- 6 Inflationary Expectations, Taxes, and the Demand for Money in the United States
- 7 Inflation, Taxation, and the Rate of Interest in Eight Industrial Countries, 1961–82
- 8 Financial Market Taxation and International Capital Flows
- 9 Impact of Taxation on International Capital Flows: Some Empirical Results

# 8 Financial Market Taxation and International Capital Flows

- Editor(s):
- Vito Tanzi
- Published Date:
- June 1984

Extensive literature exists on the various relationships between inflation, interest rates, and exchange rates and on their interconnections in an integrated international capital market. Very little of this literature, however, deals with the impact of taxes and the effects that differential taxation across countries have on international financial markets.

## I. THEORETICAL BACKGROUND

In a world without taxes, theory suggests the simultaneous holding of the Fisher effect (linking domestic inflation and interest rates), purchasing power parity (relating the exchange rate to domestic and foreign inflation), and interest rate parity (linking domestic and foreign interest rates).^{1} The introduction of taxes tends to prevent the simultaneous fulfillment of these three propositions. The nature and direction of the departures from these propositions, as well as their consequences, can be traced to the type of taxes imposed and to their relationships across countries. However, most of the specific analyses of taxation have centered on the effects of taxes on the Fisher equation within the framework of a closed economy. Darby (1975) and Tanzi (1976) modified the Fisher relationship and established that, when taxes are considered, nominal interest rates tend to be affected more than proportionally by changes in the expected rate of inflation.^{2} Although Makin (1978) extended the treatment by Tanzi and Darby to an open economy, most of the empirical and theoretical studies on interest rate parity and on purchasing power parity have ignored the effects of taxes on international financial equilibrium. The omission of taxes from the analysis can be justified by assuming that capital flows are not affected by taxation or, alternatively, that taxes affect both sides of interest rate parity proportionally. However, tax practices in Western countries appear to contradict both of these implicit assumptions.^{3}

The importance of differences in taxation practices for the analysis of international financial relationships and some of the implications of incorporating tax factors in the context of an open economy are studied by Levi (1977).^{4} He considers the tax rules of Canada and of the United States regarding foreign-generated income and also the tax treatment of capital gains. His analysis proves that differential taxation plays a central role in explaining deviations from pretax interest rate parity. Levi’s study also sheds light on the motivation for what seems, in the absence of tax factors, “abnormal” capital movements, such as two-way capital flows.

In a related paper, Hartman (1979) adds to the analysis the effects of inflation. He is particularly concerned with the effects of different tax arrangements applying to income that is generated domestically and to income that is generated abroad. In his model, taxation leads to the reallocation of real capital stock across countries.

Other recent contributions in this area include Tanzi and Blejer (1982), in which tax considerations are used to explain capital movements between developing and developed countries, and Ben-Zion and Weinblatt (1982), where it is shown that tax differentials may provide a significant incentive for short-term capital flows that may affect real interest rates and real exchange rates. Similar non neutralities arising from variations in the rate of inflation are obtained by Howard and Johnson (1982). They explain those effects as arising from the taxation of nominal, instead of real, interest income.

These contributions are indicative of the importance of considering taxes specifically in the context of open and integrated economies. These elements are incorporated below into a unified analytical framework that deals with the interactions between the modified Fisher effect, purchasing power parity, and interest rate parity and that provides a starting point for the study of the implications of alternative-assumptions about the tax treatment of financial instruments.

## II. ANALYTICAL FRAMEWORK

To assess the nature of the nonneutralities arising from alternative tax treatments of financial assets, a simple analytical framework is considered, so that it is possible to evaluate the conditions under which these nonneutralities arise, as well as their expected consequences and implications. Within this framework, a number of simplifying assumptions are made in order to center the discussion around the main influence that taxation has on domestic capital markets as well as on international capital movements.

A two-country setting is assumed, where neither country is large with respect to the other. It is assumed that the whole spectrum of financial assets available in each country can be subsumed by a characteristic asset, called here a bond, and that these bonds are perfect substitutes across countries in the sense that they are identical in all respects except for the currency of denomination. This assumption rules out the presence of country-specific financial risks, with the exception of differential exchange risks. However, in order to concentrate on the effects of taxation, the role of exchange risk premiums is not considered. Instead, it is assumed that all operations can be covered in the forward exchange market or, alternatively, that expectations about the future course of the exchange rate are held with perfect certainty.^{5} It is also assumed that marginal tax rates can be described by a single characteristic rate that applies in each country to the characteristic financial asset described above. Finally, the actual and expected rates of inflation are taken as exogenous, and transaction costs are ignored.

The following notation is used for the analytical framework:

*i _{A}* = nominal interest rate on Country A bonds

*i _{B}* = nominal interest rate on Country B bonds

π_{a} = expected rate of inflation in Country A

π_{b} = expected rate of inflation in Country B

*t _{a}* = marginal tax rate on interest income in Country A

*t _{b}* = marginal tax rate on interest income in Country B

^{6}

*R ^{A}* = nominal after-tax rate of return on Country A bonds for residents of Country A

*R ^{B}* = nominal after-tax rate of return on Country B bonds for residents of Country B

*e*= expected rate of devaluation^{7}

With respect to the structure of taxation, three specific assumptions are made: (1) tax treaties assure that residents of each country pay taxes only in their own country, (2) the same tax rate applies to interest income generated by domestic and by foreign instruments, and (3) capital gains, other than exchange gains, are not taxed in either of the two countries. These three assumptions are made in older to limit the number of cases discussed, but the impact of changing these assumptions can be easily analyzed within the same framework.

The two building blocks of the analysis are the Fisher equation and the interest rate parity hypothesis. Although they are reduced forms derived from unspecified behavioral relationships, they are widely taken as a representation of equilibrium conditions consistent with a variety of adjustment models.

As nominal interest is fully taxed (but there is no taxation on capital gains), the Fisher equation, as modified by Tanzi (1976) and Darby (1975), indicates that expected inflation is bound to affect the nominal interest rate more than proportionally in order to leave the real after-tax rate of return unchanged:

In the open economy, however, domestic interest rates may be affected also by the level of foreign exchange rates, because they are interconnected across countries by the interest rate parity condition. If there are no interferences to the free international flow of capital, portfolio considerations require that

Equations (3) and (4) indicate that, in equilibrium, the after-tax nominal returns of domestic and foreign assets should be equal within each country.

The third element, closing the system formed by equations (1) through (4), is the exchange rate rule. It is postulated here that there are two alternative patterns of exchange rate behavior: (1) the exchange rate follows purchasing power parity, and (2) the exchange rate adjusts in order to ensure the fulfillment of the interest rate parity conditions.^{8}

The analysis is conducted under two different assumptions about the tax treatment of gains and losses arising from foreign currency transactions. First, it is assumed that all foreign exchange gains and losses are treated as regular revenue and therefore are subjected to the same tax treatment (i.e., *t*_{a} and *t*_{b} apply to foreign exchange gains and losses as well as to interest income). Second, it is assumed that foreign exchange gains and losses are taxed at a lower rate than interest income. Also considered is the case in which an additional asymmetry arises when tax deductions for foreign exchange losses are fully claimed while foreign exchange gains are effectively tax exempt owing to widespread tax evasion.

### Equal tax treatment of interest income and foreign exchange transactions^{9}

Considering first the case in which foreign exchange gains and losses are treated as regular revenue for tax purposes, portfolio equilibrium becomes, as expressed in equations (3) and (4),

### Purchasing power parity

Under purchasing power parity, the rate of change of the exchange rate is determined by the differential rate of inflation and, therefore, the expected rate of devaluation will follow:

Using equations (1), (2), and (7), the interest rate parity conditions, as represented by equations (5) and (6), require that

Clearly, unless *t _{a}* =

*t*

_{b}*and*π

_{a}= π

_{b}, the simultaneous emergence of purchasing power parity, interest rate parity, and the revised Fisher relationship will imply different real rates of interest across countries. Consider an initial equilibrium position in which

*t*=

_{a}*t*, π

_{b}_{a}= π

_{b}, and

*i*=

_{A}*i*and

_{B}*e*= 0). If the income lax rate in Country A,

*t*, is now increased above

_{a}*t*, this will result in a higher after-tax nominal return on Country A’s bonds for residents of both countries. The difference in relative return, as obtained from equations (1), (2), (5), and (6), is

_{b}The above differentials indicate that the increase in the rate of interest income tax in Country A creates an incentive for capital flows from the lower-income-tax Country B to the higher-income-tax Country A. These flows will result in a reduction in the after-tax real interest rate of Country A and an increase in the corresponding rate of Country B. Equilibrium will be restored when real rates have changed sufficiently to satisfy^{10}

The transfer of capital and the changes in real interest rates will be larger, the higher the difference is between tax rates in the two countries and the higher the rates of inflation (equal across countries) are. Furthermore, from equations (9) and (10) it is clear that, under differential taxation, an equal increase in the rate of expected in inflation in both countries will not be neutral with respect to the level of real interest rates but will tend to reduce the real interest rate in the higher-tax country and increase it in the lower-tax country.

From equations (9) and (10), it is also seen that differential rates of in inflation give rise to capital flows, even if the rates of income tax across countries are identical. An increase of π_{a} over π_{b} raises the expected rate of devaluation according to purchasing power parity and, with *t _{a}* =

*t*, yields

_{b}Clearly, the differential in real returns will he larger, the larger the differential is in expected inflation rates and the higher the common rate of income tax is. However, the extent to which

### Interest rate parity

If the exchange rate is determined by the interest rate parity condition, its rate of change, as obtained from equations (1), (2), (5), and (6), is

Equation (13) indicates that for equal after-tax real interest rates, the exchange rate that preserves portfolio equilibrium will generally depart from purchasing power parity if rates of taxation *and/or* rates of inflation differ across countries. If

When both countries experience the same rate of inflation, equation (13′) becomes

indicating that, even if rates of inflation are identical, the exchange rate will devalue in the higher-tax country (unless _{j}, is negative). The extent of the exchange rate devaluation will depend on the level of (equal) inflation and real interest rates, as well as on the difference between tax rates.

From equation (13′) it is also observed that equality of lax rates does not eliminate the nonneutralily with respect to the exchange rate. In the presence of differential inflation and with *t _{a}* =

*t*, the equilibrating exchange rate is

_{b}Thus, if π_{a} > π_{b}, the exchange rate depreciates more than conventional purchasing power parity, with the extent of the depreciation being inversely proportional to the rate of the tax on interest income.

In general, for π_{a} ≠ π_{b}, and *t*_{a} ≠ *t*_{b}, the exchange rate may depreciate more or less than indicated by purchasing power parity. However, *e* will vary less (or be exactly equal) than implied by purchasing power parity only when the higher-inflation country has the lower tax rate.^{11} As a whole, for given changes in inflationary expectations, taxation will lead to an exchange rate that displays more variability than it does under purchasing power parity, and the degree of variability will be larger, the higher the rates of taxation are in both countries.

### Differential tax treatment of interest income and foreign exchange transactions

Next, consider the case in which interest income is taxed at a higher rate than foreign exchange gains and losses. For simplicity, the tax rate on foreign exchange gains is normalized to zero; however, the conclusions apply to all the cases where this rate is lower than the regular tax on interest income. Given this assumption, the interest rate parity conditions, as expressed in equations (3) and (4), result in

Once again, in order to evaluate the effects of taxation on the standard equilibrium propositions, the two alternative exchange rate rules should be imposed, as discussed below.

### Purchasing power parity

Imposing *e* = π_{a}–π_{b} on equations (14) and (15) and substituting equations (1) and (2) into those equations, the difference between domestic and foreign yields in each country that would give rise to capital flows can be written as follows;

Equations (16) and (17) indicate that, in the presence of differential taxation (*t _{a}* ≠

*t*), the Fisher relationship and purchasing power parity are not consistent with the absence of capital flows or with the simultaneous holding of interest rate parity in both countries unless both are experiencing the same rate of inflation. Consider, for example, an initial equilibrium position in which

_{b}*t*=

_{a}*t*and π

_{b}_{a}> π

_{b}. This implies a higher nominal interest rate in the higher-inflation country (

*i*>

_{A}*i*), the difference being matched by an equivalent devaluation. Therefore, real after-tax interest rates are equalized

_{B}*t*, such as

_{a}*t*>

_{a}*t*. Clearly, incentives for capital flows from Country B to Country A are generated in both countries

_{b}The capital flow generated by the tax differential tends to depress the real interest rate of the higher-tax country and to raise that of the lower-tax country. To satisfy parity conditions in Country A, it is observed from equation (16) that the new level of real after-tax interest rate in that country has to fulfill

However, given π_{a} > π_{b}, the volume of capital flows and the consequent change in real interest rates that equalize domestic and foreign returns for residents in Country A will not be sufficient to eliminate the return differential for residents of Country B. From equation (17), it is observed that

If, however, capital flows out from Country B to Country A to satisfy equation (19)—instead of equation (18)—an inverse incentive would arise in Country A, since the lower relative level of

Thus, the nonharmonization of taxes (across countries and domestically in terms of income and foreign exchange gains taxes) gives rise, in the presence of purchasing power parity and with differential inflation, to incentives for two-way capital flows of the nature discussed by Levi (1977). The new equilibrium position requires _{a} < π_{b}, the new equilibrium position will induce capital imports in both countries _{a} = π_{b} and therefore equations (18) and (19) are identical, ensuring that two-way capital flows do not take place, the unidirectional capital flows arising from differential taxation result in the same type of relationship between relative taxation and real interest rates: the higher-tax country will, in equilibrium, have a lower real after-tax interest rate.

### Interest rate parity

When the exchange rate adjusts to maintain interest rate parity, the variation required to equalize, within Country A, the returns on domestic and foreign bonds is, from equations (1), (2), and (14),

However, from equations (1), (2), and (15), interest rate parity in Country B requires

Two observations can be made from equations (20) and (21). As in the case of symmetrical taxation of interest and exchange gains, the exchange rate departs more from purchasing power parity as differences between taxes across countries become greater. In addition, it is clear that, even when inflation rates are equal, differential taxation induces capital flows and changes in real interest rates. That is true because, unless *t*_{a} = *t*_{b}, equations (20) and (21) cannot hold simultaneously for ^{12}

Assuming again an initial equilibrium where *t*_{a} = *t*_{b}, the Fisher effect, interest rate parity, and purchasing power parity will hold simultaneously, regardless of the relative rates of inflation. If *t*_{a} is set higher than *t*_{b}, the exchange rate will, according to both equations (20) and (21), devalue more (or appreciate less) than purchasing power parity. Assuming that the exchange rate adjusts according to equation (20), its value may be replaced in equations (14) and (15), and, as obtained from equations (1) and (2),

Thus, the exchange rate change required to preserve portfolio equilibrium in Country A does not maintain interest rate parity in Country B as long as *t*_{a} ≠ *t*_{b}. If π_{a} ≥ π_{b}, the returns on foreign investments for residents in Country B are higher than the returns on domestic investments, and capital will flow from Country B to Country A, reducing ^{13} From equation (23) the new relationship between real after-tax interest rates that will equalize returns for residents of Country B is obtained as^{14}

By substituting equation (24) into equations (20) or (21), *e* = 0 is obtained in both equations. Substituting equation (24) into equation (1) and using equations (14) and (15), it is observed that. when *e* = 0, parity conditions are maintained in both countries. This result indicates that, under differential taxation and differential treatment of interest and exchange gains, interest rate parity in both countries is only consistent with a constant exchange rate and therefore requires equality of *nominal* rates of interest. When inflation rates differ, equalization of nominal rates is attained by changes in the after-tax real rates that are brought about by international capital flows. This mechanism of equalization of nominal interest rates through capital mobility has a number of implications. For example, an acceleration in Country A’s inflation induces a capital inflow into Country A, raising the real interest rate in Country B and reducing its own real rate. Such an effect will be magnified by increases in the tax rate of the inflationary country.

### Tax evasion on exchange gains

The results obtained above in relation to differential tax treatment of interest income and foreign exchange transactions are based on the assumption that both countries differentiate in their tax treatment between interest income and the value changes arising from exchange fluctuations. It can be shown that similar qualitative results arise from the alternative assumption that exchange variations and interest income are formally taxed at the same rate within each country, but, while tax deductions for foreign exchange losses are fully claimed. the effective tax rate on foreign exchange gains is much lower (or zero) because of the widespread incidence of tax evasion. This assumption imposes an additional asymmetry on the system—between devaluing and revaluing countries—which, although not based on legal considerations, appears to be an economic fact widely observed in practice.

With this assumption, interest rate parity conditions, as expressed by equations (3) and (4), result in

where

Imposing purchasing power parity and using equations (1) and (2), the differential in returns within each country between domestic and foreign bonds becomes, assuming π_{a} > π_{b}, or *e* > 0,

Again, as under previous assumptions, purchasing power parity and the presence of taxation result in incentives for two-way capital flows and changes in the real rates of interest. If inflation rates differ, this result will emerge even if *t*_{a} = *t*_{b}. Consider the case where π_{a} > π_{b}. From equations (27) and (28), it is observed that capital will flow from Country B to Country A and the real after-tax interest rate in the higher-inflation country will therefore tend to fall. The differential in real rates that will restore portfolio equilibrium in Country A is equal to the rate of devaluation:

This differential, however, provides an incentive in Country B for capital flows in the opposite direction, since, by substituting into equation (28) the equilibrium condition for Country A obtained in equation (29), the result is

which implies that *t*_{a} > *t*_{b}), but it can be offset by *t*_{b} > *t*_{a}. These results are similar to the ones obtained in the preceding subsection under full tax exemption of foreign exchange gains and losses.^{15}

## III. SUMMARY AND CONCLUSIONS

The Fisher effect, purchasing power parity, and interest rate parity are equilibrium relationships that are taken to hold simultaneously in the absence of exogenous interferences. Taxation in the financial market constitutes one such interference, because it may introduce a wedge between the returns on domestic and foreign assets (when the various components of those assets are taxed differently) and/or between the returns of a given asset according to the residence of the holder (when taxation differs across countries). Therefore, taxation often prevents the simultaneous emergence of the three basic propositions and induces departures from their conventional formulations.

This paper has considered the nature of these departures and has discussed their implications. The basic premise of the analysis is that the introduction of taxes induces portfolio shifts aimed at restoring equality between the returns on domestic and foreign assets. These shifts result in interest rate and exchange rate nonneutralities that can be traced to the types and the combinations of taxes used. Some conclusions obtained from the analysis can be summarized as follows:

(1) Identical rates of taxation across countries will not prevent the emergence of nonneutralities when rates of inflation differ and interest income and foreign exchange gains are taxed at the same rate. Differences in inflation rates (with equal tax rates across countries) do *not* result in international capital flows, with the consequent changes in real interest rates and/or in departures from purchasing power parity, only in the case where exchange gains are not taxed at the same rate as interest income.

(2) Differences in tax rates between countries are conducive to differentials in real after-tax rates of interest (except when the exchange rate can depart from purchasing power parity in order to maintain interest rate parity and equal taxation applies to interest income and foreign exchange changes). In general, higher tax rates result in lower real interest rates, even if rates of inflation are identical.

(3) Under purchasing power parity, increases in the rate of inflation of a high-tax country result in a capital inflow and in a reduction of its real rate of interest. If the increase in inflation is not matched by an equivalent increase in other countries, the new equilibrium will induce two-way capital flows when exchange gains are taxed at a lower rate than interest income.

(4) When the exchange rate is determined by interest rate parity, the departures from purchasing power parity and the variability of the exchange rate are proportional to the differences between tax rates. However, when foreign exchange gains are not taxed, interest rate parity is consistent only with constant exchange rates, which implies equality of nominal interest rates. Such an equality in nominal rates is brought about by capital flows from the low-tax country to the high-tax country, with consequent adjustment in real after-tax rates.

Although some of the conclusions obtained here are dependent on the assumptions made, it is clear that the introduction of tax considerations provides an additional dimension to the analysis of interest rate determination in an open economy. One of the aspects of that dimension relates to the relationships between domestic inflation and the real rate of interest. The effects of expected inflation on the real rate of interest have been extensively analyzed in the context of a closed economy as arising from the domestic: substitutions between real and nominal assets (the Mundell and Tobin effects). In an open economy, the presence of taxation appears to provide an additional rationale for the relationship between inflation and real rates—a rationale based on the response of capital flows to differential inflation and the consequent relocation of the international capital stock.

Ben-ZionUri and J.Weinblatt“Purchasing Power Interest Rate Parity and the Modified Fisher Effect in the Presence of Tax Agreements” (unpublishedAugust1982).

DarbyMichael R.“The Financial and Tax Effects of Monetary Policy on Interest Rates,”Economic Inquiry (Long BeachCalifornia) Vol. 13 (June1975) pp. 266–74.

GandolfiArthur E.“Inflation, Taxation, and Interest Rates,”Journal of Finance (New York) Vol. 37 (June1982) pp. 797–807.

HartmanDavid G.“Taxation and the Effects of Inflation on the Real Capital Stock in an Open Economy,”International Economic Review (Osaka, Japan) Vol. 20 (June1979) pp. 417–25.

HowardDavid H. and Karen H.Johnson“Interest Rates, Inflation, and Taxes: The Foreign Connection,”Economics Letters (Amsterdam)Vol. 9No. 2 (1982) pp. 181–84.

LeviMaurice D.“Taxation and ‘Abnormal’ International Capital Flows,”Journal of Political Economy (Chicago) Vol. 85 (June1977) pp. 635–46.

MakinJohn H.“Anticipated Inflation and Interest Rates in an Open Economy,”Journal of Money Credit and Banking (ColumbusOhio) Vol. 10 (August1978) pp. 275–89.

MilesJames A.“Taxes and the Fisher Effect: A Clarifying Analysis,”Journal of Finance (New York) Vol. 38 (March1983) pp. 67–77.

ModiJitendra R.“Survey of Tax Treatment of Investment Income and Payments in Selected Industrial Countries” (unpublishedFiscal Affairs Department, International Monetary FundMay271983).

PeekJoe“Interest Rates, Income Taxes, and Anticipated Inflation,”American Economic Review (Nashville Tennessee)Vol. 72 (December1982) pp. 980–91.

RollRichard and BrunoSolnik“On Some Parity Conditions Encountered Frequently in International Economics,”Journal of Macroeconomics (Detroit Michigan)Vol. 1 (Summer1979) pp. 267–83.

TanziVito“Inflation Indexation and Interest Income Taxation” Quarterly ReviewBanca Nazionale del Lavoro (Rome) No. 116 (March1976) pp. 64–76.

TanziVito and Mario I.Blejer“Inflation, Interest Rate Policy, and Currency Substitution in Developing Economies: A Discussion of Some Major Issues,”World Development (OxfordEngland) Vol. 10 (September1982) pp. 781–89.

See, for example, Roll and Solaik (1979).

Extensions of the Darby-Tanzi treatment are presented by Gandolfi (1982) and Miles (1983). For a recent empirical implementation supporting the revised Fisher effect, see Peek (1982).

As the tax practices of major industrial counries with respect to interest income and payments, as well as dividends and capital gains (including foreign exchange gains), are very different, it is difficult to generalize regarding their points of contact and differences. For a review, see Modi (1983).

See Chapter 3 (paper by Ben-Zion) for a survey of the literature dealing with the effects of taxation on the international capital market.

When both alternative assumptions coincide, there is no risk premium in the forward exchange market—that is, the forward rare and the expected future exchange rate are the same.

The after-tax real rate of interest is defined as *r*^{*} = (1-*t*)*i*–π. As π is the expected rate of inflation, *r*^{*} refers to the ex ante or expected real rate of return.

The exchange rate is expressed in terms of units of the currency of Country A per unit of the currency of Country B. Therefore, *e* stands for the expected percentage change of the value of Country A’s currency in terms of Country B’s currency.

The first assurmption is based on the premise that the volume of trade and commodity prices are the main determinants of exchange rates, even in the short run, while the second considers the capital account as the main force driving exchange rates.

Japan, the Netherlands, and some other industrial countries do not distinguish for lax purposes between regular income and foreign exchange gains. The United States, Canada, and the United Kingdom apply the tax rates of capital gains to foreign exchange transactions (and therefore these rates differ from income tax rates). The assumption here of equal lax treatment of interest income and capital gains transactions is, therefore, appropriate for the first group of countries while the assumption of differential tax treatment of interest income and foreign exchange transactions is relevant for the second group. An additional distinction refers to the liming of taxation. While most countries tax foreign exchange gains and losses when they are realized, the United States, Japan, Canada, and the United Kingdom also tax accrued gains and losses. In the Federal Republic of Germany, unrealized gains are not taxable until they are realized whereas unrealized losses are deductible as they are incurred.

Replacement of

The condition for the exchange rate to follow purchasing power parity (i.e., *e* = π_{a}, – π_{b}, in equation (13′)) is that _{b} > π_{a}.

For _{a} = π_{b}, equations (20) and (21) result in

Clearly, unless *t*_{a} = *t*_{b} the exchange rate change required by equation (20′) differs from that required by equation (21′).

It should be noted that, if

When real rates differ across countries, equation (23) becomes

It can also be shown that, as in the previous case, if the exchange rate adjusts in order to maintain interest rate parity, real after-tax interest rates will change, so that *e* = 0 will be the only exchange rate change consistent with interest rate parity in both countries.