Monetary Models

The monetary models are based on parity conditions linking interest rates on assets denominated in different currencies (equations (1) and (2)); a parity condition linking domestic and foreign prices, with the long-run real exchange rate constant under purchasing power parity (equation (3)); and equilibrium in financial markets, as represented by equality between the stock demands and supplies of domestic and foreign money (equations (4) and (5)). ⁴ The monetary models assume residents of each country hold only their own money:

Monetary models have been applied in different ways over the recent floating rate period.⁵ In “flexible-price” applications, the models determine nominal exchange rates on the basis of given values for real variables such as relative outputs and the real exchange rate. ⁶ In “sticky-price” applications, equations (1)–(5) describe the long-run behavior of nominal exchange rates, with departures from these relationships in the short run. ⁷

Under the assumption of flexible prices, the solution to equations (1)-(5) has a familiar form in which the nominal exchange rate depends on the current and expected future values of its determinants: here, relative money supplies, relative outputs, and the real exchange rate: ⁸

The significance of events in the future in equation (6) is determined by the size of the discount factor a/(1 + a), which depends on the (semi-interest) elasticity of money demand. If the elasticity of money demand is very high, events expected far into the future have almost the same weight as those in the present.

The forward-looking nature of equation (6) does not mean that exchange rates should be entirely unpredictable and follow driftless random walks (or martingales)—conditions that are sometimes regarded as an implication of market efficiency.⁹ The only efficiency conditions in the monetary model relate to the forward exchange market and imply that forward exchange rates should be unbiased predictors of expected future spot exchange rates. Given the assumption that residents in each country only hold their own money, there is no efficiency condition for the spot exchange market and exchange rates can be characterized by systematic movements in the monetary model.

The amount of systematic exchange rate movement in the flexible-price model is determined by the size of the (semi-) interest elasticity of money demand and the systematic movement in the forcing variables. Table 1 shows the solutions for the nominal exchange rate in the flexible-price model under alternative assumptions about the stochastic processes for relative money supplies, including those with intertemporally correlated components.¹⁰ Interpretation of the solutions is facilitated by decomposing the change in the nominal exchange rate between any two periods into a component that is unpredictable as of period t –1 (unsystematic component) and a component that is predictable (systematic component):¹¹

Given that there are no intrinsic dynamics in the flexible-price monetary model, systematic exchange rate movements only occur when there are predictable elements in the forcing variables. The most obvious source of systematic movements arises from differences in monetary growth rates across countries: countries with faster monetary growth rates will tend to have continuously depreciating currencies (Table 1). ¹² But systematic movement can also be the result of temporary or intertemporally correlated shocks to money supplies.¹³ For example, a temporary increase in the domestic money supply will lead to a temporary depreciation of the domestic currency, followed by expectations of appreciation.

The only case in which the nominal exchange rate follows a driftless random walk in the flexible-price model is when the forcing variables are described by driftless random walks. ¹⁴

The solutions in Table 1 also imply that any nonstationarity in nominal exchange rates derives from nonstationarity in the forcing variables. In each solution, the order of integration of the exchange rate matches that of the money supply process. ¹⁵ If, for example, relative money supplies are integrated of order two, exchange rates have the same order of integration. The real exchange rate is exogenous in this version of the flexible-price monetary model. If purchasing power parity holds in the long run, the real exchange rate will be a stationary variable. However, if there are permanent real shocks, such as changes in tastes or productivity, it will be nonstationary.

An important extension of the monetary model allows for short- run price rigidities. In an early paper, Dornbusch (1976) modified the flexible-price monetary model to allow for price stickiness and attempted to rationalize the high degree of volatility in nominal exchange rates over the floating rate period. The major features of Dornbusch’s sticky-price model are illustrated in Table 2 using a simple stochastic model that includes a Phillips curve and an aggregate demand equation. ¹⁶ Solutions to the model are shown for the case in which the domestic money supply follows a first-order autoregressive process, and all other forcing variables are constant. ¹⁷ In contrast to the flexible-price model, the Dornbusch model includes intrinsic dynamics associated with slowly adjusting commodity prices.

Table 1.

Flexible-Price Monetary Model

Note: For each money supply process, the systematic (or predictable) change in the exchange rate over the next period is given by (E_t, s_t+1 - s_t).

Table 1.

Flexible-Price Monetary Model

General solution

$s_t=\frac{1}{(1+\alpha )}\underset{i=0}{\overset{\infty }{\mathrm{\Sigma }}}{E}_t[{\frac{\alpha }{(1+\alpha )}]}^i\bullet [(m_{t+i}-m_{t+i}^{*})-\beta (y_{t+i}-y_{t+i}^{*})+q_{t+i}]$

In what follows it is assumed that

$y_{t+i}-y_{t+i}^{*}=q_{t+i}=0\text{for all}i\mathrm{.}$

Case I:Money supplies follow random walks around random trends (money supplies are I(2)processes):

$m_t=m_{t-i}+g_t+d_t{m}_t^{*}=m_{t-1}^{*}+g_t^{*}+d_t^{*}$

$g_t=g_{t-i}+k_t{g}_t^{*}=g_{t-1}^{*}+k_t^{*},$

where dt, dt*, kt, kt*are white-noise errors.

$s_t=(m_{t-1}-m_{t-1}^{*})+(d_t-d_t^{*})+(1+\alpha )(g_{t-1}+g_{t-1}^{*})+(1+\alpha )(k_t-k_t^{*})$

$E_t{s}_{t+i}=(m_{t-1}-m_{t-1}^{*})+(d_t-d_t^{*})+(i+1+\alpha )(g_{t-1}-g_{t-1}^{*})+(i+1+\alpha )(k_t-k_t^{*})$

$E_t{s}_{t+1}-s_t=(g_{t-1}-g_{t-1}^{*})+(k_t-k_t^{*})$

Case II: Money supplies are subject to temporary and permanent shocks (money supplies are I(1)processes):

$m_t={\overline{m}}_t+v_t{m}_t^{*}={\overline{m}}_t^{*}+v_t^{*}$

${\overline{m}}_t={\overline{m}}_{t-1}+{\mu }_t{\overline{m}}_t^{*}={\overline{m}}_{t-1}^{*}+m_t^{*},$

where vt, vt* ,μt,μt* are white-noise errors.

$s_t=({\overline{m}}_{t-1}-{\overline{m}}_{t-1}^{*})+({\mu }_t-{\mu }_t^{*})+(1+\alpha {)}^{-1}(v_t-v_t^{*})$

$E_t{s}_{t+i}=({\overline{m}}_{t-1}-{\overline{m}}_{t-1}^{*})+({\mu }_t-{\mu }_t^{*})$

$E_t{s}_{t+1}-s_t=-(1+\alpha {)}^{-1}(v_t-v_t^{*})$

Case III: Money supplies are subject to intertemporally correlated shocks (money supplies are I(0)processes):

$m_t=\overline{m}+v_t{m}_t^{*}={\overline{m}}^{*}+v_t^{*}$

$v_t=\rho v_{t-1}+{\mathrm{\Sigma }}_t{v}_t^{*}=\rho v_{t-1}^{*}+{\mathrm{\Sigma }}_t^{*}$

$\rho \in (0,1),$

where vt, vt* ,∑t,∑t* are white-noise errors

$s_t=(\overline{m}-{\overline{m}}^{*})+\lambda (v_t-v_t^{*})$

$E_t{s}_{t+1}=(\overline{m}-{\overline{m}}^{*})+\lambda \rho \bullet (v_t-v_t^{*}),$

$\text{where}\lambda =[1+\alpha (1-\rho ){]}^{-1}\mathrm{.}$

$E_t{s}_{t+1}-s_t=\lambda (\rho -1)(v_t-v_t^{*})$

Note: For each money supply process, the systematic (or predictable) change in the exchange rate over the next period is given by (E_t, s_t+1 - s_t).

View Table

Table 1.

Flexible-Price Monetary Model

General solution

$s_t=\frac{1}{(1+\alpha )}\underset{i=0}{\overset{\infty }{\mathrm{\Sigma }}}{E}_t[{\frac{\alpha }{(1+\alpha )}]}^i\bullet [(m_{t+i}-m_{t+i}^{*})-\beta (y_{t+i}-y_{t+i}^{*})+q_{t+i}]$

In what follows it is assumed that

$y_{t+i}-y_{t+i}^{*}=q_{t+i}=0\text{for all}i\mathrm{.}$

Case I:Money supplies follow random walks around random trends (money supplies are I(2)processes):

$m_t=m_{t-i}+g_t+d_t{m}_t^{*}=m_{t-1}^{*}+g_t^{*}+d_t^{*}$

$g_t=g_{t-i}+k_t{g}_t^{*}=g_{t-1}^{*}+k_t^{*},$

where dt, dt*, kt, kt*are white-noise errors.

$s_t=(m_{t-1}-m_{t-1}^{*})+(d_t-d_t^{*})+(1+\alpha )(g_{t-1}+g_{t-1}^{*})+(1+\alpha )(k_t-k_t^{*})$

$E_t{s}_{t+i}=(m_{t-1}-m_{t-1}^{*})+(d_t-d_t^{*})+(i+1+\alpha )(g_{t-1}-g_{t-1}^{*})+(i+1+\alpha )(k_t-k_t^{*})$

$E_t{s}_{t+1}-s_t=(g_{t-1}-g_{t-1}^{*})+(k_t-k_t^{*})$

Case II: Money supplies are subject to temporary and permanent shocks (money supplies are I(1)processes):

$m_t={\overline{m}}_t+v_t{m}_t^{*}={\overline{m}}_t^{*}+v_t^{*}$

${\overline{m}}_t={\overline{m}}_{t-1}+{\mu }_t{\overline{m}}_t^{*}={\overline{m}}_{t-1}^{*}+m_t^{*},$

where vt, vt* ,μt,μt* are white-noise errors.

$s_t=({\overline{m}}_{t-1}-{\overline{m}}_{t-1}^{*})+({\mu }_t-{\mu }_t^{*})+(1+\alpha {)}^{-1}(v_t-v_t^{*})$

$E_t{s}_{t+i}=({\overline{m}}_{t-1}-{\overline{m}}_{t-1}^{*})+({\mu }_t-{\mu }_t^{*})$

$E_t{s}_{t+1}-s_t=-(1+\alpha {)}^{-1}(v_t-v_t^{*})$

Case III: Money supplies are subject to intertemporally correlated shocks (money supplies are I(0)processes):

$m_t=\overline{m}+v_t{m}_t^{*}={\overline{m}}^{*}+v_t^{*}$

$v_t=\rho v_{t-1}+{\mathrm{\Sigma }}_t{v}_t^{*}=\rho v_{t-1}^{*}+{\mathrm{\Sigma }}_t^{*}$

$\rho \in (0,1),$

where vt, vt* ,∑t,∑t* are white-noise errors

$s_t=(\overline{m}-{\overline{m}}^{*})+\lambda (v_t-v_t^{*})$

$E_t{s}_{t+1}=(\overline{m}-{\overline{m}}^{*})+\lambda \rho \bullet (v_t-v_t^{*}),$

$\text{where}\lambda =[1+\alpha (1-\rho ){]}^{-1}\mathrm{.}$

$E_t{s}_{t+1}-s_t=\lambda (\rho -1)(v_t-v_t^{*})$

Note: For each money supply process, the systematic (or predictable) change in the exchange rate over the next period is given by (E_t, s_t+1 - s_t).

The solutions to the sticky-price model presented in Table 2 have two important implications for short-run exchange rate behavior. The first is that nominal and real exchange rates may overshoot in response to innovations in the money supply. A sufficient condition for nominal exchange rate overshooting in the model presented in Table 2 is that the money supply follows a random walk, so that all changes in the money supply are expected to be permanent.¹⁸ The second implication is that there can be predictable nominal and real exchange rate movements associated with the slow adjustment of commodity prices. These can arise with the money supply itself following a random walk (with p = 1 in Table 2), implying that deviations of the exchange rate from a random walk occur even if forcing variables are entirely unpredictable, and reflect the intrinsic dynamics of the model.

Table 2.

Sticky-Price Monetary Model

Table 2.

Sticky-Price Monetary Model

$m_t-p_t=\beta {\overline{y}}_t-\alpha R_t$

$R_t=R_t^{*}+E_t{s}_{t+1}-s_t$

$P_{t+1}=P_t+\pi (y_t-{\overline{y}}_t)$

$y_t=\theta (s_t-p_t),p_t^{*}=0$

The solutions when the money supply follows an AR1 process with a coefficient of p E [0,1] and capacity output and the foreign interest rate are constant and set to zero are given by

$(s_t-p_t)=\frac{({\lambda }_1-1)}{\pi \theta }{p}_t+\frac{{\alpha }^{-1}}{{\lambda }_2-\rho }{m}_t$

$p_t={\lambda }_1{p}_{t-1}+\frac{\pi \theta {\alpha }^{-1}}{{\lambda }_2-\rho }{m}_{t-1}$

$s_t=\frac{({\lambda }_1-1)}{\pi \theta }{p}_t+{\lambda }_1{P}_{t-1}+{\alpha }^{-1}\frac{(\rho +\pi \theta )}{{\lambda }_2-\rho }{m}_{t-1}+\frac{{\alpha }^{-1}}{{\lambda }_2-\rho }{\mu }_r,$

where λ1 and λ2 are the eigenvalues given by

${\lambda }_1=1-\frac{\pi \theta }{2}-\frac{1}{2}\left[\right(\pi \theta {)}^2+40\pi {\alpha }^{-1}{]}^{1/2}$

${\lambda }_2=1-\frac{\pi \theta }{2}+\frac{1}{2}\left[\right(\pi \theta {)}^2+40\pi {\alpha }^{-1}{]}^{1/2},$

so that λ2 > λ1. It is assumed that λ1 lies inside the unit circle and that λ2 lies outside it.

View Table

Table 2.

Sticky-Price Monetary Model

$m_t-p_t=\beta {\overline{y}}_t-\alpha R_t$

$R_t=R_t^{*}+E_t{s}_{t+1}-s_t$

$P_{t+1}=P_t+\pi (y_t-{\overline{y}}_t)$

$y_t=\theta (s_t-p_t),p_t^{*}=0$

The solutions when the money supply follows an AR1 process with a coefficient of p E [0,1] and capacity output and the foreign interest rate are constant and set to zero are given by

$(s_t-p_t)=\frac{({\lambda }_1-1)}{\pi \theta }{p}_t+\frac{{\alpha }^{-1}}{{\lambda }_2-\rho }{m}_t$

$p_t={\lambda }_1{p}_{t-1}+\frac{\pi \theta {\alpha }^{-1}}{{\lambda }_2-\rho }{m}_{t-1}$

$s_t=\frac{({\lambda }_1-1)}{\pi \theta }{p}_t+{\lambda }_1{P}_{t-1}+{\alpha }^{-1}\frac{(\rho +\pi \theta )}{{\lambda }_2-\rho }{m}_{t-1}+\frac{{\alpha }^{-1}}{{\lambda }_2-\rho }{\mu }_r,$

where λ1 and λ2 are the eigenvalues given by

${\lambda }_1=1-\frac{\pi \theta }{2}-\frac{1}{2}\left[\right(\pi \theta {)}^2+40\pi {\alpha }^{-1}{]}^{1/2}$

${\lambda }_2=1-\frac{\pi \theta }{2}+\frac{1}{2}\left[\right(\pi \theta {)}^2+40\pi {\alpha }^{-1}{]}^{1/2},$

so that λ2 > λ1. It is assumed that λ1 lies inside the unit circle and that λ2 lies outside it.

If purchasing power parity is assumed to hold in the long run, the real exchange rate is a stationary variable in the sticky-price model. It will, however, be closely correlated with the nominal exchange rate in the short run and display predictable movements. If there are permanent real shocks, changing, for example, the level of potential output, the real exchange rate will be nonstationary in this model.

Given the potential importance of real shocks for exchange rates, the flexible-price monetary model has been adapted to allow a more explicit role for real factors, giving rise to intertemporal equilibrium models in which nominal and real exchange rates are simultaneously determined.¹⁹ These applications continue to assume price flexibility but allow for shocks, such as productivity improvements or increases in real government spending, to explicitly influence nominal and real exchange rates. In addition, the models include intrinsic dynamics associated with factors such as slow adjustment of investment spending and intertemporal labor substitution.

Portfolio Models

The key assumptions of the portfolio models are that (interest-earning) assets denominated in different currencies are imperfect substitutes, and residents of each country have a preference at the margin for assets denominated in their own currencies.²⁰ An important source of systematic movement in nominal and real exchange rates in these models derives from intrinsic dynamics associated with current account imbalances. When residents of a country have a preference at the margin for their own commodities and assets, countries with current account surpluses may have continually appreciating real and nominal exchange rates due to the need to offset incipient excess demands for domestic assets and goods.²¹ More generally, these models suggest that nominal exchange rates will depend on the accumulation of outside domestic assets (money and bonds) and (net) foreign bonds. Given that outside domestic assets arc created through the fiscal balance, while (net) foreign assets are acquired by running current account surpluses, the models imply that the evolution of the exchange rate may be associated with fiscal and current account balances. The stochastic processes that nominal and real exchange rates follow then depend on the stochastic processes describing asset supplies—money, public interest-bearing debt, and foreign assets—as well as the intrinsic dynamics of the models.²²

Neither the monetary nor portfolio models reviewed in this section imply that nominal or real exchange rates should be characterized by random walks. Predictable movements in exchange rates can arise either from extrinsic dynamics in the forcing variables or from intrinsic dynamics that spread the adjustment to shocks over time. If purchasing power parity is assumed to hold in the long run, real exchange rates will display predictable fluctuations around a fixed mean. But if allowance is made for permanent real shocks, such as changes in tastes or technology, real exchange rates will be nonstationary. The stochastic process for nominal exchange rates also displays predictable movements in these models, which can either be around a fixed or a varying mean, depending on the processes followed by forcing variables. Differences in monetary policy across countries are one reason why nominal exchange rates might be nonstationary. Divergences in fiscal policy as well as current account imbalances are another potential source of nonstationarity. Both nominal and real exchange rates are expected, therefore, to deviate from random walks on the basis of standard exchange rate models.²³

The only models that predict a (driftless) random walk for nominal or real exchange rates, regardless of the stochastic process describing forcing variables, are the currency substitution and financial arbitrage models. Neither of these models, however, appears to have found its way into the mainstream of structural exchange rate models. In currency substitution models, the requirement that the expected pecuniary returns from holding domestic and foreign monies be equal when monies are perfect substitutes implies that the change in the nominal exchange rate should be unpredictable (see Adams and Boyer (1986)). In financial arbitrage models, the equalization of expected real interest rates on domestic and foreign bonds is interpreted as requiring that the expected change in the real exchange rate be unpredictable.²⁴ It is unclear, however, why real interest rates measured in terms of different consumption baskets should be equalized, suggesting that the financial arbitrage models may not provide a firm basis for expecting real exchange rate changes to be entirely unpredictable.