~19-

where ia and r& are defined in equations (31) and (33) above.
Analogously to Theorem 1, we can prove that the value of t which

minimizes As, mn, is in general greater than zero but less than

infinity. There is, however, one special case where min =o, That

is the special parameter configuration for which fully discretionary
monetary policy and nominal GNP targeting imply exactly the same re-
sponse to unanticipated supply shocks. (Note that nominal GNP targeting
and fully discretionary monetary policy always imply the same response
to aggregate demand disturbances; in both cases the central bank tries
to stabilize the price level.)

The remainder of this subsection is devoted to discussing how to
prove the propositions discussed in the paragraph above. Because the
theorem proved here is, for the most part, conceptually identical to
Theorem 1, the reader is advised to skip to the next subsection unless
he is specifically interested in the small differences in the two proofs.

~2= in > o> i
Theorem 2. For i 2n, 0< 1TM" <o unless i = 1, in which case 7TMTM

= 0, or unless 9 = o> for all 1 2 0, in which case quin =o,

Proof: It is possible, but algebraically cumbersome, to prove Theorem
2 by exactly the same method as was used to prove Theorem 1. A simpler
though less direct method is discussed below:

In Section IV, it was easy to establish by simple inspection the
fact that art /ae is zero at ¢ = 0, and strictly positive for 0 <¢e<o.
Here, are/ar has the same properties (except in one special case), but
it is somewhat more work to establish this fact by direct differentia-
tion of (33a). We can alternatively make use of the following Lemma:

‘ define r4 as in equation (18) and

For an arbitrary regime


-All-

to compare regimes when the central bank places an infinite weight on
its target, and pays no direct attention to the social objective func-
tion, A.

Equations (Fl) through (F4) give the expected value of the social
loss function (12) under fully discretionary monetary policy, and rigid

50/

inflation-rate, nominal-GNP and money-supply targeting:—'

P= MIX / (0 +0] + xbED/I’, cA)

A |e = eins (#2)

AS gg = Og/n’ + only? + x ~ 2v(1-Bta8) /n)/ (73)
(- 6 +08)",

A oe = tlm? + osE(y? + I? - 2vI/n] (4)

+ (v7 + lo? + (A/8)7021/,

where J = (E-1)/6a&.

In (F2)-(F4), rigid targeting reduces the average rate of
inflation, 7, to 7; the common term Gm? has been omitted from all
four equations, as it is independent of monetary policy and therefore
does not affect comparisons across regimes.

A comparison of equation (Fl) with equations (F2)-(F4) reveals
that the larger the size of the labor market distortion, fi-n, and the
smaller the variance of the supply shock, o, the more likely it is
that any given form of rigid targeting will be superior to fully dis--
eretionary monetary policy. Note also that the higher the variances of
the two aggregate demand shocks, oo and o, the worse money supply

targeting works relative to other regimes. The other regimes allow


-AT-

targets (say, the inflation rate and nominal GNP) differ, depending on
whether the intermediate targets are to be given infinite or optimal
weights in the central bank's objective function.

To develop an algebraic example, we shall consider the special

case where 8 =w = 0. In this case, equations (24) and (33) become
T= (2a) (q + ©)? + xl la? + x + 6)", (24")
B= (or/ayty? + x(t) 7/07 1/ A/a? + t/a” + x)”. (33")

It is easy to check that as € + ©, rie ola’; and as
TH %, re > yor. Using expressions (22) and (31) of the text, we see
that 1! = 1° when ec = t/a. (Note that 140 = 1/a when 8 =w =0). So
by setting ¢ = t/a in (24'), we can confirm the possibility of rank
reversal if we can show that r& - rt changes signs as T varies. This
is sufficient because we know that we can generate any value of quin by
varying the ratio of (-a)*/02.

After dividing out the common factor fa from both (24') and
(33'), form re - rl. Note that the denominator of both expressions is

positive. After multiplying the numerator of ré by the denominator of

r, and visa versa, and after massive cancellation, we find that

@1)

Av

xl x? + la? + 2tx/a + 7/07]

Loft + yo? + 20 + 17).

Both sides of (Dl) are polynomials in T; we see that as T becomes large,
2
the inequality approaches x % 1/02. This is exactly what one would

expect given the limits of (24') and (33') as « and tT go to infinity.


-Fl-

Footnotes

* Board of Governors of the Federal Reserve System. Part of the
work on this paper was completed while the author was on leave at the
Research Department of the International Monetary Fund. The views
expressed here are the author's own, and should not be interpreted as
the official views of either institution. Mathew Canzoneri, David
Folkerts-Landau, Maurice Obstfeld, Alessandro Penati, Franco Spinelli,
Lawrence Summers and Clifford Wymer provided helpful comments on an
earlier draft.

1/ See, for example, Sargent and Wallace (1976), and Fischer
(1977).

2/ For a discussion of the early monetary targeting experience,

see (Organization for Economic Co-operation and Development (1979). For

a discussion of more recent experiences, See Howard and Johnson (1982).
3/ The time consistency component of the analysis is based
heavily on Barro and Gordon (1983), who in turn have extended the
work of Kydland and Prescott (1977). An example of a standard rational
expectations cum wage contracting analysis is Canzoneri, Henderson
and Rogoff (1982), whose work is, in part, an update of Poole (1970).
4/ Fischer (1977), Phelps and Taylor (1977), Canzoneri (1980)
and Taylor (1980) demonstrate that monetary policy can stabilize the
leve!. of output in the presence of nominal wage contracts, even when
private agents have rational expectations.
5/ Section V contains a discussion of the underlying incentives

which might work to make such a commitment credible (to a degree).


-13-

Suppose, for example, that in period t-1 society selects an
agent to head the central bank in period t who is known to have the
following objective function (henceforth, time t subscripts are omitted

where the meaning is obvious):
x12 x2 .
I= (n-fi')” + (xte (1-7),  € > - X- (21)

When ¢€ is strictly greater than zero, then this agent places a greater
relative weight on inflation stabilization than society does 20/

The algorithm for deriving the time consistent equilibrium is
exactly the same as in the previous section. Equations (22), (23),

and (24) are the "I" regime counterparts of equations (17), (19) and

(20) respectively:

w= Gt - #? = [W@wI17/O¢e)?, (22)
[p, - By (p,)}" = doe = -v2,/n(v? + x + €)s (23)
rt = (oI) Lcte)? + vx +x+ e)?. (24)

(Note that rt is obtained by plugging apy into r4, as defined in

equation cas) 22/4 The reader can confirm that these equations are the
same as (17), (19) and (20) when ¢ = 0. The (expected value of the)

social loss function under "inflation-rate targeting," as we shall term

this regime is
ae Gm? + yar. (25)

To solve for the value of ¢ which minimizes a, differentiate

(25) with respect to e:


-Fll-

49/ A more formal proof involves subtracting the right-hand
side of (Dl) from the left-hand side and equating to zero. It is
then possible to choose x and a so that the resulting quadratic
equation in tT has two real roots, one positive and one negative. The
positive root is the t at which rank reversal occurs.

50/ Expressions (F2), (F3) and (F4) may be derived by taking
the limits of equations (24), (33) and (44) with respect to the index
variables. It is also fairly easy to derive them directly from the
model, by alternately holding p, pty, and m fixed. The second method,
incidentally, provides an algebra check on expressions (24), (33) and
(44).

51/ Rigid inflation-rate targeting is always better than fully
discretionary monetary policy when the labor market distortion is
relatively large, and is similarly always better than rigid money
supply targeting when either or both of the aggregate demand shocks
have large variance relative to the aggregate supply shock.

52/ Wage setters, who are concerned with employment stabiliza-

tion, have no reason to index in this case anyway.


-F9-

one point on a mutually consistent money supply path; otherwise the
price level is indeterminate.

40/ The expressions for ap® and T* are omitted as they are not
essential to the main argument here. But it is relevant to note that
the Lemma of Section V.2 assures that rR 2 r.

41/ It can be shown that low nominal interest-rate targeting
is counterproductive when the regime is expected to last for any
finite number of periods, if in the final period the expected infla-
tion cate is the one consistent with a return to fully discretionary
monetary policy. The regime fails because the central bank cannot
systematically achieve a below-market real interest rate for any future
period.

42/ "Step one" is executed in Appendix F, where rigid versions
of the regimes are compared. That analysis makes even more obvious
the fact that any of the targeting regimes considered (except interest-
rate <argeting) works best for some parameter configurations.

43/ In the absence of a fully-specified microeconomic model,
one cannot be sure how much the parameters of the model change when
policy regimes change; (Lucas 1976). Another caveat to the present
analysis is that we have not considered the possibility of adopting
more than one target; this is fodder for future research.

44/ This is the standard result; see Friedman (1975).

45/ Nor can one necessarily choose the best intermediate target
solely on the basis of which target can be made the most credible.
(This argument is sometimes given for why the monetary base is the
ideal intermediate monetary target.) The stabilization properties

of a <arget also need to be taken into account.


-F7-

PLO Ey (Pp) = -2,/a(1+0) under rigid nominal GNP targeting

(since u and v shocks are fully offset). It is easy to confirm
that by plugging the above expression into r4 in equation (18), one
obtains the same expression as by taking the limit of (33) as T+.

33/ In the special case where w = 0, the numerator of (38) is
negative, zero or positive as x 2 vB. It is possible to show that
the rumerator of (38) is always negative when 8 = 0. In other words,
rigic. GNP targeting is never optimal in the absence of wage indexa-
tion.

34/ It is not possible to prove, in general, that ymin is the
sole real positive root of the polynomial equation ane yar = 0. How-
ever, one can prove that anS yar has at most three positive real roots.
A sufficient condition for min to be the only positive real root
(and thus for ie to be strictly concave in T) is that (1-8)/a(1-a) > x.
See Appendix B for a proof. Note that the above uniqueness (con-
cavity) condition holds as long as society does not place a very large
weight on inflation-rate stabilization relative to unemployment rate
stabilization. Recall that 4 is capital's share of output, when
a= ,25, then 1/a(1-0) = 5.33. Thus when there is no wage indexation
(so that B = 0), uniqueness requires only that society not prefer
av5 percent fall in employment below its socially desired level to
a one percent increase in the inflation rate above its desired level.
In fact, it is likely that Xx is much smaller than one in most
societies. When the uniqueness condition fails, it is still possible
to reliably compute caSymin by computing all the roots of aneyar = 0,

since for given values of the parameters, one can use a computer to

solve for all the roots of a polynomial.


-23-

Employing the macro-model of equations (9), (10) and (11), one
can obtain the equation that will be used to substitute out for nm. in

(39) 333/
m= WM + ep, + CHE We C/Su, + EW)e,/8a + ves (40)

where — = 1+ 0/6 + 60>. Substituting into (39) using equations (40),

(8) and (13) yields:

—M.

= _ 2
M= [z./n + v@, ve

y= Gm)? + xtp, - Pp tH yy

+ ulep, + (e-em - O/é)u, + (E-Dz,/0a + vy - Ppp > H?.

Once more following the algorithm of Section III for obtaining a

time-consistent rational expectations equilibrium, one obtains26/

ts GMa? = Pam Ix + uel? ; (42)

M
dp,

{-[(v/n) + HE(E—1)/oalz, + vedu,/d - vey, }/ (43a)

tv? + ue? + x], or

apy = ove, + oyu, + ony, 3 (43b)
mM. {ott + x05] + T4l0 + o1sy?o21 WT , where (44a)
T, = {yelv(1-€)/00 + (E/n)] + x/n} (44b)
T= v/n + vE(E-1) /80 , (44c)
T3 = (wv? + x) ue)” , and (44d)
Tt, =v + ue? +X. (44e)


-Al2-

the monetary authorities to fully offset the combined effect of aggre-
gate demand shocks on the price level; full neutralization is optimal
because there is no trade-off between price stabilization and employ-
ment stabilization in the face of an aggregate demand shock.

In proving Theorem 2, we derived a condition under which the
best form of nominal GNP targeting is rigid nominal GNP targeting. In
this special case, rigid nominal GNP targeting brings the inflation rate
down to its optimal level without any sacrifice in stabilization rela-
tive to fully discretionary monetary policy. Thus, nominal GNP tar-
geting is clearly better than other forms of targeting for at least
one special parameter configuration. Analogously, in proving Theorem
3, we found one special case where rigid money-supply targeting is opti-
mal. (The necessary condition is given by equation (46); o = o = 0 is
also required.) Examination of one further special case allows us to
definitely conclude that any of the four regimes given by equations
(F1)-(F4) may be best, depending on the structure of the economy.

Setting the wage indexation parameter B and real wage elasticity of

labor supply w equal to zero, (F2) and (F3) simplify to

I) _ 2,2 '

A le = oF /a', and (F2")

G,  _. 2 '

Nl co = XFz° (F3")
Inspection of (F2') and (F3') 1s that At| AS | Lor 2

nspection 0: an reveals a e= T= as a < xX:

2
<
a

Thus when x is very large, so that society places a great weight on

inflation stabilization relative to employment stabilization, rigid
51/

inflation-rate targeting is superior to rigid nominal GNP targeting.—

Incidentally, the reason that (F3') gives such a simple expression in


-4-
II. The Macroeconomic Model

This section is devoted to developing a stochastic rational
expectations IS-LM mode1.©/ Monetary policy can have short-term teal
effects in this model because nominal wage contracts are set a period
in advance. Due to high administrative and negotiation costs, these
contracts are not indexed fully against all possible disturbances Lv
But while the central bank can affect the variance of employment, it
cannot systematically affect the average level of employment. The

reason for this will become apparent when in Section III we study wage

setters' expectations about future monetary policy.

II.1. Aggregate Supply

Each of the large number of identical firms in the economy has

a Cobb-Douglass production function. In the aggregate,

y= £9 + ok + (-a)n, +z, qd)

t t

where y is output, k is the fixed capital stock, n is labor, fo is a
constant term and z is a serially uncorrelated aggregate productivity
disturbance; 2N(0,02). Throughout, lower case letters denote natural
logarithms and the period t value of a variable is denoted by a "i:"
subscript. All coefficients are non-negative.

Firms hire labor until the marginal product of labor equals the

real wage:

ey t+ log(1-a) + ak - ond tz =w (2)

t 7 Pee

where w is the nominal wage, p is the aggregate price level, and nt is

the quantity of labor demand.


-A5-

and (b) The Lemma of Section V.2 implies that arS/ar > 0 for 0<1t<»,
and equals zero for t = 0. It is convenient to redefine b' = b"/ (146)?
and h' = (y + v2) / (+0) so that ar fat is of the form b'e/[h' + q(149) 3.
The fact that amo/at is of the form -a'/[y + r(+e) 1? is confirmed by
direct differentiation of (31).

The above discussion implies that ano/ar = 0 has exactly the
form of equation (B2) with ¢ replaced by t(1+0), a/b replaced by a'/b'
and h replaced by h'. (Note that @ > 0.) Therefore ano /ar has at most
three positive roots and definitely has only one positive root if h' =
(+ v7 /(146) 2x. The foregoing condition can be rewritten, using the
definitions of y and 6, as (1-g)/a(l-a) 2 y-

Proof of Proposition B.3.

The same as the proof of Proposition B.2.

Appendix C: The Incomplete Contemporaneous Information Case.

In the text, the monetary authorities and investors are ass.imed
to have full knowledge of the period t disturbances when making their
period t money supply and investment decisions. It is extremely straight-
forward, though in some cases algebraically cumbersome, to extend the

46/

analysis to the case of incomplete contemporaneous information.— All
of the theorems can be generalized, but the ranking of regimes does
depend on the characteristics of the information set.

To sketch an example, consider the case where the monetary
authorities and investors observe price level and interest rate data
contemporaneously, but only observe real output with a one-period lag.

They can use their knowledge of the price level and the interest rate

to make imperfect estimates of the three contemporaneous disturbance


-20-

A_ OA A A,

apy = Pye, + Pot, + P3YE° (35)
Lemma. rA is minimized at (fs ops mn = (p?,0,0). Furthermore,
r4 is a strictly increasing function in all elements of (et - |.

A A
legl> 193|)- ;
; A_ A A. A

Proof: Substitute dp, EP, - E,_ (Pe) into [ as defined in equation
(18):

A Al2 A\2

rv = E,_,{lz,/n + vap}” + x(dp,)}- (36)

Substituting (35) into (36), and making use of the fact that z, u, and

v are independent by assumption yields

A 2
tA = f/m + yop)? + x(op)7] (37)
2, Av2 2, A\2,, 2 2
+ [a (py)” + o(03) 10" + Xx )-
By inspection of (37), TM" = (ohyTMP = 0. Differentiating the
term multiplied by oo and equating to zero confirms that on = o>

as defined in equation (19). Note also that A is quadratic in all
elements of (ets on ro) and that the second order conditions for a
minimum are satisfied. Therefore rs is strictly increasing in all

elements of (et - ol, logl> |os| ). Q.E.D.

The proposition underlying the Lemma is entirely sensible.
Under fully discretionary monetary policy, the central bank is not
encumbered by any commitment. Therefore it is free to respond to
unanticipated disturbances in a way which minimizes the social loss
function. The more that response is distorted by a commitment to a

particular monetary target, the greater 4, the weighted average


-26-

indeed cause the market-determined interest rate r, to be less than
can suggests a serious credibility problem. The central bank has to
target high interest rates if it wants low interest rates.

The underlying problem is that, given Eee Pray - Pi)» announ-
cing a target for ur is, in fact, tantamount to targeting the real
interest rate. For the regimes analyzed earlier, targeting succeeds
in at least temporarily lowering the inflation rate regardless of how
long the targeting regime is expected to last. This is no longer =rue
when the nominal interest rate is used as a target 41/

The nominal interest rate would not appear to be a suitable
instrument for precommitment. This conclusion, of course, does not

imply that the interest rate should not be used as a information vari-

able in setting monetary policy, as in Poole (1970).

VI. On Comparing Alternative Targeting Regimes

We have shown how the framework developed in this paper can be
used to analyze a variety of widely-discussed intermediate monetary
targeting regimes. The general result is that "flexible" intermediate
monetary targeting almost always dominates both rigid targeting and
fully discretionary monetary policy. The reader is unlikely to be
stunned if he is now informed that the choice of intermediate monetary
target (between nominal GNP, the money supply, etc.) should depend on
all the parameters of the model as well as on the relative sizes of
the disturbances; Poole's (1970) result is now standard.

The qualitative properties of the various intermediate tar-
geting regimes are well known: Money-supply targeting works poorly

when the monetary authorities have information on how aggregate demand


-22-

v.3. Money-Supply Targeting

Given the widespread adoption of money-supply targeting over the
past decade, this case should require little motivation.
The central bank's objective function under money-supply tar~-

geting is assumed to be given by:

M

(as')? + xr)? + wm, - 97, U2 0, (39)

where n, i+ Pry + am; the " superscript denotes the money-
supply targeting regime. Employing the macro-model of equations (9),
(10) and (11), one can easily demonstrate that the target level of the

money supply, fi,, is the level of mn which would be consistent with

te
society's desired inflation rate 7 provided that (a) there are no dis-
turbances in period t, and (b) 7 is the expected inflation rate in
period t.

The choice of the money supply target requires some comment.
If the central bank is going to select the money supply as an inter-
mediate target, it is certainly logical for it to commit to a rate of
money growth consistent with the socially-desired inflation rate. But
because the demand for real balances is interest-elastic, the central
bank needs to form an estimate at time t-1 of E, iad - P,) in order
to select the appropriate value of i: The time consistent value of
E ray - P,) depends, in turn, on the policy regime expected to pre-
vail in period ttl. Here we have assumed that everyone expects the
money-supply targeting regime to remain in place. This assumption is
not crucial to the results below. However, the assumption that the
central bank chooses its money supply target based on a realistic

assessment of next period's expected inflation rate, is quite important.


-A6-

terms. Canzoneri et al. (1982) develop an algorithm for calculating
these estimates and prove that the disturbance term prediction errors
are independent of systematic monetary policy. 2/

Using these results, it is possible to decompose the expected

value of the social loss function AL under an arbitrary monetary policy

regime "A" as follows (compare with equation (18) of the text):
ew 2_ 2,2 A :
Epp) = Gea)” + (2 - of )/n? + at + ar’, qs")

2 2 2, =E - . gf
where Jay = E,_, dz.) 3 dz, = EL1 (2) EF, @,)3 E

t-1 stands for

the expectations operator conditional on all period t-1 information plus
the period t interest rate and price level. i is defined as in equation
(18), and ar* is the same as r4 in equation (18), except that z is re-
placed by dz.- In deriving (18'), we have used the fact that dz, is
independent of (@, - dz.) under rational expectations. The first two
terms on the right-hand side of (18') are independent of monetary
policy, so that monetary policy regimes should be compared on the basis
of rf and ar* only. For the information structure assumed in this
appendix, all the results and theorems of the text on inflation-rate

and nominal GNP targeting go through with the trivial modification that

z, and a are replaced everywhere by dz, and Ped While straightforward

t dz*
in principle, it is more work to modify the results on money supply tar-
geting. The reason is that although ue Vy, and 2Z, are uncorrelated by

assumption, du,» dv, and dz, are in general correlated.

Appendix D: Rank Reversal
This appendix gives both algebraic and numerical examples of

“yank reversal." This is the phenomenon where the rankings of two


-17-

somewhat contrived, it does capture the basic mechanism underlying the
monetary targeting regimes implemented in numerous countries over the
past decade 20!

Adopting an intermediate targeting interpretation of the analysis
of Section IV, we observe that inflation-rate targeting succeeds in
reducing the rate of wage inflation. But this gain is not without
stabilization cost, as inflation-rate targeting causes the monetary
authorities to place too much emphasis on reducing the variance of
inflation and not enough emphasis on reducing the variance of employ-
ment. It is natural to ask whether the central bank can find other
intermediate targets with which to bring down the inflation rate at
lower stabilization cost. We shall now consider, in turn, nominal GNP
targeting, money-supply targeting, and interest-rate targeting. Because
the derivations of the results for these regimes are basically analogous

to the algorithm of Sections III and IV, we shall omit some of the

details.

v.2. Nominal GNP Targeting

The drawback to inflation-rate targeting is that it does not
allow real wages to properly adjust in response to supply shocks.
Because a positive supply shock tends to raise output and lower the
price level, one would expect there to be some circumstances in which
nominal GNP targeting is more appropriate. Suppose then that the cen-
tral bank adopts nominal GNP targeting so that its objective function is

given by

wy\2 = wy 2
= (nett)? + yet)? + ty, HPL By (29)
t t t-1

where y = f(m) according to the production function qa .3l/


-~9-

respect to a variable but perfectly-anticipated rate of inflation. And
of course, high rates of inflation force agents to economize on their
hold:ings of non-interest-bearing money--the so-called "shoe leather cost
of inflation." Despite the foregoing considerations, the optimal target
inflation rate 7} may be non-zero. It is sometimes optimal to use the
seignorage tax at a non-zero level when alternative taxes also generate

17/

dead--weight costs through distortions .—

III. Time-Consistent Equilibrium Under Fully
Discretionary Monetary Policy.

Here, stochastic equilibrium is derived under the assumption
that the monetary authorities attempt to minimize the social loss
function A, given by equation (12) above 28/

In the present model, as in most rational expectations monetary
models, today's prices and interest rates depend in part on agents'
expectations about the future path of the money supply. But here this
path is not exogenously given; expectations about future money-supply
growth depend endogenously on expectations about the monetary authori-
ties' future short-run stabilization objectives. Agents will not, in
fact, fully believe promised future paths for the money supply which
are time inconsistent; i.e., promises on which the central bank will
have a predictable incentive to reneg12/ Wage setters could protect
themselves against systematic inflation by fully indexing wage con-
tracts to the price level, but this would leave them with no insulation

against supply shocks 20/

Instead, as we demonstrate formally below,
wage setters can set nominal wage increases at a sufficiently high

level. so that, in the absence of disturbances, the central bank will


-A3-

The following coefficient definitions are used:
n = a(1-cw)
v = (1-8)/o
6 = (1-8) (l-a) /a

W

1 + 6A/8 + 6O

my
i

All parameters are defined as positive numbers.

Appendix B: Proofs of Minor Propositions which are of Computational

Convenience in Finding the Optimal Weight to Assign a Target.

In this Appendix, we prove the following three propositions:
Proposition B.1. enin is the unique real positive root of alt /9e = 0.
Proposition B.2. aac/ar has at most three positive real roots; a
sufficient condition for min to be the only positive real root is that
(1-8) /a(1-a) > x.

Proposition B.3. an you = 0 has at most three positive real roots; a
sufficient condition for ymin to be the sole real positive root is that
(1-8) /a(l-a) 6 + 2/8] > x

These three propositions are of no qualitative significance for any
of the results in the text. They are relevant only for the choice of
computational algorithms available to numerically calculate emin qin
and ymin for specific values of the model's parameters. When the poly-
nomial equations specified in the propositions have only one real pcsi-
tive root, one can use a local gradient algorithm to find the global
minimum, (Since the objective function is then concave in the index
variable.) Otherwise, it is necessary (but not much more difficult) to

calculate all the positive roots of the polynomial and check to see which

one is the minimum.


-29-

the presence of a labor market distortion, society can actually make
itself better off by having a central bank which does not directly

attempt to maximize the social welfare function.


where y = (1 -8)/a-

In choosing the base wage rate W,

t? Wase setters seek to mini-

mize Ey, - a), where Eee denotes expectations based on period
t-1 information and a is the level of employment that would arise if
contracts could be negotiated after observing the productivity dis-

turbance z and all other period t information. ne is found using the

labor supply and demand equations (2) and (3):
BL = n+ wz /(1 tau). @)
From equations (6) and (7),
-n'= -W
n ne z,/n + vv, Ww)» (8)

where n = a(l tow). It is clear from equation (8) that E 2

Ly

-7
1, BD)
is minimized by setting wee E._y P,)-

Using equations (1) and (6), together with the analytically con-

venient assumption (at no sacrifice in generality) that “Cy = ok + (1-a)n

so that Ei = 0, the aggregate supply equation can be written as

ye = 00, - 4) + 2/0, (9)

where 6 = (l-a)(1-8)/a. Since Ww = E._1 (P,)> equation (9) can be

written in the standard rational expectations "price level surprise’

form:
ye = lp, - E,_y(P,)] + z,/0- Q')

It is very important to note from (9') and (6) that output and employ-
ment stabilization are not equivalent to price prediction error minimiza-

tion in the presence of a productivity shock (z).


-=5-
Labor supply, n’, is an upward-sloping function of the real wage:
n° = 7+ ww, - p,) 3
t Owe 7 Pee (3)

To simplify algebra without loss of generality, n is set equal to
EK + (1/a) [log (1-0) + Col-

The above labor supply curve (3) is assumed to embody some dis-
tortions which raise the real wage required to induce a given level of
labor supply. Factors which might distort the labor-leisure decision
towards leisure include income taxation and high levels of unemployment
penefits.2/ Monopolistic unions might also succeed in driving up real

wage demands. The labor supply curve in the absence of distortions is

given by
s_s _
a ent dy + ww, Pd» (4)

where ~dy is the distortion in the actual labor supply curve (3) 2/
Price-indexed wage contracts for period t are negotiated at
the end of period t-1. The base wage rate is We and the indexation

parameter is gld/ :

WA

(5)

wew + BCP, - wi) 0

The nature of the employment contract is that laborers agree to supply
whatever amount of labor is demanded by firms in period t, provided
firms pay the negotiated wage rate. The level of employment in period t
is thus found by substituting the wage rate equation (5) into the

labor demand equation (2):

n= nt vp, - wy) + z,/a» (6)


-18-

G stands for "nominal GNP targeting regime," and the index parameter
t gives the weight that the central bank places on achieving its int:er-
mediate target relative to the weight it places on directly maximiz:ing
the social objective function A.

We can rewrite the objective function G by using equations (8),

(9), (13), and (29):

6 = [z,/n + vp, - WE) - Gm)? + xlp, - PL - #1? (30)

—G ~42
+ tlz/o + O(p, - Wi) + Pp - Py - mi, T2 0.

To find the time consistent path of the economy under nominal
GNP targeting, again follow the alogrithm of Section r11.32/ Equat:ions
(31), (32) and (33) below correspond to equations (17), (19) and (20) of

Section III:

To = Go - %? = v2 @em2/[rC140) + I, G1)

dpe = -2,[v/n + 1(148)/a}/[v? + (148)? + x] = pe, (32)

re = res "S, where (33a)

Te = o2{E(/n) + t(140)2/n - t(148)v/al? (3p)
+ xLv/n + 1(140)/a)7}, and

TS = [y? + (140)? + yI”. (33c)

The expected value of the social loss function under nominal GNP

targeting is

Ao = Gam)? + yn? + 7S, (34)


-8-

this potential source of tension is fundamental to the interplay between
stabilization policy and wage-setting. Indeed, if the monetary autchori-
ties do target the same level of employment as wage setters, then there
is no reason to consider any regime other than fully discretionary mone-
tary policy. (Though, for information-theoretic reasons, the optimal
fully-discretionary monetary policy may involve using a feedback rule.)
Here is is assumed that the social loss function A depends on
deviations of employment and inflation from their optimal (socially-

desired) levels A4/

A, = (a, - 81)? + x(n, - 7)”, (12)

where TM, = Pe Peep jw is the socially-desired trend inflation rate,
ay is the socially-desired level of employment in period t, and X is the
relative weight society places on inflation stabilization versus employ-
ment stabilization. at is found by using the labor demand schedule (2)
together with the "undistorted" labor supply schedule (4), and by

assuming that wage setters have full period t information:£2/

toate =a-n
7 BE dy/Q +a) =na-n (13)
We have already discussed why fi might be greater than n.

It is somewhat difficult, in the context of a rational expecta-
tions model, to justify incorporating the level of the inflation rate
directly into the social loss function. The costs of perfectly antici-

16/

pated inflation are imperfectly understood.— Some costs which are
known include the administrative costs of posting new prices and the

costs of adjusting the entire tax system to be fully neutral with


-24-

The expected value of the social objective function under mone-

tary targeting is

Me Gm? + ytts (45)
Defining ymin as the value of , which minimizes a, one can’ prove
Theorem 3. For fi 2 n, 0 < yon < o, unless oF = o> for all y and
o = on = 0, in which case ymin =o, or unless fi = n in which case
ymin =0.

Proof: The method of proof is exactly the same as for Theorem 2 and
makes use of the Lemma developed in Section V.2. Note that the

condition for oF al o> for all uy is thar2Z/

w+ ye? = &)/ea = vers. (46)

Under full information, rigid money-supply targeting qin =o)

is only optimal in the unlikely event that there are no aggregate demand
shocks and condition (46) holds. However, the assumption that there
are no aggregate demand shocks can be relaxed when the monetary authori-

38/

ties have incomplete contemporaneous information.—’ Appendix C dis-
cusses how to extend the present analysis to the incomplete information

case.

v.4. Nominal Interest Rate Targeting

The monetary authorities cannot systematically raise or lower
the average value of the real interest rate in the rational expecte-
tions model employed in this paper. But it might seem reasonable for
the central bank to try to bring down the inflation rate by committing

itself to achieving a low nominal interest rate.22/ Here we argue that

this method of intermediate monetary targeting is counterproductive.


International Finance Discussion Papers
Number 230

September 1983

THE OPTIMAL DEGREE OF COMMITMENT TO AN INTERMEDIATE MONETARY TARGET:
INFLATION GAINS VERSUS STABILIZATION COSTS

by

Kenneth Rogoff

NOTE: International Finance Discussion Papers are preliminary materials
circulated to stimulate discussion and critical comment. References in
publications to International Finance Discussion Papers (other than an
ackncwledgment by a writer that he has had access to unpublished
material) should be cleared with the author or authors.


-F3-

implies that the monetary authorities have no information advantage
whatsoever over investors.

13/ The social objective function used here is a slight general-
ization of the one employed by Kydland and Prescott (1977) in their
deterministic model. Kydland and Prescott demonstrate how activist
monetary policy can result in high average levels of inflation. Barro
and Gordon (1983) also employ a similar specification. In contrast to
the present paper, Barro and Gordon prefer not to incorporate Gray-
Fischer-Taylor type nominal wage contracts into their model. For this
reason, Barro and Gordon choose not to emphasize the potential use of
activist monetary policy to offset random disturbances.

14/ The analysis below would be exactly the same if the social
loss function were instead given by the present discounted value

functional At = E{(a, - ay? + xq, - 71a +R).

s

15/ The value of at may be derived from equations (7) and (13).
Note that at - at is independent of the productivity shock Zee

16/ Fischer and Modigliani (1978) catalogue the economic costs
of both anticipated and unanticipated inflation.

17/ See, for example, Phelps (1973).

18/ Throughout the text, the monetary authorities and investors
are assumed to have complete contemporaneous information. The
generalization of this analysis to the case of incomplete contempor—
aneous information is for the most part straightforward; see Appendix C.

19/ Kydland and Prescott (1977) develop the notion of time

consistency in a macroeconomic context.


-1ll-
P = via) /y +7. (17)

Thus, as Kydland and Prescott as well as Barro and Gordon have
pointed out, the time consistent rate of inflation is too high when
i>. Society suffers from stagflation.22/

We are now prepared to evaluate the social welfare function, A,
uncer fully discretionary monetary policy. But to facilitate exposi-
tion in later sections, we shall first develop a notation for evalua-
ting social welfare under any given monetary policy regime. Obviously,
it does not make sense to evaluate A for a particular realization of the
disturbances, given that society must decide a period in advance which
regime to choose. Rather, the relevant consideration is Ei Ay)» the
expected value of the social welfare function at the time the choice of
regime is made. Defining Weezy (A,) under an arbitrary monetary

policy regime "A",
Ah = Gem? + at + 34, as)

where ts = Ge - wp? and

A A A 2 A A\,2
rs Ey_y{lz./n + veep - Ey @,)1" + xlp, - E,_@,)1°}, where we

have made use of the fact that Eel @,) = Wee The first component of
i is nonstochastic and invariant across monetary regimes. It represents
the deadweight loss due to the labor market distortion. This loss can-
not: be reduced through monetary policy in a time consistent rational
expectations equilibrium. The second term is also nonstochastic but
does depend on the choice of monetary policy regime. The average

(expected) inflation rate depends, in part, on the weight the monetary

authorities place on inflation stabilization versus employment


-10-

not choose to inflate the money supply beyond the point consistent with

21/

wage setters' desired real wage.—’ At this high level of inflation,
the central bank finds that the marginal utility gain from the higher
employment induced by a small further price increase is fully offset by
the marginal disutility from still higher inflation.

By substituting equations (8) and (13) into equation (12), the

central bank's objective function under fully discretionary monetary

policy may be written as:
= ~-Dpy_ Ke 2 _ _ x2
Da AL = f2,/n + v@, - 8) -— @- Oy + xEP, - Pp 7 AT"
(14)

("D" superscripts stand for "fully discretionary regime.")
The central bank maximizes social welfare by choosing a level of the

money supply consistent with Pps the period t price level which mini-
22/

mizes AY _—

D

Pp = vw + VR - 24/n) + xO) + 1/0? +H. (1s)

Recall that (the logarithm of) wage setters' target real wage is zero.

Thus wage setters select ca by taking expectations across (15) and

setting * = E,_ y) ,23/

—D D. ~ ~ os
= Ey Py) = Peay +7 +H - n)/x. (16)

By choosing we according to (16), wage setters assure themselves that
the monetary authorities will not attempt to systematically drive down
the real wage.

Investors similarly use (16) to solve for ?, the expected rate

24/

of inflation under fully discretionary monetary policy:—


-R1-
References
Barro, Robert J. and David B. Gordon, "A Positive Theory of Monetary

Policy in a Natural-Rate Model," Journal of Political Economy,
Vol. 91 (August 1983).

Canzoneri, Matthew B., "Labor Contracts and Monetary Policy," Journal
of Monetary Economics, Vol. 6, No. 2 (April, 1980), pp. 241-255.

Canzoneri, Matthew B., Henderson, Dale W. and Kenneth Rogoff, "The
Information Content of the Interest Rate and Optimal Monetary
Policy," (1982), forthcoming in The Quarterly Journal of Economics.

Fischer, Stanley, "Long-Term Contracts, Rational Expectations, and the
Optimal Money Supply Rule," Journal of Political Economy, Vol.
85 (February 1977), pp- 191-206.

Fischer, Stanley and Franco Modigliani, "Towards an Understanding of the
Real Effects and Costs of Inflation," Weltwirtschaftliches Archiv,
Vol. 114, No. 4 (1978), pp. 810-833.

Friedman, Benjamen M., "Targets, Instruments, and Indicators of Monetary
Policy," Journal of Monetary Economics, Vol. 1, No. 4 (October
1975), pp. 443-473.

Gray, Jo Anna, "Wage Indexation: A Macroeconomic Approach," Journal of
Monetary Economics, Vol. 2 (April 1976), pp- 221-235.

Howard, David H. and Karen H. Johnson, "Financial Innovation, Deregula-
tion, and Monetary Policy: The Foreign Experience," (unpublished,
Board of Governors of the Federal Reserve System, 1982).

Kydland, Finn E. and Edward C. Prescott, "Rules Rather than Indiscretion:
The Incon$istency of Optimal Plans," Journal of Political Economy,
Vol. 85, No. 3 (June, 1977), pp. 473-92.

Lucas, Robert E., "Econometric Policy Evaluation: A Critique," in Karl
Brunner, ed., The Phillips Curve and Labor Markets, Supplement to
the Journal of Monetary Economics, (1976), pp. 19-46.

Obstfeld, Maurice and Kenneth Rogoff, "Speculative Hyperinflations in
Maximizing Models: Can We Rule Them Out?" Journal of Political
Economy, Vol. 91 (August 1983).

Organization for Economic Cooperation and Development, Monetary Targets
and Inflation Control (Paris: Organization for Economic Coopera-
tion and Development, 1979).

Phelps, Edmund S., "Inflation in the Theory of Public Finance," Swedish
Journal of Economics, Vol 75, No. 1 (1973), pp- 68-82.

Phelps, Edmund S. and John B. Taylor, "Stabliizing Powers of Monetary
Policy under Rational Expectations," Journal of Political Economy,
Vol. 85, No. 1 (February 1977), pp- 163-90.



~A2-

Il = squared deviation of the mean market-determined inflation rate from
the socially-optimal inflation rate.

D = fully discretionary monetary policy regime

I = inflation-rate/price-level targeting regime

G = nominal GNP targeting regime

M = money-supply targeting regime

R = interest-rate targeting regime

A = arbitrary regime: D,1I,G,R, or M.

A"—" above a variable denotes its average "market-determined"
value and a "~" denotes its average socially-preferred value; a "t"
subscript denotes the time t value of a given variable; the value of a
variable corresponding to a given regime is superscripted by the letter

corresponding to that regime (D,1I,G,M,R or A).

€,) Ts ly U = weights placed on the intermediate monetary targets rela-
tive to direct employment and inflation-rate stabilization objectives

in the central bank's loss function in regimes 1,G,M, and R respectively.

w = real wage elasticity of labor supply
oa = coefficient on capital in the Cobb-Douglass production function
8 = waze indexation parameter

-§ = real interest rate semi-elasticity of aggregate demand

1
>»
"

aominal interest rate semi-elasticity of real money demand
= real income elasticity of real money demand
x = the relative weight on squared inflation deviations relative to

squared employment deviations in the social loss function.


-F4-

20/ Gray (1976) demonstrates that full price level indexation
is suboptimal in the presence of supply shocks.

21/ Note that the individual groups of wage setters are, in
fact, concerned with the aggregate inflation rate and not just with
the level of employment at their own firms. But because each indi-
vidual firm has only a small impact on the aggregate price level, an
individual group of wage setters has little incentive to take into
account the effects of their contract on the aggregate inflation rate.

22/ Pe is found by setting aD, /8P, = 0. Note that the second
order conditions for a mininum are met; given the quadratic form of
D, the minimum is global.

23/ The reader is again reminded that quadratic form of (14)
implies certainty equivalence.

24/ Assuming that the monetary authorities are expected to mini-
mize the same objective function (14) in all future periods, the same
algorithm can be applied repeatedly to derive a time consistent path

for all future prices:

tiv

E,1 Preys) = v(i-n) (s+1)/x + (stl)# +p._y, 5 20. (£1)

Note that in the text, we treat the price level as if the monetary
authorities controlled it directly, ignoring the fact that the central
bank only directly controls the money supply. The anticipated future
path of the money supply consistent with the time consistent path of
prices (f1) can be found by working through the macro-model of equa-

tions (9), (10) and(11), and by imposing saddlepath stability:


-Al-

Appendix A: Notation

y = logarithm of output

n = logarithm of employment

p = logarithm of the price level

w = logarithm of the wage rate

m = logarithm of the nominal money supply
r = level of the interest rate

z = productivity (supply) disturbance

v = money demand disturbance

u = goods market demand disturbance

o, o, o = The variances of the three serially-uncorrelated, zero-

mean disturbances.

n = level of employment targeted by wage setters

n' = level of employment which wage setters would target if they knew
the aggregate productivity disturbance.

fl = average level of employment targeted by the monetary authorities.

n' = level of employment targeted by the monetary authorities after
receiving information about the productivity disturbance.

a = inflation rate

7 = socially optimal inflation rate

Ea = expectations operator, based on period t-1 information.

dp, = Py By

A = Social loss function

T = A weighted-average of deviations of inflation and employment around
their respective market-determined values; the weight is the

same as in the social loss function.


-7-

Il.2. Aggregate Demand
Demand for the good which firms produce is a decreasing function

of the real interest rate:

ye 7-6 (r- [E (yy) - PA) tu, (10)

where r is the level of the nominal interest rate and EL up ~ PE

represents the rate of inflation expected by investors, based on period
1
t ir formation.22/ The serially uncorrelated goods market demand dis-
turbance u,v N(0,0°).
t u

The demand for real money balances is a decreasing function of

the nominal interest rate and an increasing function of output:

m7 P= “ar, + oy, + Vs (11)

where m is the logarithm of the nominal money supply and v is the money
market disturbance term; vN(O,02). To simplify exposition, the dis-

turbances, v, u and z, are assumed to be independent.

II.3. The Social Loss Function

The principal differences between the present paper and previous
rational expectations cum wage contracting analyses of monetary stabili-
zation policy derive from the specification of the social objective
function.22/ Because it is generally agreed that the monetary authori-
ties cannot systematically affect the level of employment when wage
setters have rational expectations, the issue of whether or not the
central bank wishes it could lower the average level of employment is
commonly ignored. (It is ignored by assuming that the monetary authori-

ties target the same level of employment as do the wage setters.) But


-21-

variance of employment and inflation around their respective market-
deternined levels.

From the Lemma, it follows immediately that are/ar is zero at
t= 0. To prove that re is strictly increasing in t, the Lemma tells
us that we only need to prove that o° - pe is monotonically’ increasing
or monotonically decreasing in t. Using the definition of o° given in

equation (32), and differentiating yields

apo/ar = 2{-[ (148) /al (v2 + y) + v(140)2/n}/ (38)

tv? + (40)? + I?

Examining the right-hand side of (38), the denominator is posi-

>
tive since t = 0, and the numerator is a constant which can be negative,

33/

zero, or positive.— When the numerator is strictly negative or posi-
tive, 30°/ at is strictly monotonic and therefore according to the
Lemna, re is strictly monotonically increasing. In the special case
where the numerator of (38) is zero, 0° is independent of t and, since

of = o> at T=0, ré equals r for all t. In this special case, ra

in
=o since that is the level of t which minimizes n° in equation (31).
This completes our discussion of the proof of Theorem 2. All
other aspects of the proof are the same as for Theorem 1 .34/
Comparing Theorem 1 and Theorem 2, we see that the qualitative
difference between them is that totally rigid nominal GNP targeting is
optimal in one special case. For one special parameter configuration,
fixing nominal GNP is exactly equivalent to what the central bank would

do in the absence of any commitments. Further comparison of the two

regimes will be postponed until Section VI.


-A9-

Appendix E: An Analysis of the Inefficient Form of Nominal GNP Targeting.

In Section V.2 of the text, we discussed a potential implementa-
tion problem with nominal. GNP targeting. A nominal GNP target has two
components: price and output. Clearly there should be no problem announ-
cing a price level consistent with the desired rate of inflation 7. But
it may be very difficult for the central bank to admit that its out-
put target is consistent with the distorted market-determined level of
employment n rather than the higher socially preferred level A. Announ-
cing a nominal GNP target consistent with ¥ = f(ii) will have no effect
on the average level of employment and will only serve to raise the
average level of inflation. But, as we shall demonstrate below, there
are cases where even the suboptimal form of nominal GNP targeting domi-
nates money-supply and inflation-rate targeting.

A formal analysis of the case where the central bank targets the
"suboptimal" (¥ + Pry + it) is essentially the same as the case consid-
ered in section V.2 of the text, where the target is (y + Pry + 7).

But there are a few differences. Equation (29) becomes

¥ = (nit)? + xi)? + 5, + PL - F- Py 77s (29")

t-

equation (30) is modified similarly. Equations (32) and (33) are the

"

same; ape apy, re = ry. But equation (31) becomes

nr

@ - %? =v + 5140) C-a) 12m)? /Iccte) + 12. G31")

r is always greater than n°; therefore ie is always greater than ae,
In fact, the suboptimal form of nominal GNP targeting will not neces--

sarily even produce a lower (average) inflation rate than fully


2
D 2
TI as (l-a) < v/y . But it is not

AN

discretionary monetary policy; iw
difficult to produce reasonable numerical examples where ac < i < a,
Tr D
NN?

Example: Set q = .25, fin = .02, B= 0, xy=1, o = .0l, o = .00001,

Q
"

-005, w=0, A =5, > = 1, 6 = 3. Then, at their respective minima,

Zz

Dn 6.4, = 4.7, = 11.1; m= 9, rh = 5.3, AT = 6.2, for em =
1.68; 1 = .2, P= 5.0, AY = 5.2, for cM" = 53; 1° = .05, 1° = 4.85,
\S = 4.9, for 1TM = 2.673 P= 1.2, = 5.9, AM = 7.1, for pm = 214.

All tke numbers above are multiplied by 10? except for €, T, 7, and us
the ccmmon term (an)” is omitted from all the A's.

Because the average inflation rate F does not converge to 7 as
t + © under the suboptimal version of nominal GNP targeting, the argu-

ment used below equation (27) does not go through; it is possible for

min

c to equal infinity under much weaker conditions than were necessary
for nin =o, For the special case, 8 = w= 0, it is possible to prove
that ¢TM" =0 if

222 2 ~ =y\2

Xavo, < [-(l-a) “ya + (1-0) ] Gi-n)”. (El)

(Note that when 8 = 0, x(l-a)a < 1 is the necessary condition for sub-
optimal nominal GNP targeting to be superior to fully discretionary

monetary policy.) Condition (El) implies that lim xan /ac}/Lart/3c] > le
inal

Appendix F: A Comparison of Rigid Targeting Regimes

In general, it is difficult to derive tractable closed-form
expressions for the social loss function under any of the monetary tar-
geting regimes considered in Sections IV and V, provided that the tar—-

gets are assigned optimal weights. But it is easy and illustrative


-F10-

46/ The analysis can also be extended to the case where the
monetary authorities have only a subset of the information available
to investors, or visa-versa. Weiss (1980) and Canzoneri et al. (1982)
show how the monetary authorities can systematically exploit any
superior information available to investors by making promises of
future money supply infusions contingent on variables which only
investors can respond to contemporaneously (but which the central bank
can respond to with a lag.) This "prospective feedback channel" does
not function smoothly in a model where the monetary authorities and
wage setters have different target employment rates. Investors will
have no reason to believe contingent money supply promises which are
not time consistent. But it may be possible to restore some role
for prospective feedback rules by directly or indirectly introducing
investors' inflation rate prediction errors into the social utility
function. (The variance of these errors may affect the equilibriun
level of the capital stock.)

47/ Strictly speaking, Canzoneri et al. derive these results
for the case where the monetary authorities and investors observe
only the contemporaneous interest rate. But their methods and results
are easily extended to the case at hand. Also, it is necessary to
assume that the monetary authorities in no case completely reduce the
variance of prices or interest rates to exactly zero, as this would
change the information structure. (Alternatively, we may assume that
agents also observe the contemporaneous money supply.)

48/ Note that a, the coefficient on capital in the Cobb-Douglas

production function, is less than one.


-27-

shocks are affecting the price level. Inflation-rate targeting works
poorly when supply shocks are large, etc. And we have demonstrated that
while the nominal interest rate may be a useful information feedback
variable, it is not useful for the monetary authorities to precommit

to achieving a low nominal interest rate.

Algebraic comparisons of the alternative regimes, however, are
complicated considerably by the fact that the optimally flexible re-
gimes of Section IV and V do not have tractable closed-form solutions.
This would not be such a problem if the following two-step procedure
were entirely valid: First, calculate the values of the social objec-
tive function attained when each regime is employed rigidly; i.e., when
the weight on the target goes to infinity in the central bank's objec-
tive function. The resulting closed-form expressions are reasonably easy

to compare; (see Appendix F) A2/

Select the target which performs

best. The second step would then be to find the optimal weight to
place on the selected target. (This second step would require esti-
mates of the parameters of the model.) Unfortunately, this procedure is
not fool-proof because of the possibility of "rank-reversal." In
Appendix D, we demonstrate that a comparison of rigid targeting regimes
does not necessarily yield the same target rankings as a comparison of
optimally flexible regimes. The problem of rank reversal is mitigated
by the fact that it is reasonably easy to use a computer to quantita-
tively compare alternative optimally-flexible targeting regimes for any

43/

given set of parameter values.—'


3+

of the underlying labor market distortion which causes the market-
determined average level of employment to be too low. If this distor-
tion can be removed at low social cost, then it would be possible to
both raise employment and lower inflation. A second-best solution,
which does nothing to raise the average level of employment, would be
for tke central bank to somehow guarantee to wage setters that it will
not systematically inflate for the purpose of raising employment to its
socially optimal level. Such a guarantee would bring down the rate of
wage inflation without limiting the scope of the central bank to use
monetery policy to offset unanticipated disturbances. (Unfortunately,
it is hard to envision how such a guarantee could be enforced.) Thus
it is only when the first and second-best solutions are too costly or
unachievable that monetary targeting should be used as a "third-best"
solution to the problem of stagflation.

Section II of the text describes a stochastic rational expec-
tations macroeconomic model in which, because of wage contracting, there
is a well-defined role for central bank stabilization policy. Section
III derives the ("time consistent") equilibrium under fully discretion-
ary monetary policy. Section IV shows how society can make itself
better off by appointing an agent to head the central bank whose dis-
like for inflation relative to unemployment is stronger than average.
Section V reinterprets the formal analysis of Section IV as a model of
inflation-rate targeting, and demonstrates how to extend the framework
to encompass nominal GNP targeting, money supply targeting and nominal
interest rate targeting. Section VI discusses comparisons across re-

gimes, and the main results are summarized in the conclusions.


-F2-

6/ The aggregate demand specification is the same as in Canzoneri,
Henderson and Rogoff (1982). The aggregate supply specification is
based on Gray (1976) and Fischer (1977).

7/ The model does incorporate price-level indexation.

8/ Barro and Gordon (1983) mention these as two possible factors
which could distort the labor supply curve.

9/ Modeling the labor supply distortion as a shift in the inter-
cept of the labor supply curve is analytically convenient; allowing the
slope coefficient to change instead should not require altering any of
the theorems proved later in an important way.

10/ Note that wages are indexed to the contemporaneous price
level as in Gray (1976). The type of wage indexation considered here

is not of the "

catch-up" variety. The wage indexation parameter B
will be treated as a fixed parameter throughout the analysis. Modifying
this assumption should not alter the main conclusions though it could,
in principle, modify the comparison of regimes in Section VI. We
shall return to this issue below.

il/ This is the first of many times throughout the paper where
use is made of the fact that certainty equivalence holds when the loss
function is quadratic; here the base nominal wage is set in exactly
the same way as it would be if the price level had zero variance
around its expected value. See Sargent (1979).

12/ Investors have "rational expectations." They choose

aoa 2 .
Ee ea) to minimize E, [p - Ey)! » None of the main results

ttl

would be affected in any important way if investors instead based their

expectations on t-l information. The assumption employed in the text


=R2-

Poole, William, "Optimal Choice of Monetary Instruments in a Simple

Stochastic Macro Model," Quarterly Journal of Economics, Vo... 84
(May, 1970), pp. 197-216.

Sargent, Thomas J., Macroeconomic Theory (New York, 1979).

Sargent, Thomas J. and Neil Wallace, "Rational Expectations and the
Theory of Economic Policy," Journal of Monetary Economics, Vol. 2,
(April 1976), pp. 169-183. .

Taylor, John B., "Aggregate Dynamics and Staggered Contracts," Journal of
Political Economy, Vol. 88, No. 1 (February 1980), pp. 1-23.

Weiss, Larry, "The Role of Active Monetary Policy in a Rational Expecta-
tions Model," Journal of Political Economy, Vol. 88, No. 2 (April
1980), pp. 221-33.



~14-

anise = y(amt/oe) + art/ae . (26a)
2 -

art/ae = 2loa/n Mev MW? + x + e)°I, (26b)
I 92 3 ;

al /de = -2[v(i-n)]"/(yte)TM. (26c)
Define cTM" as the value of ¢ which minimizes jl. We are now

ready to prove:

Theorem 1. For fi > n, 0 <etTM, oO.

Proof: Note that ¢ > -y by assumption. Thus, by inspection of (26c),

amt /ae is strictly negative. Note also, by inspection of (26b), that

art /3e is strictly negative for -y < « < 0, zero when ce = 0, and positive
for ¢ > 0. Therefore ahi /9e is strictly negative for ¢ £0. 9At/3e
must change from negative to positive at some sufficiently large value

of ¢, since as ¢ approaches positive infinity, ort /3e converges to zero

~2 I -
at rate ¢ -, whereas 91 /de converges to zero at rate ¢ 3. Therefore

et co, QLE.D.

It follows immediately that for i =n, € = 0.

Theorem 1 states that in the presence of a labor market dis-
tortion, it is optimal to choose an agent to head the central bank who
places a greater, but not infinitely greater, weight on inflation than
society does. To intuitively interpret Theorem 1, consider the effects
of raising e« from zero. By increasing the central bank's commitment to
fighting inflation, the time-consistent average rate of wage inflation
is reduced. But the relative weight the central bank places on inflation
versus employment stabilization is altered, and this distorts the mone-

tary authorities' responses to unanticipated shocks. To see why the


-16-

the money supply, the interest rate, nominal GNP, or the inflation rate.
The commitment may be enforced directly through a system of rewards and
punishments. Or it may derive credibility from the agent's known
integrity and dedication as a public servant; the agent's reputation
and therefore expected future income could be affected by his success
in meeting his targets. The relevant degree of commitment should be
measured, of course, only by these underlying incentives, and not by

how loudly promises are proclaimed.

v.1. Inflation - Rate (Price Level) Targeting

According to the above interpretation, the analysis of Section
IV may be viewed as an analysis of inflation-rate (price-level) tar-

29/

geting.—' It is easiest to see this by rewriting the central banker's

objective function (21) as
T= (na'y)? + ye)? +e (nity? (28)

The first two terms in equation (28) comprise A, the social
objective function (12). The final term represents the additional
weight which the central bank places on the inflation rate because it
has chosen the inflation rate (or price level) as an intermediate tar-
get. Unless the index parameter ¢ is infinite, the central bank cannot
be relied on to religiously hit its intermediate target, because that policy
generally conflicts with the policy which would maximize the social
utility function A. € measures the extra (artificial) incentives the
central bank has for fulfilling its inflation rate target. These incen-
tives are additional to the fact that the inflation rate enters directly

into A. While the formalism embodied in equation (28) may appear


-2-

rate of employment. By setting nominal wages high enough, they are able
to place the monetary authorities in a position where the social cost of
inflating beyond the level consistent with wage setters' desired real
wages begins to outweigh the benefits of reduced unemployment. Thus,
as in virtually all rational expectations models, the monetary authori-
ties cannot systematically raise the level of employment. But since
wage contracts are not fully indexed to the price level, monetary
policy can still be used to improve social welfare by stabilizing
employment and inflation around their market-determined teveis.4/

By implementing monetary targeting, the central bank can con-
vince wage setters that it is not as likely to inflate, but only at the
cost of constraining its ability to respond to future unanticipated
disturbances»! The optimal degree to which the monetary authorities
should credibly commit themselves to their intermediate monetary target
trades off the inflation gains against the stabilization costs. Rigid
targeting is appropriate only in certain very special cases. A number
of types of targets for monetary commitment are considered: the money
supply, nominal GNP, the inflation rate (price level) and the shor+-
term nominal interest rate. Which one of these works best depends, of
course, on the structure of the economy and the nature of the under-
lying disturbances. (Though the interest rate generally appears to be
an unsatisfactory tool.)

It is important to stress that while flexible monetary targeting
is preferable to either fully discretionary monetary policy or rigid
monetary targeting, it is not necessarily the first-best solution to

the problem of stagflation in this model. That depends on the source


-A8-

For x > Lo, a term by term comparison reveals that ré > rt for all
7.48/ However, for y = Woe - A, it is possible to choose A suffi-
ciently small so that the first three terms on the left-hand side of
(D1) are all larger than the first three terms on the right-hand side,
although the term in 2 on the right-hand side is larger than the term
in ~ on the left-hand side 42/

This implies that although r& < rt for large t, re > rt for
small ~. (For small enough t, the a terms are almost irrelevant.)

Given propositions B.1, B.2 and B.3, it is easy to use a com-
puter to find numerical examples of rank reversal. Rank reversal
appears to occur frequently, but cases where it is very wrong to choose
the target that would be best for a rigid regime are difficult to find.

Two numerical examples:

(1) Set a = .25, BH = .02, x= 1, oF = .020, w= 0 and B= 0.
I

then IP = 6.4, T= 12.0, = 18.4; mi = 1.4, ri = 12.8, At = 14.2,

for c= 2.16; 1° = 1.4, ro = 13.0, AS = 14.4, for TP = .287,
where all numbers have been multiplied by 1000 except for ¢ and 1; the

terms in (aeny’, which are independent of regime choice, are ignored.

Note that (AT)TMTM < (AS)TMTM. However, AT|__, = 200 > 32.8 = ASI.

(2) Set a= .25, B= .03, B= 0, X= 1, oF = «I, oe = .00001,
o = 02, 0= 1, A=5, 623,621. Then P=1.4, r= 1.2, 0 = 2.6;
wee .2, 21.3, ab = 1.6, eM = 2.6; me 7, a .4, aM = 2.1,
min = .043, where all terms except ¢€ and y have been multiplied by 100,
and again the @ny? component of A is ignored. Note that cyyymin <
(aMyTM "But 20.4 = A" es > Naw = 9.6.


-25-
Consider the objective function:
R= (n-f') + y(n)? + uer-#)*, Og U< om. (47)
where ~ is the nominal interest rate target. Using equations (8),

(9), (10) and (13), the objective function (47) can be written as:

R= [2,/n + v(p, - We) - GRD] + XO, - Py? (48)

A x2
+ ult, + (6/8), - PL) + Cu, - 2/0/65 - FI”.
As we shall see, the value of 7 =E (p* ) - p, is crucial to the
, ttl = Pet t

analysis; it depends on the monetary policy regime expected to pre-

vail in period t+l. Following the algorithm of Section III for de-
40/

riving the time consistent equilibrium, one obtains:—

Re ~ x
7 - 7 = [vG@i-n) + (v0/6) Te 7 E)I/x> (49)
—A . —A

where Tee has been substituted in for Teel * (Recall that, given the

assumptions we have made about the parameters of the macro-model, the
average real interest rate under any monetary regime is zero.)

2—2
<7 as

A comparison of equations (17) and (49) reveals that 7

A

<
Tet"

E In other words, suppose a central banker is appointed for
one period, but this appointment has no effect on future central bank
regimes. Suppose this central banker announces that he is going to
try to bring interest rates below their trend rate, so that ¥ < Fa .
Then, instead of falling, the expected inflation rate and expected nomi-
nal interest rate rise. They rise because wage setters recognize that
once wages are set, the central bank can lower interest rates through
money growth.. While it is true that the central bank could try to bring

down inflation by setting f greater than Th the fact that this would


THE OPTIMAL DEGREE OF COMMITMENT TO AN INTERMEDIATE MONETARY TARGET:
INFLATION GAINS VERSUS STABILIZATION COSTS, by Kenneth Rogoff*

I. Introduction

Curiously, despite all the emphasis on thinking about monetary
policy in terms of rules,_/ no regime analyzed in the extant theoret-
ical literature can be said to closely describe the monetary targeting

2/

regimes observed over the past decade.~’ The real-world feature of
these regimes which is absent from the theoretical models is the ability
of the central bank to credibly commit itself to placing great weight
on an intermediate monetary target (relative to direct social objectives),
while retaining some scope to respond to new information. "Flexible"
monetary targeting constrains the discretionary actions of the central
bank but still leaves some room for the monetary authorities to use the
money supply as an instrument of stabilization policy.

The present paper attempts to provide a rationale for flexible
monetary targeting by synthesizing elements of the recent time consis-
tency literature into a standard rational expectations cum wage con-

3/

tracting macroeconomic model." The possible need for monetary tar-
geting arises as equilibrium under fully discretionary monetary policy
is characterized by stagflation. Either because of monopolistic unions,
or because income taxes and unemployment insurance distort the labor-
leisure decision, the equilibrium level of employment is assumed to lie
below the level that would arise in the absence of labor market distor-
tions, As in Barro and Gordon (1983), inflation is generated because

wage setters rationally fear that the central bank will inflate to try

to drive down real wages to a level consistent with the socially desired


-F5-

E,_ (mp) = py_y + CAD EVGi-n)/x + i], (£2)

t-

q
°

Ey teyg) = Egy (mp) + slvG-a/y +H, s 2 (£3)

tts

(Obsi:feld and Rogoff (1983) provide microeconomic justification for
ruling out speculative hyperinflations and hyperdeflations ‘unrelated
to the underlying fundamentals.)

25/ Note that the time-consistent rate of inflation is a
decreasing function of the indexation parameter 8, since
y = (1-8)/a. Obviously this result would not necessarily hold if
indexation were of the "catch-up" variety. Also, one can imagine
models in which indexation raises the time consistent rate of infla-
tion. Suppose that the indexation scheme involved giving all workers
the same absolute rather than percentage raise. (An example of such
a scheme is the scala mobile in Italy.) Then a government interested
in income redistribution might have a stronger incentive to inflate
than it would in the absence of indexation.

26/ The assumption that e€ > -x implies that the agent gets
disutility from inflation. We shall see in the next section how this
regime may also be thought of as one in which the central bank targets
the inflation rate.

27/ We want to evaluate the regime on the basis of the expected
value of the social loss function, not the expected value of the
central banker's loss function.

28/ Although it is extremely difficult to write down a closed-
form. solution for emin, we are able to prove in Appendix B that emin

is the unique real positive root of aat/ae = 0, so that at is concave ine.


-A4~

The proofs employ Descartes' Law of Signs, which states that the
number of positive real roots of a polynomial is at most equal to the
number of times that the coefficients of the polynomial alternate in
sign.

Proof of Proposition B.1. From equations (26),

anT/de = [be(y + €)? - ah +) W/ty + €) 3th + €)3, (1)

xv? Gen), and h = «? +x).

where b = vn)’, a
A necessary and sufficient condition for at /ae to equal zero is for

the numerator of (Bl) to equal zero:

ce” + (3y - alb)e? + (3y? = 3ha/bye? + (2 - 3n2a/d)e - n2a/b = 0.

(B2)

The fact that the first and last terms of (B2) are of opposite
signs immediately implies that the coefficients of the polynomial alter-
nate in sign at most three times. And in fact, because h = (x + vy >
X» we can deduce that there is at most one sign change. (Since if a/b
é sh, then a/b < 3y > yh and therefore the first three terms are posi-
tive. Since the final term is negative, there can only be one sign
change. Or if instead a/b > ln, then the fact that h > y implies that
a/b > 3e/3n2 and therefore the last three terms are negative. Since the
first term is positive, there can again be only one sign change.

Proof of Proposition B.2.

The fact that af°/9t is of the form b"c/[y + v2 + t(146)"]>,
with b" > 0, can be confirmed by direct differentiation of equation (33)
or can be deduced by noting that (a) any expression of the form

(dtet+ £17) (h + qu has a derivative of the form (q + rt)(h + quo


-A13-

comparison with (F3), is that rigid nominal GNP targeting is exactly

equivalent to rigid employment targeting when 8 = w = 0.22/


-28-

VII. Conclusions

This paper extends the standard rational expectations cum wage
contracting model to an environment in which the central bank has
explicit motivation for precommitting to an intermediate monetary tar-
get. We have demonstrated how this framework can be used to analyze
money supply targeting, nominal GNP targeting, inflation-rate tar-
geting and nominal interest rate targeting. The analysis can easily
be extended to consider other targets, such as the exchange rate, or to
encompass multiple intermediate targets.

A major conclusion of the analysis is that, except in certain
special cases, the type of flexible monetary targeting used by many
industrial-country central banks since the mid-seventies is preferable
to both rigid commitment and to fully discretionary policy. The optimal
weight to place on an intermediate monetary target (relative to direct
stabilization goals) trades off the gain from a lower average inflation
rate against the cost of reducing the central bank's capacity to stabi-
lize employment and inflation around their market-determined average
rates. One would expect that the best monetary target would be the one
most highly correlated with the central bank's ultimate objective func-
tion 44/ But while this is a useful rule of thumb, the situation is
actually somewhat more complicated. If we compare how each of the tar-
gets would work if used rigidly, we do not necessarily get the same
ranking as when we compare how each target works when the central bank
gives it an optimal weight (relative to direct social objectives.) 42/

Viewed from a broader perspective, the analysis of this paper
demonstrates conditions under which society should appoint a "conserva-

tive" (but not too conservative) agent to head the central bank. In


-12-

stabilization. The final term, ra represents the "stabilization"
component of the loss function. It measures how successfully the cen-
tral bank offsets disturbances to stabilize employment and inflation
around their average market-determined values.

We have already solved for the average level of inflation under
fully discretionary monetary policy, RP, see equation (17). To derive
r, first take t-l expectations across (15) and use (16) to substitute

for oe One obtains

[py - E,_,(p,)1? = dp? =Hv2,/n(v? + x) = 02, (19)

t

Note that u and v do not enter the expression for the price prediction
error that the central bank allows to occur. The central bank offsets
the price level effects of aggregate demand shocks to the best of its
ability (here perfect, because of the complete information assumption),
because offsetting these shocks is consistent with both employment
stabilization and inflation-rate stabilization. Substituting (19) into

the expression for r4 given in (18), and simplifying, yields
D 2,2
1? = (otsn*ytxs/(v? + x). (20)

IV. Social Welfare under a "Conservative" Central Banker.

The main result of this section is the following: When hor,
society can make itself better off by selecting an agent to head the
independent central bank who does not minimize the social loss function
A, but instead prefers to minimize a loss function which places a
greater weight on inflation relative to unemployment. But, within this
class of regimes, it is never optimal to choose an individual who pays
“too little" attention to unemployment, in a sense that will be made

precise below.


-F6-

29/ Price-level targeting and inflation-rate targeting are
equivalent here, since Prey is known at the time the central bank
commits itself to achieving a target for Pe - Peoye

30/ The standard literature on intermediate monetary target-
ing (Poole (1970), Friedman (1975) and Canzoneri et al. (1982), for
example) takes limited information rather than credibility as the
rationale for intermediate monetary targeting. The present model
can incorporate both rationales; see Appendix C.

31/ The nominal GNP target embodied in the final term of
equation (29) is consistent with the socially desired rate of infla-
tion and the average market-determined level of employment. However,
it may be politically difficult for the central bank to announce a
nominal GNP target consistent with a level of employment lower than
the level which would prevail in the absence of labor-market distor-
tions. The case where y is replaced by ¥ = £(f) will be discussed
in Appendix E.

32/ To obtain equations (31) - (33), first differentiate equa-
tion (30) with respect to Py Pe solves 9G, / ap, = 0. Take t-l
expectations across the equation for Pes and note that we = pe ina
time consistent equilibrium. (We have constructed the model so that
wage setters' target (for the logarithm of the) real wage is zero.)
Equations (31) and (32) follow immediately. Equation (33) is obtaired
by plugging (32) into the expression for ra in (18).

One check on the algebra is to consider the case of rigid

nominal GNP targeting (see Appendix F). Equation (9') implies that


-F8-

35/ In deriving (40), use is made of the fact that investors'
inflation-rate expectations (in period t) depend only on the time-
consistent solution to (39) and not on period t disturbances. It is
implicitly assumed that investors expect that the same regime will
be in place in period t+l as in period t.

36/ Again, one can check the algebra in expression (44) for rt
by considering the limiting case of rigid money supply targeting. It
is straightforward to work through the model of equations (9), (10)
and (11) to derive an expression for PRT E,_1 ,) under a fixed
money rule. Plugging this expression into ra in equation (18) gives
the same result as taking the limit of (44) as U goes to infinity.

37/ For the special case 8 = w=0,condition (46) reduces to
xe -Elx+ (1/a2)] = 0. (Note from (40) that > 1.) It is possible
to prove that an" yay = 0 has at most three positive real roots; a
sufficient condition for pin to be the sole real positive root is
that (1-8)/a(l-a)[¢ + A/5] >x. See Appendix B for details.

38/ There are (special) cases where, although the authorities
learn something each period about the aggregate demand shocks u and
v (say, by observing the interest rate), it never pays to react. This
happens when the knowledge about u and the knowledge about v are
negatively correlated in such a way that nothing is known about price
level movements. See, for example, Canzoneri et al. (1982). The
reader will recognize that we are describing the special case where
Poole's (1970) "combination" interest-rate feedback rule reduces to
a fixed money supply rule.

39/ As Canzoneriet al. demonstrate, the central bank can only

literally peg the nominal interest rate if it also announces at least


-15-

benefit outweighs the cost at € = 0, it is suggestive to expand (26a)

as follows:
aAtyae = xan! yam!) (amt/ae) + (aAt arty (art/ae). (27)

In the neighborhood of € = 0, the monetary authorities are
ninimizing rt (they are stabilizing optimally), so that at /3e is zero.
But inflation is not being minimized, so that am /3e is not zero. We
car. argue similarly to suggest why emin, eo. As € becomes large, 7
goes to 7 and both of the terms in ra go to zero in (27). But rt rises
to an upper bound as € goes to infinity. (It is easy to calculate from
(24) that this upper bound is o2/n*). Therefore ant yart does not go
to zero. Since in equation (27) only one of the terms in rt goes to
zero and both the terms in its go to zero as € becomes large, it is
intuitively plausible that the loss eventually outweighs the gain.28/
In fact, Theorem 1 assures us this is true. Of course, when there is no
labor market distortion, so that i = fT, then 2? = Ts, and it does not

pay to appoint a central banker who minimizes anything other than the

social loss function.

V. Intermediate Monetary Targeting.

In the previous section we demonstrated conditions under which
society can make itself better off by appointing a central banker who
places greater weight on inflation stabilization than society itself
does. Here we reinterpret the formal analysis of Section IV as a model
of intermediate monetary targeting.

Suppose society selects an individual to head the central bank who

commits himself to achieving an intermediate monetary target, such as