8.3. A STOCHASTIC MUNDELL—FLEMING MODEL

241

tals even when agents have perfect foresight. The implied dynamics are illustrated in Figure 8.7.

If there were instantaneous adjustment (π = ∞), we would immediately go to the long run and would continuously be in equilibrium. So long as π < ∞, the goods market spends some time in disequilibrium and the economy-wide adjustment to the long-run equilibrium occurs gradually. The transition paths, which we did not solve for explicitly but is treated in the chapter appendix, describe the disequilibrium dynamics. It is in comparison to the ßexible-price (long-run) equilibrium that the transitional values are viewed to be in disequilibrium.

There is no overshooting nor associated excess volatility in response to Þscal policy shocks. You are invited to explore this further in the end-of-chapter problems.

8.3A Stochastic Mundell—Fleming Model

Let’s extend the Mundell-Fleming model to a stochastic environment following Obstfeld [111]. Let ytd be aggregate demand, st be the nominal exchange rate, pt be the domestic price level, it be the domestic nominal interest rate, mt be the nominal money stock, and Et(Xt) be the mathematical expectation of the random variable Xt conditioned on date—t information. All variables except interest rates are in natural logarithms. Foreign variables are taken as given so without loss of generality we set p = 0 and i = 0.

The IS curve in the stochastic Mundell-Fleming model is

ytd = η(st − pt) − σ[it − Et(pt+1 − pt)] + dt,

(8.17)

where dt is an aggregate demand shock and it − Et(pt+1 − pt) is the ex ante real interest rate. The LM curve is

mt − pt = ytd − λit,

(8.18)

where the income elasticity of money demand is assumed to be 1. Capital market equilibrium is given by uncovered interest parity

it − i = Et(st+1 − st).

(8.19)

242	CHAPTER 8. THE MUNDELL-FLEMING MODEL

The long-run or the steady-state is not conveniently characterized in a stochastic environment because the economy is constantly being hit by shocks to the non-stationary exogenous state variables. Instead of a long-run equilibrium, we will work with an equilibrium concept given by the solution formed under hypothetically fully ßexible prices. Then as long as there is some degree of price-level stickiness that prevents complete instantaneous adjustment, the disequilibium can be characterized by the gap between sticky-price solution and the shadow ßexible-price equilibrium.

Let the shadow values associated with the ßexible-price equilibrium be denoted with a ‘tilde.’ The predetermined part of the price level is Et−1p˜t which is a function of time t-1 information. Let θ(˜pt − Et−1p˜t) represent the extent to which the actual price level pt responds at date t to new information where θ is an adjustment coe cient. The stickyprice adjustment rule is

pt = Et−1p˜t + θ(˜pt − Et−1p˜t).

(8.20)

According to this rule, goods prices display rigidity for at most one period. Prices are instantaneously perfectly ßexible if θ = 1 and they are completely Þxed one-period in advance if θ = 0. Intermediate degrees of price Þxity are characterized by 0 < θ < 1 which allow the price level at t to partially adjust from its one-period-in-advance predetermined value Et−1(˜pt) in response to period t news, p˜t −Et−1p˜t.

The exogenous state variables are output, money, and the aggregate demand shock and they are governed by unit root processes. Output and the money supply are driven by the driftless random walks

	yt	=	yt−1 + zt,	(8.21)
	mt	=	mt−1 + vt,	(8.22)
iid	iid
where zt N(0, σz2) and vt N(0, σv2). The demand shock dt also is a
unit-root process
	dt = dt−1 + δt − γδt−1,			(8.23)

iid 2

where δt N(0, σδ ). Demand shocks are permanent, as represented by dt−1 but also display transitory dynamics where some portion 0 < γ < 1

8.3. A STOCHASTIC MUNDELL—FLEMING MODEL

243

of any shock δt is reversed in the next period.7 To solve the model, the Þrst thing you need is to get the shadow ßexible-price solution.

Flexible Price Solution

Under fully-ßexible prices, θ = 1 and the goods market is continuously in equilibrium yt = ytd. Let qt = st − pt be the real exchange rate. Substitute (8.19) into the IS curve (8.17), and re-arrange to get

Ã!

	q˜t =	yt − dt	+	σ	Etq˜t+1.	(8.24)
		η + σ		η + σ

This is a stochastic di erence equation in q˜. It follows that the solution for the ßexible-price equilibrium real exchange rate is given by the present value formula which you can get by iterating forward on (8.24). But we won’t do that here. Instead, we will use the method of undetermined coe cients. We begin by conjecturing a guess solution in which q˜ depends linearly on the available date t information

q˜t = a1yt + a2mt + a3dt + a4δt.

(8.25)

We then deduce conditions on the a−coe cients such that (8.25) solves the model. Since mt does not appear explicitly in (8.24), it probably is the case that a2 = 0. To see if this is correct, take time t conditional expectations on both sides of (8.25) to get

Etq˜t+1 = a1yt + a2mt + a3(dt − γδt).							(8.26)
Substitute (8.25) and (8.26) into (8.24) to get							(139)
a1yt + a2mt + a3dt + a4δt
=	yt − dt	+	σ	[a	y	+ a m + a (d γδ )]
	η + σ		η + σ 1		t	2 t 3 t −	t

7Recursive backward substitution in (8.23) gives, dt = δt + (1 − γ)δt−1 γ)δt−2 + · · · . Thus the demand shock is a quasi-random walk without drift a shock δt has a permanent e ect on dt, but the e ect on future values (1 smaller than the current e ect.

+ (1 − in that − γ) is

244

CHAPTER 8. THE MUNDELL-FLEMING MODEL

Now equate the coe cients on the variables to get

a1

=

1

= −a3,

η

a2

=

0,

Ãη + σ ! .

a4

=

η

γ

σ

The ßexible-price solution for the real exchange rate is

q˜ =

yt − dt

+

γ

σ

δ

,

(8.27)

η

Ãη + σ !

t

η

t

where indeed nominal (monetary) shocks have no e ect on q˜t. The real exchange rate is driven only by real factors—supply and demand shocks.

Since both of these shocks were assumed to evolve according to unit root process, there is a presumption that q˜t also is a unit root process. A permanent shock to supply yt leads to a real depreciation. Since γσ/(η(η + σ)) < (1/η), a permanent shock to demand δt leads to a real appreciation.8

To get the shadow price level, start from (8.18) and (8.19) to get p˜t = mt − yt + λEt(st+1 − st). If you add λp˜t to both sides, add and

subtract λEtp˜t+1 to the right side and rearrange, you get
(1 + λ)˜pt = mt − yt + λEt(˜qt+1 − q˜t) + λEtp˜t+1.	(8.28)

By (8.27), Et(˜qt+1 − q˜t) = [γ/(η + σ)]δt, which you can substitute back into (8.28) to obtain the stochastic di erence equation

p˜t =	mt − yt	+	λγ	δt +	λ		Etp˜t+1.	(8.29)
					1 + λ
	1 + λ (η + σ)(1 + λ)
Now solve (8.29) by the MUC. Let
	p˜t = b1mt + b2yt + b3dt + b4δt,							(8.30)

be the guess solution. Taking expectations conditional on time-t information gives

Etp˜t+1 = b1mt + b2yt + b3(dt − γδt).

(8.31)

8Here is another way to motivate the null hypothesis that the real exchange rate follows a unit root process in tests of long-run PPP covered in Chapter 7.

8.3. A STOCHASTIC MUNDELL—FLEMING MODEL

245

Substitute (8.31) and (8.30) into (8.29) to get

b1mt + b2yt + b3dt + b4δt

=	mt − yt		+	λγ		δt
	1 + λ (1 + λ)(η + σ)
	λ					− γδt)].
+		[b1mt + b2yt + b3(dt
	1 + λ

Equate coe cients on the variables to get

b1	=	1 = −b2,
b3	=	0,
			λγ
b4	=			.
			(1 + λ)(η + σ)

(8.32)

(8.33)

Write the ßexible-price equilibrium solution for the price level as

p˜t = mt − yt + αδt,

(8.34)

where

λγ α = (1 + λ)(η + σ).

A supply shock yt generates shadow deßationary pressure whereas demand shocks δt and money shocks mt generate shadow inßationary pressure.

The shadow nominal exchange rate can now be obtained by adding q˜t + p˜t

t t	Ã	η	!	t − η			Ãη(η + σ)		!	t
s˜ = m +		1 − η	y		dt	+		γσ	+ α	δ .	(8.35)

Positive monetary shocks unambiguously lead to a nominal depreciation but the e ect of a supply shock on the shadow nominal exchange rate depends on the magnitude of the expenditure switching elasticity, η. You are invited to verify that a positive demand shock δt lowers the nominal exchange rate.

246	CHAPTER 8. THE MUNDELL-FLEMING MODEL

Collecting the equations that form the ßexible-price solution we have

yt	= yt−1 + zt = y(zt),
q˜t	=		yt − dt	+	γσ	δt = q˜(zt, δt),
					η(η + σ)
			η
p˜t	=	mt − yt + αδt = p˜(zt, δt, vt).

The system displays a triangular structure in the exogenous shocks. Only supply shocks a ect output, demand and supply shocks a ect the real exchange rate, while supply, demand, and monetary shocks a ect the price level. We will revisit the implications of this triangular structure in Chapter 8.4.

Disequilibrium Dynamics

To obtain the sticky-price solution with 0 < θ < 1, substitute the solution (8.34) for p˜t into the price adjustment rule (8.20), to get pt = mt−1 −yt−1 +θ[vt −zt +αδt]. Next, add and subtract (vt −zt +αδt) to the right side and rearrange to get

pt = p˜t − (1 − θ)[vt − zt + αδt].

(8.36)

The gap between pt and p˜t is proportional to current information (vt − zt + αδt), which we’ll call news. You will see below that the gap between all disequilibrium values and their shadow values are proportional to this news variable. Monetary shocks vt and demand shocks δt cause the price level to lie below its equilibrium value p˜t while supply shocks zt cause the current price level to lie above its equilibrium value.9 Since the solution for pt does not depend on lagged values of the shocks, the deviation from full-price ßexibility values generated by current period shocks last for only one period.

Next, solve for the real exchange rate. Substitute (8.36) and aggregate demand from the IS curve (8.17) into the LM curve (8.18) to

9The price-level responses to the various shocks conform precisely to the predictions from the aggregate-demand, aggregate-supply model as taught in principles of macroeconomics.

8.3. A STOCHASTIC MUNDELL—FLEMING MODEL

247

get

mt−p˜t+(1−θ)[vt−zt+αδt] = dt+ηqt−(σ+λ)(Etqt+1−qt)−λEt(pt+1−pt). (8.37)

By (8.36) and (8.34) you know that

Et(pt+1 − pt) = −αδt + (1 − θ)[vt − zt + αδt].

(8.38)

Substitute (8.38) and p˜t into (8.37) to get the stochastic di erence equation in qt

(η+σ+λ)qt = yt −dt +(1−θ)(1+λ)(vt −zt)−θ(1+λ)αδt +(σ+λ)Etqt+1.

	(8.39)
Let the conjectured solution be
qt = c1yt + c2dt + c3δt + c4vt + c5zt.	(8.40)
It follows that
Etqt+1 = c1yt + c2(dt − γδt).	(8.41)
Substitute (8.40) and (8.41) into (8.39) to get	(140)

(η + σ + λ)[c1yt + c2dt + c3δt + c4vt + c5zt]

= yt − dt + (1 − θ)(1 + λ)(vt − zt)

−θ(1 + λ)αδt + (σ + λ)[c1yt + c2(dt − γδt)].

Equating coe cients gives

c1

=

1

= −c2

,

η

c3

=

γ(σ + λ) − ηαθ(1 + λ)

,

η(η + σ + λ)

c

=

(1 − θ)(1 + λ)

=

c ,

4

η + σ + λ

− 5

and the solution is

(141)

q

=

yt − dt

+

γ(σ + λ) − αηθ(1 + λ)

d

+

(1 − θ)(1 + λ)

(v

z

).

t

η

η(η + σ + λ)

t

η + σ + λ

t −

t

248 CHAPTER 8. THE MUNDELL-FLEMING MODEL

Using the deÞnition of α and (8.27) to eliminate (yt −dt)/η, rewrite the solution in terms of q˜t and news

q	= q˜ +	(1 + λ)(1 − θ)	[v	t −	z	+ αδ ].	(8.42)
t	t	η + σ + λ			t	t

Nominal shocks have an e ect on the real exchange rate due to the rigidity in price adjustment. Disequilibrium adjustment in the real exchange rate runs in the opposite direction of price level adjustment. Monetary shocks and demand shocks cause the real exchange rate to temporarily rise above its equilibrium value whereas supply shocks cause the real exchange rate to temporarily fall below its equilibrium value.

To get the nominal exchange rate st = qt + pt, add the solutions for

qt and pt
s	= s˜ + (1	−	η	−	σ)	(1 − θ)	[v	t −	z	+ αδ ].	(8.43)
t	t					(η + σ + λ)			t	t

The solution displays a modiÞed form of exchange-rate overshooting under the presumption that η + σ < 1 in that a monetary shock causes the exchange rate to rise above its shadow value s˜t. In contrast to the Dornbusch model, both nominal and real shocks generate modiÞed exchange-rate overshooting. Positive demand shocks cause st to rise above s˜t whereas supply shocks cause st to fall below s˜t.

To determine excess goods demand, you know that aggregate de-

mand is

ytd = ηqt − σEt(∆qt+1) + dt.

(142) Taking expectations of (8.42) yields

γ (1 + λ)(1 − θ)

Et(∆qt+1) = η + σ δt − (η + σ + λ) [vt − zt + αδt].

Substitute this and qt from (8.42) back into aggregate demand and rearrange to get

yd = y	+	(1 + λ)(1 − θ)(η + σ)	[v	t −	z	+ αδ ].	(8.44)
t t		(η + σ + λ)			t	t

Goods market disequilibrium is proportional to the news vt − zt + αδt. Monetary shocks have a short-run e ect on aggregate demand, which is the stochastic counterpart to the statement that monetary policy is an e ective stabilization tool under ßexible exchange rates.