This chapter seeks to provide a general overview of the respective theories of exchange rate determination and includes a review of the empirical evidence that seeks to test these theories.^{1} The derivation of a model or theory that could be used to explain exchange rate behaviour is of importance, as exchange rates affect a country’s level of trade, which influences the income that is earned by the residents of a country. In addition, exchange rates would also affect the income of multinational firms, the income of those firms that make use of imports or exports, and the real rate of return on a global portfolio (or at least one that has some form of geographic diversification). Here, we look at some of the major forces behind exchange rate movements.

To a certain extent we have already covered one of the oldest theories of exchange-rate determination, which is the purchasing power-parity (henceforth PPP) theory that is commonly attributed to Cassel (1918), although noteworthy precursors also exist. When discussing the theory of PPP we noted that one is able to make a distinction between two versions of the theory, which include the *absolute* and the *relative* version.

According to the *absolute* version, the exchange rate between two currencies equals the ratio between the values, expressed in the two currencies considered, of the same typical basket containing the same amounts of the same commodities. For example, if such a basket is worth US $10 000 in the United States and R150 000 in the South Africa, the $/R exchange rate will be 1:15 (ie. R15 per dollar). According to the *relative* version, the percentage *variations* in the exchange rate equal the percentage variations in the ratio of the price levels of the two countries (the percentage variations in this ratio are approximately equal to the difference between the percentage variations in the two price levels, or *inflation differential*).

In both versions, PPP is put forward as a *long-run* theory of the equilibrium exchange rate, in the sense that in the short-run there may be marked deviations from PPP, which set into motion forces capable of bringing the exchange rate back to its PPP value in the long term. The problems arise when one wants to specify this theory, which implies both a precise identification of the price indexes that are to be used and the exact description of the forces that act to restore PPP; where the two questions are related.

Those who suggest using a price-index based on internationally traded commodities only, believe that PPP is restored by international commodity arbitrage, which arises as soon as the internal price of a traded good deviates from that prevailing on international markets, when both prices are expressed in a common unit (i.e. the *law of one price*).

On the contrary, those who maintain that a general price-index should be used, think that people appraise the various currencies essentially for what these can buy, so that (within a free market system) people exchange them in proportion to the respective purchasing power.

Others suggest using cost-of-production indexes, in the belief that international competition among firms in open economies are the main forces that should maintain PPP.

A fourth proposal suggests that one should use domestic inflation rates, since they assume that:

- the real interest rates are equalized among countries;
- in any country the nominal interest rate equals the sum of the real interest rate and the rate of inflation (the Fisher equation);
- the differential between the nominal interest rates of any two countries is equal (if one assumes risk-neutral agents) to the expected percentage variation in the exchange rate

From these assumptions it follows that there is equality between the expected percentage variations (which, with the further assumption of perfect foresight, will coincide with the actual ones) in the exchange rate and the inflation differential.

None of these proposals is without its drawbacks and each has been subjected to serious criticism. For example, the commodity-arbitrage idea was criticized on the grounds that it presupposes free mobility of goods (absence of tariffs and other restrictions to trade) and a constant ratio, within each country, between the prices of traded and non-traded goods: the inexistence, even in the long run, of these conditions, is a well-known fact. Besides, the law of one price presupposes that traded goods are highly homogeneous, another assumption often contradicted by fact and by the theories of international trade, which stress the role of product differentiation.

The same idea of free markets, of both goods and capital, lies at the basis of the other proposals, which run into trouble because this freedom does not actually exist. Cassel himself, it should be noted, had already singled out these problems and stated that they were responsible for the deviations of the exchange-rate from the PPP. For a survey of the PPP debate, see A. M. Taylor and Taylor (2004).

A good explanation of why PPP does not necessarily hold in terms of aggregate price indices was separately provided by Balassa (1964) and Samuelson (1964), and Harrod (1933). Briefly speaking, the Harrod-Balassa-Samuelson effect is the tendency for the consumer prices in fast-growing (rich) countries to be high (after adjusting for exchange rates) relative to the consumer prices in slow-growing (poor) countries. For example, as we have seen the cost of a McDonald’s burger is much more expensive in the United States, when compared to the US$ price of a similar burger in South Africa. To understand this, the distinction between tradables (produced by a traded goods sector) and nontradables (produced by a non-traded goods sector) is essential.

The model is largely based on the premise that labour productivity in rich countries is higher than in slow-growing poor countries, and that this productivity differential occurs predominantly in the traded goods sector (perhaps because this sector is exposed to international competition and biased towards high-innovation goods). The result is that with higher wage levels in rich countries the price of goods in those countries would be more expensive. The explanation for this phenomena is as follows:

First, assume that wages are the same in both sectors. Second, assume that prices are directly related to wages and inversely to productivity. Now, a rise in productivity in the traded goods sector will cause an increase in wages in the entire economy. Firms in the non-traded goods sector will only survive if the wage rises for such goods, despite the fact that it is unmatched by a productivity rise. This would imply that there would be an increase in the relative price of non-traded goods. Given the presence of both tradables and nontradables in a broad consumption index, and assuming that PPP holds for tradables, it follows that - after adjusting for the exchange rate - the consumer prices in rich countries (where productivity grows in the tradables sector) is higher than the consumer prices in poor countries (with no, or much lower, productivity growth).

The Harrod-Balassa-Samuelson effect does help explain why there could be large deviations from PPP, particularly with regards to aggregate price indices, but by definition it does not help to explain the deviations from PPP with regards to traded goods. These deviations, and the fact that the PPP theory is unable to explain the behaviour of exchange rates in the short-run, were among some of the reasons that induced many economists to abandon it in favour of other approaches.

It should however be noted that the PPP theory has been taken up again by the monetary approach (which will be dealt with in a later section), and has again been used as an indicator of the long-run trend in the exchange rate. However, the empirical evidence in favour of long-run PPP is often the subject of debate (Froot and Rogoff (1996); O’Connell (1998); for a contrary view see Klaassen (1999)). In addition, in the cases in which the exchange rate shows a tendency to revert to its long-run PPP value after a shock, it has been found that this reversion is not monotonic (Cheung and Lai 2000).

This approach, also called the balance-of-payments or the exchange-market approach, suggests that the exchange rate is the price of a currency that is determined in the foreign exchange market by the demand and supply forces for foreign exchange. Hence, this approach assumes that it moves (if free to do so) to equate the demand and supply for foreign exchange (if no intervention is assumed) to restore equilibrium in the balance of payments.

That the exchange rate is determined in the foreign exchange market by demand and supply forces for foreign exchange is an irrefutable fact (we do, of course, exclude economies with administrative exchange controls), however, the manner in which one seeks to determine the extent of these demand and supply forces is a source of disagreement. The traditional flow approach regards the demand and supply of foreign exchange as pure flows, which are largely derived from imports and exports of goods, which in turn depend on the exchange rate. In addition, such flows could also be influenced by national income. The introduction of capital movements as a further component of the demand and supply forces for foreign exchange does not alter this view insofar as these movements are also regarded as pure flows.

This approach has been criticized for several shortcomings, amongst which the fact that it neglects stock adjustments. For example, when the price of a currency changes then this could affect the wealth of an investor or country, which could influence the relative demand and supply for a particular currency. In addition, when the central bank adds to the monetary stock then the price of the currency would also need to adjust. It should however be stressed that while such criticism must induce us to consider the traditional approach inadequate in its specification of the determinants for the demand and supply of foreign exchange, it does not affect the fact that it is the interaction between these demand and supply forces that do ultimately determine the value of the exchange rate.

The asset-market approach (which is also termed the modern approach) takes the exchange rate as the relative price of a commodity. Such a commodity could take the form of monies, which is a concept that is consistent with the monetary approach of exchange rate determination. Alternatively, it could be regarded as the relative price of bonds, which would be consistent with the portfolio approach. The two views differ with regards to the assumptions made on the substitutability between domestic and foreign bonds, given the common hypothesis of perfect capital mobility. The monetary approach assumes perfect substitutability between domestic and foreign bonds, so that asset holders are indifferent as to which they hold, and bond supplies become irrelevant. Conversely, in the portfolio approach domestic and foreign bonds are imperfect substitutes, which implies that their supplies become relevant.

To avoid confusion it would be worthwhile to make a clear distinction between (perfect) capital mobility and (perfect) substitutability. Perfect capital mobility means that the actual portfolio composition instantaneously adjusts to the desired one. This would imply that if we assume that there is no risk of default or future capital controls then *covered interest parity* must hold. Perfect substitutability is a stronger assumption, as it means that asset holders are indifferent as to the composition of their bond portfolios (provided of course that both domestic and foreign bonds have the same expected rate of return expressed in a common numéraire). This indifference between holding bonds in different countries would imply that *uncovered interest parity* would hold.

It is important to note that according to some writers (see, for example, Helliwell and Boothe (1983)), the condition of covered interest parity itself becomes a theory of exchange-rate determination (the *interest parity model*, where interest parity may be expressed either in nominal or real terms), if one assumes that the forward exchange rate is an accurate and unbiased predictor of the future spot rate: it would in fact suffice, in this case, to find the determinants of the expected future spot exchange rate to be able to determine, given the interest rates, the current spot rate.

Since classifications are largely a matter of convenience (and perhaps of personal taste) we have chosen to follow the dichotomy based on the perfect or imperfect substitutability between domestic and foreign bonds within the common assumption of perfect capital mobility.

The monetary approach to the balance of payments assumes the validity of PPP as a long-run theory. In the case of a flexible exchange rate, where we assume that the monetary stock is exogenous, the equilibrium relations may be described as follows:

\[\begin{eqnarray} M_d = k_d \cdot P_d \cdot Y_d, \;\;\;\; M_{f}=k_{f} \cdot P_{f} \cdot Y_{f}, \;\;\text{and} \;\; P_d= S \cdot P_{f} \tag{3.1} \end{eqnarray}\]where as these are equilibrium relationships, we omit the time subscript and the adjustment process that is set in motion as a result of discrepancies between money demand and supply. In terms of notation, \(M_d\) and \(M_f\) denote the domestic and foreign monetary supplies, while \(P_d\) and \(P_f\) denote the respective price indices. Income in the two countries are denoted \(Y_d\) and \(Y_f\), while the reciprocal of the velocity of the circulation of money (which is assumed to remain constant) are \(k_d\) and \(k_f\). As before, the spot exchange rate is denoted \(S\).

The first two equations in (3.1) suggest that monetary equilibrium in the two countries, while the third refers to the PPP equation. For simplicity’s sake, we assume the rest-of-the-world variables are exogenously given. Then with a flexible exchange rate, the domestic stock of money becomes exogenous, so that the first equation determines \(P_d\) and the third determines the exchange rate. After simple manipulations we are able to provide an expression for the exchange rate,

\[\begin{eqnarray} S = \frac{M_d}{M_{f}} \cdot \frac{k_{f}Y_{f}}{k_d Y_d}, \tag{3.2} \end{eqnarray}\]This suggests that the exchange rate is the relative price of two monetary stocks \(M_d\) and \(M_{f}\), since the other variables are given exogenously. In this model the interest rate is not explicitly present, but they could be introduced by assuming that the demand for money in real terms is also a function of \(i\), that is

\[\begin{eqnarray} M_d = P_d \cdot g_d(y_d, i_d), \;\;\;\; M_{f}=P_{f} \cdot g_{f}(y_{f}, i_{f}),\; P_d=S \cdot P_{f} \tag{3.3} \end{eqnarray}\]where

\[\begin{eqnarray} S = \frac{M_d}{M_{f}} \cdot \frac{g_{f}(y_{f},i_{f})}{g_d(y_d,i_d)} \tag{3.4} \end{eqnarray}\]From this equation it can clearly be seen that, *ceteris paribus*, an increase in the domestic money-stock brings about a depreciation in the exchange rate, while an increase in domestic national income or interest rates causes an appreciation. These conclusions are perfectly consistent with the vision of the monetary approach to the Balance fo Payments. For example an increase in income raises the demand for money; given the money stock and the price level, the public will try to get the desired additional liquidity by reducing absorption, which causes a balance-of-payments surplus, hence the appreciation. This appreciation, by simultaneously reducing the domestic price level \(P_d\) (given that PPP holds), raises the value of the real money-stock (\(M_d/P_d\) increases), and so restores monetary equilibrium. Similarly, an increase in the interest rate, by raising capital inflows (or reducing capital outflows), brings about an appreciation in the exchange rate.

The monetary approach to the exchange rate can be made more sophisticated by introducing additional elements, such as sticky-prices which do not immediately reflect PPP arbitrage relations between \(i_d\) and \(i_{f}\), and the possibility that domestic agents hold foreign money.

Let us make use of a small open-economy model that follows Dornbusch (1976), where we assume flexible exchange rates, perfect capital mobility, flexible prices in the long-term and a given full-employment level of output. We note at the outset that the exchange rate is a typical *jump* variable, since it is free to make jumps in response to ‘*news*’. In this setting, prices are sticky in the short-term and take the form of predetermined variables, where they are assumed to adjust slowly to their long-run equilibrium value. Hence this model structure may replicate some of the features of a *sticky-price monetary model* that may be used for exchange rate determination.

Perfect capital mobility coupled with perfect asset substitutability implies that the domestic and foreign nominal interest rates are related through the *uncovered interest rate parity* (UIP) condition. In this case we can summarise this condition as

where \(x\) is the expected rate of change in the exchange rate. Since we are in a small open-economy, the foreign interest rate is taken as given. In addition, under conditions of rational expectations in such a deterministic setting, it would imply that the agents have perfect foresight, so that the expected and actual rate of depreciation of the exchange rate coincide. Letting \(s\) denote the rate of depreciation of the nominal spot exchange rate \(S\), we have

\[\begin{eqnarray} i_d=i_{f}+{s} \tag{3.6} \end{eqnarray}\]Let us now consider money market equilibrium,

\[\begin{eqnarray*} \frac{M_d}{P_d}={e}^{-\lambda i_d}Y_d^{\phi} \end{eqnarray*}\]where \(M_d\) is the money supply, \(P_d\) the price level, \(\lambda\) the semi-elasticity of money demand with respect to the interest rate, and \(\phi\) the elasticity of money demand with respect to income \(Y_d\). Taking logs (lower-case letters denote the logs of the corresponding upper-case-letter variables) and rearranging terms we have

\[\begin{eqnarray} p_d-m_d=\lambda i_d-\phi y_d \tag{3.7} \end{eqnarray}\]Combining equation (3.7) with the UIP condition that incorporates rational expectations in (3.6), we obtain the condition for *asset market equilibrium*

where output has been taken as that which is given by the full-employment level of \(y_d\). In long-run equilibrium with a stationary money supply, we have

\[\begin{eqnarray} \bar{p}_d-m_d=-\phi \bar{y}_d+\lambda i_{f} \tag{3.9} \end{eqnarray}\]since actual and expected depreciation are assumed to be zero in long-run equilibrium. Hence, equation (3.9) determines the long-run equilibrium for the price level. Subtracting equation (3.9) from equation (3.8) we obtain:

\[\begin{eqnarray*} p_d- \bar{p}_d=\lambda {s}, \end{eqnarray*}\]or

\[\begin{eqnarray} {s}=\frac{1}{\lambda}(p_d-\bar{p}_d) \tag{3.10} \end{eqnarray}\]This is one of the key equations of the model, as it expresses the dynamics of the current spot exchange rate in terms of the deviations of the current price level from its long-run equilibrium level.

We now turn to the goods market, which will allow us to determine the price level, given the assumption of a constant level of output. Since output is given, excess demand for goods will cause an increase in prices. If we initially assume that aggregate demand for domestic output depends on the relative price of domestic goods with respect to foreign goods, \(S \cdot P_{f}/P_d\), which in log terms would be \([(s+p_{f})-p_d]\). In addition, aggregate demand for domestic output would also depend on the interest rate and real income. Thus we have

\[\begin{eqnarray} d = u + \delta (s-p_d) +\gamma y_d - \sigma i_d \tag{3.11} \end{eqnarray}\]where \(d\) represents the change in demand and the given foreign price level \(P_{f}\) has been normalised to unity (i.e. \(p_{f}=\log P_{f}=0\)). The term \(u\) represents a constant or shift parameter. Hence, the change in prices that gives rise to inflationary pressure could be described as:

\[\begin{eqnarray} \pi_d = \theta(d-y_d) , \;\;\; \theta>0, \tag{3.12} \end{eqnarray}\]from which

\[\begin{eqnarray} \pi_d = \theta \left[ u + (s-p_d) + (\gamma - 1) y_d - \sigma i_d \right] \tag{3.13} \end{eqnarray}\]In long-run equilibrium, \(\pi_d=0\) and \(p=\bar{p}\), such that equation (3.13) may be used to derive the long-run equilibrium exchange rate

\[\begin{eqnarray} \bar{s} = \bar{p}_d + (1/\delta) \left[ \sigma i_{f} + (1-\gamma) \bar{y}_d - u \right], \tag{3.14} \end{eqnarray}\]where \(\bar{p}_d\) is given by equation (3.9) and \({i_d}=i_{f}\) since the long-run equilibrium exchange rate is expected to remain constant.

Let us now turn back to the disequilibrium dynamics of equation (3.13). Solving equation (3.7) for the interest rate and substituting into equation (3.13), we have

\[\begin{eqnarray} \pi_d = \theta \left[ u + \delta (s-p_d)+\frac{\sigma}{\lambda} (m_d-p_d)-\rho \bar{y}_d \right] \tag{3.15} \end{eqnarray}\]where \(\rho \equiv \phi \sigma / \lambda+1 - \gamma\). We now subtract from the right-hand-side of (3.15) its long-run equilibrium value, which is zero, namely \(0= \theta[ u + {\delta}(\bar{s}-\bar{p_d})+\frac{\sigma}{\lambda}(m_d-\bar{p}_d)-\rho \bar{y} ]\). Since \(\pi_d=(p_d-\bar{p}_d)\) for constant \(\bar{p}_d\), we can thus express price dynamics in terms of deviations from the long-run equilibrium values of the exchange rate and the price level, that is

\[\begin{eqnarray} \pi_d = - \theta \left( {\delta}+\frac{\sigma}{\lambda} \right) \left( p_d-\bar{p}_d \right)+\theta{\delta} (s-\bar{s}) \tag{3.16} \end{eqnarray}\]Equations (3.16) and (3.10) govern the dynamics of our system.

The dynamics of the system are rather complex and here we limit ourselves to observing that there exists only one trajectory along which the movement of the system converges to the equilibrium point (which is in the nature of a *saddle point* concept), where any point outside of this trajectory will move one further away from equilibrium.

Let us now consider the phenomenon of exchange-rate overshooting in response to ‘*news*’ (i.e. to an unanticipated event such as a monetary shock). This can be examined by means of Figure 1. Suppose that the economic system is initially at its long-run equilibrium (point \(Q\)). By an appropriate choice of units we can set \({p_d}={s}\), so that \(OR\) is a \(45^{o}\) line. Suppose now that the nominal monetary stock permanently increases. Economic agents immediately recognise that the long-run equilibrium price level and exchange rate will increase in the same proportion, as money is neutral in the long-run. In terms of Figure 1, this means that economic agents recognise that the economy will move from \(Q\) to \(Q_{1}\).

Due to the rigidities in the economy, it cannot instantaneously jump from \(Q\) to \(Q_{1}\), as the prices are sticky in the short-run. However, the exchange rate is able to move freely (as it is characterised as a jump variable), which would allow the economy to move from \(Q\) to \(Q_{2}\) on the (unique) stable trajectory \(AA\). From the economic point of view, the current exchange rate will depreciate because an increase in the money supply, given the stickiness of prices in the short run, will cause an increase in the real money supply and hence a fall in the domestic nominal interest rate. Since the UIP condition is assumed to hold instantaneously, and the nominal foreign interest rate is given, the exchange rate immediately depreciates by more than the increase in the long-run equilibrium value to create the expectation of an appreciation. This is required from the UIP condition, where the interest-rate differential equals the expected rate of appreciation. In fact, given \(i_d=i_{f}\) in the initial long-run equilibrium, the sudden decrease in \(i_d\) requires \(x< 0\) in equation (3.5), which describes the anticipated appreciation.

Thus the exchange rate initially overshoots its (new) long-run equilibrium level \((s_{0}>{s}_{1})\), after which it will gradually appreciate with the increase in the price level, following the path from \(Q_{2}\) to \(Q_{1}\) on the \(AA\) line. Overshooting results from the requirement that the system possesses ‘*saddle-path*’ stability, which in turn is a typical feature of the dynamics of rational-expectation models.

This approach, in its simplest version, is based on a model of portfolio choice between domestic and foreign assets. According to the theory of portfolio selection, asset holders will determine the composition of their portfolios,^{2} which could include the share of domestic and foreign bonds on the basis of the expected return and risk of such assets. If perfect substitutability between domestic and foreign assets exists, then uncovered interest parity should hold, that is

where \({s}\) denotes the expected change in the exchange rate over a given time interval; \({i}_d\) and \(i_{f}\) are to be taken as referring to the same interval. In the case of imperfect substitutability this relation becomes

\[\begin{eqnarray} i_d=i_{f}+{s}+{\delta} \tag{3.18} \end{eqnarray}\]Hence, with imperfect substitutability a divergence may exist between \(i_d\) and \((i_{f} + s)\); the extent of this divergence will, *ceteris paribus*, determine the allocation of wealth (\(W\)) between domestic (\(B_d\)) and foreign (\(B_f\)) bonds. For simplicity’s sake we make use of a model for a very small economy, where we assume that domestic bonds are held solely by residents, as the country’s assets are not of interest to foreign investors. The model can be extended to consider the general case without substantially altering the results, provided that residents of any country wish to hold a greater proportion of their wealth as domestic bonds (the so-called *preferred local habitat* hypothesis).^{3}

Given our simplifying assumption we can write

\[\begin{eqnarray} W=B_{d}^d + S \cdot B_{f}^d \tag{3.19} \end{eqnarray}\]where the superscript signifies that these represent the quantities that are demanded, in accordance with portfolio selection theory. In addition, we may expand these conditions to describe the demand for each instrument.

\[\begin{eqnarray} B_{d}^d=g({i}_d-i_{f}-s)W, \tag{3.20} \end{eqnarray}\] \[\begin{eqnarray*} S \cdot B_f^d=h({i}_d-i_{f}-s)W, \end{eqnarray*}\]where \(g(\ldots)+h(\ldots)=1\) because of (3.19). If we impose the equilibrium condition that the amounts demanded should be equal to the given quantities that are supplied, we get

\[\begin{eqnarray} B_{d}^d=B_{d}^s,\;\;\; B_{f}^d=B_{f}^s \tag{3.21} \end{eqnarray}\]and so, by substituting into (3.20) and dividing the second by the first equation there, we obtain

\[\begin{eqnarray} \frac{S \cdot B_f^{s}}{B_d^{s}}= \varphi (i_d - i_{f}-s) \tag{3.22} \end{eqnarray}\]where \(\varphi(\ldots)\) denotes the ratio between the \(h(\ldots)\) and \(g(\ldots)\) functions. From equation (3.22) we can express the exchange rate as a function of the other variables:

\[\begin{eqnarray} S= \frac{B_f^{s}}{B_d^{s}} \varphi (i_d - i_{f}-s) \tag{3.23} \end{eqnarray}\]Equation (3.23) shows that the exchange rate can be considered as the relative price of two stocks of assets, since it is determined (given the interest differential corrected for the expectations of exchange-rate variations) by the relative quantities of \(B_d^{s}\) and \(B_f^{s}\).

The basic idea behind all this is that the exchange rate is the variable that adjusts instantaneously so as to keep the (international) asset markets in equilibrium. For example, let us assume that an increase occurs in the supply of foreign bonds from abroad to domestic residents (in exchange for domestic currency) and that expectations are static \((s=0)\), to simplify matters. This increase, *ceteris paribus*, causes an instantaneous appreciation in the exchange rate. To understand this apparently counter-intuitive result, let us begin with the observation that, given the foreign-currency price of foreign bonds, their domestic-currency price will be determined by the exchange rate. Now, residents will be willing to hold (demand) a higher amount of foreign bonds, *ceteris paribus*, only if the domestic price that they have to pay for these bonds (i.e., the exchange rate) is lower. In this way the value of \(S \cdot B_f^{d}= S \cdot B_f^{s}\) remains unchanged, as it should remain, since all the magnitudes present on the right-hand side of equation (3.20) are unchanged, and the market for foreign assets remains in equilibrium \((B_f^{d}=B_f^{s})\) at a higher level of \(B_f\) and a lower level of \(S\).

The simplified model that we have described is a partial equilibrium model, as it does not consider the determinants of the interest rate(s) and the possible interaction between the current account and the financial account in the determination of the exchange rate. With regards to this interaction, we consider a model that was proposed by Kouri (1983). In this exposition, we simplify the model by assuming static expectations, the absence of domestic bonds in foreign investors’ portfolio, and an exogenously given interest differential. Thanks to these assumptions, the stock of foreign bonds held in domestic investors portfolios becomes an inverse function of the exchange rate. In fact, when at a point of equilibrium such a portfolio may be described as:

\[\begin{eqnarray} S \cdot B_f=h(i_d-i_{f}-s)W \tag{3.24} \end{eqnarray}\]so that, given \(i_d, i_{f}, W\), and letting \(s=0\) (where we assume static expectations), the right-hand side of equation (3.24) becomes a constant, hence the relationship is of inverse proportionality between \(S\) and \(B_f\). In what follows we use \(B_f\) to denote the equilibrium stock of foreign bonds held by residents, \(B_f=B_f^{d}=B_f^{s}\).

With regards to the current account, Kouri (1983) assumes that its balance is an increasing function of the exchange rate. The long-run equilibrium can prevail only when both the current account and the financial account are in equilibrium. In the opposite case, in fact, since any current-account disequilibrium is matched by a financial-account disequilibrium in the opposite sense.^{4} Hence a variation in the stock of foreign assets held by residents, will result in a variation in the exchange rate, which will feed back on the current account until this is brought back to equilibrium.

To illustrate this mechanism we can use Figure 2, where the left-hand panel shows the current account \((CA)\) as a function of the exchange rate. The right-hand panel shows the relation between \(S\) and \(B_f\), which is a rectangular hyperbola, when given the equation in (3.24).

The long-run equilibrium corresponds to the exchange-rate \(S_{E}\), where both the current account and the financial account are in equilibrium. For example, let us assume that the exchange rate happens to be \(S_{0}\), hence a current account surplus \(0C\); the initial stock of foreign assets (which corresponds to \(S_{0}\)) is \(B_{f,0}\). The current-account surplus is matched by a capital outflow (i.e. by an increase in foreign assets) as the domestic residents stock of foreign bonds increases (from \(B_{f,0}\) towards \(B_{f,E}\), which is the equilibrium stock). This results in an exchange rate appreciation (point \(A\) moves towards point \(E\)). This appreciation reduces the current-account surplus until the process restores equilibrium.

In the case where we have an initial exchange rate that is lower than \(S_{E}\) we would have a current-account deficit and a decrease in the stock of domestically held foreign assets, which would result in an exchange rate depreciation. Kouri (1983) concludes that this explains why the exchange rate of a country with a current account surplus tends to appreciate, while those with deficits tend to depreciate. This phenomenon is termed the *acceleration hypothesis* in the literature.

Although this is a highly simplified model, it does highlight the importance of the interaction between the current account and the financial account. It should also be emphasized that this interaction does not alter the fact that in the short-run, the exchange rate is always determined by asset market(s), even if it tends towards a long-run value \((S_{E})\) which corresponds to the point of equilibrium in both the current and financial accounts. Note also that all the above reasoning is based on the assumption that equation (3.24) holds instantaneously as an equilibrium relation, so that any change in \(B_f\) *immediately* gives rise to a change in \(S\), which feeds back to the current account and responds to the exchange-rate behaviour. The assumption of instantaneous equilibrium in asset markets, or (at any rate) of *a much higher adjustment speed of asset markets compared to that of goods markets*, is thus essential to the approach under consideration. As a result of this assumption, the introduction of goods markets and the current account does not alter the nature of the relative price of two assets attributed to the exchange rate in the short run.

More sophisticated models, in which the simplifying assumptions adopted here are dropped do not alter the fundamental conclusion that is stated above.

It is worth noting that if the exchange rate is defined as the relative price of two monies, as per the asset-market approach, it would be vacuous to state that it is only determined by the relative demand and supply of two monies, where there is no consideration for the demand and supply of foreign exchange in the foreign exchange market. This lack of, or insufficient, consideration of the foreign exchange market is one of the main shortcomings of the asset-market approach, according to Kouri (1983). In fact, no theory of exchange-rate determination can be deemed satisfactory if it does not explain how the variables that it considers crucial (whether they are the stocks of money or the stocks of assets or expectations or whatever) actually translate into supply and demand in the foreign exchange market, which together with supplies and demand coming from other sources, determine the exchange rate.

However, this would not imply that the asset-market approach is of no use. When considering the relative merits of the traditional and asset-market approaches, one would conclude that neither is by itself satisfactory, or according to Dornbusch “I regard any one of these views as a partial picture of exchange rate determination, although each may be especially important in explaining a particular historical episode” (Dornbusch 1983). In fact, as we have already observed, the determinants that we are looking for are both real and financial, derive from both pure flows and stock adjustments, with a network of reciprocal interrelationships in a disequilibrium setting. It follows that only an eclectic approach is capable of tackling the problem satisfactorily and more complex models would be necessary, since the exchange rate is only one of many endogenous variables in a complex economic system. Hence, simple models like those that have been described above would no longer be sufficient.^{5} To accomplish this task, one should move towards an economy-wide macroeconom(etr)ic models. When doing this, however, one should pay particular attention to the way in which exchange-rate determination is dealt with.

In order to put this important topic into proper perspective, we first need to introduce the distinction between models where there is a specific equation for the exchange rate and models where the exchange rate is implicitly determined by the balance-of-payments equation (thus the exchange rate is obtained by solving out this equation). From a mathematical point of view the two approaches are equivalent, as can be seen from the following considerations.

Let us consider the typical aggregate foreign sector of any economy-wide macroeconomic model, and let \(CA\) denote the current account, \(NFA\) the stock of net foreign assets of the private sector, \(R\) the stock of international reserves. Then the balance-of-payments equation simply states that

\[\begin{eqnarray} CA-\triangle NFA-\triangle R=0 \tag{4.1} \end{eqnarray}\]We could then introduce the following functional relations:

\[\begin{eqnarray} CA=f(S, \ldots ) \tag{4.2} \\ \triangle NFA=g(S, \ldots) \tag{4.3} \\ \triangle R = h(S,\ldots) \tag{4.4} \\ S= \varphi (\ldots) \tag{4.5} \end{eqnarray}\]where \(S\) is the exchange rate and the dots indicate all the other potential explanatory variables, that for the present purposes can be considered as exogenous. These relations do not require particular explanation. We only observe that the reserve-variation equation, which is numbered (4.4), represents the (possible) monetary authorities’ intervention in the foreign exchange market, also called the monetary authorities’ *reaction function*. System (4.1) - (4.5) contains five equations with four unknowns, but one of the equations, between (4.2) and (4.5), can be eliminated given the constraint (4.1). Which equation we drop is irrelevant from the mathematical point of view but not from the point of view of economic theory. From the economic point of view there are three possibilities:

- we drop the capital-movement equation (4.3) and use the balance-of-payments equation (4.1) to determine capital movements
*residually*(i.e., once the rest of the model has determined the exchange rate, the reserve variation, and the current account). - we drop the reserve-variation equation (4.4) and use the balance-of-payments equation to residually determine the change in international reserves.
- we drop the exchange-rate equation (4.5) and use the balance-of-payments equation as an implicit equation that determines the exchange rate. The exchange-rate, in other words, is determined by solving it out of the implicit equation (4.1). More precisely, if we substitute from equation (4.2) to (4.4) into equation (4.1), we get

which can be considered as an implicit equation in \(S\). This can be in principle solved to determine \(S\), which will of course be a function of all the other variables represented by the dots.

Since the functions \(f\), \(g\) and \(h\) represent the excess demands for foreign exchange coming from commercial operators (the current account), financial private operators (private capital flows), and monetary authorities (the change in international reserves), what we are doing under approach (3) *is simply determine the exchange rate through the equilibrium condition in the foreign exchange market* (*the equality between demand and supply of foreign exchange*).

It should be stressed that if one uses the balance-of-payments definition to determine the exchange rate one is not necessarily adhering to the traditional or ‘*flow*’ approach to the exchange rate, as was once incorrectly believed. A few words are in order to clarify this point. Under approach (3) one is simply using the fact that the exchange rate is determined in the foreign exchange market, which is reflected in the balance-of-payments equation, under the assumption that this market clears instantaneously. This assumption is actually true, if we include the (possible) monetary authorities’ demand or supply of foreign exchange as an item in this market; this item is given by equation (4.4), which defines the monetary authorities reaction function. As we have already observed above, in fact, no theory of exchange-rate determination can be deemed satisfactory if it does not explain how the variables that it considers crucial (whether they are the stocks of assets or the flows of goods or expectations or whatever) actually translate into supply and demand in the foreign exchange market, which together with supplies and demands coming from other sources, determine the exchange rate. When all these sources, including the monetary authorities reaction function that could also influence the foreign exchange market, through equation (4.4), are present in the balance-of-payments equation. Note that this equation is no longer an identity as it becomes a *market-clearing condition*. Thus it is perfectly legitimate (and consistent with any theory of exchange-rate determination) to use the balance-of-payments equation to calculate the exchange rate once one has specified behavioural equations for *all* the items included in the balance of payments.

As we have noted above, the various cases are equivalent mathematically but not from the economic point of view. In models of type (1) and (2), in fact, it is in any case necessary to specify an equation for exchange-rate determination, and hence adhere to one or the other theory explained in the previous sections. Besides, these models leave the foreign exchange market (of which the balance-of-payments equation is the mirror) out of the picture. Contrary to this position, the foreign exchange market is put at the centre of the stage in models that are of type (3). Hence, these models are not sensitive to possible theoretical errors made in the specification of the exchange-rate equation (4.6).

In this chapter we reviewed some of the major theories for exchange rate determination, which include purchasing power parity, the traditional flow approach, and various versions of the asset market approach (such as the monetary and portfolio mechanisms). We note that no single theory provides a comprehensive explanation for the movements that we observe in the exchange rate. As such, it may be to necessary to make use of a macroeconomic model that is able to incorporate a number of features from different theories to explain the behaviour of the exchange rate over a period of time.

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Dornbusch, Rudiger. 1976. “Expectations and Exchange-Rate Dynamics.” *Journal of Political Economy* 84: 1161–76.

———. 1983. “Economic Interdependence and Flexible Exchange Rates.” In, edited by J. S. Bhandari and B. H. Putnam, 45–83. Cambridge, MA: MIT Press.

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For general surveys of the problems treated in this chapter see, for example, MacDonald and Taylor (1992), Isard (1995), De Jong (1997), MacDonald and Stein (1999), and MacDonald (2007).↩

To simplify to the utmost, we follow Frankel (1983) where we only consider bond holdings; the introduction of money would not alter the substance of these results, as shown in Krueger (1983). Alternatively, one could assume that asset holders make a two-stage decision, by first establishing the allocation of their wealth between money and bonds, and then allocating this second part between domestic and foreign bonds.↩

As we know from balance-of-payments accounting, the algebraic sum of the current account and the financial account is necessarily zero. Here the implicit assumption is that there are no compensatory capital movements (i.e. there is no official intervention and the exchange rate is perfectly flexible). Under this assumption all capital movements are autonomous and originate from private agents.↩

In the sense that, notwithstanding the formal difficulty of many of these, they can account for only a very limited number of variables and aspects of the real world, and must ignore others which may have an essential importance (as, for example, the distinction between consumption and investment, which are usually lumped together in an aggregate expenditure function).↩