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Working Paper Series
Wicksell's Monetary Framework and
Dynamic Stability
WP 90-07
Robert F. Graboyes
Federal Reserve Bank of Richmond
Thomas M. Humphrey
Federal Reserve Bank of Richmond
This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/
Working Paper 90-7
WICKSELL'S MONETARY FRAMEWORK
AND DYNAMIC STABILITY
Robert F. Graboyes
and
Thomas M. Humphrey
Federal Reserve Bank of Richmond
September 1990
The views expressed here are solely those of the authors and do
not necessarily reflect the views of the Federal Reserve Bank of
Richmond or the Federal Reserve System.
INTRODUCTION
Traditionally, central banks seeking to stabilize general
prices have followed policies similar to those advocated by Knut
Wicksell: when prices are higher than desired, raise interest
rates to exert downward pressure on prices, and conversely.
Despite the historical predominance of interest rate-based
monetary policies, analysts frequently focus on how prices are
affected by control of the money stock (or its high-powered
base). In those cases where they do examine the relationship
between interest rates and prices, they mostly do so in a
Keynesian framework rather than a Wicksellian one. For several
reasons, Wicksellls analysis deserves renewed attention. Here, we
examine whether his interest rate-adjustment rule, coupled with
his famous cumulative process mechanism of price level change,
can stabilize prices (and interest rates). We .find that if the
interest rate rule is properly specified, it can.
Wicksell's cumulative process analysis assumes the existence
of two real interest rates: the bank lending rate and the
equilibrium or natural rate corresponding to the marginal
productivity of capital. He did not distinguish between real and
nominal rates; no expected inflation premia enter his analysis.
Wicksell believed prices rose (fell) when the bank rate was below
(above) the natural rate. By contrast, Keynesian IS-LM models
assume only one real interest rate, namely the one that
simultaneously equilibrates the goods and money markets.
Wicksell's analysis is of timely importance. For the past
three decades, the goal of price stability has been seen by many
as conflicting with the goals of robust real activity and strong
growth, at least in the short run. Now, the notion of a stable
price level is once again within the bounds of serious policy
debate. This much is evident in the attention being given the
Neal Resolution which would require the Fed to eliminate
inflation within five years'.
Wicksell's logic is as familiar as the daily business pages:
lead to inflation or deflation,
interest rates set '@incorrectlytl
so if prices are rising or falling, use interest rates as an
instrument to stop their movement and return them to their fixed
target level. The central question here is whether this logic can
translate into a practical policy rule capable of delivering
price stability.
I. PRICE STABILITY VS. ZERO INFLATION
Wicksellian Price stability is not just the absence of
persistent inflation, but also the absence of price level drift.
The Neal Resolution requires that the Federal Reserve attain
"zero inflation" within five years of passage. It goes'farther,
saying:
' For a discussion of the Neal Bill, see Black (1990).
3
inflation will be deemed to be eliminated when the expected
rate of change of the general level of prices ceases to be a
factor in individual and business decisionmaking;
Compare this with Wicksell's definition of price stability:
the problem of keeping the value of money steady, the
average level of money prices at a constant height ...
evidently is to be regarded as the fundamental problem of
monetary science ... (Wicksell, 1907. p. 553)
These two definitions are close in meaning. The price level only
ceases influencing the real economy when no one expects the
return on eff.ortor investments to be affected by changes in the
price level. There is a view which holds that policymakers should
be content merely to halt inflation at a higher price level
rather than roll back prices to their pre-inflation level.
Whatever the merits of this argument, it does not accord with
Wicksell. Expected upward price shocks will discourage investment
in long bonds or fixed income pensions just as surely as will
expected continual inflation. To Wicksell, the ultimate goal of
monetary policy was to remove the general price level from
decisions about investment and production. He was aware that
merely stopping inflation once it starts without rolling back
prices to some fixed target level gives politicians strong
incentive to tax through price rises because they need never fear
that a future deflation will remove the proceeds of the tax.
After a long absence, the notion of Wicksellian price
stability has returned to the realm of policy debate. Because
monetary authorities have consistently favored interest rates
over the monetary base as a policy tool, it is a good idea to
4
know whether Wicksell's mechanism can, in fact, achieve its
desired end.
Unfortunately, Wicksell does not provide easy proof of the
efficacy of his policy recommendations. Though he was a trained
mathematician, Wicksell wrote his price stabilization analysis in
prose, not in equations. The translation of his prose into
mathematics is not always straightforward nor entirely consistent
from one passage to another.
Wicksell was clearest in explaining how prices respond to
deviations of the bank rate from the natural rate of interest. He
was not so clear in his statement of the interest rate rule the
monetary authorities should follow to achieve price stability.
Due to its ambiguity, WicksellIs writing can be read in different
ways. Accordingly, the remainder of this paper looks at five
different two-equation inflation models, each using a different
interpretation of Wicksellls interest rate rule. The first three
models exhibit price instability. By contrast, the final two
models deliver full price stability.
The first model, from a paper by Federal Reserve Board
economists Jeff Fuhrer and George Moore, gives an initially
plausible (though, we feel, doctrinally inaccurate) twodimensional version of Wicksell's system. We feel this model
accords with many current observers' interpretation of Wicksell's
5
policy prescription. The remaining four models are of our design,
all four sharing the same price adjustment equation, and
differing only in the way we specify the interest rate reaction
function. The paper is organized as follows:
Model #cl (Fuhrer and Moore's): Here, the rate of inflation (not
the level of prices) responds to deviations of the real
interest rate from the natural rate. Interest rates are
adjusted to keep the inflation rate (not the level of
prices) at a target level. The result is an explosive
system.
Model #2: Monetary authorities adjust bank interest rates in
response to deviations of the price level from its desired
target level. This rule anchors the price level, on average,
but permits perpetual oscillations of constant amplitude
about that average level.
Model #3: Monetary authorities adjust bank rates in response to
price changes, not to gaps between actual and target price
levels. This rule always halts price movements at a
different price level than prevailing before.
Model #4: Here, monetary authorities adjust bank rates in
response to both price gaps and price movements. Under this
rule, interest rate adjustments alternate between strong and
weak in response to deviations from the price level. This
version is algebraically cumbersome, but provides an
intuitive geometric analysis.
Model #5: Again, monetary authorities respond to both price gaps
and to price movements. This version is analytically simpler
than Model #4. Under this rule, the strength of interest
rate changes is the sum of changes under the rules in Models
#l and #2.
Appendix: As much as possible, the mathematical analyses are
relegated to the Appendix.
II. MODEL #l: FUHRER AND MOORE'S FIRST MODEL
In this model, the rate of inflation rises when the market
interest rate falls below the natural rate. The central bank then
6
adjusts the market rate in response to deviations from the
policymakers' target inflation rate. This model proves to be
dynamically unstable -- adjusting the market interest rate to
counter undesired movements in the inflation rate destabilizes
the inflation rate rather than stabilizing it. Both inflation and
interest rates either rise monotonically or rise and fall in
cycles of increasing amplitude. Following, in our notation, are
Fuhrer and Moore's two equations:
(1)
*t
(2)
*, = P P$-~rfl
= a[+{rt-7rt}]
where
r
I?
7r
2
E
= the nominal interest rate (which Fuhrer and Moore
identify as the Fed Funds rate)
= dr/dt: change in nominal interest rates
= the inflation rate
= the target inflation rate
= dn/dt: change in the inflation rate
= the natural rate of interest (unobservable)
a,/3are parameters, p being the monetary policy control
variable;
On historico/doctrinal grounds, we have several objections
to this interpretation of the Wicksellian system:
[l] Including the inflation rate as an argument in equation
(1) assumes a Fisher effect (an inflation premium in nominal
interest rates). Wicksell was a contemporary of Irving Fisher's,
but incorporated no such effect in his thinking.
7
[Z] It is because the model is stated in inflation rates
rather than in price levels that there is a Fisher effect. With
only transient deviations from the target price level, the Fisher
effect would be of little or no importance.
[3] It is this non-Wicksellian element (an inflation premium
in nominal interest rates) that causes the model to be
dynamically unstable. Dropping the inflation premium T from
equation (1) yields not the monotonically explosive path for
prices (or inflation) and interest rates found by Fuhrer and
Moore, but rather a cyclical path of constant amplitude. (see
Appendix, i and ii).
III.
STABILIZING
WICKSELL'S
MODEL
In this section, we present four alternative interpretations
of the Wicksellian system, each represented by a two-equation
model. We believe our price-change equation [equation (3)] is
correct and can be derived unambiguously from Wicksell's own
complete structural model of the inflationary process (see
Appendix iii). And we believe Wicksell formulated at least two
alternative versions of his monetary policy rule and not just the
single version suggested by equation (2).
III.1
Model
#2
We begin by eliminating the Fisher Effect from equation (1)
and by directing the policymakers to achieve a target price level
8
rather than a target rate of inflation. In equation (3), prices
rise (fall) when market interest rates are above (below) the
natural rate of interest. In equation (4), policymakers raise
(lower) market interest rates when the price level is above
(below) the target level.
(3) ISt= a[E-rt]
(4) et
= P [P,-P,l
where
= the price level
= changes in the price level
=
the target price level
PT
The Appendix (iii) shows how reduced-form equation (3) is derived
from Wicksell's complete structural model of the inflation
process. Equation (4) is equivalent to equation (2), except the
arguments are price levels rather than inflation rates.
In his writings, Wicksell alternates between adjusting the
market rate in response first, to price movements, and second, to
gaps between actual and target price levels. In so doing, he
leaves doubts as to the exact specification of his monetary
policy rule. In a fuller version of the passage quoted on page 3
above, he seems to favor a policy rule targeting the price level:
[Under a fiat paper standard,] the problem of
keeping...the average level of money prices at a
constant height, which evidently is to be regarded as
the fundamental problem of monetary science, would be
solvable [through] proper regulation of general bankrates, lowering them when prices are getting low, and
raising them when prices are getting high. (Wicksell,
9
1907, pp. 553)
Wicksell's wording is ambiguous; it is unclear whether
interest rates should respond to the price level (relative to the
target) or to the direction of change of the price level. Our
examination of the full quote leads us to believe that Wicksell
meant that monetary authorities should focus on the price level,
as shown in equation (4). In other words, we interpret "prices
are getting lowI1to mean the price level is below the target,
rather than meaning that prices are declining. Evidence
supporting this interpretation comes from page 223 of the second
volume of his Lectures on Political Economy. There, he refers to
a policy consisting of Itaraising or lowering of bank rates ...
in order to depress the commodity price level when it showed a
tendency to rise and to raise it when it showed a tendency to
fall." Obviously, he means that prices should be rolled back to
their former levels after inflation is stopped. On the other
hand, quotes from Wicksell's other writings appear to lean toward
adjusting rates in response to inflationary and deflationary
price changes, such as the rule shown in equation (5); one such
quote is the following:
So long as prices remain unaltered, the banks' rate of
interest is to remain unaltered. If prices rise, the
rate of interest is to be raised, and if prices fall,
the rate of interest is to be lowered... (Wicksell,
1898, p. 189)
Unlike Fuhrer and Moore's equations (1) and (2), equations
(3) and (4) do not result in an explosive system, though neither
10
is the system stable in a strong sense. Rather, the system yields
perpetual oscillations about equilibrium with no convergence
toward it. From any point, the path through (r,p)-space will
follow one of a family of geometrically similar, endless ellipses
as shown in Figure 1. In other words, prices will cycle
ceaselessly about their target equilibrium level, and market
interest rates will cycle about the natural rate. True, prices
are stable on average over the whole cycle, but they are forever
rising and falling. The same is true of interest rates. Such is
not the sort of stability Wicksell envisioned.
III.2 Model #3
If Wicksell intended for the monetary authorities to respond
to price changes, rather than to deviations from the target price
level, then his model would be composed of equations (3) and (5),
below.
Equation (3), repeated here, says that prices rise (decline)
if the market interest rate is below (above) the natural rate.
Equation (5) says that policymakers should raise (lower) market
interest rates proportionally with the rise (drop) in the price
level.
(3) Pt = aCE-ql
(5) *t = rP
11
(3) 15,= aWrti
price
level
(4)
Pl
if
Figure 1
interest rate
t’t
= P[P,-P,l
12
Substituting equation (4) into equation (5) gives equation
(5l), which says that the change in market interest rates is
proportional to the difference between the market interest rate
and the natural rate. This formulation can be depicted in the
phase diagram of Figure 2a. For practical policy purposes,
however, central bankers must rely on (5), since (5') contains
the unobservable natural rate and is therefore nonoperational.
(5') f-,= ra(E-r)
By adjusting the market rate in the same direction that prices
are moving, this rule eventually halts such movements and brings
price changes to a standstill. The system reaches a new
equilibrium price level and interest rate (see Figure 2a). But,
the new equilibrium price level is not the same as the
preexisting one. For example, in Figure 2b, assume that a real
economic shock causes the natural rate to shift from E to rl,
thus introducing a divergence between the market and natural
rates. Prices begin rising, so the monetary authorities respond
by raising the market rate. Eventually, price movements will
cease at,the new equilibrium price level pt. This upward drift in
the price level violates the notion of absolute price stability
and contrasts sharply with Wicksell's statement that the
"fundamental problem of monetary science" is to stabilize the
price level.
P
price
level
interest rate
E
Figure 2a
price
level
i
I
I
P’
7
P
I
I
I
I
I
I
I
I
E
1
PInterest rate
Figure 2b
14
The model composed of equations (3) and (4) "stabilizesItthe
price level (on average)
but produces a perpetual cycling of
prices and interest rates. In the alternative model composed of
equations (3) and (5), inflation or deflation disappear over
time, but the equilibrium price level can drift about,
anchorless. Wicksell was clearly concerned both with attaining
zero inflation and with stabilizing the price level, so it is
unlikely that he would be satisfied with either of these policy
rules.
III.3 Model #4
A clue to how one might make the model stable (in the sense
of always returning prices to a fixed target level) can be found
in the geometry of the system formed by equations (3) and (4).
Figure 3a shows a family of paths through (r,p)-space, given the
of the ellipse is an
parameters a and p(=pl). The tlflatnesstl
increasing function of the ratio P/a. Therefore, assuming a is
so
unchanged, Figure 3b shows a family of paths where p=/32</31,
the ellipses are less flat in Figure 3b than they are in Figure
3a.
One way to stabilize the model is to incorporate a switching
rule that directs policymakers to switch back and forth between
two values of p. Consider the four quadrants formed by the pT and
r lines. In the northwest quadrant, two problems exist: prices
15
(3) P = a[%rt]
(4) p = P2 [P-PTI
where p2</II
(3) p = a(E-rt]
(4) 2 = PI CP-P,l
I
I
I
I
I
I
I
I
I
PTI
I
I
I
I
I
I
f
interest
rate
Figure 3b
Figure 3a
rs
t=
t=
interest
rate
Figure 3c
interest
rate
= a[+r)
p,[p-p,) in NW 61SE
p,[p-p,) in NE L SW
16
are above the target level and they are rising even higher.
Moving into the northeast quadrant, prices are still too high,
but they are declining toward the target level. Similarly, there
is a double problem in the southeast quadrant (prices too low and
getting lower) but only one problem in the southwest (prices too
low). Intuitively, this suggests a new policy rule: the monetary
authorities should react strongly (j3relatively high) in the
northwest and southeast quadrants, and they should react less
strongly (p relatively low) in the northeast and southwest. In
other words, the strength of the interest rate response should
depend on whether the deviation from the target price level has
the same sign as the direction of price changes. Algebraically,
this rule can be represented as follows:
(6)
li- = P(P,W(P-P,)
where P[sgn(p-p,)=sgn(P)l > P[sgn(p-pT)+sgn(P)1
In Figure 3c, the time paths shown in 3a and 3b are
superimposed. By switching the monetary policy reaction parameter
each time F or pT is passed, movement switches back and forth
between the two families of ellipses. Such switching leads to an
orbit (shown in bold) that decays inward toward the equilibrium.
Of course policymakers can't observe the natural rate E, so they
use the direction of price change as a proxy to signal when to
change p.
17
III.4 Model
#S
Another version of the preceding model is produced by
allowing both the gap between actual and target price levels and
the change in prices to enter the policy rule additively. Now,
the change in interest rates equals the sum of changes under
equations (4) and (5). The complete system now consists of
equations (3) and (7):
(3) Pt = a[E-rt]
(7) F = /UP-P,) + rp
Note that the right-hand side of equation (7) consists of
two terms. The last term rp directs the policymakers to adjust
the market rate to halt the price change. The first term p(p-p,)
directs them to undo damage already done by rolling back prices
to target. In other words, equation (7) says that market rates
should be adjusted both to arrest and to reverse price changes.
This model yields a dynamically stable solution [see
Appendix for proof]. Moreover, the rule is operational, and
rather than being saddlepoint stable, will cause the system to
converge to equilibrium from any point, as long as a, /3,and 7
are positive. To map out a phase diagram, as in Figure 4, we
substitute equation (4) into equation (7), yielding:
(7') 2 = P(P-P,) + 7Wf-r)
18
(3) P = a[+rt]
(7) I?=
PT
interest
rate
Figure 4
19
V. Conclusion
We have shown that Wicksell's simple two-equation model can
yield dynamic stability such that policymakers can always restore
prices to target. However, this result requires a fully specified
version of the policy response function. That particular response
function requires authorities to adjust the market rate of
interest in response to two variables, namely the price gap p-p=
and the movement or change in prices dp/dt. Previous versions of
the policy rule have incompletely specified Wicksell's policy
response function, calling for policymakers to react to one
variable or the other, but not to both. Since neither variable
alone is sufficient to ensure stability, it is not surprising
that Wicksell's model has been judged dynamically unstable by
some observers. A close reading of Wicksell, however, suggests
that while he postulated two separate policy reaction or
interest-rate adjustment functions -- one containing the price
gap p-pT as an argument and the other containing dp/dt -- he
never took the final step of incorporating both in a single
function. Apparently realizing that a price-change feedback rule
would not stabilize the price level, he then postulated the
price-gap feedback rule, evidently thinking the latter rule would
stabilize prices. He did not realize that both the price-change
and price-gap variables were necessary to render the feedback
rule powerful enough to stabilize prices. Though Wicksell never
explicitly recommended incorporating both variables into a single
20
policy reaction function, we can do so here without having to
look beyond Wicksell's own words.
21
APPENDIX
.
1.
Why Model #1 Explodes
Formal analysis of the model's stability properties requires
expressing it in matrix form and then examining the signs of the
determinant and the trace of the coefficient matrix. In matrix
notation, Fuhrer and Moore's two equation system [equations (1)
and (2)] is:
1I 1I
aE
+
-PKf
1
J
Stability of equilibrium requires that the determinant of
the coefficient matrix be positive and the trace negative. This
model passes the first test, but not the second. The determinant
of the coefficient matrix /3a is positive, but the trace a is
positive, not negative. A positive determinant and positive trace
can mean two things: Either [l] the roots of the system's
characteristic equation are real and positive implying
monotonically explosive paths; or [2] the roots are imaginary
with real parts positive, implying explosive cycles. Either way,
the system diverges progressively from equilibrium. If the trace
were zero, the system would orbit ceaselessly, but not
explosively. The trace a consists of the coefficient on the
inflation rate in the price response equation. That inflation-
22
rate variable is only present because Fuhrer and Moore have
assumed,
contrary to Wicksell, the existence of a Fisher Effect.
In other words, without that assumption, the system would not
explode.
ii.
Why
Model
#2 Yields
a Steady,
Nonconvergent
Ellipse
Model #2 [equations (3) and (4)] is shown here in matrix
form:
‘1
1 -al1 11
1 1I
*I, =I0
IPI
+ I
I?
P
0
.1 1 1 11 1 1
aF
(24.2)
r
-PP,
L
Here, we use the price level instead of the inflation rate.
As in (A.l), the determinant of the coefficient matrix is
positive. However, since the matrix contains no term equivalent
to Fuhrer and Moore's Fisher Effect, the trace is zero, so this
system loops around endlessly, neither damping nor exploding.
Following is a more -intuitive geometric exposition:
To find the slope, or direction of motion at a given point,
we divide Equation (3) by Equation (4), so that:
(A.3) P/t
= dp/dr
= [~(~-r)llUW?-pT)l
23
I
I
I
I
I
I
2
Figure
A.1
24
In Figure A.l, point [l] is arbitrarily chosen. Points [l]
and [Z] are equidistant from the vertical line at r. Points [l]
and (33 are equidistant from the horizontal line at pT, as are
points [2] and [4]. Thus, these four points form a rectangle
which is symmetric with respect to the lines intersecting at the
equilibrium point. Algebraically, these points are: [l] (r,p);
[2] (2%r,p);
[3] (r,2p,-p); and [4] (2r-r,2p,-p). Evaluating
Equation (A.3) at any ‘of these four points, the absolute value of
dp/dr is the same. That is, the phase diagrams are symmetrical
across the E and pT lines. A upward path in the NW quadrant
(defined by these two lines) will be mirrored in the downward
path through the NE quadrant, and so on. Further, the family of
ellipses are geometrically similar. This is true because for
@>O,
the slope at [5] (%c$(r-r),p,+@(p-p,)) is the same as at [l].
iii.
Wicksell's
Full
Structural
Model
of Price
Stabilization
The two-equation model dp/dt = a(E-r) and dr/dt = @(p-p,) +
r(dp/dt) [equations (3) and (7)] is but the condensed or reducedform version of Wicksell's complete structural model of the
inflationary process -- his famous cumulative process model. The
purpose of this section is to spell out that model in some
detail.
Wicksell's cumulative process model assumes full employment
and describes the interaction of the markets for goods, credit,
and money. The model consists of 13 equations linking the
25
variables investment,I, saving S (both planned or ex ante real
magnitudes), market (loan) rate r, natural rate E, loan demand
I&
loan supply L,, excess supply of money X, excess aggregate
demand E, money-stock change dM/dt, price-level change dp/dt, and
market rate change dr/dt. Of these, saving and investment are
taken to be increasing and decreasing linear functions of the
market rate of interest, the presumption being that higher rates
encourage thrift but discourage capital formation.
Equation A.3 states that real investment I exceeds saving S
when the market rate of interest falls below its natural
equilibrium level E (the level that equilibrates saving and
investment):
(A.4) I-S = a(F-r)
Here the coefficient a relates the investment-saving gap to the
rate differential that creates it. Since Wicksell assumed that
banks lend only to investors and that all investment is financed
by bank loans, equation (A.5) states the (investment) demand for
loans Ln as:
(A.5) Ln = I(r)
where I(r) is the schedule relating desired investment spending
to the market or loan rate. Equation (A.6) expresses loan supply
26
L, as the sum of household saving S(r)--all of which Wicksell
assumes is deposited in banks--plus new money dM/dt created by
banks in accommodating loan demands:
(A-6)
L, = S(r) + dM/dt.
Equation (A.7) states the market-clearing condition in the credit
market (i.e., the market for bank loans):
(A.7) L,.,
= L,.
Substituting (A.5) and (A.6) into (A.7) yields:
(A.8) I-S = dM/dt
which says that, assuming banks create money by way of loan,
monetary expansion occurs when they lend more to investors than
they receive in deposit from savers. Thus the investment-saving
gap is matched by new money created to finance it.
But since the demand for money to hold at existing prices
and real incomes has not changed, the newly created money dM/dt
represents an equivalent excess supply of money X according to
the expression
(A.9) dM/dt = X.
27
Cash-holders then attempt to get rid of their excess cash
holdings by spending them on goods. As a result, the excess
supply of money X then spills over into the commodity market in
the form of an excess demand E for goods as aggregate expenditure
at full employment outruns real supply:
(A.lO) X = E.
This excess demand bids up prices, which rise by an amount dp/dt
proportionate to the excess demand,
(A.ll) dp/dt = kE.
Substituting equations (A.4), (A.8), (A.9), and (A.lO) into
(A.ll) yields:
(A.12) dp/dt = ka(r-r)
or
(A.12') dp/dt = a(E-r) where a = ka.
Equation (A.12') says that price-level changes stem from the
discrepancy between the natural and market rates of interest.
Adding the interest-rate adjustment or policy-reaction function
28
(A.13) dr/dt = /UP-P,) +
r(dpldt)
yields the second reduced-form equation of Wicksell's model.
In sum, equations (A.4)-(A.13) constitute the full
structural model underlying the reduced-form model consisting of
equations (A.12') and (A.13).
iv. Stability
of Wicksellls
Rule
#4
Wicksell System #4, composed of equations (3) and (7), is
represented here in matrix form:
(A.14)
. .
=
The determinant of the coefficient matrix is positive and
the trace is negative, yielding convergent cycles or even
monotonic paths to equilibrium.
29
References
Black, Robert P. (1990). "In Support of Price Stability." Federal
Reserve Bank of Richmond Economic Review. (January/February)
76(l).
Chiang, Alpha C. (1984) Fundamental Methods of Mathematical
Economics, Third Edition. New York: McGraw-Hill.
Fuhrer, Jeff and George Moore (1989). "The Stability of
Wicksell's Monetary Policy Rule.tt (November) Paper #96,
Finance and Economics Discussion Series. Division of
Research and Statistics, Division of Monetary Affairs.
Washington, DC: Federal Reserve Board.
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