Dornbusch or Frankel ? : the Same Model

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(1)DORNBUSCH OR FRANKEL ? : THE SAME MODEL by Gilles Desvilles 1. I) I N T R O D U C T I O N It is generally acknowledged that the Dornbusch2 sticky price monetarist model — also called overshooting model — represents a major contribution to exchange rate theory in the sense that departures from purchasing power parity and volatile currency levels can be explained rationally. C o n v e r s e l y , i t i s a l s o a d v o c a t e d t h a t t h e m o d e l e x h i b i t s s e v e r a l s h o r t co mings.. H. Bourguinat3 quotes that a weighted average of domestic and. import prices would have been a more appropriate deflator but would have weakened Dornbusch’s results.. He also points out that domestic agents. don’t hold external money balances, which limits the scope of the model. He asserts that agents are more and more able to foresee monetary shocks and adjust accordingly their understanding of the long run monetary path. But the most frequent critique is that Dornbusch’s model doens’t properly handle inflation.. In a classic article, J. Frankel 4 combines sticky prices with. secular rates of inflation, and finds that the exchange rate between two countries is driven by their real interest rates differential, and not by their nominal one as in Dornbusch. T h e p u r p o s e o f t h i s a r t i c l e i s t o s h o w t h a t D o r n b u s c h ’ s a n d F r a n k e l ’ s mo dels are in fact identical.. Consequently, they bring the same results and. 1 Maître de Conférences at CNAM, and Member of the Centre de Recherches sur la Gestion (Cer e g ) , P a r i s - D a u p h i n e U n i v e r s it y . 2 ‘‘Expectations and Exchange Rate Dynamics’’, Rudiger Dornbusch, Journal of Political Economy, Volume 84, n°6, 1976. 3 ‘‘Finance internationale’’, Henri Bourguinat, Thémis-PUF, May 1992. See also ‘‘International Finance’’, Keith Pilbeam, Macmillan, 1992, for a discussion about the Dornbusch model..

(2) bear the same drawbacks.. Part. I I s e t s u p t h e o v e r s h o o t i n g m o d e l i n a se -. cular inflationary dual framework by relaxing the two original assumptions of constant prices and small country.. Part. III shows that the exchange. rate disequilibrium depends on the same real interest rate differential as with Frankel.. Part. IV points out the main theoretical pitfall met by both. models.. II) T H E M O D E L The model we develop here draws much on the sticky price model.. But. unlike Dornbusch, we don’t assume that the domestic country is small, so t h a t t h e r e s t o f t h e w o r l d , r e p r e s e n t e d a s a f o r e i g n c o u n t r y , f a c e s c o m pa rable money and goods markets.. Furthermore, we will not suppose that the. exogeneous variables are constant.. However investors will still be indif-. f e r e n t b e t w e e n h o m e a n d a b r o a d i n t h e i r m o n e y l e n d i n g o r b o r r o w i n g be c a u s e t h e y a r e s u p p o s e d n e u t r a l w i t h r e s p e c t t o t h e r i s k o f c u r r e n c y d e p r eciation.. II.1. Uncovered Interest Parity. Capital is perfectly mobile so that investors can instantly alter the co m position of their international investments.. Besides they regard domestic. and foreign interest bearing assets, that we will name bonds, as equally risky.. Because in our simple framework there is one horizon length, one natio-. nal rate and one national issuer, equal riskiness boils down to investors’ risk neutrality with respect to exchange rate volatility.. Hence we can say that. domestic and foreign bonds are perfect substitutes.. This yields the follo-. wing relationship: r = r* + E ( x ). (1). where r and r* are the domestic and foreign nominal interest rates, and E(x) is the expected depreciation rate of the domestic currency. I f e i s t h e l o g o f t h e e x c h a n g e r a t e o f t h e d o m e s t i c c u r r e n c y , i . e . t h e do m e s t i c p r i c e o f o n e u n i t o f f o r e i g n c u r r e n c y , E ( x ) c a n b e ex p r e s s e d a s. 4 ‘‘On the Mark: A Theory of Floating Exchange Rates Based on Real Interest Differentials’’, Jeffrey A. Frankel, The American Economic Review, September 1979.. 2.

(3) E(x ) = E(. de ) = E(e) dt. provided that the interest rates are instantaneous. (1) is known as the Uncovered Interest Parity (UIP), and derives its quali fier uncovered from the fact that risk neutral investors don’t hedge — cover — their foreign belongings. UIP holds continuously in the model.. II.2. Purchasing. Power. Parity. Departure. and. Expecta-. tions Formation Contrary to UIP, the Purchasing Power Parity (PPP) holds only in the long run.. It doesn’t in the short because, for example after a monetary shock,. prices and wages adjust much slower than the exchange rate. Hence we first need to posit (absolute) PPP in the long run: e = p – p*. (PPP). where e, p and p* are the logs of the equilibrium spot rate, price at home and price abroad.. Note that e, p and p* are not held constant as in Dorn∗. b u s c h , a n d a r e i m p l i c i t l y i n d e x e d w i t h t i m e , i . e . e t , p t a n d pt . Differentiating (PPP) provides: x = Π – Π* (PPP’) w h e r e x = e& , Π = p& , a n d Π * = p& * .. This means that the currency depreci a-. tion is equal to the inflation differential. Although x, Π and Π* need not be constant, it may be useful to look at the stationary case where e, p and p* evolve at constant growth rates to better understand how they can be non constant in the long run. Now suppose that for some reason PPP is violated.. We can measure the. d ep a r t u r e f r o m P P P a s : e – p + p* which is frequently called the PPP gap or real exchange rate or else terms of trade, and that can be expressed with the variables’ rates of increase:. 3.

(4) x – Π + Π* W e p o s t u l a t e t h a t i n t h e l o n g r u n t h e e c o n o m y i s s t a b l e a n d P P P g a p va nishes through time under price pressure from imported goods. We then relate the short term track of the exchange rate to the anticipated closure of PPP gap: E(x – Π + Π*) = θ(e – e). (2). The larger the departure from PPP, the wider the spread between spot and equilibrium rates — which we name EER gap in this paper 5 — .. θ must. be positive for stability concern, for example home currency must be cheap t o o f f s e t a n e x c e s s d o m e s t i c i n f l a t i o n 6. Because in the long-run agents perfectly understand the economy, E(Π) = Π and E(Π*) = Π*, and (2) becomes E(x ) = Π – Π* + θ(e – e) We can read this relationship in terms of expectations formation.. (2’) Agents. expect the home currency depreciation to fit a long term trend plus any corrective path due to some disturbance.. The long term trend obtains. when e = e and goes with the perfect foresight E(x) = x: E(x) = x = Π – Π*. (2’’). (2’’) is simply the long term (PPP). Note that in the long range we can write (UIP) as: r = r* + x. (1). w h i c h c o m b i n e d w i t h (2 ’ ’ ) p r o v i d e s t h e r e a l i n t e r e s t r a t e p a r i t y ( R I P ) i n the long-run: r – Π = r* – Π*. (RIP). Combining (1) and (2’) gives the equation:. 5 EER stands for Equilibrium Exchange Rate. The EER achieves internal and external balances. More refined versions are developed by John Williamson, ‘‘Estimates of FEERs’’, Estimating Equilibrium Exchange Rates, Institute for International Economics, 1994, and Michael J. Artis & Mark P. Taylor, ‘‘DEER Hunting: Misalignment, Debt Accumulation, and Desired Equilibrium Exchange Ra tes’’, Working Paper, IMF, 1993. 6 (2) can be rewritten: E(x ) = θ (e – e) + E(Π ) – E ( Π * ) which is exactly what Frankel proposes in his article.. 4.

(5) e – e = - 1 [ ( r – Π) – (r* – Π*)] ?. (3). from which Frankel’s work has been labelled ‘‘real interest differential model’’. Note however that r – Π is not exactly the real interest rate since Π is not the expected current inflation E(p) but is the underlying long run inflation. We will address this issue further down in this paper.. II.3. The Money Market. The domestic demand to hold real money balance is given continuously by the conventional logarithmic function φy – λr which, equated to the real money supply from the central bank, yields: m – p = φy – λr. (4). w h e r e m , p a n d y d e n o t e t h e l o g s o f r e s p e c t i v e l y t h e n o m i n a l m o n e y is sued by the central bank, the price level and real income or output.. (4 ) i s. a classic (LM) identity. Domestic demand is supposed independent of the foreign demand for home money to settle imports. Here again, yet exogeneous variables, m and y are nevertheless implicitly indexed with time for they are not held constant.. For example, we can. imagine a case where long run inflation Π is fed by a steady monetary g r o w t h o f gM = d m / d t .. As m and y are exogenous, their long run and short. term values are identical, so we leave them without upbar.. Conversely, p. and r are endogenous and deserves their short- and long-term represent ations. In the long range, (4) becomes: m – p = φy – λr. (4). The same money function applies abroad: m * – p * = φy * – λr *. (4*). m * – p * = φy * – λr *. (4*). Subtracting (4) from (4), and (4*) from (4*), brings two equalities: 5.

(6) λ( r – r ). = p – p. λ( r * – r * ) = p * – p * whose difference gives: λ[ r – r – ( r * – r * ) ] = p – p – ( p * – p * ). (5). R e p l a c i n g r* – r by Π – Π* i n t h e a b o v e e q u a l i t y y i e l d s : λ[ r – Π – ( r * – Π * ) ] = p – p – ( p * – p * ) a n d w i t h ( 3) we gets: e = e –. 1 [ p – p – ( p* – p*)] λ?. (6). This is a key equation as it relates the exchange rate to the relative price l ev e l d e p a r t u r e f r o m i t s l o n g r u n t r e n d .. II.4. The Goods Market. The function depicting the demand for domestic goods in an open economy is classic: d = u + γ y – σr + δ( e + p * – p ). (7). where d is the logarithm of the demand D and u embodies the autonomous spending. At equilibrium supply Y meets demand D, which is the (IS) basics. reaching equilibrium takes time in our model where prices are sticky.. But. Hence. after a disturbance, (7) is violated in the short run and generates an excess demand.. We suppose that excess demand D – Y gives rise to an instant a-. neous short-term inflation p in the following way: p = π l n ( D / Y ) = π[ u + ( γ – 1 )y – σr + δ( e + p * – p ) ]. (8). A t e q u i l i b r i u m , w e g e t p& = Π a n d h e n c e Π = π[ u + ( γ – 1 )y – σr ]. (8). p = δπ( e + p * – p ) + Π – σπ( r – r ). ( 8'). so we can rewrite (8) as. Price disequilibrium depends on both PPP gap and interest rate disequilibrium.. This results differs from Frankel’s assumption where excess inflation is. r el a t e d s o l e l y t o P P P g a p . 6.

(7) Now the same demand function prevails abroad and so we get similar ident ities: p * = π[ u + ( γ – 1 )y * – σr * + δ( p – e – p * ) ] Π * = π[ u + ( γ – 1 )y * – σr * ] p * = - δπ( e + p * – p ) + Π * – σπ( r * – r * ). (8*) (8*) ( 8*'). S u b t r a c t i n g ( 8*') f r o m ( 8) p r o v i d e s : p – p * = 2 δπ( e + p * – p ) + Π – Π * – σπ[ r – r – ( r * – r * ) ]. ( 8*''). w h i c h w i t h ( 5) becomes: p – p * = 2 δπ( e + p * – p ) + Π – Π * – σπ [ p – p – ( p * – p * ) ] λ p – p& – ( p * – p& * ) = 2δπ( e + p * – p ) – σπ [ p – p – ( p * – p * ) ] λ . p – p – ( p * – p * ) = 2δπ( e + p * – p ) – σπ [ p – p – ( p * – p * ) ] λ. (9). We need to relate the PPP gap to the relative price disequilibrium to get a simple differential equation. bra.. T h i s c a n b e a c h i e v e d t h r o u g h a l i t t l e a l g e-. Subtracting (4*) from (4) brings: m – m * = p – p * + φ( y – y * ) – λ( r – r * ). and in the long run: m – m * = p – p * + φ( y – y * ) – λ( r – r * ) and by subtracting the latter from the former: p – p * = p – p * – λ[ ( r – r * – ( r – r * ) ] Using (PPP) and (RIP) in order to substitute p – p* w i t h e and r – r* w i t h Π – Π*, we get: e + p * – p = – λ[ ( r – Π ) – ( r * – Π * ) ] D i v i d i n g t h e t w o h a n d s i d e s o f t h i s e q u a t i o n w i t h t h e r e s p e c t i v e h a n d si des of (3) yields:. e + p*− p e− e. = λθ. or equivalently e + p * – p = ( 1 + λθ )( e – e ) w h i c h w i t h ( 6) gives the promised relation:. 7. (10).

(8) e + p * – p = - 1+ λ? [ p – p – ( p * – p * ) ] λ?. (10’). R e p l a c i n g P P P g a p i n (9 ) w i t h i t s r e l a t i v e p r i c e d e p a r t u r e e q u i v a l e n t , w e obtain a simple differential equation: . p – p – ( p * – p * ) = -υ [ p – p – ( p * – p * ) ]. (11). 2pd (1+ ?? )+ sp? ??. (12). where υ = and whose solution is: p – p * = p – p * + [ p0 – p 0 – ( p 0 – p 0 ) ] e - υ t ∗. ∗. (13). υ c a n b e i n t e r p r e t e d a s a r a t e o f c o n v e r g e n c e o f t h e r e l a t i v e p r i c e to wards its equilibrium path.. II.5. Consistency Condition. (6) shows that the currency follows the same type of convergence: ∗ ∗ e = e – 1 [ p0 – p 0 – ( p 0 – p 0 ) ] e - υ t λ?. e = e + 1 ( e0 – e0)e-υ t λ?. (14). Home currency depreciates to catch up its long-term path as long as the price differential is above its equilibrium level. T a k i n g t h e t i m e d e r i v a t i v e o f (1 4 ) — e = e – υ / λθ ( e 0 – e 0 ) e - υ t — , r e c a l l i n g t h a t e = x a n d e& = x , a n d u s i n g ( P P P ’ ) , w e e n d u p w i t h : x = Π – Π* – υ (e – e) C o m p a r i n g t h e l a t t e r e q u a l i t y w i t h e x p e c t a t i o n p a t h d e s c r i p t i o n (2 ’ ) en tails that expectations fit realizations only when θ equals υ. Consistent and stable expectations are then reached when θ verifies: θ =. 2 pd (1+ ?? ) + sp? and θ > 0 ??. (15). Solving the quadratic equation gives: θ = p ( σ + 2 δ) + [ p² ( σ + 2 δ) ² + π 2δ ] ½ 2 λ 4 λ λ The only difference with Dornbusch’s result is that which is the effect of our two-country framework. 8. (16). δ i s r e p l a c e d b y 2 δ,. It is worth noting that.

(9) r e l a x i n g t h e h y p o t h e s i s o f c o n s t a n t e q u i l i b r i u m v a l u e s h a s n o i n c i d e n c e t he re. When. expectations. meet. have a perfect foresight.. realizations,. it. means. that. economic. agents. This strong assumption reduces the attractiveness. of the model as it doesn’t show very much realistic.. A m o r e a p p e a l i n g mo -. del would allow some error in predictions, but without persistent bias rel ative to outcomes.. Agents would then feed rational expectations.. To our. knowledge, no stochastic setting of the Dornbusch or Frankel’s models has b e e n p r o p o s e d s o f a r 7. L a s t , i f (1 6 ) i s a n e c e s s a r y c o n d i t i o n f o r m o d e l c o n s i s t e n c y , n o t h i n g t e l l s us that it is also sufficient. issue in Part. II.6. We will see a complete discussion about this. IV.. Conclusion. W e h a v e g e n e r a l i z e d D o r n b u s c h ’ s o v e r s h o o t i n g m o d e l b y r e l a x i n g t h e hypotheses of a small country with constant secular macroeconomic variables, and by assuming a force pulling back the exchange rate towards not a given level but a dynamic equilibrium trajectory.. Yet we have found the. same kind of behaviour for the relative price and the exchange rate, that is to say an exponentional convergence (in log terms) towards the long-run underlying equilibrium path. However we will see in Part. IV that Dornbusch’s original model is not so. robust when facing a more realistic stance.. III) D O R N B U S C H ’ S I M P L I E D R E A L I N T E R E S T R A T E S D I F F E R E N T I A L The real interest differential model shown in: e – e = - 1 [ ( r – Π) – (r* – Π*)] ?. (3). may not deserve its name since the real rates are not computed with the actual inflations p and p*, but with the underlying long-term inflations. Π. and Π*.. 7 Frankel only hints at it in ‘‘Monetary and Portfolio Balance Models of Exchange Rate Determina tion’’, Economic Interdepedence and Flexible Exchange, MIT Press, 1984.. 9.

(10) Nevertheless after handling out a set of equations Frankel derives from his m o d e l t h a t t h e e q u i l i b r i u m e x c h a n g e r a t e d e p a r t u r e — o r E E R g a p — i s li nearly dependent on the real interest differential involving current inflations. We will see below that Dornbusch’s generalized model leads to the same kind of relationship.. We warn the reader that the algebra involved is sim-. ple but somewhat cumbersome.. III.1 Linking. the. Real. Rates. Differential. to. the. EER. and PPP Gaps Our first computational objective is to gather the EER gap, PPP gap and real interest differential into a single identity. First recall the long-term (RIP) : r – Π = r* – Π*. (RIP). and combine it with (3) to get: e – e = - 1 [ ( r – r*) – (Π – Π*)] θ. (3’). Plugging (3’) into (10) yields: e + p * – p = - 1+ λθ [ ( r – r * ) – ( Π – Π * ) ] λθ. (17). that is, PPP gap is linearly related to the pseudo-real rates difference. N o w u s e a g a i n ( R I P ) w i t h e q u a t i o n ( 8*'') b y s u b s i t u t i n g r – r* w i t h Π – Π * : p – p * = 2 δπ( e + p * – p ) + Π – Π * – σπ[ ( r – r * ) – ( Π – Π * ) ]. ( 8*''). and use (3’) to express Π – Π* without the pseudo-real rates spread: Π – Π * = p – p * – 2 δπ( e + p * – p ) – σπθ ( e – e ). (18). Then replace this equivalent value of Π – Π* in (18) : e + p * – p = - 1+ λθ [ ( r – p ) – ( r * – p * ) + 2 δπ( e + p * – p ) + σπθ ( e – e ) ] λθ Collecting PPP gap terms gives: e + p* – p = -. 1+ λθ [ ( r – p ) – ( r * – p * ) + σπθ ( e – e ) ] θ + 2 π(1+ λθ ). This is the first identity we were looking for. b u t w i t h o u t t h e σπθ ( e – e ) t e r m .. (19). Frankel finds a similar result. T h i s d i f f e r e n c e s t e m s f r o m h i s s i m p l e r as 10.

(11) sumption about the goods market.. Indeed Frankel’s pull back forces follow. the pro cess: p = πδ( e + p * – p ) + Π w h i c h i s s i m p l e r t h a n (8 ) s i n c e c u r r e n t i n f l a t i o n d o e s n ’ t d e p e n d d i r e c t l y on the endogenous current interest rate, contrary to what we have allow e d 8.. III.2 The Real Interest Rate Differential at Work E l i m i n a t i n g Π – Π* f r o m ( 3’) by using its value from (18), we find: e – e = - 1 [ ( r – r * ) – ( p – p * ) – 2δπ( e + p * – p ) – σπθ ( e – e ) ] θ a n d r e p l a c i n g t h e P P P g a p b y i t s v a l u e f r o m (1 9 ) b r i n g s , a f t e r r e a r r a n g i n g the real rate differential and the EER gap terms in the right hand side: e – e = - 1 θ. [. θ [ ( r – p) – (r* – p*)] – θ + 2 π(1+ λθ ). πσθ ² ( e – e )] θ + 2 π(1+ λθ ). F i n a l l y , w e g e t t h e l i n e a r r e l a t i o n s h i p 9: e – e = -. 1 [ ( r – p) – (r* – p*)] θ(1+ πσ ) + 2δπ (1+ λθ ). F r a n k e l ’ s s i m i l a r r e l a t i o n s h i p c o e f f i c i e n t i s e q u a l t o 1 / [ θ + 2 δπ( 1 + λθ ) ] a n d c l e a r l y a d d i n g u p ( 1 + πσ) r e f l e c t s o u r t a k i n g a c c o u n t o f t h e c u r r e n t i n t e r e s t r a t e i n t h e g o o d s m a r k e t s h o r t - t e r m d yn a m i c s . The main finding in Part. III is that the enlarged Dornbusch model del i-. vers one of the main result from Frankel, i.e. that the real interest rates differential drives the currency spot.. IV) S T A B I L I T Y D I S C U S S I O N A m a j o r w e a k n e s s i n o u r a p p r o a c h i s t h a t i t h a s i n j e c t e d a a d h o c l i n k (2 ) between PPP gap and EER gap into a classic IS-LM model in an open eco-. 8 Π i n v o l v e s l o n g - r u n r — see (8) — n o t i n s t a n t r . 9 Notice that (r – p) is not per say the real interest rate since p is not an inflation expectation but a realized figure. But remind that we are looking for a stable path for the economy and that this requires a perfect foresight from the agents. Hence p = E ( p).. 11.

(12) n o m y a n d w i t h p e r f e c t l y m o b i l e c a p i t a l 10, w i t h o u t c h e c k i n g w h e t h e r t h e model is by itself sufficient to provide such a link or an alternative one. In what follows we get a sense of our model from a Control Science per spective.. W i t h o u t s u p p o s i n g t h a t t h e e x p e c t a t i o n f o r m a t i o n (2 ) h o l d s e x. ante, we will analyze how an economy whose agents have perfect foresight can be st able.. IV.1. The State and Control Variables of the Model. We can think of the long-term variables, depicted with an upbar, as being the engine of the world economy and as well as a planner constraining it to stick to some route.. The short-term variables, they, then describe. how the economy reacts to once-and-for-all external shocks. The economy will be considered as a system described by a subset of nonredundant short-term variables — termed the state variables — and forced by fundamental variables — termed the control variables — to follow some t r a j ect o r y . S t a t e v a r i a b l e s a r e t o b e d r a w n f r o m e q u a t i o n s (1 ) , (4 ) & (4 * ) , ( 8 ) & (8 * ) , and from the perfect foresight assumption, all holding continuously.. I n ge -. neral for a given problem, the appropriate definition of the state is not unique, there being alternative ways of completely describing the current p o s i t i o n o f t h e s ys t e m . We collect (1), (4) & (4*), and (8) & (8*) to set up the system: r = r* + E ( x ). (1). m – p = φy – λr. (4). m * – p * = φy * – λr *. (4*). p = π[ u + ( γ – 1 )y – σr + δ( e + p * – p ) ]. (8). p * = π[ u + ( γ – 1 )y * – σr * + δ( p – e – p * ) ]. (8*). and recall that perfect foresight means that E(x) = x. T h e v e c t o r c o m p o s e d o f (r ; r * ; p ; p * ; e ) i s a s t r a i g h t f o r w a r d c a n d i d a t e t o become state variables, but by first collapsing the above system we will. 10 See for example the Mundell-Fleming model.. 12.

(13) then solve it more easily.. We observe that the system can be expressed in. terms of current exchange rate and price relative by eliminating the nominal interest rate differential.. Indeed:. (8) – (8*), (4) – (4*) and (1) yield respectively: ( p – p * ) = π[ ( γ – 1 ) ( y – y * ) – σ( r – r * ) + 2 δ( e + p * – p ) ]. (20). ( m – m * ) – ( p – p * ) = φ( y – y * ) – λ( r – r * ). (21). ( r – r*) = e. (22). ( 21) allows to express r – r* in terms of prices and exogenous variables: r – r* =. 1 [ ( p – p * ) – ( m – m * ) + φ( y – y * ) ] λ. which, when replaced in (22) and (20) provides: e =. 1 φ 1 (p – p*) + (y – y*) – (m – m*) λ λ λ. (23). . σ σφ σπ p – p * = 2 δπe – ( + 2 δ)π( p – p * ) + ( γ – 1 – )π( y – y * ) + (m – m*) λ λ λ. (24). Now restate this set of two equations in the long-run:. 1 φ 1 e& = (p – p*) + (y – y*) – (m – m*) λ λ λ . p – p * = 2 δπe – (. σ σφ σπ + 2 δ)π( p – p * ) + ( γ – 1 – )π( y – y * ) + (m – m*) λ λ λ. Subtracting the equations in underlying long-run variables from the co rresponding equations in current variables gives: . 1 e – e = [ p – p + ( p* – p*)] λ . p – p – ( p * – p * ) = 2δπ( e – e ) – (. σ + 2 δ)π[ p – p – ( p * – p * ) ] λ. This is a system that can be represented in matrix form: X( t ) = AX( t ). (25). where:. e−e   X( t ) =   p − p − ( p * − p *)  . 13.  0 A =  2δπ. 1/ λ.   - (σ/λ + 2 δ)π.

(14) X( t ) i s t h e v e c t o r o f s t a t e v a r i a b l e s a n d A i s a t i m e i n v a r i a n t m a t r i x .. N o t i c e t h a t X( t ) i s a v e c t o r o f g a p v a r i a -. is also called the state equation. bles.. (2 5 ). We now turn to so lving the equation.. IV.2. Solving for the State Equation. The state equation reveals a linear continuous-time system free of constaint and whose state variables can be readily expressed as functions of t i m e 11.. The solut i o n o f ( 25) is: X( t ) = e A t X( 0 ). eAt c a n b e c o m p u t e d w i t h t h e S y l v e s t e r m e t h o d w h i c h c o n s i s t s i n s e t t i n g the fo llowing determinant equal to zero:. -θ t e 1 -θ t e 2 At e. 1 - ?1 1 - ?2 Ι. A. (26). w h e r e -θ 1 a n d -θ 2 a r e t h e e i g e n v a l u e s o f m a t r i x A , t h u s s a t i s f y i n g :. -θ. - 1/ λ. - 2 δπ. - θ + (σ/λ + 2δ)π. = 0. (27). (27) leads to the same equation as (15) whose roots are: θ 1 = p ( σ + 2 δ) + [ p² ( σ + 2 δ) ² + π 2δ ] ½ 2 λ 4 λ λ. θ 2 = p ( σ + 2 δ) – [ p² ( σ + 2 δ) ² + π 2δ ] ½ 2 λ 4 λ λ. and. Hence (26) yields: ( - θ 1 e - θ 2 t + θ 2 e - θ 1 t )Ι – ( e - θ 2 t – e - θ 1 t ) A + ( -θ 2 + θ 1 ) e A t = 0 and therefore we have:. 1 eAt = ? 2 −? 1.  - θ2 t - θ1t + θ 2e  - θ1e  -θ t -θ t - 2 δπ (e 2 − e 1 ) . 1 - θ 2t - θ1t  (e −e )  λ - θ2 t - θ1t σ - θ2 t - θ1t  - θ1e + θ 2e + ( + 2δ) π (e −e ) λ  -. The free state solution is then of the form: e – e = a 1e-θ 1t + a 2e-θ 2t. and. p – p – ( p* – p*) = b1e-θ 1t + b 2e-θ 2t. 11 ‘‘Théorie de la commande temporelle’’, C.A. Darmon, Ecole Supérieure d’Electricité, 1980.. 14.

(15) w h e r e a1, a 2, b 1, b 2 a r e c o n s t a n t s w h o s e v a l u e s d e p e n d o n θ 1, θ 2, e 0 – e 0, ∗. ∗. p 0 – p 0 – ( p 0 – p 0 ) a n d t h e s t r u c t u r a l p a r a m e t e r s δ, λ, π, σ: a1 =. 1 ∗ ∗ [ θ 2 ( e 0 – e 0 ) + 1 ( p 0 – p 0 – ( p 0 – p 0 ))] λ ? 2 −? 1. a2 =. 1 ∗ ∗ [ - θ 1 ( e 0 – e 0 ) – 1 ( p 0 – p 0 – ( p 0 – p 0 ))] λ ? 2 −? 1. b1 =. 1 ∗ ∗ [ 2 δπ( e 0 – e 0 ) + ( θ 2 – ( σ + 2 δ)π)( p 0 – p 0 – ( p 0 – p 0 ))] λ ? 2 −? 1. b2 =. 1 ∗ ∗ [ - 2 δπ( e 0 – e 0 ) + ( - θ 1 + ( σ + 2 δ)π)( p 0 – p 0 – ( p 0 – p 0 ))] λ ? 2 −? 1. It is clear that the economy is stable if and only if a2 = b2 = 0.. This condi-. tion summarizes as: e0 = e0 –. 1 [ ( p – p – ( p ∗ – p ∗ ))] 0 0 0 0 λ ?1. e0 = e0 –. 1 ( θ 1 – ( σ + 2 δ)π) [ ( p 0 – p 0 – ( p ∗ – p ∗ ))] 0 0 2δπ λ. (28). T h e t w o e q u a t i o n s c o l l a p s e i n t o o n e b e c a u s e w e k n o w f r o m (2 7 ) t h a t θ 1 ver i fies. 1 = 1 ( θ – ( σ + 2 δ)π) . 1 λ ?1 2δπ λ. L e t u s k e e p t h e f i r s t o n e (2 8 ) . = 0.. W e n o t i c e t h a t t h i s i s i d e n t i t y (6 ) a t t i m e t. I t i s e a s y t o s e e t h a t h a v i n g ( 6 ) a t t i m e t = 0 i s e q u i v a l e n t t o h a v i n g (2 ). at time t = 0.. Hence to get a stable economy the initial shock needs to lie. on the expectation path that we did not assume to hold in this part.. Once. this condition is fulfilled, we also know that the logs of EER and relative prices gaps then follow decreasing exponential functions of time with rate of convergence θ 1.. F r o m t h i s r e s u l t 12 w e c a n i n f e r e a s i l y t h a t (2 ) h o l d s c o n -. tinuously. At this stage it may be tempting to claim that the model endogenously p r o v i d e s t h e e x p e c t a t i o n p a t h (2 ) .. But this would overlook that the initial. c o n d i t i o n (2 8 ) i s a l l b u t c o m m o n .. I n d e e d t h e p r i c e s h o c k m u s t e x a c t l y re -. l a t e t o E E R g a p i n a u n i q u e w a y , i . e . l i n e a r l y w i t h c o e f f i c i e n t - 1 / λθ 1 . doesn’t then the economy is unstable and (2) doesn’t hold.. 12 and from perfect foresight.. 15. If it.

(16) F r o m a m a t h e m a t i c a l s t a n d p o i n t , (2 8 ) i s w e a k e r t h a n (6 ) , a n d t h e r e f o r e t h e m o d e l g a i n s i n g e n e r a l i t y b y a s s u m i n g (2 8 ) r a t h e r t h a n (6 ) o r e q u i v a l en t l y (2 ) .. But this doesn’t make sense from an economic point of view, for. (28) is too specific.. Indeed, what is the economic rationale for assuming (6). or equivalently (2) only at time t = 0 and not later ?. We don’t see any, and. so probably did Dornbusch when he posits (2) at anytime.. IV.3. About Frankel’s approach. F o l l o w i n g F r a n k e l w e c o u l d h a v e d e s c r i b e d t h e e c o n o m y w i t h (2 3 ) a n d (24), that is with a set of variables which are not gaps.. The economy can. then be repres e n t e d i n t h e f o l l o w i n g m a t r i x f o r m : X( t ) = AX( t ) + B ( t )U ( t ). (29). where:.  e  X( t ) =   p − p ∗.  0 A =  2δπ. 1/ λ   - (σ/λ + 2 δ)π. φ/λ - 1/ λ   B =   ( γ 1 σφ / λ ) π σπ /λ  . y-y*  U( t ) =   m - m *. X( t ) i s t h e v e c t o r o f s t a t e v a r i a b l e s , U ( t ) t h e v e c t o r o f c o n t r o l v a r i a b l e s , U( t ) depends only on exogenous. and A and B are time invariant matrices.. variables and deserves to be defined as the control vector. We have already said that Frankel supposes that the dynamics of the goods market does not depend on the contemporanous interest rate.. T h e-. refore he gets different values of A, B and U(t):.  0 A =  2δπ. 1/ λ   - 2δπ . - 1/λ 1 B =    0 1.  p -p *  U( t ) =   Π - Π *. In both cases, the state equation is typical of a constrained — or forced — linear continuous-time system.. T h e g en e r a l s o l u t i o n o f ( 29) is:. X( t ) = e A t X( 0 ) +. ∫. t 0. e. A(t − τ). B( τ)U( τ)d τ. e A t X( 0 ) i s t h e s o l u t i o n o f t h e f r e e s y s t e m a n d i s e x p l i c i t e d i n Hence Frankel’s solution is the sum of a vector of the type. - θ1t -θ t  + a 2e 2  a1e b1e- θ1t + b 2e- θ2 t   . 16. IV.3..

(17) and of the vector. ∫. t 0. e. A(t − τ). B( τ)U( τ)d τ .. ∫ e A( t - τ) B ( τ)U( τ)d τ c a n p r o v e t o b e d i f f i c u l t t o o b t a i n a n d c h i e f l y h a s n o r e a s o n t o b e f u l l y m a d e o f t e r m s i n e- θ 1 t a n d e- θ 2 t .. For instance, suppose. t h e l o n g - r u n e c o n o m y g r o w s a t s t e a d y r a t e s , a n d i n p a r t i c u l a r e x h i b i t s s t ea dy inflations Π and Π*.. T h e n p a n d p * a r e l i n e a r f u n c t i o n s o f t i m e a n d ∫ e A( t -. τ) B ( τ)U( τ)d τ c a r r i e s t e r m s i n t e - θ 1 t a n d t e - θ 2 t . Therefore approaching the problem with non-gap variables, as Frankel did, doesn’t allow to assess that the endogenous variables — in logs for the exchange rate and prices — converge exponentially to their long-term equil i b r i u m v a l u e s 13. Yet there exists one special case where it is allowed: genous variables are held constant.. this is when all exo-. But we then get back to Dornbusch’s. fram ew o r k .. V.Conclusion In this paper we have shown that by using the same model as Dornbusch’s but wi thout its restrictive assumptions of no growth in the long-run macroeconomic variables and of a small country, we keep the same kind of results as with the original mo del and moreover as with Frankel’s ‘‘real interest rate model’’.. Briefly speaking, set -. ting Dornbusch model in motion brings Frankel mo del. Unfortunately, we also have proved that the expectations formation process at the heart of both models is unlikely to hold endogenously.. 13 p r o v i d e d t h e i n i t i a l c o n d i t i o n ( 28) be met.. 17.

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