Lecture 9: The monetary theory of the exchange rate

Transcription

1 Lecture 9: The monetary theory of the exchange rate Open Economy Macroeconomics, Fall 2006 Ida Wolden Bache October 24, 2006

2 Macroeconomic models of exchange rate determination Useful reference: Chapter 4 of Sarno&Taylor (2002) Some in uential classes of models Portfolio balance models The monetary approach Flexible-price model Sticky-price model New Open Economy Macroeconomics

3 Portfolio balance models Useful reference: Branson&Henderson (985) "The speci cation and in uence of asset markets" in Handbook of International Economics, vol 2. Imperfect substitutability between di erent assets Wealth enters asset demand equations

4 The monetary approach Perfect substitutability between foreign and domestic assets (UIP) Exogenous money supply Flexible price models: Prices respond instantly to shocks Output is at its natural level at all times Purchasing power parity (PPP) holds continuously (P = SP )

5 Sticky price models (e.g., Dornbush (976) overshooting model): Prices respond sluggishly to shocks Output deviates from natural level in the short run Short-run deviations from PPP Money neutral in the long run

6 New open economy macroeconomics (NOEM) Seminal contribution: Obstfeld&Rogo (995) "Exchange Rate Economics Redux", Journal of Political Economy vol 03 Useful references: chapter 0 in Obstfeld&Rogo (996), chapter 5 in Sarno&Taylor (2002) Key features: Dynamic general equilibrium model Explicit microfoundations (intertemporally optimising agents) Imperfect competition and nominal rigidities (sticky prices and/or sticky wages) Changes in the money supply a ect real variables in the short run

7 The monetary theory of the exchange rate Required readings: Chapter 4 in Rødseth (2000) and chapter in Obstfeld&Rogo (996) Preliminaries on rst-order linear di erential equations (see e.g., ch in Sydsæter et al. (2002) Matematisk Analyse Bind 2). De ne _x(t) = dx=dt Consider the equation _x(t) + ax(t) = b(t) For a given initial solution x(t 0 ); the general solution is x(t) = x(t 0 ) exp( a(t t 0 )) + Z t t 0 b() exp( a(t ))d

8 Assumptions International goods markets are perfectly integrated PPP holds continuously Perfect capital mobility UIP Rational expectations (perfect foresight) Flexible prices and wages output is always at its natural level Money supply is exogenous Focus on log-linearised model: lower case letters denote variables in natural logs (x = ln X and _x = _X=X)

9 Model equations p(t) = s(t) + p (t) _s(t) = i(t) i (t) m(t) p(t) = i(t) + y(t); where p is the domestic price level, s is the nominal exchange rate, p is the foreign price level, m is the money supply, i is the nominal interest rate, i is the foreign nominal interest rate and y is the domestic output level Endogenous variables: s; p; i Exogenous variables: p ; i ; m; y

10 Deriving the di erential equation for the exchange rate. Solve out for the domestic interest rate from the money market equilibrium condition i(t) = (m(t) p(t) y(t)) 2. Substitute in for the domestic price level from the PPP condition i(t) = (m(t) s(t) p (t) y(t)) 3. Substitute into the UIP condition _s(t) = (m(t) s(t) p (t) y(t)) i (t) _s(t) = s(t) (m(t) p (t) y(t)) i (t) {z } z(t) _s(t) = s(t) z(t)

11 The di erential equation for the exchange rate is fundamentally unstable: the higher is the level of the exchange rate, the higher is the rate of depreciation Intuition: s " p " (from PPP) m p # i " (from the money market equilibrium condition) _s " (from UIP)

12 The exchange rate is free to jump to any level without an initial condition there is an in nite number of solutions for the exchange rate The initial exchange rate is determined by the requirement that the exchange rate should tend to a strictly positive value when t goes to in nity Solution in the special case where the exogenous variables are expected to remain constant: exchange rate jumps immediately to the stationary value; i.e, the value corresponding to _s(t) = 0 _s(t) = 0 =) i = i =) s = m p y + i

13 General solution s(t) = s(t 0 ) exp (t t 0) = s(t 0 ) exp (t t 0) Z t Z t z() exp (t ) t 0 z() exp t 0 (t ) (t t 0) + (t t 0) = s(t 0 ) exp (t t 0) = " exp (t t 0) s(t 0 ) Z t Z t t 0 z() exp d d z() exp t 0 (t ) (t t 0) ( t 0) d # exp (t t 0) d

14 For the exchange rate not to explode the term in brackets must go to zero as t goes to in nity, i.e., Z s(t 0 ) = z() exp ( t 0) d t 0 Recall that which implies Z b a f(x)dx Z b Z a f(x)dx = f(x)dx a b Z c Z b f(x)dx = f(x)dx + a a Z b Z c Z a f(x)dx = a c f(x)dx + Z b c f(x)dx a f(x)dx = Z b c f(x)dx

15 Insert into the general solution s(t) = = = = = 2 4 " Z t Z t Z t Z t R t0 z() exp ( t 0 ) d R t t0 z() exp ( t 0 ) d z() exp z() exp z() exp # 3 5 exp (t t 0) ( t 0) d exp (t t 0) ( t 0) + (t t 0) d ( t) d " (m() p () y()) + i () # exp ( t) d The equilibrium exchange rate is the discounted sum of the whole (expected) future path of the exogenous variables where the discount rate is the inverse of the interest elasticity of money demand.

16 Quali cations For the integral to converge z must grow at an absolute value smaller than = as t The above solution is the fundamental solution, denoted s (t). All solutions are related to s (t) by s(t) = s (t) + c exp t ; where c is a constant and there is one such solution for every real number c. The last term is a rational bubble and is unrelated to the underlying exogenous variables.

17 When the exogenous variables are expected to remain constant the solution simpli es to s(t) = = = " (m p y) + i " (m p y) + i " # (m p y) + i = m p y + i # Z t # " 8 >< >: exp exp exp ( t) ( t) ( t) d # t {z } 0 +exp (t t) {z } 9 >= >;

18 An unanticipated temporary increase in z Assume that z(t) = ( z0 + z if t 0 < t < t z 0 if t > t where z 0 and z are positive constants Then in period t 0 < t < t s(t) = = = Z t Z t t Z z() exp ( t) d (z 0 + z) exp ( t) z 0 exp ( t) d + t {z } s 0 d + Z t t Z t z exp z 0 exp ( t) ( t) d d

19 Recall that Z exp(ax)dx = a exp(ax) + C which implies s(t) = s 0 + z " 0 B exp ( t) = s 0 + z B (t t) = s 0 + exp # t (t t) z t (t t) C A {z } + exp If z = m s(t) = s 0 + exp (t t) m

20 E ect of a permanent change is found by letting t s(t) = s 0 + m

21 Empirical evidence Weak empirical evidence for the monetary model as a model of short-run exchange rate movements Empirical tests tend to reject both the joint hypothesis of UIP and rational expectations; and the assumption of continuous PPP (see graph) Some evidence supporting the model in periods when in ation is high (e.g., Frenkel s (976) study of the German hyperin ation in the 920s) as a model of long-run exchange rate determination

22 0, 0,08 0,06 Nominal Broad Dollar Index Price adjusted Broad Dollar Index 0,04 0,02 0 0,02 0,04 0,06 0,08 0, may.7 nov.75 may.7 nov.78 may.8 nov.8 may.8 nov.84 may.8 nov.87 may.8 nov.90 may.9 nov.93 may.9 nov.96 may.9 nov.99 may.0 nov.02 may.0 nov.05 Figure : US nominal and real e ective (trade-weighted) exchange rate (log di erences)