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Lecture 8: Exchange-Rate Dynamics Gregory Corcos Eco572: International Economics
19 November 2014
Mathematical Appendix
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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
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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 real models (see next slide). They also seem to be disconnected from macro fundamentals (output, employment, inflation...) Meese and Rogoff (JIE 1983): random walk explains short term 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 and RER volatility without increasing the volatility of fundamentals
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Figure: Nominal DEM/USD exchange rate and ratio of German/US CPIs: in logs (left panel) and in log differences (right panel). Source: Obstfeld and Rogoff (1996).
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Transitory shocks to real variables can explain some part of with ER volatility. Monetary shocks and nominal rigidities add some ER volatility and can explain part of the disconnect. However, even a combination of transitory shocks and price stickiness cannot explain that ERs converge to PPP so slowly (the ’PPP puzzle’ in Rogoff, JEL 1996). An additional explanation is that producers absorb some shocks in setting prices (Goldberg and Knetter JEL 1995). This is rationalized in ’pricing-to-market’ models.
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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: they depend on expectations on future monetary policy Volatility comes from: changes in these expectations possibility of bubbles
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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 UIP e it − it∗ = st − st,t+1
PPP pt = pt∗ − st Exogenous supply: yt = 0 Small open economy, exogenous world price: p ∗ = 0. Rational expectations e st,t+1 = Et st+1
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Solving for st : ∀t, st =
mt + βit∗ β Et st+1 − 1+β 1+β | {z }
(1)
zt
Rational expectations imply: Et st+1 =
β β Et Et+1 st+2 + Et zt+1 = Et st+2 + Et zt+1 1+β 1+β
so that, by iterating, " # l k ∞ X β β st = lim Et st+l + Et zt+k l→∞ 1+β 1+β k=0
l β Transversality condition: liml→∞ 1+β Et st+l = 0. Exchange rates depend on the expected future path of zt . k ∞ X β st = Et zt+k 1+β
k=0
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Implications of the Monetarist Model
Changes in expectations over all future monetary policy changes create ER volatility. Central Banks announcements will affect ERs: to let the ER depreciate, a CB should announce permanently higher money supply growth or lower interest rates. to reverse an ER depreciation, a CB should announce a permanent reduction in money supply growth or a permanent rise in future interest rates
In a two-country framework, dissimilar monetary policies would lead to exchange rate instability and there would be an argument for monetary policy coordination.
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Bubble Solutions In addition to the fundamental solution P∞ β k f st ≡ k=0 1+β Et zt+k there is a bubble solution. Define bt = of (1) since
β 1+β Et bt+1 .
stf + bt =
Then stb ≡ stf + bt is also a solution
β β f Et st+1 + zt + Et bt+1 1+β 1+β
The bubble solution is explosive. Iterating as before yields: " # l k ∞ X β β st = lim Et bt+l + Et zt+k l→∞ 1+β 1+β k=0
The transversality condition is met only if bubbles are expected to burst with some probability at each point in time.
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Figure: EUR/USD nominal exchange rate, 1980-2010. Source: Banque de France, Reuters
Some sharp changes may be interpreted as bubble bursts.
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The Dornbusch (JPE 1976) Overshooting Model
As in the monetarist model, asset and FX markets clear instantaneously, and the long-term ER is driven by PPP. However prices are sticky in the short-run. Main result: ERs may ’overshoot’, ie have magnified short-term responses to shocks. Intuition: suppose there is a permanent rise in the money supply since prices are sticky and output is exogenous, interest rates fall immediately due to UIP the spot ER depreciates immediately as prices increase, interest rates rise and the ER appreciates at steady-state, the ER reaches a lower value according to PPP
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All variables in logs and functions of time, t, with x˙ ≡
Mathematical Appendix
dx(t) . dt
Money market m − p = φy − βi, φ > 0, β > 0 UIP and perfect foresight (s e (t + 1|t) = s(t + 1)) i = i ∗ − s˙ Demand y d = 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 y d − y¯ : p˙ = λ(y d − y¯ ) = −λσ(p + s) with λ > 0
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Short-run equilibrium p = m + βi = m + β(i ∗ − s˙ ) p˙ s = −p − λσ Long-run steady-state equilibrium: s˙ = p˙ = 0. Denoting by ¯s and p¯ steady-state values: p¯ = m + βi ∗ ¯s = −¯ p = −(m + βi ∗ )
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During the transition p − p¯ = −β s˙ s − ¯s = −(p − p¯) −
p˙ λσ
⇔
1 s˙ = − (p − p¯) β
⇔
p˙ = −λσ [(s − ¯s ) + (p − p¯)]
or in matrix form 0 − β1 s˙ s − ¯s = p˙ p − p¯ −λσ −λσ Differential equation of the form X˙ = AX with det(A) = − λσ β < 0 and Tr (A) = −λσ < 0. Eigenvalues have opposite signs, saddle-path: only one path leads to the steady state (see mathematical appendix).
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1 s˙ = 0 ⇒ − (p − p¯) = 0 β p˙ = 0 ⇒ −λσ (s − ¯s + p − p¯) = 0
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Figure: Impulse responses of prices, ER and interest rates to a permanent and unexpected rise in the money supply.
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Dornbusch Model: Empirical Evidence
The Dornbusch model rationalizes ER overshooting which can explain ER instability. However two empirical results are at odds with the model: domestic currency depreciates on impact, but then keeps depreciating overshooting delayed by several months (Eichenbaum and Evans QJE 1995)
Evidence consistent with a modified overshooting model (Gourinchas and Tornell JIE 2004) the persistence of monetary shocks is ex ante unknown agents learn about interest rate shock persistence and update their priors when agents underestimate persistence, overshooting is delayed
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Real Exchange Rate Fluctuations and the Trade Balance How do RER fluctuations affect the trade balance? A RER depreciation increases net exports quantity, but reduces the relative price of exports. To determine which effect dominates we must discuss export and import price elasticities. Trade balance in domestic currency: B = PX X − PM M Total differentiation dB = XdPX + PX dX − PM dM − MdPM dB dPX dX PM M dPM dM = + − + pX X PX X PX X PM M
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No nontradables, isoelastic demand: −σX X X = SP , PX = P P∗ −σM ∗ , PM = PS M = PPM Then
dX X
= −σX dRER RER and
dM M
= σM dRER RER with RER =
SP P∗ .
Starting from balanced trade, we can rewrite dP ∗ dS dB dP = −(σX + σM − 1) − ∗ + pX X P P S A RER depreciation raises the trade balance if and only if σX + σM > 1
(Marshall-Lerner)
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No nontradables, isoelastic demand, all home and foreign goods used as inputs in Cobb-Douglas production function: −σX ∗ γ X X = SP , PX = P 1−γX PS X ∗ P −σM ∗ 1−γM , PM = P γM PS M = PPM Starting from balanced trade, similar calculations yield dB = − ((σX − 1)(1 − γX ) + (σM − 1)(1 − γM ) − 1) pX X
dP dP ∗ dS − ∗ + P P S
A RER depreciation raises the trade balance if and only if (σX −1)(1−γX )+(σM −1)(1−γM ) > 1 (Marshall-Lerner-Robinson)
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The J-Curve Price elasticities must be high enough for quantity effects to dominate price effects. Evidence suggests that quantity effects dominate, but only in the long-term. J-curve pattern: a RER depreciation causes first a fall in the CA balance, then a sharp rise
Figure: US REER depreciation and CA variation, 1985-1990
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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 markets with increasing elasticities, a cost decrease raises the markup
price rigidities in importer currency some costs (distribution) paid in importer currency intrafirm trade endogenous quality differences across countries
In all cases, exchange rate movements are not fully passed on to the final consumer (incomplete pass-through). Possible explanation for the relative failure of PPP.
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Conclusions ERs are very volatile and disconnected with macro fundamentals with slow convergence to PPP. Exchange rate volatility can be explained by: changes in expectations on future monetary policy overshooting due to sticky prices
CA are affected by RERs through price and quantity effects. the latter dominates the former under the Marshall-Lerner-Robinson condition empirically quantity effects are lagged (J-curve) lags may come from foreign consumers’ search, long-term quantity contracts
The slow convergence to PPP can be partly explained by pricing-to-market strategies.
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Mathematical Appendix
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Basic Linear Algebra
a b Consider matrix A = and assume c d 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 a unique solution y = 0 (when det(B) 6= 0) or infinitely many solutions (when det(B) = 0). Therefore eigenvalues r are such that det(A − rI ) = 0 which is equivalent to the characteristic equation: (a − r )(d − r ) − bc = 0 ⇔ r 2 − Tr (A)r + det(A) = 0 √ Tr (A)± Tr (A)2 −4det(A) which has roots r = if det(A) < 0. 2
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Linear Homogenous First-Order Differential Equations A system of two linear homogenous first-order differential equations 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) = e rt k, where k is a vector of coefficients, then Z˙ (t) = re rt k which implies e rt Ak = re rt k ⇔ Ak = rk r is an eigenvalue and k is an eigenvector of A.
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det(A) < 0 implies Tr (A)2 − det(A) > 0, therefore there are two real distinct roots (eigenvalues). Theorem: if A has 2 linearly independent eigenvectors k1 , k2 with eigenvalues r1 , r2 , then the solution has the form Z (t) = c1 e r1 t k1 + c2 e r2 t k2 where c1 and c2 are arbitrary scalars. The sign of the eigenvalues is crucial to determine the stability of the solution. det(A) < 0 implies r1 < 0, r2 > 0. All trajectories with c2 6= 0 will diverge, but there is one trajectory (saddle path, c2 = 0) that converges.
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Figure: Only one trajectory, along the negative root’s eigenvector, leads to the critical point. All other trajectories lead to the positive root’s eigenvector, away from the critical point.
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Discrete Time Dynamics Consider a similar system in discrete time: Zt+1 − Zt = AZt Solutions takes the general form: Zt = (I + A)t Z0 If det(A) < 0 then as earlier then A has 2 real eigenvalues r1 < 0 and r2 > 0. Denote by k1 and k2 the associated eigenvectors. Then Z0 = k1 ⇒ Zt = (1 + r1 )t k1 Z0 = k2 ⇒ Zt = (1 + r2 )t k2 The first path converges to zero while the second diverges. The solution has the form Zt = c1 (1 + r1 )t k1 + c2 (1 + r2 )t k2 .
