-17-

Table 2. Real Interest Trajectories (percent points)

Predetermined Dynamic Time Inconsistent
Path for c Programming Optimal Policy
Time q) (2) (3)
0 9.3 8.6 8.3 (0.83)
1 3.6 3.5 3.7 (0.47)
2 1.4 1.4 1.5 (1.11)
3 0.5 0.6 0.6 (1.31)
4 0.2 0.2 0.2 (1.65)

Note (1) Real interest rate trajectories assuming z(o) - 10 percent

points.

(2) The ratio of r to z is constant for (1) and (2), at 0.93 and
Values for r/z for (3) vary over time and
are shown explicitly in parentheses.

0.86 respectively.



-ll-

To obtain the dynamic Programming solution, Cohen and Michel
show that policy makers must optimize subject to linear constraint
which in this case is that the real exchange rate be a linear function
of the state variable (i.e. c = 6z). If one one finds the correct
parameters (the right value for 6), so that the real exchange rate
chosen is actually the integral of future real interest differentials,
one will satisfy the Principle of Optimality: policy makers will be
optimizing subject to a constraint which captures market expectations
of continuous future optimization.

Formal details are given in the Appendix. We remark here that
the solution for the infinite horizon case can be obtained by an
iterative scheme involving the elements of the stable eigen vector.
The stable eigen vector associated with the solution of the last
section may be used to provide a first estimate of the value of 6:
using that as a constraint one can solve to obtain a new stable eigen
vector, and continue until convergence is achieved.

The numerical results for such time consistent optimal policy
are shown in the third column of table 3, Compared to the second
column, the initial recession is yet further reduced and the length of
the recession correspondingly prolonged (the stable root falls to 0.905
for the dynamic programming solution). To see why this is so, it is
worth considering how the Cohen/Michel constraint affects the "shadow
cost of cumulated output," P,, since the path of output is determined

by this shadow cost (y = - P,)-


-21-

To avoid excessive dependence on particular initial conditions,
one may alternatively consider the asymptotic properties of the system
in choosing a simple rule (to minimise some function of the asymptotic
variance-covariance matrix for example). This is the approach adopted
by John Taylor (1985) who observes that: "By focussing on the
stochastic equilibrium we are implicitly assuming an infinite time
horizon for policy choice with no discounting. This seems appropriate
for macroeconomic policy. We are also assuming that time consistency
will not be a problem: once a policy rule is chosen we assume that it
will remain in force with no attempt by the policy makers to exploit
the past committments of economic agents" (p. 69).

As Kydland and Prescott forewarned, the search for simple and
sustainable feedback rules in an environment where expectations are
formed rationally appears to lead one away from explicit use of optimal
control and dynamic programming techniques for short run stabilisation.
Whether the alternative, longer-run focus of policy meets the
criticisms levelled against floating (for the incentives it offers to
use competitiveness for short-run stabilisation policy, cf. Williamson

(1985)) is an issue worth further discussion.


The focus of attention here however, is, on alternative
stabilisation policies which treat expectations in the foreign
exchange market in two sharply contrasting ways, either by explicitly
accomodating the effect that continuous policy reoptimisation will have
on the market; or by ignoring the impact of interest rate policy on the
market altogether. The former is achieved by "dynamic programming"
using an elegant technique recently developed by Cohen and Michel
(1984). The latter approach, suggested by William Buiter (1983), turns
out here to produce a more activist policy. Both policies take the
form of constant linear feedback rules for interest rate setting. So
the question of how they compare with the optimal linear feedback rule
is also addressed. (As a preliminary the policy problem in a closed
economy briefly considered).

Oudiz and Sachs (1985) have already carried out such an
investigation using a discrete-time model of inflation in a small open
economy and a clear treatment of some of these issues is to be found in
Turner (1984). In what follows, therefore, the intention is not to
Present novel results, but, by using a continuous time framework and
applying the technique of Cohen and Michel, to make known results more
analytically accessible and transparent.

In conclusion the case for choosing a simple rule with reference

to more long run properties of the model is briefly discussed.


- 18 -

One may be prompted to ask whether, if real interest policy were

restricted to the class of linear feedback rules, one could improve on

those already calculated. One can, of course, never do worse, and in
general, as Cohen and Michel show, one can do better. Continued
adherence to such an "optimal rule” would, doubtless, require some form
of precommitment, though it seems much more likely that governments
could credibly adopt such a simple feedback rule than precommit
themselves to the time-varying trajectory required for the time
inconsistent optimal policy earlier described.

For the particular coefficients and welfare weights used here,
however, it so happens that the feedback coefficient of 0.926 obtained
when the real exchange rate is taken as predetermined is approximately
the optimal linear feedback policy. Table 3 which shows the integrated
costs generated by various feedback rules, running from 0.80 to 1.0 (so
real interest rates are set either just below, or equal to, inherited
core inflation, measured by Ez where — = $= 1). While, in this
particular instance, taking account of future optimizing policy by
dynamic programming leads to under-responding, it turns out that
ignoring the exchange rate channel generates the optimal feedback
policy! That this happy coincidence is not in general true, even for
the given economic structure, is shown by doubling the welfare weight
on inflation (8 = 2). While the dynamic programming response is once
again smaller than when one optimizes treating the exchange rate as

given, the optimal feedback coefficient lies between the two, see

Table.


-A3-

III Time Inconsistent Optimal Policy

2
fry +p, Dz + Pp, De the First Order

From the Hamiltonian H =

Conditions are
oH/ ar = -Yy -YP, + P, =o
-Dp, = oH/ oz = BEOT
-Dp, = QH/dc = BEon + Sy + P,)

which, together with the equations of the model imply

[.. | ; 7 8 He

| de | 0 ys yi - c |

vp, |= | -ek -éok 9 0 P,

Dp, -gok -o*k 0 -6y) Py where k = ge?
Lok It J

The unique solution is obtained by restricting the dynamic
path to the stable manifold subject to the historically given value of
z(o) and the optimality conditions that p,(0) = 0, i.e. the shadow

cost of the real exchange rate be minimized at time t = 0.


- 20 -

Conclusion

When high real interest rates are used to combat inflation, one
can expect to find the exchange rate overvalued when inflation is high.
Paradoxically, however, the time-consistent interest rate policy which
recognises this, promptly chooses to underrespond in its choice of
interest rates. (The perception that future policy makers will use
high interest rates reduces the incentive to implement them now, since
the rise in the exchange rate reduces the inflation problem in the
first place!)

In the economic setting considered in this paper it happens that
turning a blind eye to the exchange rate effects generates a more
appropriate, activist response. Indeed for the initial set of
parameters, such a Nelsonian policy resulted in the optimal linear
feedback response! However, it is not in general true that the optimal
linear feedback rule can be found by ignoring the effects of policy on
forward-looking variables, so it will have to be determined by explicit
minimisation of welfare costs, see Appendix.

Since it was not selected by dynamic programming, the viability
of such an optimal rule depends on precommittment. As Sachs and Oudiz
observe, it is easier to see policy makers adopting a simple rule than
adopting a complicated trajectory for real interest rates. On the
other hand, the independence of the rule and the initial conditions
(characteristic of the scalar case) fails when there is more than one

state variable (as Cohen and Michel have pointed out).


- 15 -

cumulated output

z

A

abe Rfey

aed
competitiveness

(a) Treating c as predetermined

(b) Time inconsistent optimal

(c) Dynamic programming

Figure 2. Varieties of stabilisation policy


Small Open Economy

How is the analysis affected by the floating exchange rate?
First of all, at the positive level, the movements in the (real)

exchange rate alter the rate of inflation, so

i= oy ++ 0 De

where c is the real exchange rate which we take to be a forward-looking
variable, see (3) below. Thus, with the same exponential process for TM,

it turns out that, in the open economy,

(2)' w= Egz + Eoc.

so core inflation is a weighted sum of the backward-looking variable,
z, and the forward looking variable, c.

The idea that inflation is a centered moving-average of past and
(expected) future movements in wages is of course a familiar one, cf.
for example the overlapping-contracts model of John Taylor (1979). It
is evident, however, that openness of the economy induces a forward-
locking element into core inflation even when this was not present from
the contractual structure (we began with a simple Phillips curve).

To proceed with the problem of policy, one needs to know more
precisely how the real exchange is determined. As is common in
theoretical analysis, we assume that expected movements of the real

exchange rate match real interest differentials, so

(3) De =r


References

Austin, G.P. and W.H. Buiter (1982). “Saddlepoint -- A Programme for
Solving Continuous Time Linear Rational Expectation Models.” LSE

Econometrics Programme DP No. 37, November.

Buiter, Willem (1983). "Optimal and Time-Constant Policies in
Continuous Time Rational Expectations Models.” LSE Econometrics
Programme DP No. 39, June.

Buiter, Willem and Marcus Miller (1982). "Real Exchange Rate
Overshooting and the Ouput Cost of Bringing Down Inflation.” European
Economic Review, 18, 85-123.

Cohen, Daniel and Phillippe Michel (1984). "Towards a Theory of Optimal
Commitment I: An Analysis of Time Consistent Equilibria.” CEPREMAP DP
No 8412, May.

Dornbusch, Rudiger (1976) “Expectations and Exchange Rate Dynamics.”
Journal of Political Economy, 84, December, 261-83.

Driffill, John (1982) “Optimal Money and Exchange Rate Policies.” Greek
Economic Review, 4, December, 261-83.

Kydland, Finn and Edward Prescott (1977) “Rules Rather Than Discretion:
The inconsistency of Optimal Plans.” Journal of Political Economy, 3.

Miller, Marcus and Mark Salmon (1985a) “Dynamic Games and the Time
Inconsistency of Optimal Policy in Open Economies." Economic Journal,
Supplement.

(1985b) “Policy Co-ordination and Dynamic
Games" in W.H. Buiter and R.C. Marston (eds.) International Economic
Policy Coordination, N.Y.: Cambridge University Press.

Oudiz, Gilles and Jeffrey Sachs (1985). “International Policy .
Co-ordination in Dynamic Macroeconomic Models” in W.H. Buiter and R.C.
Marston (op. cit.).

Taylor, J. (1979). “Staggered Wage Setting in a Macro Model." American
Economic Review, 69, 108-113.

(1985). “International Co-ordination in the Design of
Macroeconomic Policy Rules." European Economic Review, 28, 53-81.

Williamson, John (1985). “On the System in Bretton Woods.” American
Economic Review, 75, 74-79.

Turner, Paul (1984). “Real Wage Resistance and Optimal Disinflation in
Open Economies." University of Southampton (mimeo).


where D = d/dt is the differential operator.

Then, by defining z as the integral of y, so z measures cumulated

excess demand and

(1) Dz = y,
core inflation can be written as a multiple of z, namely

(2) m= Ez.

If the policy-maker penalizes the weighted but undiscounted
squared deviations of core inflation? and output, then the infinite

horizon policy problem can be expressed as that of minimizing V(o) with

respect to y subject to equations (1) and (2) above, given z(o),

where V(o) =} f Bm(e) + y(a) ds.
°

Details of the ‘derivation are given in the Appendix. What

emerges is that the optimal policy is to set
ys-p,

where p, is the "shadow cost" of z, cumulated excess demand. (This PL
measures the welfare cost imposed by a unit increase in z along an
optimal trajectory.) This shadow cost is found to evolve according to

Dp, = ~ BEon.


Closed Economy

The analysis of policy in an open economy with a floating

exchange rate influenced by expected future policy may be assisted by

focussing first

absence of such

on the nature of “optimal” stabilization policy in the

forward-looking behaviour. That is the purpose of this

section, cf. Turner (1984).

Consider

for example a closed economy where the inflationary

process is described by a Phillips curve "augmented" by a term

representing backward-looking "core" inflation.? We write this as

i=dy

where

y is

Let core

inflation, so

Drs

+7

the rate of price inflation
the log of output (measured from its "natural" rate)

core inflation.

inflation be an exponential moving average of past

E(i-m)


- 16 -

Summary

For a linear augmented Phillips curve, the underlying rate of
inflation depends on the simple integral of output and not on the
detailed trajectory (thus, for the closed economy, 1 = Ez where z
denotes the integral). Use of a quadratic penalty function allows one
to discriminate between trajectories on welfare grounds, however, and
"optimal" policies may then be derived. The openness of an economy
helps to reduce the welfare costs of an anti-inflationary policy led by
high real interest rates, as the early appreciation that this induces -
even though it is reversed later - shifts inflation into the future.

Relative to the other policies examined, the optimal (but time
inconsistent) trajectory of real interest rates for the open economy is
alow start" policy. This can be seen in Table 2 where the time
inconsistent trajectory exhibits the lowest real interest rate at time
zero but the highest levels over the next two years. Note that
although real interest rates in column 3 are falling throughout, the
ratio of r to z, shown in parentheses, does not. (Precommitment is of
course required to add credibility to this threat -- that "tomorrow we
will act tougher on inflation than we are acting today.") For the
other two policies, however, the ratio of real interest rate to the
integral of real output remains constant over time, so both imply fixed
linear feedback rules, with dynamic programming having the smaller
feedback coefficient as each government passes on some of the pain of

fighting inflation to it successors.


The nature of optimal policy can be shown by the saddlepoint
phase diagram, figure 1. There the horizontal axis measures both
output (increasing to the right) and the shadow cost PL (increasing to
the left), while the vertical axis measurés both the integral of
output, and the rate of core inflation, a positive multiple of z.

The dynamics of z are obvious as it is the integral of y. The

adjustment of y over time follows from the first order conditions that

y= -P, and that Dp, = - BEom; hence Dy = BEom = Bt g72,
so y rises for positive z and falls when z is negative. For the
infinite horizon problem being treated here, the unique optimal
trajectory is that lying along the "stable manifold" leading from A to
the origin. For the case when f= € = $= 1 it turns out that the
initial values are y(o) = - p, (o) = 2(o) and the stable root is unity.
Given the linear quadratic structure, the minimal integrated cost
Vis given by 0.5 Piz = 0.527.

Thus for an economy with a positive level of core inflation,
1 > 0, the optimal policy is to initiate a recession followed by a
gradual recovery towards the equilibrium (where inflation is zero and

output at its non-inflationary equilibrium, y = 0).


Appendix

CLOSED ECONOMY

©
To minimize $ f pr2(s) + y2(s) ds given z(o) and
wor.tey °

subject to (1) Dz =y and (2) m= &Gz, form the Hamiltonian

pre + y-
2

He +P, Dz with First Order Conditions for as minimum

follows:
aH/ay = y +p, = 0
“Dp, = oH/ dz = BEOT.

The evolution of the state and costate variables (z, P,) is thus

| [fe [[. |

oe, | ~ | -#8°# 0 | >, |
Jt

which, when restricted to the stable manifold, implies
At
x(t) =e” 2(0) A= -EOVB

y= -P,* -yz where p= -A.


International Finance Discussion Papers

Number 262

August 1985

MONETARY STABILIZATION POLICY IN AN OPEN ECONOMY

by

Marcus H. Miller

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.


paths consistent with continous reoptimization, The techniques of
dynamic programming can be used to find such a time consistent
solution.

We begin with a third approach, proposed by William Buiter
(1983) as a means of avoiding time inconsistent policy. The idea is
that the policy-maker chooses the trajectory of real interest rates
taking the path of the real exchange rate as exogenous. As the real
exchange rate is a forward-looking variable, it will consequently
anticipate a. future in which policy is constrained by this
“self-denying ordinance." The outcome of such a policy is easy to
calculate -and we find that it dominates the dynamic programming
solution! (Later we compare this with the optimal linear feedback
rule).

(1) Policy when the exchange rate is treated as predetermined

Formally the policy-maker chooses r so as to minimize the
integral of costs, V, subject to the equations of the model, (1), (2)',
(3), (4), treating the path of ¢ as exogenous. The formal derivation
of the solution which ensures that the assumed path currently discounts
future interest differentials is given in the Appendix. The numerical®
outcomes (for parameter values $= — = 8= 1, ¢ = 0.1) are given in
table 1, column 2. Here we consider the changes from those obtained
for the closed economy, shown in the first column of the table.

It is worth remarking ex ante that the integral of (net) lost
output to reduce inflation to zero should Rot be affected by the
openness of the economy, provided that the real exchange rate begins

and ends at the same level, a point discussed in some detail in Buiter

and Miller (1982). We observe, however, that for the open economy the


where r is the real interest rate (measured as a deviation from the
constant world level) and competitiveness, c, is measured as a
deviation from equilibriun.

The model may be completed by specifying how output is
determined. Specifically let output be determined by the demand side
where the latter depends on the real interest rate and on the level of

competitiveness (which is a forward-looking integral of future expected

real interest rates) so

(4) y = -yr + 6c.

The policy problem can now be formally expressed as that of
minimizing V with respect to r given equations (1), (2)', (3) and (4)
above. (Note that r may be eliminated by substitution, and the problem
reduced to one of choosing an output path so as to minimize V.)

In choosing the path for real interest rates, however, the
policy maker has to take a view as to what future path of real interest
rates the market expects. One possibility, perhaps the most obvious,
is that the market should presently expect that path of real interest
rates which generates the minimum cost for the policy maker at the
present time. This idea begins to lose its attractiveness when one
observes that the policy which minimizes V at some later date will not
in fact be a continuation of the policy chosen today - such policy
design is said to be time inconsistent.

A second approach is to require that the Principle of Optimality

be satisfied, so that what the market expects must be interest rate


Z integral of y

TW core inflation

Figure 1.

Optimum Policy in the Closed Economy

y, output


- 19 -

Table 3. Welfare Costs of Various Linear Feedback Rules

Low weight on inflation (8 =1) High weight on inflation (8 = 2)
Feedback Integrated Feedback Integrated
Coefficient Cost Coefficient Cost
0.80 0.45423 1.404 0.628351
0.862 0.45250 1.45 0.627708
0.90 0.46197 1.49 0.627499
0.926 »¢ 0.45187 1.50° 0.627486
0.95 0.45194 1.51° 0.627489
1.00 0.45250 1.52 0.627506

Notes a Dynamic programming solution.
b Optimal feedback rule (approximate).
c Result of treating real exchange rate as predeterminent.
Feedback rule relates the real rate of interest r to z, integrated
output.
Cost shown as multiple of 22,


- AA -

The optimal linear feedback rule

The problem is to minimise V(o) with respect to p subject to
(1), (2)", (3), (4) and (5) r = pz, given z(0). Under such a feedback

rule, it follows that

Dz -yp 6 Zz

De p ° c

- + sz =
So Aus ps, 5 uy Ye and 4,8 Py

, being the stable root, an the unstable root and c = 6z on the
1,2 2
stable manifold. Consequently V(o) = tz fBn°(s) + y“(s) ds
°

2
= zie {8 (&6 + E00)? + (60 - yo)?}.
8

Hence the First Order Condition to be satisfied, namely

dv(o) av av 36) av aa,

=~=0

dp "%p * 38 Bp * Bh, Bp

00 ar :
(where Tp and om follow from the relations given above) will be

independent of z(o).


"I do believe firmly that by monetary means exercised promptly and
courageously we can greatly mitigate the worst evils of inflation and
deflation ... one thing is certain: we will not obtain stability

unless we work for it."

Marriner S. Eccles, Federal Reserve Board Chairman 1934-48.

Introduction

The purpose of this paper is to study the design of monetary
stabilization policy in an open economy with a floating exchange rate,
ignoring feedbacks from the rest of the world.! The impact of monetary
policy on both output and inflation is assumed to operate via real
interest rates and the real exchange rate: so the problem posed is
that of choosing the optimal setting of real interest rates so as to
minimise quadratic costs attached to fluctuations in output and
inflation.

The model used here, cf. Buiter and Miller (1982), features a
substantial autoregressive element in the inflationary process, as well
as the rational expectations, asset-price approach to the determination
of the exchange rate of Dornbusch (1976). Consequently the rate of
inflation is influenced both by its own past history (via the
autoregressive structure) and by current and expected future monetary
policy (via the exchange rate).

The design of optimal open-loop monetary policy in such an
economy has been studied by John Driffill (1982). He notes the
systematic temptation upon the policy maker to depart from the plan
first announced if ever policy can subsequently be reset; so the
optimal trajectory is not "time consistent," to use the terminology
suggested by Kydland and Prescott (1977) in their celebrated critique

of optimal control techniques when agents form rational expectations.


-12-

The constraint characterizing this dynamic programming solution
is c = - 0.95 z; international competitiveness is negatively correlated
with cumulated output when high real interest rates are used to check
inflation. The shadow cost of a cumulated boom is consequently reduced
since the inflationary consequences are seen to be damped
by the counter-cyclical behaviour of competitiveness. Policy makers
are thus less inclined to sacrifice output to check inherited
inflation; so the size of an initial recession is lower and core
inflation higher than under previous policy (both are 0.9050 z(o)).

While this behaviour seems perfectly reasonable, it is
interesting to see that the integral of costs rises marginally relative
to what was obtained under the previous self-denying policy of treating

the exchange rate as given.

(III) Time inconsistent optimal policy

What trajectory of real interest rates would the policy maker
choose at time zero, knowing that the announcement of these rates will
influence the foreign exchange rate? Note that the so-called time
inconsistent trajectory which (if credible) minimizes V(o) will require
some form of precommitment since subsequent reoptimization would not
lead to a continuation of the announced policy. (In the absence of
precommitment - or other factors like that of acquiring a reputation -
the situation is likely to revert to the time consistent outcome
obtained from dynamic programming.) How will the policy maker choose
to use his or her influence over the foreign exchange market to alter

the shape of an inflationary recession?


-4-

Table 1. Varieties of Stabilisation Policy
Three Policies for the Open Economy
Closed Dynamic Time
Economy Exogenous c Programming | Inconsistent
Symbol (1d) (2) (3) (4)
Stable
eigen value a -1 -0.950 -0.905 -0.954 + 0.3i
Stable 8 nea. -0.9744 -0.95 Conjugate
eigen vector!) y 1 0.9500 0.905 Complex
Starting y(o) -1 -0.9500 -0.9050 -0.9026
values for mo) 1 -0.9026 0.9050 0.9026
endogenous e(o) nea. -0.9744 -0.9500 -0.9737
variables?) r(o) | ma. 0.9257 0.8599 0.83151
Integrated v0o) | 0.5 22(o)| 0.4518 22(o)] 0.4525 22(0)| 0.4513 22(0)
costs*>
Footnotes

(1) 6 gives the ratio of c to z on the stable manifold.
y gives the ratio of p, to z on the stable manifold.
(2) reported as multiple of z(o0).
B20) z(o)/2 for columns (1), (3) and (4) and as

62 + y*)z2(0)/4 A for column (2).

(3) measured as
(1 + 200+ 6

Parameters values used $= 6 = B= 1, y= 6 = 0.05, o = 0.1


Abstract

This paper investigates optimal stabilization policy in a small
open economy using a continuous time model in which inflation depends on
future monetary policy as well as past inflation. The impact of monetary
policy is assumed to operate via real interest rates and the real
exchange rate and the setting of real interest rates is chosen so as to
minimise quadratic costs of fluctuations in output and inflation, subject
to varying expectations in the foreign exchange market. Analytical
expressions and simulation results are presented for "time inconsistent"
optimal policy, the “dynamic programming" solution, for policy which
ignores the exchange rate effects when setting real interest rates, and

for the “optimal linear feedback" rule.


OPEN ECONOMY

To minimize 4 f Br(s)” + y%(s) de = Wo)

wor.t.r °

subject to (1) Dz

(3) De

y

r

(2)' 1 = Egz + Eoc

(4) y = -yr + 6c

given z(o), we consider three approaches.

I (c_ assumed predetermined) and II (assuming c = 6z)

are

From the Hamiltonian H = fi +P, Dz the First Order Conditions are

dH/dr = -yy - yp, = 0

-Dp, = dH/dz = BEC > + Oo)T + Oly + P,)

which, together with the equations (1) to (4) imply

fe] foe alle |

6 |

ee tee

c where h = BE? ( $+00) .

For Case I, the solution is obtained by setting ® = o before computing

the stable eigen value and vector. For Case II, however, 6 must be

determined so as to equal the second element of the stable eigen

vector. This may be achieved by an iterative procedure, as discussed

in the text.


MONETARY STABILIZATION POLICY IN AN OPEN ECONOMY

Marcus H. Miller*
University of Warwick
Coventry CV47AL, England

August 1985

* Thanks are due to Willem Buiter, Matt Canzoneri, Hali Edison and
Robert Johnson for constructive suggestions and programming support.
Responsibility for errors remains with the author. This paper was
Prepared while the author was a visiting scholar in the Division of
International Finance. This paper represents the views of the author
and should not be interpreted as representing the view of the Board of
Governors of the Federal Reserve System or other members of its staff.



Footnotes

These issues are discussed in a two-country, dynamic-game setting by
Miller and Salmon (1985a,b) and Oudiz and Sachs (1985).

Note, however, that the problem of designing policy in the face of
rational expectations will not be avoided, even for a closed economy,
where the wage level depends on averaging forward-looking contracts,
as in Taylor (1979).

For analytical simplicity and not i appears in the welfare
function. The analysis may, however, be repeated using i instead if
this is thought desirable.

All the numerical results reported here were obtained using
“Saddlepoint”, see Austin and Buiter (1982), to whom we are grateful
for making this programme available.


-10-

initial size of the recession falls (from z(o) to 0.95 z(o)) and the
rate of recovery is slower (the stable root likewise falls from unity
to 0.95 in absolute value).

In a setting such as this it is pretty obvious that the
reduction of inflation requires a recession, but why should the policy
maker in the open economy opt for a shallow initial cutback in output
followed by a slower recovery? In order to induce the recession real
interest rates have to be increased, but this necessarily means that
the initial values of both competitiveness, c, and core inflation, 7
will be low (as the expected path of real rates lifts the external
value of the currency): later, however, the recovery of
competitiveness will put back into core inflation what this initial
appreciation takes out. Without actually planning it this way,
inflation in the open economy will be seen as less of a problem early
on than more of a problem later on - which provides the rationale for
having less of a recession now and more later.

It is evident that this "reshaping" of inflation and recession
is attractive to the policy maker, since the minimized integral of

costs, v, falls from 0.5 270) to 0.45 2(0), see Table 1.

(II) The dynamic programming solution

So far we have considered the outcome where policy-making is
optimized subject to an "exogenous" path for the real exchange rate - a
path which is solved to forecast correctly the continuation of such
constrained policy optimization. Recently Cohen and Michel (1984) have
shown that a similar, simple characterization of policy which satisfies
the Principle of Optimality is available for a linear structure with

quadratic costs.


-13-

Formally the optimal trajectory can be found by setting up a
shadow price for the real exchange rate, Po, and assuming that
p,(o) = 0, see Appendix. The ability to precommit oneself to future
policy actions permits a more interesting convergence to equilibrium,
an approach described by two eigen values. In this particular case the
eigen values turn out to be complex conjugate so the convergence is
cyclical, see table. Under this policy, the initial values for core
inflation and recession are both reduced compared to the time
consistent outcome, falling from 0.9050z(0) in column 3 to 0.9026z(o)
in column 4.

How is this possible - to increase output while reducing
inflation? The trick is to announce a reduced real interest rate
spread for the early part of the period (cf. the initial values for
r(o) shown in the table) but a wider spread for later. This shifts
inflation and recession away from early period when the squared
deviations are heavily penalized towards a later period when they are
less "costly". The integral of costs consequently falls to the lowest
value shown in the table.

These fluctuations over time in the real interest differential
show up in the path followed by the real exchange rate, see figure 2,
where the time inconsistent optimal trajectory is labelled (b). ‘The
other two trajectories have a constant ratio of competitiveness to
cumulated output. This ratio (labelled 6) is, of course, explicitly
taken into account in choosing the dynamic programming trajectory: but
note that the outcome (a), when competitiveness is taken as
predetermined, resembles the time inconsistent optimal trajectory more

closely.