Back Matter

### Appendix I: Derivation of the Household’s Decision Rules

To illustrate the agent’s maximization problem, let us first consider the end of the period when he already knows his labor supply.

At the end of period 1, the agent will choose his current and future consumption levels of tradable and nontradable goods. He knows that in period 2, he will have a nominal wealth: “A2” and will have to choose Y2 and ${\mathrm{Y}}_{2}^{\mathrm{*}}$ in order to:

$\begin{array}{cc}\mathrm{\left(}\mathrm{1.1}\mathrm{\right)}& \mathrm{M}\mathrm{ax}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{Y}}_{\mathrm{2}}^{\mathrm{*}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{Y}}_{\mathrm{2}}^{\mathrm{*}}\mathrm{\left(}\mathrm{1}\mathrm{-}\mathrm{\tau }\mathrm{\right)}\end{array}$

subject to:

$\begin{array}{cc}\mathrm{\left(}\mathrm{1.2}\mathrm{\right)}& {\mathrm{A}}_{\mathrm{2}}\mathrm{=}{\mathrm{P}}_{\mathrm{2}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{Y}}_{\mathrm{2}}^{\mathrm{d}}\mathrm{+}\mathrm{\left(}{\mathrm{p}}_{\mathrm{2}}^{\mathrm{*}}{\mathrm{S}}_{\mathrm{2}}\mathrm{\right)}{\mathrm{\left(}\mathrm{Y}\mathrm{*}\mathrm{\right)}}_{\mathrm{2}}^{\mathrm{d}}\end{array}$

From equations (1.1) and (1.2), the demand functions for the agent’s second period of life are obtained:

$\begin{array}{cc}\mathrm{\left(}\mathrm{1.3}\mathrm{\right)}& {\mathrm{Y}}_{\mathrm{2}}^{\mathrm{d}}\mathrm{=}{\mathrm{\tau A}}_{\mathrm{2}}\mathrm{/}{\mathrm{p}}_{\mathrm{2}}\end{array}$
$\begin{array}{cc}\mathrm{\left(}\mathrm{1.4}\mathrm{\right)}& {\mathrm{\left(}\mathrm{Y}\mathrm{*}\mathrm{\right)}}_{\mathrm{2}}^{\mathrm{d}}\mathrm{=}\mathrm{\left(}\mathrm{1}\mathrm{-}\mathrm{\tau }\mathrm{\right)}{\mathrm{A}}_{\mathrm{2}}\mathrm{/}{\mathrm{P}}_{\mathrm{2}}^{\mathrm{*}}{\mathrm{S}}_{\mathrm{2}}\end{array}$

As stated above, these demand functions show a unitary elasticity of demand with respect to expenditure on future consumption (A).

Given (1.3) and (1.4), the maximized value of ${\mathrm{Y}}_{\mathrm{2}}^{\mathrm{\tau }}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{\left(}\mathrm{Y}\mathrm{*}\mathrm{\right)}}_{\mathrm{2}}\mathrm{\left(}\mathrm{1}\mathrm{-}\mathrm{\tau }\mathrm{\right)}$ is:

$\begin{array}{cc}\mathrm{\left(}\mathrm{1.5}\mathrm{\right)}& {\mathrm{\left(}{\mathrm{Y}}_{\mathrm{2}}^{\mathrm{\tau }}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{{\mathrm{\left(}\mathrm{Y}\mathrm{*}\mathrm{\right)}}_{\mathrm{2}}}^{\mathrm{\left(}\mathrm{1}\mathrm{-}\mathrm{\tau }\mathrm{\right)}}\mathrm{\right)}}_{\mathrm{max}}\mathrm{=}{\mathrm{A}}_{\mathrm{2}}\mathrm{/}\mathrm{\left(}{\mathrm{\tau }}^{\mathrm{-}\mathrm{\tau }}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{\left(}\mathrm{1}\mathrm{-}\mathrm{\tau }\mathrm{\right)}}^{\mathrm{-}\mathrm{\left(}\mathrm{1}\mathrm{-}\mathrm{\tau }\mathrm{\right)}}{\mathrm{P}}_{\mathrm{2}}^{\mathrm{\tau }}{\mathrm{\left(}{\mathrm{P}}_{\mathrm{2}}^{\mathrm{*}}{\mathrm{S}}_{\mathrm{2}}\mathrm{\right)}}^{\mathrm{1}\mathrm{-}\mathrm{\tau }}\mathrm{\right)}\mathrm{\equiv }{\mathrm{A}}_{\mathrm{2}}\mathrm{/}\mathrm{\left(}\mathrm{j}\mathrm{\prime }{\mathrm{PI}}_{\mathrm{2}}\mathrm{\right)}\end{array}$

where: j’ ≡ τ(1-τ)-(1-τ)

PI2 ≡ price index in period $\mathrm{2}\mathrm{\equiv }{\mathrm{P}}_{\mathrm{2}}^{\mathrm{\tau }}{\mathrm{\left(}{\mathrm{P}}_{\mathrm{2}}^{\mathrm{*}}{\mathrm{S}}_{\mathrm{2}}\mathrm{\right)}}^{\mathrm{1}\mathrm{-}\mathrm{\tau }}$

The price index PI is homogenous of degree one, due to the Cobb-Douglas specification of the utility function. Substituting (1.5) into equation (17) of the main text and noticing that the budget constraint (equation (18) in the main text) requires:

$\begin{array}{cc}\mathrm{\left(}\mathrm{1.6}\mathrm{\right)}& {\mathrm{A}}_{\mathrm{2}}\mathrm{=}{\mathrm{W}}_{\mathrm{1}}{\mathrm{L}}_{\mathrm{1}}\mathrm{+}{\mathrm{Tr}}_{\mathrm{2}}\mathrm{+}{{}^{\mathrm{h}}\mathrm{\pi }}_{\mathrm{2}}\mathrm{-}{\mathrm{P}}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{1}}^{\mathrm{d}}\mathrm{-}\mathrm{\left(}{\mathrm{P}}_{\mathrm{1}}^{\mathrm{*}}{\mathrm{S}}_{\mathrm{1}}\mathrm{\right)}{\mathrm{\left(}\mathrm{Y}\mathrm{*}\mathrm{\right)}}_{\mathrm{1}}^{\mathrm{d}}\mathrm{\equiv }{\mathrm{a}}^{\mathrm{H}}\mathrm{2}\end{array}$

The first period problem is obtained. This requires the agent to choose ${\mathrm{Y}}_{\mathrm{1}}^{\mathrm{d}}\mathrm{,}{\mathrm{\left(}\mathrm{Y}\mathrm{*}\mathrm{\right)}}_{\mathrm{1}}^{\mathrm{d}}$ to:

$\begin{array}{lll}\mathrm{\left(}\mathrm{1.7}\mathrm{\right)}& \underset{\mathrm{\left\{}{\mathrm{Y}}_{\mathrm{1}}^{\mathrm{d}}\mathrm{,}{\mathrm{\left(}\mathrm{Y}\mathrm{*}\mathrm{\right)}}_{\mathrm{1}}^{\mathrm{d}}\mathrm{,}{\mathrm{\lambda }}_{\mathrm{2}}\mathrm{\right\}}}{\begin{array}{c}\mathrm{M}\mathrm{ax}\end{array}}\mathrm{=}& {\mathrm{\mu }}_{\mathrm{1}}\mathrm{1}\mathrm{n}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{\left[}{\mathrm{Y}}_{\mathrm{1}}^{\mathrm{\tau }}{\mathrm{\left(}\mathrm{Y}\mathrm{*}\mathrm{\right)}}_{\mathrm{1}}^{\mathrm{\left(}\mathrm{1}\mathrm{-}\mathrm{\tau }\mathrm{\right)}}\mathrm{\right]}\mathrm{+}\mathrm{1}\mathrm{n}\mathrm{\left(}\overline{\mathrm{L}}\mathrm{-}{\mathrm{L}}_{\mathrm{1}}\mathrm{\right)}\\ & & \mathrm{+}\frac{\mathrm{\beta }}{\mathrm{j}\mathrm{\prime }}\mathrm{b}{\mathrm{1}}^{\mathrm{E}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{\left[}\frac{{\mathrm{W}}_{\mathrm{1}}{\mathrm{L}}_{\mathrm{1}}\mathrm{+}{\mathrm{Tr}}_{\mathrm{2}}\mathrm{+}{{}^{\mathrm{h}}\mathrm{\pi }}_{\mathrm{2}}\mathrm{-}{\mathrm{P}}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{1}}^{\mathrm{d}}\mathrm{-}\mathrm{\left(}{\mathrm{P}}_{\mathrm{1}}^{\mathrm{*}}{\mathrm{S}}_{\mathrm{1}}\mathrm{\right)}{\mathrm{\left(}\mathrm{Y}\mathrm{*}\mathrm{\right)}}_{\mathrm{1}}^{\mathrm{d}}}{{\mathrm{PI}}_{\mathrm{2}}}\mathrm{\right]}\\ & & \mathrm{+}\mathrm{\lambda }\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{\left[}{\mathrm{W}}_{\mathrm{1}}{\mathrm{L}}_{\mathrm{1}}^{\mathrm{s}}\mathrm{-}{\mathrm{P}}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{1}}^{\mathrm{d}}\mathrm{-}{\mathrm{P}}_{\mathrm{1}}^{\mathrm{*}}{\mathrm{S}}_{\mathrm{1}}{\mathrm{\left(}\mathrm{Y}\mathrm{*}\mathrm{\right)}}_{\mathrm{1}}^{\mathrm{d}}\mathrm{\right]}\end{array}$

where λ is the Kuhn-Tucker multiplier associated with the constraint (19) in the main text.

Imposing the restrictions that consumption of both commodities be positive every period and that the young individual’s demand for money be positive, the maximization procedure leads to the following decision rules:

$\begin{array}{cc}\mathrm{\left(}\mathrm{1.8}\mathrm{\right)}& {\mathrm{Y}}_{\mathrm{1}}^{\mathrm{d}}\mathrm{=}\frac{\mathrm{j}\mathrm{\prime }\mathrm{\tau }}{\mathrm{\beta }}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{\mu }}_{\mathrm{1}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\frac{\mathrm{b}{\mathrm{1}}^{\mathrm{E}}\mathrm{\left(}{\mathrm{PI}}_{\mathrm{2}}\mathrm{\right)}}{{\mathrm{P}}_{\mathrm{1}}}\end{array}$
$\begin{array}{cc}\mathrm{\left(}\mathrm{1.9}\mathrm{\right)}& {\mathrm{\left(}\mathrm{Y}\mathrm{*}\mathrm{\right)}}_{\mathrm{1}}^{\mathrm{d}}\mathrm{=}\frac{\mathrm{j}\mathrm{\prime }\mathrm{\left(}\mathrm{1}\mathrm{-}\mathrm{\tau }\mathrm{\right)}}{\mathrm{\beta }}{\mathrm{\mu }}_{\mathrm{1}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\frac{\mathrm{b}{\mathrm{1}}^{\mathrm{E}}\mathrm{\left(}{\mathrm{PI}}_{\mathrm{2}}\mathrm{\right)}}{{\mathrm{P}}_{\mathrm{1}}^{\mathrm{*}}{\mathrm{S}}_{\mathrm{1}}}\mathrm{.}\end{array}$

Now, we can turn to the young agent’s maximization problem at the beginning of the period. At that time, the individual’s demands for commodities are stochastic variables. The agent’s problem is:

$\begin{array}{cc}\mathrm{\left(}\mathrm{1.10}\mathrm{\right)}& \underset{\mathrm{\left\{}{\mathrm{L}}_{\mathrm{1}}^{\mathrm{s}}\mathrm{\right\}}}{\begin{array}{c}\mathrm{M}\mathrm{ax}\end{array}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{\mu }}_{\mathrm{1}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{{\mathrm{a}}_{\mathrm{1}}}^{\mathrm{E}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{\left[}\mathrm{1}\mathrm{n}\mathrm{\left(}{\mathrm{\left(}\stackrel{\mathrm{˜}}{\mathrm{Y}}\mathrm{\right)}}_{\mathrm{1}}^{\mathrm{\tau }}{\mathrm{\left(}\stackrel{\mathrm{˜}}{\mathrm{Y}}\mathrm{*}\mathrm{\right)}}_{\mathrm{1}}^{\mathrm{\left(}\mathrm{1}\mathrm{-}\mathrm{\tau }\mathrm{\right)}}\mathrm{\right)}\mathrm{\right]}\mathrm{+}\mathrm{1}\mathrm{n}\mathrm{\left(}\overline{\mathrm{L}}\mathrm{-}{\mathrm{L}}_{\mathrm{1}}\mathrm{\right)}\mathrm{+}\frac{\mathrm{\beta }}{\mathrm{j}\mathrm{\prime }}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{al}}^{\mathrm{E}}\mathrm{\left(}\frac{{\mathrm{W}}_{\mathrm{1}}{\mathrm{L}}_{\mathrm{1}}\mathrm{+}{\mathrm{Tr}}_{\mathrm{2}}\mathrm{+}{{}^{\mathrm{h}}\mathrm{\pi }}_{\mathrm{2}}\mathrm{-}{\mathrm{P}}_{\mathrm{1}}{\stackrel{\mathrm{˜}}{\mathrm{Y}}}_{\mathrm{1}}\mathrm{-}{\mathrm{P}}_{\mathrm{1}}^{\mathrm{*}}{\mathrm{S}}_{\mathrm{1}}{\stackrel{\mathrm{˜}}{\mathrm{Y}}}_{\mathrm{1}}^{\mathrm{*}}}{{\mathrm{PI}}_{\mathrm{2}}}\mathrm{\right)}\end{array}$

where the “~” on top of (Y1) and $\mathrm{\left(}{\mathrm{Y}}_{\mathrm{1}}^{\mathrm{*}}\mathrm{\right)}$ is used to indicate the stochastic nature of these variables.

Notice that the inequality constraint (19) in the main text does not apply at the beginning of the period, because the demands for consumption goods and money are made effective at the end of the period. The maximization procedure leads to the following labor supply decision rule:

$\begin{array}{cc}\mathrm{\left(}\mathrm{1.11}\mathrm{\right)}& {\mathrm{L}}_{\mathrm{1}}^{\mathrm{s}}\mathrm{=}\overline{\mathrm{L}}\mathrm{-}\frac{\mathrm{j}\mathrm{\prime }}{\mathrm{\beta }}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\frac{{{\mathrm{a}}_{\mathrm{1}}}^{\mathrm{E}}\mathrm{\left(}{\mathrm{PI}}_{\mathrm{2}}\mathrm{\right)}}{{\mathrm{W}}_{\mathrm{1}}}\mathrm{.}\end{array}$

### Appendix II: Derivation of the Final Solution for the Price and Output Levels of the Nontradable Good and the Level of Foreign Reserves

The macro-model of Section V can be solved to yield the following set of semi-reduced forms for the price level of the nontradable good and the end of period level of foreign exchange reserves.

$\begin{array}{cc}\mathrm{\left(}\mathrm{2.1}\mathrm{\right)}& {\mathrm{p}}_{\mathrm{t}}\mathrm{=}{\mathrm{X}}_{\mathrm{0}}\mathrm{+}{\mathrm{X}}_{\mathrm{1}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{f}}_{\mathrm{t}\mathrm{-}\mathrm{1}}\mathrm{+}{\mathrm{X}}_{\mathrm{2}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{gc}}_{\mathrm{t}}\mathrm{+}{\mathrm{X}}_{\mathrm{3}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{at}}^{\mathrm{E}}\mathrm{\left(}{\mathrm{p}}_{\mathrm{t}\mathrm{+}\mathrm{1}}\mathrm{\right)}\mathrm{+}{\mathrm{X}}_{\mathrm{4}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{\left(}{\mathrm{p}}_{\mathrm{t}}^{\mathrm{*}}\mathrm{+}{\mathrm{s}}_{\mathrm{t}}\mathrm{\right)}\mathrm{+}{\mathrm{X}}_{\mathrm{5}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{\left(}{\mathrm{at}}^{\mathrm{E}}\mathrm{\left(}{\mathrm{p}}_{\mathrm{t}\mathrm{+}\mathrm{1}}^{\mathrm{*}}\mathrm{+}{\mathrm{s}}_{\mathrm{t}\mathrm{+}\mathrm{1}}\mathrm{\right)}\mathrm{\right)}\mathrm{+}{\mathrm{X}}_{\mathrm{6}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{u}}_{\mathrm{t}}\mathrm{+}{\mathrm{X}}_{\mathrm{7}}{\mathrm{\epsilon }}_{\mathrm{t}}\end{array}$
$\begin{array}{cc}\mathrm{\left(}\mathrm{2.2}\mathrm{\right)}& {\mathrm{f}}_{\mathrm{t}}\mathrm{=}{\mathrm{Y}}_{\mathrm{0}}\mathrm{+}{\mathrm{Y}}_{\mathrm{1}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{f}}_{\mathrm{t}\mathrm{-}\mathrm{1}}\mathrm{+}{\mathrm{Y}}_{\mathrm{2}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{gc}}_{\mathrm{t}}\mathrm{+}{\mathrm{Y}}_{\mathrm{3}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{at}}^{\mathrm{E}}\mathrm{\left(}{\mathrm{p}}_{\mathrm{t}\mathrm{+}\mathrm{1}}\mathrm{\right)}\mathrm{+}{\mathrm{Y}}_{\mathrm{4}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{\left(}{\mathrm{p}}_{\mathrm{t}}^{\mathrm{*}}\mathrm{+}{\mathrm{s}}_{\mathrm{t}}\mathrm{\right)}\mathrm{+}{\mathrm{Y}}_{\mathrm{5}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{\left(}{\mathrm{at}}^{\mathrm{E}}\mathrm{\left(}{\mathrm{p}}_{\mathrm{t}\mathrm{+}\mathrm{1}}^{\mathrm{*}}\mathrm{+}{\mathrm{s}}_{\mathrm{t}\mathrm{+}\mathrm{1}}\mathrm{\right)}\mathrm{\right)}\mathrm{+}{\mathrm{Y}}_{\mathrm{6}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{u}}_{\mathrm{t}}\mathrm{+}{\mathrm{Y}}_{\mathrm{7}}{\mathrm{\epsilon }}_{\mathrm{t}}\end{array}$

and:

and: det = 1/(1-τ + δ1 + δ2)

For stability purposes, it will be assumed that δ2 > a1 and b2 > a1. Also, from equation (2.2) it is clear that (leaving aside the constant d3) the effects of gct and ft-1 on the level of foreign reserves are similar. This is, of course, due to the fact that both variables are components of the beginning-of-period monetary base. Hence, the usual requirement for the stability of equation (2.2):

$\mathrm{|}\frac{{\mathrm{\delta f}}_{\mathrm{t}}}{{\mathrm{\delta f}}_{\mathrm{t}\mathrm{-}\mathrm{1}}}\mathrm{|}\mathrm{<}\mathrm{1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{is}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{related}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{to}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{the}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{value}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{of}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{|}\frac{{\mathrm{\delta f}}_{\mathrm{t}}}{{\mathrm{\delta gc}}_{\mathrm{t}}}\mathrm{|}\phantom{\rule{0ex}{0ex}}$

In general, 46/ in this model |δft/δgct| < 1 and, 1 > (δft/δft-1) > 0. Hence the model will be stable. 47/

In order to proceed towards a final solution of the model, that is, taking into account the endogenous property of price expectations, it is important to realize that equations (2.1) and (2.2) are not independent from each other. The method of undetermined coefficients is used here to obtain the final solution of the model (see Lucas (1972), Barro (1976, 1978)). Following McCallum (1983), notice that the solution for the price level of nontradables can be written as a linear function of the predetermined state variables ${\mathrm{f}}_{\mathrm{t}\mathrm{-}\mathrm{1}}\mathrm{,}{\mathrm{gc}}_{\mathrm{t}\mathrm{-}\mathrm{1}}\mathrm{,}\mathrm{m}\mathrm{,}{\mathrm{v}}_{\mathrm{t}}\mathrm{,}{\mathrm{x}}_{\mathrm{t}\mathrm{-}\mathrm{1}}\mathrm{,}{\mathrm{p}}_{\mathrm{t}\mathrm{-}\mathrm{1}}^{\mathrm{*}}$, j*, nt, tt, tt-1, st-1, ξt, ut, εt, and the constant 1. Hence:

$\begin{array}{lll}\left(2.3\right)& {\mathrm{p}}_{\mathrm{t}}& \mathrm{=}{\mathrm{\theta }}_{\mathrm{o}}\mathrm{+}{\mathrm{\theta }}_{\mathrm{1}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{f}}_{\mathrm{t}\mathrm{-}\mathrm{1}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{\theta }}_{\mathrm{2}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{gc}}_{\mathrm{t}\mathrm{-}\mathrm{1}}\mathrm{+}{\mathrm{\theta }}_{\mathrm{3}}\mathrm{m}\mathrm{+}{\mathrm{\theta }}_{\mathrm{4}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{v}}_{\mathrm{t}}\mathrm{+}{\mathrm{\theta }}_{\mathrm{5}}{\mathrm{x}}_{\mathrm{t}}\\ & & \mathrm{+}{\mathrm{\theta }}_{\mathrm{6}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{x}}_{\mathrm{t}\mathrm{-}\mathrm{1}}\mathrm{+}{\mathrm{\theta }}_{\mathrm{7}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{\left(}{\mathrm{p}}_{\mathrm{t}\mathrm{-}\mathrm{1}}^{\mathrm{*}}\mathrm{+}{\mathrm{s}}_{\mathrm{t}\mathrm{-}\mathrm{1}}\mathrm{\right)}\mathrm{+}{\mathrm{\theta }}_{\mathrm{8}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{j}\mathrm{*}\mathrm{+}{\mathrm{\theta }}_{\mathrm{9}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{\left(}{\mathrm{n}}_{\mathrm{t}}\mathrm{+}{\mathrm{\xi }}_{\mathrm{t}}\mathrm{\right)}\\ & & \mathrm{+}{\mathrm{\theta }}_{\mathrm{10}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{t}}_{\mathrm{t}}\mathrm{+}{\mathrm{\theta }}_{\mathrm{11}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{t}}_{\mathrm{t}\mathrm{-}\mathrm{1}}\mathrm{+}{\mathrm{\theta }}_{\mathrm{12}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{u}}_{\mathrm{t}}\mathrm{+}{\mathrm{\theta }}_{\mathrm{13}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{\epsilon }}_{\mathrm{t}}\end{array}$

When the θs are the unknown coefficients.

Leading equation (2.3) once, taking beginning-of-period t expectations, and using equation (2.2), we obtain:

$\begin{array}{lll}\mathrm{\left(}\mathrm{2.4}\mathrm{\right)}& {\mathrm{at}}^{\mathrm{E}}\mathrm{\left(}{\mathrm{p}}_{\mathrm{t}\mathrm{+}\mathrm{1}}\mathrm{\right)}\mathrm{=}\frac{\mathrm{1}}{\mathrm{1}\mathrm{-}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{3}}}& \mathrm{\left\{}\mathrm{\left(}{\mathrm{\theta }}_{\mathrm{0}}\mathrm{+}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{0}}\mathrm{\right)}\mathrm{+}\mathrm{\left(}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{1}}\mathrm{\right)}{\mathrm{f}}_{\mathrm{t}\mathrm{-}\mathrm{1}}\\ & & \mathrm{+}\mathrm{\left(}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{2}}\mathrm{+}{\mathrm{\theta }}_{\mathrm{2}}\mathrm{\right)}\mathrm{\left(}{\mathrm{gc}}_{\mathrm{t}\mathrm{-}\mathrm{1}}\mathrm{+}{\mathrm{v}}_{\mathrm{t}}\mathrm{-}{\mathrm{x}}_{\mathrm{t}\mathrm{-}\mathrm{1}}\mathrm{\right)}\\ & & \mathrm{+}\mathrm{\left(}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{2}}\mathrm{+}{\mathrm{\theta }}_{\mathrm{2}}\mathrm{+}{\mathrm{\theta }}_{\mathrm{3}}\mathrm{\right)}\mathrm{m}\\ & & \mathrm{+}\mathrm{\left(}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{2}}\mathrm{+}{\mathrm{\theta }}_{\mathrm{2}}\mathrm{+}{\mathrm{\theta }}_{\mathrm{6}}\mathrm{\right)}{\mathrm{x}}_{\mathrm{t}}\\ & & \mathrm{+}\mathrm{\left(}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{4}}\mathrm{+}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{5}}\mathrm{+}{\mathrm{\theta }}_{\mathrm{7}}\mathrm{\right)}\mathrm{\left(}{\mathrm{p}}_{\mathrm{t}\mathrm{-}\mathrm{1}}^{\mathrm{*}}\mathrm{+}{\mathrm{s}}_{\mathrm{t}\mathrm{-}\mathrm{1}}\mathrm{+}{\mathrm{\xi }}_{\mathrm{t}}\mathrm{+}{\mathrm{n}}_{\mathrm{t}}\mathrm{-}{\mathrm{t}}_{\mathrm{t}\mathrm{-}\mathrm{1}}\mathrm{\right)}\\ & & \mathrm{+}\mathrm{\left(}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{4}}\mathrm{+}\mathrm{2}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{5}}\mathrm{+}{\mathrm{\theta }}_{\mathrm{7}}\mathrm{+}{\mathrm{\theta }}_{\mathrm{8}}\mathrm{\right)}\mathrm{j}\mathrm{*}\\ & & \mathrm{+}\mathrm{\left(}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{4}}\mathrm{+}{\mathrm{\theta }}_{\mathrm{7}}^{\mathrm{N}}\mathrm{+}{\mathrm{\theta }}_{\mathrm{11}}\mathrm{\right)}{\mathrm{t}}_{\mathrm{t}}\\ & & \mathrm{+}\mathrm{\left(}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{6}}\mathrm{\right)}{\mathrm{u}}_{\mathrm{t}}\\ & & \begin{array}{c}\mathrm{+}\mathrm{\left(}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{7}}\mathrm{\right)}{\mathrm{\epsilon }}_{\mathrm{t}}\end{array}\mathrm{\right\}}\end{array}$

Substituting equation (2.4) into equation (2.1), the final solution for the price level of nontradables is obtained:

$\begin{array}{lll}\mathrm{\left(}\mathrm{2.5}\mathrm{\right)}& {\mathrm{p}}_{\mathrm{t}}\mathrm{=}\frac{\mathrm{1}}{\mathrm{1}\mathrm{-}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{3}}}& \mathrm{\left\{}\mathrm{\left[}{\mathrm{X}}_{\mathrm{0}}-{\mathrm{X}}_{0}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{3}}\mathrm{+}{\mathrm{X}}_{\mathrm{3}}{\mathrm{\theta }}_{\mathrm{0}}\mathrm{+}{\mathrm{X}}_{\mathrm{3}}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{0}}\mathrm{\right]}\\ & & \mathrm{+}\mathrm{\left[}{\mathrm{X}}_{\mathrm{1}}\mathrm{-}{\mathrm{X}}_{\mathrm{1}}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{3}}\mathrm{+}{\mathrm{X}}_{\mathrm{3}}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{1}}\mathrm{\right]}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{f}}_{\mathrm{t}\mathrm{-}\mathrm{1}}\\ & & \mathrm{+}\mathrm{\left[}{\mathrm{X}}_{\mathrm{2}}\mathrm{-}{\mathrm{X}}_{\mathrm{2}}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{3}}\mathrm{+}{\mathrm{X}}_{\mathrm{3}}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{2}}\mathrm{+}{\mathrm{X}}_{\mathrm{3}}{\mathrm{\theta }}_{\mathrm{2}}\mathrm{\right]}\mathrm{\left(}{\mathrm{gc}}_{\mathrm{t}\mathrm{-}\mathrm{1}}\mathrm{+}{\mathrm{v}}_{\mathrm{t}}\mathrm{-}{\mathrm{x}}_{\mathrm{t}\mathrm{-}\mathrm{1}}\mathrm{\right)}\\ & & \mathrm{+}\mathrm{\left[}{\mathrm{X}}_{\mathrm{2}}\mathrm{-}{\mathrm{X}}_{\mathrm{2}}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{3}}\mathrm{+}{\mathrm{X}}_{\mathrm{3}}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{2}}\mathrm{+}{\mathrm{X}}_{\mathrm{3}}{\mathrm{\theta }}_{\mathrm{2}}\mathrm{+}{\mathrm{X}}_{\mathrm{3}}{\mathrm{\theta }}_{\mathrm{3}}\mathrm{\right]}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{m}\\ & & \mathrm{+}\mathrm{\left[}{\mathrm{X}}_{\mathrm{2}}\mathrm{-}{\mathrm{X}}_{\mathrm{2}}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{3}}\mathrm{+}{\mathrm{X}}_{\mathrm{3}}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{2}}\mathrm{+}{\mathrm{X}}_{\mathrm{3}}{\mathrm{\theta }}_{\mathrm{2}}\mathrm{+}{\mathrm{X}}_{\mathrm{3}}{\mathrm{\theta }}_{\mathrm{6}}\mathrm{\right]}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{x}}_{\mathrm{t}}\\ & & \mathrm{+}\mathrm{\left[}\mathrm{\left(}{\mathrm{X}}_{\mathrm{4}}\mathrm{+}{\mathrm{X}}_{\mathrm{5}}\mathrm{\right)}\mathrm{\left(}\mathrm{1}\mathrm{-}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{3}}\mathrm{\right)}\mathrm{+}{\mathrm{X}}_{\mathrm{3}}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{4}}\mathrm{+}{\mathrm{X}}_{\mathrm{3}}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{5}}\mathrm{+}{\mathrm{X}}_{\mathrm{3}}{\mathrm{\theta }}_{\mathrm{7}}\mathrm{\right]}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{\left(}{\mathrm{p}}_{\mathrm{t}\mathrm{-}\mathrm{1}}^{\mathrm{*}}\mathrm{+}{\mathrm{s}}_{\mathrm{t}\mathrm{-}\mathrm{1}}\mathrm{+}{\mathrm{\xi }}_{\mathrm{t}}\mathrm{+}{\mathrm{n}}_{\mathrm{t}}\mathrm{-}{\mathrm{t}}_{\mathrm{t}\mathrm{-}\mathrm{1}}\mathrm{\right)}\\ & & \mathrm{+}\mathrm{\left[}\mathrm{\left(}{\mathrm{X}}_{\mathrm{4}}\mathrm{+}\mathrm{2}{\mathrm{X}}_{\mathrm{5}}\mathrm{\right)}\mathrm{\left(}\mathrm{1}\mathrm{-}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{3}}\mathrm{\right)}\mathrm{+}{\mathrm{X}}_{\mathrm{3}}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{4}}\mathrm{+}\mathrm{2}{\mathrm{X}}_{\mathrm{3}}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{5}}\mathrm{+}{\mathrm{X}}_{\mathrm{3}}\mathrm{\left(}{\mathrm{\theta }}_{\mathrm{7}}\mathrm{+}{\mathrm{\theta }}_{\mathrm{8}}\mathrm{\right)}\mathrm{\right]}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{j}\mathrm{*}\\ & & \mathrm{+}\mathrm{\left[}{\mathrm{X}}_{\mathrm{4}}\mathrm{-}{\mathrm{X}}_{\mathrm{4}}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{3}}\mathrm{+}{\mathrm{X}}_{\mathrm{3}}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{4}}\mathrm{+}{\mathrm{X}}_{\mathrm{3}}\mathrm{\left(}{\mathrm{\theta }}_{\mathrm{7}}\mathrm{+}{\mathrm{\theta }}_{\mathrm{11}}\mathrm{\right)}\mathrm{\right]}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{t}}_{\mathrm{t}}\\ & & \mathrm{+}\mathrm{\left[}{\mathrm{X}}_{\mathrm{6}}\mathrm{-}{\mathrm{X}}_{\mathrm{6}}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{3}}\mathrm{+}{\mathrm{X}}_{\mathrm{3}}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{6}}\mathrm{\right]}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{u}}_{\mathrm{t}}\\ & & \mathrm{+}\mathrm{\left[}{\mathrm{X}}_{\mathrm{7}}\mathrm{-}{\mathrm{X}}_{\mathrm{7}}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{3}}\mathrm{+}{\mathrm{X}}_{\mathrm{3}}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{7}}\mathrm{\right]}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{\epsilon }}_{\mathrm{t}}\mathrm{\right\}}\end{array}$

Equating coefficients among equations (2.3) and (2.5), the solution for the θs are obtained. In particular:

$\begin{array}{cc}\mathrm{\left(}\mathrm{2.6}\mathrm{\right)}& {\mathrm{\theta }}_{\mathrm{1}}\mathrm{=}\frac{\mathrm{-}\mathrm{\left(}{\mathrm{X}}_{\mathrm{3}}{\mathrm{Y}}_{\mathrm{1}}\mathrm{-}{\mathrm{X}}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{3}}\mathrm{-}\mathrm{1}\mathrm{\right)}\mathrm{±}\sqrt{{\mathrm{\left(}{\mathrm{X}}_{\mathrm{3}}{\mathrm{Y}}_{\mathrm{1}}\mathrm{-}{\mathrm{X}}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{3}}\mathrm{-}\mathrm{1}\mathrm{\right)}}^{\mathrm{2}}\mathrm{-}\mathrm{4}{\mathrm{Y}}_{\mathrm{3}}{\mathrm{X}}_{\mathrm{1}}}}{\mathrm{2}{\mathrm{Y}}_{\mathrm{3}}}\end{array}$

There are two possible solutions 48/ for θ1. To choose between them, we will follow McCallum (1983) by imposing the requirement that the solution for θ1 must be valid for all admissible values of the structural parameters. In particular, ft-1 appears in the solution for Pt because it forms part of the system (equation (2.1)). In the special case in which X1 = 0, ft-1 would not be an argument for pt and hence, would not be included in the “minimal set of state variables.” Thus, θ1 would be equal to zero. But from equation (2.6) it is clear that θ1 = 0 would be obtained (under the assumption X1 = 0) only if the negative root is used.

The parameters of the model implies that θ1 has a positive value 49/ and hence allows us to sign the rest of θs since they are functions of θ1. Thus:

$\begin{array}{cc}\mathrm{\left(}\mathrm{2.7}\mathrm{\right)}& {\mathrm{\theta }}_{\mathrm{2}}\mathrm{=}\frac{{\mathrm{\theta }}_{\mathrm{1}}\mathrm{\left(}{\mathrm{X}}_{\mathrm{3}}{\mathrm{Y}}_{\mathrm{2}}\mathrm{-}{\mathrm{X}}_{\mathrm{2}}{\mathrm{Y}}_{\mathrm{3}}\mathrm{\right)}\mathrm{+}{\mathrm{X}}_{\mathrm{2}}}{\mathrm{1}\mathrm{-}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{3}}\mathrm{-}{\mathrm{X}}_{\mathrm{3}}}\mathrm{>}\mathrm{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{and}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{less}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{than}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{one}\mathrm{.}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}50\mathrm{/}\end{array}$
$\begin{array}{cc}\mathrm{\left(}\mathrm{2.8}\mathrm{\right)}& {\mathrm{\theta }}_{\mathrm{3}}\mathrm{=}\frac{{\mathrm{\theta }}_{\mathrm{2}}\mathrm{\left(}\mathrm{1}\mathrm{-}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{3}}\mathrm{\right)}}{\mathrm{1}\mathrm{-}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{3}}\mathrm{-}{\mathrm{X}}_{\mathrm{3}}}\mathrm{>}\mathrm{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{and}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{\theta }}_{\mathrm{3}}\mathrm{>}{\mathrm{\theta }}_{\mathrm{2}}\end{array}$
$\begin{array}{cc}\mathrm{\left(}\mathrm{2.9}\mathrm{\right)}& {\mathrm{\theta }}_{\mathrm{4}}\mathrm{=}\mathrm{-}{\mathrm{\theta }}_{\mathrm{6}}\mathrm{=}{\mathrm{\theta }}_{\mathrm{2}}\end{array}$
$\begin{array}{cc}\mathrm{\left(}\mathrm{2.10}\mathrm{\right)}& {\mathrm{\theta }}_{\mathrm{5}}\mathrm{=}\frac{{\mathrm{\theta }}_{\mathrm{2}}\mathrm{\left(}\mathrm{1}\mathrm{-}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{3}}\mathrm{-}{\mathrm{X}}_{\mathrm{3}}\mathrm{\right)}}{\mathrm{1}\mathrm{-}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{3}}}\mathrm{>}\mathrm{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{and}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{\theta }}_{\mathrm{5}}\mathrm{<}{\mathrm{\theta }}_{\mathrm{2}}\end{array}$
$\begin{array}{cc}\mathrm{\left(}\mathrm{2.11}\mathrm{\right)}& {\mathrm{\theta }}_{\mathrm{7}}\mathrm{=}\frac{\mathrm{\left(}\mathrm{1}\mathrm{-}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{3}}\mathrm{\right)}\mathrm{\left(}{\mathrm{X}}_{\mathrm{4}}\mathrm{+}{\mathrm{X}}_{\mathrm{5}}\mathrm{\right)}\mathrm{+}{\mathrm{X}}_{\mathrm{3}}{\mathrm{\theta }}_{\mathrm{1}}\mathrm{\left(}{\mathrm{Y}}_{\mathrm{4}}\mathrm{+}{\mathrm{Y}}_{\mathrm{5}}\mathrm{\right)}}{\mathrm{1}\mathrm{-}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{3}}\mathrm{-}{\mathrm{X}}_{\mathrm{3}}}\mathrm{>}\mathrm{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{since}\mathrm{|}{\mathrm{Y}}_{\mathrm{4}}\mathrm{|}\mathrm{>}\mathrm{|}{\mathrm{Y}}_{\mathrm{5}}\mathrm{|}\mathrm{;}\mathrm{and}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{\theta }}_{\mathrm{7}}\mathrm{<}\mathrm{1}\end{array}$
$\begin{array}{cc}\mathrm{\left(}\mathrm{2.12}\mathrm{\right)}& {\mathrm{\theta }}_{\mathrm{8}}\mathrm{=}{\mathrm{\theta }}_{\mathrm{7}}\mathrm{+}\frac{{\mathrm{X}}_{\mathrm{5}}}{\mathrm{1}\mathrm{-}{\mathrm{\theta }}_{\mathrm{1}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{Y}}_{\mathrm{3}}\mathrm{-}{\mathrm{X}}_{\mathrm{3}}}\mathrm{>}\mathrm{0}\mathrm{;}\mathrm{and}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{\theta }}_{\mathrm{8}}\mathrm{>}{\mathrm{\theta }}_{\mathrm{7}}\end{array}$
$\begin{array}{cc}\mathrm{\left(}\mathrm{2.13}\mathrm{\right)}& {\mathrm{\theta }}_{\mathrm{9}}\mathrm{=}\mathrm{-}{\mathrm{\theta }}_{\mathrm{11}}\mathrm{=}{\mathrm{\theta }}_{\mathrm{7}}\end{array}$
$\begin{array}{cc}\mathrm{\left(}\mathrm{2.14}\mathrm{\right)}& {\mathrm{\theta }}_{\mathrm{10}}\mathrm{=}\frac{{\mathrm{X}}_{\mathrm{4}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{\left(}\mathrm{1}\mathrm{-}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{3}}\mathrm{\right)}\mathrm{+}{\mathrm{X}}_{\mathrm{3}}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{4}}}{\mathrm{1}\mathrm{-}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{3}}}\mathrm{>}\mathrm{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{since}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{Y}}_{\mathrm{4}}\mathrm{>}\mathrm{0}\mathrm{;}\mathrm{and}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{\theta }}_{\mathrm{10}}\mathrm{<}\mathrm{1}\end{array}$
$\begin{array}{cc}\mathrm{\left(}\mathrm{2.15}\mathrm{\right)}& {\mathrm{\theta }}_{\mathrm{12}}\mathrm{=}\frac{{\mathrm{X}}_{\mathrm{6}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{\left(}\mathrm{1}\mathrm{-}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{3}}\mathrm{\right)}\mathrm{+}{\mathrm{X}}_{\mathrm{3}}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{6}}}{\mathrm{1}\mathrm{-}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{3}}}\mathrm{<}\mathrm{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{since}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{X}}_{\mathrm{6}}\mathrm{<}\mathrm{0}\end{array}$
$\begin{array}{cc}\mathrm{\left(}\mathrm{2.16}\mathrm{\right)}& {\mathrm{\theta }}_{\mathrm{13}}\mathrm{=}\frac{{\mathrm{X}}_{\mathrm{7}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{\left(}\mathrm{1}\mathrm{-}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{3}}\mathrm{\right)}\mathrm{+}{\mathrm{X}}_{\mathrm{3}}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{7}}}{\mathrm{1}\mathrm{-}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{3}}}\mathrm{>}\mathrm{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{since}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{X}}_{\mathrm{7}}\mathrm{>}\mathrm{0}\end{array}$
$\begin{array}{cc}\mathrm{\left(}\mathrm{2.17}\mathrm{\right)}& {\mathrm{\theta }}_{\mathrm{0}}\mathrm{=}\frac{{\mathrm{X}}_{\mathrm{0}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{\left(}\mathrm{1}\mathrm{-}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{3}}\mathrm{\right)}\mathrm{+}{\mathrm{X}}_{\mathrm{3}}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{0}}}{\mathrm{1}\mathrm{-}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{3}}\mathrm{-}{\mathrm{X}}_{\mathrm{3}}}\mathrm{>}\mathrm{0}\end{array}$

The final solution for the levels of output of both commodities are found by substituting equations (2.3) and (2.4) into equations (41) and (42) of the main text. Those reduced form solutions are presented in Table 1.

Finally, the solution for the level of foreign reserves can be easily found by substituting equation (2.4) into equation (2.2).

$\begin{array}{lll}\mathrm{\left(}\mathrm{2.18}\mathrm{\right)}& {\mathrm{f}}_{\mathrm{t}}\mathrm{=}\frac{\mathrm{1}}{\mathrm{1}\mathrm{-}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{3}}}& \mathrm{\left\{}\mathrm{\left[}{\mathrm{Y}}_{\mathrm{0}}\mathrm{+}{\mathrm{Y}}_{\mathrm{3}}{\mathrm{\theta }}_{\mathrm{0}}\mathrm{\right]}\mathrm{+}\mathrm{\left[}{\mathrm{Y}}_{\mathrm{1}}\mathrm{\right]}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{f}}_{\mathrm{t}\mathrm{-}\mathrm{1}}\\ & & \mathrm{+}\mathrm{\left[}{\mathrm{Y}}_{\mathrm{2}}\mathrm{+}{\mathrm{Y}}_{\mathrm{3}}{\mathrm{\theta }}_{\mathrm{2}}\mathrm{\right]}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{\left(}{\mathrm{gc}}_{\mathrm{t}\mathrm{-}\mathrm{1}}\mathrm{+}{\mathrm{v}}_{\mathrm{t}}\mathrm{-}{\mathrm{x}}_{\mathrm{t}\mathrm{-}\mathrm{1}}\mathrm{\right)}\\ & & \mathrm{+}\mathrm{\left[}{\mathrm{Y}}_{\mathrm{2}}\mathrm{+}{\mathrm{Y}}_{\mathrm{3}}\mathrm{\left(}{\mathrm{\theta }}_{\mathrm{2}}\mathrm{+}{\mathrm{\theta }}_{\mathrm{3}}\mathrm{\right)}\mathrm{\right]}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{m}\\ & & \mathrm{+}\mathrm{\left[}{\mathrm{Y}}_{\mathrm{2}}\mathrm{\right]}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{x}}_{\mathrm{t}}\\ & & \mathrm{+}\mathrm{\left[}{\mathrm{Y}}_{\mathrm{4}}\mathrm{+}{\mathrm{Y}}_{\mathrm{5}}\mathrm{+}{\mathrm{Y}}_{\mathrm{3}}{\mathrm{\theta }}_{\mathrm{7}}\mathrm{\right]}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{\left(}{\mathrm{p}}_{\mathrm{t}\mathrm{-}\mathrm{1}}^{\mathrm{*}}\mathrm{+}{\mathrm{n}}_{\mathrm{t}}\mathrm{+}{\mathrm{s}}_{\mathrm{t}\mathrm{-}\mathrm{1}}\mathrm{+}{\mathrm{\xi }}_{\mathrm{t}}\mathrm{-}{\mathrm{t}}_{\mathrm{t}\mathrm{-}\mathrm{1}}\mathrm{\right)}\\ & & \mathrm{+}\mathrm{\left[}{\mathrm{Y}}_{\mathrm{4}}\mathrm{+}\mathrm{2}{\mathrm{Y}}_{\mathrm{5}}\mathrm{+}{\mathrm{Y}}_{\mathrm{3}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{\left(}{\mathrm{\theta }}_{\mathrm{7}}\mathrm{+}{\mathrm{\theta }}_{\mathrm{8}}\mathrm{\right)}\mathrm{\right]}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{j}\mathrm{*}\\ & & \mathrm{+}\mathrm{\left[}{\mathrm{Y}}_{\mathrm{4}}\mathrm{\right]}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{t}}_{\mathrm{t}}\\ & & \begin{array}{c}\mathrm{+}\mathrm{\left[}{\mathrm{Y}}_{\mathrm{6}}\mathrm{\right]}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{u}}_{\mathrm{t}}\mathrm{+}\mathrm{\left[}{\mathrm{Y}}_{\mathrm{7}}\mathrm{\right]}{\mathrm{\epsilon }}_{\mathrm{t}}\end{array}\mathrm{\right\}}\mathrm{.}\end{array}$

## References

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• Barro, Robert J.,Money and Output in Mexico, Colombia and Brazil,Short Term Macroeconomic Policy in Latin America, edited by J. Behrman and J. Hanson, The National Bureau of Economic Research (1979) pp. 177200.

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• Clower, Robert W.,A Reconsideration of the Microfoundations of Monetary Theory,Western Economic Journal, Vol. 6, No. 4 (December 1967), pp. 19.

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• Dornbusch, Rudiger,Money, Devaluation and Nontraded Goods,American Economic Review, Vol. 63 (December 1973), pp. 87180.

• Edwards, Sebastian,The Short-Run Relation Between Growth and Inflation in Latin America: Comment,American Economic Review, Vol. 73 (June 1983), pp. 47782.

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• Fleming, Marcus J.,Domestic Financial Policies Under Fixed and Floating Exchange Rates,International Monetary Fund, Staff Papers (Washington), Vol. 9 (March 1962), pp. 36977.

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• Fry, Maxwell J.,Models of Financially Repressed Developing Economies,World Development, Vol. 10, No. 2 (1982) pp. 73150.

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• Lucas, Robert E. Jr.,Equilibrium in a Pure Currency Economy,Economic Inquiry, Vol. 18, No. 2 (April 1980), pp. 20320.

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• Mathieson, Donald J.,Financial Reform and Capital Flows in a Developing Economy,Staff Papers, International Monetary Fund (Washington) Vol. 26, No. 3 (September 1979), pp. 45089.

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• McCallum, Bennett T.,On Non-Uniqueness in Rational Expectations Models: An Attempt at Perspective,Journal of Monetary Economics (1983), pp. 13968.

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• McKinnon, Ronald, Money and Capital in Economic Development (Washington, The Brookings Institution, 1973).

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• Muth, John R.,Rational Expectations and the Theory of Price Movements,Econometrica, Vol. 29, No. 3 (July 1961), pp. 31535.

• Samuelson, Paul A.,An Exact Concumption-loan Model of Interest with or without the Social Contrivance of Money,Journal of Political Economy, Vol. 66 (December 1958), pp. 46782.

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• Stockman, Allan,Anticipated Inflation and the Capital Stock in a Cash-in-Advance Economy,Journal of Monetary Economics, Vol. 8 (1981), pp. 38793.

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• Stockman, Allan,A Theory of Exchange Rate Determination,Journal of Political Economy, Vol. 88, No. 4 (August 1980), pp. 67398

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I wish to thank Peter Howitt, David Laidler, and Michael Parkin for their contributions to this paper. Useful comments were also provided by Donald J. Mathieson and Daniel Gros.

Regarding the analysis of expansions in the exogenous component of the money supply, models that assume some kind of price stickiness in the short run conclude that the initial increase in aggregate demand will stimulate economic activity, while those models that assume price flexibility conclude that the resulting increase in the relative price of nontradables will temporarily shift resources from the production of tradables towards the production of nontradables with no change in the aggregate level of output.

The study by Edwards covers the period 1965-80 and focuses on the following countries: India, Malaysia, Philippines, Sri Lanka, Thailand, Greece, Israel, Brazil, Colombia, El Salvador, South Africa, and Yugoslavia.

The study by Gomez-Oliver includes: Bolivia, Costa Rica, Ecuador, and Mexico. Gomez-Oliver found that devaluations during the 1970s were accompanied by a slowdown in real GDP for one or two years and that devaluations in the early 1980s were accompanied by actual contractions in real GDP.

The study by Barro covers the period 1959-74. During most of that period (1959-70), the Mexican exchange rate was fixed at 12.50 pesos per U.S. dollar.

The data for Chile covers the period 1952-70 and the data for Brazil covers the period 1952-74.

Hanson disregarded this result since he interpreted it as inconsistent with rational expectations. His argument was that in the context of a Lucas (1973) aggregate supply function, the actual rate of change of the money supply will affect output only if money is a random walk. Since Hanson found that the Brazilean and Chilean money supply processes are significantly different from a random walk, he concluded that a regression of output on actual money growth “imposes an incorrect money supply process and therefore irrational expectations” (p. 980). However, in this paper we show that such a relation between output and money growth is perfectly consistent with the rational expectations hypothesis in the context of an output supply equation alternative to that of Lucas.

It is important to note that the positive relationship between output and real money in this model does not arise from introducing money as a wealth variable. Instead, it is due to the financial constraint assumption which in turn is closely related to the standard cash-in-advance assumption used in the open economy models of Stockman (1980), Helpman (1981), and Aschauer and Greenwood (1983).

For a review of this literature see Edwards (1985).

There are also some models that advance supply-side arguments to explain the contractionary effects of a devaluation. The most common argument is that a devaluation increases the cost of imported intermediate inputs and hence results in a decrease in aggregate supply.

There are, however, several models that produce an upward slowing Phillips curve (a negative correlation between output and inflation) in the long run. See for example Fry (1978), Mathieson (1980), and Stockman (1980).

It is assumed that, in the initial period of the life of this economy, an old generation owned all relevant property rights. Since every individual is assumed to maximize utility over only his own lifetime, it would have been in the best interest of the old generation to sell these property rights at the beginning of this period, because they were to die at the end of it. However, in this economy, there is no demand for property rights at the beginning of any period. Other old people obviously do not want to buy them, and young people are not able to buy them. They begin their life with no money and are unable to borrow. Hence, there is no market for property rights.

It is also assumed that imperfections in the capital market prevent households from obtaining bank loans.

It is assumed here that government regulations prevent perfect competition in the banking system. Fry (1982) argues that this is a mechanism through which the government can expropriate a large seigniorage.

In general, the subscript “a” refers to “beginning of period” and the subscript “b” refers to “end of period.”

The productivity terra in the denominator means that a positive random increase in productivity will result in a lower amount of labor required to produce a given amount of output. The sequence {ϕt} is assumed to be a strictly positive stationary stochastic process and its distribution will be specified in Section IV.

Notice that the assumption of joint production allows us to treat the demand for credit as a single variable, without having to decompose it according to its uses (the production of tradable or nontradable goods).

The information set at the beginning of the period also includes the random term μt which affects the household’s utility function, and will be considered below.

In addition, the use of Cobb-Douglas functions implies that the derived elasticities of demand for current consumption of both tradable and nontradable commodities with respect to total expenditure on current consumption is equal to one. By the same token, the elasticities of demand for future consumption of both tradable and nontradable commodities with respect to total expenditure on future consumption is equal to one.

It should be recalled that the wage rate and the interest rate solutions are not final solutions because the price level (and its expectations) are taken as exogenous for the time being.

The solution for the interest rate is:

${\mathrm{i}}_{\mathrm{t}}\mathrm{=}{\mathrm{\gamma }}_{\mathrm{0}}^{\mathrm{\prime }}\mathrm{-}{\mathrm{\gamma }}_{\mathrm{1}}^{\mathrm{\prime }}\mathrm{\left(}{\mathrm{a}}^{\mathrm{h}}\mathrm{t}\mathrm{-}{\mathrm{at}}^{\mathrm{E}}\mathrm{\left(}{\mathrm{pi}}_{\mathrm{t}}\mathrm{\right)}\mathrm{\right)}\mathrm{+}{\mathrm{\gamma }}_{\mathrm{2}}^{\mathrm{\prime }}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{\left(}{\mathrm{at}}^{\mathrm{E}}\mathrm{\left(}{\mathrm{pi}}_{\mathrm{t}\mathrm{+}\mathrm{1}}\mathrm{-}{\mathrm{pi}}_{\mathrm{t}}\mathrm{\right)}\mathrm{\right)}\mathrm{+}{\mathrm{\gamma }}_{\mathrm{3}}^{\mathrm{\prime }}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{u}}_{\mathrm{t}}$

where:

${\mathrm{\gamma }}_{\mathrm{0}}^{\mathrm{\prime }}\mathrm{=}\frac{{\mathrm{\gamma }}_{\mathrm{1}}{\mathrm{\beta }}_{\mathrm{o}}\mathrm{+}{\mathrm{\alpha }}_{\mathrm{1}}{\mathrm{\beta }}_{\mathrm{o}}\mathrm{-}{\mathrm{\gamma }}_{\mathrm{o}}\mathrm{+}{\mathrm{\gamma }}_{\mathrm{o}}{\mathrm{\alpha }}_{\mathrm{1}}\mathrm{+}{\mathrm{\alpha }}_{\mathrm{0}}\mathrm{-}{\mathrm{\alpha }}_{\mathrm{o}}{\mathrm{\alpha }}_{\mathrm{1}}}{{\mathrm{\alpha }}_{\mathrm{1}}\mathrm{\left(}\mathrm{1}\mathrm{+}{\mathrm{\gamma }}_{\mathrm{1}}\mathrm{\right)}}{\mathrm{\gamma }}_{\mathrm{1}}^{\mathrm{\prime }}\mathrm{k}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{\gamma }}_{\mathrm{1}}^{\mathrm{\prime }}\mathrm{=}\frac{{\mathrm{\gamma }}_{\mathrm{1}}\mathrm{+}{\mathrm{\alpha }}_{\mathrm{1}}}{{\mathrm{\alpha }}_{\mathrm{1}}\mathrm{\left(}\mathrm{1}\mathrm{+}{\mathrm{\gamma }}_{\mathrm{1}}\mathrm{\right)}}\mathrm{;}{\mathrm{\gamma }}_{\mathrm{2}}^{\mathrm{\prime }}\mathrm{=}\frac{\mathrm{\left(}\mathrm{1}\mathrm{-}{\mathrm{\alpha }}_{\mathrm{1}}\mathrm{\right)}{\mathrm{\gamma }}_{\mathrm{1}}}{{\mathrm{\alpha }}_{\mathrm{1}}\mathrm{\left(}\mathrm{1}\mathrm{+}{\mathrm{\gamma }}_{\mathrm{1}}\mathrm{\right)}}\mathrm{;}{\mathrm{\gamma }}_{\mathrm{3}}^{\mathrm{\prime }}\mathrm{=}\frac{\mathrm{1}}{{\mathrm{\alpha }}_{\mathrm{1}}}$

and the solution for the wage rate is:

${\mathrm{w}}_{\mathrm{t}}\mathrm{=}{\mathrm{at}}^{\mathrm{E}}\mathrm{\left(}{\mathrm{pi}}_{\mathrm{t}}\mathrm{\right)}\mathrm{+}{\mathrm{\beta }}_{\mathrm{o}}^{\mathrm{\prime }}\mathrm{+}{\mathrm{\beta }}_{\mathrm{1}}^{\mathrm{\prime }}\mathrm{\left(}{\mathrm{at}}^{\mathrm{E}}\mathrm{\left(}{\mathrm{pi}}_{\mathrm{t}\mathrm{+}\mathrm{1}}\mathrm{-}{\mathrm{pi}}_{\mathrm{t}}\mathrm{\right)}\mathrm{\right)}\mathrm{+}{\mathrm{\beta }}_{\mathrm{2}}^{\mathrm{\prime }}\mathrm{\left(}{\mathrm{a}}^{\mathrm{h}}\mathrm{t}\mathrm{-}{\mathrm{at}}^{\mathrm{E}}\mathrm{\left(}{\mathrm{pi}}_{\mathrm{t}}\mathrm{\right)}\mathrm{\right)}$

where: ${\mathrm{\beta }}_{\mathrm{o}}^{\mathrm{\prime }}\mathrm{=}\frac{{\mathrm{\alpha }}_{\mathrm{o}}\mathrm{-}{\mathrm{\gamma }}_{\mathrm{o}}\mathrm{-}{\mathrm{\beta }}_{\mathrm{o}}\mathrm{+}\mathrm{k}}{\mathrm{\left(}\mathrm{1}\mathrm{+}{\mathrm{\gamma }}_{\mathrm{1}}\mathrm{\right)}}\mathrm{;}{\mathrm{\beta }}_{\mathrm{1}}^{\mathrm{\prime }}\mathrm{=}\frac{{\mathrm{\gamma }}_{\mathrm{1}}}{\mathrm{\left(}\mathrm{1}\mathrm{+}{\mathrm{\gamma }}_{\mathrm{1}}\mathrm{\right)}}\mathrm{;}{\mathrm{\beta }}_{\mathrm{2}}^{\mathrm{\prime }}\mathrm{=}\frac{\mathrm{1}}{\mathrm{\left(}\mathrm{1}\mathrm{+}{\mathrm{\gamma }}_{\mathrm{1}}\mathrm{\right)}}$

Imposing equilibrium in the markets for tradable and nontradable commodities imply equilibrium in the money market.

With the obvious allowance made for the constant of linearization: ω1.

Equation (46) is a log approximation of equation (38).

In the remainder of this paper, ft will always stand for the end of period level of foreign reserves since that is the relevant endogenous variable. In addition, the predetermined level of reserves at the beginning of every period: atft equals the end of the previous period level of reserves: ft-1.

This process is identical to the one presented in Barro (1978).

This is so because (from Appendix II): θ2 < 1 and (in equation (2.18) of Appendix II): |δft/δgct-1| < 1 since|Y2| < 1.

Since ${\mathrm{p}}_{\mathrm{t}}^{\mathrm{*}}$ is an exogenous variable, the analysis of the expected general inflation rate can be conducted by solely considering the expected inflation rate of the nontradable good.

$\frac{\mathrm{\delta }\mathrm{\left(}\mathrm{X}\mathrm{\right)}}{\mathrm{\delta }\mathrm{\left(}\mathrm{Y}\mathrm{\right)}}$ refers to the partial derivative of variable X with respect to variable Y.

From Appendix II, θ3 > θ2 and (in equation (2.18) of Appendix II)

$\mathrm{|}\frac{{\mathrm{\delta f}}_{\mathrm{t}}}{\mathrm{\delta m}}\mathrm{|}\mathrm{>}\mathrm{|}\frac{{\mathrm{\delta f}}_{\mathrm{t}}}{{\mathrm{\delta v}}_{\mathrm{t}}}\mathrm{|}\mathrm{.}$

This is so because when m increases, there will be an additional increase in the next period level of government credit, which generates additional pressure on the “potential” current excess demands for both commodities.

That is, if the financial constraint effect outweighs the output effects of an appreciation of the real exchange rate.

From equation (2.10) of Appendix II: θ5 < θ2; and (in equation (2.18) of Appendix II):

$\mathrm{|}\frac{{\mathrm{\delta f}}_{\mathrm{t}}}{{\mathrm{\delta x}}_{\mathrm{t}}}\mathrm{|}\mathrm{<}\mathrm{|}\frac{{\mathrm{\delta f}}_{\mathrm{t}}}{{\mathrm{\delta gc}}_{\mathrm{t}\mathrm{-}\mathrm{1}}}\mathrm{|}\mathrm{.}$

From Appendix II: θ7 > 0 and (in equation (2.18) of Appendix II):

$\frac{{\mathrm{\delta f}}_{\mathrm{t}}}{{\mathrm{\delta \xi }}_{\mathrm{t}}}\mathrm{=}\frac{{\mathrm{Y}}_{\mathrm{4}}\mathrm{+}{\mathrm{Y}}_{\mathrm{5}}\mathrm{+}{\mathrm{Y}}_{\mathrm{3}}{\mathrm{\theta }}_{\mathrm{7}}}{\mathrm{1}\mathrm{-}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{3}}}\mathrm{>}\mathrm{0}$

because |Y4| > |Y5+Y3| and θ7 < 1.

Since ${\mathrm{p}}_{\mathrm{t}+1}^{\mathrm{*}}$ is a component of pit+1 it is no longer possible to solely concentrate on the effects of a rise in ${\mathrm{p}}_{\mathrm{t}}^{\mathrm{*}}$ on the inflation rate of the nontradable good.

From equation (2.12) of Appendix II: θ8 > θ7.

From equation (2.18) of Appendix II:

$\frac{{\mathrm{\delta f}}_{\mathrm{t}}}{\mathrm{\delta j}\mathrm{*}}\mathrm{=}\frac{{\mathrm{Y}}_{\mathrm{4}}\mathrm{+}\mathrm{2}{\mathrm{Y}}_{\mathrm{5}}\mathrm{+}{\mathrm{Y}}_{\mathrm{3}}\mathrm{\left(}{\mathrm{\theta }}_{\mathrm{7}}\mathrm{+}{\mathrm{\theta }}_{\mathrm{3}}\mathrm{\right)}}{\mathrm{1}\mathrm{-}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{3}}}$

which will be negative if | Y4| < |2Y5 + Y37 + θ8)|.

From Appendix II: θ8 < 1.

Notice from equation (2.18) of Appendix II that:

$\mathrm{|}\frac{{\mathrm{\delta f}}_{\mathrm{t}}}{{\mathrm{\delta t}}_{\mathrm{t}}}\mathrm{|}\mathrm{>}\mathrm{|}\frac{{\mathrm{\delta f}}_{\mathrm{t}}}{{\mathrm{\delta v}}_{\mathrm{t}}}\mathrm{|}\mathrm{.}$

From Appendix II: $\frac{\mathrm{\delta }\mathrm{\left(}{\mathrm{at}}^{\mathrm{E}}\mathrm{\left(}{\mathrm{pi}}_{\mathrm{t}\mathrm{+}\mathrm{1}}\mathrm{\right)}\mathrm{\right)}}{{\mathrm{\delta t}}_{\mathrm{t}}}\mathrm{=}\frac{{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{4}}}{\mathrm{1}\mathrm{-}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{3}}}$ which is positive.

From Appendix II: θ12 < 0; θ13 > 0, and in equation (2.18) of Appendix II:

$\frac{{\mathrm{\delta f}}_{\mathrm{t}}}{{\mathrm{\delta u}}_{\mathrm{t}}}\mathrm{=}\frac{{\mathrm{Y}}_{\mathrm{6}}}{\mathrm{1}\mathrm{-}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{3}}}\mathrm{>}\mathrm{0}\mathrm{;}\mathrm{and}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\frac{{\mathrm{\delta f}}_{\mathrm{t}}}{{\mathrm{\delta \epsilon }}_{\mathrm{t}}}\mathrm{=}\frac{{\mathrm{Y}}_{\mathrm{7}}}{\mathrm{1}\mathrm{-}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{3}}}\mathrm{.}$

That is, if: $\mathrm{|}{\mathrm{a}}_{\mathrm{2}}\mathrm{|}\mathrm{<}\mathrm{|}\frac{\mathrm{-}{\mathrm{a}}_{\mathrm{1}}{\mathrm{\tau \theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{6}}}{\mathrm{1}\mathrm{-}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{3}}}\mathrm{+}{\mathrm{\theta }}_{\mathrm{12}}\mathrm{\left(}\mathrm{1}\mathrm{-}\mathrm{\tau }\mathrm{\right)}\mathrm{|}$ (see Table 1 and Appendix I).

The confusion between vt and xt does not generate a discrepancy between beginning and end-of-period expectations because the distinction between the shocks is only known with one period lag.

Comparing equations (50) and (52), it can be concluded that the decline in the expected inflation rate following a monetary shock was larger if the shock was temporary than if it was permanent. Hence, the increase in the real monetary base was larger under a temporary monetary shock.

Which in the present case are equal to:

${\mathrm{at}}^{\mathrm{E}}\mathrm{\left(}{\mathrm{p}}_{\mathrm{t}}^{\mathrm{*}}\mathrm{\right)}\mathrm{=}{\mathrm{p}}_{\mathrm{t}\mathrm{-}\mathrm{1}}^{\mathrm{*}}\mathrm{+}\mathrm{j}\mathrm{*}\mathrm{-}{\mathrm{t}}_{\mathrm{t}\mathrm{-}\mathrm{1}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{and}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{at}}^{\mathrm{E}}\mathrm{\left(}{\mathrm{s}}_{\mathrm{t}}\mathrm{\right)}\mathrm{=}{\mathrm{s}}_{\mathrm{t}\mathrm{-}\mathrm{1}}\mathrm{.}$

It is important to recall that, in this model, price misperceptions of the Lucas type are not an argument in the supply functions.

The demand for yt will increase because of a rise in ${\mathrm{bt}}^{\mathrm{E}}\mathrm{\left(}{\mathrm{p}}_{\mathrm{t}\mathrm{+}\mathrm{1}}^{\mathrm{*}}\mathrm{\right)}$. The demand for ${\mathrm{y}}_{\mathrm{t}}^{\mathrm{*}}$ will decrease because of a rise in ${\mathrm{p}}_{\mathrm{t}}^{\mathrm{*}}$.

The restrictions imposed by the microfoundations in Section III of the main text imply the following: First, δ2 and b2 are respectively the elasticities of the demand for nontradables and tradables, with respect to the real monetary base. From equations (24) and (25) it is clear that both elasticities are less than one, and that δ2 = b2. Second, δ1 and b1 are respectively the elasticities of the demand for nontradables and tradables with respect to the expected inflation rate, which again from equations (24) and (25) are less than one and δ1 = b1. In addition, (δ12) = (b1+b2) = 1. Thus, from equation (2.2) δft/δgct is less than one in absolute terms for most plausible values of the model’s parameters.

In order to avoid a bias in the model arising from the constant of linearization, d3, the remainder of this paper will assume d3 = 1. Under that assumption, |δft/δft-1| < 1 and the model will be stable.

Notice that if d3 < 1 the model will also be stable; but then (δft/δft-1) might be less than zero, which would imply that the convergence pattern would exhibit dampened cycles.

The roots from equation (2.6) are real since the term inside the square root is always positive. This is so because Y3 < 0 and X1 > 0. The fact that the roots are real is an indication that the model posses an economically sensible solution.

(X3Y1-X1Y3-1) will be negative under the simplifying assumptions: d1 = d2 and d3 = 1. Since a requirement from the microfoundations of the model is: δ2 = b2; then:

$\mathrm{\left(}{\mathrm{X}}_{\mathrm{3}}{\mathrm{Y}}_{\mathrm{1}}\mathrm{-}{\mathrm{X}}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{3}}\mathrm{-}\mathrm{1}\mathrm{\right)}\mathrm{=}\frac{\mathrm{\tau }\mathrm{\left(}{\mathrm{\delta }}_{\mathrm{1}}\mathrm{+}{\mathrm{a}}_{\mathrm{1}}\mathrm{\right)}}{\mathrm{1}\mathrm{-}\mathrm{\tau }\mathrm{+}{\mathrm{\delta }}_{\mathrm{1}}\mathrm{+}{\mathrm{\delta }}_{\mathrm{2}}}\mathrm{-}\mathrm{1}$ which is negative and less than

one in absolute terms.

Next, consider the term under the square root:

${\mathrm{\left(}{\mathrm{X}}_{\mathrm{3}}{\mathrm{Y}}_{\mathrm{1}}\mathrm{-}{\mathrm{X}}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{3}}\mathrm{-}\mathrm{1}\mathrm{\right)}}^{\mathrm{2}}\mathrm{-}\mathrm{4}{\mathrm{Y}}_{\mathrm{3}}{\mathrm{X}}_{\mathrm{1}}\mathrm{=}\mathrm{1}\mathrm{+}\frac{{\mathrm{\tau }}^{\mathrm{2}}{\mathrm{\left(}{\mathrm{\delta }}_{\mathrm{1}}\mathrm{+}{\mathrm{a}}_{\mathrm{1}}\mathrm{\right)}}^{\mathrm{2}}}{{\mathrm{\left(}\mathrm{1}\mathrm{-}\mathrm{\tau }\mathrm{+}{\mathrm{\delta }}_{\mathrm{1}}\mathrm{+}{\mathrm{\delta }}_{\mathrm{2}}\mathrm{\right)}}^{\mathrm{2}}}\mathrm{-}\frac{\mathrm{2}\mathrm{\tau }\mathrm{\left(}{\mathrm{\delta }}_{\mathrm{1}}\mathrm{+}{\mathrm{\alpha }}_{\mathrm{1}}\mathrm{\right)}}{\mathrm{1}\mathrm{-}\mathrm{\tau }\mathrm{+}{\mathrm{\delta }}_{\mathrm{1}}\mathrm{+}{\mathrm{\delta }}_{\mathrm{2}}}\mathrm{+}\frac{\mathrm{4}\mathrm{\left(}\mathrm{1}\mathrm{-}{\mathrm{\omega }}_{\mathrm{1}}\mathrm{\right)}\mathrm{\left(}{\mathrm{\delta }}_{\mathrm{2}}\mathrm{-}{\mathrm{a}}_{\mathrm{1}}\mathrm{\right)}\mathrm{\left(}{\mathrm{d}}_{\mathrm{1}}\mathrm{\tau }\mathrm{\right)}\mathrm{\left(}{\mathrm{a}}_{\mathrm{1}}\mathrm{+}{\mathrm{\delta }}_{\mathrm{1}}\mathrm{\right)}\mathrm{\left(}\mathrm{1}\mathrm{+}{\mathrm{\delta }}_{\mathrm{1}}\mathrm{+}{\mathrm{\delta }}_{\mathrm{2}}\mathrm{\right)}}{{\mathrm{\left(}\mathrm{1}\mathrm{-}\mathrm{\tau }\mathrm{+}{\mathrm{\delta }}_{\mathrm{1}}\mathrm{+}{\mathrm{\delta }}_{\mathrm{2}}\mathrm{\right)}}^{\mathrm{2}}}$

which is positive and greater than one. Since the relevant root in equation (2.6) is the negative root and since Y3 < 0, θ1 will be positive.

50/

Under the restrictions imposed by the microfoundations, X3Y2=X2Y3.

Thus, ${\mathrm{\theta }}_{\mathrm{2}}\mathrm{=}\frac{{\mathrm{X}}_{\mathrm{2}}}{\mathrm{1}\mathrm{-}{\mathrm{\theta }}_{\mathrm{1}}{\mathrm{Y}}_{\mathrm{3}}\mathrm{-}{\mathrm{X}}_{\mathrm{3}}}$, which is positive but less than one.

Devaluation and Monetary Policy in Developing Countries: A General Equilibrium Model for Economies Facing Financial Constraints
Author: International Monetary Fund