%slides_rer.tex
%Slides for Chapter 8 of ``Open Economy Macroeconomics'' by Martín Uribe and Stephanie Schmitt-Grohé, Princeton University Press 2017. 
%© Martín Uribe and Stephanie Schmitt-Grohé.

\input{slides_preamble}

\begin{slide}
\begin{minipage}{5in}
\rightline{\includegraphics[width=4.5in]{USG_cover.eps}}

\centerline{\tiny  Princeton University Press,  2017}
\end{minipage}
\begin{minipage}{4.5in}
\centering 
{\small 
slides 

chapter 8   

Nontradable Goods And 

The Real Exchange Rate

}
\vspace{1in}

\end{minipage}
\end{slide}

\topleft{\textit{Open Economy Macroeconomics, Chapter 8} }
\topright{M.\ Uribe and S.\ Schmitt-Groh\'e}

\begin{slide}
{\bf \large Introduction}

The {\bf MX} model of Chapter 7 overestimates the role of terms-of-trade shocks as drivers of the business cycles: they explain 20\%  of the variance of output and other macro indicators  in the MX model  versus only 10\% in an estimated SVAR model. 

One possible explanation is that the MX model unrealistically assumes that all goods are internationally traded (either importable or exportable). 

In reality, the bulk of goods falls into a third category: nontradables. 

Nontradable goods are goods that are exclusively produced and absorbed domestically. Reasons for nontradability include transportation costs and trade barriers. 

In this chapter we study open economies with nontradable goods and explore the role of this type of goods in transmitting aggregate disturbances. 

\end{slide}


\begin{slide}
{\bf \large The Real Exchange Rate}

The real exchange rate, denoted $RER_t$,  is the relative price of  consumption goods baskets  across countries.  Formally, 
\begin{equation}
\label{eq:rer_RER}
RER_t \equiv 
\frac{\mathcal{E}_tP^*_t}{P_t},
\end{equation}
 where $P^*_t$
is  the nominal price of consumption in the foreign country in units of foreign currency,  
$P_t$ is
the nominal  price of consumption in the domestic country in units of domestic currency, and $\mathcal{E}_t$ is  the nominal exchange rate, defined as the price of one unit of foreign currency in terms of domestic currency. 

$\bullet$ When $RER_t$ increases, the domestic economy becomes  cheaper relative to the foreign economy, and we say that the real exchange rate depreciates. 


$\bullet$ When $RER_t$ falls, the domestic economy becomes  more expensive  relative to the foreign economy, and we say that the real exchange rate appreciates. 
\end{slide}

\begin{slide}
{\bf \large The TNT Model}

The tradable-nontradable (TNT) model is an endowment open economy model with one
 good that is fully imported (not domestically produced), one good that is fully exported (not domestically consumed), and one nontradable good. 


It is a useful structure because it allows us to understand in a simple environment how different disturbances affect the real exchange rate. 

The predictions of the TNT model extend to more complex environments, which adds to its theoretical appeal. 

\end{slide}

\begin{slide}
{\bf The Representative Household}\\
Households are infinitely lived and 
choose sequences $\{c^m_t$,$c^n_t$,$d_{t+1}\}_{t=0}^{\infty}$ to maximize
\[
\sum_{t=0}^{\infty} \beta^t
U(c_t),
\]
subject to 
\[
c_t = A(c^m_t,c^n_t),
\]
\[
c^m_t + p^n_t c^n_t + d_t
= \frac{d_{t+1}}{1+r} + tot_t y^x + p^n_t y^n,
\]
\[
\lim_{j\rightarrow \infty}(1+r)^{-j}d_{t+j}\le 0,
\]
where $c_t$ is consumption; 
 $c^m_t$ and $c^n_t$
are consumption of importables and nontradables, respectively and  $A(\cdot,\cdot)$  is increasing, concave,  and HD1;  $d_t$ is external debt maturing in $t$; $r>0$ is the interest rate; $tot_t$ is  the terms of trade; $y^x$ and $y^n$ are  constant endowments of exportables and nontradables, respectively. 

Note that $c^m_t$ is fully imported and $y^x$ is fully exported. 
\end{slide}

\begin{slide}
{\bf Optimality Conditions}\\
The  FOCs are the sequential budget constraint and 
\begin{equation}
\label{eq:rer_endowment_cm}
U'(c_t) A_1(c^m_t,c^n_t) = \lambda_t, 
\end{equation}
\begin{equation}
\label{eq:rer_endowment_Euler}
\lambda_t = \beta(1+r) \lambda_{t+1},
\end{equation}
\begin{equation}
\label{eq:rer_endowment_cn}
p^n_t = 
\frac{A_2(c^m_t,c^n_t)}
{A_1(c^m_t,c^n_t)},
\end{equation}
\begin{equation}
\label{eq:rer_endowment_transversality}
\lim_{j\rightarrow \infty}(1+r)^{-j}d_{t+j}
=0.
\end{equation}
Because  $A(\cdot,\cdot)$ is increasing, HD1, and concave, we can rewrite  the optimality condition~\eqref{eq:rer_endowment_cn} as
\begin{equation}
\label{eq:rer_endowment_pn_foc}
p^n_t =  P\left(\frac{c^m_t}{c^n_t}\right); \quad \mbox{ with } P'(\cdot)>0.
\end{equation}
This expression is intuitive. It says that  if  nontradables become more expensive relative to importables, households consume relatively less nontradables  and more importables. 

\end{slide}


\begin{slide}
{\bf \large The   Real Exchange Rate and the Relative Price of Nontradables}\\
In the present model there is a one-to-one  relationship between the relative price of nontradables,  $p^n_t$, and the real exchange rate, $RER_t$. 
 To see this, divide the numerator and denominator of the RHS of~\eqref{eq:rer_RER}  by the  nominal  price of the importable good, denoted $P^m_t$, to obtain
\[
RER_t  = 
\frac{\mathcal{E}_tP^*_t/P^m_t}{P_t/P^m_t}.
\]
Assume that the {\it law of one price} holds for importable goods\\
\centerline{$
P^m_t = \mathcal{E}_tP^{m*}_t.
$}\\
Then,  letting $p^c_t\equiv P_t/P^m_t$ and $p^{c*}_t\equiv P^*_t/P^{m*}_t$  denote the relative prices of consumption domestically and abroad, we can write $RER_t = p^{c*}_t/p^{c}_t$.  	Note that  $p^{c*}_t$ is exogenously determined; assume that it is constant and normalized to unity, $p^{c*}_t=1$. Then we have 
\begin{equation}
\label{eq:rer_rer_pc}
RER_t = \frac1{p^c_t}.
\end{equation}
(The argument continues on the next slide.)
\end{slide}
\begin{slide}
It remains to link $p^c_t$ to $p^n_t$. To this end, let's {\bf decentralize the market for final consumption goods.}Firms producing $c_t$  $c^m_t$ and $c^n_t$ to maximize profits
\[
p^c_t A(c^m_t,c^n_t) - c^m_t-p^n_tc^n_t.
\]
The FOC with respect to $c^m_t$ is
\[
p^c_t A_1(c^m_t,c^n_t) = 1.
\]
Finally, combine this with~\eqref{eq:rer_endowment_pn_foc} and~\eqref{eq:rer_rer_pc}
to get
$
RER_t = A_1\left(
P^{-1}(p^n_t),1
\right) 
$. So we can write
\[
RER_t =
 e(p^n_t); \quad \mbox{ with }
e'(\cdot)<0. 
\]
This expression says that \\
$\bullet$  The real exchange rate appreciates if and only if the relative price of nontradables increases. And\\ 
$\bullet$ 
The real exchange rate depreciates if and only if the relative price of nontradables decreases. \\
For this reason, many economists  use the terms real exchange rate and  relative price of nontradables interchangeably.  
\end{slide}

\begin{slide}
{\bf \large Market Clearing}

The quintessential property of nontradable goods is that  they are produced and consumed domestically: 
\begin{equation}
\label{eq:rer_endowment_cn=yn}
c^n_t = y^n,
\end{equation}
 By contrast, the gap between production and consumption of tradables is bridged by foreign trade in goods and assets:
\begin{equation}
\label{eq:rer_endowment_resource_constraint}
c^m_t + d_t = \frac{d_{t+1}}{1+r} + tot_t y^x.
\end{equation}
\end{slide}

\begin{slide}
{\bf \large The Equilibrium Real Exchange Rate}

Iterating the resource constraint~\eqref{eq:rer_endowment_cn=yn}  forward ad infinitum and using the transversality 
condition~\eqref{eq:rer_endowment_transversality} yields
\begin{equation}
\label{eq:rer_endowment_cm_eqm}
c^m = -\frac r{1+r} d_0 + 
\frac r{1+r} \sum_{t=0}^{\infty}
\frac{tot_{t}y^x}{(1+r)^t}.
\end{equation}
Finally, assume that $\beta(1+r)=1$. Then, 
\eqref{eq:rer_endowment_cm},
\eqref{eq:rer_endowment_Euler}, and \eqref{eq:rer_endowment_cn=yn} imply that  importable consumption is constant, $c^m_t=c^m$. Thus we can write 
\begin{equation}
\label{eq:rer_endowment_pn}
p^n
 =  P\left(\frac{c^m}{y^n}\right)
\end{equation}
Taken together, \eqref{eq:rer_endowment_cm_eqm} and \eqref{eq:rer_endowment_pn}   say that the equilibrium real exchange rate
 appreciates (i.e., the economy becomes more expensive vis-\`a-vis the rest of the world) when: (a) The supply of nontradables falls; (b) The current terms of trade improve or the current supply of tradables increases; (c) The terms of trade are expected to improve or the supply of tradables expected to increase in the future. 
\end{slide}

\begin{slide}
{\bf \large Adjustment of the Real Exchange Rate to \\Temporary Terms-of-Trade  Shocks}

Assume that $tot_0$ increases, while  $tot_t$ remains unchanged for all $t>0$. From \eqref{eq:rer_endowment_cm_eqm} and \eqref{eq:rer_endowment_pn}  we have that 
\[
\left.
\frac{\partial p^n}{\partial tot}
\right|_{\mbox{temporary}} =
\frac r{1+r}
\frac{y^x}{y^n}
P'\left(\frac{c^m}{y^n}\right)>0.
\]
{\bf Intuition:} the increase in the relative price of the exportable endowment creates a positive income effect $\Rightarrow$ households increase their demand for all consumption goods, importables and nontradables. But the supply of nontradables is fixed at $y^n$ $\Rightarrow$  $p^n$ increases  to eliminate the excess demand. 

Because the relative price of nontradables increases, the real exchange rate, $RER_t$,  appreciates in response to the temporary improvement in the terms of trade (i.e., the domestic economy becomes more expensive relative to the rest of the world). 
\end{slide}

\begin{slide}

{\bf \large Adjustment of the Real Exchange Rate to \\Permanent  Terms-of-Trade  Shocks}

Assume now that  $tot_t$ increases for all $t\ge0$. Then equilibrium conditions \eqref{eq:rer_endowment_cm_eqm}
and 
  \eqref{eq:rer_endowment_pn} imply that 
\[
\left.
\frac{\partial p^n}{\partial tot}
\right|_{\mbox{permanent}} =
\frac{y^x}{y^n}
P'\left(\frac{c^m}{y^n}\right)>0.
\]
The intuition is the same as with temporary shocks. However, 
permanent changes in the terms of trade have a larger effect on the relative price of nontradables than temporary ones
\[
\left.
\frac{\partial p^n}{\partial tot}
\right|_{\mbox{permanent}}
>
\left.
\frac{\partial p^n}{\partial tot}
\right|_{\mbox{temporary}}>0
.
\]
{\bf Intuition:} The more permanent is the increase in the terms of trade, the larger the income effect it generates will be, and therefore the larger the increase will be in the desired demand for nontradables, which in turn requires a larger increase in the relative price of nontradables to clear the market. 
\end{slide}

\begin{slide}
{\bf \large Adjustment of the Real Exchange Rate to\\ Interest-Rate  Shocks}

Suppose that at time $0$ the interest rate experiences a one-time increase to $r_0>r$ and then returns to is normal value $r$. 
It can be shown that this shock causes a contraction in consumption  of importables,
\[
\frac{\partial c^m_0}
{\partial r_0} <0.
\]
This is intuitive. When the interest rate is high it is a good time to save more and consume less. 
This expression together with~\eqref{eq:rer_endowment_pn_foc}  and~\eqref{eq:rer_endowment_cn=yn} implies that 
\[
\frac{\partial p^n_0}
{\partial r_0}
=P'\left(
\frac{c^m_0}{y^n}
\right)\frac1{y^n}
\frac{\partial c^m_0}
{\partial r_0}
 <0.
\]
$\Rightarrow$ the real exchange rate depreciates in response to a temporary increase in the interest rate. Intuitively,
households want to consume less of all goods. 
But because the supply of nontradables is fixed, the price of nontradables must fall to clear the market.
\end{slide}

\begin{slide}
{\bf \large Adjustment of  Output to Terms-of-Trade Shocks }

Chapter 7 shows that the MX model overestimates the role of TOT shocks as drivers of output movements. Can introducing nontradables ameliorate this problem?
Output, denoted $y_t$,  is  given by 
\[
y_t = \frac{P^x_ty^x+P^n_ty^n}{P_t},
\]
where $P^x_t$, $P^n_t$, and $P_t$ denote, respectively, the nominal prices of exportables, nontradables, and final goods.  Divide the numerator and denominator by the nominal price of importables  to get
\[
y_t = \frac{tot_ty^x+p^n_ty^n}{p^c_t}. 
\]
Recalling that $p^c_t =1/ A_1(c^m_t,y^n)$ and that $p^n_t =A_2(c^m_t,y^n)/ A_1(c^m_t,y^n)$, we can write
\begin{equation}
\label{eq:rer_output_eqm}
y_t  = A_1(c^m_t,y^n)tot_ty^x+ A_2(c^m_t,y^n)y^n.
\end{equation}
(The argument continues in the next slide.)
\end{slide}
\begin{slide}
{\bf  Adjustment of  Output to Terms-of-Trade Shocks (continued)}

Assume that there is an unexpected permanent increase in the terms of trade in period 0. 




Assume that the aggregator function takes the Cobb-Douglas form 
$A(c^m,y^n) = (c^m)^{\alpha}(y^n)^{1-\alpha},$
 with $\alpha\in(0,1)$. Then equation~\eqref{eq:rer_output_eqm} becomes
\[
y_t = c_t 
\left[
\alpha \frac{tot_ty^x}{c^m_t}+(1-\alpha)
\right]
\]
\[
\mbox{which implies that }
\left.
\frac{\partial y_0}{\partial tot}
\right|_{1-\alpha=0}
= y^x>0 \mbox{ and }
\left.
\frac{\partial y_0}{\partial tot}
\right|_{1-\alpha=1}
= 0,
\]
$\Rightarrow$ as the share of nontradables in total 
expenditure increases from zero to 100 percent, the output effect of an improvement in the  terms of trade 
falls from a positive value to zero. The relationship between the share of nontradables in consumption and the effect in the terms of trade is in general nonmonotonic.

\end{slide}




\begin{slide}
\begin{center}
{\bf \Large Empirical Evidence on  the Effects of Terms-of-Trade Shocks on  the Real Exchange Rate and Aggregate Activity}
\end{center}
\end{slide}



\begin{slide}
\centerline{\bf A Structural Vector Autoregressive (SVAR) Model}

$\bullet$ Let $x_t = 
\left[
\begin{array}{cccccc}
\widehat{tot}_t,&
  \widehat{tb}_t,&
 \widehat{y}_t,&
  \widehat{c}_t,&
  \widehat{i}_t,&
  \widehat{RER}_t
\end{array}
\right]'. 
$

$\bullet$ VAR: $x_t = \mathbf{h_x} \, x_{t-1} + u_t$

$\bullet$ {\bf Identification of TOT shocks:} \hspace{1cm}
\begin{tabular}{l}
$u_t = \Pi \epsilon_t$\\
$\epsilon_t \sim (0, I)$\\
$\Pi_{1,j} = 0$ for $j=2,\dots,6$
\end{tabular} 

$\bullet$ ToT process is univariate:  $h_{x,1,j} = 0$ for $j=2,\dots, 6$ 

$\bullet$ All variables (except $tb_t$) are log-quadratically detrended. 

$\bullet$ Estimate the SVAR country-by-country using OLS. 

$\bullet$ This is the natural extension of the SVAR model of Chapter 7, to include the real exchange rate. 

\end{slide}









\begin{slide}
{\bf Data:}   



$\bullet$ Include all poor and emerging countries that have at least 30 consecutive annual observations of output, consumption, investment, net exports, the terms of trade, and the real exchange rate in the World Bank's WDI database. 

$\bullet$ Poor and emerging countries are defined as countries  with average PPP-converted GDP per capita in U.S. dollars of 2005 over the period 1990 to 2009 below 25,000 dollars.

$\bullet$ 38 countries satisfy both criteria. \\ 
    {\tiny Algeria,
    Argentina,
    Bolivia,
    Botswana,
    Brazil,
    Burundi,
    Cameroon,
    Central African Republic,
    Colombia,
    Congo, Dem. Rep.,
    Costa Rica,
    Cote d'Ivoire,
    Dominican Republic,
    Egypt, Arab Rep.,
    El Salvador,
    Ghana,
    Guatemala,
    Honduras,
    India,
    Indonesia,
    Jordan,
    Kenya,
    Korea, Rep.,
    Madagascar,
    Malaysia,
    Mauritius,
    Mexico,
    Morocco,
    Pakistan,
    Paraguay,
    Peru,
    Philippines,
    Senegal,
    South Africa,
    Sudan,
    Thailand,
    Turkey,
and
    Uruguay.}
\end{slide}\begin{slide}

$\bullet$ Sample period: 1980-2011  (32 years).  

$\bullet$  Our sample of 38 countries.  
\[
\widehat{tot}_t = \rho \widehat{tot}_{t-1}
 + \sigma_{tot} \epsilon^{tot}_t; \quad \quad \epsilon^{tot}_t\sim (0,1)
\]
\centerline{Estimate $\rho$ and $\sigma_{tot}$ country by country}
\centerline{\begin{tabular}{lcc}
\hline
\hline 
&
$\rho$
&$\frac{\sigma_{tot}}{\sqrt{1-\rho^2}}$
\\
\hline
Median&0.52&0.10\\
%Median Absolute Deviation&0.11&0.03\\
Interquartile Range&$[0.41, 0.61]$&$[0.09,0.13]$\\
\hline \hline
\end{tabular}
}

%$\bullet$ Not very different than the estimates of Mendoza (1995), which were for the period 1960 to 1990. 

\end{slide}

\begin{slide}

\centerline{Impulse Response to A 10\% Increase in the Terms of Trade}
\centerline{SVAR Evidence, Median across 38 countries}


\centerline{\includegraphics[width=9in]{rer_ir_slides.eps}}
%produced by running rer_ir.m in directory: 
%z:\uribe\book\rer\data. 

\end{slide}



\begin{slide}
{\bf Observations on the Estimation Results}

$\bullet$ TOT shocks are short lived: half-life of TOT shock is just 1 year. 

$\bullet$ $R^2$ of tot equations is modest on average, 30 percent

$\bullet$ The trade balance improves in response to a TOT shock (i.e. the data lends support to the  Harberger-Metzler-Laursen (HLM) effect). 

$\bullet$  Terms-of-trade improvements are expansionary: On average,  
a 10\% increase in the TOT causes a 0.4\% increase n GDP.

$\bullet$ On average, $c$ and $i$ increase, but  with a one-year delay.  It is puzzling---and indeed an issue worth exploring both empirically and theoretically. 

$\bullet$ A TOT improvement leads to an appreciation of the real exchange rate. That is, when $tot_t\uparrow\Rightarrow$ the country becomes more expensive vis-\`a-vis the rest of the world. Thus the SVAR model lends support to the predictions of the TNT model. 
\end{slide}

\begin{slide}
\centering 

{\bf Share of Variance Explained by Terms of Trade Shocks: SVAR Evidence}

\vspace{1in}

\centerline{\begin{tabular}{llcccccc}
\hline \hline
 & $tot$& $tb$&$y$&  $c$&  $i$& $RER$\\ 
\hline
Median &  100&  12&  \fbox{\bf 10}&  9&  10&  14\\
Median Absolute Deviation&  0&  7&  7&  6&  7&  11\\
\hline \hline
\end{tabular} 
}
\end{slide}




\begin{slide}
\begin{center}
{\bf \large Country-Level Shares of Variance of Output Explained by Terms of Trade Shocks}

\centerline{\includegraphics[width=7in]{rer_vox_eu_38_countries_slides.eps}}
\end{center}
%produced with vox_eu_38_countries.m in directory Z:\uribe\book\rer\data
\end{slide}

\begin{slide}
{\bf \large Summary of Variance Decomposition}


$\bullet$  On average, TOT shocks explain 10 percent of the variance of output in poor and emerging countries. 

$\bullet$ In only 5 countries (Botswana, Egypt, Cote d'Ivoire, Sudan, and Uruguay) do ToT shocks explain more than 30 percent of the variance of output. 

$\bullet$ Thus, the SVAR evidence is at odds with the conventional wisdom according to which ToT shocks account for a large share of output variability in poor and emerging markets. 
\end{slide}

\begin{slide}

\centerline{\bf \Large The MXN Model}
\end{slide}






\begin{slide}

\centerline{\bf \large  Elements of the MXN Model}

$\bullet$  small open economy that takes terms of trade as given.

$\bullet$  Three sectors of production: exportable goods, importable goods, nontradable goods $\Rightarrow $ Natural extension of the MX model to include nontradable sector. 

$\bullet$  All goods are produced and consumed (difference with the TNT model). 

$\bullet$ All goods are produced using capital and labor. 

$\bullet$ Sector specificity of  capital and labor. 

\end{slide}

\begin{slide}
{\bf The Household Problem: Maximize}

\centerline{$
E_0 \sum_{t=0}^{\infty} \beta^t
U(c_t,h^m_t,h^x_t,h^n_t)
$}

subject to the period budget constraint

$
c_t + i^m_t + i^x_t + i^n_t 
+  p^{\tau}_t d_t
+ \Phi_m(k^m_{t+1}-k^m_t) 
+ \Phi_x(k^x_{t+1}-k^x_t) 
+ \Phi_n(k^n_{t+1}-k^n_t) 
=\frac{p^{\tau}_t d_{t+1}}{1+r_t} 
+ w^m_t h^m_t 
+ w^x_t h^x_t 
+ w^n_t h^n_t 
+ u^m_t k^m_t 
+ u^x_t k^x_t
+ u^n_t k^n_t,
$

and to the laws of motion for physical capital

\centerline{$k^m_{t+1} = (1-\delta) k^m_t + i^m_t$;\, $k^x_{t+1} = (1-\delta) k^x_t + i^x_t$;\, $k^n_{t+1} = (1-\delta) k^n_t + i^n_t.$}
Notation: $c_t$ consumption;
$h^i_t,k^i_t,i^i_t,w^i_t,u^i_t$ hours, capital, investment, wages, rental rates of capital in sector $i=m,x,n$ (importable, exportable, and nontradable sector); 
$d_t$ debt due in $t$;
 $p^{\tau}_t$ relative price of tradables in terms of final goods. All terms in the budget constraint are expressed in units of final goods. 
Parameter $\delta\in(0,1)$  is the depreciaiton rate of capital. 
Functions $\Phi_j(x) = \frac{\phi_j}{2}x^2; \quad j=m, x, n$ introduce capital adjustment costs. 
\end{slide}

\begin{slide}
\centerline{\bf Firms Producing Final Goods}
\[\max\{ a_t -p^{\tau}_ta^{\tau}_t-p^n_ta^n_t\}\]
subject to
\[
a_t = 
\left[
\chi_{\tau} 
\left(a^{\tau}_t\right)^{1-\frac1{\mu_{\tau n}}}
+ (1-\chi_{\tau })
\left(a^n_t\right)^{1-\frac1{\mu_{\tau n}}}
\right]^{\frac{1}{1-\frac{1}{\mu_{\tau n}}}},
\]

Notation:\\
$a_t = $  domestic absorption of  final goods a.\\
$a^{\tau}_t = $  domestic absorption of a composite of traded goods. \\
$a^n_t = $ (domestic) absorption of nontraded goods. \\
$\mu_{\tau n} = $ elasticity of substitution between T and N goods. \\
$\chi_{\tau}=$ expenditure share on tradables if $\mu_{\tau n}$.
\end{slide}


\begin{slide}
\centerline{\bf Production of the Tradable Composite Good}

Firms producing tradable goods  solve the problem
\[\max\{p^{\tau}_t a^{\tau}_t - p^m_t a^m_t - p^x_t a^x_t\}
\]
\[
a^{\tau}_t = 
\left[
\chi_m 
\left(a^m_t\right)^{1-\frac1{\mu_{mx}}}
+ (1-\chi_m)
\left(a^x_t\right)^{1-\frac1{\mu_{mx}}}
\right]^{\frac{1}{1-\frac{1}{\mu_{mx}}}}
\]

Notation:\\
$a^{\tau}_t = $ domestic absorption of  tradable goods. \\
$a^m_t = $ domestic absorption of importable goods. \\
$a^x_t = $ domestic absorption exportable goods. \\
$\mu_{mx}=$ elasticity of substitution between importables and exportables.\\
$\chi_m =$ expenditure share if $\mu_{mx}=1$.
\end{slide}

\begin{slide}
\centerline{\bf Production of Importable Goods}

Firms producing importable goods  solve the problem
\[
\max  \{ p^m_t y^m_t - w^m_th^m_t-u^m_tk^m_t\}
\]
subject to the production technology 
\[ 
y^m_t = A^m \left(k^m\right)^{\alpha_m}\left(h^m\right)^{1-\alpha_m}
\]
Notation:\\
$y^m_t = $ quantity of importable goods produced domestically. \\
$A^m =$ level of productivity in the importable sector.\\
$k^m_t = $ capital input in the importable sector. \\
$h^m_t = $ labor input in the importable sector. \\
$w^m_t =$ wage rate in the importable sector.\\
$u^m_t =$ rental rate of capital in  the importable sector.\\
$1-\alpha_m = $ labor share in the importable sector.
\end{slide}

\begin{slide}
\centerline{\bf Production of Exportable Goods}

Firms producing exportable goods  solve the problem
\[
\max  \{ p^x_t y^x_t - w^x_th^x_t-u^x_tk^x_t\}
\]
subject to the production technology 
\[ 
y^x_t = A^x \left(k^x\right)^{\alpha_x}\left(h^x\right)^{1-\alpha_x}
\]
Notation:\\
$y^x_t = $ quantity of exportable goods produced. \\
$A^x =$ level of productivity in the exportable sector.\\
$k^x_t = $ capital input in the exportable sector. \\
$h^x_t = $ labor input in the exportable sector. \\
$w^x_t =$ wage rate in the exportable sector.\\
$u^x_t =$ rental rate of capital in  the exportable sector.\\
$1-\alpha_x = $ labor share in the exportable sector.
\end{slide}

\begin{slide}
\centerline{\bf Production of Nontradable Goods}

Firms producing nontradable goods  solve the problem
\[
\max  \{ p^n_t y^n_t - w^n_th^n_t-u^n_tk^n_t\}
\]
subject to the production technology 
\[
y^n_t = A^n \left(k^n\right)^{\alpha_n}\left(h^n\right)^{1-\alpha_n}
\]

$y^n_t = $ quantity of nontraded goods produced. \\
$A^n =$ level of productivity in the nontradable sector.\\
$k^n_t = $ capital input in the nontradable sector. \\
$h^n_t = $ labor input in the nontradable  sector. \\
$w^n_t =$ wage rate in the nontradable sector.\\
$u^n_t =$ rental rate of capital in  the nontradable sector.
\\
$1-\alpha_n = $ labor share in the nontraded sector.


\end{slide}

\begin{slide}
{\bf \large Debt Elastic Interest-Rate Premium}

To ensure a stationary equilibrium process for 
external debt,  we assume that
the country interest-rate premium 
is  debt elastic,
\[
r_t = r^*+ p(d_{t+1})
\]
where $p(\cdot)$ is assumed to be increasing taking the form
\[
p(d) = \psi \left(e^{d-\bar{d}}-1 \right)
\]

\end{slide}

\begin{slide}
\centerline{\bf The Terms of Trade Process} 
As in the empirical SVAR analysis, we assume that the terms of trade follow a univariate first-order  autoregressive (AR(1)) process of  the form 
\[
\ln
\left(\frac{tot_t}{\overline{tot}}\right)
= \rho
\ln
\left(\frac{tot_{t-1}}{\overline{tot}}\right)
+ \sigma_{tot} \epsilon^{tot}_t;
\quad \epsilon^{tot}_t \sim (0,1)
\]
where 
$\overline{tot}>0$ is the deterministic level of the terms of trade.\\ 
$\rho\in(-1,1)$ is the serial correlation.\\
$\sigma_{tot}>0$ is the standard deviation of the innovation to the terms of trade.



\end{slide}

\begin{slide}
\centerline{\bf Calibrated and Estimated Parameters}

\vspace{2cm}

{\tiny

\centerline{\begin{tabular}{*{13}{c}}
\hline \hline 
\multicolumn{13}{l}{\underline{Calibrated Structural Parameters}}
\\
$\rho$&
$\sigma_{tot}$&
$\alpha_m,\alpha_x$&
$\alpha_n$&
$\omega_m,\omega_x,\omega_n$&
$\mu_{mx}$&
$\mu_{\tau n}$&
$\overline{tot}$&
$A^m, A^n$&
$\beta$&
$\sigma$&
$\delta$&
$r^*$
\\
$*$&
$*$&
0.35&
0.25&
1.455&
1&
0.5&
1&
1&
$1/(1+r^*)$&
2 &
0.1&
0.11
\\
\hline
\multicolumn{12}{l}{\underline{Moment Restrictions}}
\\
$\frac{\sigma_{i}}{\sigma_{y}}$&
$\frac{\sigma_{tb}}{\sigma_{y}}$&
$\frac{\sigma_{i_m+i_x}}{\sigma_{i_n}}$&
$s_n$&
$s_x$&
$s_{tb}$&
$\frac{p^m y^m}{p^x y^x}$&
\\
$*$&
$*$&
1.5&
0.5&
0.2&
0.01&
1
\\
\hline 
\multicolumn{13}{l}{\underline{Implied Structural Parameter Values }}
\\
$\phi_m$&
$\phi_x$&
$\phi_n$&
$\psi$&
$\chi_m$&
$\chi_{\tau}$&
$\overline{d}$&
$A^x$&
$\beta$
\\ 
${}^{*}$&
${}^{*}$&
${}^{*}$&
${}^{*}$&
0.8980&
0.4360&
0.0078&
1&
0.9009
\\
\hline
\hline 
\end{tabular}
}
Notes.\\
${}^*$Country-specific estimates. 
\\
$\frac{\sigma_{i}}{\sigma_{y}}$ and $\frac{\sigma_{tb}}{\sigma_{y}}$ are conditional on $tot$ shocks
\\
$s_n\equiv p^ny^n/y$,\\ $s_x \equiv  x/y$,\\ $s_{tb}\equiv (x-m)/y$, where $y\equiv p^my^m+p^xy^x+p^ny^n$.\\ 
%The steady state is produced  in the file mxn_ss.m in the directory
%z:\uribe\book\rer\mxn
}
\end{slide}
\begin{slide}
Key parameters determining the importance of terms of trade shocks: 


$\bullet$ $\rho$ and $\sigma_{tot}$, the more volatile and the more persistent are terms of trade shocks, the more volatile is output. 

$\bullet$ The size of the nontraded sector: $\frac{p^n y^n}{y} (=50\%)$. The larger the nontraded sector, the smaller the output effects of  tot shocks. 

$\bullet$ The steady-state trade share: $\frac{x+m}{y} (=39\%)$. The larger the trade share, the larger the output effects of  tot shocks. 


\end{slide}

\begin{slide}
Estimate capital adjustment cost  parameters and the debt elasticity of the interest rate, $\phi_m$, $\phi_x$, $\phi_n$, $\psi$, $\chi_m$, to match country-by-country the relative standard deviations 
\[
\sigma_{i}/\sigma_{y}  \quad \mbox{ and } \sigma_{tb}/\sigma_{y}
\]
conditional on terms of trade shocks and $\sigma_{i_x+i_m}/\sigma_{i_n}=1.5$. 



\end{slide}
\begin{slide}
{Medians of Country-Specific Estimates of the Capital Adjustment Cost  Parameters and the Debt Elasticity of the Interest Rate}

\begin{tabular}{l *{8}{r}}
\hline \hline
&&&&& \multicolumn{2}{c}{$\sigma_i/\sigma_y$}&
\multicolumn{2}{c}{$\sigma_{tb}/\sigma_y$}\\
& $\phi_m$&$\phi_x$&$\phi_n$&$\psi$&  Data &Model&Data&Model\\
\hline
Median&  1.13&  1.40&  0.69&  0.84&  3.36&  3.00&  0.64&  0.74\\
MAD&  1.13&  1.40&  0.69&  0.77&  1.42&  0.52&  0.33&  0.34\\
\hline 
\hline
\end{tabular}

%produced by running  paasche_phi_pssi_cbc.m in directory: 
%Z:\uribe\book\rer\mfiles\paasche

\end{slide}

\begin{slide}
\begin{center}
{\bf 
Median of Country-Specific Impulse Response to a Ten-Percent Terms-of-Trade Shock\\ Predicted by the MXN Model

\centerline{\includegraphics[width=9in]{plot_mxn_ir_slides.eps}}
}
\end{center}
%produced by running 
%plot_mxn_ir_slides.m in
%z:\uribe\book\rer\mxn 
\end{slide}

\begin{slide}
Observations on Impulse Responses to a Terms-of-Trade Shock Implied by the MXN Model
{\small \begin{itemize}
  \item {\it Substitution effect of an increase in $tot_t$ on supply side:} Firms produce more exportables and less importables, and,  given $p^n_t$, would produce less nontradables. 
  \item {\it Substitution effect of an increase in $tot_t$ on demand side:}  
  Demand for importable goods and nontraded goods rises, domestic demand for exportable goods falls. 
The improvement in the terms of trade generates a positive wealth effect   increasing the demand for all goods. 

\item The price of nontradables, $p^n_t$,  rises and the real exchange rate appreciates. 
\item Both exports and imports increase. Net effect on trade balance turns out to be positive. Thus model impulse response is consistent with Harberger-Laursen-Metzler effect. 
\item Aggregate investment increases by less than  10\% on impact. But investment in the exportable sector rises by 61\%  while in decreases by over 40\% in the importable sector. 	
\end{itemize}
}
\end{slide}

\begin{slide}
{\bf \large Comparison of SVAR and MXN  Model}\\
Proper comparison of empirical and theoretical models requires measuring macroeconomic indicators ($y_t$, $c_t$, $i_t$, $tb_t$) in the same units. 


To this end, we construct the theoretical  counterpart to the observable variable  `GDP at constant LCU'  used in the SVAR analysis. This requires deflating nominal variables by a Paasche GDP deflator index. This yields 
\[
y^{\mbox{constant prices}}_t = p^x_{ss} y^x_t + p^m_{ss} y^m_t + p^n_{ss} y^n_t, 
\]
Similarly, to obtain total consumption expenditure  at constant prices, we must divide $c_t$ by the factor $ \frac{p^x_t y^x_t + p^m_t y^m_t + p^n_t y^n_t}{p^x_{ss} y^x_t + p^m_{ss} y^m_t + p^n_{ss} y^n_t}
$, which yields
\[
c^{\mbox{constant prices}}_t =
c_t 
\frac{ p^x_{ss} y^x_t + p^m_{ss} y^m_t + p^n_{ss} y^n_t}
{ p^x_t y^x_t + p^m_t y^m_t + p^n_t y^n_t}
\]
Similar expressions obtain for the observable versions of $i_t$ and $tb_t$. 
\end{slide}

\begin{slide}
{\bf \large Share of Variances  of Output and Other Indicators Explained by Terms-of-Trade Shocks According to the MXN Model}

To compute the share of the variance of output explained by TOT shocks in the MXN model, we proceed as follows:

(1) With $tot_t$ as the sole driving process, compute the variance of output predicted by the MXN model.

(2) Divide this value obtained in (1)  by the observed unconditional  variance of output  predicted by the SVAR model (i.e., the variance of output when all shocks in the SVAR are active). 

We do the same with the other macroeconomic indicators of interest (consumption, investment, the trade balance, and the real exchange rate). 

\end{slide}

\begin{slide}
\begin{center}
{\bf Share of Variance Explained by Terms of Trade Shocks: SVAR Versus MXN Predictions}

\bigskip 
 
\centerline{\begin{tabular}{lccccc}
\hline \hline
 &  $tb$&$y$&  $c$&  $i$& $RER$\\ 
\hline
MXN  Model& 21&  13&  18&  11&  1\\
SVAR Model& 12&   10&  9&  10&  14\\
\hline \hline
\end{tabular} 
}

Note. Cross-country medians.
%produced by running
% edit paasche_tnt_var_cbc.m in Z:\uribe\book\rer\mfiles\paasche

\end{center}
{\bf Finding:} \\
$\bullet$ The median share of the variance of output explained by tot shocks, according
to the MXN model is close to the one predicted by the SVAR model (13 vs. 10 percent).\\
$\bullet$ Thus both the MXN  and SVAR models concur in that, contrary to the conventional wisdom, terms-of-trade shocks are not a major driver of the business cycle. \\
$\bullet$
This result extends to the other macroeconomic indicators included in the SVAR, although the MXN model overestimates the role of TOT shocks for the trade balance and consumption and underestimates it for the real exchange rate. 

\setcounter{page}{\theslide}
\label{slide:share_of_variances_explained_by_tot_shocks}
\end{slide}







\begin{slide}
{\bf  The Importance of Measuring Variables Consistently in the SVAR and MXN Models}\\
{\small One possible reason for the  discrepancy between our findings of a small role for TOT shocks as drivers of the business cycle and the conventional view that assigns them a major role is measurement. \\
\indent Suppose that in computing  the predictions of the MXN model for the importance of TOT shocks, we had measured output in units of current consumption 
($ p^x_t y^x_t + p^m_t y^m_t + p^n_t y^n_t$)---a common practice---instead of  at constant prices using a Paasche price deflator ($ p^x_{ss} y^x_t + p^m_{ss} y^m_t + p^n_{ss} y^n_t$).\\
Let $x^{CPI}_t$ denote variable $x_t$ measured in {\bf units of consumption}. \\
Let $x^{YPI}_t$ denote variable $x_t$ measured in {\bf units of output}. }
\begin{center}
{\bf Ratio of Variances Predicted by Theoretical Model}
\begin{tabular}{lccc}
\hline \hline
 &  $y$&  $c$&  $i$\\ 
\hline
$\frac{\mbox{var}(x^{CPI}_t)}{\mbox{var}(x^{YPI}_t)}$&   2.4&    1.7& 1.2\\
\hline \hline
\end{tabular}
\end{center} 
$\bullet$ Predicted variances due to ToT shocks of output, consumption, and investment are {\bf larger} when these aggregates are expressed in units of consumption goods as opposed to units of output. 
\end{slide}

\begin{slide}
{\bf \large The Terms-of-Trade Disconnect}

The table in slide 
 slide~\pageref{slide:share_of_variances_explained_by_tot_shocks}
conveys the impression that the MXN 
and SVAR model speak with the same voice with respect to the importance of terms-of-trade shocks for aggregate movements in output. 


This is certainly the case on average across the 38 countries in the panel. However, the picture that emerges when one looks at the predictions of the MXN and SVAR models at the country level is quite different. 

This is the point of the  graphs on the next  slide. 

\end{slide}


\begin{slide}
\begin{center}
{\bf \large Evaluating The Conventional Wisdom\\ How Does the Model Fit Country-Level Data? }
\end{center}

\centerline{
\includegraphics[width=4.5in]{rer_disconnect_gdp.eps}}
%Produced with
%mxn_versus_svar.m
%z:\uribe\book\rer\mxn. 
\end{slide}

\begin{slide}
{\bf Observations on the Figure}

$\bullet$ If the  MXN model were to fit the data well, all circles in the figure would line up close to the 45-degree line. 

$\bullet$ The cloud of circles, however, displays no discernible pattern. 

$\bullet$ Thus although the predictions of the MXN model are in line with the SVAR model on average across countries, the match is quite poor when we look at countries individually. 

$\bullet$ This result extends to the other macroeconomic indicators included in the SVAR model, as the figure in the next slides shows. 
\end{slide}

\begin{slide}
\centerline{\tiny Variance of Consumption, Investment, the Trade Balance, and the Real Exchange Rate}\centerline{\tiny Explained By Terms-of-Trade Shocks: SVAR   Versus  Model}
\centerline{\includegraphics[width=6in]{rer_disconnect_c_i_tb_rer.eps}}


%{\small Note. Variances explained by terms-of-trade shocks in the SVAR and the theoretical models are expressed as a percentages of the respective observed total variances.}
%produced by running 
%mxn_versus_svar.m in
%z:\uribe\book\rer\mxn. 

\end{slide}

\begin{slide}
{\bf  Conclusions:  Empirical Analysis}

{\small 1. Conventional wisdom has it that terms of trade shocks represent a major source of fluctuations for emerging countries. 

2.  Using  SVAR analysis, 
we have found  a modest  role for TOT shocks. 
 
3. Does this mean that world prices are not important transmitters of aggregate disturbances? No. It  simply indicates that ToT shocks, being a highly aggregated summary of world prices faced by a country, might be a poor way of capturing the mechanism through which external shocks transmit to the domestic economy. 

4. The typical country trades  internationally in a large number of goods and services. Thus, world shocks are likely to be transmitted through a large number of world prices.  Using a single summary statistic, such as the terms of trade, may not capture all of the domestic effect of external disturbance. 

5.  The above observations suggest using more disaggregated world price data. Fern\'andez, Schmitt-Groh\'e, and Uribe (JIE, forthcoming) follow this route.  They estimate  a variation of the SVAR model studied here in which the terms of trade is replaced by 3 world commodity prices. They find that jointly, these prices explain about 30 percent of aggregate fluctuations in emerging and poor countries. 


}
\end{slide}

\begin{slide}
{\bf  Conclusions: Theoretical Analysis}

1. The analysis suggests that the theoretical framework on which the conventional wisdom is based, a multi-sector,  open economy version of the RBC  model,  which we called the MXN model,  
 captures well the transmission mechanism  of TOT shocks
on average, but does a poor job 
 at the country level. 

2. One promising avenue is to expand the MXN model to allow for multiple importable and exportable goods (as opposed to just one importable and one exportable good). In this setting, just as in the empirical analysis, the terms of trade will not suffice to capture the whole mechanism through which world shocks transmit to the domestic economy. 


3.  An important byproduct of the theoretical analysis presented in this chapter (as well as in Chapter 7) is the need to be careful about measuring variables in the same units in the theoretical and the empirical models. Failing to do so can lead to spurious differences between the predictions of the theoretical model and the data. 

\end{slide}
\end{document}



