A macroeconomic analysis of foreign assets and foreign liabilities

Abstract

Although an increase in foreign assets and a decrease in foreign liabilities both increase a nation’s net foreign assets (NFA), they have alternative macroeconomic transmission mechanisms: while an increase in foreign assets is expansionary, the effect of a decrease in foreign liabilities is mixed due to the asymmetry between its income effect and wealth effect on aggregate demand. It is the relative strengths of the NFA’s wealth effect and income effect that determine the existence and natures of a saddle-point equilibrium in the NFA-real balance space as well as its comparative statics. The cointegration analysis suggests that in the 1990s, foreign liabilities bear more weight than foreign assets in the US NFA movement whereas the opposite holds for the case of Japan; therefore, correcting NFA imbalances calls for accelerated money growth and fiscal expenditure pruning in the U.S. but for the opposite policy responses in Japan.

Appendices

Appendix 1 Determination of the saddle-point equilibrium

First, consider the case in which dK* _d  ≠ 0 and dK _f    = 0. Let J ₁ be the Jacobian matrix of Eq. 6 for the case in which foreign assets drive changes in NFA. According to Eq. 6, the determinant of J ₁ can be written as

$$ {\left| {J_{1} } \right|} = m{C}\ifmmode{'}\else$'$\fi_{1} {\left( {{h}\ifmmode{'}\else$'$\fi_{1} - {h}\ifmmode{'}\else$'$\fi_{2} } \right)} < 0, $$

(A1)

where ${h}\ifmmode{'}\else$'$\fi_{1} = L^{{ - 1}}_{1} \prime \frac{{K_{d} + {\text{NFA}}}}{{{\left( {K_{d} + {\text{NFA}} + m} \right)}^{2} }} < 0$ and ${h}\ifmmode{'}\else$'$\fi_{2} = L^{{ - 1}}_{1} \prime {\left[ { - \frac{m}{{{\left( {K_{d} + {\text{NFA}} + m} \right)}^{2} }}} \right]} > 0$.

The sign for the trace of J ₁ is undetermined since

$$ Tr{\left( {J_{1} } \right)} = - C'_{\kern2pt1} - mh'_{\kern2pt1} . $$

(A2)

It follows that the system has a saddle-point equilibrium when changes in NFA totally stem from foreign assets only.

Next, turning to the case in which dK _f    ≠ 0 and dK* _d  = 0, the determinant of the Jacobian matrix J ₂ in Eq. 9 is

$$ {\left| {J_{2} } \right|} = m{h}\ifmmode{'}\else$'$\fi_{2} {\left[ {{F}\ifmmode{'}\else$'$\fi{\left( {1 - {C}\ifmmode{'}\else$'$\fi_{2} } \right)} + {C}\ifmmode{'}\else$'$\fi_{1} } \right]}{\left[ { - \frac{{{C}\ifmmode{'}\else$'$\fi_{1} }} {{{F}\ifmmode{'}\else$'$\fi{\left( {1 - {C}\ifmmode{'}\else$'$\fi_{2} } \right)} + {C}\ifmmode{'}\else$'$\fi_{1} }} - {\left( { - \frac{{{h}\ifmmode{'}\else$'$\fi_{1} }} {{{h}\ifmmode{'}\else$'$\fi_{2} }}} \right)}} \right]}, $$

(A3)

where h′₁ remains the same as above but $h^{\prime } _{2} = - {\left[ {L^{{ - 1}}_{2} \prime {F}\ifmmode{'}\else$'$\fi + L^{{ - 1}}_{1} \prime \frac{m}{{{\left( {K_{d} + {\text{NFA}} + m} \right)}^{2} }}} \right]} < 0$ if and only if the first term in the bracket (the income effect) is greater than the second term (the wealth effect). ∣J ₂∣is negative either when h′ ₂ > 0, or when h′₂ < 0 and the term in the second bracket of Eq. A3 has a positive sign, which actually means that the ${\mathop m\limits^. } = 0$ schedule is steeper than the ${\text{N}}{\mathop {\text{F}}\limits^{\text{.}} }{\text{A}} = 0$ schedule. Additionally, the trace of J ₂ remains unsigned:

$$ Tr{\left( {J_{2} } \right)} = {\left[ { - {F}\ifmmode{'}\else$'$\fi{\left( {1 - {C}\ifmmode{'}\else$'$\fi_{2} } \right)} - {C}\ifmmode{'}\else$'$\fi_{1} } \right]} - m{h}\ifmmode{'}\else$'$\fi_{1} . $$

(A4)

Therefore, under these conditions, the system continues to have a saddle-point equilibrium.

Appendix 2 Slope of the stable branch in the saddle-point equilibrium

Again, first consider the case in which dK* _d  ≠ 0 and dK _f    = 0. The characteristic equation associated with the dynamic system (6) has two characteristic roots:

$$ \lambda {\kern 1pt} {\kern 1pt} _{1} = - \frac{1} {2}{\left( {{C}\ifmmode{'}\else$'$\fi_{1} + m{h}\ifmmode{'}\else$'$\fi_{1} } \right)} - \frac{1} {2}{\sqrt {{\left( {{C}\ifmmode{'}\else$'$\fi_{1} + m{h}\ifmmode{'}\else$'$\fi_{1} } \right)}^{2} - 4{\left( {{C}\ifmmode{'}\else$'$\fi_{1} m{h}\ifmmode{'}\else$'$\fi_{1} - {C}\ifmmode{'}\else$'$\fi_{1} m{h}\ifmmode{'}\else$'$\fi_{2} } \right)}} } < 0, $$

and

$$ \lambda {\kern 1pt} {\kern 1pt} _{2} = - \frac{1} {2}{\left( {{C}\ifmmode{'}\else$'$\fi_{1} + m{h}\ifmmode{'}\else$'$\fi_{1} } \right)} + \frac{1} {2}{\sqrt {{\left( {{C}\ifmmode{'}\else$'$\fi_{1} + m{h}\ifmmode{'}\else$'$\fi_{1} } \right)}^{2} - 4{\left( {{C}\ifmmode{'}\else$'$\fi_{1} m{h}\ifmmode{'}\else$'$\fi_{1} - {C}\ifmmode{'}\else$'$\fi_{1} m{h}\ifmmode{'}\else$'$\fi_{2} } \right)}} } > 0. $$

A solution must assume the form

$$ {\left( {\begin{array}{*{20}c} {{{\text{NFA}} - {\text{\ifmmode\expandafter\bar\else\expandafter\=\fi{N}\ifmmode\expandafter\bar\else\expandafter\=\fi{F}\ifmmode\expandafter\bar\else\expandafter\=\fi{A}}}}} \\ {{m - \ifmmode\expandafter\bar\else\expandafter\=\fi{m}}} \\ \end{array} } \right)} = {\left( {\begin{array}{*{20}c} {{b_{{11}} }} & {{b_{{12}} }} \\ {{b_{{21}} }} & {{b_{{22}} }} \\ \end{array} } \right)}{\left( {\begin{array}{*{20}c} {{e^{{\lambda _{1} t}} }} \\ {{e^{{\lambda _{2} t}} }} \\ \end{array} } \right)}, $$

(A5)

where b _ij are determined by the characteristic vectors of the Jacobian matrix in Eq. 6 and the initial values of NFA and m. For the system to be convergent to its steady state, the initial conditions must be such that b ₁₂ = b ₂₂ = 0, so NFA and m converge exponentially to $\overline{{{\text{NFA}}}}$ and $\overline{m}$. Then, from Eq. A5, we have ${\mathop m\limits^. } = \lambda {\kern 1pt} _{1} {\left( {m - \ifmmode\expandafter\bar\else\expandafter\=\fi{m}} \right)}$, which can be combined with the equation for ${\mathop m\limits^. }$ in Eq. 6 to yield

$$ {\text{NFA}} - {\text{\ifmmode\expandafter\bar\else\expandafter\=\fi{N}\ifmmode\expandafter\bar\else\expandafter\=\fi{F}\ifmmode\expandafter\bar\else\expandafter\=\fi{A}}} = - \frac{{\lambda {\kern 1pt} _{1} + m{h}\ifmmode{'}\else$'$\fi_{1} }} {{m{h}\ifmmode{'}\else$'$\fi_{2} }}{\left( {m - \ifmmode\expandafter\bar\else\expandafter\=\fi{m}} \right)}. $$

(A6)

Eq. A6 defines the positively sloped stable branch for the case in which changes in NFA are produced by changes in foreign assets only.

Likewise, for the other case in which dK _f  ≠ 0 and dK* _d  = 0, the characteristic equation associated with the dynamic system (9) has the following two characteristic roots:

$$\mu {\kern 1pt} _{1} = - \frac{1}{2}{\left( {\phi {\kern 1pt} {\kern 1pt} + mh^{\prime } _{1} } \right)} - \frac{1}{2}{\sqrt {{\left( {\phi {\kern 1pt} {\kern 1pt} {\kern 1pt} + mh^{\prime } _{1} } \right)}^{2} - 4{\left( {\phi {\kern 1pt} {\kern 1pt} {\kern 1pt} mh^{\prime } _{1} - {C}\ifmmode{'}\else$'$\fi_{1} mh^{\prime } _{2} } \right)}} } < 0,$$

where $\phi \equiv {F}\ifmmode{'}\else$'$\fi{\left( {1 - {C}\ifmmode{'}\else$'$\fi_{1} } \right)} + {C}\ifmmode{'}\else$'$\fi_{1}$, and

$$ \mu {\kern 1pt} _{2} = - \frac{1} {2}{\left( {\phi {\kern 1pt} {\kern 1pt} + m{h}\ifmmode{'}\else$'$\fi_{1} } \right)} + \frac{1} {2}{\sqrt {{\left( {\phi {\kern 1pt} {\kern 1pt} {\kern 1pt} + m{h}\ifmmode{'}\else$'$\fi_{1} } \right)}^{2} - 4{\left( {\phi {\kern 1pt} {\kern 1pt} {\kern 1pt} m{h}\ifmmode{'}\else$'$\fi_{1} - {C}\ifmmode{'}\else$'$\fi_{1} m{h}\ifmmode{'}\else$'$\fi_{2} } \right)}} } > 0. $$

Recall that h′₂ < 0 if the income effect on the demand for real balances outweighs the wealth effect. Then, the same analysis as in the former case leads to the following equation that determines the slope of the stable arm in the saddle-point equilibrium:

$$ {\text{NFA}} - {\text{\ifmmode\expandafter\bar\else\expandafter\=\fi{N}\ifmmode\expandafter\bar\else\expandafter\=\fi{F}\ifmmode\expandafter\bar\else\expandafter\=\fi{A}}} = - \frac{{\mu {\kern 1pt} {\kern 1pt} _{1} + m{h}\ifmmode{'}\else$'$\fi_{1} }} {{m{h}\ifmmode{'}\else$'$\fi_{2} }}{\left( {m - \ifmmode\expandafter\bar\else\expandafter\=\fi{m}} \right)}. $$

(A7)

Therefore, the stable arm is downward sloping for the case in which changes in NFA are produced by changes in foreign liabilities only.

Appendix 3 Comparative static analysis of macroeconomic policies

Setting (2) and (5) to zero and totally differentiating them with respect to $\overline{{{\text{NFA}}}} ,\,\ifmmode\expandafter\bar\else\expandafter\=\fi{m}$, g _M , and G produces

$$ {\left( {\begin{array}{*{20}c} {{ - {C}\ifmmode{'}\else$'$\fi_{1} }} & {{ - {C}\ifmmode{'}\else$'$\fi_{1} }} \\ {{ - m{h}\ifmmode{'}\else$'$\fi_{2} }} & {{ - m{h}\ifmmode{'}\else$'$\fi_{1} }} \\ \end{array} } \right)}{\left( {\begin{array}{*{20}c} {{d{\text{\ifmmode\expandafter\bar\else\expandafter\=\fi{N}\ifmmode\expandafter\bar\else\expandafter\=\fi{F}\ifmmode\expandafter\bar\else\expandafter\=\fi{A}}}}} \\ {{d\ifmmode\expandafter\bar\else\expandafter\=\fi{m}}} \\ \end{array} } \right)} = {\left( {\begin{array}{*{20}c} {{dG}} \\ {{ - mdg_{M} }} \\ \end{array} } \right)}.\, $$

(A8)

Applying the Cramer’s rule yields

$$ \frac{{\partial {\text{\ifmmode\expandafter\bar\else\expandafter\=\fi{N}\ifmmode\expandafter\bar\else\expandafter\=\fi{F}\ifmmode\expandafter\bar\else\expandafter\=\fi{A}}}}} {{\partial g_{M} }} = \frac{1} {{{h}\ifmmode{'}\else$'$\fi_{2} - {h}\ifmmode{'}\else$'$\fi_{1} }} > 0,\,\frac{{\partial \ifmmode\expandafter\bar\else\expandafter\=\fi{m}}} {{\partial g_{M} }} = \frac{1} {{{h}\ifmmode{'}\else$'$\fi_{1} - {h}\ifmmode{'}\else$'$\fi_{2} }} < 0; $$

(A9)

$$ \frac{{\partial {\text{\ifmmode\expandafter\bar\else\expandafter\=\fi{N}\ifmmode\expandafter\bar\else\expandafter\=\fi{F}\ifmmode\expandafter\bar\else\expandafter\=\fi{A}}}}} {{\partial G}} = - \frac{{{h}\ifmmode{'}\else$'$\fi_{1} }} {{{C}\ifmmode{'}\else$'$\fi_{1} {\left( {{h}\ifmmode{'}\else$'$\fi_{1} - {h}\ifmmode{'}\else$'$\fi_{2} } \right)}}} < 0,\,\frac{{\partial \ifmmode\expandafter\bar\else\expandafter\=\fi{m}}} {{\partial G}} = \frac{{{h}\ifmmode{'}\else$'$\fi_{2} }} {{{C}\ifmmode{'}\else$'$\fi_{1} {\left( {{h}\ifmmode{'}\else$'$\fi_{1} - {h}\ifmmode{'}\else$'$\fi_{2} } \right)}}} < 0. $$

(A10)

Similarly, following the same analytical procedure for the case in which foreign liabilities rather than foreign assets drive changes in NFA yields

$$ {\left( {\begin{array}{*{20}c} {{ - {F}\ifmmode{'}\else$'$\fi{\left( {1 - {C}\ifmmode{'}\else$'$\fi_{2} } \right)} - {C}\ifmmode{'}\else$'$\fi_{1} }} & {{ - {C}\ifmmode{'}\else$'$\fi_{1} }} \\ {{ - m{h}\ifmmode{'}\else$'$\fi_{2} }} & {{ - m{h}\ifmmode{'}\else$'$\fi_{1} }} \\ \end{array} } \right)}{\left( {\begin{array}{*{20}c} {{d{\text{\ifmmode\expandafter\bar\else\expandafter\=\fi{N}\ifmmode\expandafter\bar\else\expandafter\=\fi{F}\ifmmode\expandafter\bar\else\expandafter\=\fi{A}}}}} \\ {{d\ifmmode\expandafter\bar\else\expandafter\=\fi{m}}} \\ \end{array} } \right)} = {\left( {\begin{array}{*{20}c} {{dG}} \\ {{ - mdg_{M} }} \\ \end{array} } \right)}. $$

(A11)

Applying the Cramer’s rule produces

$$ \frac{{\partial {\text{\ifmmode\expandafter\bar\else\expandafter\=\fi{N}\ifmmode\expandafter\bar\else\expandafter\=\fi{F}\ifmmode\expandafter\bar\else\expandafter\=\fi{A}}}}} {{\partial g_{M} }} = - \frac{{m{C}\ifmmode{'}\else$'$\fi_{1} }} {{{\left| {J_{2} } \right|}}} > 0,\,\frac{{\partial \ifmmode\expandafter\bar\else\expandafter\=\fi{m}}} {{\partial g_{M} }} = \frac{{m{\left[ {{F}\ifmmode{'}\else$'$\fi{\left( {1 - {C}\ifmmode{'}\else$'$\fi_{2} } \right)} + {C}\ifmmode{'}\else$'$\fi_{1} } \right]}}} {{{\left| {J_{2} } \right|}}} < 0; $$

(A12)

$$ \frac{{\partial {\text{\ifmmode\expandafter\bar\else\expandafter\=\fi{N}\ifmmode\expandafter\bar\else\expandafter\=\fi{F}\ifmmode\expandafter\bar\else\expandafter\=\fi{A}}}}} {{\partial G}} = - \frac{{m{h}\ifmmode{'}\else$'$\fi_{1} }} {{{\left| {J_{2} } \right|}}} < 0,\,\frac{{\partial \ifmmode\expandafter\bar\else\expandafter\=\fi{m}}} {{\partial G}} = \frac{{m{h}\ifmmode{'}\else$'$\fi_{2} }} {{{\left| {J_{2} } \right|}}} > 0, $$

(A13)

where ∣J ₂∣ is defined as in Eq. A3.