-19-

constraining the stabilization effort. The choice of $ in the last

example would involve a similar calculation if productivity disturbances
were added to the model. Rogoff (1983a) considers the relative merits of
money, interest rate, inflation rate and nominal income targeting, all in

a much more interesting control setting than was practical here.

A Reputation Building Approach:

The credibility problem in the Kydland-Prescott game is one of
Pprecommitment; the Fed would like to be able to promise the wage setters
that it will run the ideal policy, and then have no reason for reneging
on its promise later. Barro and Gordon (1983a) show that Friedman's
(1971) solution concept for repeated games offers a resolution to this
problem.

Suppose the Fed and the wage setters have arrived at the
following set up: Wage setters believe the Fed's promise to run the
ideal policy in the current period if the Fed did not renege on its
promise last period. (Assume for the moment that there is no private
information.) If however the Fed reneged last period, then wage setters
revert to their non-cooperative wage settings this period; in subsequent
periods, they again believe the Fed's promise, until shown otherwise.
Barro and Gordon (1983a) show that the wage setters are not irrational in
going along with this set up, because the Fed will never have an

13/

incentive to renege.—

To see this, note that the Fed's temptation to renege in period

t #s measured by


-R2-

Kydland, F. and E. Prescott, "Rules Rather than Discretion: the
Inconsistency of Optimal Plans," Journal of Political Economy, 85,
June 1977, p. 4730-91.

Lee, W., Optimal Policy Selection and Evaluation with Rational
Expectations, Columbia University, Ph.D. dissertation, 1981.

Leijonhufvud, A. "Constitutional Constraints on the Monetary Powers of
Goverment," presented at the Conference on Constitutional Economics:
The Emerging Debate sponsored by The Heritage Foundation, Nov. 1982.

Phelps, E. "Phillips Curves, Expectations of Inflation and Optimal
Employment Over Time," Economica, Aug. 1967, p. 254-281.

Radner, R., "Monitoring Cooperative Agreements in a Repeated Principal -
Agent Relationship," Econometrica II, 1981, p. 1127 - 1148.

Rogoff, K. "The Optimal Degree of Commitment to an Intermediate Monetary
Traget: Inflation Gains Versus Stabilization Costs," unpublished
manuscript, June 1983a.

Rogoff, K. "Cooperative Monetary Policies with Flexible Exchange
Rates," unpublished manuscript, Sept. 1983b.

Roth, A. and F. Schoumaker, “Expectations and Reputations in Bargaining:
An Experimental Study," American Economic Review, June 1983, p. 362-
372.

Taylor, J. "Comments on ‘Rules, Discretion and Reputation in a Model of
Monetary Policy'", Journal of Monetary Economics 12, July 1983, p.
123-125,

Thompson, E. "Who Should Control the Money Supply," American Eccnomic
Review, May 1981, p. 356-361.

Townsend, R. "Optimal Multiperiod Contracts and the Gain from Erduring
Relationships under Private Information," Journal of Political
Economy, vol. 90, no. 6, 1982.

Waldo, D., “Sticky Nominal Wages and the Optimal Employment Rule",
Journal of Monetary Economics 7, 1981, p. 339-353.



-13-

that the Fed's policy is time inconsistent. Once the wage setters have
committed themselves to aes oe will no longer be the Fed's utility
maximizing action. Letting of = . in (12), and maximizing EUF with

respect to Spe the best Cheating Solution for the Fed is

(15) gf =a" +e, +e y"/(1 + F), ge" = at

ch

EUF,

= EUR + y 271 + #), Ewch

*2
= EUW - yy /(1 +f)

Wage setters cannot set Wy on the expectation that inflation will equal
. and then count on the Fed to follow through with oe The Fed will be
tempted to inflate a little more than r, lowering the real wage and
inflating away some of gy. Unless the Fed can be credibly preconmitted
to the ideal policy, wage setters will almost surely play non-
cooperative.

Can the Fed be credibly committed to the ideal policy at the
time that wage setters have to set We in their contract? Views seem to
differ as to whether this is a real world problem. Taylor (1983),
commenting on Barro and Gordon (1983a and b), points out that society has
found sensible ways out of other obvious time inconsistency problems; he
cites the example of patent law. Taylor seems skeptical that the
Kydland-Prescott game really captures the credibility problem the Fed is
generally thought to face.

My own view is that Taylor would probably be right were it not

for private information. Absent private information, resolutions of the


-23-

is large, then from (26), a big ‘e can be specified; consequently, the
probability of a reversion is small, and so is the frequency of
inflationary episodes.

So the modification for private information allows the model to
explain periodic inflationary reversions .22/ It also suggests that,
despite the apparent instability, Taylor (1983) might be right; the Fed
and the private sector may have already hit upon a rather good resolution
of the problem. In fact, the worse is the problem to begin with, the
better is the resolution; 6y* is the difference between the players'
output goals, and if y*/f (= ey*/(s/e)) is big, a large © can be
specified, and the frequency of inflationary reversions is small. It
should also be noted that in this resolution of the credibility problem,
the Fed retains full flexibility for stabilization.

There still remain some weaknesses or unanswered questions:

(1) Tne game cannot be finite, or as Barro and Gordon (1983a) have noted,
the solution unravels backward. It is not clear how this fact jibes with
the finite terms of office of Fed policy makers. (2) It is not clear
that the inflationary reversions modeled here correspond to the policy
scenario outlined in the introduction. It is tempting to interpret the
triggering of reversions as credibility breakdowns, and the inflationary
bouts as times of credibility building or learning. But this would not
be aporopriate. The reversionary periods modeled here are necessary
evils that make the Fed's promise of a non-inflationary policy credible.
Their existence are evidence that the credibility problem of
Precommitment has been resolved. The instability inherent in a
straightforward interpretation of Rogoff's (1983a) model may more closely

correspond to the policy scenario outlined in the introduction.


-4-

when the Fed has been honest. However, as Green and Porter (1983) have
noted in a different context, this may be a blessing in disguise. The
Friedman solution, like Rogoff's (1983a), is too stable to explain the
episodic periods of breakdown that are actually observed. A
reformation of the model accommodates private information and actually
predicts periodic bouts of inflation (though it would be inappropriate to
think of these episodes as credibility breakdowns). It also suggests
that, despite these periodic difficulties, the Fed and the wage setters
may have already found a rather satisfactory resolution to the
credibility problem.

Section IV discusses the legislative approach in a more
sophisticated way than was possible in section II; the proper legislation
can also make the Fed's opportunities in future periods depend upon
honest behavior in the present. There are monetary policy rules that
produce a better game outcome (generally at some expense to the
stabilization effort) and are incentive compatible, even if the Fed's
forecast of money demand is private information. These rules are,
however, less intuitive than the typical stabilization rule; they are
more easily motivated as the Fed's optimal response to a Congressionally
mandated targeting procedure.</ Various targeting procedures are
considered, some leaving the Fed with more latitude than others. Which
is best depends on the Fed's ability to predict money demand disturbances
and on the importance society attaches to the benefits of stabilization
relative to the costs of the inflation bias resulting from the non-
cooperative behavior of the players.

Section V summarizes the basic results of the paper and

suggests where further research is needed.


-F3-

22/ This can be verified by substituting 9) = res pe} +y Jer and

92 = Qn’ - G1 into the Fed's two period optimization problem and
maximizing wer.t. ef.

23/ To prove this, first calculate the non-cooperative solution with no
private information; then replace e, and eo with efand e3 in the Fed's
policy rules for g; and go, and calculate the Fed's utility maximizing
values for ef and e9.

24/ This can be verified by substituting 9] = r+ et + y*/2F and

92 = Qn* + 51 - 91 into the Fed's two period optimization problem and
maximizing wer.t. ef.

25/ The general Flexible Targeting Solution is

git =a + LO +f + ch)/(1 + 2F)Jeq + (1/2F)yTM
oft =n + coy - gft
Eurft = - (1 - c)@ER(1 + £)/2(1 + 2F) 02 - 62 (F/2)02

- CO) + £)/21( 0% + of) ~ [1 + (1/4F) I"?


-20-

ch

(20) Eure". Eure = y 7. +f)

while its reward in period t + 1 for not reneging in period t is

c nc _ .*2
(21) EUFE,, - EUFES) sy /f

Since the reward for honesty outweighs the temptation to cheat, it pays
the Fed to invest in its reputation; it has no incentive to renege L4/

There are two weaknesses in this resolution of the credibility
problem that can be dealt with in ways suggested by Green and Porter
(1983). (There are some others that are not so easily remedied; they
will be discussed at the end of the section.) The first weakness is
that the Barro - Gordon solution is, like Rogoff's (1983a), too stable.
If taken literally, it implies that one should never expect to see
reversions to inflationary periods. The second weakness is that if the
Fed's forecast of money demand is private information, then the wage
setters can never know for sure when the Fed has reneged. The wage
setters do not observe the Fed's forecast at the end of the period, and
as was shown in section II, it would do no good to have the Fed announce
a forecast, for it would not be believed.

The model can be reformulated to deal with both weaknesses as
follows: The wage setters do see the money demand disturbance, Sp. at
the end of the period; they just can't decompose it into the Fed's
forecast, ep and the residual, ee Thus while it is not feasible to

have the wage setters revert to their inflationary wage settings when 9p

is greater than a = ort + ep it is feasible to have a reversionary


-26-

* * sat t
gs = - pe, -y af grt = 98
*
Eurat = - (1 + f)Lo% - (0/2) 00] - C1 + (1/4t) Iv"?
2
EUWat = - 02 - (1/2)E1 + (1 - 9)? ]00 < Eu’

where p = (1 + f)/(1 + 2f) < 1, and curt and ena are expected
utilities averaged over the two periods.

Unlike the ideal rule of = x + ep, the policy rules in (29)
are incentive compatible. Congress could legislate them directly,
instead of the targeting procedure; the Fed would announce et at the time
it set 9y> and would have no incentive to make et greater than e, Ae

Here, the Fed will do some stabilizing, but the stabilization
will only be partial, and it will only occur in the first period. In
addition, the new flexibility results in some inflation bias in the first
period; the averaging requires the bias to be reversed in the second
period, and this curbs the Fed's temptation to inflate. Compared with
strict targeting, average targeting's advantage is in its flexibility for
stabilization; compared with no targeting, average targeting's advantage
is in its reduced inflation bias.

In fact, it can be shown that

(30) eure > euret if R> 6


-22-

The first term is the expected gain in the current period from inflating
away some of the output differential, ey sal/ the second term is the
expected loss next period from increasing the probability of a
reversion. Note that the increase in the probability of a reversion

(evaluated at 5 = of) is just the probability that ee equals ©;
(25) P'(0 - €) = p(-e)

where p(+) is the p.d.f. of the forecast error, e¢ From (13) and (14),

t°
Cc nc _  *2 56,
CUFF a1 - EUF ae y “/f3 so

(26) U, (g5,=) < 0 iff p(-e) > 2/y"/F)
Ot

If yt is greater than 2 (so that 0 < ary" /f) <1), then © can be
chosen to eliminate any incentive for the Fed to renege.

The trigger mechanism for a reversionary period, % >
r + 5 +e, was designed to be applicable in the private information
case, where wage setters cannot tell whether or not the Fed has reneged.
As a result, it has a very interesting property. Even if the Fed is
always running the ideal policy, there will still be periodic
inflationary episodes associated with large negative prediction
errors 28/ Wage setters will see a large oy and a small Spe and they

will revert to inflationary wage settings. The frequency of these

* *
inflationary reversions depends inversely upon the size of y /f. If y /f


REFERENCES

Aizenman, J. and J. Frenkel, "Wage Indexation and the Optimal Exchange
Rate Regime," unpublished manuscript, Aug. 1983.

Baccus, D. and J. Driffill, “Rational Expectations and Policy Credibility
Following a Change in Regime," unpublished manuscript, May 1984.

Barro, R. and D. Gordon, "Rules, Discretion and Reputation in a Model of
Monetary Policy," Journal of Monetary Economics, July 1983a, p. 101-
122.

Barro, R and D. Gordon, "A Positive Theory of Monetary Policy in a
Natural-Rate Model," Journal of Political Economy 91, Aug. 1983b,
p. 589-610,

Canzoneri, M. "The Intermediate Control Problem," Journal of Money Credit
and Banking, May 1977, p. 368-371.

Canzoneri, M. "The Time Inconsistency Problem," unpublished manuscript,
Aug. 1980.

Canzoneri, M. and D. Henderson, Strategic Aspects of Policy Making in
Open Economies, monograph forthcoming in 1985.

Fischer, S. "Long-Term Contracts, Rational Expectation and the Optimal
Money Supply Rule," Journal of Political Economics, Feb. 1977, p.
191-205,

Fischer, S. "Dynamics Inconsistency, Cooperation and the Benevolent
Dissembling Government," Journal of Economic Dynamics and Centrol,
Feb. 1980, p. 93-108.

Friedman, J. "A Noncooperative Equilibrium for Supergames," Review of
Economic Studies, 38, Jan. 1971, p. 861-74.

Gray, J.A., "Wage Indexation: A Macroeconomic Approach," Journal of
Monetary Economics 2, 1976, p. 221-235.

Green, E. and R. Porter, "Noncooperative Collusion Under Imperfect Price
Information," Econometrica, Jan. 1984,

Green, S. "Unattainable Objectives, Incomplete Information and
Regrettable Macroeconomic Policy," unpublished manuscript, April
1983,

Hart, 0. "Optimal Labor Contracts Under Asymmetric Information: An
Introduction," Review of Economic Studies, Jan. 1983, p. 3-36.

Hurwicz, L. "On Informationally Decentralized Systems," in Decision and
Organization: A Volume in Honor of Jacob Marschak, edited by C.R.
McGuire and Ray RAdner, Amsterdam: North-Holland, 1972.



-17-

his preferences.
If ee is private information, this is not possible. The policy

maker could announce a forecast, ee, when he set Gh» but once again the
12/

mechanism would not be incentive compatible.. To see this, suppose the
policy maker represents his preferences as UP, (with ¢ > 1), but
actually his preferences are UF ye Then, substituting of = ol + y sof and
9, = 1 + ef + y* of into (12) and maximizing EUF, with respect to ef
the policy maker's optimal misrepresentation, et - @y5 can be seen to be

[lo-1)/o(1 + fy’. It can be shown that a marginal increase in $ over
unity will raise EUF, above EUFyS and lower EW, below ews.

Thus a policy maker with true social preferences has an
incentive to look conservative, and if the Fed's forecast of money demand
is private information, wage setters have no way of telling for sure.

The policy maker's actions and announcements will not reveal his actual
preferences. The implications of this observation would appear to be two
fold. The model may now provide a more plausible interpretation of
history; without private information, Rogoff's resolution of the
credibility problem is too stable to explain the comings and goings of
individual policy makers, with all of the attendant hoopla, posturing and
changes in public perception. However, the very same instability that
may help explain history also detracts from the appeal of this approach
as a final resolution of the credibility problem.

Rogoff (1983a) does not consider private information, but he

does offer an alternative interpretation of his formal analysis that may

be helpful in addressing the issue. He suggests that the added weight on


Suppose a stochastic disturbance is added to money demand in

(7). If this disturbance follows a random walk, then (8) becomes

(9) gp - m = 6,

where 5 is the white noise innovation. Suppose further that wage
setters do not see 8 at the time they have to specify Wee but that the
Fed does have & (or some forecast of it) when it sets Ee More

specifically, let ory be decomposed into ee and fps

where e is the Fed's forecast of 6,5 and ee is its forecast error. If
the “ed is left with sufficient flexibility, it can accommodate ae before
it passes on to prices and output; this benefits both wage setters and
society as a whole. The problem now is one of trading off the
flexibility needed for stabilization with the constraint required for
eliminating the inflation bias; this problem is touched upon in Barro and
Gordon (1983a), and it is the main subject of Rogoff (1983a).

Two quite different information structures will be considered
below, and it is convenient to be more explicit about them at this point.
With "symmetric information," both wage setters and the Fed are assumed
to observe ee at the same time, though it is too late for the wage
setters to incorporate this new information into the contract wage, We
With "private information," either the Fed has superior information that

cannot be revealed or the policy making process is such that the private

sector cannot reconstruct the Fed's forecast of money demand. At the end


Footnotes

*/ I would like to thank J. Friedman, E. Green, D. Henderson, K. Rogoff,
J. Taylor and an anonymous referee for their helpful comments. However,
the views presented here are my own. They are not necessarily shared by
the Federal Reserve Board or any other member of its staff.

1/ Targeting money or interest rates is not likely to be efficient in
terms of meeting final targets; see, for example, Canzoneri (1977). This
observation is the principal theme of Rogoff (1983a).

2/ See Barro and Gordon (1983a, b), Taylor (1983), Green (1983), Rogoff
(1983a, b), Baccus and Driffill (1984), and more generally, Lee (1981).
Phelps (1967) had many of the elements of this game.

3/ This paper borrows heavily from Hurwicz (1972), as does much of
the recent contracting literature; see Townsend (1983), Hart (1983), and
the Quarterly Journal of Economics, 98, (supplement, 1983).

4/ They could also be motivated along the lines of Peel's Bank Act of
1844; see Leijonhufvud's (1982) discussion of the Peel-Friedman system.

5/ This assumption has been criticized; see the exchange between Barro
and Fischer in Journal of Monetary Economics 3, 1977, and also Waldo
(1981).

6/ Aizenman and Frenkel (1983) provide a rationale for this utility
function. The analysis is simplified by abstracting from productivity
disturbances; see Gray (1976) and Rogoff (1983a).

7/ It is not necessary to specify conflicting utility functions in order
to generate the time inconsistency on which this game rides; see Fischer
(1980) and Canzoneri and Henderson (1985) for expositions of the time
consistency problem that are mathematically unencumbered.

8/ While the two views are equivalent for present purposes, they do have
‘different implications for the first-best solution to the policy problem
posed here. Canzoneri's (1980) view suggests labor legislation, while
Barro and Gordon's (1983a, b) points to fiscal policy reform.

9/ Some experimental studies suggest that players may be willing to go
‘along in this way; see Roth and Schoumaker (1983).

10/ In a more realistic setting, the ideal policy rule could well be more
complicated, but this should not in and of itself create a credibility
problem. Credibility problems arise only when scope has been left for
cheating.

l1/ Wage setters can tell if the Fed is telling the truth "on average",
For then the sum (5, - ef) = Ze, should tend to zero. If the
relationship is an enduring one, and if utility is not discounted over
time, then this approach might work. See Radner (1981).


-29-

stabilize fully in the first period. However, it turns out that this is

*
again a polar case. Suppose c is chosen optimally; that is, let c he
the value of c that maximizes eur{*, It can be shown that

(34) +) as oo /o

*
2706 + 0 and c +0 as o +

2
e
and that if c is less than one, the Fed will not find it optimal to fully
accommodate ey 25/ Full stabilization in the first period should be
allowed only if the Fed is very good at forecasting money demand. In
fact, with c set equal to one, flexible targeting beats average targeting
only if o/oe is less than (1 + f)/(1 + 2f); with c set optimally,
flexible targeting always beats average targeting.

It is beyond the scope of the present paper to try to identify
the most efficient form of targeting. That would require a more
interesting control setting (with dynamics, productivity shocks, etc.)
and a multi-period targeting horizon (presumably corresponding to the
policy maker's term of office). In addition, Townsend's (1983) simple
example of multi-period contracting suggests that a complete
specification of the most efficient social contract would be difficult,
if not impossible.

The examples provided here do however provide two insights that
should be helpful in designing efficient targeting procedures. (1) The
procedure should be defined in terms of variables that are observed by
all narket participants; actual money demand was used in the examples
above, but actual inflation would have served just as well. Credibility

problems will remain if latitude for stabilization is incorporated in any


-6-

obtain in the contract period. The wage setters' play is, essentially,
their prediction.

Substituting (3) into (1) results in a familiar prediction
error model of supply. Note also that (1) and (2) can be expressec in

terms of inflation and expected inflation:

(4) -y, = ¥ + (0, = PE) = ¥ + O(n ~ aE)

(5) UW, = = (my = 98)? = 170") y, - 9)?

. . e e
where Tm = Pe ~ Pez is the actual rate of inflation, and Te = Pe Peay

is expected inflation.

Equation (2) (or (5)) gives the wage setters' utility. “he Fed
maximizes social utility, which is given by

(6) US, = = (yy - KV)? - s(n - 0)?

where k > 1, and r is the optimal rate of inflation. y, the rate of
output sought by wage setters, is too small froma social point of
view _/ This assumption can be motivated in several ways. Canzoneri -
(1980) assumes that the labor market is dominated by large unions. The
labor supply curve in Figure 1 includes only union members, and wage
setters' behavior systematically excludes other workers. By contrast,
the social utility function includes all workers. Barro and Gordon
(1983a, b) have a different view. They assert that tax policy and

unemployment compensation distort the labor supply curve in Figure 1 and

lead to a socially inefficient equilibrium. (Under this interpretation,


-5-

II. The Game and the Importance of Private Information

Consider a wage contracting model like those of Gray (1976) and
Fischer (1977). A typical firm's notional supply and demand for labor
are illustrated in Figure 1. The basic assumption of this model is that
the nominal wage must be set in a labor contract prior to the setting of
the money supply, and thus the realization of the price level. Later,
during the contract period, firms learn the price from the market, and
the contract provides that they are allowed to maximize profits; they
hire along the demand curve in Figure 1.2/ The output supply function

can be expressed as
(1) yy, =y + 0(Py - 4)

where Yer Pt and Wy, are the logs of output, the price level and the
contract wage; y is the equilibrium rate of output, corresponding to the
rate of employment n.

Wage setters want to achieve the equilibrium pictured in Figure

1. Their utility can be represented pyS/

2 2

(2) WW, = = (Pp = We)? = = (1/8) = ¥)

Of course, Py is not known when the contract is made; the expected

utility maximizing strategy for wage setters is thus

= pe
(3) mw, = PE

where PF is the wage setters’ prediction of the price level that will


-18-

inflation in (16) may be viewed as a commitment to targeting; in this
case, the policy maker commits himself to targeting the inflation rate.
Rogoff also suggests that the policy maker's commitment should be
enforced through a system of rewards and punishments; Congress, for
example, might be able to achieve an institutional reform by legis’ating
appropriate bureaucratic incentives. If Fed forecasts are indeed private
information, then incentive compatibility is one consideration that will
make the design of such a system of rewards and punishments more
difficult, but the general approach seems worth pursuing. The targeting
discussion in section IV may be viewed in this light.

With this alternative interpretation, Rogoff (1983a) also has a
plausible explanation for the existence of other forms of targeting.
Suppose the policy maker's perversity is a penchant for targeting J, on

*
nm, So that
=. . e . . * 2 . . . *
(18) UP, = - (9, - of - 5 -y Vo - Fl, - 8 8
In this case, the EUP, maximizing policy is
Ply *
(19) gh = +E + f)/(1 +f + thley ty /(F + 4)

and rt can be chosen to maximize EUF,-

The size of + determines the tightness of the targeting
procedure. Note that if 7 > 0, the policy maker will not try to fully
accommodate money demand distrubances. The choice of t must trade-off

the benefits of reducing the inflation bias with the costs of


(8) 9, - m = 0

where Me = ME 7 My is the growth rate of the money supply; crs is taken
to be the Fed's instrument.

The basic structure of the precommitment problem should now be
apparent. The wage setters must act first; this is an assumption of the
contracting model. They must specify ME in labor contracts, not knowing
what ot will be, but fully aware of the Fed's motivations as embodied in
the social utility function. The wage setters know that the Fed will be
tempted to inflate away their real wage and achieve its higher output
target, ky. The wage setters can make this strategy expensive for the
Fed, in terms of its inflation goal, by setting We high in the first
place; in fact, if they set We high enough, the Fed will find it optimal
to give them their preferred real wage and output, y- This is the
problem analyzed by Canzoneri (1980) and Barro and Gordon (1983a), and by
Kydland and Prescott (1977) before them. As they all point out, the non-
cooperative solution to this game is inefficient. If the Fed could
somehow be credibly committed to setting 9% equal to ry then wage
setters would not feel compelled to set a high rate of wage inflation.
Wage setters would still get their desired rate of output, y, but society
would be spared the unnecessary inflation.

The message so far is that the inefficiencies brought about by
the conflict over output goals and the timing of players' actions could
be eliminated by tying the hands of the Fed. This would be the final
message if there were no benefit to society from the Fed's retaining a
degree of flexibility. There is just such a benefit to society if the

Fed can play a stabilization role.


-24-

IV. The Legislative Approach with Private Information

The legislative approach might provide a resolution that better
fits the game at hand. The social contract enacted can have a finite
duration, matching the policy maker's term of office. And a dominant
player with something to gain, namely the government, can take the active
role in initiating the reform.

Congress could legislate incentive compatible policy rules
directly, or it could legislate new rules of the game, rules that would
constrain the strategies of the players; targeting procedures are an
example of the latter. The solutions described in this section could be
legislated either way 22/ However, as was suggested in section JI, it is
not straightforward to design incentive compatible rules for monetary
policy if the Fed's forecast of money demand is private information. It
is easier to motivate the solutions described here in terms of targeting
procedures.

The tightest form of money targeting is a k-percent rule

*
requiring ot to be equal to a each period. The non-cooperative, Strict

Targeting Solution is2l/

* *
(27) gf = gett =o

*
erst =- (14F) 0% -y?
st _ 2 nc

cust = - of < EUW"


International Finance Discussion Papers

Number 249

September 1983
Revised September 1984

MONETARY POLICY GAMES AND THE ROLE OF PRIVATE INFORMATION

by

Matthew B. Canzoneri

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
acknowledgment by a writer that he has had access to unpublished materials)
should be cleared with the author or authors.


-11-

private information, they cannot make this decomposition.

The non-cooperative solution can now be found by working
backwards through the sequence of decisions that the players have to
make. First, maximize EUF, taking ot as a fixed parameter; the first
order condition gives the Fed's decision rule as a function of the wage
setters’ action, gr Then take the expectation of the Fed's decision
rule (conditional on information available to the wage setters when they
set w.) to find the wage setters’ utility maximizing action, a The

resulting Non-cooperative Solution is

enc. ,* ty UF

ne _ _* *
(13) gf = 1 te, ty /f he

Yery- Bey. un = rk + y*/f - ee
2 2 2
EUFES = -(1 + fol - [1 + (1/F)y*, EUWTS = -oo

This is a Nash solution; neither player can expect to do better, given
the strategy of the other. And the solution holds under either
information structure; that is, one obtains the same non-cooperative
solution whether or not the Fed's forecast of money demand is private
information.

Wage setters get exactly what they want in this solution. They
set wage inflation high enough that the Fed is not tempted to try to
achieve its higher output goal, and the Fed fully accommodates

predictable money demand disturbances, stabilizing output to the fullest


-10-

of the contract period, wage setters can calculate & from (9), for they
will then observe both g,and The However, they cannot decompose 5 into

e, and e, The Fed's forecast, eps is private information.

t t’
Before going on, it is also convenient to express the players’
utility functions in terms of their actions, and to give a more formal
description of the game. Using (9), the wage setters' utility function
(5) can be expressed as

- e 2
(11) UW, = - (a, - of - 8)
where a is the wage setters’ prediction of the Fed's action at the time

they must set w In view of (5) and the fact that ot = 1 it should be

t
clear that ot can be taken to be the wage setters’ action. Using ({%) and

(1), the social utility function (6) can be expressed as
- e *\2 *\2
(12) UF, = -(94 - gf - 8-9) - Flay - b> *)

where UF, = US, /9 f= 5/6 and y = (k-1)y/0; ey” is the difference
between the players' output goals. The game proceeds as follows:

(1) wage setters form ot on the basis of their knowledge of the Fed's
utility function and their expectation that 5 (and ee and e,) is zero.

(2) The Fed sets 9 to maximize its expected value of UF, in (12),

t
knowing oe and its own forecast of Se (3) Ex-post, wage setters observe

Spe and if information is symmetric, they can decompose it into the Fed's

forecast, ee» and the forecast error, ee If the Fed's forecast is


-31-

The trick is to do this in a way that leaves the Fed with the maximum
amount. of discretion for stabilization. The modified Barro-Gordon model
resolves the credibility problem in a very efficient manner, as does
Rogoff''s perversely inflation conscious policy maker (in the absence of
productivity distrubances and private information, anyway). In either
case, the Fed is left with the latitude to fully accommodate percieved
money demand disturbances. The targeting resolutions discussed in
section V do not appear to be as successful; a decrease in the inflation
bias is achieved at the expense of discretion, the optimal amount of
discretion depending upon the ability of the Fed to stabilize and
importance of stabilization relative to the elimination of the inflation
bias.

More generally, private information should be viewed as a
constraint, much like a technology constraint. The job of welfare
economists is to develop utility maximizing mechanisms, given just such
constraints. The trade-off between the discretion needed for
stabilization and the rule required for a resolution of the credibility

problem is not well understood and deserves more attention.


-30-

other way. (2) The tightness of the targeting procedure should depend
upon the ability of the Fed to stabilize (modeled above as the Fed's

ability to predict money demand distburances that weren't reflected in
existing wage settlements) and the importance of stabilization relative
to the disutility resulting from the credibility problem (modeled above

as an inflation bias).

V. Conclusion

Kydland and Prescott's (1977) game between wage setters and the
Fed provides a plausible explanation of the scenario outlined in the
introduction, especially if the Fed's forecast of money demand is private
information. Without private information, Taylor's (1983) observations
are persuasive; it is difficult to see why straightforward resolutions of
the precommitment problem would not have already been found. But with
private information, the Fed cannot demonstrate its forebearance by
simply running a non-inflationary policy for a few periods, and it is not
feasible to simply legislate the “optimal” feedback rule. There are
incentive compatible policy rules that could be legislated, but they are
more complicated than what is usualy envisioned; the rules suggested here
can be equivalently encoded as targeting procedures. Private information
also enriches Rogoff's (1983a) model of the perverse policy maker and
Barro and Gordon's (1983a) reputation model; both are too stable, without
private information, to explain periodic inflationary breakdowns.

There is a recurring tension in this paper that provides a new
structure for the old controversy over rules versus discretion. Some
rule or discipline must be placed on the Fed's behavior to resolve the

credibility problem and achieve a better, non-inflationary, game outcome.


equaticn (2) is a guide to wage setters’ behavior, but not a measure of
their welfare.) Either view can be used to rationalize the conflict
betweer (2) and (6) over optimal levels of the real wage, employment and
output .2/ The second term in the social utility function penalizes
deviations from the optimal rate of inflation. Ina less-than fully
indexecl economy, inflation redistributes income from one group to another
and distorts incentives; in addition, there is seigniorage. Many factors
go into the calculation of 3 here, is rather cavalierly taken to be
a regime independent constant.

Some will prefer not to identify the Fed's utility function (6)
with social utility. In this view, (6) merely reflects the biases of Fed
policy makers. It is sometimes asserted for example that macro policy
makers have tended to underestimate how high the "natural rate" of
unemployment has actually been, and that while the “evils" of inflation
are poorly documented, there is clearly a political mandate to sacrifice
some employment and output to keep inflation in check. If this view of
(6) and its conflict with (2) is adopted, the normative content of what
follows must be appropriately reinterpreted.

To explain the Fed's interaction with the wage setters, the
rest of the economic environment must be modeled. This is done in the
most rudimentary way; that is, control theoretic aspects are trivialized
so as to focus upon game theoretic aspects. The model is closed with a

simple quantity equation,
(7) m - PE zy

where m, is the log of the money supply. First differencing, (7) becomes


-21-

* _— _
period triggered if 9p is greater than 7 + by +e where € is some
appropriately chosen constant. The probability of a reversion in period

t+1is 2

(22) P(g, - gf - =) = Pre <a, - gf - el

where P(+) is the c.d.f. of the Fed's forecast error, ¢ If the Fed

t
reneges and sets Sp greater than gr it increases the probability of a
reversion in period t + 1. Similarly, decreasing the constant &
increases the probability of a reversion. Wage setters are not
irrational in going along with this set up if © is small enough that the
Fed has no incentive to renege .26/

The problem now is to find an © such that if the Fed considers
raising o marginally above a> it will conclude that the expected gain
in period t is offset by the expected loss in period t + 1. Suppose 9p 21
was less than r + aT + € (so that the Fed is not currently in a

reversionary period), and Tet U(g,> ©) be the Fed's expected utility over

the next two periods;

+ (1 - P)EUFS

=) = ne
(23) U(g,, ©) = EUF, + PEUF 41

ttl

The Fed has no incentive to renege if U (sf |) < 0, where

9

*
(24) U, (gf, =) = ay - PY(-E)EUFE,) - EUFIS)]

St


-27-

curét > Burne if R< (lL tno

t
where
(4f)oe 6 = (1 + 2F)/(14f)
ie and n= 2/(1 + 3f)

R measures the utility of stabilization relative to the disutility
resulting from non-cooperative behavior (that is, the inflation bias).
Note that R depends upon the ability of the Fed to predict money demand
disturbances (as measured by o), as well as preferences. If R < 9, the
Fed's hands should be tied completely with the k-percent rule. If
¢<R < (1 + n)$, the Fed should only be constrained by average
targeting. If R > (1 + n)$, then the Fed's stabilization efforts should
not be constrained at all, and the full inflation bias should be
tolerated.

Even here, the full range of posibilities has only begun to be
considered. Average targeting is a rather heavy handed way of
achieving flexibility; it is inefficient because it does not link
flexibility directly to the size of the disturbances that the Fed ought
to be allowed to accommodate. An ideal form of targeting would replace

(28) with
*
(31) 9) +95 = 2 +e, +e,
However, legislating (31) would not provide an incentive compatible

resclution to the credibility problem if the Fed's forecast is private

information; the Fed would have an incentive to announce ef and e) larger


money demand is indeed private information. No stable resolution of the
credibility problem can rely on the Fed's own announcement of its
forecast if the Fed has an incentive to misrepresent; the resolution must
be, in Hurwicz's (1972) terminology, incentive compatible.2/

The present paper proceeds as follows:

Section II outlines the Kydland-Prescott game in some detail
and describes the modifications necessary to give the Fed a stabilization
role. The importance of private information is highlighted. Without
private information, Taylor's (1983) observations seem justified;
resolutions of the precommitment problem would be too easy to come by for
this game to be taken seriously as a real world problem.

Section III reviews the work of Rogoff (1983a) and Barro and
Gordon (1983a) in the light of private information. Rogoff's (1983a)
theory of the perverse policy maker is shown to have incentive
compatidility problems since the policy maker's actions do not reveal his
true preferences (or whether he is responding to an incentive structure
laid down by Congress). However, this fact may only serve to make the
theory more compelling as an explanation of the often dramatic comings
and goings of individual policy makers. The rest of section III (and
section IV) focuses exclusively on repeated game solution concepts. Ina
repeated game, honest behavior in the present period can be made to yield
rewards or expanded opportunities in the next. Thus, there is an added
strategic dimension; a player can invest in his reputation. Using
Friedman's (1971) notion of reversionary periods, Barro and Gordon
(1983a) develop a model that resolves the credibility problem in the
absence of private information. Their model falls apart if the Fed's

forecast of money demand is private, because wage setters cannot tell


-14-

precommitment problem would be relatively easy to find. Suppose
information is symmetric, and wage setters observe e at the same time
the Fed does. The Fed might be able to establish credibility in the
ideal policy by simply running it for a number of periods. If wage

setters actually observe e, each period, they can verify that the Fed is

t

implementing the policy o> and they might well be willing to go along.2/
Alternatively, Congress could legislate the ideal policy rule and police
the Fed's adherence to it dO/

If the Fed's forecast of money demand is private information, a
resolution of the precommitment problem is much more difficult to come
by. If e is not observed by the wage setters or Congress, then direct
verification of the Fed's adherence to the ideal policy rule is not
possible. The Fed is often urged to announce a forecast, ef, at the time
it sets 9%: However, this would be of little use in the present context,
for in the language of Hurwicz (1972), the mechanism would not be
incentive compatible; the Fed would have an incentive to misrepresent its
information ./ By letting its announcement, e?, be equal to the true
forecast, e> plus yy( + £), the Fed can make the cheating policy (15)
ch r +e, Straightforward

t t
reputational or legislative approaches to the problem seem more likely to

look like the ideal policy; that is, g

result in the scenario outlined in the introduction. The Fed sets a high
Ses claiming to be accommodating an unexpectedly high money demand; the
private sector charges that it is instead running an inflationary pclicy
to expand output, and a credibility breakdown ensues.

The results of this section may be summarized as follows. The


-16-

Proclivity towards monetary targets. Rogoff (1983a) suggests that this
might be one of society's ways of trying to improve upon the Non-
Cooperative Solution. Society may have found that choosing a policy
maker with preferences other than those expressed in (12) leads to a non-
cooperative solution that is closer to the Ideal Solution.

Suppose the Fed is run by a perverse policy maker with the

utility function
*\2 *\2
(16) UP, == (a, - of - Oy) - oflg, = 8-1)

UP, puts more weight on inflation than UF. if ¢> 1. The Non-Cooperative

Solution with a Perverse Policy Maker is

* * * *
(17) hem te, ty set gh P = a ty sor

EUPR = -(1 + of)oe - C1 + (I/oF)

» EUFP = Euro ~ y "#74?

The more perverse is the policy maker (that is, the bigger is 9), the
closer is solution (17) to the optimal solution (14). The policy maker's
sensitivity to inflation allows the wage setters to get their way with
less inflation.

Here again, adding private information both complicates and
enriches the model. If e, were directly observable at the end of the
period, then wage setters could verify that the policy ab had indeed been
implemented, and that UPy truly represented the policy maker's

Preferences, rather than UF, The policy maker's actions would reveal


-2-

setters, but it is also inflation conscious. The wage setters kriow that
the Fed will be tempted to inflate away some of their real wage, to
achieve a higher level of employment, so they purposely set the wage
high. The Fed weighs its employment and inflation objectives, and only
finds it optimal to inflate the real wage down to the level wanted by the
wage setters. The non-cooperative solution has an inefficient inflation
bias, simply because the Fed has no credible way of precommitting itself
to a non-inflationary policy. Adding a stabilization role for monetary
policy, and private information, the Kydland-Prescott game can be used to
model the scenario outlined above.

There are a variety of ways in which the non-cooperative
solution described above might be improved upon. Congress could
legislate institutional reforms. It could legislate a monetary dolicy
rule directly, or it could constrain the behavior of the players by
changing the rules of the game; wage-price guidelines and legislated
targeting procedures are examples of the later. Thompson (1981) and
Rogoff (1983a) suggest changing the incentive structure for Fed policy
makers, or simply choosing a policy maker with "perverse" preferences.
And Barro and Gordon (1983a) suggest that the players themselves may have
already found a resolution to the problem in a reputation building
mechanism.

An efficient resolution of the credibility problem must leave
the Fed with the latitude or discretion to perform its stabilization
role. Rogoff (1983a) and Barro and Gordon (1983a) discuss the trade-off
between flexibility, which is required for stabilization, and constraint,
which is required for a resolution of the credibility problem, but
neither of their papers incorporates private information. An efficient

resolution is much more difficult to achieve if the Fed's forecast of


FIGURE 1

labor supply

labor demand



-25-

Comparing (27) and (13), it turns out that eure > Eur
. *2 2 *2 . Cc nc .
ify “/f > (V+ F)o,- y “/f is equal to EUFY - EUF, » the social

disutility resulting from the non-cooperative behavior of the players;
(1 + foe is the social utility derived from having the Fed fully
accommodate perceived money demand disturbances.

Legislating a k-percent rule is obviously incentive compatible,
and it is better than doing nothing if the disutility of the inflation
bias due to non-cooperative behavior outweighs the utility of
stabilization. Of course, legislating a k-percent rule and doing
nothing are the two polar extremes. There are better, less rigid, forms
of targeting that limit the Fed's temptation to inflate, but leave it
with some latitude for stabilization.

For example, Congress might just require the Fed to meet its

target on average. In a two period social contract, beginning in period

t = 1, average targeting requires that
i *
(28) 9, + Go = an

This constrains the Fed's behavior intertemporally; in fact, once 9} is
set, Py is known with certainty, by everyone, including the wage setters

writing contracts for period t = 2. The non-cooperative, Average

Targeting Solution is

(29) 8 =n" + pe, ty "af gGtt = a yar


Monetary Policy Games and the Role of Private Information

by

Matthew B. Canzoneri*

I, Introduction

Consider the following scenario: Monetary aggregates surge
ahead of expected or targeted rates; the Fed claims that it is
accommodating a perceived increase in money demand in order to stabilize
the price level; the private sector (or the Administration, or Congress)
counters that the Fed is running an inflationary policy to expand
employment; a period of credibility building ensues, which may focus upon
the personalities of policy makers, targeting procedures, or even
proposals for legislative reform. A key element in this scenario is that
the private sector cannot verify the Fed's claim; the Fed's forecast of
money demand is private information.

A plausible explanation for such a scenario is that the Fed is
involved in a non-cooperative game with certain agents in the private
sector, or with some other branch of government. Why else for example
would the Fed seek to constrain its own behavior, with say a targeting
procedure, unless it thought that this would bring a resolution to a
credibility problem resulting from a non-cooperative game.2/ And why
does the Fed have a "credibility" problem, as opposed to an "information"
problem; that is, why are its announcements not accepted at face value?

A game suggested by Kydland and Prescott (1977) has been
receiving a lot of attention recent ly =! In this game, wage setters have
to specify the nominal wage in a labor contract before the Fed sets the

money supply. The Fed wants a higher level of employment than the wage


-F2-

12/ Footnote 11 applies here as well.
13/ Actually, Barro and Gordon (1983a) postulate a different utility
function for the Fed, one that is linear (rather than quadratic) in the
output term, and consequently, for them the Ideal Solution is not an
achievable outcome. However, they can improve upon the Non-Cooperative
Solution. See also the next footnote.
14/ For simplicity, the Fed is assumed not to discount future utility.
If future utility is heavily discounted, two or more reversionary periods
might be required; if future utility is very heavily discounted, the
approach may not work at all. See Friedman (1971).

~ : * — * _
15/ ~o see this, note that gg > 1 + 6 +e =m ter te, te
= g& + e + © is equivalent to ey < gp - gf - ©.

16/ “he refurmulation of the model can also be explained in terms of
incentive compatibility. Suppose the Fed announces

ef and sets gp = re e@. Since gf = r+ eb Gt = OE if and only if
e? = @p3 No reneging, as defined above, is equivalent to incentive
compatibility. Suppose the wage setters revert to inflationary wage
settings if e? > b+ ‘. It is easy to show that e? > 64 + © if and
only if gy > r+ oe + , and that (22) can be expressed as

P(e? -e - ©) = Prlep < ef -e - €]. If the Fed sets ef > ez, it

increases the probability of a reversion in period t + 1. The arguement
exactly parallels that in the text.

17/ It is calculated by substituting gf = gf°° = 1 into UF, in (12),
differentiating EUF, wer.t. gp, and evaluating the result at g, = gf.
18/ gf = rt ey > r+ bp tee +e, + e¢ + € is equivalent to
ee <i e

19/ This is the basic insight of Green and Porter (1983).

20/ They could also be achieved by institutional reform; see footnote 4.

21/ All of the non-cooperative solutions in this section are calculated
by working backwards, as in section II.


-28-

than ey and 2 8 Legislating the ideal form of targeting would do no
more good than legislating the ideal rule itself.
There is however an incentive compatible form of targeting that

is more flexible than average targeting. Flexible targeting requires
that

*
(32) 9) +95 = 2m + cb) c > 0
where 8) is of course observed by the wage setters at the end of the

first period. With c set equal to one, the non-cooperative Flexible

Targeting Solution (c = 1) is

(33) oft = +e, + y" /2f aor" a+ y"/2f
* * of f
ap =n te, -y sar ghrft = oft
ft _ 1 2.2, 1,2 *2
BUF, = 3 (1 + F)(og +o, ) - fo, - C1 + (1/4)

where eur and ent are expected utlities averaged over the two

periods. The policy rules in (33) are incentive compatible and could be
legislated directly; the Fed would have no incentive to misrepresent its
forecast 24/

With c set equal to one, the Fed would find it optimal to


-12-

extent possible about the wage setters' preferred rate. There is, 9f

course, an inflation bias equal to y*/f. Society is left (on average)
with a level of employment that is too low and a rate of inflation that
is too high; as Barro and Gordon (1983b) and Rogoff (1983a) point out,
this is a model of stag-flation.

Now, it may be thought that the non-cooperative solution just
described is a rather silly, destructive situation for the Fed and the
wage setters to allow themselves to fall into. The players have
conflicting output goals, but the inflation bias benefits no one. An
Ideal Solution would eliminate the inflation bias without changing the

(average) rate of output;

*
(14) ot zm +e, cane =r
— *
Ye ry > Se, T] FT > ey
cL 2 *2 nc (a nc
EUFE = - (1+ f)oc-y > EUFES, EWE = EUW?

Here, the Fed fully accommodates the predictable part of the money
*
demand, and it does not try to inflate away ey . The solution is

efficient in the sense that no other set of strategies, g,and oe can

St
make one player better off without making the other worse off; thus the
Ideal Solution can be thought of as a cooperative solution. (That is why
the superscript "c" is used to identify it.)

As Kydland and Prescott (1977), Canzoneri (1980) and Barro and

Gordon (1983a) have all pointed out, the problem with this solution is


-15-

Kydland-Prescott game of precommitment has a simple resolution if there
is no benefit to the Fed's retaining discretionary powers; the Fed's
hands should be tied with a k-percent rule. If the Fed can play a useful
stabilization role, then an efficient resolution of the precommitment
problem requires that the Fed should retain some discretionary power.
However, straightforward resolutions still exist if information is
symmetric; the Fed might be able to build a reputation for itself by
simply running the ideal policy for a few periods, or Congress could
legislate the ideal policy rule and police the Fed's adherence to it. If
the Fed's forecast of money demand is private information, then the Fed's
adherence to the ideal policy rule cannot be verified directly, and
efficient resolutions of the precommitment problem are much more

difficult to find.

III. Adding Private Information to Rogoff (1983a) and
Barro and Gordon (1983a)

Rogoff (1983a) and Barro and Gordon (1983a) discuss more
sophisticated approaches to the precommitment problem. However, neither
allows for private information. If the Fed's forecast of money demand is
private information, both approaches yield more interesting
interpretations of the credibility problem, but Rogoff's does not offer a
resolution and Barro and Gordon's does not explain the scenario outlined

in the introduction.

The Perverse Policy Maker:
It is sometimes asserted that monetary policy makers have a

conservative or anti-inflation bias, or that they have an unnatural