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Optimal target zones: How an exchange rate mechanism can improve upon discretion
e*” & Control arnlcs
Journal
ELSEVIER
of Economic Dynamics and Control 20 (1996) 1641-1660
Optimal target zones: How an exchange rate mechanism can improve upon discretion Marcus Miller *A~, Lei Zhang ” “Deprrrtment
qf Economics, lJnioer.sir)~ of Wantick, C’ormtr~ bCEPR. Londorl WIX 1 LB. UK
(Received August 1994; final version received October
C‘V4 7AL,
UK
1995)
Abstract Using Krugman’s (1991) target zone model, we find an explicit, subgame-perfect solution for a central bank wishing to stabilise the exchange rate given proportional costs of intervention. We demonstrate, however, that precommitment to narrower bands would yield a welfare gain - which provides a theoretical rationale for an Exchange Rate Mechanism (ERM). Numerical simulations suggest that the optimal currency band with precommitment via an ERM is only half as wide as that under discretion, and that the welfare benefits are sufficient to sustain the system. Kq words: Target zones; Regulated rate mechanism JEL
cluss~fi~ution:
E42;
F31;
F33;
Brownian
motion; Time-consistent
policy; Exchange
F42
1. Introduction If agents are rational, a policy of intervention at the edges of a currency band should help to stabilise the rate inside the band, even before the intervention takes place. This effect - the ‘honeymoon’ effect - was demonstrated by Krugman (1991) in a simple monetary model where the velocity of money was assumed to follow a Brownian motion process.
* Corresponding
author.
While remaining responsible for any errors, we would like to thank Avinash Dixit, Laura Papi, Jonathan Thomas, Steve Tumovsky, and an anonymous referee for helpful discussions and comments. 0165-1889/96/$15.00 0 1996 Elsevier Science B.V. All rights reserved SSlII 0165-1889(95)00908-E
1642 M. Miller, L. ZhanglJournal
of‘ Economic Dynamics and Control 20 (1996) 1641-1660
Why should the authorities want to intervene? And, if so, how wide should the band be? These are the questions we tackle in this paper, using the same canonical target zone model. As the price of foreign currency is the principal determinant of domestic prices in this model, it is natural to assume that the monetary authorities will want to stabilise the exchange rate. But, even when there is no shortage of reserves, allowing for some fluctuations within a band can be optimal because of positive costs of intervening (see Avesani, 1990). It is well known from studies on the regulation of Brownian motion processes that positive intervention costs strictly proportional to the change in the fundamental imply an optimal policy of instantaneous control at predetermined intervention points, with the fundamental being ‘regulated’ by reflecting barriers (Harrison and Taksar, 1983). (Lump sum costs induce discrete intervention; see Harrison et al., 1983.) So it might seem that the answer to the questions posed involve the straightforward application of existing results in optimal control. This is not the case, however, because the exchange rate which the authorities are to stabilise is not a simple Brownian motion process; it is the discounted present value of future fundamentals, which include the effects of future intervention! There is a risk that the authorities may be tempted to announce intervention barriers which are time-inconsistent in the sense that, when the time comes to intervene, the authorities will choose not to do so. To ensure that the announced barriers are credible, one needs to restrict the choice of policy to those which are subgame-perfect. This can be achieved using the technique of dynamic programming. Alternatively - and this is the technique used here - one can find the time-consistent optimal barriers as the Nash equilibrium of a game between the authorities and the public, as in Cohen and Michel (1988). In our application, the private sector determines the behaviour of the exchange rate as a function of fundamentals and the intervention trigger of the authorities, while the latter choose the optimal intervention trigger conditional on given private sector expectations. The intersection of these two ‘reaction functions’ determines the optimal triggers and so the width of the target zone. After characterising the time-consistent equilibrium in this way, Cohen and Michel went on to observe that other, nonsubgame-perfect, outcomes could generate lower costs for the authorities - if only they could be rendered credible by precommitment (with the reduction of costs representing the value of precommitment). In this application, we find likewise that triggers that are closer together (narrower bands) could improve upon optimal discretion if they could be made credible, and we compute the optimal band in that case. We argue, specifically, that an Exchange Rate Mechanism (ERM) where two (or more) countries publicly announce intervention bands, support of which is compulsory and not discretionary, may well represent just such a commitment. We first examine numerically how much narrower a band can be supported as optimal by such a mechanism, relative to the optimum time-consistent target zone
M. Miller, L. Zhunyi Journul of‘ Economic Dynamics and Control 20 (1996)
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computed as a Nash equilibrium. Then, using the welfare functions under the optimal rule and with discretion, we quantify the extent to which the stabilisation gains of membership (relative to the costs of intervention) can help to make the rule sustainable. The paper is constructed as follows. In the next section the monetary model and the objectives of the authorities are defined and the optimal discretionary bands are determined as a Nash equilibrium. We go on to discuss the link between the second-order smooth pasting required as a condition of optimality and the smooth pasting implied by arbitrage in the foreign exchange market using the Harrison diagram. (Optimal subgame-perfect intervention given lump sum costs is derived in Appendix A.) In Section 3, we demonstrate the value of precommitment to narrower bands and derive the optimal rule. In Section 4, we report the results of simulation exercises to measure the gains in quantitative terms. What we find is that an Exchange Rate Mechanism could sustain as optimal bands which are half as wide as those under discretion. The actual gain in welfare associated with this increased stabilisation is about 20%. If deviations result in an immediate shift to the discretionary equilibrium, this is sufficient to ensure that such an optimal rule is self-sustaining. After a discussion of qualifications and possible extensions to include other costs and benefits and other sources of disturbances as in Svensson ( 1994) Section 6 concludes. 2. 2.1.
infinitesimal
intervention
Tim+consistent
barriers
Let the exchange rate be the expected velocity-adjusted money, so s1 =
E, , , X /X,-e-B(x-t)dx, I’
present
discounted
value
of future
(1)
where sy is the logarithm of exchange rate at time t, k, is the velocity-adjusted money stock (in log form), E, is an expectation operator conditional on information available at t, and /? is the discount factor (the inverse of the semi-elasticity of demand for money). Let fundamentals evolve as the resultant of two influences: first, a random walk in velocity itself and, second, official interventions which include buying and selling foreign currency in exchange for domestic money, so dk, = odW, + dR, - dL,,
(2)
where W, is a standard Brownian motion, CJis a parameter measuring the volatility of fundamentals, RI/L, are two right continuous processes which represent marginal intervention to buy/sell foreign currency.
1644 M. Miller. L. Zhangl Journal of Economic Dynamics and Control 20 (1996)
1641-1660
Assuming that the monetary authorities seek to minimize the squared deviation of the exchange rate from a target value (constant and normalized to zero) subject to intervention costs proportional to the size of intervention, we can write the value function as O3s:e-P’dt
V(k) = minR,LEo
+
[J 0
ceeP’(dR, + dL,)
1 ,
(3)
subject to (1) and (2), where p is the discount factor and c is the constant unit cost of intervention in either direction. The form of the cost function has been selected so as to rule out continuous intervention; i.e., it has been chosen to make reflecting barriers the optimal policy. But inside these barriers the value function will be a function of unregulated Brownian motion. Hence it will satisfy the Hamilton-Jacobi-Bellman (HJB) equation pV(k)
= ;a*V&)
+ s2,
(4)
which can be derived directly by differentiating (3) to obtain E(dV)/dt = pV -s2, and noting that if V is a stationary function of k, then E(dV)/dt = ((7*/2)Vkk(k) by Ito’s lemma. The symmetric cost structure ensures that the value function is symmetric, so we need consider only one of the reflecting barriers. The two boundary conditions needed to solve (4) are thus provided by the conditions for optimal intervention at, say, the upper barrier k, specifically
V&c)= c,
(5)
Vkk(k) = 0.
(6)
The first of these is the condition that the marginal welfare cost must match the (constant) unit cost of intervention; the other is the ‘second-order smooth-pasting’ required of an optimum (see Harrison and Taksar, 1983; Dixit, 1991). To find the solution V(k), we first solve for the exchange rate s. Assuming a stationary mapping from k to s(k) and applying Ito’s lemma to s(k) using (1) and (2) yields a following HJB equation within the intervention barriers, pas(k) = ;a2s&k> which has a general
+ Bk, solution
s(k) = Cle” + Cze-i.k + k, where subscripts denote partial derivatives, and parameter 3, is defined by 1 = m.
Cl and C2 are two arbitrary constants,
M. Miller,
L. Zhunyl Journul of Economic
Dynamics
und Control 20 (1996)
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If the market expects fundamentals to be ‘regulated’ at fixed barrier points which are symmetric around the origin, say k and -k, then applying the ‘smoothpasting’ conditions given by arbitrage
the exchange rate we obtain is the hyperbolic described by Krugman (1991), i.e., )+k
sine function
of fundamentals
=Asinh(ik)+k.
(7)
Note that the ‘honeymoon effect’ parameter, pattern for the exchange rate is given by
-A,
which generates
an S-shaped
-A = I/[i. cosh@k)],
(8)
where k, -k are the reflecting barriers for the fundamental k; see Krugman ( 199 1, 1992) or Svensson (1991). Eq. (8) shows how expectations of marginal intervention at k affects the behaviour of exchange rate inside the band, so it can be interpreted as an ‘expectations constraint’. It is shown by the curve EE in Fig. 1 where the intervention point, k, is plotted on the horizontal axis and the honeymoon effect parameter, -A, on the vertical. The curve EE slopes downward to the right because wider intervention points are associated with a weaker honeymoon effect. Substitution of (7) into (4) provides an explicit representation of the value function which, given the symmetry of the problem and the conditioning on il, can be written as A2 V(k;.4) = sinh*(ik) P - 48 +jk’+;
2A + ----k P-B
($$+$)
sinh(lk)
2A,&s’ + ~ cosh( ik ) (P - m2
+Bcosh@k),
(9)
where i. and A are as defined above, and p = &$?. Note that expected costs in the absence of any intervention are given simply by (k2 + 02/p)lp. The other terms involving A and B in Eq. (9) capture the effects of anticipated intervention at any barrier k. Determining the optimal barrier involves differentiating the value function partially with respect to k, treating A as predetermined, and applying the boundary conditions already described. This will, of course, define the choice of the optimal barrier giuen market expectations of intervention. So the conditions for the optimal barrier are found by the application of (5) and (6) to the partial derivatives
1646 M. Miller, L. ZhanglJournal
of Economic Dynamics and Control 20 (1996)
1641-1660
of v, AA2
-
2M + -k P-P
-
cosh(;lk)
+ $
. smh(M)
2(PfP)A (P _ p)2
sinh(2Ak) + BP sinh(PLk) + P - 4a
(10)
= c,
4pM 211=A2 cosh(2lk) + (p _ p)2 cosh(;lk) + P-48 -
By2 cosh(&)
2i2A + -ksinh(lk)
+ 3 = 0.
P-P
(11)
Combining ( 10) and (11) to eliminate for the monetary authorities:
B yields the following
- i tanh(&)
2(~ + P)A
sinh(llk)
_ 4
cosh(23,k)
2~
PP+P
+ (P - n2
reaction
function
1
tanh($)
cosh(l,&)
I
2M k +P-B This is shown in Fig. 1, labelled TC. As it is quadratic in A, given any c > 0, it has two roots: one negative and the other positive. Since from (8) it is clear that A < 0, we choose the negative root. It can be shown that this negative root is decreasing in k and asymptotically tends to l/n when k --f +oo. This reaction function which maps A to k is the ‘time-consistent constraint’ on policy choice (i.e., the optimal selection of the intervention barrier k is conditional on the market expectation A). Clearly, the actual time-consistent intervention barrier k,v must be a fixedpoint, where market conjectures are satisfied by the optimal choice of barrier. Specifically, combining (8) and (12) to eliminate A yields an explicit fixed-point equation for the optimal time-consistent barrier as follows: (13) p-’ tanh(,ui)
= -
2
P - 48
2?b tanh2(nk) - -ktanh(lk)-a P-B
M. Miller. L. Zhany I Journal of’ Economic Dynumics und Cbnrrol20 Honeymoon
11996) 1641.~1660 1637
effect
-A t
I
Iso-value
Policy
-ic ‘kP Fig.
lntervenuon pomt
k,
I, Time-consistent
and precommitted
intervention
barrier
where ~=----_----4P (P - PI2
2
2
P-48
P’
For given k, the optimal target zone will be defined by -S;S, where S = k I.-’ tanh( A?). Fig. 1 shows the same result graphically. EE shows how the market reacts to anticipated intervention, choosing A as a function of k. TC shows how optimal intervention reacts to market expectations; specifically, it is the locus of minima of the dashed iso-value contours which show all combinations of A and k which give the same expected welfare costs at k = 0. The intersection of these two schedules at point N defines the Nash equilibrium, the time-consistent barrier k,v 2.2. A quudratic approximation and the Hurrison
diqrum
The nature of the solution and how it changes in response to changes in costs etc. is more apparent when the value function (9) is somewhat simplified. This
1648 M. Miller, L. Zhany I Journal of Economic Dynamics and Control 20 (1996)
can be achieved
by replacing
the conjecture
1641-1660
of (3) above by
s = (M + l)k,
(14)
i.e., the linear approximation of Krugman’s hyperbolic solution near the origin, the approximation recommended for small target zones by Delgado and Dumas (1992). Using (14) we find the value function simplifies to
V(k;A) =
(’+A3L)2 P
(15)
i.e., it is the sum of two terms, a quadratic form and a symmetric term in exponentials. The boundary conditions specified above require that, at the upper barrier k,
Vk(k;A) = 2(1 +Ai)‘i 2(1 +A/?)2 Vf!f(k;A) =
+@sinh(&) + p2B cosh(/&)
(16)
= 0,
(17)
P
which can be used to determine tuting for B and A, we find ,u-’ tanh(&)
= c,
= k -
I$ and B, conditional
on A. Specifically,
substi-
CP 2( 1 - sech(Ak))2 ’
where Al, has been replaced by -sech(?bk) according to (8) above. To an approximation, this defines the optimal subgame-perfect target zone, given the constant marginal (and average) intervention cost c. The link between the optimizing conditions (16) and (17) and the tangency condition derived by Krugman can be seen most easily using the diagrammatic approach employed by Harrison et al. (1983) and Dixit (1993). Given the conjecture of A, one can plot the derivative of the value function as a function of k. If, for example, B = 0 so the (approximated) value function (10) is quadratic, then Vk(k) is a linear function of k, V, = [2( 1 +A2)2/p]k, as shown by the line UU in Fig. 2. For B < 0, the expectation of future intervention lowers the cost function and generates solutions such as that labelled WW. The schedule WW has been chosen so as to satisfy the two boundary conditions that V,(k) = c and V,,(k) = 0, so it is tangent to c at k (and -k). The fixed point argument of the previous section ensures that the intervention point identified by the optimality conditions is indeed consistent with the conjecture embedded in A. Hence the solution for the exchange rate, s = k +A sinh(Ak), reaches a maximum at k (minimum at -k) as is shown in the figure where s is tangent to S at k. The
M. Miller.
L. ZhunylJournal
01 Economic
Dwamics
tmd Control 21) (19%)
1641-1660
1640
(I+;iA)k
/
ARBITRAGE
-k k
k
c
Approximation
Fig. 2. Optimal
time-consistent
target zone
time-consistent solution requires both these ‘tangency conditions’ to be satisfied at the same value of k. Arbitrage arguments show that the exchange rate will be tangent to its band at any intervention point, no matter how it is chosen - provided it is fully credible. In the next section, indeed, we discuss how a credible intervention ‘rule’, where the second-order smooth-pasting of the value function is not satisfied, might be preferred to the time-consistent optimum just derived. The quadratic approximation of Eq. (18) suggests clearly how the band will change in response to changes in costs condition and the discount factor. Raising the discount factor will tend to flatten the schedule UZJ implying wider barriers for intervention and a wider band. For a given discount factor, raising c will also widen the barriers and the band. While the qualitative nature of these changes is clear enough, the quantitative answers provided by ( 17) and (18) will not of course be exact. In Appendix A, the same graphical device is used to show how optimal barrier will be chosen when intervention involves lump sum costs (as well as proportional costs).
1650 M. Miller, L. ZhanglJournal of‘ Economic Dynamicsand Control 20 (1996) 1641-1660
3. Rules rather than discretion The intervention policies derived above have been obtained using the techniques of dynamic programming. But in a context like this, where expectations play a crucial role, it is well known that such ‘discretionary’ equilibria can be improved upon by adopting rules - if only some precommitment mechanism is available to enforce these rules (Kydland and Prescott, 1977). In this section, we assume that an Exchange Rate Mechanism (ERM) with prespecified intervention points acts as a precommitment; and we show in the next section how the increased exchange rate stability generated by the ERM may well produce the incentive needed to ensure sustainability. Why should the rules enforced by an ERM offer room for improvement over the dynamic programming outcome? It is because, in choosing the width of ERM bands, the effect on market expectations is explicitly taken into account. This is in contrast to the time-consistent bands where the conjecture about the exchange rate was taken as predetermined when applying the optimality conditions. That welfare improvement is possible by choosing a narrower band is evident from Fig. 1; where moving to the left along EE reduces welfare costs. Taking explicit account of the effect of the barriers on exchange rate expectations will evidently lead an ERM with narrower barriers. But how narrow should these barriers be? Assume there is to be marginal infinitesimal intervention at the ERM barriers. If so, the Krugman conjecture (7) will still apply to the exchange rate, and the derivative of the value function should still equal c (-c) at upper (lower) barrier for the fundamental. But, at such predetermined barriers, secondorder smooth-pasting is no longer appropriate (see Whittle, 1983), so we drop the condition L’kk(k;A) = 0 that signifies the optimal choice of discretionary barriers. What are we to put in its place? Consider for instance the ‘state-dependent’ criterion, that the barriers be chosen so as to rnp V(0; A(k), B), subject to
Vk(k;A,B) = c.
(19) (20)
Together with the consistency condition (8), the rule is chosen so as to minimize discounted costs conditional on starting at the middle of the band. Consider the iso-value contours shown as the dashed curves in Fig. 1. The optimal rule is found where the contour LL is tangent to the expectations constraint EE. As the welfare costs decrease with a stronger honeymoon effect (i.e., moving vertically upwards in the figure), it is clear that intervention at P will incur lower costs than at the Nash equilibrium IV. The narrower bands selected this way (which we denote -k~, k~) not only reduce the value function conditional on starting at k = 0, but the integrated costs implied by -kR, XR are in fact lower than those associated with the discretionary
M. Miller. L. Zhang
I Journal oJ‘ Economic D_vnamics and Control -70 (I 996) 1641-1660
165 I
Fig. 3. A rule which dominating the HJB solution
barriers for all starting values of k such that 0 5 k 5 k~; so the rule chosen strictly dominates the dynamic programming outcome. The strict dominance of this rule over discretion is shown in Fig. 3 (cf. Fig. 2 in Constantinides and Richard, 1978). There the value function implied by the Hamilton-Jacobi-Bellman (HJB) equations of dynamic programming is labelled an as shown, it satisfies first- and second-order smooth pasting at _J’%+&)); d kD, -kD (where there are points of inflexion and slopes of c, -c, respectively). The narrower barriers selected by our state-dependent criterion are shown as k~, -kR and the associated value function is labelled VR(k;A(kk)) with a slope of 0, -c at these barriers (but no point of intlexion). That VR(O;A(k8)) < VD(O;.4(k,)) should hardly be surprising, given the assumption of costless commitment; what is remarkable is that VR lies strictly below V D for all values of k between the barriers. (The proof is given in Appendix B.)
4. The gains from precommitment
- Some numerical examples
While we have confirmed theoretically that a precommitment mechanism can improve upon discretion, but one could well ask how quantitatively significant
1652 M. Miller, L. ZhanylJournal
of’ Economic Dynamics and Control 20 (1996)
1641-1660
is the improvement. Assuming, for example, that a &2;% band of the European ERM was an optimal choice under precommitment, what bandwidth would be intervention? And optimal without the precommitment, i.e., with time-consistent what would the loss of welfare be? We seek answers by illustrative numerical examples. Our benchmark case has parameters chosen in line with Svensson (1994) where p = 0.1, cr2 = 0.1, and /!J = 1. For these parameters, we first seek the intervention cost coefficient c which would generate (as the precommitted outcome) the f2$% bandwidth, a characteristic of the European ERM until 1992. This is shown in the middle row of Table 1, 4.09 x 10e3. In the fourth column we show the bandwidth (sd) which would be optimal given these costs, but without precommitment. Thus we find that a narrow band of 2$% under precommitment gives way to a band of about 4.5% under discretion (sd = 4.44%). In other words, with these parameters, the ERM could, via precommitment, sustuin a bundwidth only half as wide us thut under discretion. Welfare will not, however, be affected so dramatically: in the last column we find the reduction of welfare costs is about 20% before taking account of any ‘costs of precommitment’. The top and bottom rows of Table 1 show how optimal bandwidth and welfare vary with the change of p. What we note is that the ratio of bandwidths (sP/sd) and of welfare costs (VJVd) are insensitive to the changes in p. Varying the volatility in the fundamentals (g2) or the discount rate (p) does not significantly alter these results (see Tables 2 and 3); nor, we found, does setting a wider precommitted bandwidth (sP = 6%, for example). The key feature of these simulations (that a precommitted target zone could halve the bandwidth and gain 20% welfare) seems robust to variations in the parameters of the theoretical model. We have seen that there are substantial gains to precommitment; but are these sufficient to sustain the ERM itself? This depends on whether the perceived benefits of intervening in support of the optimal rule exceed the costs of doing so. In what follows, we measure the unit cost of intervention by parameter c; as for the benefits, let us assume that failing to intervene in support of the optimal precommitment band R, leads to an instant and permanent loss of credibility, and agents thenceforth expect discretionary intervention at k. On this, admittedly strong assumption of a permanent switch to the Nash equilibrium, we first derive a sufficient condition under which the optimal precommitment rule is selfsustaining; then in Tables 1 to 3, we examine whether this condition is satisfied in our numerical examples. Proposition
Jqi)
I. A su&ient
condition for the rule k~ to be self-sustaining
- V&R) 2 CCL - Gd,
where Vd and precommitment)
VP denote value function rule respectively.
is that
(21) under
discretion
and the (optimul
M. Miller. L. Zhang IJournal of’ Economic Dynamics and Control 20 (1996)
Table
1641-1660
1653
I
Response of bandwidth and welfare costs to changes in /I (p = 0. I, o2 := 0. I ) Unit intervention
Bandwidth
Ratio
Welfare
Unit benefits ( x IO- i ) _ __
cost (’ (x 10-3)
SP
&I
.ypisd
VpiVd
(V,l(k)-
113
5.92
2.25%
4.58%
0.492
0.776
72.4
1
4.09
2.25%
4.44%
0.507
0.792
87.7
3
2.28
2.25%
4.28%
0.526
0.809
92.0
P
V,(ke))l(k
-
kio
Table 2 Response of bandwidth and welfare costs to changes in 0’ (p = 0.1,
0.3 0. I .I.‘3
/r -= I )
Unit intervention
Bandwidth
Ratio
Welfare
Unit benefits ( x IO-’
cost (’ (X 10-3)
SP
SC/
.FPiSd
V,/l;/
(b;/(k)
2.01
2.25%
4.52%
0.497
0.785
67.1
4.09
2.25%
4.44%
0.507
0.792
x7.7
8.20
2.25%
4.32%
0.52 I
0.805
I IO.1
-
V[>(kR))!(k
) l/O
Table 3 Response of bandwidth and welfare costs to changes in 0 (/I = I, CT’ = 0.1) Umt intervention
Bandwidth
cost (’ (x10-3)
SP
SC/
Ratio
Welfare
Unit benefits (x10-‘)
.S,,&
VP / V‘/
(r;,(k) - V,(kN))/(x-- kR)
0.2
4.02
2.25%
4.48%
0.503
0.786
0.1
4.09
2.25%
4.44%
0.507
0.792
87.7
4.1 I
2.25%
4.42%
0.509
0.795
170.0
.05
46.4
To see this, consider the benefits and costs of intervening in response to various velocity shocks occurring at the edge of the precommitment band. Let E denote the size of the shock measured from k~. The condition is obvious enough when I: is the same as I? - 1~. For, given this shock, the left-hand side indicates the permanent reputation losses if the monetary authorities were not to intervene at the edge of the precommitted band c R, i.e., the benqjit of defending the precommitted band, and the right-hand side is the cost of doing so. But what if c is larger than k - k~? Here we note that, since the monetary authorities operating a discretionary band will have to prevent the fundamentals going beyond k, so i-:- (k - kR) of the shock has to be stabilised irrespective of whether or not the authorities choose to defend the precommitment band. Since the extra cost of maintaining the precommitment band is only c(g - k~ ), condition (21) is still sufficient for size of the shock larger than k - 6~. Finally, we look at the case where the size of the shock is smaller than I? - k~. In this case, (21) becomes
V(,(kR + E)
-
VP(&)
2
As Vd(k), for k 5 k, is a convex for (22).
(22)
(‘E.
function
of k, so (21 ) is a sufficient
condition
1654 M. Miller, L. ZhanglJournal of Economic Dynamics and Control 20 (1996) 1641-1660
Turning to see whether this condition is fulfilled in our numerical examples, we report in the last_ columns in Tables 1 to 3 the value of (V,(i)_V,(i~))/(k-i~), which for E 2 k - kR is the reputational benefit of defending kR, measured per unit of extra intervention required. For the base case (p = 0.1, G* = 0.1, and p = I), this per unit benefit amounts to 87.7x 10P3, more than 20 times the unit intervention cost c. Varying parameter values does not change this result very much, as the unit benefits in all our numerical examples are more than 10 times the unit intervention cost. The conclusion that the narrow bands chosen under an ERM are self-sustaining is evidently robust to parameter variations; but it does of course depend on the strong assumption that failing to intervene carries the penalty of an irreversible loss of credibility.
5. Qualifications and possible extensions To see how central banks might determine an optimal target zone we have _ as is standard in inventory theory - assumed that intervention is costly; and, assuming costs are strictly proportional to the size of intervention, we have determined the cost parameter c corresponding to a narrow band using an inverse optimal argument. How should this parameter c be interpreted? In discussing intervention costs in a recent paper entitled ‘Why exchange rate bands?‘, Lar Svensson (1994) comments that ‘the weight put on minimising interventions can either be interpreted as representing an actual intervention cost (in which case the weight is bound to be very small) or just representing the central bank’s aversion to intervention (in which case the weight could be large)‘. It should be stressed that, in the formal model of this paper, the parameter c, designed to capture transactions costs, is in fact a subjective measure and not a cash cost. (This is because, in deriving c, the coefficients appearing in the objective function have both been set at unity - implicitly giving cost minimisation and stabilisation equal weight. If, in reality, the central bank was to weight stabilisation with the parameter 6 # 1, then c is to be interpreted as cash costs divided by 6. So if stabilisation was given less weight (6 < l), c would be bigger than the actual cash costs, and vice versa.) Secondly we observe that c represents costs net of any benefits - such as the profit margin attached to stabilising intervention. It is true that stabilisation profits are typically transferred to the Treasury and not retained by the monetary authorities, so the latter are not strictly speaking driven by the motive for profit: but it could be interesting nevertheless to allow for the bandwidth to have some effect on the net costs of intervention. Finally, we note that there may of course be other costs to stabilising the exchange rate (and other shocks hitting the economy) not taken account of in our analysis. In Svensson (1994), for example, the objective function for the monetary authority includes the cost of interest rate variations as well as intervention costs.
M.
Miller,
L. ZhangIJournal
of Economic
Dynamics
nnd Control
20 (1996)
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Where the latter are set to zero, he finds that it is not velocity shocks which lead to a currency band but stochastic variations in foreign interest rates. Letting the domestic currency fluctuate permits some smoothing of domestic interest rates, i.e., for some monetary independence. ’ Strictly speaking what Svensson derives from a linear quadratic structure is a managed float with no actual band on the exchange rate. (The ‘band’ he refers to is the region containing 99% of the probability weight.) To pursue his argument a step further along the lines explored in this paper, one could see whether attaching proportional rather than quadratic costs to interest rate variations would lead to bands for exchange rates (when foreign interest rates vary) even when there are no explicit intervention costs.
6. Conclusion
For a single monetary authority aiming to stabilise the exchange rate with positive proportional costs of intervention, the optimal policy is a target zone with a reflecting barrier on fundamentals. Assuming these fundamentals follow a Wiener process we solve for the ‘time-consistent’ optimal barrier at which marginal intervention will take place. We then use our results to answer the inverse optimal question: How big would proportional intervention costs have to be to make existing exchange rates bands optimal? The analysis, which uses the popular monetary model, shows the link between the arbitrage conditions for the exchange rates derived by Krugman and the ‘smooth-pasting’ conditions that apply to the value function. Not all observed intervention takes place at the edge of exchange rate bands: one explanation for intra-marginal intervention is that there may be lurrrp SZU~I costs involved. We show, in Appendix A, how the time-consistent optimal barriers for the fundamental are affected by such lumpy costs, and how the inverse optimal problem for a given band width can be solved by combining fixed and variable costs. Because the exchange rate is assumed to discount future policy, the optimal ‘discretionary’ equilibrium can in principle be improved by precommitment to a rule, imposed for example by an Exchange Rate Mechanism. We show theoretically that narrower bands dominate the optimal ‘discretionary’ outcome, and we solve for the optimal band width under precommitment. Illustrative calculations suggest that the narrowing of the bands which precommitment makes possible is quite substantial. Specifically, an Exchange Rate Mechanism with precommitment to an optimal rule can narrow by half the width
’ From his numerical analysis he concludes that ‘the amount of monetary independence appears sizeable. For instance, an increase in the Swedish Krona band from zero to 32% may reduce the Krona interest rates deviation by about a half’ (Svensson. 1994, p. 157).
1656 M. Miller, L. Zhangl Journal of’ Economic Dynamics and Control 20 (I996)
1641-1660
of the exchange rate band chosen under discretion. 2 The proportionate reduction of welfare costs achieved by the narrower band may be less, about one fifth, but this is significant enough in terms of helping to sustain optimal intervention.
Appendix A: Lump sum costs and discrete intervention The target zone model has been generalized to include the case where the policy authority may use discrete intervention. Flood and Garber (1992) show that in this case the exchange rate is still related to fundamentals as in Eq. (4) above, but that intervention takes place inside the currency band. They also note that in the limit as the size of intervention is reduced, the outcome approaches the infinitesimal marginal intervention described by Krugman. In this section we use the Harrison diagram to indicate graphically how lump sum costs of intervening lead to discrete intervention as the optimal policy; and how infinitesimal intervention emerges in the limit as these lumpy costs vanish. If there are lump sum costs C (in addition to the proportional costs c), then the value function becomes 33
V(k) = n&~ E. [i
’
0
s2eKP’dt + F(C f
+ clS;l)e-Pr’
i=l
1 ,
(23)
subject to (1) and (2) above. Here ri denotes ‘stopping time’ when intervention takes place, l&l denotes the size of the intervention which is equal to D given below. Given C > 0, so-called S,s policy will be optimal, i.e., there will be intervention at k which reduces the fundamental to k-D, where k, D depend on C, c. The behaviour of the exchange rate between I? and -I? will be as in Eq. (7), as Flood and Garber pointed out, so the value function will take the form given in Eq. (9); but the boundary conditions for an optimal policy are now V&A)
= c,
(24)
V& - D;A) = c,
(25)
V&A)=
(26)
V(k-D,A)+C+cD,
where A is defined by k + A sinh(l,k)
= i - D + A sinh(A(k - D)).
(27)
Consider for simplicity the approximated value function of (23) above. The implications of applying the boundary conditions given above, subject to the consistency requirement that the barriers so selected match the Flood-Garber conjecture embedded in the value function, can be seen clearly in Fig. 4 where 2 Gains for precommitment also apply to ‘dirty’ floating, no band; see Svensson (1994) and Papi (1993).
where there is continuous
intervention
but
M. Miller. L. Zhany IJournal qf Economic
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cmd Control 20 (1996)
1641 1660 1657
the derivative of the value function and the exchange rate are shown in relation to the fundamental k. The optimal discrete intervention requires the derivative of the value function to match the proportional cost c as at the points X and Y in the figure; in addition, the integral between k - D and k must match the total intervention costs of C + CD, as shown. Since the area defined by X, Y, k, and 4 - D is CD, the shaded area must be equal to the lumpy cost C. The ‘no profitable arbitrage’ condition implies that s(k) = s(k - D), as Flood and Garber pointed out (and consistency requires that the conjecture embedded in the value function is the optimal policy chosen). So once again one finds a close analogy between the condition on the exchange rate implied by arbitrage and the optimality condition required of the marginal value function. That the policy of discrete intervention rrithirr the currency band will give way to a policy of infinitesimal marginal intervention when C tends to zero is evident from the figure; as C + 0, so D + 0. Thus, in the limit, one obtains the smoothpasting solution shown earlier in Fig. 2. [ This can, of course, be demonstrated more formally by showing how the three boundary conditions listed as (23)-( 26) above tend to the two conditions (5) and (6) as C 4 0.1
k k
Fig. 4. Lump sum costs and discrete
intervention
1658 M. Miller. L. ZhanglJournal
of Economic Dynamics and Control 20 (1996)
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Appendix B: The optimal rule First we show that for k E (O,~D), where I?D denotes time-consistent intervention barrier, V(O;A(k)) attains minimum at an interior. Then using quadratic approximation for value function (given that ,&D < < 1), we show that, for such choice of in, the corresponding value function indeed dominates those generated by k where k E [0, I?D] and k # LR. From (9) together with consistency condition, the value function at the central parity (k = 0) can be written as
VO;A(k))
where B(k) is rearranged
using Eq. (lo), 1 c - (P-4P
B(k) = &sch(@) Differentiating
(=“‘::I’+j),
2cr2sech(Ak) +; = B(k) (P _ 8)2
V(O;A(k))
so
p + ,!I ) 2 tanh(lk) - ___ (P-N2 J.
coth(,&)
P(P - B) 2
dim
wo;46) al
1.(29)
(P - B2 1
1 P+B -p _ 48 - (p
+2a2isech(lk)tanh(Ak)
kAcD
2pk P(P - B)
P+P
[(
+/,-ksch(&)
at time-consistent
1 --___ P - 48
( 1
c -
2P
2sech2( Ai) +
text to above equation,
+
with respect to k yields
av(o;A(k)) = -csch($) al
Evaluating
(28)
(p
1
tanh2(lk) sech(Ak) - p(p _ 48)
barrier I$ = ko, by substituting
-
- a
1.
I (30)
Eq. (13) from the
yields
= -&csch(&)
-
2(P + B)
tanh2(Ai,)
(P - LQ2
22 + -ko P-P
-
+
tanh(i&o)
1
1
2c21sech( AiD) tanh( Ai,)
> 0,
(Pfor
LD > 0.
sech(/?lfo) P(P -4P)
1 (31)
M. Miller, L. Zhang I Journal of Economic Dynamics and Control 20 (1996
Evaluating Tim k10
i 1641-1660
1659
at i = 0 yields
WO; A(Q) = c?k
-0c.
(32)
Since aV(O; A(k))/% is continuous in (0,k~], it certainly attains minimum at an interior point 2,. To find everywhere dominant value function within (0.k~) we rewrite the value function (9) incorporating (28): V(k;A) = V(O;A) + B(cosh($)
f-
- 1) + s(cosh(ik)
A2
2A sinh2( nk) + ---ksinh(ik) P - 48 P - B
For kD << 1, k 5 ED, we can approximate quadratic term, V(k; A) = QO;A)+-$ Substituting
the quadratic
B+
- 1) + F.
the above value function
2a2i_A --+;($$+;)]k’. (P_8)2
approximation
(33)
of Eq. (28) to the equation
up to the
(34) above yields
V(k;A) = V(O;A) (1 + Sk’) It is obvious that for any number k, k E [O,k], V(k; A) attains minimum if and only if V(O;A) is minimum. Thus, the choice of the barrier k~ generates an everywhere dominant value function for k E (0,k~).
References Avesani, Renzo G., 1990, Endogenously determined target zone and central bank optimal policy with limited reserves, Mimeo., Presented at CEPR / NBER conference on exchange rates and currency bands (University of Warwick, Coventry). Cohen, Daniel and Philippe Michel, 1988, How should control theory be used to calculate a time consistent government policy?, Review of Economic Studies 55, 2633274. Constantinides, George M. and Scott F. Richard, 1978, Existence of optimal simple policy for discounted-costs inventory and cash management in continuous time, Operations Research 26, 620636. Delgado, Francisco and Bernard Dumas, 1992, Target zones, broad and narrow, in: P. Krugman and M. Miller, eds., Exchange rate targets and currency bands (Cambridge University Press, Cambridge) 17-27. Dixit, Avinash K., 1991, A simplified exposition of the theory of optimal control of Brownian motion. Journal of Economic Dynamics and Control 15, 657-673. Dixit, Avinash K., 1993, The art of smooth pasting (Hanvood Academtc Publishers, Chur).
1660 M. Miller, L. ZhanylJournal
of Economic Dynumics and Control 20 (1996)
1641-1660
Flood, Robert and Peter Garber, 1992, The linkage between speculative attack and target zone models of exchange rates: Some extended results, in: P. Krugman and M. Miller, eds., Exchange rate targets and currency bands (Cambridge University Press, Cambridge) 17-27. Harrison, J. Michael and Michael 1. Taksar, 1983, Instantaneous control of Brownian motion, Mathematics of Operations Research 8, 4399453. Harrison, J. Michael, Thomas M. Sellke, and Allison J. Taylor, 1983, Impulse control of Brownian motion, Mathematics of Operations Research 8, 4544466. Krugman, Paul. 1991, Target zones and exchange rate dynamics, Quarterly Journal of Economics 163, 669-682. Krugman, Paul, 1992, Exchange rates in a currency band: A sketch of the new approach, in: P. Krugman and M. Miller, eds., Exchange rate targets and currency bands (Cambridge University Press, Cambridge) 9-14. Kydland, Finn E. and Edward C. Prescott, 1977, Rules rather than discretion: The inconsistency of optimal plans, Journal of Political Economy 85, 619-637. Papi, Laura, 1993, Ph.D. thesis (University of Warwick, Coventry). Svensson, Lars. 1991, Target zones and interest rate variability, Journal of International Economics 31, 27-54. Svensson, Lars, 1994, Why exchange rate bands? Monetary independence in spite of fixed exchange rates, Journal of Monetary Economics 33, 157-199. Whittle, Peter, 1983, Optimisation over time, dynamic programming and stochastic control, Vol. 2 (Wiley, Chichester).
Journal
ELSEVIER
of Economic Dynamics and Control 20 (1996) 1641-1660
Optimal target zones: How an exchange rate mechanism can improve upon discretion Marcus Miller *A~, Lei Zhang ” “Deprrrtment
qf Economics, lJnioer.sir)~ of Wantick, C’ormtr~ bCEPR. Londorl WIX 1 LB. UK
(Received August 1994; final version received October
C‘V4 7AL,
UK
1995)
Abstract Using Krugman’s (1991) target zone model, we find an explicit, subgame-perfect solution for a central bank wishing to stabilise the exchange rate given proportional costs of intervention. We demonstrate, however, that precommitment to narrower bands would yield a welfare gain - which provides a theoretical rationale for an Exchange Rate Mechanism (ERM). Numerical simulations suggest that the optimal currency band with precommitment via an ERM is only half as wide as that under discretion, and that the welfare benefits are sufficient to sustain the system. Kq words: Target zones; Regulated rate mechanism JEL
cluss~fi~ution:
E42;
F31;
F33;
Brownian
motion; Time-consistent
policy; Exchange
F42
1. Introduction If agents are rational, a policy of intervention at the edges of a currency band should help to stabilise the rate inside the band, even before the intervention takes place. This effect - the ‘honeymoon’ effect - was demonstrated by Krugman (1991) in a simple monetary model where the velocity of money was assumed to follow a Brownian motion process.
* Corresponding
author.
While remaining responsible for any errors, we would like to thank Avinash Dixit, Laura Papi, Jonathan Thomas, Steve Tumovsky, and an anonymous referee for helpful discussions and comments. 0165-1889/96/$15.00 0 1996 Elsevier Science B.V. All rights reserved SSlII 0165-1889(95)00908-E
1642 M. Miller, L. ZhanglJournal
of‘ Economic Dynamics and Control 20 (1996) 1641-1660
Why should the authorities want to intervene? And, if so, how wide should the band be? These are the questions we tackle in this paper, using the same canonical target zone model. As the price of foreign currency is the principal determinant of domestic prices in this model, it is natural to assume that the monetary authorities will want to stabilise the exchange rate. But, even when there is no shortage of reserves, allowing for some fluctuations within a band can be optimal because of positive costs of intervening (see Avesani, 1990). It is well known from studies on the regulation of Brownian motion processes that positive intervention costs strictly proportional to the change in the fundamental imply an optimal policy of instantaneous control at predetermined intervention points, with the fundamental being ‘regulated’ by reflecting barriers (Harrison and Taksar, 1983). (Lump sum costs induce discrete intervention; see Harrison et al., 1983.) So it might seem that the answer to the questions posed involve the straightforward application of existing results in optimal control. This is not the case, however, because the exchange rate which the authorities are to stabilise is not a simple Brownian motion process; it is the discounted present value of future fundamentals, which include the effects of future intervention! There is a risk that the authorities may be tempted to announce intervention barriers which are time-inconsistent in the sense that, when the time comes to intervene, the authorities will choose not to do so. To ensure that the announced barriers are credible, one needs to restrict the choice of policy to those which are subgame-perfect. This can be achieved using the technique of dynamic programming. Alternatively - and this is the technique used here - one can find the time-consistent optimal barriers as the Nash equilibrium of a game between the authorities and the public, as in Cohen and Michel (1988). In our application, the private sector determines the behaviour of the exchange rate as a function of fundamentals and the intervention trigger of the authorities, while the latter choose the optimal intervention trigger conditional on given private sector expectations. The intersection of these two ‘reaction functions’ determines the optimal triggers and so the width of the target zone. After characterising the time-consistent equilibrium in this way, Cohen and Michel went on to observe that other, nonsubgame-perfect, outcomes could generate lower costs for the authorities - if only they could be rendered credible by precommitment (with the reduction of costs representing the value of precommitment). In this application, we find likewise that triggers that are closer together (narrower bands) could improve upon optimal discretion if they could be made credible, and we compute the optimal band in that case. We argue, specifically, that an Exchange Rate Mechanism (ERM) where two (or more) countries publicly announce intervention bands, support of which is compulsory and not discretionary, may well represent just such a commitment. We first examine numerically how much narrower a band can be supported as optimal by such a mechanism, relative to the optimum time-consistent target zone
M. Miller, L. Zhunyi Journul of‘ Economic Dynamics and Control 20 (1996)
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1643
computed as a Nash equilibrium. Then, using the welfare functions under the optimal rule and with discretion, we quantify the extent to which the stabilisation gains of membership (relative to the costs of intervention) can help to make the rule sustainable. The paper is constructed as follows. In the next section the monetary model and the objectives of the authorities are defined and the optimal discretionary bands are determined as a Nash equilibrium. We go on to discuss the link between the second-order smooth pasting required as a condition of optimality and the smooth pasting implied by arbitrage in the foreign exchange market using the Harrison diagram. (Optimal subgame-perfect intervention given lump sum costs is derived in Appendix A.) In Section 3, we demonstrate the value of precommitment to narrower bands and derive the optimal rule. In Section 4, we report the results of simulation exercises to measure the gains in quantitative terms. What we find is that an Exchange Rate Mechanism could sustain as optimal bands which are half as wide as those under discretion. The actual gain in welfare associated with this increased stabilisation is about 20%. If deviations result in an immediate shift to the discretionary equilibrium, this is sufficient to ensure that such an optimal rule is self-sustaining. After a discussion of qualifications and possible extensions to include other costs and benefits and other sources of disturbances as in Svensson ( 1994) Section 6 concludes. 2. 2.1.
infinitesimal
intervention
Tim+consistent
barriers
Let the exchange rate be the expected velocity-adjusted money, so s1 =
E, , , X /X,-e-B(x-t)dx, I’
present
discounted
value
of future
(1)
where sy is the logarithm of exchange rate at time t, k, is the velocity-adjusted money stock (in log form), E, is an expectation operator conditional on information available at t, and /? is the discount factor (the inverse of the semi-elasticity of demand for money). Let fundamentals evolve as the resultant of two influences: first, a random walk in velocity itself and, second, official interventions which include buying and selling foreign currency in exchange for domestic money, so dk, = odW, + dR, - dL,,
(2)
where W, is a standard Brownian motion, CJis a parameter measuring the volatility of fundamentals, RI/L, are two right continuous processes which represent marginal intervention to buy/sell foreign currency.
1644 M. Miller. L. Zhangl Journal of Economic Dynamics and Control 20 (1996)
1641-1660
Assuming that the monetary authorities seek to minimize the squared deviation of the exchange rate from a target value (constant and normalized to zero) subject to intervention costs proportional to the size of intervention, we can write the value function as O3s:e-P’dt
V(k) = minR,LEo
+
[J 0
ceeP’(dR, + dL,)
1 ,
(3)
subject to (1) and (2), where p is the discount factor and c is the constant unit cost of intervention in either direction. The form of the cost function has been selected so as to rule out continuous intervention; i.e., it has been chosen to make reflecting barriers the optimal policy. But inside these barriers the value function will be a function of unregulated Brownian motion. Hence it will satisfy the Hamilton-Jacobi-Bellman (HJB) equation pV(k)
= ;a*V&)
+ s2,
(4)
which can be derived directly by differentiating (3) to obtain E(dV)/dt = pV -s2, and noting that if V is a stationary function of k, then E(dV)/dt = ((7*/2)Vkk(k) by Ito’s lemma. The symmetric cost structure ensures that the value function is symmetric, so we need consider only one of the reflecting barriers. The two boundary conditions needed to solve (4) are thus provided by the conditions for optimal intervention at, say, the upper barrier k, specifically
V&c)= c,
(5)
Vkk(k) = 0.
(6)
The first of these is the condition that the marginal welfare cost must match the (constant) unit cost of intervention; the other is the ‘second-order smooth-pasting’ required of an optimum (see Harrison and Taksar, 1983; Dixit, 1991). To find the solution V(k), we first solve for the exchange rate s. Assuming a stationary mapping from k to s(k) and applying Ito’s lemma to s(k) using (1) and (2) yields a following HJB equation within the intervention barriers, pas(k) = ;a2s&k> which has a general
+ Bk, solution
s(k) = Cle” + Cze-i.k + k, where subscripts denote partial derivatives, and parameter 3, is defined by 1 = m.
Cl and C2 are two arbitrary constants,
M. Miller,
L. Zhunyl Journul of Economic
Dynamics
und Control 20 (1996)
1541-1660
1645
If the market expects fundamentals to be ‘regulated’ at fixed barrier points which are symmetric around the origin, say k and -k, then applying the ‘smoothpasting’ conditions given by arbitrage
the exchange rate we obtain is the hyperbolic described by Krugman (1991), i.e., )+k
sine function
of fundamentals
=Asinh(ik)+k.
(7)
Note that the ‘honeymoon effect’ parameter, pattern for the exchange rate is given by
-A,
which generates
an S-shaped
-A = I/[i. cosh@k)],
(8)
where k, -k are the reflecting barriers for the fundamental k; see Krugman ( 199 1, 1992) or Svensson (1991). Eq. (8) shows how expectations of marginal intervention at k affects the behaviour of exchange rate inside the band, so it can be interpreted as an ‘expectations constraint’. It is shown by the curve EE in Fig. 1 where the intervention point, k, is plotted on the horizontal axis and the honeymoon effect parameter, -A, on the vertical. The curve EE slopes downward to the right because wider intervention points are associated with a weaker honeymoon effect. Substitution of (7) into (4) provides an explicit representation of the value function which, given the symmetry of the problem and the conditioning on il, can be written as A2 V(k;.4) = sinh*(ik) P - 48 +jk’+;
2A + ----k P-B
($$+$)
sinh(lk)
2A,&s’ + ~ cosh( ik ) (P - m2
+Bcosh@k),
(9)
where i. and A are as defined above, and p = &$?. Note that expected costs in the absence of any intervention are given simply by (k2 + 02/p)lp. The other terms involving A and B in Eq. (9) capture the effects of anticipated intervention at any barrier k. Determining the optimal barrier involves differentiating the value function partially with respect to k, treating A as predetermined, and applying the boundary conditions already described. This will, of course, define the choice of the optimal barrier giuen market expectations of intervention. So the conditions for the optimal barrier are found by the application of (5) and (6) to the partial derivatives
1646 M. Miller, L. ZhanglJournal
of Economic Dynamics and Control 20 (1996)
1641-1660
of v, AA2
-
2M + -k P-P
-
cosh(;lk)
+ $
. smh(M)
2(PfP)A (P _ p)2
sinh(2Ak) + BP sinh(PLk) + P - 4a
(10)
= c,
4pM 211=A2 cosh(2lk) + (p _ p)2 cosh(;lk) + P-48 -
By2 cosh(&)
2i2A + -ksinh(lk)
+ 3 = 0.
P-P
(11)
Combining ( 10) and (11) to eliminate for the monetary authorities:
B yields the following
- i tanh(&)
2(~ + P)A
sinh(llk)
_ 4
cosh(23,k)
2~
PP+P
+ (P - n2
reaction
function
1
tanh($)
cosh(l,&)
I
2M k +P-B This is shown in Fig. 1, labelled TC. As it is quadratic in A, given any c > 0, it has two roots: one negative and the other positive. Since from (8) it is clear that A < 0, we choose the negative root. It can be shown that this negative root is decreasing in k and asymptotically tends to l/n when k --f +oo. This reaction function which maps A to k is the ‘time-consistent constraint’ on policy choice (i.e., the optimal selection of the intervention barrier k is conditional on the market expectation A). Clearly, the actual time-consistent intervention barrier k,v must be a fixedpoint, where market conjectures are satisfied by the optimal choice of barrier. Specifically, combining (8) and (12) to eliminate A yields an explicit fixed-point equation for the optimal time-consistent barrier as follows: (13) p-’ tanh(,ui)
= -
2
P - 48
2?b tanh2(nk) - -ktanh(lk)-a P-B
M. Miller. L. Zhany I Journal of’ Economic Dynumics und Cbnrrol20 Honeymoon
11996) 1641.~1660 1637
effect
-A t
I
Iso-value
Policy
-ic ‘kP Fig.
lntervenuon pomt
k,
I, Time-consistent
and precommitted
intervention
barrier
where ~=----_----4P (P - PI2
2
2
P-48
P’
For given k, the optimal target zone will be defined by -S;S, where S = k I.-’ tanh( A?). Fig. 1 shows the same result graphically. EE shows how the market reacts to anticipated intervention, choosing A as a function of k. TC shows how optimal intervention reacts to market expectations; specifically, it is the locus of minima of the dashed iso-value contours which show all combinations of A and k which give the same expected welfare costs at k = 0. The intersection of these two schedules at point N defines the Nash equilibrium, the time-consistent barrier k,v 2.2. A quudratic approximation and the Hurrison
diqrum
The nature of the solution and how it changes in response to changes in costs etc. is more apparent when the value function (9) is somewhat simplified. This
1648 M. Miller, L. Zhany I Journal of Economic Dynamics and Control 20 (1996)
can be achieved
by replacing
the conjecture
1641-1660
of (3) above by
s = (M + l)k,
(14)
i.e., the linear approximation of Krugman’s hyperbolic solution near the origin, the approximation recommended for small target zones by Delgado and Dumas (1992). Using (14) we find the value function simplifies to
V(k;A) =
(’+A3L)2 P
(15)
i.e., it is the sum of two terms, a quadratic form and a symmetric term in exponentials. The boundary conditions specified above require that, at the upper barrier k,
Vk(k;A) = 2(1 +Ai)‘i 2(1 +A/?)2 Vf!f(k;A) =
+@sinh(&) + p2B cosh(/&)
(16)
= 0,
(17)
P
which can be used to determine tuting for B and A, we find ,u-’ tanh(&)
= c,
= k -
I$ and B, conditional
on A. Specifically,
substi-
CP 2( 1 - sech(Ak))2 ’
where Al, has been replaced by -sech(?bk) according to (8) above. To an approximation, this defines the optimal subgame-perfect target zone, given the constant marginal (and average) intervention cost c. The link between the optimizing conditions (16) and (17) and the tangency condition derived by Krugman can be seen most easily using the diagrammatic approach employed by Harrison et al. (1983) and Dixit (1993). Given the conjecture of A, one can plot the derivative of the value function as a function of k. If, for example, B = 0 so the (approximated) value function (10) is quadratic, then Vk(k) is a linear function of k, V, = [2( 1 +A2)2/p]k, as shown by the line UU in Fig. 2. For B < 0, the expectation of future intervention lowers the cost function and generates solutions such as that labelled WW. The schedule WW has been chosen so as to satisfy the two boundary conditions that V,(k) = c and V,,(k) = 0, so it is tangent to c at k (and -k). The fixed point argument of the previous section ensures that the intervention point identified by the optimality conditions is indeed consistent with the conjecture embedded in A. Hence the solution for the exchange rate, s = k +A sinh(Ak), reaches a maximum at k (minimum at -k) as is shown in the figure where s is tangent to S at k. The
M. Miller.
L. ZhunylJournal
01 Economic
Dwamics
tmd Control 21) (19%)
1641-1660
1640
(I+;iA)k
/
ARBITRAGE
-k k
k
c
Approximation
Fig. 2. Optimal
time-consistent
target zone
time-consistent solution requires both these ‘tangency conditions’ to be satisfied at the same value of k. Arbitrage arguments show that the exchange rate will be tangent to its band at any intervention point, no matter how it is chosen - provided it is fully credible. In the next section, indeed, we discuss how a credible intervention ‘rule’, where the second-order smooth-pasting of the value function is not satisfied, might be preferred to the time-consistent optimum just derived. The quadratic approximation of Eq. (18) suggests clearly how the band will change in response to changes in costs condition and the discount factor. Raising the discount factor will tend to flatten the schedule UZJ implying wider barriers for intervention and a wider band. For a given discount factor, raising c will also widen the barriers and the band. While the qualitative nature of these changes is clear enough, the quantitative answers provided by ( 17) and (18) will not of course be exact. In Appendix A, the same graphical device is used to show how optimal barrier will be chosen when intervention involves lump sum costs (as well as proportional costs).
1650 M. Miller, L. ZhanglJournal of‘ Economic Dynamicsand Control 20 (1996) 1641-1660
3. Rules rather than discretion The intervention policies derived above have been obtained using the techniques of dynamic programming. But in a context like this, where expectations play a crucial role, it is well known that such ‘discretionary’ equilibria can be improved upon by adopting rules - if only some precommitment mechanism is available to enforce these rules (Kydland and Prescott, 1977). In this section, we assume that an Exchange Rate Mechanism (ERM) with prespecified intervention points acts as a precommitment; and we show in the next section how the increased exchange rate stability generated by the ERM may well produce the incentive needed to ensure sustainability. Why should the rules enforced by an ERM offer room for improvement over the dynamic programming outcome? It is because, in choosing the width of ERM bands, the effect on market expectations is explicitly taken into account. This is in contrast to the time-consistent bands where the conjecture about the exchange rate was taken as predetermined when applying the optimality conditions. That welfare improvement is possible by choosing a narrower band is evident from Fig. 1; where moving to the left along EE reduces welfare costs. Taking explicit account of the effect of the barriers on exchange rate expectations will evidently lead an ERM with narrower barriers. But how narrow should these barriers be? Assume there is to be marginal infinitesimal intervention at the ERM barriers. If so, the Krugman conjecture (7) will still apply to the exchange rate, and the derivative of the value function should still equal c (-c) at upper (lower) barrier for the fundamental. But, at such predetermined barriers, secondorder smooth-pasting is no longer appropriate (see Whittle, 1983), so we drop the condition L’kk(k;A) = 0 that signifies the optimal choice of discretionary barriers. What are we to put in its place? Consider for instance the ‘state-dependent’ criterion, that the barriers be chosen so as to rnp V(0; A(k), B), subject to
Vk(k;A,B) = c.
(19) (20)
Together with the consistency condition (8), the rule is chosen so as to minimize discounted costs conditional on starting at the middle of the band. Consider the iso-value contours shown as the dashed curves in Fig. 1. The optimal rule is found where the contour LL is tangent to the expectations constraint EE. As the welfare costs decrease with a stronger honeymoon effect (i.e., moving vertically upwards in the figure), it is clear that intervention at P will incur lower costs than at the Nash equilibrium IV. The narrower bands selected this way (which we denote -k~, k~) not only reduce the value function conditional on starting at k = 0, but the integrated costs implied by -kR, XR are in fact lower than those associated with the discretionary
M. Miller. L. Zhang
I Journal oJ‘ Economic D_vnamics and Control -70 (I 996) 1641-1660
165 I
Fig. 3. A rule which dominating the HJB solution
barriers for all starting values of k such that 0 5 k 5 k~; so the rule chosen strictly dominates the dynamic programming outcome. The strict dominance of this rule over discretion is shown in Fig. 3 (cf. Fig. 2 in Constantinides and Richard, 1978). There the value function implied by the Hamilton-Jacobi-Bellman (HJB) equations of dynamic programming is labelled an as shown, it satisfies first- and second-order smooth pasting at _J’%+&)); d kD, -kD (where there are points of inflexion and slopes of c, -c, respectively). The narrower barriers selected by our state-dependent criterion are shown as k~, -kR and the associated value function is labelled VR(k;A(kk)) with a slope of 0, -c at these barriers (but no point of intlexion). That VR(O;A(k8)) < VD(O;.4(k,)) should hardly be surprising, given the assumption of costless commitment; what is remarkable is that VR lies strictly below V D for all values of k between the barriers. (The proof is given in Appendix B.)
4. The gains from precommitment
- Some numerical examples
While we have confirmed theoretically that a precommitment mechanism can improve upon discretion, but one could well ask how quantitatively significant
1652 M. Miller, L. ZhanylJournal
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is the improvement. Assuming, for example, that a &2;% band of the European ERM was an optimal choice under precommitment, what bandwidth would be intervention? And optimal without the precommitment, i.e., with time-consistent what would the loss of welfare be? We seek answers by illustrative numerical examples. Our benchmark case has parameters chosen in line with Svensson (1994) where p = 0.1, cr2 = 0.1, and /!J = 1. For these parameters, we first seek the intervention cost coefficient c which would generate (as the precommitted outcome) the f2$% bandwidth, a characteristic of the European ERM until 1992. This is shown in the middle row of Table 1, 4.09 x 10e3. In the fourth column we show the bandwidth (sd) which would be optimal given these costs, but without precommitment. Thus we find that a narrow band of 2$% under precommitment gives way to a band of about 4.5% under discretion (sd = 4.44%). In other words, with these parameters, the ERM could, via precommitment, sustuin a bundwidth only half as wide us thut under discretion. Welfare will not, however, be affected so dramatically: in the last column we find the reduction of welfare costs is about 20% before taking account of any ‘costs of precommitment’. The top and bottom rows of Table 1 show how optimal bandwidth and welfare vary with the change of p. What we note is that the ratio of bandwidths (sP/sd) and of welfare costs (VJVd) are insensitive to the changes in p. Varying the volatility in the fundamentals (g2) or the discount rate (p) does not significantly alter these results (see Tables 2 and 3); nor, we found, does setting a wider precommitted bandwidth (sP = 6%, for example). The key feature of these simulations (that a precommitted target zone could halve the bandwidth and gain 20% welfare) seems robust to variations in the parameters of the theoretical model. We have seen that there are substantial gains to precommitment; but are these sufficient to sustain the ERM itself? This depends on whether the perceived benefits of intervening in support of the optimal rule exceed the costs of doing so. In what follows, we measure the unit cost of intervention by parameter c; as for the benefits, let us assume that failing to intervene in support of the optimal precommitment band R, leads to an instant and permanent loss of credibility, and agents thenceforth expect discretionary intervention at k. On this, admittedly strong assumption of a permanent switch to the Nash equilibrium, we first derive a sufficient condition under which the optimal precommitment rule is selfsustaining; then in Tables 1 to 3, we examine whether this condition is satisfied in our numerical examples. Proposition
Jqi)
I. A su&ient
condition for the rule k~ to be self-sustaining
- V&R) 2 CCL - Gd,
where Vd and precommitment)
VP denote value function rule respectively.
is that
(21) under
discretion
and the (optimul
M. Miller. L. Zhang IJournal of’ Economic Dynamics and Control 20 (1996)
Table
1641-1660
1653
I
Response of bandwidth and welfare costs to changes in /I (p = 0. I, o2 := 0. I ) Unit intervention
Bandwidth
Ratio
Welfare
Unit benefits ( x IO- i ) _ __
cost (’ (x 10-3)
SP
&I
.ypisd
VpiVd
(V,l(k)-
113
5.92
2.25%
4.58%
0.492
0.776
72.4
1
4.09
2.25%
4.44%
0.507
0.792
87.7
3
2.28
2.25%
4.28%
0.526
0.809
92.0
P
V,(ke))l(k
-
kio
Table 2 Response of bandwidth and welfare costs to changes in 0’ (p = 0.1,
0.3 0. I .I.‘3
/r -= I )
Unit intervention
Bandwidth
Ratio
Welfare
Unit benefits ( x IO-’
cost (’ (X 10-3)
SP
SC/
.FPiSd
V,/l;/
(b;/(k)
2.01
2.25%
4.52%
0.497
0.785
67.1
4.09
2.25%
4.44%
0.507
0.792
x7.7
8.20
2.25%
4.32%
0.52 I
0.805
I IO.1
-
V[>(kR))!(k
) l/O
Table 3 Response of bandwidth and welfare costs to changes in 0 (/I = I, CT’ = 0.1) Umt intervention
Bandwidth
cost (’ (x10-3)
SP
SC/
Ratio
Welfare
Unit benefits (x10-‘)
.S,,&
VP / V‘/
(r;,(k) - V,(kN))/(x-- kR)
0.2
4.02
2.25%
4.48%
0.503
0.786
0.1
4.09
2.25%
4.44%
0.507
0.792
87.7
4.1 I
2.25%
4.42%
0.509
0.795
170.0
.05
46.4
To see this, consider the benefits and costs of intervening in response to various velocity shocks occurring at the edge of the precommitment band. Let E denote the size of the shock measured from k~. The condition is obvious enough when I: is the same as I? - 1~. For, given this shock, the left-hand side indicates the permanent reputation losses if the monetary authorities were not to intervene at the edge of the precommitted band c R, i.e., the benqjit of defending the precommitted band, and the right-hand side is the cost of doing so. But what if c is larger than k - k~? Here we note that, since the monetary authorities operating a discretionary band will have to prevent the fundamentals going beyond k, so i-:- (k - kR) of the shock has to be stabilised irrespective of whether or not the authorities choose to defend the precommitment band. Since the extra cost of maintaining the precommitment band is only c(g - k~ ), condition (21) is still sufficient for size of the shock larger than k - 6~. Finally, we look at the case where the size of the shock is smaller than I? - k~. In this case, (21) becomes
V(,(kR + E)
-
VP(&)
2
As Vd(k), for k 5 k, is a convex for (22).
(22)
(‘E.
function
of k, so (21 ) is a sufficient
condition
1654 M. Miller, L. ZhanglJournal of Economic Dynamics and Control 20 (1996) 1641-1660
Turning to see whether this condition is fulfilled in our numerical examples, we report in the last_ columns in Tables 1 to 3 the value of (V,(i)_V,(i~))/(k-i~), which for E 2 k - kR is the reputational benefit of defending kR, measured per unit of extra intervention required. For the base case (p = 0.1, G* = 0.1, and p = I), this per unit benefit amounts to 87.7x 10P3, more than 20 times the unit intervention cost c. Varying parameter values does not change this result very much, as the unit benefits in all our numerical examples are more than 10 times the unit intervention cost. The conclusion that the narrow bands chosen under an ERM are self-sustaining is evidently robust to parameter variations; but it does of course depend on the strong assumption that failing to intervene carries the penalty of an irreversible loss of credibility.
5. Qualifications and possible extensions To see how central banks might determine an optimal target zone we have _ as is standard in inventory theory - assumed that intervention is costly; and, assuming costs are strictly proportional to the size of intervention, we have determined the cost parameter c corresponding to a narrow band using an inverse optimal argument. How should this parameter c be interpreted? In discussing intervention costs in a recent paper entitled ‘Why exchange rate bands?‘, Lar Svensson (1994) comments that ‘the weight put on minimising interventions can either be interpreted as representing an actual intervention cost (in which case the weight is bound to be very small) or just representing the central bank’s aversion to intervention (in which case the weight could be large)‘. It should be stressed that, in the formal model of this paper, the parameter c, designed to capture transactions costs, is in fact a subjective measure and not a cash cost. (This is because, in deriving c, the coefficients appearing in the objective function have both been set at unity - implicitly giving cost minimisation and stabilisation equal weight. If, in reality, the central bank was to weight stabilisation with the parameter 6 # 1, then c is to be interpreted as cash costs divided by 6. So if stabilisation was given less weight (6 < l), c would be bigger than the actual cash costs, and vice versa.) Secondly we observe that c represents costs net of any benefits - such as the profit margin attached to stabilising intervention. It is true that stabilisation profits are typically transferred to the Treasury and not retained by the monetary authorities, so the latter are not strictly speaking driven by the motive for profit: but it could be interesting nevertheless to allow for the bandwidth to have some effect on the net costs of intervention. Finally, we note that there may of course be other costs to stabilising the exchange rate (and other shocks hitting the economy) not taken account of in our analysis. In Svensson (1994), for example, the objective function for the monetary authority includes the cost of interest rate variations as well as intervention costs.
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L. ZhangIJournal
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nnd Control
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Where the latter are set to zero, he finds that it is not velocity shocks which lead to a currency band but stochastic variations in foreign interest rates. Letting the domestic currency fluctuate permits some smoothing of domestic interest rates, i.e., for some monetary independence. ’ Strictly speaking what Svensson derives from a linear quadratic structure is a managed float with no actual band on the exchange rate. (The ‘band’ he refers to is the region containing 99% of the probability weight.) To pursue his argument a step further along the lines explored in this paper, one could see whether attaching proportional rather than quadratic costs to interest rate variations would lead to bands for exchange rates (when foreign interest rates vary) even when there are no explicit intervention costs.
6. Conclusion
For a single monetary authority aiming to stabilise the exchange rate with positive proportional costs of intervention, the optimal policy is a target zone with a reflecting barrier on fundamentals. Assuming these fundamentals follow a Wiener process we solve for the ‘time-consistent’ optimal barrier at which marginal intervention will take place. We then use our results to answer the inverse optimal question: How big would proportional intervention costs have to be to make existing exchange rates bands optimal? The analysis, which uses the popular monetary model, shows the link between the arbitrage conditions for the exchange rates derived by Krugman and the ‘smooth-pasting’ conditions that apply to the value function. Not all observed intervention takes place at the edge of exchange rate bands: one explanation for intra-marginal intervention is that there may be lurrrp SZU~I costs involved. We show, in Appendix A, how the time-consistent optimal barriers for the fundamental are affected by such lumpy costs, and how the inverse optimal problem for a given band width can be solved by combining fixed and variable costs. Because the exchange rate is assumed to discount future policy, the optimal ‘discretionary’ equilibrium can in principle be improved by precommitment to a rule, imposed for example by an Exchange Rate Mechanism. We show theoretically that narrower bands dominate the optimal ‘discretionary’ outcome, and we solve for the optimal band width under precommitment. Illustrative calculations suggest that the narrowing of the bands which precommitment makes possible is quite substantial. Specifically, an Exchange Rate Mechanism with precommitment to an optimal rule can narrow by half the width
’ From his numerical analysis he concludes that ‘the amount of monetary independence appears sizeable. For instance, an increase in the Swedish Krona band from zero to 32% may reduce the Krona interest rates deviation by about a half’ (Svensson. 1994, p. 157).
1656 M. Miller, L. Zhangl Journal of’ Economic Dynamics and Control 20 (I996)
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of the exchange rate band chosen under discretion. 2 The proportionate reduction of welfare costs achieved by the narrower band may be less, about one fifth, but this is significant enough in terms of helping to sustain optimal intervention.
Appendix A: Lump sum costs and discrete intervention The target zone model has been generalized to include the case where the policy authority may use discrete intervention. Flood and Garber (1992) show that in this case the exchange rate is still related to fundamentals as in Eq. (4) above, but that intervention takes place inside the currency band. They also note that in the limit as the size of intervention is reduced, the outcome approaches the infinitesimal marginal intervention described by Krugman. In this section we use the Harrison diagram to indicate graphically how lump sum costs of intervening lead to discrete intervention as the optimal policy; and how infinitesimal intervention emerges in the limit as these lumpy costs vanish. If there are lump sum costs C (in addition to the proportional costs c), then the value function becomes 33
V(k) = n&~ E. [i
’
0
s2eKP’dt + F(C f
+ clS;l)e-Pr’
i=l
1 ,
(23)
subject to (1) and (2) above. Here ri denotes ‘stopping time’ when intervention takes place, l&l denotes the size of the intervention which is equal to D given below. Given C > 0, so-called S,s policy will be optimal, i.e., there will be intervention at k which reduces the fundamental to k-D, where k, D depend on C, c. The behaviour of the exchange rate between I? and -I? will be as in Eq. (7), as Flood and Garber pointed out, so the value function will take the form given in Eq. (9); but the boundary conditions for an optimal policy are now V&A)
= c,
(24)
V& - D;A) = c,
(25)
V&A)=
(26)
V(k-D,A)+C+cD,
where A is defined by k + A sinh(l,k)
= i - D + A sinh(A(k - D)).
(27)
Consider for simplicity the approximated value function of (23) above. The implications of applying the boundary conditions given above, subject to the consistency requirement that the barriers so selected match the Flood-Garber conjecture embedded in the value function, can be seen clearly in Fig. 4 where 2 Gains for precommitment also apply to ‘dirty’ floating, no band; see Svensson (1994) and Papi (1993).
where there is continuous
intervention
but
M. Miller. L. Zhany IJournal qf Economic
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cmd Control 20 (1996)
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the derivative of the value function and the exchange rate are shown in relation to the fundamental k. The optimal discrete intervention requires the derivative of the value function to match the proportional cost c as at the points X and Y in the figure; in addition, the integral between k - D and k must match the total intervention costs of C + CD, as shown. Since the area defined by X, Y, k, and 4 - D is CD, the shaded area must be equal to the lumpy cost C. The ‘no profitable arbitrage’ condition implies that s(k) = s(k - D), as Flood and Garber pointed out (and consistency requires that the conjecture embedded in the value function is the optimal policy chosen). So once again one finds a close analogy between the condition on the exchange rate implied by arbitrage and the optimality condition required of the marginal value function. That the policy of discrete intervention rrithirr the currency band will give way to a policy of infinitesimal marginal intervention when C tends to zero is evident from the figure; as C + 0, so D + 0. Thus, in the limit, one obtains the smoothpasting solution shown earlier in Fig. 2. [ This can, of course, be demonstrated more formally by showing how the three boundary conditions listed as (23)-( 26) above tend to the two conditions (5) and (6) as C 4 0.1
k k
Fig. 4. Lump sum costs and discrete
intervention
1658 M. Miller. L. ZhanglJournal
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Appendix B: The optimal rule First we show that for k E (O,~D), where I?D denotes time-consistent intervention barrier, V(O;A(k)) attains minimum at an interior. Then using quadratic approximation for value function (given that ,&D < < 1), we show that, for such choice of in, the corresponding value function indeed dominates those generated by k where k E [0, I?D] and k # LR. From (9) together with consistency condition, the value function at the central parity (k = 0) can be written as
VO;A(k))
where B(k) is rearranged
using Eq. (lo), 1 c - (P-4P
B(k) = &sch(@) Differentiating
(=“‘::I’+j),
2cr2sech(Ak) +; = B(k) (P _ 8)2
V(O;A(k))
so
p + ,!I ) 2 tanh(lk) - ___ (P-N2 J.
coth(,&)
P(P - B) 2
dim
wo;46) al
1.(29)
(P - B2 1
1 P+B -p _ 48 - (p
+2a2isech(lk)tanh(Ak)
kAcD
2pk P(P - B)
P+P
[(
+/,-ksch(&)
at time-consistent
1 --___ P - 48
( 1
c -
2P
2sech2( Ai) +
text to above equation,
+
with respect to k yields
av(o;A(k)) = -csch($) al
Evaluating
(28)
(p
1
tanh2(lk) sech(Ak) - p(p _ 48)
barrier I$ = ko, by substituting
-
- a
1.
I (30)
Eq. (13) from the
yields
= -&csch(&)
-
2(P + B)
tanh2(Ai,)
(P - LQ2
22 + -ko P-P
-
+
tanh(i&o)
1
1
2c21sech( AiD) tanh( Ai,)
> 0,
(Pfor
LD > 0.
sech(/?lfo) P(P -4P)
1 (31)
M. Miller, L. Zhang I Journal of Economic Dynamics and Control 20 (1996
Evaluating Tim k10
i 1641-1660
1659
at i = 0 yields
WO; A(Q) = c?k
-0c.
(32)
Since aV(O; A(k))/% is continuous in (0,k~], it certainly attains minimum at an interior point 2,. To find everywhere dominant value function within (0.k~) we rewrite the value function (9) incorporating (28): V(k;A) = V(O;A) + B(cosh($)
f-
- 1) + s(cosh(ik)
A2
2A sinh2( nk) + ---ksinh(ik) P - 48 P - B
For kD << 1, k 5 ED, we can approximate quadratic term, V(k; A) = QO;A)+-$ Substituting
the quadratic
B+
- 1) + F.
the above value function
2a2i_A --+;($$+;)]k’. (P_8)2
approximation
(33)
of Eq. (28) to the equation
up to the
(34) above yields
V(k;A) = V(O;A) (1 + Sk’) It is obvious that for any number k, k E [O,k], V(k; A) attains minimum if and only if V(O;A) is minimum. Thus, the choice of the barrier k~ generates an everywhere dominant value function for k E (0,k~).
References Avesani, Renzo G., 1990, Endogenously determined target zone and central bank optimal policy with limited reserves, Mimeo., Presented at CEPR / NBER conference on exchange rates and currency bands (University of Warwick, Coventry). Cohen, Daniel and Philippe Michel, 1988, How should control theory be used to calculate a time consistent government policy?, Review of Economic Studies 55, 2633274. Constantinides, George M. and Scott F. Richard, 1978, Existence of optimal simple policy for discounted-costs inventory and cash management in continuous time, Operations Research 26, 620636. Delgado, Francisco and Bernard Dumas, 1992, Target zones, broad and narrow, in: P. Krugman and M. Miller, eds., Exchange rate targets and currency bands (Cambridge University Press, Cambridge) 17-27. Dixit, Avinash K., 1991, A simplified exposition of the theory of optimal control of Brownian motion. Journal of Economic Dynamics and Control 15, 657-673. Dixit, Avinash K., 1993, The art of smooth pasting (Hanvood Academtc Publishers, Chur).
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Flood, Robert and Peter Garber, 1992, The linkage between speculative attack and target zone models of exchange rates: Some extended results, in: P. Krugman and M. Miller, eds., Exchange rate targets and currency bands (Cambridge University Press, Cambridge) 17-27. Harrison, J. Michael and Michael 1. Taksar, 1983, Instantaneous control of Brownian motion, Mathematics of Operations Research 8, 4399453. Harrison, J. Michael, Thomas M. Sellke, and Allison J. Taylor, 1983, Impulse control of Brownian motion, Mathematics of Operations Research 8, 4544466. Krugman, Paul. 1991, Target zones and exchange rate dynamics, Quarterly Journal of Economics 163, 669-682. Krugman, Paul, 1992, Exchange rates in a currency band: A sketch of the new approach, in: P. Krugman and M. Miller, eds., Exchange rate targets and currency bands (Cambridge University Press, Cambridge) 9-14. Kydland, Finn E. and Edward C. Prescott, 1977, Rules rather than discretion: The inconsistency of optimal plans, Journal of Political Economy 85, 619-637. Papi, Laura, 1993, Ph.D. thesis (University of Warwick, Coventry). Svensson, Lars. 1991, Target zones and interest rate variability, Journal of International Economics 31, 27-54. Svensson, Lars, 1994, Why exchange rate bands? Monetary independence in spite of fixed exchange rates, Journal of Monetary Economics 33, 157-199. Whittle, Peter, 1983, Optimisation over time, dynamic programming and stochastic control, Vol. 2 (Wiley, Chichester).