Marshall-Lerner condition and economic globalization

Abstract

The analysis considers the impact of FDI inflows and FDI outflows and shows that the presence of (cumulated) FDI requires higher import elasticities in absolute terms than stated in the standard Marshall Lerner condition. One may derive a range for the elasticity of the ratio of exports to imports with respect to the real exchange rate, namely that the sum of the absolute import elasticities at home and abroad must exceed unity plus an additional parameter—for standard special cases the sum of both elasticities must exceed 2 if a real depreciation is to improve the real current account. Not only can one determine a modified Marshall Lerner condition for a world economy with economic globalization, rather one also can get new insights from considering a broader macroeconomic perspective. The insights obtained are highly relevant for the discussion about high current account deficits of the US and high surplus positions of countries such as Japan, China and Germany—adjustment could be more complex than suggested by traditional models.

1 Introduction

High current account deficits have been a major challenge for US policy makers since the 1990s; many other countries have also occasionally faced problems with the current account deficit. Such deficits often give rise to protectionism, and it is therefore quite important to understand which policy options exist for correcting a trade balance deficit or a current account deficit. With respect to the US, Obstfeld and Rogoff (2005) have raised the problem that even a considerable adjustment of the real exchange rate might be insufficient to cope with the high current account deficit. Instead, the internal relative price—the ratio of tradables to non-tradables prices—would also have to adjust in a way which stimulates net exports of goods and services. The authors also emphasize that the asymmetric structure of the capital account is important, since US investment abroad is mainly in the form of equity investment and foreign direct investment, while the share of foreign investment in the US is dominated by investment in bonds and other assets, which are less risky than equities and therefore carry a relatively low yield. The rate of return from US foreign investment is higher than the average return foreign investors in the US have obtained. While foreign direct investment (FDI) has become a common fact of economic reality, economists have not fully taken account of its implications; one crucial implication concerns international profit transfers from subsidiaries to the parent company. One may add than for many countries in Europe, Asia and Latin America, high cumulated foreign direct investment contributes to a considerable difference between GDP and gross national income—e.g., Ireland transfers more than 1/10 of its GDP in the form of profits to parent companies of foreign MNC subsidiaries. Subsequently, we will show that (cumulated) foreign direct investment affects the reaction of both the trade balance and the current account balance. Modern globalization—characterized by high FDI flows and increasing activities of multinational companies—affects the impact exchange rates have on the external balance.

While economic globalization and FDI have become a common trait of the world economy, economic modeling largely ignores the distinction between GDP and GNP (as an exception see e.g. Welfens 2008), and this is not likely only to bring about inconsistent multiplier results, but it also implies a bias in many empirical analysis of import functions, export functions and whole macro systems. (For example, Germany’s export function can be estimated with standard approaches in a satisfactory way; however, for many years it its well-known that estimating Germany’s import function yields rather poor results.) It is straightforward to show that the traditional derivation of the Marshall-Lerner condition brings inconsistent results for a world economy characterized by economic globalization. One should, however, not overlook that in a broader macroeconomic approach both the real exchange rate and real income will affect the current account—as has been emphasized by McKinnon (2005) in his analysis on the current account of Japan and China. Subsequently it will be argued that the real exchange rate indeed has a direct impact on real income in the home country and the foreign country; with trade depending on GNP at home and abroad on the one hand and, on the other hand, savings and consumption depending on GNP and investment depending on the real exchange rate we get new multiplier results for the case of an FDI-augmented macro model (see the appendix).

The well-known Marshall-Lerner condition states that a real devaluation (a rise of the real exchange rate q*:= eP*/P where e is the nominal exchange rate, P the price level and * denotes foreign variables) will improve the real current account if the absolute sum of the import elasticities at home and abroad exceed unity. Alternatively, one may define a Marshall-Lerner condition as the reaction of the nominal trade balance to a change in the nominal exchange rate. This definition is useful if one wants to study pricing to market behavior which concerns the case that markups that exporting firms make vary with movements in the exchange rate—the market structure (Dornbusch 1987), the product-demand curve (Marston 1990) and the technology of the firm (Bughin and Monfort 1993) imply different degrees of exchange rate pass-through so that a change in the home currency leads to between zero and a more-than-proportionate change in the foreign-currency price (Bughin 1996).

Denoting real exports, the real exchange rate, and foreign GDP as X, q* and Y*, respectively, and assuming—with φ*being the elasticity of exports with respect to q* and x representing a positive parameter—the Marshall-Lerner condition can be derived by considering a standard export function ${\hbox{X}} = {\hbox{q}}{ *^{{\varphi * }}}{\hbox{x}}\,{\hbox{Y}} *$ and a standard import function ${\hbox{J}} = {\hbox{q}}{ *^{{ - \varphi }}}{\hbox{jY}}$ (J is real imports, φ is the absolute elasticity of J with respect to the real exchange rate and j is a positive parameter). The current account in real terms is X′ = X−q*J. It is convenient to derive the Marshall-Lerner condition by taking a look at the equivalent ratio ${\hbox{X}}\prime \prime = {\hbox{X}}/[{\hbox{q}} * {\hbox{J}}]$: We get the familiar result that ${\hbox{dX}}\prime \prime /{\hbox{dq}} * > 0$ only if φ*+φ > 1. In the standard two-country model (with home country 1 and foreign country 2), the export elasticity φ* is, of course, equal to the foreign import elasticity.

In the modern world economy, one should, however, not overlook that both foreign direct investment (FDI) flows and multinational companies can considerably change the picture, since a realistic approach will have to consider that savings are proportionate to gross national income (and not to GDP). Moreover, imports of goods and services are proportionate to gross national income and not to GDP—the latter being the standard assumption in models without foreign direct investment. The standard literature indeed puts emphasis on the GDP. It suffices to point out that the IMF-methodology used for calculating equilibrium real exchange rates (Faruqee 1998, p. 56) clearly states that one considers the equilibrium savings-investment balance and the current account balance as a share of GDP and thus implicitly ignores the distinction between GDP and GNP. With foreign direct investment inflows strongly rising relative to GDP in many Newly Industrialized Countries and in many OECD countries since the mid-1980s, one should carefully consider the distinction between GDP and GNP; many OECD countries are both major source countries of foreign direct investment and host countries of FDI at the same time.

These aspects imply that the modified Marshall-Lerner condition stated here—for the case of an economy with FDI and multinational companies—is stricter than the traditional Marshall-Lerner condition. It will be useful to define X″ as the ratio of real exports to real imports and to denote the elasticity of X″ with respect to the real exchange rate q* as E _X″,q*. For a two-country setup—with both countries producing in accordance with a Cobb-Douglas production function and with two-way foreign direct investment—, one may express the modified condition for an improvement of the current account (with φ, φ* denoting import elasticities in the home country and the foreign country) indirectly as $\varphi * + \varphi - 1 > {E_{{X\prime \prime, q * }}} > \varphi * + \varphi - 2$. One should note that with foreign direct investment, one has to make a distinction between the reaction of the trade balance with respect to the real exchange rate and the reaction of the current account balance with respect to the real exchange rate as profit income accruing from abroad (or paid to parent companies abroad) is part of the current account; only under the special condition that all profits earned abroad are reinvested could one ignore this distinction. One should also note that the relative size of the home country to the foreign country affects the results. In order to shed more light on the sometimes complex formula, we will consider interesting limit conditions.

With both multinational companies and FDI playing an increasing role in the world economy since the mid-1980s, one may conclude that real exchange rate adjustments might have become rather ineffective in correcting current account imbalances. Given the fact that multinational companies are typically active in technology intensive production in OECD countries, it is almost natural that a real depreciation hardly helps in reducing a current account deficit. This is already obvious if one assumes that multinational companies—dominating trade within the OECD—have specialized on exports of Schumpeterian technology-intensive goods whose price elasticity is relatively low. An offsetting effect might work on the import side if there is an increasing range of intermediate products which can be imported from many producers in many countries worldwide. (If we additionally consider pricing to market, namely that that the foreign price level/the import price P* is a negative function of e, because exporters in country II will want to dampen the loss in market share—associated with a fall of the nominal exchange rate—by lowering the national offer price in national currency units, the problem looks slightly different; considering the negative link between P* and e and the positive link between P and e the nominal depreciation rate required to correct a current account deficit would rise even more.)

The following analysis considers the impact of FDI inflows and FDI outflows and shows that the presence of FDI requires higher import elasticities in absolute terms than stated in the standard Marshall Lerner condition; in the simple setup presented we focus on a two country model where β stands for the share of profits in GDP of country 1 and β* for the share of profits in GDP of country 2—using a Cobb-Douglas function in both countries facilitates the analysis considerably, but the arguments presented can easily be generalized. Following the mathematical derivation in Section 2, the final section presents key policy conclusions.

2 FDI, the external balance and the real exchange rate

With inward FDI and outward FDI, one has to make a distinction between real gross domestic product (Y) and real gross national product (Z). For ease of exposition, the following analysis assumes that both the home country and the foreign country produce on the basis of a Cobb-Douglas function where K denotes capital, L labor and A knowledge. Hence we have ${\hbox{Y}} = {{\hbox{K}}^{\beta }}{({\hbox{AL}})^{{{1} - \beta }}}$ in country 1 and ${\hbox{Y}} * = {\hbox{K}}{ *^{{\beta * }}}{({\hbox{A}} * {\hbox{L}} * )^{{{1} - \beta * }}}$ in country 2 (* denotes foreign variables; 0 < β < 1). It also is assumed that there is competition in goods market and factor markets so that profit income in country 1 (home country) is equal to βY, in country 2 equal to β*Y*. Thus we can—denoting the share of country 1′ capital stock owned by foreign investors as α* and the share of country 2′ capital stock owned by investors from country 1 as α—express Z and Z*, respectively, as follows:

$$ {\hbox{Z}} * = {\hbox{Y}} * ({1} - \alpha \beta * ) + \alpha * \beta {\hbox{Y}}/{\hbox{q}} * $$

(1)

$$ {\hbox{Z}} = {\hbox{Y}}({1} - \alpha * \beta ) + {\hbox{q}} * \alpha \beta * {\hbox{Y}} * $$

(2)

As βY stands for profits in country 1 the amount α*βY will accrue to investors in country 2; profits accruing from 1—expressed in units of country 2′s output—thus is given by α*βY/q*. A real devaluation will reduce the real GNP in country 2, as real profits (expressed in units of country 2′s output) accruing from country 1 have reduced. As regards real gross national income in country 1, we clearly see that a real devaluation will raise Z which in turn stimulates imports of goods if real imports J are proportionate to Z. Note that α is in the interval 0,1; and the same holds for α*.

The next analytical step to consider is the assumption that exports of goods and services depend on the real exchange rate and the real foreign GNP—not on GDP as stated traditionally. Similarly, imports depend on real GNP; we will assume imports are proportionate to real GNP. Thus we will derive a Modified Marshall-Lerner Condition (as a first step in a setup without international profit transfers):

$$ {\hbox{X}} = {\hbox{X}}({\hbox{q}} *, ({1} - \alpha \beta * ){\hbox{Y}} * + \alpha * \beta {\hbox{Y}}/{\hbox{q}} * ); $$

(3)

$$ {\hbox{J}} = {\hbox{J}}({\hbox{q}} *, {\hbox{Y}}({1} - \alpha * \beta ) + {\hbox{q}} * \alpha \beta * {\hbox{Y}} * ); $$

(4)

We specify the following modified equations for exports and imports, respectively (x and j are positive parameters as are φ and φ*):

$$ {\hbox{X}} = {\hbox{q}}{ *^{{\varphi * }}}{\hbox{x}}[{\hbox{Y}} * ({1} - \alpha \beta * ) + \alpha * \beta {\hbox{Y}}/{\hbox{q}} * ] $$

(5)

$$ {\hbox{J}} = {\hbox{q}}{ *^{ - }}^{\varphi }{\hbox{j}}[{\hbox{Y}}({1} - \alpha * \beta ) + {\hbox{q}} * \alpha \beta * {\hbox{Y}} * ] $$

(6)

If one computes ${\hbox{X}}\prime \prime = {\hbox{X}}/[{\hbox{q}} * {\hbox{J}}]$ and calculates ${\hbox{dX}}\prime \prime {\hbox{/dq}}*$ we get a modified Marshall-Lerner condition. Here it should be noted that a real depreciation reduces real income of the foreign source country and thus dampens exports of goods; conversely, the denominator in X″ is raised through the term q* αβ*Y*. It therefore will not come as a surprise that the following mathematical derivation shows that the import elasticities in the foreign country and the home countries have to exceed the range defined by the standard Marshall-Lerner condition. It will be straightforward to show that for the special case of a small open home economy and for the case of a dominant home economy the Marshall-Lerner condition reads ${\hbox{dX}}\prime \prime /{\hbox{dq}}* > 0\;{\hbox{if}}\;\varphi * + \varphi > {2}$; the same condition is obtained for the case of “symmetrical long term FDI involvement” in the sense of αβ* = α*β (while not considering the additional constraint that Y = Y*).

For the special case of a symmetric world economy—with equal size of countries (Y = Y* and the initial q* being unity) and symmetric cumulated FDI involvement of both countries (αβ* = α*β)—the modified Marshall-Lerner condition looks rather similar to the standard Marshall-Lerner condition: The elasticity ${{\hbox{E}}_{{{\rm{X}}\prime \prime, {\rm{q}} * }}} = \varphi * + \varphi - {1} - {2}\alpha * \beta$. Considering that the output elasticity of capital typically is 1/3 and that the ratio of α (or of α*) rarely will exceed 0.5 the implication is that in this extreme case we have to consider that E_X″,q* will be positive if φ* + φ > 1,33.

Before we turn to the formal analysis, one should note that Froot and Stein (1991) have argued that foreign direct investment—in a world of imperfect capital markets—should positively depend on q*; the authors have presented empirical evidence for the US that the ratio of FDI inflows to GDP positively depends on the real exchange rate. A fortiori one may consider the case that α and α* depend on q*. In the formal analysis, we will ignore the FROOT/STEIN argument at first and only later we will take into account that α* is a positive function of the real exchange rate while α is a negative function of q*.

2.1 Formal analysis

The formal analysis starts by considering the ratio of real exports to real imports and taking logarithms. We then calculate the elasticity of X″ with respect to q* (see equation 10). It should be noted that we will refer to the relative size of the foreign economy through the parameter Ω:= Y*/Y. One may indeed use such a parameter here while a general equilibrium approach—in which Y and Y* also would depend on q*—does not require us to stipulate any given ratio Y*/Y. The interesting point in deriving the subsequent modified Marshall-Lerner condition is that the limit case of Ω approaching infinity represents the case of a small of economy (Y relative to Y* is very small), while the case of Ω approaching zero means that we consider a setup in which the home economy is dominant (Y* is very small relative to Y). The first step in our analysis is to consider the effect that real exports are proportionate to foreign GNP; and that real imports are proportionate to real GDP—an aspect which already has been raised in Welfens (2008) in the context of an asymmetric general equilibrium model (FDI flows only in one direction). As ${\hbox{X}}\prime \prime : = {\hbox{X}}/({\hbox{q}}*{\hbox{J}})$ we have:

$$ \eqalign{ X\prime \prime = \frac{{{\text{q}}{ *^{{\varphi * }}}{\text{x[Y}} * {(1 - }\alpha \beta * {) + }\alpha * \beta {\text{Y/q}} * {]}}}{{{\hbox{q}}{ *^{{{1 - }\varphi }}}{\hbox{j [Y(1 - }}\alpha * \beta {\hbox{) + q}} * \alpha \beta * {\hbox{Y}} * {]}}} \cr = {\hbox{q}}{ *^{{\varphi * { + }\varphi { - 2}}}}\frac{{{\text{ x[q}} * {\text{Y}} * {(1 - }\alpha \beta * {) + }\alpha * \beta {\text{Y]}}}}{{{\hbox{j[Y(1 - }}\alpha * \beta {\hbox{) + q}} * \alpha \beta * {\hbox{Y}} * {]}}} < 0 \cr }<!endgathered> $$

(7)

Using the parameter Ω from the implicit definition Y* = ΩY gives:

$$ \eqalign{ \ln (X\prime \prime ) = \ln \left( {{\hbox{q}}{ *^{{\varphi * { + }\varphi { - 2}}}}} \right) + \ln \left( {{\hbox{x [q}} * {\hbox{Y}} * {(}1 - \alpha \beta * {) + }\alpha * \beta {\hbox{Y]}}} \right) - \ln \left( {{\hbox{j[Y(}}1 - \alpha * \beta {\hbox{) + q}} * \alpha \beta * {\hbox{Y}} * {]}} \right) \cr = (\varphi * { + }\varphi - {2})\ln (q * ) + \ln (x) - \ln (j) + \ln \left( {{\hbox{q}} * {\hbox{Y}} * {(1} - \alpha \beta * {) + }\alpha * \beta {\hbox{Y}}} \right) - \ln \left( {{\hbox{Y(1 - }}\alpha * \beta {\hbox{) + q}} * \alpha \beta * {\hbox{Y}} * } \right) \cr = (\varphi * { + }\varphi - {2})\ln (q * ) + \ln (x) - \ln (j) + \ln \left( {{\hbox{q}} * \Omega {\hbox{Y(1 - }}\alpha \beta * {) + }\alpha * \beta {\hbox{Y}}} \right) - \ln \left( {{\hbox{Y(1 - }}\alpha * \beta {\hbox{) + q}} * \alpha \beta * \Omega {\hbox{Y}}} \right) \cr = (\varphi * { + }\varphi - {2})\ln (q * ) + \ln (x) - \ln (j) + \ln ({\hbox{q}} * \Omega - \alpha \beta * {\hbox{q}} * \Omega { + }\alpha * \beta ) - \ln ({1 - }\alpha * \beta {\hbox{ + q}} * \alpha \beta * \Omega ) \cr }<!endgathered> $$

(8)

The elasticity ${E_{{X\prime \prime, q * }}}$ is calculated using the equivalence:

$$ {E_{{X\prime \prime, q * }}} = \frac{{d\ln (X\prime \prime )}}{{d\ln (q * )}} = \frac{{d\ln (X\prime \prime )}}{{dq * }}{\left( {\frac{{d\ln (q * )}}{{dq * }}} \right)^{{ - 1}}} = \frac{{d\ln (X\prime \prime )}}{{dq * }}q * $$

(9)

$$ {E_{{X\prime \prime, q * }}} = \varphi * { + }\varphi { - 1 + }\left[ {\frac{{1 - \alpha \beta * }}{{1 - \alpha \beta * { + }\frac{{\alpha * \beta }}{{{\hbox{q}} * \Omega }}}} - \frac{{\alpha \beta * }}{{\frac{{{1} - \alpha * \beta }}{{{\hbox{q}} * \Omega }}{ + }\alpha \beta * }} - 1} \right] < \varphi * { + }\varphi - {1} $$

(10)

The inequality holds because [.] is negative:

$$ \matrix{ {\frac{{1 - \alpha \beta * { }}}{{1 - \alpha \beta * { + }\frac{{\alpha * \beta }}{{{\hbox{q}} * \Omega }}}} - \frac{{\alpha \beta * }}{{\frac{{{1} - \alpha * \beta }}{{{\hbox{q}} * \Omega }}{ + }\alpha \beta * }} - 1 < 0} \hfill \cr {\frac{{1 - \alpha \beta * }}{{1 - \alpha \beta * { + }\frac{{{ }\alpha * \beta }}{{{\hbox{q}} * \Omega }}}} < 1 + \frac{{\alpha \beta * }}{{\frac{{{1} - \alpha * \beta }}{{{\hbox{q}} * \Omega }}{ + }\alpha \beta * }}} \hfill \cr {\frac{{{1} - \alpha * \beta }}{{{\hbox{q}} * \Omega }}{ + }\alpha \beta * - \frac{{{1} - \alpha * \beta }}{{{\hbox{q}} * \Omega }}\alpha \beta * - \alpha \beta * \alpha \beta * { < }\left( {\frac{{{1} - \alpha * \beta }}{{{\hbox{q}} * \Omega }}{ + 2}\alpha \beta * } \right)\left( {1 - \alpha \beta * { + }\frac{{{ }\alpha * \beta }}{{{\hbox{q}} * \Omega }}} \right)} \hfill \cr {\alpha \beta * - 1{ < }\frac{{{1} - \alpha * \beta }}{{{{\left( {{\hbox{q}} * \Omega } \right)}^2}}}{ + 2}\frac{{\alpha * \beta }}{{{\hbox{q}} * \Omega }}} \hfill \cr {0 < \frac{{{1} - \alpha * \beta }}{{{{\left( {{\hbox{q}} * \Omega } \right)}^2}}}{ + 2}\frac{{\alpha * \beta }}{{{\hbox{q}} * \Omega }} + \left( {1 - \alpha \beta * } \right)} \hfill \cr }<!end array> $$

$$ {\hbox{With}}\;\alpha = 0\;{\hbox{this}}\;{\hbox{gives:}}{E_{{X\prime \prime, q * }}} = \varphi * { + }\varphi - {1} - \frac{1}{{1{ + }\frac{{{\text{q}} * \Omega }}{{\alpha * \beta }}}} $$

(10′)

$$ {\hbox{With}}\;\alpha * = 0\;{\hbox{this}}\;{\hbox{gives}}:{E_{{X\prime \prime, q * }}} = \varphi * { + }\varphi - 1 - \frac{1}{{1 + \frac{{1}}{{\alpha \beta * {\hbox{q}} * \Omega }}}} $$

(10″)

For $1 - \alpha * \beta - \alpha \beta * > 0$ it also holds that $\varphi * + \varphi - 1 > {E_{{X\prime \prime, q * }}} > \varphi * + \varphi - 2$ as:

$$ \eqalign{ \frac{{{1} - \alpha * \beta }}{{{\hbox{q}} * \Omega \alpha \beta * }} > \frac{{\alpha * \beta }}{{\left( {1 - \alpha \beta * } \right){\hbox{q}} * \Omega }} \hfill \cr {(1} - \alpha * \beta {)(}1 - \alpha \beta * {\hbox{)q}} * \Omega > {\hbox{q}} * \Omega \alpha \beta * \alpha * \beta \hfill \cr {\hbox{q}} * \Omega - \alpha * \beta {\hbox{q}} * \Omega - \alpha \beta * {\hbox{q}} * \Omega > 0 \hfill \cr 1 - \alpha * \beta - \alpha \beta * > 0 \hfill \cr }<!endgathered> $$

For the limit processes Ω → ∞ and Ω → 0 it holds that:

$$ {E_{{X\prime \prime, q * }}} = \varphi * { + }\varphi - {2} $$

(11)

In the case of αβ* = α* β = 0.5 the elasticity is as well given as:

$$ {E_{{X\prime \prime, q * }}} = \varphi * { + }\varphi - {2} $$

(12)

Let us also consider the special case of two rather similar countries with symmetrical two-way FDI. In the case of q*Ω = 1 and αβ* = α*β the elasticity is given by:

$$ {E_{{X\prime \prime, q * }}} = \varphi * { + }\varphi - {1} - {2}\alpha \beta * $$

(13)

So far we have analyzed a setup in which there is a distinction between GDP and GNP, namely because there is foreign direct investment. Implicitly, however, we have focussed on a situation in which there are no effective international profit transfers (so all profits obtained abroad are invested in the host country). Alternatively, we could say that the conditions derived so far refer to the reaction of the trade balance to the real exchange rate. Let us now consider the current account in the case that there are international profit transfers. In the following analysis, we want to consider profit transfers received (and transferred to the source country) as an explicit element of overall exports X (and similarly on the import side) so that X is substituted by $\widetilde{X}$

$$ \widetilde{X} = {\hbox{q}}{ *^{{\varphi * }}}{\hbox{x}}\left[ {{\hbox{Y}} * ({1} - \alpha {\ss}*) + \alpha *\beta {\hbox{Y}}/{\hbox{q}} * } \right] + \alpha \beta * {\hbox{Y}} * /{\hbox{q}} * $$

(14)

$$ \widetilde{\hbox{J}} = {\hbox{q}}{ *^{{ - \varphi }}}{\hbox{j}}\left[ {{\hbox{Y}}\left( {{1} - \alpha * \beta } \right) + {\hbox{q}} * \alpha \beta * {\hbox{Y}} * } \right] + {\hbox{q}} * \alpha * \beta {\hbox{Y}} * $$

(15)

This gives a new condition for the ratio of overall exports to overall imports:

$$ X\prime \prime \prime = \frac{{{\text{q}}{ *^{{1 + \varphi * }}}{\text{x[Y}} * {(1} - \alpha \beta * {) + }\alpha * \beta {\text{Y/q}} * {] + }\alpha \beta * {\text{Y}} * }}{{{\hbox{q}}{ *^{{{2 - }\varphi }}}{\hbox{j [Y(1}} - \alpha * \beta {\hbox{) + q}} * \alpha \beta * {\hbox{Y}} * {\hbox{] + q}}{ *^{{3}}}\alpha * \beta {\hbox{Y}} * }} > 0 $$

(16)

$$ \eqalign{ {\hbox{ln(X}}\prime \prime \prime {)} = \ln \left( {{\hbox{q}}{ *^{{1 + \varphi * }}}{\hbox{x[Y}} * {(1} - \alpha \beta * {) + }\alpha * \beta {\hbox{Y/q}} * {] + }\alpha \beta * {\hbox{Y}} * } \right) \cr - \ln \left( {{\hbox{q}}{ *^{{{2} - \varphi }}}{\hbox{j [Y(1}} - \alpha * \beta {\hbox{) + q}} * \alpha \beta * {\hbox{Y}} * {\hbox{] + q}}{ *^{{3}}}\alpha * \beta {\hbox{Y}} * } \right) \cr }<!endgathered> $$

(17)

$$ \eqalign{ {E_{{{\rm{X}}\prime \prime \prime {\rm{,q}} * }}} = \frac{{{(}1 + \varphi * {\text{)q}}{ *^{{\varphi * }}}{\text{xY}} * {(1} - - \alpha \beta * {) + }\varphi * {\text{q}}{ *^{{\varphi * - {1}}}}\alpha * \beta {\text{Y}}}}{{{\hbox{q}}{ *^{{1 + \varphi * }}}{\hbox{x[Y}} * {(1} - \alpha \beta * {) + }\alpha * \beta {\hbox{Y/q}} * {] + }\alpha \beta * {\hbox{Y}} * }} \cr - \frac{{\left( {{2} - \varphi } \right){\text{q}}{ *^{{{1} - \varphi }}}{\text{j[Y(1}} - \alpha * \beta {\text{) + q}} * \alpha \beta * {\text{Y}} * {\text{] + q}}{ *^{{{2} - \varphi }}}{\text{j}}\alpha \beta * {\text{Y}} * {\text{ + 3q}}{ *^{{2}}}\alpha * \beta {\text{Y}} * }}{{{\hbox{q}}{ *^{{{2} - \varphi }}}{\hbox{j[Y(1}} - \alpha * \beta {\hbox{) + q}} * \alpha \beta * {\hbox{Y}} * {\hbox{] + q}}{ *^{{3}}}\alpha * \beta {\hbox{Y}} * }} \cr }<!endgathered> $$

(18)

In the case of Y* = ΩY and α* = 0 this gives:

$$ {E_{{X\prime \prime \prime, q * }}} = (1 + \varphi * )q{ *^{{ - 1}}}\frac{1}{{1{ + }\frac{{\alpha \beta * }}{{{\hbox{q}}{ *^{{\varphi * { - 1}}}}{\hbox{x(1 - }}\alpha \beta * {)}}}}} - \left( {{2 - }\varphi } \right){\hbox{q}}{ *^{{ - 1}}} + q{ *^{{ - 1}}}\frac{1}{{\frac{{1}}{{\alpha \beta * {\hbox{q}} * \Omega }}{ + 1}}} $$

(19)

If Y* = ΩY and α = 0 this gives:

$$ {E_{{X\prime \prime \prime,q*}}} = \frac{{\left( {1 + \varphi * } \right){\text{x }}\Omega { + }\varphi * {\text{q}}{ *^{{{ - 1}}}}\alpha * \beta }}{{{\hbox{ x [q}} * \Omega { + }\alpha * \beta {]}}} - \frac{{\left( {{2 - }\varphi } \right){\text{q}}{ *^{{{ - 1 - }\varphi }}}{\text{j(1 - }}\alpha * \beta {) + 3}\alpha * \beta \Omega }}{{{\hbox{q}}{ *^{{{ - }\varphi }}}{\hbox{j(1 - }}\alpha * \beta {) + }\alpha * \beta \Omega }} $$

(20)

Even considering purely asymmetric cases does not yield clear results.

2.2 The FROOT-STEIN aspect

Froot and Stein (1991) have argued that in a world of imperfect international capital markets, foreign direct investment inflows will be a positive function of the real exchange rate, and international bidding games are influenced by the amount of equity capital a foreign bidder can put up—the higher the amount is, expressed in units of the host country’s currency, the higher the loan which can be obtained for a leveraged international mergers & acquisition. A real depreciation of the host country currency implies that foreign bidders will have more equity capital and thus are more likely to take over firms in country 1 (host country). Let us go back to equation (10) and assume that indeed α and α* are a function of q*. Note that considering the Froot and Stein (1991) effect, namely that α is a negative function of q* and α* is a positive function of q* only slightly modifies the result obtained.

$$ {E_{{X\prime \prime, q * }}} = \varphi * { + }\varphi - {2 + }\frac{{\Omega - \alpha \beta * \Omega - {\alpha_{{q * }}}\beta * {\text{q}} * \Omega + {\alpha_{{q * }}}\beta }}{{\Omega {\hbox{q}} * - \alpha \beta * {\hbox{q}} * \Omega { + }\alpha * \beta }} - \frac{{ - \alpha_{{{\rm{q}} * }}^{ * }\beta { + }\alpha \beta * \Omega + {\text{q}} * {\alpha_{{q * }}}\beta * \Omega }}{{{1} - \alpha * \beta {\hbox{ + q}} * \alpha \beta * \Omega }} $$

(21)

The general case with FDI inflows and FDI outflows is, however, quite cumbersome. Even the simple case of asymmetric FDI outflows does not yield straightforward analytical results so that empirical analysis will have to shed more light on the results derived.

3 Policy conclusions

Taking into account foreign direct investment and the role of multinational companies implies a modified Marshall-Lerner condition. Compared to the original Marshall-Lerner condition one may state the general finding that the sum of the absolute elasticities must exceed unity by an additional term; this holds at least in a world of asymmetric cumulated foreign direct investment. This is an interesting insight, but it should not be considered a general implication that real exchange rate movements will be less powerful in a world of globalization—a two country model with FDI—than in a world without factor mobility. No such implication can be stated without looking at least a simple macro model (this is done in the appendix, namely for a specific parameter constellation). However, it is clear that in a consistent Mundell-Fleming model with FDI a real depreciation (appreciation) would shift the IS curve less to the right (left) than the traditional Mundell-Fleming approach suggests. The effectiveness of fiscal policy and monetary policy under alternative exchange rate regimes thus will be affected. The international debate about current account imbalances should take into account the role of cumulated FDI: There will be no adequate resolution of the US-China dispute about current account imbalances and the role of nominal and real exchange rate adjustment if the modified Marshall Lerner condition is not taken into account.

As regards the need to modify the Marshall Lerner condition in a world with foreign direct investment—and international profit transfers—a general analysis with inward FDI and outward FDI does not easily yield unambiguous results. More straightforward are the analytical findings for certain special parameter constellations.

The special case of asymmetric pure FDI outflow (and no profit repatriation) has shown that the requirements for the sum of the import elasticities in the home and the foreign countries are sharper than the original Marshall Lerner condition. This case is approximately represented by the combination US and non-OECD countries (as country 2) so that a real depreciation of the US currency would have to be stronger than the traditional Marshall Lerner condition suggests (also the EU in combination with non-OECD countries are a similar case). Interestingly, a sharper condition than the original Marshall Lerner requirement also holds in the case of a rough internationalization symmetry, which roughly is the US and the EU. If the US current account deficit is largely vis-à-vis Asia and the EU—as is the case in reality—, the elasticity requirements are clearly sharper than Marshall Lerner suggests. There is, however, no reason for a general elasticity pessimism. While it is true that modern economic globalization brings about a larger role for both FDI and multinational companies, one should also consider the growing role of an international splitting up of the value-added chain, which includes a rising share of intermediate products exported by Newly Industrialized Countries (e.g. Mexico, Brazil, Asian NICs). Since exporters from developing countries and NICs can hardly pursue pricing to market behavior—as opposed to innovative exporters of final products in OECD countries—, the changing international division of labor could lead to an increase in import price elasticities. With respect to the US, there is considerable evidence that pricing to market behavior has weakened—and US import price elasticities have increased—in the context of rising imports from Mexico in the period 1985–2005 (Bussiere and Peltonen 2008, 24). The Bussiere and Peltonen analysis highlights several influences on the import elasticities and comes up with several key insights, namely that the elasticity of trade prices in NICs is not significantly higher than in OECD countries; elasticities are mainly determined by macroeconomic influences such as the inflationary environment and the exchange rate regime—product differentiation also plays a role. Moreover, elasticities are found to be strongly correlated across countries. Exchange rate pass-through to import prices has reduced in the US and in some other countries, which is consistent with an increase in pricing-to-market in several NICs and particularly with a change in the geographical composition of imports of the US.

As regards the role of international profit transfers, one may, on the one hand, consider periods of tranquility in which a normal share of profits made abroad is repatriated. On the other hand, periods of crisis in which firms in FDI source countries—facing a liquidity shock—tend to reduce new outward FDI might therefore want to repatriate a larger share of profits. (In periods of tranquility profits abroad, this will typically be reinvested to a large extent.)

The case of asymmetric pure FDI inflow (and not profit repatriation) implies weaker elasticity requirements than the Marshall Lerner condition. A typical case of such an asymmetric FDI inflow pattern is China. By implication, real exchange rate adjustments by China should be a relatively powerful instrument to affect China’s current account.

There are several issues which could be crucial in future research: (i) Calculating a fundamental equilibrium exchange rate will have to take into account—current and cumulated—FDI inflows and FDI outflows. (ii) In a macro model with rational expectations and a fixed exchange rate regime economic agents will want to anticipate the time path of all policy variables. This, however, creates a difficult problem, since it is doubtful that nominal exchange rates can be credibly fixed on the one hand. On the other hand, one would have to anticipate both the time path of the foreign price level relative to the domestic price level, which in turn puts the focus on domestic monetary policy and monetary policy in the anchor country. Moreover, there also could be a certain probability of a change in the exchange regime, which makes modeling rather complex—at the same time this raises new empirical issues. (iii) The enormous asymmetry of the world economy in terms of inward (cumulated) FDI-GNP shares and outward (cumulated) FDI-GNP shares raises some doubts about the notion of a real effective exchange rate, since this is related only to geographical patterns of trade but does not consider relative FDI stocks.

Key concepts in international macroeconomics—e.g., the fundamental real equilibrium exchange rate—will have to reconsidered in the modified context emphasized here. Export functions should be stated in a way that the quantity of exports depends of real foreign GNP (not on foreign GDP) and imports should be considered as depending on real GNP. Finally, one may note that FDI is typically linked to international technology transfer and future research might look into these aspects. For example, it is quite interesting to consider an asymmetric, two country model—with country 1 being the only source country—in which the level of technology abroad (A*) is a positive function of α. Such a perspective corresponds to the FDI approach of Dunning (1977) and is also in line with empirical findings for some countries (e.g. Görg et al. 2006). As FDI outflows will reflect ownership-specific advantages (read: technology advantages) of the respective firms in the source country, FDI inflows in country 2 will bring about an international technology transfer. Finally, one may note that within a macro model, capital flows and certain model parameters affect trade dynamics in several important ways: (1) the net asset position of the private sector may affect savings, imports and exports; (2) the structure of capital flows—the share of FDI in total capital flows—will be affected by the real exchange rate (FROOT-STEIN argument); and the parameters of the money demand function and the adjustment speed of learning in the equation describing exchange rate expectations—e.g., in the context of the DORNBUSCH model—will affect the degree of nominal and real exchange rate overshooting. If there is an excessive, temporary depreciation, this will bring about higher FDI inflows, which in turn will affect both GDP and GNP. Thus there is a broad array of new issues to be considered in further theoretical and empirical research.

It is well known that the real exchange rate is not an ideal instrument to bring about short-run changes of the current account—adjustment in the current account can, of course, also result from changes in domestic absorption: As the current account position X′ (net exports of goods and services) is equal to Y – [C(…) + I(…) + G] a rise of the production potential Ypot and a fall of domestic absorption—the sum of consumption C(…), investment I(…) and government consumption—will raise net exports of goods and services; a consistent formulation of X′(…) should reflect the arguments of the consumption function and the investment function; e.g. if consumption is proportionate to real GNP and if the share of the foreign capital stock owned by investors of country I is α the consumption function reads: ${\hbox{C}} = {\hbox{c(1}} - \tau {)}[{\hbox{Y}} + \alpha \beta {\hbox{q}} * {\hbox{Y}} * ]$; moreover, if the investment function can simply be stated as ${\hbox{I}} = {\hbox{I(r}},{\hbox{q}}*{)}$ where r is real investment and q* is the real exchange rate we would have ${\hbox{X}}\prime = {\hbox{X}}\prime ({\hbox{Y}},{\hbox{r}},{\hbox{q}} * {\hbox{Y}} * )$ which slightly differs from the standard equation ${\hbox{X}}\prime ({\hbox{Y}},{\hbox{Y}}*,{\hbox{q}}*)$—it should, however, not be difficult to empirically verify that trade is related to the real interest rate as trade financing plays a role for both exports and imports. One may, however, raise the question whether or not the structure of FDI inflows will affect the elasticity of exports and imports of goods and services.

• If FDI inflows are mainly within the nontradables sector it is clear that the reaction of exports with respect to the real exchange rate will not be affected.

• If FDI inflows are mainly in the tradables sector one may assume that the export elasticity will increase in the medium term as a rising share of the additional production of the tradables sector will be directed towards the world market. However, in the long run one may witness that product upgrading and quality improvements are strong enough to bring about a reduction of the export elasticity. The import elasticity should increase over time as subsidiaries of MNCs will increasingly try to exploit their experience with international trading partners.

• A specific issue concerns developing countries which often obtain tied aid from OECD countries; in such an institutional setting one will not expect that the import elasticity will increase over time if there are FDI inflows. The impact of tied aid programs often will be strong enough to avoid a significant rise of the import elasticity of goods and services.

As regards the impact of capital flows on domestic investment there is empirical evidence for eastern European transition countries which shows that capital inflows can have contemporaneous and long-term effects on investment; in countries with weakly developed financial markets FDI inflows stimulate investment in other sectors of the economy so that there are spillover effects—in countries with better financial governance indicators and stronger institutions long term loans raise domestic investment and FDI inflows generate small spillover effects in the long run.

Appendix: macro model with FDI

The role of multinational companies has increased in the global economy over many years, and hence foreign direct investment (FDI) has therefore become an increasingly important aspect of real economic life. As multinational companies are major drivers of technological progress, FDI is not only affecting capital formation in host countries but also international technology transfer and supply-side dynamics. In the presence of FDI, one has to make a distinction between real gross domestic product (Y) and real gross national income (Z). The difference can be considerable as is known from countries such as Ireland, Hungary or China, which represent small countries and a large country, respectively, ranking among leading host countries—with not much outward FDI. The US, Germany, France, the UK and the Netherlands as well as Switzerland are key players in both a host country perspective and a source country perspective. A simplified model with asymmetric FDI (the home country is the only source country) brings out some new ambiguities with respect to the multiplier analysis: The modified Mundell Fleming model with FDI looks more complicated than the traditional setup.

In a simple, two-country model—where both countries produce according to a Cobb-Douglas function, namely ${\hbox{Y}} = {{\hbox{K}}^{\beta }}{({\hbox{AL}})^{{{1} - }}}^{\beta }$ and Y* = – with K, A, L and *denoting capital, knowledge, labor and foreign variables, respectively (0 < β < 1)—the GNP of the single source country 1 (home country) is ${\hbox{Z}} = {\hbox{Y}} + \alpha \beta * {\hbox{Y}} * {\hbox{q}} *$; here α is the share of country 2’s capital stock owned by investors from country 1 and q* is the real exchange rate eP*/P where e is the exchange rate and P the price level. It has been assumed that in both countries there is competition in goods markets and labor markets so that production factors are rewarded in accordance with the marginal product rule, and hence profits from abroad, expressed in units of country 1’s output, are β*Y*q*. If one wants to understand the role of FDI inflows, it is crucial to point out that GNP is given by ${\hbox{Z = Y}}({1} - \alpha * \beta )$, where α* is the share of the host country’s capital stock owned by investors from country 2; again for the sake of simplicity the setup is asymmetric so that we consider only capital inflows in country 1. Does FDI make a difference in terms of the Mundell Fleming model? It does for various reasons where one assumption adopted here is that investment is a positive function of the real exchange rate q*; this link between the real exchange rate and FDI inflows was pointed out in the context of an approach with imperfect capital markets by Froot and Stein (1991) who presented positive empirical evidence for the US. Moreover, one has to consider that consumption is proportionate to GNP and that imports are proportionate to GNP; and one has to add an additional element, namely that net capital imports are a positive function of q* so that the augmented Mundell-Fleming model reads for the case of a non-inflationary world (τ is the tax rate, b, b′ and b″ as wll as j and x are positive parameters, G is government consumption; by assumption real exports X = jZ*/q* so that the elasticity of X with respect to q* is unity and real imports J = jZ/q* so that the respective elasticity is minus one:

$$ \matrix{ {{\hbox{Y}} = {\hbox{c(1}} - \tau {)(1} - \alpha * \beta {\hbox{)Y}} + \left[ {{\hbox{bq}} * --{\hbox{b}}\prime {\hbox{r}} + {\hbox{b}}\prime \prime \left( {\beta {\hbox{Y}}/{\hbox{K}} - \beta * {\hbox{Y}} * /{\hbox{K}} * } \right)} \right] + } \hfill \cr { + {\hbox{G}} - {\hbox{q}} * {\hbox{jY}}({1} - \alpha * \beta )/{\hbox{q}} * + {\hbox{xq}}*[{\hbox{Y}}* + \alpha *\beta {\hbox{Y}}/{\hbox{q}}*]({\hbox{goods}}\;{\hbox{market}}\;{\hbox{equilibrium}})} \hfill \cr }<!end array> $$

(I)

$$ {\hbox{M}}/{\hbox{P}} = {\hbox{hY}}--{\hbox{h}}\prime {\hbox{r(money}}\;{\hbox{market}}\;{\hbox{equilibrium)}} $$

(II)

$$ {\hbox{n}}\prime {\hbox{r}} - {\hbox{n}}\prime {\hbox{r}} * \prime + [{\hbox{b}}\prime \prime \left( {\beta {\hbox{Y}}/{\hbox{K}}--\beta * {\hbox{Y}} * /{\hbox{K}} * } \right) + {\hbox{n}}\prime \prime {\hbox{q}} * \left] { = {\hbox{q}} * {\hbox{jY}}/{\hbox{q}} * - {\hbox{xq}} * } \right[{\hbox{Y}} * + \alpha * * \beta {\hbox{Y}}]\;({\hbox{equilibrium}}\;{\hbox{condition}}\;{\hbox{for}}\;{\hbox{the}}\;{\hbox{foreign}}\;{\hbox{exchange}}\;{\hbox{market}}/{\hbox{external}}\;{\hbox{equilibrium}}) $$

(III)

The consumption function thus is ${\hbox{C}} = {\hbox{c}}({1} - \tau )({1} - \alpha * \beta ){\hbox{Y}}$, the investment function I = I(q*,r,Ω) where r is the real interest rate and Ω the differential between the domestic marginal product of capita ${{\hbox{Y}}_{\rm{K}}}({{\hbox{Y}}_{\rm{K}}} = \beta {\hbox{Y}}/{\hbox{K}})$ and Y*_K* (abroad ${\hbox{Y}}_{\rm{K}}^{ * } = \beta * {\hbox{Y}} * /{\hbox{K}} *$). As regards the demand for money, it has been assumed that the real demand for money is proportionate to real GDP and a negative function of the interest rate; an alternative specification would be the case that the money demand is not proportionate to Y but to Z. The modified investment function—taking into account the arguments of FROOT/STEIN with respect to the link between FDI inflows and q*—implies that a real depreciation will bring about a stronger rightward shift of the new IS curve than the standard Mundell Fleming model; a real depreciation will not only affect net exports of goods but also raise FDI inflows. With respect to the medium term there is, however, a theoretical caveat which concerns the distinction between international mergers and acquisitions—the type of FDI emphasized by FROOT/STEIN—and greenfield investment. (q* also will affect FDI inflows here because acquiring land for building a new factory is facilitated if there is a real depreciation of the currency of the host country, but the medium term effect could be rather a future excess supply in the goods market—if the capacity effect of investment were considered.)

Real net capital inflows are assumed to be a positive function of r−r*′ (r*′:= r* + a′ where a′ is the expected depreciation rate of the currency which is assumed to be zero; n′ and n′ are positive parameters) and a positive function of both q* and Ω; the square bracketed term on the left hand side of the equilibrium condition for the foreign exchange market stands for FDI inflows while n′(r−r*) represents portfolio capital inflows.

It is only for the sake of simplicity that we consider the simple case that the absolute elasticity of imports with respect to q* and the elasticity of goods exports with respect to q* are assumed to be unity. Next let us solve for the special case of exports inelastic with respect to q* for the real exchange rate (in this case any real exchange rate change will leave net exports of goods unaffected) so that we get:

$$ {\hbox{q}} * = \left[ {{\hbox{n}}\prime {\hbox{r}} * \prime - {\hbox{n}}\prime {\hbox{r}} + {\hbox{b}}\prime \prime \beta * {\hbox{Y}} * /{\hbox{K}} * - {\hbox{xY}} * \left] {/{\hbox{n}}\prime \prime + } \right[\left( {{\hbox{j}} + {\hbox{x}}\alpha * \beta - {\hbox{b}}\prime \prime \beta /{\hbox{K}}} \right)/{\hbox{n}}\prime \prime } \right]{\hbox{Y}} $$

(IV)

If one considers only the foreign exchange market and ignores the goods market equilibrium condition, the implication is that the real exchange rate is a positive function of Y—provided that j + xα*β > b″β/K—and a negative function of Y*; the latter holds if x > b″β*Y*/K*. It is noteworthy that the slope of the balance of payments equilibrium line EE—not shown here—could have a zero slope in r-Y space, namely if n″ approaches infinity (the traditional case) and if j + xα*β = b″β/K. The latter case implies that a marginal rise in goods imports generated by a rise in Y—net of the effect of higher exports generated by the rise of GNP abroad—would automatically be offset by rising FDI inflows, which will react to a higher Y as (at given K) this represents a rise in the marginal product of capital.

Note that in principle, a real depreciation not only affects net goods imports but also net capital inflows. If the parameter n″ is relatively large, there is a strong financing impact of a real depreciation, since not only a rise in q* will dampen net imports of goods and services, but net imports of capital will increase at the same time through higher FDI inflows. Subsequently we summarize the whole set of equations and derive the multipliers.

$$ Y = cY - b\prime r * G + xq * Y * - jY $$

(V)

$$ \frac{M}{P} = hY - h\prime r $$

(VI)

$$ u\prime r - {\hbox{u}}\prime {\hbox{r}} * \prime = {\hbox{jY}} - {\hbox{xq}} * {\hbox{Y}} * $$

(VII)

$$ \left( {\matrix{ {1 - c + j} & {b\prime } & { - xY * } \cr h & { - h\prime } & 0 \cr { - j} & {u\prime } & {xY * } \cr }<!end array> } \right)\left( {\matrix{ {dY} \cr {dr} \cr {dq * } \cr }<!end array> } \right) = \left( {\matrix{ 1 \hfill & 0 \hfill & {xq * } \hfill & 0 \hfill \cr 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill \cr 0 \hfill & 0 \hfill & { - xq * } \hfill & {u\prime } \hfill \cr }<!end array> } \right)\left( {\matrix{ {dG} \cr {dM/P} \cr {dY * } \cr {dr * \prime } \cr }<!end array> } \right) $$

(VIII)

$$ \eqalign{ U = \det \left( {\matrix{ {1 - c + j} & {b\prime } & { - xY*} \cr h & { - h\prime } & 0 \cr { - j} & {u\prime } & {xY*} \cr }<!end array> } \right) = - xY*\left( {\matrix{ h & { - h\prime } \cr { - j} & {u\prime } \cr }<!end array> } \right) + xY * \left( {\matrix{ {1 - c + j} & {b\prime } \cr h & { - h\prime } \cr }<!end array> } \right) \cr = - xY*\left( {h(u\prime + b) + (1 - c)h\prime } \right) < 0 \cr }<!endgathered> $$

(IX)

$$ \frac{{dY}}{{dG}} = - \frac{{h\prime xY * }}{U} > 0 $$

(X)

$$ \frac{{dY}}{{dM/P}} = - \frac{{xY * }}{U}(b\prime + u\prime ) > 0 $$

(XI)

$$ \frac{{dY}}{{dY * }} = 0 $$

(XII)

$$ \frac{{dY}}{{dr * \prime }} = - \frac{{u\prime h\prime xY * }}{U} > 0 $$

(XIII)

$$ \frac{{dr}}{{dG}} = - \frac{{hxY * }}{U} > 0 $$

(XIV)

$$ \frac{{dr}}{{dM/P}} = \frac{{(1 - c)xY * }}{U} < 0 $$

(XV)

$$ \frac{{dr}}{{dY * }} = 0 $$

(XVI)

$$ \frac{{dr}}{{dr * \prime }} = - \frac{{u\prime hxY * }}{U} < 0 $$

(XVII)

$$ \frac{{dq * }}{{dG}} = \frac{{hu\prime - h\prime j}}{U} $$

(XVIII)

$$ \frac{{dq * }}{{dM/P}} = - \frac{{(1 - c + j)u\prime + b\prime j}}{U} > 0 $$

(XIX)

$$ \frac{{dq * }}{{dY * }} = - \frac{{q * }}{{Y * }} < 0 $$

(XX)

$$ \frac{{dq * }}{{dr * \prime }} = - \frac{{u\prime }}{U}\left( {(1 - c + j)h\prime + b\prime h} \right) > 0 $$

(XXI)

$$ \eqalign{ Y = c\left( {Y(1 - \alpha * \beta ) + \alpha \beta * q * Y * } \right) + c\prime \prime \tau \prime (G\prime, G\prime \prime )Y\# + I\left( {r,q *, \beta Y/K - \beta * Y * /K *, \Omega \prime } \right) \hfill \cr { } + G\prime + G\prime \prime + x(q*)\left( {Y(1 - \alpha * \beta ) + \alpha \beta * q * Y * } \right) - q * j(q * )\left( {Y(1 - \alpha * \beta ) + \alpha \beta * q * Y * } \right) \hfill \cr }<!endgathered> $$

(XXII)

$$ \frac{M}{P} = h\left( {Y(1 - \alpha * \beta ) + \alpha \beta * q * Y * } \right) - h\prime r $$

(XXIII)

$$ Q(r,r * \prime, q *, \beta Y/K - \beta * Y * /K *, \Omega \prime ) = q * j(q * )\left( {Y(1 - \alpha * \beta ) + \alpha \beta * q * Y * } \right) - x(q * )\left( {Y(1 - \alpha * \beta ) + \alpha \beta * q * Y * } \right) $$

(XXIV)

$$ \eqalign{ \left( {\matrix{ {{k_3}} & { - {I_r}} & {{k_4}} \cr { - h(1 - \alpha * \beta )} & {h\prime} & { - h\alpha \beta * Y * } \cr {{k_5}} & {{Q_r}} & {{k_6}} \cr }<!end array> } \right)\left( {\matrix{ {dY} \cr {dr} \cr {dq * } \cr }<!end array> } \right) = \left( {\matrix{ {c\prime \prime \tau_{{G\prime }}^{\prime } + 1} \hfill & {c\prime \prime \tau_{{G\prime \prime }}^{\prime } + 1} \hfill & {{k_1}} \hfill \cr 0 \hfill & 0 \hfill & {h\alpha \beta *q*} \hfill \cr 0 \hfill & 0 \hfill & {{k_2}} \hfill \cr }<!end array> } \right)\left( {\matrix{ {dG\prime } \cr {dG\prime \prime } \cr {dY * } \cr }<!end array> } \right) \hfill \cr {k_1} = c\alpha \beta * q * - \beta * {I_{{Y * }}}/K * + x(q * )\alpha \beta * q * - q * j(q * )\alpha \beta * q * \hfill \cr {k_2} = \beta * {Q_{{Y * }}}/K * - x(q * )\alpha \beta * q * + q * j(q * )\alpha \beta * q * \hfill \cr {k_3} = 1 - c\left( {1 - \alpha *\beta } \right) - {I_Y}\beta /K - x\left( {q*} \right)\left( {1 - \alpha * \beta } \right) + q*j\left( {q*} \right)\left( {1 - \alpha *\beta } \right) \hfill \cr {k_4} = - c\alpha \beta * Y * - {I_{{q * }}} - \left( {{x_{{q * }}}\alpha \beta * q * Y * + x(q * )\alpha \beta * Y * } \right) \hfill \cr + \left( {j(q * )\left( {Y(1 - \alpha * \beta ) + \alpha \beta * q * Y * } \right) + q * {j_{{q * }}}\left( {Y\left( {1 - \alpha * \beta } \right) + \alpha \beta * q * Y * } \right) + q * j(q * )\alpha \beta * Y * } \right) \hfill \cr {k_5} = {Q_Y}\beta /K - q*j(q * )\left( {1 - \alpha * \beta } \right) + x\left( {q * } \right)\left( {1 - \alpha * \beta } \right) \hfill \cr {k_6} = {Q_{{q*}}} + \left( {{x_{{q*}}}\alpha \beta * q * Y * + x(q * )\alpha \beta * Y * } \right) \hfill \cr - \left( {j(q * )\left( {Y(1 - \alpha * \beta ) + \alpha \beta * q * Y * } \right) + q * {j_{{q * }}}\left( {Y\left( {1 - \alpha * \beta } \right) + \alpha \beta * q * Y * } \right) + q * j(q * )\alpha \beta * Y * } \right) \hfill \cr }<!endgathered> $$

(XXV)

$$ \eqalign{ U = \det \left( {\matrix{ {{k_3}} & { - {I_r}} & {{k_4}} \cr { - h\left( {1 - \alpha * \beta } \right)} & {h\prime } & { - h\alpha \beta * Y * } \cr {{k_5}} & {{Q_r}} & {{k_6}} \cr }<!end array> } \right) = \cr = {k_3}h\prime {k_6} + {I_r}h\alpha \beta *Y*{k_5} - {k_4}{Q_r}h\left( {1 - \alpha *\beta } \right) \cr + {k_3}h\alpha \beta *Y*{Q_r} - {k_4}{k_5}h\prime - {I_r}h\left( {1 - \alpha *\beta } \right){k_6} \cr }<!endgathered> $$

(XXVI)

We assume U is positive.

$$ \frac{{dY}}{{dG\prime }} = \frac{1}{U}\det \left( {\matrix{ {c\prime \prime \tau_{{G\prime }}^{\prime } + 1} & { - {I_r}} & {{k_4}} \cr 0 & {h\prime } & { - h\alpha \beta * Y * } \cr 0 & {{Q_r}} & {{k_6}} \cr }<!end array> } \right) = \frac{{\left( {c\prime \prime \tau_{{G\prime}}^{\prime} + 1} \right)\left( {h\prime {k_6} + h\alpha \beta * Y * {Q_r}} \right)}}{U} > 0 $$

(XXVII)

$$ \frac{{dY}}{{dG\prime \prime }} = \frac{1}{U}\det \left( {\matrix{ {c\prime \prime \tau_{{G\prime \prime }}^{\prime } + 1} & { - {I_r}} & {{k_4}} \cr 0 & {h\prime } & { - h\alpha \beta * Y * } \cr 0 & {{Q_r}} & {{k_6}} \cr }<!end array> } \right) = \frac{{\left( {c\prime \prime \tau_{{G\prime }}^{\prime } + 1} \right)\left( {h\prime {k_6} + h\alpha \beta * Y * {Q_r}} \right)}}{U} > 0 $$

(XXVIII)

$$ \eqalign{ \frac{{dY}}{{dY * }} = \frac{1}{U}\det \left( {\matrix{ {{k_1}} & { - {I_r}} & {{k_4}} \cr {h\alpha \beta * q * } & {h\prime } & { - h\alpha \beta * Y * } \cr {{k_2}} & {{Q_r}} & {{k_6}} \cr }<!end array> } \right) = \hfill \cr = \frac{{{k_1}h\prime {k_6} + {I_r}h\alpha \beta * Y * {k_2} + {k_4}{Q_r}h\alpha \beta * q * + {k_1}h\alpha \beta * Y * {Q_r} - {k_4}{k_2}h\prime + {I_r}h\alpha \beta * q * {k_6}}}{U} < 0{ } \hfill \cr {\hbox{if}}\;{k_1}h\prime {k_6} < 0 \hfill \cr }<!endgathered> $$

(XXIX)