Real Exchange Rate and Output Variability The Role of Sticky Prices

Bankim Chadha

doi:10.5089/9781451973068.024.A006

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Real Exchange Rate and Output Variability The Role of Sticky Prices

Author: Mr. Bankim Chadha

Publication Date:: 01 Jan 1990

eISBN:: 9781451973068

Language:: English

Keywords:: SP; exchange rate; money stock; real exchange rate; aggregate price level; price flexibility; price level term; Inflation; Sticky prices; Exchange rates; Real interest rates

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The relationships between the degree of price stickiness and the variability of output and the real exchange rate are investigated in an open economy with flexible exchange rates and capital mobility. A critical degree of price inflexibility is shown to exist below which increased inflexibility reduces the variability of output. Also, as prices become more inflexible, the relationship between the variability of the real exchange rate and that of output will be nonmonotonic; that is, as the variability of the real exchange rate increases, the variability of output first declines and then increases.

Abstract

The considerable variability of real exchange rates since the advent of floating rates has been extensively documented and analyzed. The large and persistent movement of real exchange rates has been characterized by substantially greater movements in nominal exchange rates than in national price levels.¹ Such exchange rate behavior is consistent with the popular view—as exemplified by the Dornbusch (1976) model—that national price levels are sticky, often reflecting the existence of past contracts, whereas exchange rates, like other asset prices, are flexible and forward looking. Numerous models incorporating these stylized facts have been developed.² However, despite the widespread incorporation of the assumption of sticky goods price adjustment in models of exchange rate determination, little effort has been devoted to examining the relationship between measures of the degree of price stickiness, or the speed of goods price adjustment, and the variability of the real exchange rate and, hence, of output.³ Such a link is important, since a plausible inference often drawn is that the greater the degree of price stickiness, the more variable are the real exchange rate and output. Indeed, fluctuations in aggregate output are often attributed to the short-run rigidity of wages and prices, and various proposals and pleas to make wages and prices more flexible have often been advanced.

The link between the degree of price flexibility and the variability of output presents a fundamental question. The two standard textbook static models are the classical model and the Keynesian model. In the Keynesian model prices are assumed to be perfectly rigid, and output adjusts. In the classical model prices are completely flexible and output is perfectly stable. The two models suggest that as prices become more flexible, output should become less variable. Since Mundell (1963) pointed it out, it has been well known that the price level and the rate of change of the price level—that is, the expected inflation rate—exert opposing forces on the level of output in a static Keynesian model.⁴ Because inflation is essentially a dynamic phenomenon, the static Mundell effect prompts a natural consideration of the link between the degree of price flexibility and the variability of output in a dynamic context. Tobin (1975) presented a formal model where lower prices work to move the economy toward full employment, but an expectation of falling prices raises the real interest rate and moves the economy away from full employment.

DeLong and Summers (1986a, 1986b), Driskill and Sheffrin (1986), and, more recently, King (1988) and Chadha (1989) have examined the question of whether increased price flexibility or inflexibility will stabilize or destabilize output in a closed economy. Whereas Driskill and Sheffrin (1986) and King (1988) find no possibility of destabilizing price flexibility, DeLong and Summers (1986b) present simulations of Taylor’s (1979, 1980) staggered wages model to show that increased flexibility could be destabilizing. This paper examines the question for an open economy with flexible exchange rates and perfect capital mobility. It establishes that, in general, for such an economy a critical degree of price inflexibility exists, below which increased inflexibility of prices is stabilizing in that it reduces the variance of output around capacity, and above which increased inflexibility is destabilizing. It also shows that, as prices become more inflexible, the relationship between the variability of the real exchange rate and that of output will be nonmonotonic; that is, as the variability of the real exchange rate increases, the variability of output will decline up to a point and only then increase. Moreover, it is shown that the existence of a nonmonotonic relationship between the variability of output and the degree of price stickiness does not depend solely on the presence of the real interest rate in the aggregate demand function, as is implied in the closed economy literature.

Section I develops a simple and traditional sticky goods price, flexible exchange rate model with capital mobility. In Section II the model is solved for the time paths of the price level and the exchange rate, and the response of these variables to innovations in the money supply is examined. Section III then examines the variability of the real exchange rate and output as a function of the degree of price stickiness. Section IV contains concluding remarks.

I. The Model

Consider a country producing a basket of goods that are, in principle, entirely tradable in world markets. The domestically produced (composite) good is assumed to be differentiated from, and an imperfect substitute in consumption for, the rest of the world’s (composite) good. Domestic producers are hence assumed to be able to actively set the price of their good. The country is, however, assumed to be small, in the sense of being a price-taker in the market for the imported good. Further, domestic producers are assumed to price their good solely in terms of domestic currency.

The level of output produced is assumed to equal the demand for output. The demand for domestic output is posited to be a positive function of the real exchange rate, representing the ratio of competitors’ output price in the world market to the price of domestic output, and a negative function of the real interest rate:

y_{t} = y_{t}^{d} = d [e_{t} + P_{t} - P_{t}^{*}] - σ [i_{t} - (E_{t} P_{t + 1} - P_{t})], (1)

where d > 0 and σ > 0; y_t denotes (the log of) output at time t; e_t is the (log of the) spot exchange rate—that is, the number of units of the domestic currency for one unit of the foreign currency; P*_t is the (log of the) foreign currency price of the rest of the world’s good; P_t is the (log of the) price of the domestically produced good; is the domestic nominal interest rate; and E_t represents the mathematical expectations operator, conditional on information available at time t.

The equilibrium condition in the money market, which is assumed always to hold, is

M_{t} - P_{t} = - α i_{t} + Φ y_{t}, (2)

where α and Ф are > 0, and M_t is the (log of the) money supply at time t. With perfect capital mobility and perfect substitutability between domestic and foreign currency assets, the interest rate at home is equal to the foreign interest rate, adjusted for the expected rate of depreciation of the home currency:

i_{t} = i_{t}^{*} + [E_{t} (e_{t + 1}) - e_{t}] . (3)

The money supply process is assumed to be given by

M_{t} = M_{t - 1} + є_{t}, (4)

where e_t is a white noise innovation in the money supply process and

E_{t} (є_{t + s}) = 0 \forall s = 1, 2, \dots; V (є) = σ_{є}^{2} .

Note that the only shocks to this economy are monetary. It would be straightforward to introduce a real shock to the goods market in equation (1). However, little would be added to the analysis, since the monetary shock is a demand shock, and the results below are not affected in any significant manner.

Sticky Goods Price Adjustment

The model is closed by specifying the dynamic adjustment process for the aggregate price level at home. The price of the domestically produced good is assumed to be sticky, in that it is completely predetermined at a point in time and adjusts only slowly over time to equilibrate the goods market. The particular price adjustment rule employed is that posited by Mussa (1981a, 1981b):⁵

D P_{t} = E_{t} [D {\bar{P}}_{t}] + (1 - τ) [{\bar{P}}_{t} - P_{t}], (5)

where

0 < τ < 1

The forward-difference operator is denoted by D; that is. DX_s= X_s+1—X₅ and P, represents the equilibrium price level. The rate of inflation, DP_t, is therefore posited to equal the sum of the expected rate of change of the equilibrium price level and a fraction of disequilibrium in the goods market. Here, the extent of disequilibrium in the goods market is measured as the difference between the equilibrium price level and the actual price level. In general, the extent of disequilibrium in the goods market could be measured in price or quantity terms. The formulation in equation (5) has been chosen primarily for analytical tractability.⁶

In equation (5), (1 - τ) represents the proportion of disequilibrium in the goods market that is dissipated by changes in the price level in one time period. It therefore represents a measure of the speed of adjustment of prices in the economy, or a measure of the degree of price flexibility. As is brought out more clearly below, τ then represents a measure of the degree of price stickiness or inflexibility.

The equilibrium price level, P_t is defined as the price level that solves

y_{t} = y_{t}^{d} = d (e_{t} + P_{t}^{*} - P_{t}) - σ [i_{t} - (E_{t} P_{t + 1} - P_{t})] = \bar{y}, (6)

where $\bar{y}$ denotes fixed capacity output. The unique, convergent solution for $\bar{P_{t}}$ is

{\bar{P}}_{t} = \frac{[d + σ / α]}{[d + σ + σ / α]} σ_{i = 0}^{\infty} [\frac{d + σ / α}{d + σ + σ / α}]^{i} E_{t} M_{t + 1}, (7)

where (the log of) fixed capacity output, $\bar{y}$ , the foreign nominal interest rate, i*, and the (log of the) foreign price level, P*, have all been “normalized” to equal zero. Under the assumption that the process generating the money supply is that given by equation (4), it follows, as would be expected, that the equilibrium price level is simply equal to the money supply:

{\bar{P}}_{t} = M_{t} . (7')

II. Solution

The Appendix establishes the existence of a unique, convergent solution. It is then possible to solve the model by the method of undetermined coefficients. The solution is

P_{t} = τ P_{t - 1} + (1 - τ) M_{t - 1}, (8)

e_{t} = (1 - θ) P_{t} + θ M_{t - 1} + θ ξ_{t}, (9)

where

θ = \frac{1 + α (1 - τ)}{α (1 - τ) + Φ α (1 - τ) + Φ d} > 0. (10)

The aggregate price level in equation (8) is, therefore, a weighted average of last period’s price level and last period’s money stock. An increase in τ increases the weight given to last period’s price level and reduces the weight given to last period’s money stock. Maintaining the assumption that prices are predetermined, consider the maximum degree of price flexibility permitted; that is. when (1 - τ) = 1, or τ = 0. In this case prices are flexible with a one-period lag; that is, prices are set prior to the opening of goods markets in each period. The solution for the current price level is then the lagged value of the money stock, which is the time t— 1 expectation of the steady-state price level. In this sense, an increase in τ implies slower adjustment of the price level to monetary innovations and, hence, provides a dynamic measure of the degree of price stickiness in the economy.

The solution for the exchange rate in equation (9), noting (10), shows clearly that the response of the exchange rate to innovations in the money supply is in the same direction as the innovations. The exchange rate can, however, overshoot or undershoot its new long-run value, depending on whether θ is greater than unity, in which case it overshoots, or less than unity, in which case it undershoots. This, in turn, implies that overshooting or undershooting depends on whether

1_{<}^{>} Φ d + Φ σ (1 - τ) . (11)

An increase in the degree of price stickiness reduces the value of the terms in the right-hand side of equation (11); therefore, the stickier are prices, the greater is the likelihood that the exchange rate will overshoot its new long-run value in response to an innovation in the money supply.

III. Real Exchange Rate and Output Variability

This section examines the variability of the real exchange rate and output as functions of the degree of price stickiness, and the association between the variability of the real exchange rate and that of output as the degree of price stickiness changes. It is shown that the relationship between the degree of price stickiness and the variability of output is nonmonotonic. There are, however, two distinct channels through which changes in the degree of price stickiness bring about a nonmonotonic effect on the variability of output. The first relies on the presence of the real interest rate as an argument in the aggregate demand function and operates through the Mundeli-Tobin effect—that is, through the effect of expected inflation on real interest rates. This argument was first put forward by DeLong and Summers (1986a, 1986b), who presented simulations of Taylor’s (1979, 1980a) model for the U.S. economy, and was proved in Chadha (1989) in a closed economy context. The second channel requires only the presence of the real exchange rate as an argument in the aggregate demand function. It is therefore useful to consider simplified versions of the aggregate demand function in equation (1), so as to highlight the alternative channels.

The case is first considered where aggregate demand depends only on the real interest rate. It is shown in this context that there is a nonmonotonic association between the variability of the real exchange rate and that of output. The case where aggregate demand depends only on the real exchange rate is then examined. Finally, the case where aggregate demand depends on both the real exchange rate and the real interest rate is considered.

Aggregate Demand Dependent Only on Real Interest Rate (σ > 0, d = 0)

When aggregate demand depends only on the real interest rate, so that

y_{t} = y_{t}^{d} = - σ [i_{t} - (E_{t} P_{t + 1} - P_{t})], (12)

the solution to the system may be written as

P_{t} = τ P_{t - 1} + (1 - τ) M_{t - 1, (13)}

e_{t} = (1 - θ_{1}) P_{t} + θ_{1} M_{t - 1} + θ_{1} є_{t, (14)}

where

θ_{1} = \frac{1 + α (1 - τ)}{a (1 - τ) + Φ σ (1 - τ)} > 0. (15)

In this case the time paths of the real exchange rate and output may, employing (13) and (14), be written as

(e_{t} - P_{t}) = τ (e_{t - 1} - P_{t - 1}) + θ_{1} є_{t}, (16)

y_{t} = τ y_{t - 1} + θ_{1} σ (1 - τ) є_{t}, (17)

where

\frac{\partial θ_{1}}{\partial τ} = \frac{1}{(1 - τ)^{2} (α + Φ σ)} > 0. (18)

An increase in the degree of price stickiness, T, therefore, increases the persistence and impact effect on the real exchange rate. For output, however, whereas an increase in the degree of price stickiness increases persistence or inertia in consecutive levels of output, it reduces the impact effect of a monetary shock, since

\frac{Δ [θ_{1} σ (1 - τ)]}{Δ τ} = \frac{- α σ}{(α + Φ σ)} < 0. (19)

Figure 1 plots the implied impulse response of output to a positive monetary (or real) shock at time T for alternative degrees of price stickiness(τ). A greater degree of price stickiness (T₂) implies a smaller immediate jump in output in response to the monetary shock but implies a slower return of output to its capacity level. The reasoning for the opposing forces is straightforward and intuitive. For expository purposes, consider the situation where the economy is at its steady state at time T— 1, and a positive monetary shock is realized at time T. The equations describing the point-in-time equilibrium of the system, (equations (2) and (12)), can be graphed in the real interest rate and output plane as the traditional IS/LM curves, respectively, for a given rate of inflationary expectations. The impact effect of the monetary shock can be understood as follows. The increase in the nominal money stock, given the stickiness of prices, translates into a real increase in the money supply, shifting the LM curve down. Now, given that in this model expectations of inflation are formed rationally, the increase in the money stock gives rise to expectations of inflation, causing a second shift of the LM curve downward. As shown formally by equation (19). and as would be intuitively expected, the stickier are prices, the smaller is the expected inflation differential and the smaller is this second shift; hence, the higher is the real interest rate and the lower is the level of output on impact.

The variability of the real exchange rate and output are from (16) and (17):

v (e - p) = θ_{1}^{2} \frac{1}{(1 - τ^{2})} σ_{є}^{2}, (20)

v (y) = θ_{1}^{2} \frac{σ^{2} (1 - τ)^{2}}{(1 - τ^{2})} σ_{t}^{2} = v (e - P) [σ^{2} (1 - τ)^{2}] . (21)

In this case

\frac{δ v (e - p)}{δ τ} = σ_{є}^{2} [\frac{2 τ}{(1 - τ^{2})^{2}} + \frac{2}{(1 - τ^{2})} . \frac{δ θ}{δ τ} 1] > 0, (22)

so that the variability of the real exchange rate unambiguously increases as the degree of price stickiness rises. The effect of an increase in the degree of price stickiness on the variability of output is, however, ambiguous, since

\frac{δ v (y)}{δ τ} = \frac{2 σ_{є}^{2} \cdot σ^{2} [1 + α (1 - τ)]}{(α + Φ σ)^{2} (1 - τ^{2})} {τ + α τ - α} \frac{>}{<} 0, (23)

\frac{δ v (y)}{δ τ} \frac{>}{<} 0 0 a s τ \frac{>}{<} \frac{α}{1 + α} . (24)

The variance of output, therefore, declines with increases in the degree of price stickiness until a critical degree of price stickiness (τ^C), given by τ^C = α/(1 + α), after which it increases.

The above has shown that as prices get stickier, the variance of the real exchange rate unambiguously increases, but the variance of output declines until a critical degree of price stickiness is reached, after which it increases. There is therefore an implied nonmonotonic association between the variability of output and the variability of the real exchange rate as the degree of price stickiness changes.

Aggregate Demand Dependent Only on Real Exchange Rate (σ = 0, d >0)

If aggregate demand is a function simply of the level of the real exchange rate

y_{t} = y_{t}^{d} = d [e_{t} + P_{t}^{*} - P_{t}], (25)

then the solutions for the exchange rate and the aggregate price level are as in equations (8) and (9), with &₂ replacing 0, where

θ_{2} = \frac{1 + α (1 - τ)}{α (1 - τ) + Φ d} > 0 (26)

is the coefficient on the innovation in the money supply in the solution for the exchange rate. The exchange rate, therefore, overshoots or undershoots depending on whether

1 - Φ d \begin{matrix} > \\ < \end{matrix} 0. (27)

Employing the solution for the exchange rate and the price level, the time paths of the real exchange rate and output may be written as

(e_{t} - P_{t}) = τ (e_{t - 1} - P_{t - 1}) + θ 2 є_{t}, (28)

y_{t} = τ y_{t - 1} + θ_{2} d є_{t} . (29)

The real exchange rate and output again move on impact in the direction of the innovation in the money supply. The effect of an increase in the degree of price stickiness—that is, an increase in T—is, as before, to increase persistence or inertia in consecutive levels of both the real exchange rate and output. The effect of an increase in the degree of price stickiness on the impact effects depends, however, on whether the exchange rate overshoots or not, since

\frac{δ θ_{2}}{δ τ} = \frac{α (1 - Φ d)}{[α (1 - τ) + Φ d]^{2}}, (30)

and

s i g n o f \frac{δ θ_{2}}{δ τ} = s i g n o f (1 - Φ d), (31)

which is positive if the exchange rate overshoots, (see equation (27)), and negative if the exchange rate undershoots. Figure 2 plots the impulse response of output to a positive monetary shock at time 7” for alternative degrees of price stickiness.

The variability of the real exchange rate and that of output may. from equations (28) and (29), be written as

V (e - p) = θ_{2}^{2} \frac{1}{(1 - τ^{2})} σ_{є}^{2}, (32)

V (y) = θ_{2}^{2} \frac{d^{2}}{(1 - τ^{2})} σ_{є}^{2} = V (e - P) \cdot d^{2} . (33)

It follows from (33) that, in this case, the variability of output and that of the real exchange rate changes in the same direction as the degree of price stickiness changes. Now

\frac{δ V (y)}{δ τ} = \frac{2 θ_{2}^{2} d^{2} σ_{e}^{2}}{(1 - τ^{2})} {f (τ)}, (34)

where

f (τ) = \frac{τ}{(1 - τ^{2})} + \frac{α (1 - Φ d)}{[1 + α (1 - τ)] [α (1 - τ) + Φ d]} . (35)

If the exchange rate overshoots—that is, if (1 -Фd)>0—then f(τ)>0 for all values of τ, so that the variability of output is a strictly positive function of the degree of price stickiness. If, however, the exchange rate undershoots, it can be shown that

sign f (τ) = sign of g (τ), (36)

where

g (τ) = α^{2} τ^{3} - [2 α (1 + α)] τ^{2} + [(1 + α) (α + Φ d) τ + α (1 - Φ d) . (36')

Proposition. If the exchange rate undershoots, then there exists a degree of price stickiness τ = τ₁, such that g(τ) < 0 (that is, δV(y)/δτ< 0) for all τ < τ₁; similarly, there exists a τ < τ₂, such that g (τ) > 0 (that is, δV(y)/δτ< 0) for all τ > τ₂, and where τ₂ ≥ τ₁.

Proof: g(0) = α(l - Фd) <0; g(1) = Фd >0. Given that g(τ) is a continuous, single-valued function, the proposition follows.

A corollary of the above proposition is that for “small” values of τ, g(τ) < 0 and for “large” values of τ, g(τ) > 0. In other words, when the exchange rate undershoots, increases in the degree of price stickiness reduce the variability of output and the real exchange rate up to a critical degree of price stickiness, τ₁. It is only after the degree of price stickiness exceeds a crticial value, (τ₂ ≥ τ₁), that increases in the degree of price stickiness increase the variability of the real exchange rate and output.⁷

Although it is difficult to establish intuitive reasoning for the result, it is possible to show that it can be expected to hold in general, that it does not depend on the specific form of the pricing rule employed in equation (5), and that it is a consequence of forward-looking behavior in the foreign exchange market. Substituting the expression for output in equation (25) into the condition of money market equilibrium in equation (2) and rearranging, the exchange rate may be written as

e_{t} = \frac{1}{(α + Φ d)} σ_{i = 0}^{\infty} [\frac{α}{α + Φ d}] E_{t} {[M}_{t + i} - (1 - Φ d) P_{t + i}] . (37)

Recalling equations (2) and (3), if one is to think of the exchange rate as moving to continuously equilibrate the money market, note that this equilibrium is conditional on the expected future path of the exchange rate. Since the future exchange rate depends on future money market equilibrium, the level of the current exchange rate is, as equation (37) clearly brings out, a function of the expectation of all future variables affecting money market equilibrium. To bring out the effect of an increase in the degree of price stickiness on the exchange rate (and consequently, output), it is useful to consider the impact of a positive monetary shock at time t, where the economy is assumed to be in equilibrium at time r—1. The expectation of all future money supplies in equation (37) is, recalling equation (4), simply the current level of the money supply. If prices are stickier, in the sense that the general price level tends to adjust more slowly to the current level of the money stock (the new long-run, expected steady-state value of the price level), then this would imply that each expected future price level term in (37) would be smaller. Therefore, (1 - Фd) > 0 would imply a larger impact effect on the exchange rate and (1 - Фd) < 0, a smaller impact effect on the exchange rate. Since the price level is predetermined and output depends on the real exchange rate, the magnitude of the impact effect on the nominal exchange rate translates directly to the magnitude of the impact effect on output.

General Case

In the general case, the variability of the real exchange rate and output may be written

V (e - P) = \frac{θ^{2}}{(1 - γ^{2})} σ_{є}^{2}, (38)

V (y) = V (e - P) [d + σ (1 - τ)]^{2}, (39)

where θ is defined as in equation (10).

It can be shown that the derivatives of both the real exchange rate and output are ambiguous with respect to τ. Little more is gained by an examination of the conditions that determine the signs, so they are not presented here. Instead, simulations are presented of the variability of the real exchange rate and that of output, and the implied association between the variability of the real exchange rate and output for reasonable parameter values. The parameter values employed are d=0.5; σ = 3; α = 3; and Ф = 1. The variance of innovations in the money supply has been normalized to unity. Figures 3 and 4 plot the variance of output and the real exchange rate as functions of the degree of price stickiness, respectively. Initially, the variance of output declines substantially in percentage terms as the degree of price stickiness increases. It starts to rise only after a certain degree of price stickiness. For the parameter values here, it turns out that the variability of the real exchange rate increases monotonicaily with increases in the degree of price stickiness. Figure 5 plots the variability of output against the variability of the real exchange rate for the values in Figures 3 and 4. In Figure 5 the variance of output declines as the variance of the real exchange rate increases, up to a critical value; only beyond this point does it start to rise.

IV. Conclusion

This paper has investigated the links between the degree of price stickiness or (in)flexibility and the variability of output and that of the real exchange rate in an open economy under flexible exchange rates and perfect capital mobility. A critical degree of price inflexibility is shown to exist, below which increased inflexibility of prices reduces the variability of output; there is, furthermore, a nonmonotonic association between the variability of output and the variability of the real exchange rate as the degree of price stickiness changes. In addition, the existence of a nonmonotonic relationship between the variability of output and the degree of price stickiness does not require the dependence of aggregate demand on the real interest rate, as has been argued in the context of a closed economy. A nonmonotonic relationship is shown to exist here when aggregate demand depends only on the real exchange rate and if the economy is characterized by nominal exchange rate undershooting in response to a monetary shock.

These findings have important policy implications. It has frequently been argued that fluctuations in aggregate output are caused by the short-run rigidity of wages and prices, and that wages and prices should hence be made more flexible in order to reduce the variability of output. Given the analysis here, and as has been previously pointed out in the context of a closed economy, such proposals may not be appropriate at all times. In order to determine their appropriateness, it is important to determine the degree of price stickiness in the economy. If it is less than a critical value (τ^c), increased flexibility of prices would increase the variability of output. Only if it is greater than τ^C would proposals for increased flexibility be appropriate for the purpose of reducing the variability of output.

APPENDIX

Proof of Unique, Convergent Solution of the Model

This Appendix provides a proof for the existence of a unique, convergent solution of the model developed in the text. The dynamics of the system may be described by a pair of stochastic difference equations in the price level and the exchange rate:

[\begin{matrix} P_{t + 1} \\ e_{t + 1} \end{matrix}] = [\begin{matrix} τ, \\ \frac{[1 - Φ (d + σ) + Φ σ τ]}{α + Φ σ} \end{matrix}, \begin{matrix} 0 \\ \frac{\begin{matrix} [α + Φ (d + σ)] \end{matrix}}{α + Φ σ} \end{matrix}] [\begin{matrix} P_{t} \\ e_{t} \end{matrix}] + [\begin{matrix} (1 - τ) M_{t} \\ \frac{- [1 - Φ σ (1 - τ)]}{α + Φ σ} \end{matrix} M_{t}], (40)

or simply

S_{t + 1} = A S_{t} + F_{t} . (41)

In solving the model, as is traditional in rational expectations models, the condition is imposed that the solution be convergent in the expected value sense, constrained on information when the forecast is made. The existence of a unique, convergent solution to (41) requires that matrix A possess exactly one characteristic root inside the unit circle, given that there is one predetermined variable, the domestic price level, in the system.

Proposition. The matrix A possesses exactly one characteristic root inside the unit circle.

Proof. The characteristic equation of the matrix A may be written as

X^{2} - [τ + 1 + \frac{Φ d}{α + Φ σ}] X + τ [1 + \frac{Φ d}{α + Φ σ}] = 0.

Denoting the roots by X_t, note that

X_{1} = τ < 1,

and

X_{2} = 1 + \frac{Φ d}{α + Φ α} > 1.

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Bankim Chadha is an Economist in the External Adjustment Division of the Research Department. He holds a doctorate from Columbia University. The author thanks Richard Barth, Guillermo Calvo, Fabrizio Coricelli, Jeffrey Davis, Susan Jones, Reva Krieger, Ichiro Otani, and Ranjit Teja for their comments.

For a wide-ranging review of the experience under floating rates, see Goldstein (1984) and Obstfeld (1985). For a recent extensive empirical study of the behavior of real exchange rates, see Mussa (1986).

Obstfeld and Rogoff (1984) discuss various contributions.

An exception is Calvo (1987), who links real exchange rate overshooting to the average length of a price quotation in the economy.

An increase in the price level lowers real money supply, creating an excess demand for money and resulting in higher interest rates and lower output. An increase in the rate of inflation, through the Fisher effect, lowers money demand, creating an excess supply of money, lowering interest rates, and raising output.

See Obstfeld and Rogoff (1984) for a discussion of appropriate, forward-looking, sticky goods price adjustment rules.

The reduced-form solution for the aggregate price level generated as a consequence of adopting (5) is exactly the solution in Chadha (1987), where a much more complicated and appealing two-part price-setting mechanism is employed; it corresponds to the reduced-form solution in Rotemberg (1982), where the aggregate price adjustment rule is derived from microfoundations.

Whether the exchange rate overshoots or undershoots is an empirical matter. Papell (1985) finds empirical cases of exchange rate overshooting and undershooting.

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