108
5 The Stalic Classical and Keynesian Models
production function is initially y = 2nO. 5 , but then a technological
innovation shifts it to y = 3nO. 5 • For concreteness, also suppose that
labor supply behavior is described by n = 100 WIP. Determine
algebraically the equilibrium values of n* and y* before and after
the technological improvement. Also, represent the change graphically, using diagrams like those of Figure 5-14. What would be the
qualitative effects on r * , p "', and (WI P) * according to the classical
model?
•
Steady Inflation
References
Bailey, Martin J., National Income and the Price Level, 2nd ed. (New York:
McGraw-Hill Book Company, 1971) Chaps. 2, 3.
Barro, Robert J., Macroeconomics. (New York: John Wiley & Sons, Inc.,
1984) Chaps. 6-8, 19.
Brunner, Karl, and Allan H. Meltzer, "Money and Credit in the Monetary
Transmission Process," American Economic Review Papers and Proceedings
78 (May 1988), 446-5l.
Dornbusch, Rudiger, and Stanley Fischer, Macroeconomics, 3rd ed. (New
York: McGraw-Hill Book Company, 1984) Chaps. 11, 12.
Henderson, James M., and Richard E. Quandt, Microeconomic Theory, 3rd
ed. (New York: McGraw-Hill Book Company, 1980).
Keynes, John M., The General Theory of Employment, Interest and Money.
(London: Macmillan Publishing Company, 1936).
Patinkin, Don, Money, Interest, and Prices, 2nd ed. (New York: Harper &
Row, Publishers, 1965), Chaps. 9-15.
Sargent, Thomas J., "Beyond Demand and Supply Curves in Macroeconomics," American Economic Review Papers and Proceedings 72 (May 1982),
382-89.
Sargent, Thomas J., Macroeconomic Theory. (New York: Academic Press,
Inc. 1979), Chaps. 1,2.
6.1
Introduction
At this point we begin the transition from static to dynamic analysis. In
Chapter 5 the analytical method was to compare static equilibrium
positions of the economy prevailing before and after some postulated
change in an exogenous variable (or some postulated shift in a behavioral relation). The dynamic behavior of the variables in the time
interval after the postulated change, but before the attainment of the
new equilibrium position, was not considered. In effect, then, this
analysis was timeless; we simply did not keep track of the passage of
time. But it would clearly be extremely desirable to be able to describe
the behavior of an economy functioning in real time-to explain movements in variables from quarter to quarter, for example. Full-fledged
dynamic analysis is much more difficult than comparative statics, however, so we shall have to proceed slowly. The first step, accordingly,
will be to develop the concept of a steady-state equilibrium, in which
some of the system's variables change over time but in a restrictive
manner. In particular, in steady-state analysis all variables are required
to be growing at a constant rate-at a rate that is unchanging as time
passes. For some of the variables that rate will be zero; that is, the
values of the variables themselves will be constant. The values of the
other variables, by contrast, will be changing through time, but will be
doing so at a rate that is constant. This rate may in principle differ from
variable to variable; all that the steady-state concept requires is that
each variable be growing at some constant rate. In many cases, however, the same rate will apply to all variables that are growing at nonzero rates.
By a "growth rate" what is meant here is the relative rate of change
(through time) of a variable. Thus in terms of variables that change
continuously through time, the measure of the growth rate is not the
109
110
6
Steady Inflation
6.1
time derivative, but that derivative divided by the value of the variable
itself. For the variable x, the growth rate would be (1/x) dx/dt, not
dx/dt, for instance. The amount of the change per unit time is measured relative to the magnitude prevailing.
In our discussion we shall not, however, treat our variables as changing continuously through time. Instead, we will work with discrete
periods of time-periods such as months or quarters or years-pretending that the value of any variable is the same throughout each
period but liable to change "between" periods. From this perspective,
the growth rate of x between periods t - 1 and t might possibly be
defined as (x, - X'_I)/ Xt-b where XI is the value ofx in period t and Xt-l
its value in period t
1. It would be just as logical, however, to define
the growth rate of x as (x t XI_I)/X t, or as (x t - Xt_I)/O.S(x, + Xl-I),
instead of the expression in the preceding sentence. Indeed, reference
to the continuous-time concept (l/x) dx/dt suggests that a definition
preferable to all of the foregoing would be
growth rate of X between t - 1 and t = .:l log X t
log X t - log X,-I
=
X
(1)
log-' .
Xt-I
Here log X t means the natural logarithm, that is, the logarithm to the
base e = 2.713, of the variable Xl' The symbol.:l attached to any dated
variable indicates the difference between the dated value and that of
the previous period, so that (for example) .:lz, means z, - Z,-l' In the
present case, then, .:llog x, means log x, log X,-I. Finally, the last
equality in (1) obtains as a result of the properties of logarithms: for
any number A and B, log (A/B) = log A - log B.
That the definition of the growth rate given by expression (1) is
consistent with the continuous-time concept may be seen by recalling
that the derivative (w.r.t. x) of log X is 1/x. Thus in continuous time the
concept (l/x) dx/ dt is equivalent to d log x/ dt, while the latter has
.:l log x, as its discrete-time analog.
The main advantage of working with expression (1) rather than the
other discrete-time possibilities stems from the following fact: because
of the properties of logarithms, the growth rate of a product such as YZ
is equal to the sum of the growth rates of Yand Z. To prove that, note
that if X t == YrZt, then
.:llog XI
log XI
log X t- 1
= log Yr + log Z,
log (YrZ,) - log (Yr-IZt-l)
log Yr-l - log Z'-l
Introduction
111
o' - - - - " - - - - - - - - - - - - - T i m e
o
Figure 6·1
Similarly, it is true that if Xl
Y;/ Z" then .:l log X t = .:l log Yr
.:llog Z,. That is, the growth rate of a ratio of two variables is the
difference in the growth rates of these variables. These identities,
which would not hold under other definitions, are highly useful.
Given our definition of growth rates, the concept of a steady-state
equilibrium implies that the steady-state time path of a variable Yl must
conform to an equation of the form
.:llog y, = b,
(3)
where b is some constant. This in turn implies that the steady-state
values of y, are describable by an expression of the form
log y, = a + bt,
(4)
where b is the same constant as in (3) and where a is a constant that
equals the logarithm of the value of Yt in period t
O. (That is,
a = log Yo.) Consequently, any steady-state' time path can be represented graphically as in Figure 6-1. The slope of the line representing
log Yt gives the growth rate of Yt. 1
The logarithmic definition of growth rates (or rates of change) that
we have adopted for discrete-time variables may seem strange to some
readers, who are familiar with some alternative concept such as
(x,
X,-l)/XI-l' The main reason for our choice is that algebraic manipulations with the .:llog x, definition are simpler and clearer. Also, our
concept accords well with the continuous-time measure (l/x) dx/dt .
These reasons seem adequate to justify our convention, especially since
log Y; - log Y;-l
+ log Zt - log Z'_I = .:llog Y; + .:llog Zt.
Strictly speaking, we should plot values of log y, only at discrete points for
0,1,2, ... when the discrete-time representation is being used. For convenience,
however, we will use continuous graphs to depict discrete-time variables.
I
(2)
t
112
6
6.3 Inflation in the Classical Model
Steady Inflation
II log X t is numerically close to (x t Xt-l)/Xt-l whenever the latter
for growth rates of
number is reasonably small relative to 1.0. That
the order of magnitude of 0.05,0.10, or 0.15 (in percentage terms these
are growth rates of 5, 10, and 15 percent), the values of (XI Xt-l)/X1-l
and ~ log X t are approximately the same. That this must be true can be
verified by noting that our concept lliog X t equals
t
"" 10g(_X_t
10g_X_
Xt-l
Xt-l
X_t-_l + 1) =
10g(1 + -,Xt_-_X.::-t--=-J)
Xt-l
(5)
Xt-l
and recalling that, for small values of the number z, log (1 + z) is
approximately equal to Z.2 Thus ~ log X t is approximately equal to
(x t - Xt-l)/Xt-l when the latter is small.
6.2
Real Versus Nominal Interest Rates
At this point it will be convenient to introduce the heretofore ignored
distinction between real and nominal interest rates. This distinction is
important in the context of steady-state analysis because a steady-state
equilibrium may feature a constant but nonzero rate of change of the
price level, that is, a nonzero inflation rate. Since this nonzero inflation
rate will be constant over time in a steady state, it is implausible to
believe that the economy's individuals will not recognize the existence
of inflation and correctly anticipate that it will continue to prevail in the
future. Indeed, the requirement that inflation (and other variables) be
correctly anticipated is a natural assumption for steady-state analysis,
and is one that we shall henceforth utilize. 3
But with the price level changing over time, the economically relevant rate of interest on a loan with provisions specified in monetary
terms depends on the anticipated inflation rate. Imagine, for example,
a loan of $1000 for a period of one year, with the provision that the
borrower must pay the lender $1100 at the end of the year. In monetary
terms, the rate of interest on this loan is 0.10 (or 10 percent). But if the
lender expects the price level to be 10 percent higher at the end of the
year than at the time of the loan, he expects to be repaid an amount
that is worth in real terms only just as much as the amount lent. Thus to
him the (expected) real rate of interest on the loan is zero. If, however,
he expected the price level to be only 4 percent higher, he would
2 Technically, for z close to zero the first-order Taylor series approximation to log
(1+z)is1.
3 In effect, then, our definition of a steady state is an ongoing situation in which every
variable is growing at some constant rate and in which expectations are correct.
113
anticipate receiving a payment worth in real terms 106 percent of his
loan. In this case, the real interest rate as viewed by the lender would
be 0.10 0.04 = 0.06 or 6 percent.
From this type of reasoning we see that in general the real rate of
interest on a loan specified in monetary terms is the nominal (monetary) rate of interest minus the expected inflation rate for the life of the
loan. In particular, if R, is the nominal rate of interest on one-period
loans made in t, and 7T', is the inflation rate expected between t and
t + 1, the real rate r, will be given by
(6)
The distinction represented in this equation reflects one of the most
important ideas in monetary economics. It will be utilized extensively
in the discussion that follows. For other discussions of this distinction,
and the material of this chapter more generally, the reader is referred
to Bailey (1971, Chap. 4) and Barro (1984, Chap. 8).
6.3
Inflation In the Classical Model
Having introduced some necessary tools, we are now in a position to
develop the analysis of a steady, ongoing inflation. This analysis will be
conducted in the context of the classical model. We begin with this
model, rather than the Keynesian version, for two reasons. First, it is
clear that, because of its assumption of a given money wage rate, the
Keynesian model would have to be seriously modified to permit a
sensible analysis of ongoing inflation. 4 Second, since steady-state comparisons represent a type of "long-run" analysis, designed for the
elucidation of policies maintained over a long span of time,S the
assumption of wage flexibility is not at all inappropriate-as many
economists believe it would be for "short-run" issues.
Let us begin the analysis by assuming that the monetary authority, in
a classical economy with no growth of population and no technical
progress, causes the money stock to increase period after period at the
rate po. Thus we assume that
(7)
where M, is the money stock in period t. From the discussion in Section 6.1 we know that the path of M, can then be depicted as in Figure 6-2, which is drawn under the assumption that po is positive. Now,
4 Which would presumably be accompanied by ongoing increases in the nominal wage rate.
5 That this is so will become apparent subsequently.
6.3
114
Inflation in the Classical Model
115
6 Steady Inflation
the capital stock is constant over time, an assumption that we will want
to relax subsequently.
With Yt equal to the constant value y*, determined in the manner just
described, it remains to determine the behavior of Pr and r t • But from
the discussion in Section 6.2, we know that we must consider whether it
is the real or nominal interest rate that appears in the IS and LM
equations. As it happens, the answer is that both appear-or, more
precisely, that Rt appears in one equation and rr in the other. In
particular, the analysis of Chapter 3 indicates that it is Rr that belongs in
the LM equation/ which we therefore rewrite as
M,
' - - - - - - - - - - - - - - - - - - Time
Figure 6·2
in a situation such as this, it seems natural to guess-on the basis of our
knowledge of the comparative static properties of the classical modelthat the price level Pr will also grow over time at the rate f.k. In that
case, the path of Pr would be as shown in Figure 6-2, where the slope of
the path of log Pr is the same as that of log M,. 6
To verify that Pr would in fact behave in this conjectured manner,
we must consider the complete model. In Section 5.6 the latter was
summarized with the five following equations:
Y = C(y - 7',r) + I(y,r) + g,
M
P = L(y,r),
y = fen),
- = L(y" RJ
Pr
Reflection on the nature of the IS function as discussed in Section 5.2
indicates, however, that it is the real rate r, that is relevant for saving
and investment decisions, basically because people care about real
magnitudes. The IS curve can then be represented as
Yr = C(Yr
T, rt )
+ I(y/> rt ) + g.
Here we have written the tax and government spending variables as
constants, which they must be (with no growth in population) for a
steady state. Solving this last equation for rh we then obtain
(9)
where the g and T values are suppressed for notational simplicity.
Our next task is to bring the IS and LM equations together as in the
various diagrams of Chapter 5. That task is rendered more difficult
by the appearance of R t in (8) and r, in (9), but this problem can
be overcome by means of expression (6). From (6) we· know that
R, r, + 1Tr, so we can substitute (9) into the latter and obtain
Rr = O(Yr) + 1T1•
f'(n) = ; ,
Since we are now assuming that there is no change in population or
technology as time passes, we can use the final three of these to determine the values of y t, n" and ~/ Pr that will prevail in our steady
state. Note that in doing this, however, we are implicitly assuming that
6 Note that it is unimportant whether the log P, path is drawn above or below log M, in
Figure 6-2; the relative vertical locations depend on the units of measurement of the
economy's single commodity and are therefore arbitrary.
(8)
(10)
For a given value of 1Tr, then, equations (8) and (10) may be solved for
Rr and Pr since Mr is exogenously set by the policy authorities and Yt is
given as y*.
Graphically, we proceed as shown in Figure 6-3. There the vertical
axis measures R r , so the LM curve is plotted as usual. Then the function
7 This is demonstrated formally in the analysis of Section 3.3. The basic idea can.
however, be expressed very simply as follows: since the nominal interest rate paid on
money is zero and that on "bonds" is R" the difference between these is R,. Similarly,
the real interest rate on money is 0 -1r, = -11', while the real rate on bonds is R, 1r" so
the difference between these is also R,. It is R" therefore. that measures the marginal
opponunity cost of holding wealth in the form of money, which is the relevant determinant.
116
6 Steady Inflation
6.4 Comparative Steady States
117
R,
!l(y,) + "-,
!l(y) +,,!l(y,)
L-____________________________ y,
!l(y)
L-________-L_________________ y,
y'
Figure 6-3
Figure 6-4
O( Yt) is drawn in place; it shows where the IS curve would be if 7Tt were
equal to zero. Finally, the actual IS curve is obtained by adding vertically the quantity 7Tt to this last curve at each point, as suggested by
(10). The relevant intersection occurs at point A.
To depict a steady-state equilibrium we modify the analysis of Figure 6-3 in three ways. First, we recognize that steady-state output will
be given as y*, so the relevant intersection (like point A) must lie
on a vertical line through y*. Second, the value of y, R, 7T, and M/ P
are recognized to be constant over time. 8 Third, the value of 7T must be
the same as ~ log Pr, and for M / P to be constant ~ log Pr must be the
same as f.L. Consequently, 7T = f.L in the steady state with no output
growth. 9 Recognizing these modifications, we can depict a situation of
steady-state equilibrium with an inflation rate of 7T = f.L as shown in
Figure 6-4. Here the value of R is determined by the intersection of the
IS curve R = O(y) + 7T and the vertical line y = yO. The LM curve
must of course pass through that same point. So the role of the LM
relation is--just as in the static analysis of Section 5.6-only to determine the value of real money balances M/P, given y and R. With M
exogenous, that value determines the price level.
The main conclusion of the foregoing is that we have been able to
assemble the components of our model in a way that is internally
consistent. This verifies, then, that Pr does indeed grow at the rate
f.L in the steady state, as conjectured above. Figure 6-4 provides, conse8 We have already discussed why y and 'Tr will be constant. That R must be constant
is a consequence of the fact that " must be constant-which will be explained in
Section 6.6-and that R, = r, + 'Tr,. Then with y and R constant, M/ P will be con.
stant by equation (8).
9 If y is growing because of technical progress or population growth, then p. WIll grow
less rapidly than M,. This will be explained in Section 6.6.
quently, a representation of a steady-state equilibrium. The latter
differs from a static equilibrium in that the values of M and Pare
continually changing from period to period. In particular, they are
growing at the rate f.L, that is, ~ log Pr = ~ log Mt = f.L. This fact can be
emphasized by appending to Figure 6-4 a companion diagram that
would look exactly the same as Figure 6-2.
6.4
Comparative Steady States
The main purpose of a diagram like Figure 6-4 is to facilitate the
comparison of different steady states, just as the main purpose of a
static equilibrium diagram like Figure 5-14 is to serve as a prelude to
static equilibria comparisons such as that of Figure 5-15. Accordingly,
we shall now carry out, as an example, a graphical analysis of a
comparison of two steady states.
This is done in Figure 6-5. There the initial steady-state equilibrium
is one in which the money stock growth rate is f.L0, while the second
(steady-state) equilibrium is with a higher money growth rate, f.Ll. The
. t he
value of y* is not affected by the change 0 f f.L from f.L to f.L 1 , nor IS
position of the O(y) curve. Thus the relevant IS curve in terms of the
nominal interest rate is higher in the second steady state: O(u) + 7T 1
lies above O(y) + 7T O• The equilibrium value of the nominal interest
rate is therefore higher also; Rl > RO. Indeed, since the real interest
rate r is determined by the intersection of r = O(y) and y = y*, it is
the same in the two steady-state equilibria; consequently, R 1 exceeds
RO by the same amount as 7T 1 exceeds 7T O, which is in turn the amount by
°
118
6 Steady Intlation
6.4 Comparative Steady States
119
R
L-_ _ _ _ _ _ _ _ _
~
_ _ _ _ _ _ _ _ _ _ _ _ _ Tlrne
I...-_ _ _ _ _ _...L.-_ _ _ _ _ _ _ _ _ _ _ y
y'
Figure 6-5
which JLl exceeds JL o. The higher is the money growth rate, therefore,
the higher is the nominal interest rate-and by the same amount.
One of the main implications of the analysis in Figure 6-5 is that the
value of real money balances in the second steady-state equilibrium is
smaller than in the first; that is, (M/ p)1 < (M/ P)o. That this must be
so is clear from consideration of the diagram-where fi(y*) + 71"1 is
above fi(y*) + 71"°-since the position of the LM curve is farther to
the right the higher the M / P. Thus in a comparison of steady states, the
one with the higher growth rate of the nominal money stock will feature
a smaller quantity of real money balances held by the economy's
individuals! This somewhat paradoxical conclusion is one of the key
propositions of neoclassical monetary analysis.
The comparison of steady-state equilibria in Figure 6-5 may be
further investigated by means of a companion diagram that plots time
paths of log M t and log Pr. In Figure 6-6 it is assumed that the initial
steady state (with JL
JL o) prevails for time periods up to t
t+. Then
at time t+ the growth rate of M, changes to JLl. This is represented in
Figure 6-6 by the increased slope of the log M, path for periods after
t+,IO The time paths for log F; in the initial and final steady states are
also shown in Figure 6-6. Since M/ P is constant in each of these
10 While the growth rate changes abruptly at [
[T, the value of the money stock M,
itself does not jump upward or downward at that instant in the example depicted.
Figure 6-6
steady-state equilibria, the slope of log PI is the same as the slope of log
M/ in both cases. But since the value of M / P is smaller in the final
equilibrium (as emphasized in the preceding paragraph) the height of
the log Pr path must be higher, relative to the height of the log M, path,
reflecting an increased value of P/ M.
From the way in which the log Pr path is drawn in Figure 6-6, it
appears that the price level jumps upward in a discontinuous fashion at
time t+. This aspect of the diagram should not be taken literally,
however, for the diagram is intended only to describe the comparison
of steady-state positions. Although our analytical apparatus tells us a
good deal about this comparison, it does not tell us enough to permit
any reliable description of the transition between the initial and final
steady states. In this respect, comparative steady-state analysis is similar to comparative statics: we do not know how the system behaves
dynamically during the transition from one equilibrium to another.
In particular, it is possible that the time path of F; would move
toward the new steady-state path only gradually, as indicated by the
dashed curve in Figure 6-6 rather than by jumping upward at t*, But
notice that if this is the case, then during the transition F; must grow
Jaster than Mt for some time. An increase in the money growth rate
will, therefore, require either a jump in the price level or an inflation
rate that temporarily exceeds the new (higher) money growth rate.
Analogously, a sustained decrease in an economy's money growth rate
requires either an abrupt fall in the price level or an interval of time
120
6 Steady Inflation
6.5 Analysis with Real-Balance Effects
during which the inflation rate is lower than the new (lowered) money
growth rate.
These last statements depend, it should be said, on the assumption
that there is no upward or downward jump in the level of the money
stock-no discontinuity in the MI path. For if there were an appropriate downward jump in M I , then ~ would not have to jump upward for
M/~ to fall as required.
In addition, it should be pointed out that our analysis assumes that
the change in the money growth rate that takes place at t+ comes as a
complete surprise to the economy's private individuals. In each period
before t+ they confidently believe that the money growth rate will continue to equal J.L0 indefinitely. If the change from J.L0 to J.L! at t+ were
foreseen, the price level would begin to rise toward its new steady-state
path before the change in .1 log MI actually occurs.
To conclude this section, it may be useful to emphasize the importance of distinguishing clearly between statements pertaining to the
properties of one steady state and statements comparing alternative
steady states. In particular, while growth rates of all variables in a given
steady state are constant over time, these may change if the economy
moves to a different steady state. In the first steady state of Figures 6-5
and 6-6, for example, .1 log ~ is constant at the value J.L0 . In the second
steady state it is again constant, but at a different value, namely J.Ll . If
one wants to say that some variable will have the same value in
different steady states, the proper terminology is not "constant" but
"invariant across steady states."
6.5
Analysis with Real-Balance Effects
In footnote 15 and Problem 5 of Chapter 5, the possibility was mentioned that households' consumption choices depend positively on their
real wealth, in addition to the y and r determinants. In particular, the
consumption function c = C( y - 'T,', M / P) was introduced, with M / P
reflecting the monetary component of real wealth held by households.
That change in specification gives rise, by way of analysis paralleling
that of Section 5.2 and 6.3, to an IS function of the form
'1
=
n( ~) +
Yl,
7Tt ,
121
R
Figure 6-7
different values of J.L = 1T). That is, the classical model ceases to have
the property of "superneutrality," to use a term that is often employed
to describe a model in which all real variables except M / P are independent of the inflation rate.
To demonstrate graphically that supemeutrality does not prevail
when the "real-balance effect" is important,l1 consider Figure 6-7.
There each of the IS functions includes M/P as an argument, as
required by (ll). Thus when an increase in the money growth rate
increases 7T from 7T O to 7T!, the resulting decrease in M/ P itself tends to
cause the IS curve to shift to the left. This type of shift also pertains to
n(y, M/
which locates the position of the IS cur~e in the absence of
inflation.! Thus the decrease in M/P to (M/p)1 shifts the n(y,M/p)
curve leftward and drives its intersection with y y* downward. This
implies that the real interest rate ,1 is lower than ,0, as suggested
PI'
(11)
where now n is a function of two variables. With the classical model
thus modified, it ceases to be true that the real interest rate, will be
the same in steady-state equilibria with different inflation rates (i.e.,
11 A model is said to feature a "real-balance effect" when the consumption function
includes M/ P as an argument. This term was used extensively by Patinkin (1965).
12 Note that l.1[ y', (MIP)O) in Figure 6-7 does not, however. refer to the zero
inflation equilibrium, for (M/P)O corresponds to the value of real balances that obtams
with 11. 11.0 > O.
122
6 Steady Inflation
above.13 The nominal rate increases, but by less than the difference
bet",:e~n p,°.and p,I. Whether this type of effect is of major importance
emp~n.~lly IS unclear, but the reader should be aware of the theoretical
pOSSIbIlIty.
6.6
Analysis with Output Growth
All of the foregoi~g discussion in this chapter has been conducted
under the ass~mptlOn that real output is constant over time. In most
actual economIes, b~ contrast, output tends to grow as time passes. We
need, then, to consIder how the analysis must be modified to reflect
that fact. 14
In thi.s context it is useful to distinguish three possible reasons for
growth In aggregate output. They are
~!) Growth of population (and, therefore, labor supply).
(ll) Growth of the capital stock.
(iii) Technical progress (I.e., an increase in the amount of output
possible from given inputs of labor and capital).
B~ far the simpl~st case t.o analyze is the one in which only (i)
applIes-all ~rowth IS due to Increased population. In this case we can
proceed by sImplr reinterpreting ail the equations and diagrams above
so that the quantIty measures (I.e., y, n, c, M, etc.) are understood to
refer to per capita va.lues rather than aggregates. Then, with p, being
understood as ref~rnng to t~e growth rate of the money stock per
person, t~e equatIons and dIagrams continue to be applicable. For
example, If the population (and output) growth rate were 0.02, and the
total money stock growth rate were 0.10, the per capita value for p,
would be.O.08. This would be the steady-state value of 7T and R would
exceed r In t~e steady state by 0.08 (i.e., by 8 percent).
If growth In the capital stock occurs, however, matters become
somewhat mo!e c~mphcated. Within a single steady state there is no
need for n:odlficatlOn of the analysis in the absence of technical progress, for III that case the steady-state condition requires that total
13 If we were permitting the capital stock to change across steady states, the lower
value .of r would Induce a greater value of k. This would in turn permit a greater
quantity of output for any given input of labor.
'
,
14 For a useful discussion, see Barro and Fischer (1976).
6.6
Analysis with Output Growth
123
15
capital and labor must grow at the same rate. Thus the quantity of
capital can be thought of in per capita terms and the argument of the
preceding paragraph can be repeated. But for comparisons across
steady states, this type of argument will be valid only if the values of the
capital/labor ratio are not different in the two steady-state equilibria. If
the model is of the simpler type discussed in Section 6.4, that condition
may be satisfied. But if real-balance effects are important, as discussed
in Section 6.5, the different steady states may feature different real
interest rates, and these will correspond to different values of the
capital stock per person (or per unit of labor). Consequently, valid
comparative steady-state analysis cannot be conducted by means of the
per capita reinterpretation device. A more extensive overhaul of the
analysis is needed.
When technical progress is recognized, full-fledged steady-state
analysis is still more complex. Indeed, unless the technical progress is
of a special type known as Harrod-neutral (or "labor augmenting"),
steady-state growth is not possible. For an excellent discussion of this
and related topics, the reader is referred to Solow (1970).
There is a useful result that can easily be obtained, however, without
going into the complexities of steady-state analysis that recognizes
capital accumulation and technical progress. In particular, suppose that
the aggregate (not per capita) money demand function is of the form
M,
- = L(y"R1),
(12)
r:
where Mt and YI refer to aggregate magnitudes, and suppose that
aggregate output grows steadily (for whatever reason) at the rate 11.
Then we can show that if a steady-state equilibrium prevails, the
inflation rate will be p, 1'1", where 1'1 is the elasticity of L with respect to y,. The best way to see that is to consider an approximation to
L(y" R t ) of the Cobb-Douglas form e'Y°y/1R,Y', where 1'1 is the elasticity
in question. 16 Then taking logs, we have
(13)
log M, log = 1'0 + 11 log y, + 12 log R,
or, equivalently,
Ll log M, - Lllog
11 Ll log y, + 12 Lllog R,.
(14)
r:
r:
15 Output is given by y,
F(n" k,), and the function F is typically assumed to. be
homogeneous of degree 1. That property implies tha~ t!Je rate of output growth IS a
weighted average of the growth rates of n, and k,. But It IS shown In footnote 17 that k,
must grow at the same rate as output-so fI, must do the same.
.
16 This is, of course, the form suggested by equation (20) of Chapter 3. We could Just
as well use instead the form given by equation (26) of Chapter 3, which corresponds
more closely to the analysis in Chapters 7 and 8.
124
6 Steady Inflation
6.7
Now in our steady state it is true that Illog Rr = 0,17 Illog Yt = v, and
Illog Me = I.L by assumption. Equation (14) then shows that
Il log F; = I.L
'Yl v,
The Welfare Cost of Inflation
125
R
(15)
which is the result stated above.
6.7
The Welfare Cost of Inflation
The analysis of the preceding sections suggests that the pace of a
steady, anticipated inflation has little effect on the values of most real
variables including per capita income, consumption, and the real rate
of interest. To help bring out the main points, let us ignore growth and
suppose that there is no significant real-balance effect, in which case the
above-mentioned variables are entirely unaffected by the rate at which
steady inflation proceeds. A natural question to ask, then, is whether
the rate of inflation is of any consequence in terms of the welfare of the
society's individuals. After all, these individuals care-as we have
emphasized in several places-only about real magnitudes. So why
should inflation be of any concern whatsoever, provided that it is
steady and anticipated?
From a practical perspective, actual inflation is of importance largely
because it is usually irregular in pace and largely unanticipated. But in
response to the foregoing question, which is of considerable theoretical
interest, we note that there is one real variable that is not invariant to
the inflation rate but, instead, takes on different values across alternative steady states with different rates of money creation and inflation.
In particular, the level of real money balances depends negatively, as
Figure 6-5 illustrates, on the prevailing inflation rate. But that suggests
that there are welfare effects of inflation, for a relatively small level of
real balances implies that a relatively large amount of time and energy
must be devoted to "shopping" for any given level of spending. Since
leisure is desired by individuals, it follows that higher steady inflation
rates lead to lower utility levels for society's individuals.
17 It was promised above that we would explain why the real rate of interest must be
constant in a steady state (thereby implying that R, will also be constant). Roughly
speaking, the idea is that consumption and investment must grow at the same rate for
their sum (and thus output) to grow at a constant rate. (Why? If one .grew at a faster
rate, its share in output would increase, so output growth would Increase). Also,
investment and capital must grow at the same rate, because of theIr re~atlonsh1p. Thus
capital must grow at the same rate as output-the capItal/output ratIo must be co~­
stant. But this implies a constant marginal product of cal?ltal, and the latter IS m
equilibrium equal to the real rate of interest. For more detaIls, see Solow (1970).
L - - - - -____
~~L-
__________
~
__ m
Figure 6·8
In light of the foregoing conclusion, it is of considerable interest to
develop a technique for measuring the magnitude of the welfare cost of
a steady, anticipated inflation. The approach that we shall take, which
was introduced by Bailey (1956), utilizes a type of reasoning that is
associated with "consumer surplus" analysis in the areas of public
finance and applied microeconomics. 18 For our purposes, the crucial
point is that the height of any demand curve indicates the value to
buyers of incremental units of the commodity in question. In the
context of money demand, the relevant demand curve plots real balances demanded, m = M/ P, on the horizontal axis and the nominal
interest rate, R, on the vertical axis, as in Figure 6-8. The nominal
interest rate is relevant because it represents the opportunity cost to the
holder of an incremental unit of money. If the prevailing interest rate is
RO and real money holdings are m O, for example, a one-unit increase in
m will require money holders to sacrifice RO units per period in the
form of forgone interest. 19
Rational individuals would not be holding money, thereby sacrificing
interest, unless they were receiving some benefits from the money
balances held. But our analysis in Chapter 3 tells us what these benefits
are; they are the time and energy that do not need to be devoted to
18 For a textbook exposition of consumer surplus analysis, see Varian (1987,
Chap. 15). An extension and correction of Bailey's analysis is provided by Tower
(1971 ),
19 If the price level has the units of dollars per bushel, then m = M/ P has the units
of bushels, If the interest rate R is expressed as an annual rate, then R bushels per
year must be given up for a one-bushel increase in m. Alternatively. R dollars per year
must be given up for a $1 increase in m.
126
6 Steady Inflation
6.7 The Welfare Cost of Inflation
R
Figure 6·9
shopping because of the transaction-facilitating services provided by
money. For any particular change in the level of money holdings,
moreover, the change in these benefits (the volume of these services)
can be calculated by summing the incremental benefits. Since those
incremental benefits are represented by the height of the demand
curve, the relevant sum for any change is represented by the area under
the demand curve between the initial and final positions. 2o If, for
example, money holdings increased from mO to m 1 in Figure 6-8, the
additional transaction-facilitating services to money holders would be
quantitatively represented by the shaded area under the curve between
mO and mi. Conversely, if money holdings were reduced from m 1 to m O,
that area would measure the reduction in money services (or the
increase in time and energy devoted to shopping).
N'ow consider an economy experiencing a steady inflation at the rate
TTl, as depicted in Figure 6-9, and suppose that this economy is one in
which superneutrality prevails. The quantity of real money balances is
ml, which is smaller than the quantity mO that would be held if the
inflation rate were zero. The transaction-facilitating services are accordingly less-more time and energy are devoted to shopping-and the
reduced magnitude of these services is measured by the shaded area.
20 For a smooth demand curve, the argument is like that employed in the definition of
a definite integral. Thus we visualize many narrow rectangles (under the curve) of
common width. and consider the sum of their areas in the limit as the width of each
rectangle approaches zero and the area enclosed by the rectangles approaches that
under the smooth curve.
127
Thus this area represents the social cost of steady, anticipated inflation
at the rate 7T I in the economy under consideration.
One aspect of this result needs a bit of discussion. It is fairly eas~ to
see that anticipated inflation reduces the volume of monetary servIces
and decreases the welfare of money holders. But are there no other
effects that need to be considered? In answer, it is necessary to recognize that the foregoing analysis assumes that newly created ~oney is
turned over by the monetary authority to the gov~rnment, ~hlch then
distributes it to households as transfer payments (1.e., negative taxes).
In addition, the analysis assumes that these transfers are paid i~ a
lump-sum fashion, that is, in such a :vay th.at any househol~'s behaVIOr
has no influence on the amount that It receives. An alternative assumption regarding the way that money enters the economy will be considered toward the end of this section.
For the present, let us retain the assumption that m~mey e~ters via
lump-sum transfers and carry out a numerical calcul~t.lon deSigned to
illustrate the general magnitude of welfare losses ansmg from steady
inflation according to the analysis illustrated in Figure 6-9. For the
purpose of this example, again suppose that the aggregate money
demand function has the form
log m = ')'0 + ')'1 log Y + 'Y2 log R
and that the values of the elasticities are ')'1 = 1 and 'Y2
we have, equivalently, that
(16)
-0.2. Then
which can be solved for R:
(18)
Now suppose that in the absence of inflation the rate of interest is
R = r 0.03 and the ratio Ylm is 6. 21 Then to satisfy (18), the value of
e'l1l must be 0.08266.
With this demand function, let us calculate the welfare loss from a
steady inflation rate of 10 percent per year, which makes R =
0.03 + 0.10 = 0.13. First, we find that Ylm would rise to 8.04. Then
we arbitrarily set y = 1, which means that our calculated loss will be
expressed as a fraction of one year's GNP. With this convention we
have a value of mO 0.1667 with zero inflation and m 1 = 0.1244 with
10 percent inflation. The problem, then, is to evaluate the definite
21 For the United States, the ratio of nominal GNP to nominal Ml was about 5.96 in
1987.
128
6 Steady Inflation
6.7 The Welfare Cost of Inflation
integral
0.1667 (0.08266)5 dm.
J
0.1244
(19)
m
The steps are as follows:
0.082665
0.01667
J
0.1244
m~5 dm = 0.0000039
•
[0.166T 4 - 0.1244~4]
-4
0.0000039[1295 -=-44176] = 0.0028. (20)
The answer, therefore, is that a 10 percent inflation rate leads to a
social cost equivalent to just under three-tenths of 1 percent of GNP.
If, instead, we considered an inflation rate of 100 percent per year, the
result would turn out to be 2.0 percent of GNP.
Clearly, the magnitude of the cost that we have calcul~ted is very
small for inflation rates of the magnitude experienced In most developed countries in recent years. 22 We shall return to that point later,
but first we need to consider two other topics.
The first of these is the assumption, implicit in the discussion to this
point that "money" is a homogeneous entity that pays no interest to its
hold~rs. In fact, of course, the M1 money stock in the United States
is composed of three types of assets: currency, demand deposits, and
OCDs (i.e., other checkable deposits). The last of t?e~e, mor~ov~r,
pays interest to its holders. Accordingly, a more reahstIc quantitative
analysis of the cost of a steady inflation would have to take account of
that heterogeneity. 23
The other topic concerns the way in which money enters the economy or, to express the matter differently, th~ interaction .of money
creation with fiscal policy. Instead of our prevIous assumption, let us
now suppose that the government uses money creation as a sour~e of
revenue. Thus we imagine that the government uses newly pnnted
currency not to make lump-sum transfer payments, but to finance part
of its purchases of goods and services. 24 In this case, the creation of
22 It should be kept in mind, however, that Argentina, Bolivia, Brazil, and Israel
have recently experienced much higher inflation rates (but not steadIly).
23 Since OeDs do not pay as much interest as safe short-!erm assets su,:h as Treasury
bills the difference in these rates is the relevant opportumty cost for thIS component.
Taking account of such complications would probably yield even smaller estimates of
the cost of steady inflation.
.
24 This way of describing the process presumes that ~he monetary al!thon~y turns
newly printed currency over to the government. In actuality, the process IS a little less
blatant. But the effects would be the same if the government were to sell bonds to banks
and the monetary authority were then to create high-powered money to keep bank reserves
from falling.
129
money (and the associated inflation) yields some benefit to household
(or firms) since the extra revenue enables the government to reduce its
tax collections without reducing government purchases. Inflation still
imposes a cost by bringing about a reduced level of real money holdings, but also provides a benefit in the form of reduced tax collections.
In this situation the appropriate way to formulate the cost-ofinflation question is as follows. Suppose that the rate of government
purchases is given. These purchases must be financed in some way,
either by an explicit tax of some type or by money creation (i.e.,
revenue from inflation).25 Each type of tax has some distorting effect
that imposes a cost on society, just as money creation does. The
problem is to design a package of revenue sources so as to minimize the
cost to society of raising the total revenue needed to finance government purchases. Basic marginal reasoning indicates that a necessary
condition for this minimization to occur is that, at the margin, the cost
to society per dollar of revenue be the same for each revenue source
that is utilized to any positive extent. If a possible source has a high cost
per dollar of revenue even when used to a small extent, it will be
optimal not to make use of that source.
It is possible to show that, for revenue from inflation (money creation), the marginal cost per dollar of revenue is given by the following
expression: 26
_d_c_os_t ____11-=--__
d rev
1 - 11(TT/R)
(21)
Here 11 stands for the absolute value of the elasticity of money demand
with respect to the interest rate: 1) = 1'2 = -(dm/dR)R/m. Since
TT/ R == 0 when the inflation ratio is zero, but becomes positive and rises
toward 1.0 as higher inflation rates are considered, expression (21)
equals 11 for a zero inflation rate and exceeds 11 for higher values of TT.
Accordingly, if the actual value of 11 is about 0.2, as assumed in our
numerical example, then the cost per dollar of revenue raised by
inflation is rather high. If the cost of raising government revenue is less
than 20 cents per dollar for taxes such as the income tax, the property
tax, or any specific excise tax, then it would be efficient not to use
inflation (money creation) as a source of revenue.
25 We abstract from the possibility of government borrowing, which amounts to
finance by means of taxes to be collected in the future.
26 This formula was described by Marty (1976). To show that it is valid, note that
since the cost of inflation is f R dm. the marginal value is - R dm =
+ 11') dm. The
marginal revenue from a change in the steady rate is d(m1T)
mdlT mim. Thus the
ratio, which gives the marginal cost per unit of revenue, is -(r + 1T) dm/d7r divided
by m + mlm/d7r. Inserting -mTJ/R for dm/d7r = dm/dR and rearranging, one can
obtain expression (21).
130
6 Steady Inflation
6.8
Concluding Comments
In this chapter. we have introduced tools appropriate for the analysis of
a steady, ongomg (and hence anticipated) inflation and have developed
several key results of such analysis. Although actual inflations are
usually unsteady in their pace and not completely anticipated, most of
the points developed can be extremely useful in thinking about actual
experiences. One good example is provided by the result of Section 6.4
that a sustained decrease in an economy's inflation rate requires either
~n ab.rupt fall.in the price level or an interval of time during which the
mflatlOn rate IS lower than the new money growth rate. The result is
highly pertinent to the experience of the United States during the years
1984-1986, when inflation was unusually low in relation to money
growth.
In conclusion, something should be said regarding the numerical
~esul~s of .Section 6.7, as they suggest that a moderate amount of steady
mflatlOn Imposes extremely small costs on society. Do these results
imply that it is rather foolish for people to dislike inflation intensely, as
many seem to do? A full response to that question would require a
book of its own, but three brief comments can be made. First, some of
the popular dislike of inflation probably does involve a lack of understanding, as when a worker blames inflation for rising prices of things
that he purchases but attributes increases in his nominal wages to his
own diligence and skill. This type of illusion may affect attitudes without having any impact on economic behavior. Second, in actual economies substantial inflation is (as mentioned before) almost never steady
or accurately anticipated. Consequently, there may be major distributional effects, involving wealth losses for creditors and wealth gains for
debtors, whenever debts are specified in nominal units. Such redistributions do not affect aggregate wealth or income yet may reasonably
be regarded as socially undesirable. Third, the possibility of achieving
substantial private rewards from inflation may induce both households
and fir~s to. devote valuable human resources to the task of predicting
future mflatlOn rates. Also, because tax schedules are often specified in
terms of nominal magnitudes, resources may be devoted to activities
designed to minimize tax payments, activities that would be unnecessary in the absence of inflation. 27 In both cases, these resources are
be!ng allocated to activities that are socially wasteful as they yield
neIther goods nor services that give utility to the economy's individuals.
For the reasons mentioned in these last two comments, and others
27 The specification of tax schedules in nominal terms can lead to resource misallocations even when inflation is anticipated. as Feldstein (1983) emphasizes.
References
131
discussed by Fischer and Modigliani (1978), it is likely that even mild
inflations are highly undesirable in terms of the welfare of individuals in
actual economies.
Problems
1. Consider a classical economy in which the full employment rate of
output is Yt = 200, the money demand function is MtlPt = 0.3 YtlRt,
and saving-investment behavior satisfies Y, = 250 - 1000 r" (Here
and RI are real and nominal interest rates, measured in fractional
units.) If the central bank creates money at a rate of 10 percent per
period, what will be the steady-state values of the nominal interest
rate and the real quantity of money?
2. Explain the meaning of the aphorism "the faster money is created,
the less there is." It will be helpful to utilize a plot of R versus Y and
an associated diagram with time on the horizontal axis.
3. Verify the calculated welfare cost of a 100 percent inflation rate in
the numerical example of Section 6.7.
4. Consider the economy represented in Figure 6-9. Suppose that a
slight deflation is created by the monetary authority, so that the
nominal rate of interest R falls below the real rate, r. Will this
deflation improve welfare, under the assumptions made for Figure 6-9, relative to the zero-inflation steady state? What is the
optimum rate of inflation? Compare your answer with that given by
Milton Friedman, as described by Barro and Fischer (1976, p. 144).
'I
References
Bailey, Martin J., "The Welfare Cost of Inflationary Finance," Journal of
Political Economy 64 (April 1956), 93-110.
Bailey, Martin J., National Income and the Price Level, 2nd ed. (New York:
McGraw-Hill Book Company, 1971).
Barra, Robert J., Macroeconomics. (New York: John Wiley & Sons, Inc.,
1984).
Barra, Robert J., and Stanley Fischer, "Recent Developments in Monetary
Theory," Journal of Monetary Economics 2 (April 1976), 133-67.
Feldstein, Martin, Inflation, Tax Rules, and Capital Formation. (Chicago:
University of Chicago Press, 1983).
Fischer, Stanley, and Franco Modigliani, "Towards an Understanding of the
Real Effects and Costs of Inflation," Weltwirtschaftsliches Archiv 114 (1978),
810-33.
Marty, Alvin L., "A Note on the Welfare Cost of Money Creation," Journal of
Monetary Economics 2 (January 1976), 121-24.