Lecture 8: Exchange-Rate Dynamics - Gregory...

Monetarist Model Overshooting J-curve Mathematical Appendix

Lecture 8: Exchange-Rate Dynamics

Gregory Corcos

Eco572: International Economics

19 November 2014

mailto:[email protected]

http://gregory.corcos.free.fr/epp.html


Lecture 8: Outline

1 A Monetarist Model of Exchange Rates

2 The Dornbusch overshooting model

3 Real Exchange Rate Fluctuations and the Trade Balance

4 Mathematical Appendix

2/32


Introduction: ’Puzzling’ Exchange-Rate Volatility

So far we have studied real models of the ER only.

Nominal ERs are too volatile to be explained by these realmodels (see next slide).

They also seem to be disconnected from macro fundamentals(output, employment, inflation...)

Meese and Rogoff (JIE 1983): random walk explains shortterm ER just as well as macro models, even with ex post data!Baxter and Stockman (JME 1989), Flood & Rose (JME 1995):transitions from fixed to floating ER regime increased NER andRER volatility without increasing the volatility of fundamentals

3/32


Figure: Nominal DEM/USD exchange rate and ratio of German/USCPIs: in logs (left panel) and in log differences (right panel).Source: Obstfeld and Rogoff (1996).

4/32


Transitory shocks to real variables can explain some part ofwith ER volatility.

Monetary shocks and nominal rigidities add some ER volatilityand can explain part of the disconnect.

However, even a combination of transitory shocks and pricestickiness cannot explain that ERs converge to PPP so slowly(the ’PPP puzzle’ in Rogoff, JEL 1996).

An additional explanation is that producers absorb someshocks in setting prices (Goldberg and Knetter JEL 1995).

This is rationalized in ’pricing-to-market’ models.

5/32


The Monetarist Model: Introduction

Model due to Frenkel (SJE 1976), Mussa (SJE 1976).

Small open economy with flexible prices.

Both PPP and UIP hold.

Rational expectations imply that ER are like asset prices: theydepend on expectations on future monetary policy

Volatility comes from:

changes in these expectationspossibility of bubbles

6/32


The Monetarist Model

All variables are in logs. An increase in st represents an appreciation.

Money market

mt − pt = φyt − βit , φ > 0, β > 0

UIPit − i∗t = st − set,t+1

PPPpt = p∗t − st

Exogenous supply: yt = 0

Small open economy, exogenous world price: p∗ = 0.

Rational expectations

set,t+1 = Etst+1

7/32


Solving for st :

∀t, st =β

1 + βEtst+1−

mt + βi∗t1 + β︸︷︷︸zt

(1)

Rational expectations imply:

Etst+1 =β

1 + βEtEt+1st+2 + Etzt+1 =

β

1 + βEtst+2 + Etzt+1

so that, by iterating,

st = liml→∞

[(β

1 + β

)l

Etst+l

]+∞∑k=0

(β

1 + β

)k

Etzt+k

Transversality condition: liml→∞

(β

1+β

)lEtst+l = 0.

Exchange rates depend on the expected future path of zt .

st =∞∑k=0

(β

1 + β

)k

Etzt+k

8/32


Implications of the Monetarist Model

Changes in expectations over all future monetary policychanges create ER volatility.

Central Banks announcements will affect ERs:

to let the ER depreciate, a CB should announce permanentlyhigher money supply growth or lower interest rates.to reverse an ER depreciation, a CB should announce apermanent reduction in money supply growth or a permanentrise in future interest rates

In a two-country framework, dissimilar monetary policieswould lead to exchange rate instability and there would be anargument for monetary policy coordination.

9/32


Bubble Solutions

In addition to the fundamental solution

s ft ≡∑∞

k=0

(β

1+β

)kEtzt+k there is a bubble solution.

Define bt = β1+βEtbt+1. Then sbt ≡ s ft + bt is also a solution

of (1) since

s ft + bt =β

1 + βEts

ft+1 + zt +

β

1 + βEtbt+1

The bubble solution is explosive. Iterating as before yields:

st = liml→∞

[(β

1 + β

)l

Etbt+l

]+∞∑k=0

(β

1 + β

)k

Etzt+k

The transversality condition is met only if bubbles areexpected to burst with some probability at each point in time.

10/32


Figure: EUR/USD nominal exchange rate, 1980-2010. Source: Banquede France, Reuters

Some sharp changes may be interpreted as bubble bursts.

11/32


The Dornbusch (JPE 1976) Overshooting Model

As in the monetarist model, asset and FX markets clearinstantaneously, and the long-term ER is driven by PPP.

However prices are sticky in the short-run.

Main result: ERs may ’overshoot’, ie have magnifiedshort-term responses to shocks.

Intuition:suppose there is a permanent rise in the money supplysince prices are sticky and output is exogenous, interest ratesfall immediatelydue to UIP the spot ER depreciates immediatelyas prices increase, interest rates rise and the ER appreciatesat steady-state, the ER reaches a lower value according to PPP

12/32


All variables in logs and functions of time, t, with x ≡ dx(t)dt

.

Money market

m − p = φy − βi , φ > 0, β > 0

UIP and perfect foresight (se(t + 1|t) = s(t + 1))

i = i∗ − s

Demandyd = y − σ(p + s − p∗) with

Small open economy, exogenous world price: p∗ = 0.

Exogenous supply: y = y = 0

Sticky prices vary in proportion to excess demand yd − y :

p = λ(yd − y) = −λσ(p + s) with λ > 0

13/32


Short-run equilibrium

p = m + βi = m + β(i∗ − s)

s = −p − p

λσ

Long-run steady-state equilibrium: s = p = 0. Denoting by sand p steady-state values:

p = m + βi∗

s = −p = −(m + βi∗)

14/32


During the transition

p − p = −βs ⇔ s = − 1

β(p − p)

s − s = −(p − p)− p

λσ⇔ p = −λσ [(s − s) + (p − p)]

or in matrix form(sp

)=

(0 − 1

β

−λσ −λσ

)(s − sp − p

)Differential equation of the form X = AX withdet(A) = −λσ

β < 0 and Tr(A) = −λσ < 0.

Eigenvalues have opposite signs, saddle-path: only one pathleads to the steady state (see mathematical appendix).

15/32


s = 0⇒ − 1

β(p − p) = 0

p = 0⇒ −λσ (s − s + p − p) = 0

p

s

p=plt

p=plt+slt-‐s

p>plt èds<0

p>plt+slt-‐s èdp<0

slt

plt=m+βi*

16/32


Impact of a Permanent rise in Money Supply

Short-run (on impact):

rise in real money supply (since prices are fixed), fall indomestic interest rateER falls below its value before the shock (depreciation)

Long-run:

prices increase, real money supply fallsdomestic interest rate rises, interest rate differential narrowsand currency appreciates.

17/32


Denote p′ = m′ + βi∗ and s ′ = −p′.

s = 0⇒ − 1

β(p − p′) = 0

p = 0⇒ −λσ(s − s ′ − (p − p′)

)= 0

p

s

p=plt

p=plt’+slt’-‐s

p>plt’èds<0

p>plt’+slt’-‐sèdp<0

slt

plt=m+βi*

p=plt’ plt’=m’+βi*

dm>0 E0 E’0

E1

slt’

overshoo4ng

18/32


i

s

p

"me

ilt=i*

slt’=-‐plt-‐dm

plt’=m+dm+βi*

overshoo"ng

Figure: Impulse responses of prices, ER and interest rates to a permanentand unexpected rise in the money supply.

19/32


Dornbusch Model: Empirical Evidence

The Dornbusch model rationalizes ER overshooting which canexplain ER instability.

However two empirical results are at odds with the model:

domestic currency depreciates on impact, but then keepsdepreciatingovershooting delayed by several months (Eichenbaum andEvans QJE 1995)

Evidence consistent with a modified overshooting model(Gourinchas and Tornell JIE 2004)

the persistence of monetary shocks is ex ante unknownagents learn about interest rate shock persistence and updatetheir priorswhen agents underestimate persistence, overshooting is delayed

20/32


Real Exchange Rate Fluctuations and the Trade Balance

How do RER fluctuations affect the trade balance?

A RER depreciation increases net exports quantity, butreduces the relative price of exports.

To determine which effect dominates we must discuss exportand import price elasticities.

Trade balance in domestic currency:

B = PXX − PMM

Total differentiation

dB = XdPX + PXdX − PMdM −MdPM

dB

pXX=

dPX

PX+

dX

X− PMM

PXX

(dPM

PM+

dM

M

)21/32


No nontradables, isoelastic demand:

X =(SPXP∗

)−σX, PX = P

M =(PMP

)−σM, PM = P∗

S

Then dXX = −σX dRER

RER and dMM = σM

dRERRER with RER = SP

P∗ .

Starting from balanced trade, we can rewrite

dB

pXX= −(σX + σM − 1)

(dP

P− dP∗

P∗+

dS

S

)A RER depreciation raises the trade balance if and only if

σX + σM > 1 (Marshall-Lerner)

22/32


No nontradables, isoelastic demand, all home and foreigngoods used as inputs in Cobb-Douglas production function:

X =(SPXP∗

)−σX, PX = P1−γX

(P∗

S

)γXM =

(PMP

)−σM, PM = PγM

(P∗

S

)1−γMStarting from balanced trade, similar calculations yield

dB

pXX= − ((σX − 1)(1 − γX ) + (σM − 1)(1 − γM) − 1)

(dP

P− dP∗

P∗ +dS

S

)A RER depreciation raises the trade balance if and only if

(σX−1)(1−γX )+(σM−1)(1−γM) > 1 (Marshall-Lerner-Robinson)

23/32


The J-Curve

Price elasticities must be high enough for quantity effects todominate price effects.Evidence suggests that quantity effects dominate, but only inthe long-term.J-curve pattern: a RER depreciation causes first a fall in theCA balance, then a sharp rise

Figure: US REER depreciation and CA variation, 1985-1990

24/32


Pricing-to-Market

Pricing to market: setting a different price on each market

Why would exporters price to market?variable price elasticities

optimal prices and markups will differ across marketswith increasing elasticities, a cost decrease raises the markup

price rigidities in importer currencysome costs (distribution) paid in importer currencyintrafirm tradeendogenous quality differences across countries

In all cases, exchange rate movements are not fully passed onto the final consumer (incomplete pass-through).

Possible explanation for the relative failure of PPP.

25/32


Conclusions

ERs are very volatile and disconnected with macrofundamentals with slow convergence to PPP.

Exchange rate volatility can be explained by:

changes in expectations on future monetary policyovershooting due to sticky prices

CA are affected by RERs through price and quantity effects.

the latter dominates the former under theMarshall-Lerner-Robinson conditionempirically quantity effects are lagged (J-curve)lags may come from foreign consumers’ search, long-termquantity contracts

The slow convergence to PPP can be partly explained bypricing-to-market strategies.

26/32


Mathematical Appendix

27/32


Basic Linear Algebra

Consider matrix A =

(a bc d

)and assume

det(A) ≡ ad − bc 6= 0 and Tr(A) ≡ a + d 6= 0.

Matrix A is invertible since det(A) 6= 0.

By definition eigenvalues r and eigenvectors x 6= 0 satisfy

Ax = rx ⇔ (A− rI )x = 0

Theorem: a linear homogenous system By = 0 has either aunique solution y = 0 (when det(B) 6= 0) or infinitely manysolutions (when det(B) = 0).

Therefore eigenvalues r are such that det(A− rI ) = 0 which isequivalent to the characteristic equation:

(a− r)(d − r)− bc = 0⇔ r2 − Tr(A)r + det(A) = 0

which has roots r =Tr(A)±

√Tr(A)2−4det(A)

2 if det(A) < 0.28/32


Linear Homogenous First-Order Differential Equations

A system of two linear homogenous first-order differentialequations is defined as:

x(t) = ax(t) + by(t)

y(t) = cx(t) + dy(t)

or in matrix form Z (t) = AZ (t).

Note that if Z (t) = ertk , where k is a vector of coefficients,then Z (t) = rertk which implies

ertAk = rertk ⇔ Ak = rk

r is an eigenvalue and k is an eigenvector of A.

29/32


det(A) < 0 implies Tr(A)2 − det(A) > 0, therefore there aretwo real distinct roots (eigenvalues).

Theorem: if A has 2 linearly independent eigenvectors k1, k2with eigenvalues r1, r2, then the solution has the form

Z (t) = c1er1tk1 + c2e

r2tk2

where c1 and c2 are arbitrary scalars.

The sign of the eigenvalues is crucial to determine thestability of the solution.

det(A) < 0 implies r1 < 0, r2 > 0. All trajectories with c2 6= 0will diverge, but there is one trajectory (saddle path, c2 = 0)that converges.

30/32


Figure: Only one trajectory, along the negative root’s eigenvector, leadsto the critical point. All other trajectories lead to the positive root’seigenvector, away from the critical point.

31/32


Discrete Time Dynamics

Consider a similar system in discrete time:

Zt+1 − Zt = AZt

Solutions takes the general form:

Zt = (I + A)tZ0

If det(A) < 0 then as earlier then A has 2 real eigenvaluesr1 < 0 and r2 > 0.

Denote by k1 and k2 the associated eigenvectors. Then

Z0 = k1 ⇒ Zt = (1 + r1)tk1

Z0 = k2 ⇒ Zt = (1 + r2)tk2

The first path converges to zero while the second diverges.

The solution has the form Zt = c1(1 + r1)tk1 + c2(1 + r2)tk2.32/32

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