10.3. INFINITESIMAL MARGINAL INTERVENTION

313

which follows because the integral in term (a) is Rt∞ e−x/αdx = αe−t/α and the integral in term (b) is Rt∞ xe−x/αdx = α2e−t/α(αt +1). (10.26) is the no bubbles solution for the exchange rate under a permanent freeßoat regime where the fundamentals follow the (η, σ)—di usion process (10.23) and are expected to do so forever on. This is the continuoustime analog to the solution obtained in chapter 3 when the fundamentals followed a random walk.

10.3InÞnitesimal Marginal Intervention

Consider now a small-open economy whose central bank is committed to keeping the nominal exchange rate s within the target zone, s < s < s¯. The credibility of the Þx is not in question. Krugman [88] assumes that the monetary authorities intervene whenever the exchange rate touches one of the bands in a way to prevent the exchange rate from ever moving out of the bands. In order to be e ective, the authorities must engage in unsterilized intervention, by adjusting the fundamentals f(t). As long as the exchange rate lies within the target zone, the authorities do nothing and allow the fundamentals to follow the di usion process df(t) = ηdt + σdz(t). But at those instants that the exchange rate touches one of the bands, the authorities intervene to an extent necessary to prevent the exchange rate from moving out of the band.

During times of intervention, the fundamentals do not obey the diffusion process but are following some other process. Since the forecasting rule (10.25) was derived by assuming that the fundamentals always follows the di usion it cannot be used here. To solve the model using the same technique, you need to modify the forecasting rule to account for the fact that the process governing the fundamentals switches from the di usion to the alternative process during intervention periods.

Instead, we will obtain the solution by the method of undetermined coe cients. Begin by conjecturing a solution in which the exchange rate is a time-invariant function G(·) of the current fundamentals

s(t) = G[f(t)].

(10.27)

Now to Þgure out what the function G looks like, you know by Ito’s

314	CHAPTER 10. TARGET-ZONE MODELS
lemma
	ds(t) = dG[f(t)]
					σ2
	= G0[f(t)]df(t) +					G00[f(t)]dt
				2
	= G0[f(t)][ηdt + σdz(t)] +								σ2	G00[f(t)]dt.	(10.28)
							2
Taking	expectations conditioned			on			time-t			information	you get
Et[ds(t)] = G0[f(t)]ηdt +		σ2	G00[f(t)]dt.				Dividing this result through
	2
by dt you get							σ2
	Et[sú(t)] = ηG0[f(t)] +							G00[f(t)].			(10.29)
							2

Now substitute (10.27) and (10.29) into the monetary model (10.21) and re-arrange to get the second-order di erential equation in G

00	2η 0		2		2
G [f(t)] +	σ2	G [f(t)] −	ασ2	G[f(t)] = −	ασ2	f(t).	(10.30)

Digression on second-order di erential equations. Consider the secondorder di erential equation,

y00 + a1y0 + a2y = bt	(10.31)
A trial solution to the homogeneous part (y00	+ a1y0 + a2y =	0) is
y = Aeλt, which implies y0 = λAeλt and	y00 = λ2Aeλt,	and

Aeλt(λ2 + a1λ + a2) = 0, for which there are obviously two solutions,

a1+√

√

a2

4a2

a1

a2

4a2

λ1

=

−

1

−

and λ2

=

−

−

1

−

. If you let y1 = Aeλ1t and

2

2

y2

= Beλ2t,

then clearly,

y

=

y1 + y2 also is a

solution because

(y )00 + a1(y )0 + a2(y ) = 0.

Next, you need to Þnd the particular integral, yp, which can be

obtained by undetermined coe cients.

Let yp =

β0 + β1t. Then

yp00

= 0, yp0

= β1

and yp00 + a1yp0 + a2yp

= a1β1 + a2β0 + a2β1t = bt.

b

a1b

It follows that β1 =

, and β0

= − a22 .

a2

Since each of these pieces are solutions to (10.31), the sum of the

solutions is also be a solution. Thus the general solution is,

y(t) = Aeλ1t + Beλ2t −

a1b

b

+

t.

(10.32)

a22

a2

10.3. INFINITESIMAL MARGINAL INTERVENTION

315

Solution under Krugman intervention. To solve (10.30), replace y(t) in

(10.32) with G(f), set a1

= 2η2 , a2 =

−22

, and b = a2. The result is

σ

ασ

G[f(t)] = ηα + f(t) + Aeλ1f(t) + Beλ2f(t),

(10.33)

where

−η

+ s

η

2

2

λ1

=

+

> 0,

(10.34)

2

σ

4

ασ

2

σ

η

s

η

2

2

λ2

=

−

+

< 0.

(10.35)

2

σ

4

ασ

2

σ

−

To solve for the constants A and B, you need two additional pieces of information. These are provided by the intervention rules.4 From (10.33), you can see that the function mapping f(t) into s(t) is one-to-one. This means that there is a lower and upper band on the fundamentals, [f, f¯] that corresponds to the lower and upper bands for the exchange rate [s, s¯]. When s(t) hits the upper band s¯, the authorities intervene to prevent s(t) from moving out of the band. Only inÞnitesimally small interventions are required. During instants of intervention, ds = 0 from which it follows that

G0(f¯) = 1 + λ1Aeλ1f¯ + λ2Beλ2f¯ = 0.

(10.36)

Similarly, at the instant that s touches the lower band s, ds = 0 and

G0(f) = 1 + λ1Aeλ1			f	+ λ2Beλ2	f	= 0.	(10.37)

(10.36) and (10.37) are 2 equations in the 2 unknowns A and B, which you can solve to get

eλ2f¯ − eλ2

f

A

=

< 0,

(10.38)

λ1[e(λ1f¯+λ2

+λ2f¯)]

f

) − e(λ1

f

λ1f

λ1f¯

B

=

e

− e

> 0.

(10.39)

λ2[e(λ1f¯+λ2

f

) − e(λ1

f

+λ2f¯)]

4In the case of a pure ßoat and in the absence of bubbles, you know that A = B = 0.

316	CHAPTER 10. TARGET-ZONE MODELS

The signs of A and B follow from noting that λ1 is positive and λ2 is negative so that eλ1(f¯−f) > eλ2(f¯−f). It follows that the square bracketed term in the denominator is positive.

The solution becomes simpler if you make two symmetry assumptions. First, assume that there is no drift in the fundamentals η = 0. Setting the drift to zero implies λ1 = −λ2 = λ > 0. Second, center the admissible region for the fundamentals around zero with f¯ = −f so that B = −A > 0. The solution becomes

G[f(t)] = f(t) + B[e−λf(t) − eλf(t)],								(10.40)
with
	s
λ =			2		,
			ασ	2
B =			eλf¯ − e−λf¯				.
	λ[e2λf¯ − e−2λf¯]

Figure 10.1 shows the relation between the exchange rate and the fundamentals under Krugman-style intervention. The free ßoat solution s(t) = f(t) serves as a reference point and is given by the dotted 45degree line. First, notice that G[f(t)] has the shape of an ‘S.’ The S-curve lies below the s(t) = f(t) line for positive values of f(t) and vice-versa for negative values of f(t). This means that under the targetzone arrangement, the exchange rate varies by a smaller amount in response to a given change in f(t) within [f, f¯] than it would under a free ßoat.

Second, note that by (10.21), we know that E(sú) < 0 when f > 0, and vice-versa. This means that market participants expect the exchange rate to decline when it lies above its central parity and they expect the exchange rate to rise when it lies below the central parity. The exchange rate displays mean reversion. This is potentially the explanation for why exchange rates are less volatile under a managed ßoat than they are under a free ßoat. Since market participants expect the authorities to intervene when the exchange rate heads toward the bands, the expectation of the future intervention dampens current exchange rate movements. This dampening result is called the

Honeymoon e ect.