Back Matter

APPENDIX: The Basic Model

In this appendix, I provide some guideposts to the main derivations contained in the body of the paper.

On numerous occasions in the text, use is made of the relationship between gross elasticities and Hicks-Allen elasticities of substitution. The Allen_elasticity of_substitution between goods i and j, σij = σji, equals ηijj, where ηij is the compensated elasticity demand of good i with respect to a change in price pj. Using this definition, the Slutsky decomposition of a total elasticity into its corresponding substitution and income effect components, and the homogeneity property of demand function gives a relationship between gross elasticities and an expenditure share weighted average of the elasticities of substitution and total spending (or wealth) elasticities. Under the homotheticity assumption, the elasticities of demand with respect to spending as well as of spending with respect to lifetime wealth are both unity. The following relationships used in the paper are now readily derivable. These are:

$\begin{array}{cc}\mathrm{\left(}\mathrm{A}\mathrm{-}\mathrm{1}\mathrm{\right)}& {\mathrm{\eta }}_{{\mathrm{c}}_{{\mathrm{0}}^{\mathrm{\alpha }}}}\mathrm{=}\mathrm{\gamma }\mathrm{\left(}\mathrm{\sigma }\mathrm{-}\mathrm{1}\mathrm{\right)}\end{array}$
$\begin{array}{cc}\mathrm{\left(}\mathrm{A}\mathrm{-}\mathrm{2}\mathrm{\right)}& {\mathrm{\eta }}_{{\mathrm{c}}_{{\mathrm{1}}^{\mathrm{\alpha }}}}\mathrm{=}\mathrm{-}\mathrm{\left[}\mathrm{\left(}\mathrm{1}\mathrm{-}\mathrm{\gamma }\mathrm{\right)}\mathrm{\sigma }\mathrm{+}\mathrm{\gamma }\mathrm{\right]}\end{array}$
$\begin{array}{cc}\mathrm{\left(}\mathrm{A}\mathrm{-}\mathrm{3}\mathrm{\right)}& {\mathrm{\eta }}_{{\mathrm{n}}_{{\mathrm{t}}^{{\mathrm{p}}_{\mathrm{nt}}}}}\end{array}\mathrm{=}\phantom{\rule[-0.0ex]{0.2em}{0.0ex}}\mathrm{-}{\mathrm{\beta }}_{\mathrm{mt}}\phantom{\rule[-0.0ex]{0.2em}{0.0ex}}{\mathrm{\sigma }}_{\mathrm{nm}}\phantom{\rule[-0.0ex]{0.2em}{0.0ex}}\mathrm{-}{\mathrm{\beta }}_{\mathrm{xt}}\phantom{\rule[-0.0ex]{0.2em}{0.0ex}}{\mathrm{\sigma }}_{\mathrm{nx}}\phantom{\rule[-0.0ex]{0.2em}{0.0ex}}\mathrm{-}{\mathrm{\beta }}_{\mathrm{nt}}$
$\begin{array}{cc}\mathrm{\left(}\mathrm{A}\mathrm{-}\mathrm{4}\mathrm{\right)}& {\mathrm{\eta }}_{{\mathrm{n}}_{{\mathrm{t}}^{{\mathrm{p}}_{\mathrm{mt}}}}}\end{array}\mathrm{=}{\mathrm{\beta }}_{\mathrm{mt}}\mathrm{\left(}{\mathrm{\sigma }}_{\mathrm{nm}}\mathrm{-}\mathrm{1}\mathrm{\right)}$

where σ, the intertemporal elasticity of substitution, is defined as

$\begin{array}{cc}\mathrm{\left(}\mathrm{A}\mathrm{-}\mathrm{5}\mathrm{\right)}& \mathrm{\sigma }\mathrm{=}\frac{\mathrm{\partial }\mathrm{log}\mathrm{\left(}{\mathrm{C}}_{\mathrm{1}}\mathrm{/}{\mathrm{C}}_{\mathrm{0}}\mathrm{\right)}}{\mathrm{\partial }\mathrm{log}\mathrm{\left[}\mathrm{\left(}\mathrm{\partial }\mathrm{U}\mathrm{/}\mathrm{\partial }{\mathrm{C}}_{\mathrm{0}}\mathrm{\right)}\mathrm{/}\mathrm{\left(}\mathrm{\partial }\mathrm{U}\mathrm{/}\mathrm{\partial }{\mathrm{C}}_{\mathrm{1}}\mathrm{\right)}\mathrm{\right]}}\end{array}$

Note that since σ > 0 and 0 < γ < 1, ${\mathrm{n}}_{{\mathrm{c}}_{{\mathrm{1}}^{\mathrm{\alpha }}}}$ is negative. However, whether ${\mathrm{n}}_{{\mathrm{c}}_{{0}^{\mathrm{\alpha }}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\gtrless 0$ depends on the intertemporal elasticity of substitution. If σ > 1, then a rise in the “price” of C1, αc1, raises demand for C0. In this case, real spending in the two periods are gross substitutes and conversely. Note also that since -[βmt σnmxtσnx] is a compensated effect, it is nonpositive by the negative semidefiniteness of the Slutsky substitution matrix.

The derivation of equations (7), (8), (9), (29), and (30) follows from differentiation of the trade balance equations ((5) and (6)) and the budget constraint (3) and (4). Use is made also of the relationships (A-1) and (A-2).

Many of the subsequent expressions in the paper follow from a solution to the system of market clearing conditions (10) and (11). For convenience, we rewrite these equilibrium conditions

$\begin{array}{cc}\mathrm{\left(}\mathrm{A}\mathrm{-}\mathrm{6}\mathrm{\right)}& {\mathrm{c}}_{\mathrm{n}\mathrm{0}}\end{array}\mathrm{\left(}{\mathrm{p}}_{\mathrm{n}\mathrm{0}}\mathrm{,}{\mathrm{p}}_{\mathrm{m}\mathrm{0}}\mathrm{,}{\mathrm{P}}_{\mathrm{0}}{\mathrm{C}}_{\mathrm{0}}\mathrm{\left(}{\mathrm{\alpha }}_{\mathrm{c}\mathrm{1}}\mathrm{,}{\mathrm{W}}_{\mathrm{c}\mathrm{0}}\mathrm{\right)}\mathrm{\right)}\mathrm{=}{\overline{\mathrm{Y}}}_{\mathrm{n}\mathrm{0}}\mathrm{,}$
$\begin{array}{cc}\mathrm{\left(}\mathrm{A}\mathrm{-}\mathrm{7}\mathrm{\right)}& {\mathrm{c}}_{\mathrm{n}\mathrm{1}}\end{array}\mathrm{\left(}{\mathrm{p}}_{\mathrm{n}\mathrm{1}}\mathrm{,}{\mathrm{p}}_{\mathrm{m}\mathrm{1}}\mathrm{,}{\mathrm{P}}_{\mathrm{1}}{\mathrm{C}}_{\mathrm{1}}\mathrm{\left(}{\mathrm{\alpha }}_{\mathrm{c}\mathrm{1}}\mathrm{,}{\mathrm{W}}_{\mathrm{c}\mathrm{0}}\mathrm{\right)}\mathrm{\right)}\mathrm{=}{\overline{\mathrm{Y}}}_{\mathrm{n}\mathrm{1}}\mathrm{.}$

Totally differentiating, we have

${\begin{array}{cc}\mathrm{\left(}\mathrm{A}\mathrm{-}\mathrm{8}\mathrm{\right)}& \mathrm{\left(}{\mathrm{\eta }}_{{\mathrm{n}}_{{\mathrm{0}}^{{\mathrm{p}}_{\mathrm{n}\mathrm{0}}}}}\mathrm{+}{\mathrm{\beta }}_{\mathrm{n}\mathrm{0}}\mathrm{\right)}{\stackrel{\mathrm{^}}{\mathrm{p}}}_{\mathrm{n}\mathrm{0}}\mathrm{+}\mathrm{\left(}{\mathrm{\eta }}_{{\mathrm{n}}_{{\mathrm{0}}^{{\mathrm{p}}_{\mathrm{m}\mathrm{0}}}}}\mathrm{+}{\mathrm{\beta }}_{\mathrm{m}\mathrm{0}}\mathrm{\right)}{\stackrel{\mathrm{^}}{\mathrm{p}}}_{\mathrm{m}\mathrm{0}}\mathrm{+}{\mathrm{\eta }}_{{\mathrm{c}}_{{\mathrm{0}}^{\mathrm{\alpha }}}}\stackrel{\mathrm{^}}{\mathrm{\alpha }}\end{array}}_{\mathrm{c}\mathrm{1}}\mathrm{+}{\stackrel{\mathrm{^}}{\mathrm{W}}}_{\mathrm{c}\mathrm{0}}\mathrm{=}\mathrm{0}\mathrm{,}$
${\begin{array}{cc}\mathrm{\left(}\mathrm{A}\mathrm{-}\mathrm{9}\mathrm{\right)}& \mathrm{\left(}{\mathrm{\eta }}_{{\mathrm{n}}_{{\mathrm{1}}^{{\mathrm{p}}_{\mathrm{n}\mathrm{1}}}}}\mathrm{+}{\mathrm{\beta }}_{\mathrm{n}\mathrm{1}}\mathrm{\right)}{\stackrel{\mathrm{^}}{\mathrm{p}}}_{\mathrm{n}\mathrm{1}}\mathrm{+}\mathrm{\left(}{\mathrm{\eta }}_{{\mathrm{n}}_{{\mathrm{1}}^{{\mathrm{p}}_{\mathrm{m}\mathrm{1}}}}}\mathrm{+}{\mathrm{\beta }}_{\mathrm{m}\mathrm{1}}\mathrm{\right)}{\stackrel{\mathrm{^}}{\mathrm{p}}}_{\mathrm{m}\mathrm{1}}\mathrm{+}{\mathrm{\eta }}_{{\mathrm{c}}_{{\mathrm{1}}^{\mathrm{\alpha }}}}\stackrel{\mathrm{^}}{\mathrm{\alpha }}\end{array}}_{\mathrm{c}\mathrm{1}}\mathrm{+}{\stackrel{\mathrm{^}}{\mathrm{W}}}_{\mathrm{c}\mathrm{0}}\mathrm{=}\mathrm{0.}$

where use has been made of the fact that the elasticity of the price index P with respect to a change in one of the temporal relative prices (pnt or pmt) is simply the corresponding expenditure share (βnt or βmt) We assume throughout that there are no supply shocks (endowments are constant) and that the world discount factor, αx1 is given. In this case, the discount factor relevant for domestic consumption, α, evolves according to

$\begin{array}{cc}\mathrm{\left(}\mathrm{A}\mathrm{-}\mathrm{10}\mathrm{\right)}& {\stackrel{\mathrm{^}}{\begin{array}{c}\mathrm{\alpha }\end{array}}}_{\mathrm{c}\mathrm{1}}\end{array}\mathrm{=}{\mathrm{\beta }}_{\mathrm{n}\mathrm{1}}{\stackrel{\mathrm{^}}{\mathrm{p}}}_{\mathrm{n}\mathrm{1}}\mathrm{+}{\mathrm{\beta }}_{\mathrm{m}\mathrm{1}}{\stackrel{\mathrm{^}}{\mathrm{p}}}_{\mathrm{m}\mathrm{1}}\mathrm{-}{\mathrm{\beta }}_{\mathrm{n}\mathrm{0}}{\stackrel{\mathrm{^}}{\mathrm{p}}}_{\mathrm{no}}\mathrm{-}{\mathrm{\beta }}_{\mathrm{m}\mathrm{0}}{\stackrel{\mathrm{^}}{\mathrm{p}}}_{\mathrm{m}\mathrm{0}}\mathrm{.}$

Recalling that real wealth is given by

$\begin{array}{cc}\mathrm{\left(}\mathrm{A}\mathrm{-}\mathrm{11}\mathrm{\right)}& {\mathrm{W}}_{\mathrm{c}\mathrm{0}}\end{array}\mathrm{=}\mathrm{\left\{}{\overline{\mathrm{Y}}}_{\mathrm{x}\mathrm{0}}\mathrm{+}{\mathrm{p}}_{\mathrm{m}\mathrm{0}}{\overline{\mathrm{Y}}}_{\mathrm{m}\mathrm{0}}\mathrm{+}{\mathrm{p}}_{\mathrm{n}\mathrm{0}}{\overline{\mathrm{Y}}}_{\mathrm{n}\mathrm{0}}\mathrm{+}{\mathrm{\alpha }}_{\mathrm{x}\mathrm{1}}\mathrm{\left[}{\overline{\mathrm{Y}}}_{\mathrm{x}\mathrm{1}}\mathrm{+}{\mathrm{p}}_{\mathrm{m}\mathrm{1}}{\overline{\mathrm{Y}}}_{\mathrm{m}\mathrm{1}}\mathrm{+}{\mathrm{p}}_{\mathrm{n}\mathrm{1}}{\overline{\mathrm{Y}}}_{\mathrm{n}\mathrm{1}}\mathrm{\right]}\mathrm{-}\mathrm{\left(}\mathrm{1}\mathrm{+}{\mathrm{r}}_{\mathrm{x}\mathrm{,}\mathrm{-}\mathrm{1}}\mathrm{\right)}{\mathrm{B}}_{\mathrm{-}\mathrm{1}}\mathrm{\right\}}\mathrm{/}{\mathrm{p}}_{\mathrm{0}}$

we can totally differentiate (A-11) to obtain

$\begin{array}{cc}\mathrm{\left(}\mathrm{A}\mathrm{-}\mathrm{12}\mathrm{\right)}& {\stackrel{\mathrm{^}}{\begin{array}{c}\mathrm{W}\end{array}}}_{\mathrm{c}\mathrm{0}}\end{array}\mathrm{=}\mathrm{\left[}\mathrm{\left(}\mathrm{1}\mathrm{-}\mathrm{\gamma }\mathrm{\right)}{\mathrm{\mu }}_{\mathrm{m}\mathrm{0}}\mathrm{-}\mathrm{1}\mathrm{\right]}{\mathrm{\beta }}_{\mathrm{m}\mathrm{0}}{\stackrel{\mathrm{^}}{\mathrm{p}}}_{\mathrm{m}\mathrm{0}}\mathrm{-}{\mathrm{\gamma \beta }}_{\mathrm{n}\mathrm{0}}{\stackrel{\mathrm{^}}{\mathrm{p}}}_{\mathrm{n}\mathrm{0}}\mathrm{+}{\mathrm{\gamma \mu }}_{\mathrm{m}\mathrm{1}}{\mathrm{\beta }}_{\mathrm{m}\mathrm{1}}{\stackrel{\mathrm{^}}{\mathrm{p}}}_{\mathrm{m}\mathrm{1}}\mathrm{+}{\mathrm{\gamma \beta }}_{\mathrm{n}\mathrm{1}}{\stackrel{\mathrm{^}}{\mathrm{p}}}_{\mathrm{n}\mathrm{1}}\mathrm{.}$

Substituting (A-1), (A-2), (A-3), (A-4), (A-10), and (A-12) into (A-7) and (A-8), we obtain the system

$\begin{array}{lll}\mathrm{\left(}\mathrm{A}\mathrm{.13}\mathrm{\right)}& \mathrm{\left[}\begin{array}{cc}\mathrm{-}{\mathrm{\beta }}_{\mathrm{m}\mathrm{0}}{\mathrm{\sigma }}_{\mathrm{nm}}\mathrm{-}{\mathrm{\beta }}_{\mathrm{x}\mathrm{0}}{\mathrm{\sigma }}_{\mathrm{nx}}\mathrm{-}{\mathrm{\beta }}_{\mathrm{n}\mathrm{0}}\mathrm{\gamma \sigma }& {\mathrm{\gamma \beta }}_{\mathrm{n}\mathrm{1}}\mathrm{\sigma }\\ {\mathrm{\beta }}_{\mathrm{n}\mathrm{0}}\mathrm{\left(}\mathrm{1}\mathrm{-}\mathrm{\gamma }\mathrm{\right)}\mathrm{\sigma }& \mathrm{-}{\mathrm{\beta }}_{\mathrm{m}\mathrm{1}}{\mathrm{\sigma }}_{\mathrm{nm}}\mathrm{-}{\mathrm{\beta }}_{\mathrm{x}\mathrm{1}}{\mathrm{\sigma }}_{\mathrm{nx}}\mathrm{-}{\mathrm{\beta }}_{\mathrm{n}\mathrm{1}}\mathrm{\left(}\mathrm{1}\mathrm{-}\mathrm{\gamma }\mathrm{\right)}\mathrm{\sigma }\end{array}\mathrm{\right]}\hfill & \mathrm{\left[}\begin{array}{c}{\stackrel{\mathrm{^}}{\mathrm{p}}}_{\mathrm{n}\mathrm{0}}\\ {\stackrel{\mathrm{^}}{\mathrm{p}}}_{\mathrm{n}\mathrm{1}}\end{array}\mathrm{\right]}\\ \mathrm{=}& \mathrm{\left[}\begin{array}{cc}\mathrm{-}{\mathrm{\beta }}_{\mathrm{m}\mathrm{0}}\mathrm{\left[}{\mathrm{\sigma }}_{\mathrm{nm}}\mathrm{-}\mathrm{\left\{}\mathrm{1}\mathrm{-}\mathrm{\gamma }\mathrm{\right)}\mathrm{\left(}\mathrm{1}\mathrm{-}{\mathrm{\mu }}_{\mathrm{m}\mathrm{0}}\mathrm{\right)}\mathrm{+}\mathrm{\gamma \sigma }\mathrm{\right\}}\mathrm{\right]}& \mathrm{-}{\mathrm{\beta }}_{\mathrm{m}\mathrm{1}}\mathrm{\gamma }\mathrm{\left(}\mathrm{\sigma }\mathrm{-}\mathrm{\left(}\mathrm{1}\mathrm{-}{\mathrm{\mu }}_{\mathrm{m}\mathrm{1}}\mathrm{\right)}\mathrm{\right)}\\ \mathrm{-}{\mathrm{\beta }}_{\mathrm{m}\mathrm{0}}\mathrm{\left(}\mathrm{1}\mathrm{-}\mathrm{\gamma }\mathrm{\right)}\mathrm{\left(}\mathrm{\sigma }\mathrm{-}\mathrm{\left(}\mathrm{1}\mathrm{-}{\mathrm{\mu }}_{\mathrm{m}\mathrm{0}}\mathrm{\right)}\mathrm{\right)}& \mathrm{-}{\mathrm{\beta }}_{\mathrm{m}\mathrm{1}}\mathrm{\left[}\mathrm{\left(}{\mathrm{\sigma }}_{\mathrm{nm}}\mathrm{-}\mathrm{\sigma }\mathrm{\right)}\mathrm{+}\mathrm{\gamma }\mathrm{\left(}\mathrm{\sigma }\mathrm{-}\mathrm{\left(}\mathrm{1}\mathrm{-}{\mathrm{\mu }}_{\mathrm{m}\mathrm{1}}\mathrm{\right)}\mathrm{\right)}\mathrm{\right]}\end{array}\mathrm{\right]}\hfill & \mathrm{\left[}\begin{array}{c}{\stackrel{\mathrm{^}}{\mathrm{p}}}_{\mathrm{m}\mathrm{0}}\\ {\stackrel{\mathrm{^}}{\mathrm{p}}}_{\mathrm{m}\mathrm{1}}\end{array}\mathrm{\right]}\end{array}$

This system underlies the derivation of the slopes of the N0N0 and N1N1 schedules as well as their shifts in response to various terms of trade changes.

Using (A-13), we can solve for ${\stackrel{\mathrm{^}}{\mathrm{p}}}_{\mathrm{n}\mathrm{0}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{and}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\stackrel{\mathrm{^}}{\mathrm{p}}}_{\mathrm{n}\mathrm{1}}$ in terms of ${\stackrel{\mathrm{^}}{\mathrm{p}}}_{\mathrm{m}\mathrm{0}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{and}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\stackrel{\mathrm{^}}{\mathrm{p}}}_{\mathrm{m}\mathrm{1}}$. The solutions are

$\begin{array}{ll}\mathrm{\left(}\mathrm{A}\mathrm{.14}\mathrm{\right)}& {\stackrel{\mathrm{^}}{\begin{array}{c}\mathrm{p}\end{array}}}_{\mathrm{n}\mathrm{0}}\mathrm{=}{\mathrm{\Delta }}^{\mathrm{-}\mathrm{1}}\mathrm{\left\{}{\mathrm{\beta }}_{\mathrm{m}\mathrm{0}}\mathrm{\left[}{\mathrm{\sigma }}_{\mathrm{nm}}\mathrm{-}\mathrm{\left(}\mathrm{\left(}\mathrm{1}\mathrm{-}\mathrm{\gamma }\mathrm{\right)}\mathrm{\left(}\mathrm{1}\mathrm{-}{\mathrm{\mu }}_{\mathrm{m}\mathrm{0}}\mathrm{\right)}\mathrm{+}\mathrm{\gamma \sigma }\mathrm{\right)}\mathrm{\right]}\mathrm{\left[}{\mathrm{\beta }}_{\mathrm{m}\mathrm{1}}{\mathrm{\sigma }}_{\mathrm{nm}}\mathrm{+}{\mathrm{\beta }}_{\mathrm{x}\mathrm{1}}{\mathrm{\sigma }}_{\mathrm{nx}}\mathrm{+}{\mathrm{\beta }}_{\mathrm{n}\mathrm{1}}\mathrm{\left(}\mathrm{1}\mathrm{-}\mathrm{\gamma }\mathrm{\right)}\mathrm{\sigma }\mathrm{\right]}\\ & \mathrm{+}{\mathrm{\beta }}_{\mathrm{m}\mathrm{0}}\mathrm{\left(}\mathrm{1}\mathrm{-}\mathrm{\gamma }\mathrm{\right)}\mathrm{\left(}\mathrm{\sigma }\mathrm{-}\mathrm{\left(}\mathrm{1}\mathrm{-}{\mathrm{\mu }}_{\mathrm{m}\mathrm{0}}\mathrm{\right)}\mathrm{\right)}{\mathrm{\gamma \beta }}_{\mathrm{n}\mathrm{1}}\mathrm{\sigma }\mathrm{\right\}}{\stackrel{\mathrm{^}}{\mathrm{p}}}_{\mathrm{m}\mathrm{0}}\mathrm{+}{\mathrm{\Delta }}^{\mathrm{-}\mathrm{1}}\mathrm{\left\{}{\mathrm{\gamma \beta }}_{\mathrm{m}\mathrm{1}}\mathrm{\left[}\mathrm{\sigma }\mathrm{-}\mathrm{\left(}\mathrm{1}\mathrm{-}{\mathrm{\mu }}_{\mathrm{m}\mathrm{1}}\mathrm{\right)}\mathrm{\right]}\mathrm{\left\{}{\mathrm{\beta }}_{\mathrm{m}\mathrm{1}}{\mathrm{\sigma }}_{\mathrm{nm}}\mathrm{+}{\mathrm{\beta }}_{\mathrm{nm}}\mathrm{+}{\mathrm{\beta }}_{\mathrm{x}\mathrm{1}}{\mathrm{\sigma }}_{\mathrm{nx}}\\ & \mathrm{+}{\mathrm{\beta }}_{\mathrm{n}\mathrm{1}}\mathrm{\left(}\mathrm{1}\mathrm{-}\mathrm{\gamma }\mathrm{\right)}\mathrm{\sigma }\mathrm{\right]}\mathrm{+}{\mathrm{\beta }}_{\mathrm{m}\mathrm{1}}\mathrm{\left[}\mathrm{\left(}{\mathrm{\sigma }}_{\mathrm{nm}}\mathrm{-}\mathrm{\sigma }\mathrm{\right)}\mathrm{+}\mathrm{\gamma }\mathrm{\left(}\mathrm{\alpha }\mathrm{-}\mathrm{\left(}\mathrm{1}\mathrm{-}{\mathrm{\mu }}_{\mathrm{m}\mathrm{1}}\mathrm{\right)}\mathrm{\right)}{\mathrm{\gamma \beta }}_{\mathrm{n}\mathrm{1}}\mathrm{\sigma }\mathrm{\right\}}{\stackrel{\mathrm{^}}{\mathrm{p}}}_{\mathrm{m}\mathrm{1}}\end{array}$
$\begin{array}{ll}\mathrm{\left(}\mathrm{A}\mathrm{.15}\mathrm{\right)}& {\stackrel{\mathrm{^}}{\begin{array}{c}\mathrm{p}\end{array}}}_{\mathrm{n}\mathrm{1}}\mathrm{=}{\mathrm{\Delta }}^{\mathrm{-}\mathrm{1}}\mathrm{\left\{}{\mathrm{\beta }}_{\mathrm{m}\mathrm{0}}\mathrm{\left(}\mathrm{1}\mathrm{-}\mathrm{\gamma }\mathrm{\right)}\mathrm{\left[}\mathrm{\sigma }\mathrm{-}\mathrm{\left(}\mathrm{1}\mathrm{-}{\mathrm{\mu }}_{\mathrm{m}\mathrm{0}}\mathrm{\right)}\mathrm{\right]}\mathrm{\left[}{\mathrm{\beta }}_{\mathrm{m}\mathrm{0}}{\mathrm{\sigma }}_{\mathrm{nm}}\mathrm{+}{\mathrm{\beta }}_{\mathrm{x}\mathrm{0}}{\mathrm{\sigma }}_{\mathrm{nx}}\mathrm{+}{\mathrm{\beta }}_{\mathrm{n}\mathrm{0}}\mathrm{\gamma \sigma }\mathrm{\right]}\\ & \mathrm{+}{\mathrm{\beta }}_{\mathrm{m}\mathrm{0}}\mathrm{\left(}\mathrm{1}\mathrm{-}\mathrm{\gamma }\mathrm{\right)}{\mathrm{\beta }}_{\mathrm{n}\mathrm{0}}\mathrm{\sigma }\mathrm{\left[}{\mathrm{\sigma }}_{\mathrm{nm}}\mathrm{-}\mathrm{\left(}\mathrm{\left(}\mathrm{1}\mathrm{-}\mathrm{\gamma }\mathrm{\right)}\mathrm{\left(}\mathrm{1}\mathrm{-}{\mathrm{\mu }}_{\mathrm{m}\mathrm{0}}\mathrm{\right)}\mathrm{+}\mathrm{\gamma \sigma }\mathrm{\right)}\mathrm{\right]}\mathrm{\right\}}{\stackrel{\mathrm{^}}{\mathrm{p}}}_{\mathrm{m}\mathrm{0}}\\ & \mathrm{+}{\mathrm{\Delta }}^{\mathrm{-}\mathrm{1}}\mathrm{\left\{}{\mathrm{\beta }}_{\mathrm{m}\mathrm{1}}\mathrm{\left[}\mathrm{\left(}{\mathrm{\sigma }}_{\mathrm{nm}}\mathrm{-}\mathrm{\sigma }\mathrm{\right)}\mathrm{+}\mathrm{\gamma }\mathrm{\left(}\mathrm{\sigma }\mathrm{-}\mathrm{\left(}\mathrm{1}\mathrm{-}{\mathrm{\mu }}_{\mathrm{m}\mathrm{1}}\mathrm{\right)}\mathrm{\right)}\mathrm{\right]}\mathrm{\left[}{\mathrm{\beta }}_{\mathrm{m}\mathrm{0}}{\mathrm{\sigma }}_{\mathrm{nm}}\mathrm{+}{\mathrm{\beta }}_{\mathrm{x}\mathrm{0}}{\mathrm{\sigma }}_{\mathrm{nx}}\mathrm{+}{\mathrm{\beta }}_{\mathrm{n}\mathrm{0}}\mathrm{\gamma \sigma }\mathrm{\right]}\\ & \mathrm{+}{\mathrm{\beta }}_{\mathrm{m}\mathrm{1}}\mathrm{\left(}\mathrm{1}\mathrm{-}\mathrm{\gamma }\mathrm{\right)}{\mathrm{\beta }}_{\mathrm{n}\mathrm{0}}\mathrm{\gamma \sigma }\mathrm{\left[}\mathrm{\sigma }\mathrm{-}\mathrm{\left(}\mathrm{1}\mathrm{-}{\mathrm{\mu }}_{\mathrm{m}\mathrm{1}}\mathrm{\right)}\mathrm{\right]}\mathrm{\right\}}{\stackrel{\mathrm{^}}{\mathrm{p}}}_{\mathrm{m}\mathrm{1}}\end{array}$

where Δ = [βm0σx0x0σnx][βm1σnmx1n1(1-γ)σ]+βn0γσ[βm1σnmx1σnx] ≥ 0

Equations (19) and (22) correspond to the coefficients of ${\stackrel{\mathrm{^}}{\mathrm{p}}}_{\mathrm{m}\mathrm{0}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{and}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\stackrel{\mathrm{^}}{\mathrm{p}}}_{\mathrm{m}\mathrm{I}}$ in (A-14). To determine the effects of permanent shocks, set ${\stackrel{\mathrm{^}}{\mathrm{p}}}_{\mathrm{m}\mathrm{0}}\mathrm{=}{\stackrel{\mathrm{^}}{\mathrm{p}}}_{\mathrm{m}\mathrm{I}}\mathrm{=}{\stackrel{\mathrm{^}}{\mathrm{p}}}_{\mathrm{m}}$ in (A-14). Equation (A-15) is used to determine the effect of terms of trade changes on the relative price of home goods in period one.

Substitution of the relevant terms in equations (A-14) and (A-15) into equations (26), (27) and (28) underlies the main derivations in Section IV of the paper.

References

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• Dornbusch, Rudiger, Open Economy Macroeconomics (New York: Basic Books, 1980).

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• Frenkel, Jacob A., and Assaf Razin, Fiscal Policies and the World Economy: An Intertemporal Approach (Cambridge: MIT Press), 1987 forthcoming.

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• Ostry, Jonathan,The Terms of Trade, The Real Exchange Rate and the Balance of Trade in a Two Period Three Good Optimizing Model,” Department of Economics, University of Chicago, mimeo (February 1987).

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The author, a graduate student in the Department of Economics of the University of Chicago, was an Economist Intern in the Financial Studies Division of the Research Department when this paper was written. The author thanks Jacob Frenkel and Assaf Razin for their advice and encouragement, and Joshua Aizenman, Robert Flood, and Donald J. Mathieson for helpful comments. Earlier versions of this paper were presented to the International Economics Workshop at the University of Chicago and the Research Department seminar of the IMF. I thank these workshops’ participants for their comments. Remaining errors are mine.

In a recent paper, Edwards (1987) highlights the role of anticipated future import tariffs on today’s current account.

In a forthcoming paper, I consider the effects of various terms of trade shocks under the assumption that agents have limited access to the international capital market.

The consumption based discount factor, αc1, is related to the (consumption based) real rate of interest in the usual manner, namely αc1 = (1+rc0)-1. The relationship between the real rate of interest and the exogenous world rate of interest is given by ${\mathrm{r}}_{\mathrm{c}\mathrm{0}}\mathrm{=}\frac{\mathrm{1}\mathrm{+}{\mathrm{r}}_{\mathrm{x}\mathrm{0}}}{{\mathrm{P}}_{\mathrm{1}}\mathrm{/}{\mathrm{P}}_{\mathrm{0}}}\mathrm{-}\mathrm{1.}$. Therefore a rise in P0 or a fall in P1 raises the real rate of interest.

The possibility of substituting spending between periods, independent of the time path of income, depends of course not only on σ, a parameter of the agent’s utility function, but also on institutional factors which govern the degree to which agents can borrow at a given world rate of interest in international financial markets. If the assumption of perfect capital mobility were relaxed, these other factors would come into play.

Government spending shocks could be easily accomodated in our framework as well. In the small country setting, changes in government spending do not affect the terms of trade or world interest rates, but they do alter the time path of the real exchange rate. There is, therefore, a decomposition of the effects of fiscal policy on the current account into a direct (since changes in government spending directly affect national saving) and indirect (since changes in government spending affect pn0 and pn1 which indirectly affect national savings) component. This decomposition is analagous to the decomposition of the Harberger-Laursen-Metzler effect into its direct and indirect parts.

The effect of shocks to the world rate of interest on the current account also have an interpretation in terms of direct and indirect effects since they too alter equilibrium in the home goods markets and thereby affect the path of the real exchange rate.

This diagram has had extensive previous use (see, for instance, Dornbusch (1980) and Edwards (1987)).

Edwards (1987) uses notions of stability—completely external to the model—in order to rank the two slopes. We show here that the model itself has sufficient structure to enable the two slopes to be ranked.

It is relevant to note that there are no welfare effects arising from movements along the N0N0 and N1N1 schedules. This follows from the assumption that the home goods sector clears in each period.

The result for pn1 is given in the appendix.

If σnm > max {[1-γ) (1-μm0) + γσ], (1-μm0)}, a temporary current terms of trade deterioration always causes a real appreciation.

We note that in comparison to the effect of a current deterioration in the terms of trade, there is no temporal substitution or price index effect in the case of an anticipated future disturbance.

For the response of pn1, see the appendix.

The ingredients necessary for consideration of the general case are given in the appendix.

Note that if μc0 = 1, so that the trade account is in balance, then both the direct and indirect effects are zero and a permanent terms of trade change leaves the balance of trade unaffected.

This condition states that the real income effect of a terms of trade change outweighs the temporal substitution effect. It is necessarily satisfied if importables and nontradables are Hicks complements but is compatible with their being Hicks substitutes as well. Note that if the economy does not produce importables, the condition states that σnm < 1.

Note that this is an equilibrium phenomenon and has nothing to do with lags in the adjustment of the current account to relative price changes, which underly the “J curve.”

The ingredients necessary for consideration of the general case are given in the appendix.

This role of the current account is emphasized in Sachs (1981).

In the case of logarithmic utility, σ = 1, and the trade account necessarily moves into surplus (cannot move into deficit).

The weaker condition, σnm > (1-μm) is sufficient.

The Balance of Trade, the Terms of Trade, and the Real Exchange Rate: An Intertemporal Optimizing Framework
Author: International Monetary Fund