y d t ’ d t % 0q t & Fr t . - 1 - (4.1) 4 --------------------------------- The Mundell-Fleming Model: Stochastic Dynamics The Mundell-Fleming model, which is still the workhorse model of international macroeconomics, can now be cast in a stochastic framework. Such a framework assumes a set of exogenous stochastic processes (e.g., money supply) which drives the dynamics of the equilibrium system. Since economic agents are forward looking, each short term equilibrium is based on expectations about future shocks and the resulting future short term equilibria. 4.1 The Stochastic Framework Let us begin with a description of the stochastic version of the Mundell-Fleming model. For simplicity, we express all variables in logarithmic forms (except for the interest rates) and assume all behavioral relations are linear in these log variables. This linear system (similar to the ones in Clarida and Gali (1994)) can be viewed as an approximation from an original nonlinear system. Aggregate demand in period t, y , specified as a function of an exogenous demand d t component, d , the real exchange rate, q , and the domestic real rate of interest, r , is given by t t t where 0 and F are positive elasticities. This equation is an analogue of equation (3.5) of the previous chapter. As is usual, the real variables are derived from the following nominal variables: s , the spot exchange rate (the domestic value of foreign currency); p , the foreign t *
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Appendix A: Derivation of the Full-fledged Equilibrium Solution
This appendix derives the solution for the full-fledged equilibrium (4.13)!(4.19), taking
as given the flex-price equilibrium.
1. Derivation of p (4.15)t
To get the solution for the domestic price level (4.15), we simply substitute (4.5c) and
(4.8) into (4.3).
2. Derivation of q (4.14)t
Substituting (4.4) and (4.7) into (4.1), using the definitions of real exchange rate q = st t
+ p ! p and real interest rate r = i ! E (p !p ), and subtracting (8+F)E q and adding* et t t t t+1 t t t+1
(8+F)q , we get et
Observe, from (4.5a), (4.5b), and (4.7) and the properties of , and , , that E q = q .yt+1 dt+1 t t+1 te e
We guess a solution of the form q = q + 6(, !, ) and apply it to (A2.1).t t mt yte
Since E(, !, ) = 0, we can obtain 6 = (1+8)(1!2)/(8+F+0) from the above equation.t mt+1 yt+1
This value of 6 in our guess solution yields the solution for the real exchange rate (4.14).
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3. Derivation of y (4.13) and i (4.18)t t
Substituting the solutions for p and q from (4.15) and (4.14) just derived above into thet t
aggregate demand equation (4.1) while using the interest parity (4.4) yields the solutions for yt
(4.13) and i (4.18).t
4. Derivation of B (4.16)t
Applying the definition of the expected rate of inflation, B = E (p !p ), to (4.15)t t t+1 t
derived in step 1 and (4.9) yields the solution for the inflation rate (4.16).
5. Derivation of r (4.17)t
Using i derived in step 3 and B in step 4 and the Fisher equation yields the solution fort t
the domestic real rate of interest r .t
6. Derivation of s (4.19)t
Using the solutions for p derived in step 1 and q in step 2 and applying them to thet t
definition of the real exchange rate q = s + p ! p yields a solution for the nominal exchanget t t*
rate s .t
y et ' y s
t .
q et '
1"0
(µy st &d
Xt ).
p et ' 8
1F(
(d At &y
st )%gm&gy % m s
t & y st .
Bet ' gm & gy.
r et '
1F(
(d At &y s
t ).
i et '
1F(
(d At &y
st ) % gm & gy.
' m st %
µ"0
&8%F(F(
y st &
1"0
d Xt &
8F(
d At & p ( % 8(gm&gy)
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(B.1)
(B.2)
(B.3)
(B.4)
(B.5)
(B.6)
(B.7)
Appendix B: Solutions for the Capital Controls Model
This appendix provides the solution to (4.1)', (4.1)", (4.2) and (4.3), subject to the
stochastic processes (4.5) and (4.5b)'. The flex-price equilibrium conditions are
To solve the full-fledged equilibrium, we use the flex-price solution to obtain the
equilibrium price p and inflation rate B . We then use the Fisher equation along with thet t
aggregate demand and money market equilibrium equations (4.1)" and (4.2) to get the solutions
for the real interest rate r and output y simultaneously. From the trade balance equation (4.1)',t t
we can calculate the real exchange rate q. The nominal interest rate i and the nominal exchanget t
yt ' y et %
F(8%F(
(1%8)(1&2) ,mt&8%F(F(
,yt%8F(
,Adt .
' q et %
µ"0
F(8%F(
(1%8)(1&2) ,mt&8%F(F(
,yt%8F(
,Adt
pt ' p et & (1&2) ,mt&
8%F(F(
,yt%8F(
,Adt .
Bt ' Bet % (1&2) ,mt&8%F(F(
,yt%8F(
,Adt .
rt ' r et &
18%F(
(1%8)(1&2) ,mt&8%F(F(
,yt%8F(
,Adt .
it ' i et %
8(1&F()8%F(
(1&2) ,mt&8%F(F(
,yt%8F(
,Adt .
s et %
µ"0
F(8%F(
(1%8)&1 (1&2) ,mt&8%F(F(
,yt%8F(
,
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(B.8)
(B.9)
(B.10)
(B.11)
(B.12)
(B.13)
(B.14)
rate s are then derived from the Fisher equation and the definition of the real exchange ratet
respectively. Below, we lay out the solution for the full-fledged equilibrium.
Comparing the full-fledged equilibrium under capital controls (B.8)-(B.14) with the
corresponding equilibrium under free capital flows (4.13)-(4.19), we can assess the significant
role that capital mobility plays in the Mundell-Fleming model.
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Problems
1. Consider the stochastic dynamic version of the Mundell-Fleming model with perfect
capital mobility. Introduce transitory shocks to the money supply process by adding an extra
term !N, (N > 0) to the right hand side of (4.5c). Decompose the variance of the realmt!1
exchange rate q into transitory and permanent components of the monetary shock.t
2. Consider the stochastic dynamic version of the Mundell-Fleming model with perfect
capital mobility. Introduce a correlation between the money supply process m and aggregatest
demand process d by adding Dd to the right hand side of (4.5c). One can view this as at!1 t!1
feedback rule whereby current monetary policy is conditioned on fiscal impulse in the previous
period. What value of D will minimize output variance? Inflation variance?
3. Consider the stochastic dynamic version of the Mundell-Fleming model with and without
capital controls.
(a) Compare the sensitivity of the following economic indicators to the various shocks
between the two capital mobility regimes: p , B , r , i , s .t t t t t
(b) Compare the slopes of the Phillips curves under the two regimes.
(c) Check whether the negative relation between the real exchange rate and the domestic real
rate of interest under perfect capital mobility holds also under capital controls.
- 20 -
References
Campbell, John Y., and Clarida, Richard H. 1987. The Term Structure of Euromarket Interest
Rates: An Empirical Investigation. Journal of Monetary Economics 19 (January): 25-44.
Clarida, Richard H., and Gali, Jodi. 1994. Sources of Exchnage Rate Fluctuations. NBER
Working Ppaer No. 4658.
Edison, Hali J. and Pauls, B. Dianne. 1993. A Re-assessment of the Relationship between Real
Exchange Rates and Real Interest Rates: 1974-90.
Journal of Monetary Economics 31 (April): 165-87.
Meese, Richard A. and Rogoff, Kenneth. 1988. Was It Real? The Exchange Rate-Interest
Differential Relation over the Modern Floating Rate Period. Journal of Finance 43
(September): 933-48.
Razin, Assaf, and Efraim Sadka. 1991. Efficient Investment Incentives In The Presence of
Capital Flight, Journal of International Economics 31: 171-181.
Sargent, Thomas J.. 1987. Macroeconomic Theory. Second Edition. New York: Academic
Press.
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1. To guarantee the existence of a long run (steady state) equilibrium for our system, thedeterministic growth rates of output on both the supply and demand sides (g ) are assumed toy
be identical.
2. The problem set at the end of the chapter considers also effects of transitory shocks.