 2 labor then imply
L ft = L2t< Finally, loan market clearing and (3.17) imply R^. = R ^ = R ^ . W ithout loss
of generality, we impose these conditions as a constraint on the model. In addition, we
denote L^. = L^, W^. = W ^, i = 1, 2.
Combining the financial intermediary’s cash constraint, (3.14), and the loan market
clearing condition
(A .l)
2Wt Lt = (nt + xt )Mt ,
where nt = Nt /M t . Differentiate (A .l), and take into account dnt /d x t = 0, to get
*
(A.2)
1 dL.
1
2 L. dW.
= ------ - = -------[1----- i
1
L
x’
Lt dxt
nt + xt
i
Mt d x t
Equation (3.11) and (A .l) imply
(A.3)
P t Ct = (1 + xt )Mt .
Substituting (A.3) and (A .l) into the labor supply equation, (3.21), and differentiating
while taking into account dn^/dx^ = 0, we get dW t /dx^ = M ^/(l—
7). Substituting this
into (A.2),
*
(A.4)
1
2L.
[1----!].
Lx,t =
nt + xt
1 -7
According to (A.4), Lx t is positive if, and only if, 2L^/(1— is less than one. To see that
7)
38
this condition is satisfied in our model, divide (A .l) by (A.3), then substitute out for
Wt /P t using the labor supply function, (3.21), and rearrange to get,
2L.
1
— = -----------------------------------,
1 -7
1 + 7(Mt - N t )/(M t + X t )
(A.5)
which is positive and less than one by the facts that M + Xt > 0, M —
^.
^.
2.
> 0.
Proof of Proposition 2.
Vre begin by considering the analog of (A .l) in our model:
(A.6)
Wu Ll t + W2tL2l = (n, + xt )M{.
Totally differentiating (A.6) and evaluating the result in nonstochastic steady—
state, we
obtain:
(A.7)
dL 2t
[1 — — ^
Jx,t
Ll t +L2t dxt
Mt d x t
nt + x t
Here, we have used the fact that L ^ , n^ and
cannot respond to an unanticipated
change in xt , and that W^t = W2t in nonstochastic steady state. Equations (3.11) and
(A.6) imply (A.3).
The household’s first order condition for 1 ^
is W2t / P t =
(7/ ( 1—
r))C t / ( l — ^ —
L
Lgj;). Substituting (A.3) and (A.6) into the last equation, we find
that, in equilibrium, (l/M ^dW ^/dx^. = 1/[(1—
y)(l— )]- Substituting this into (A.7),
L^t
(A.8)
1
L ot
----- [1-------- ^ --- ]
•
nt + x t
(1— l~Lit )
7)(
The value of (A.8) in nonstochastic steady-state is denoted Lx<
steady—
state, L^t and 1^
in our model coincide with
In nonstochastic
in the sluggish saving model.
Moreover, the nonstochastic steady—
state values of nt and xt also coincide in the two
models. Using these facts and 0 < (1— < 1, 0 < (1— < 1, a comparison of (A.4) and
7)
L)
*
(A.8) establishes the result that Lx > Lx.
39
The part of the proposition pertaining to u x can be obtained by using the
>
household’s first order condition for
and the fact that
and 1^ respond positively to
an unexpected jump in x^. To see that Lx does not depend on a evaluate (A.8) in
nonstochastic steady state and take into account that L, n and x are independent of the
value of a.
3.
Proof of Proposition 3.
Combining the household’s and firm’s first order condition for I ^ , we get
h t =
Thei1' V t 5 <ffl2t / dxt “ t ( i - 7)/T]{-£Hjt/ c t -
[7/(l-7)]K-2t(fHIt/ Ct) + ( 1_Ll t _L2l)fH2H2, y Ct ) I,2x1 wliere L2x,l 5 dL2t^dxf Lel ct 5
t>
exp(—
//t)C^, fg ^ denote the derivative of the product of (3.1) and exp (— with respect to
/it)
x^, and let c and fg denote the values of these variables in nonstochastic steady state.
Then f
Thus,
g
= ( l / 2)fg /c in steady state. Also, exp(—
Mt)fjj2g 21 = - f H[o + 1 /ff]/(4H).
= - ( 1/ 2)[(1- 7) / 7](£h / c){1 + [7/ ( l - 7)]R + [(l-2H )/(2H )](a+ l/ff)}L 2x. Now,
L2x= 2LLx. Then,
(A.9)
R 2x
=
" ^
1
+ I ^ 7R
” / ^ Lx’
+
where 7 = (a + 1/cr) * is the ex post interest elasticity of money demand. To get Rx , the
7
value of dRt /d x t in nonstochastic steady state, scale and rearrange (3.17), and the loan
market clearing conditions, to get Rt = [wi t k ^ R ^ + (nt— t^ lt^ R2t ^ nt' ^ en> Rx =
wl
[(n— ^Lj)/n]R2x - But, in steady-state, w^L^ = WgLg, so that w^L^ = (l/2 )(n + x ).
w
Thus,
(A. 10)
Rx = 5 iT R2x
The only way a enters Rx is via rj in R g^ Part (i) follows from the fact th at R2X is
differentiable and monotone in r) and tj is differentiable and monotone in a. P art (ii)
follows from the fact that dr//dcr > and dR g^dr/ > 0. To get part (iii) note that
(A. 11)
f„L
1-7
, 7 t . 1-2
R -----, -Hd M + I±iR> + “5H - “^L j
1
40
Comparing (A.9) and (A. 11), and taking into account a > 0 and Lx < Lx, it follows that
IU
*
< R . Note, in general it is not true that R
*
< R since (n— )/(2n) is less than one.
x
— 95
For example, when a = 20, x = .2, 8 — .0212, /? = 1.03 '
, /x = .0041, a = .346, 7 = .761,
*
then (n— )/(2n) = .35 and Rx = —
x
.415, Rx = —
.426. Part (iii) of the proposition holds
because Rx -* -® as a -* 0.
4.
Proof of Proposition 4.
The proof can be carried out in the context of the sluggish saving model, since our
assumption of ex post inflexibility is irrelevant when v = 0, given that the technology
shock is realized at the time
and
are chosen. Thus, we impose
= L^.
Equations (3.21), (A .l) and (A.3) imply [7/ ( 1—
7)](l+ x t ) + nt + xt = wt .
Since 6t
appears nowhere on the left of the equality, it follows that dw^./d^ = 0, where d ^ is an
unexpected change in 0^. But, since [7/ ( 1—
7)](1+x^) = wt ( l —
2L^), it follows that dL^/d<?t
= 0 too.
41
Table 2
The Contemporaneous Impact of a Money Growth Shock
in Three Models
Percentage Change in Hours Worked (LJ and
Percentage Point Change in the Nominal Interest Rate (R*)
in the Period of a One-Percentage-Point Surprise Increase in Money Growthf
Models
Parameters^
Money
Growth
Persistence
Basic
Cash-in-Advance
L
x
R
x
0
Sluggish Savings and
Inflexible Production
a = 10
Sluggish Savings
Sluggish Savings and
Inflexible Production
a = .5
R
x
Lx
R
x
L
x
R
x
.110
-.170
.558
-.452
.558
-.984
L
x
0
0
(2)
.81
-.197
.816
.110
-.170
.558
-.452
.558
-.984
(3)
-.81
.026
-.816
.110
-.170
.558
-.452
.558
-.984
(4)
.32
-.026
.322
.110
.558
-.452
.558
-.984
o
(1)
1
Px
tThe derivatives, Lx = d log L/dtx and R* = dR/dex, are evaluated in nonstochastic steady state. Regarding the sluggish savings model,
correspond to the L* and R* in the paper.
and R,
$The parameter \p = 0 is a curvature parameter on the utility function, u(c,L) = (c^ O -L )7]*/^, L = L, + Lj. px is the autocorrelation of
money growth; and 6 is the rate of depreciation on capital. The other parameters are set at/3* = 1.OS'0-25, /* = 0.0041, 0 = 0, x = 0.0119, p9
= 0.9857, a = 0.346, 6 = 0.0212, and y = 0.761.
Chart 1 and 2
C a s h
C h a r t 1: C a s h
F l o w
f l o w to f i r m s
in
t h e
M o d e l
E c o n o m
C h a r t 2: C a s h
i e s
f l o w b a c k to h o u s e h o l d s
Figure 1A*
Figure 1C:
R e s p o n s e of R to N B R
R e s p o n s e of R to M 1
0 0025
0 0005
/
0.0020 -
0 0000
I\
•
0005 -
- 0010 -
-.0015 -
v
0020 -
-.0025 —
i i
i i i i ii i i i i i i i i i i i i i t
0
5
10
15
20
Quarters
i' t i '
i i i" T i i i i i n ~r i i r
25
30
35
10
15
20
Quarters
Quarters
25
30
35
% Deviation from H P Trend
F i g u r e 2:
Detrended R
and NBR,
1966:1-1991:2
Figure 3: Equilibrium Response to Unanticipated M oney Shock
nominal
interest rate
hours
1 .0 2 3 0
1 . 0
worked
1 .0 0 2 5 1.0020 -
2
2
0
1.0210
1.0200
1 .0 0 1 5 1.0010-
1 .0 1 9 0
1 .0 1 8 0
1 .0 1 7 0
1 .0 1 6 0
1 .0 1 5 0
1 .0 0 0 5 1.0000-
0 .9 9 9 5 0 .9 9 9 0
8
10
12
14
8
10
investment
12
14
real w a g e
1 .1 1 7 0 -j—
1.1165-
1 .1 1 6 0 1 .1 1 5 5 1 .1 1 5 0 1 .1 1 4 5 1 .1 1 4 0 1 .1 1 3 5 1 .1 1 3 0
8
consumption
10
marginal
.2 7 5 6 0
.2 7 5 4 0
.2 7 5 2 0
.2 7 5 0 0 .2 7 4 8 0 .2 7 4 6 0
.2 7 4 4 0
.2 7 4 2 0
1 .1 3 8 5
1 .1 3 8 0
1 .1 3 7 5
1 .1 3 7 0
1.1365
1 .1 3 6 0 .
1.1355
1.1350 1.1 3 4 5
1
1.1340
^
#
$
1
\
1
1
1
\
1
1
1
\
\
\
1
\
1
14
labor cost
\
8
8
12
i
\ 1
V
------ 1
------ 1------ r --- 1
------- 1
---------
10
12
14
inflation
1 .0 1 6 0
1 .0 1 5 0 H
1 .0 1 4 0
1 .0 1 3 0 1 .0 1 2 0 1.01101 .0 1 0 0 1 .0 0 9 0
1.0080-1
1 .0 0 7 0
Figure 4: Response to .4 Percent
Innovation in M o n e y in Period 10 in
two Models
------- Basic Cash-in-Advance
Model
- -------Sluggish Saving and
Inflexible Production Model
a - .5
8
10
12
14
nominal
interest rate
hours
worked
investment
marginal
labor cost
1 .H B 8 t
1.H721.H561.M401.14241.14081.13921.1376
8
10
12
14
inflation
Figure 6: Response of Sluggish
Saving and Inflexible Production
economy to a technology shock
_____ v = 0 (no accommodation)
-----v = .3 (accommodation)
r - t l D r Ol W i ,
TsC^I
RE 5 P of W to NBR/O
Ul W IU I (u rs / II
W
r^ . -’I v n
L
j*
'■ «
RESP of W to NBR/O
RESP of W to NBR/O
RESP of W to NBR/YPRW
Q0G2S a0020 /
^
ooois aaoto0 0005n rrm uuuuu
-.0005 - \
^ ^
✓
\
-.0010 *
rrn
-«
--0015 - iim ’ »»iiiiiim - m n iiitititr n iri
J
RESP of W to NBR/YPRW
RESP of W to NBR/YPRW
RESP of W to NBR/YP
RESP of W to NBR/YP
RESP of W to NBR/YP
0
4
8
00036 *
12 16 20 21 28 32 36
*
*
0.0024 *
00012 -
lr ^
— th*— N — ^ ---- -------- E 1"=
\
-.0012 ■
\
-.0024 f
f
?
itti
*
-.0036 - r m " in tr ittir?tiiittthttt tttti- -i ■
0 1 8 12 16 20 21 28 32 36
Figure 5: Real Wage Response to M
oney Supply S hock - U.S. D
ata
@FedFRASER