%slides_dnwr.tex
%Slides for Chapter 9 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 9   

nominal rigidity 

exchange rates,  and 

unemployment

}

\vspace{1in}

\end{minipage}
\end{slide}
\topleft{Chapter 9: Nominal Rigidity, Exchange Rates,  And Unemployment}
\topright{Uribe \& Schmitt-Groh\'e}


\begin{note}{\small 
%Roadmap of the chapter

Contents\\
{9.1} {An Open Economy With Downward Nominal Wage Rigidity}
\\
{9.2} {Currency Pegs}
\\{9.3} {Optimal Exchange Rate Policy}
\\{9.4} {Empirical Evidence On Downward Nominal Wage Rigidity}
\\{9.5} {The Case of Equal Intra- And Intertemporal Elasticities of Substitution}
\\{9.6} {Approximating Equilibrium Dynamics}
\\{9.7} {Parameterization of the Model}
\\{9.8} {External Crises and Exchange-Rate Policy: A Quantitative Analysis}
\\{9.9} {Empirical Evidence On The Expansionary Effects of Devaluations}
\\{9.10} {The Welfare Costs of Currency Pegs}
\\{9.11} {Symmetric Wage Rigidity }
\\{9.12} {The Mussa Puzzle}
\\{9.13} {Endogenous Labor Supply}
\\{9.14} {Production in the Traded Sector}
\\{9.15} {Product Price Rigidity}
\\{9.16} {Staggered Price Setting: The Calvo Model}
}
\end{note}

\begin{slide}
\leftline{\bf  Roadmap}

$\bullet$ Chapter 9 develops a theoretical framework in which nominal rigidities result in inefficient adjustments to aggregate disturbances

$\bullet$ framework can be used in an intuitive graphical manner to demonstrate how nominal rigidities amplify the business cycle in open economies

$\bullet$ but framework can also be used to derive quantitative predictions useful for policy evaluation
\end{slide}
\begin{slide}
\centerline{\bf Some Motivation: Peripheral Europe and the Global Crisis of 2008}

 Take a look at the next slide. 

$\bullet$ The inception of the Euro in 1999 was followed by massive capital inflows into the region, possibly driven by expectations of quick convergence of peripheral  and core Europe. 

$\bullet$ Large current account deficits and large increases in nominal hourly  wages, with declining rates of unemployment between 2000 and 2008. 

$\bullet$ When the global crisis of 2008 starts, capital inflows dry up abruptly. Peripheral Europe suffers a severe {\bf sudden stop} (sharp reductions in current account deficits). 

$\bullet$ In spite of the collapse in aggregate demand and the lack of a devaluation, nominal hourly wages remain as high as at the peak of  the boom. 

$\bullet$ Massive unemployment affects all countries in the region. 

\end{slide}


\begin{slide}

{\bf  Figure 9.1  Boom-Bust Cycle in Peripheral Europe: 2000-2011}


\centerline{\includegraphics[angle=-90,width=10.5in]{dnwr_europe_data.eps}}
%produced with read_u.m

{\tiny Data Source: Eurostat. Labor Cost Index, Nominal, is the nominal hourly wage rate in manufacturing, construction and services (including the public sector, but for Spain.)  

Data represents arithmetic mean  of  Bulgaria, Cyprus, Estonia, Greece, Ireland, Lithuania, Latvia, Portugal, Spain, Slovenia, and Slovakia. \par}
\end{slide}


\begin{slide}
\begin{minipage}{5.5in}

\includegraphics[width=5.5in]{dnwr_jep_data_cgips.eps}
%produced with c:\data\uribe\default\ottonello\read_u_country_by_country.m

\end{minipage}
\begin{minipage}{2in}

{The Disaggregated Story:  Boom-Bust Cycles in Cyprus, Greece, Ireland, Portugal, and Spain. }

\end{minipage}

\end{slide} 

\begin{slide}
{\tiny The previous two figures  suggest the following narrative: 

Countries in the periphery of the European Union,  such as Ireland, Portugal, Greece, and a number of small eastern European countries adopted a fixed exchange rate regime by joining  the Euroarea. 
Most of these countries experienced an initial transition into the Euro characterized by low inflation, low interest rates, and economic expansion. 

{\bf However, history has shown time and again that fixed exchange rate arrangements  are easy to adopt but difficult to maintain. (Example: Argentina's 1991 convertibility plan.)}

The Achilles' heel of currency pegs is   that they hinder the efficient adjustment of the economy to negative 
external shocks, such as drops in the terms of trade or hikes in the  interest-rate. Such shocks 
produce a contraction in aggregate demand that requires a decrease in the relative price of nontradables, that is,  a real depreciation of the domestic currency, in order to bring about an expenditure switch away from tradables and toward nontradables. In turn, the required real depreciation may come about  via a nominal devaluation of the domestic currency or via a fall in nominal prices  or both. 

 The currency peg rules out a devaluation. Thus, the only way the necessary real depreciation can occur is through a decline in the  nominal price of nontradables. However, when nominal wages are downwardly rigid, producers of nontradables are reluctant to lower prices, for doing so might 
render their enterprises no longer profitable. 
As a result, the necessary  real depreciation  takes place too slowly, causing recession and unemployment along the way. 

This narrative    goes back  at least to Keynes (1925) who argued that 
Britain's  1925 decision to return to the gold standard at the 1913 parity despite the significant increase in the aggregate price level  that took place during World War I would force deflation in nominal wages with deleterious consequences for unemployment and economic activity. 
Similarly, 
Friedman's  (1953) seminal  essay  points at downward nominal wage rigidity as  the central argument against fixed exchange rates. 


}

\end{slide}


\begin{slide}

\leftline{\bf To formalize this narrative let's  build an open economy model with}
\vspace{-.3in}
$\bullet$ downward nominal wage rigidity\\
$\bullet$ a traded and a nontraded sector\\
$\bullet$ involuntary unemployment\\

\bigskip

\leftline{\bf To produce quantitative predictions  }
\vspace{-.3in}
$\bullet$ Estimate the key parameters of the  model (with particular attention on the parameter governing downward wage rigidity)  and estimate the driving forces. 
\\
$\bullet$ Characterize response to large negative external shock  under a peg and show that the model can explain the observed sudden stop. 
\\
$\bullet$ Characterize   optimal exchange rate policy. 
\\
$\bullet$ Quantify the costs of currency pegs in terms of unemployment and welfare.
 
The material is based on Schmitt-Groh\'e and Uribe (JPE, 2016).

\end{slide}

\begin{slide}
\centering

{\bf \Large   9.1  An Open Economy with 

Downward Nominal Wage Rigidity
}


(The DNWR Model) 


\end{slide}


\begin{slide}


{\bf  Downward Nominal Wage Rigidity (DNWR)}


\[
W_t \ge \gamma \,\, W_{t-1}
\tag{9.6}
\]

$W_t =$  nominal wage rate in period $t$

$\gamma =$ degree of downward wage rigidity. 

$\gamma=0\Rightarrow$ fully flexible wages. 

Think of $\gamma$ as being around 1. The empirical evidence presented later in this chapter suggests $\gamma=0.99$ at quarterly frequency. 
\end{slide}

\begin{slide}
{\bf \large Traded and Nontraded Goods}

$\bullet$ Stochastic endowment of tradable goods:\footnote{Section 9.14  relaxes this assumption.} $y^T_t$.
\\
$\bullet$ Stochastic country interest rate: $r_t$.
\\
$\bullet$
Nontraded goods,  $y^N_t$, produced with labor, $h_t$: $y^N_t = F(h_t)$
.
\\
$\bullet$
Law of one price holds for tradables: $P^T_t =  \mathcal{E}_tP^{*}_t$.
\\
$\bullet$
$P^T_t$, nominal price of tradable goods.
\\
$\bullet$
$\mathcal{E}_t$, nominal exchange rate,   domestic-currency price of one unit of foreign currency ($\mathcal{E}_t\uparrow$ depreciation of domestic currency). 
\\
$\bullet$
$P^*_t$, foreign currency price of tradable goods. 
\\
$\bullet$
Assume that $P^*_t=1$, so that $P^T_t = \mathcal{E}_t$

\end{slide}

\begin{slide}
{\bf  The Nontraded Sector }

Profits, $\Phi_t$: 
$$
\Phi_t  = P^N_t F(h_t) - W_t h_t
$$

$P^N_t$, nominal price of nontradables.

Firms maximize profits taking as given $P^N_t$ and $W_t$. 
Optimality Condition: $$P^N_t F'(h_t) = W_t $$
Divide by $P^T_t=\mathcal{E}_t$ and rearrange
\[ 
p_t  = \frac{ W_t/\mathcal{E}_t}{F'(h_t)}
\]
$p_t\equiv\frac{P^N_t}{P^T_t}$, relative price of nontradables in terms of tradables. Interpret this optimality conditions as a supply schedule for nontradables,  see next slide. 
\end{slide}

\begin{slide}
\centerline{\bf  The Supply Schedule of Nontradables}
Let's derive the supply schedule for nontradables in the space $(y^N, p)$ given the real wage, $W_t/\mathcal{E}_t$. 

Note that real marginal cost of one unit of nontraded good
\[
\mbox{marginal cost} = \frac{W_t/\mathcal{E}_t}{F'(h_t)}
\]
Use $h=F^{-1}(y^N)$ to obtain 
\[
\mbox{marginal cost} = \frac{W_t/\mathcal{E}_t}{ F'(F^{-1}(y^N)}
\]
By the profit maximization condition marginal cost equals price
or 
\[
p_t = \frac{ W_t/\mathcal{E}_t}{F'(F^{-1}(y^N_t))}
\]
Interpret this relation as a supply schedule of nontradables given the real wage. 

\end{slide}

\begin{slide}

\centerline{\bf Figure 9.3 The Supply  Of Nontradables}

\begin{minipage}{5in}{\small 
Supply schedule: $p_t = \frac{ W_t/\mathcal{E}_t}{F'(F^{-1}(y^N_t))}$

Properties: \\ 
-- upward sloping, the higher the price, the more a firm wishes to produce, given factor prices. 


-- A decrease in nominal wage from $W_1$ to $W_0<W_1$ shifts the supply schedule down and to the right. 

--  A devaluation $\mathcal{E}_t \uparrow$ (not shown) shifts the supply schedule in the same manner as a nominal wage cut. 

}
\end{minipage} \begin{minipage}{5in}
\leftline{\includegraphics[width=5in]{Z:/uribe/book/dnwr/graphical_model/dnwr_figure_9_3.eps}}
%This figure was produced on August 15, 2017 as part of the suite of figures developing the graphical apparatus underlying the dnwr model. 

%{dnwr_supply_prudential.eps}}
%produced November 5, 2014 with 
%c:\data\uribe\book\dnwr\supply_prudential.m
\end{minipage} 

\end{slide}


\begin{slide}
{\bf \large Households}
{\small \[
\max_{\{c^T_t, c^N_t, \, d_{t+1}\}} \quad \mathbb{E}_0\sum_{t=0}^{\infty}
\beta^t U(c_t)
\tag{9.1}
\]
subject to
\[
c_t =  A(c^T_t,c^N_t) \tag{9.2}
\]
\[
P^T_t c^T_t + P^N_t c^N_t + \mathcal{E}_t d_t  = 
P^T_t y^T_t + W_t h_t + \mathcal{E}_t \frac{d_{t+1}}{1+r_t} + \Phi_t  
\tag{9.3}
\]
\[
h_t\le \bar{h}
\tag{9.7}
\]

}
\noindent $\bullet$ First constraint: Consumption is a composite of traded and nontraded goods. $A(.,.)$ increasing, concave, and HD1.\\
$\bullet$ Second constraint: $d_t=$ one-period debt chosen in $t$, due in $t+1$. 
Debt is denominated in units of foreign currency $\rightarrow$ full liability dollarization. $\rightarrow${\it Original Sin:} In emerging countries almost 100\% of external debt issued in foreign currency (Eichengreen, Hausmann, and Panizza, 2005). Country interest rate, $r_t$, is stochastic.
\\
$\bullet$  Third constraint: 
 Workers supply $\bar{h}$ hours inelastically,\footnote{Section 9.13 relaxes this assumption.} but may not be able to sell them all. They take $h_t\le \bar{h}$ as given.


\end{slide}



\begin{slide}
{\bf Optimality Conditions Associated with  the Household Problem}

\[
\frac{A_2(c^T_t,c^N_t)}{A_1(c^T_t,c^N_t)} = p_t
\tag{9.5}
\]
\[
\lambda_t = U'(A(c^T_t,c^N_t))A_1(c^T_t,c^N_t)
\]
\[
\lambda_t = \beta(1+r_t)\mathbb{E}_t \lambda_{t+1}
\]
\[
P^T_t c^T_t + P^N_t c^N_t + \mathcal{E}_t d_t  = P^T_t y^T_t + W_t h_t +  \mathcal{E}_t \frac{d_{t+1}}{1+r_t} + \Phi_t  
\]
\[h_t\le \bar{h}\]
\end{slide}

\begin{slide}
{\bf \large The Demand For Nontradables}\\
Look again at the optimality condition (9.15)
\[
\frac{A_2(c^T_t,c^N_t)}{A_1(c^T_t,c^N_t)} = p_t. \tag{9.15}
\]
If $A(c^T,c^N) $ is concave and HD1, then given $c^T_t$, the left-hand side  is decreasing in $c^N_t$. This means that, all other things equal,  an increase in $p_t$ reduces the desired demand for nontradables, giving rise to the  downward sloping demand schedule shown in the next slide. 

Note that $c^T_t$ acts as a shifter of the demand schedule for nontradables: given $p_t$, an increase in  $c^T_t$ is associated with an equiproportional desired increase in $c^N_t$. Of course, this shifter is endogenously determined. 
\end{slide}


\begin{slide}
\centerline{\bf Figure 9.2 The Demand For Nontradables}

\begin{minipage}{5in}
\[
p_t= \frac{A_2(c^T_t,c^N_t)}{A_1(c^T_t,c^N_t)} \tag{9.15}
\]
\leftline{\includegraphics[width=5in]{Z:/uribe/book/dnwr/graphical_model/dnwr_figure_9_2.eps}}
%This figure was produced on August 15, 2017 as part of the suite of figures developing the graphical apparatus underlying the dnwr model. 
%{dnwr_demand_prudential_slides.eps}}
%produced with 
%c:\data\uribe\default\eg\mc_p_demand.m
\end{minipage} \begin{minipage}{5in}
\begin{itemize}
\item here we treat $c^T_t$ as a shifter of the demand schedule. 
\item  An increase in $c^T_t$ from $c^T_0$ to $c^T_1>c^T_0$, shifts the demand schedule up and to the right. 

\end{itemize}
\end{minipage} 
\end{slide}



\begin{slide}
{\bf Closing of the Labor Market}

Nominal wages are downwardly rigid
\[
W_t \ge \gamma W_{t-1} \tag{9.6}
\]
Labor demand may not exceed supply
\[
h_t \le \bar{h} \tag{9.7}
\]
Impose the following slackness condition:
\[
(\bar{h} - h_t ) \left(W_t -\gamma W_{t-1}\right) = 0 \tag{9.8}
\]
This slackness condition says that, if there is involuntary unemployment ($h_t<\bar{h}$),  then the lower bound on nominal wages must be binding. It also says that if the lower bound on nominal wages is not binding $\left(W_t >\gamma W_{t-1}\right)$, then the labor market must feature full employment. 



{\bf Market clearing in the Nontraded Sector}
\[c^N_t = y^N_t = F(h_t)\]

\end{slide}


\begin{slide}{\small 
A competitive  equilibrium is a set of stochastic processes 
$\{c^T_t,$ $h_t,$ $w_t$, $d_{t+1}$, $p_t$, $\lambda_t\}_{t=0}^{\infty}$ satisfying 
\[
\tag{9.9}%\label{eq:dnwr_market-clearing-traded-competitive}
c^T_t + d_t 
=
y^T_t + \frac{d_{t+1}}{1+r_t}
\]
\[
\tag{9.13}
%\label{eq:dnwr_lambda-competitive}
\lambda_t 
=
U'(A(c^T_t,F(h_t))) A_1(c^T_t,F(h_t))
\]
\[
%\label{eq:dnwr_Euler-competitive}
\tag{9.14}
\frac{\lambda_t}{1+r_t} 
=
\beta \mathbb{E}_t
\lambda_{t+1} 
\]
\[
%\label{eq:dnwr_p-demand-competitive}
\tag{9.15}
p_t
=\frac{A_2(c^T_t,F(h_t))}{A_1(c^T_t,F(h_t))}
\]
\[
\tag{9.16}
%\label{eq:dnwr_labor-demand-competitive}
p_t 
=
\frac{w_t}{ F'(h_t)}
\]
\[
\tag{9.17}
%\label{eq:dnwr_DNWR-competitive}
w_t 
\ge 
\gamma \frac{w_{t-1}}{\epsilon_t}
\]
\[
\tag{9.18}
%\label{eq:dnwr_labor-supply-competitive}
h_t 
\le
\bar{h}
\]
\[
\tag{9.19}
%\label{eq:dnwr_slackness-h-competitive}
(\bar{h}-h_t)\left(w_t-\gamma \frac{w_{t-1}}{\epsilon_t}\right)=0
\]
given an exchange rate policy $\{\epsilon_t\}_{t=0}^{\infty}$, initial conditions $w_{-1}$ and $d_0$, and exogenous stochastic  processes $\{r_t,$  $y^T_t\}_{t=0}^{\infty}$. 

To characterize the eqm we must specify the exchange-rate regime. We will turn to this next. 



}
\end{slide}

\begin{slide}
{\bf A Graphical Representation of (Partial) Equilibrium}

Why partial, because for the moment we take $c^T$ as given. 

In equilibrium,  $c^N=y^N=F(h)$. This means that we can draw Figures 9.2 and 9.3 in the employment-relative price of nontradables space, that is,  in the space  $(h,p)$ 

\begin{minipage}{5in}\centerline{\bf The Demand for Nontradables}

\rightline{\includegraphics[width=5in]{Z:/uribe/book/dnwr/graphical_model/dnwr_figure_9_2_h.eps}}
%This figure was produced by running figure_9_2.m in Z:\uribe\book\dnwr\graphical_model; Manually pick whether the horizontal axis is h_t or y^N_t; Also manually pick wether to show just 1 or 2 demand schedules 
\end{minipage} \begin{minipage}{5in}\centerline{\bf The Supply of Nontradables}

\leftline{\includegraphics[width=5in]{Z:/uribe/book/dnwr/graphical_model/dnwr_figure_9_3_h.eps}}
%This figure was produced by running figure_9_3.m in Z:\uribe\book\dnwr\graphical_model; Manually pick whether the horizonal axis is h_t or y^N_t. 
\end{minipage} 

\end{slide}

\begin{slide}

In eqm both (9.15) and (9.16) must hold. We refer to the  value of $h$ at which these schedules intersect for  given  $W_0/\mathcal{E}_0$ and $c^T_0$ as labor demand, and denote it $h^d$

\centerline{\includegraphics[width=5in]{Z:/uribe/book/dnwr/graphical_model/dnwr_figure_9_2_3_h.eps}}
%This figure was produced by running figure_9_2_3_h.m in Z:\uribe\book\dnwr\graphical_model 
Next add the labor supply schedule to the graph. Two generic cases emerge: at given real wages, labor demand exceeds labor supply,  or labor supply exceeds labor demand. Let's consider the first case first.

\end{slide}

\begin{slide}
\begin{minipage}{4.5in}
Suppose given $W_0/\mathcal{E}_0$, labor demand exceeds labor supply: 

\bigskip 

\centerline{\includegraphics[width=4.5in]{Z:/uribe/book/dnwr/graphical_model/dnwr_figure_9_2_3_excess_demand1.eps}}
%This figure was produced by running figure_9_2_3_excess_demand.m in Z:\uribe\book\dnwr\graphical_model 
\end{minipage}\hfill \begin{minipage}{4.5in}
$\Rightarrow$ real wage rises to clear the labor market, $W_0\uparrow$, until $h^d=h^s$. 


\centerline{\includegraphics[width=5in]{Z:/uribe/book/dnwr/graphical_model/dnwr_figure_9_2_3_excess_demand2.eps}}
%This figure was produced by running figure_9_2_3_excess_demand.m in Z:\uribe\book\dnwr\graphical_model 
\end{minipage}

\centerline{In eqm, there is full employment, $h_t= \bar{h}$}

\end{slide}


\begin{slide}
\begin{minipage}{4.5in}
Now suppose given $W_0/\mathcal{E}_0$, labor supply exceeds labor demand: 

\bigskip 

\centerline{\includegraphics[width=4.5in]{Z:/uribe/book/dnwr/graphical_model/dnwr_figure_9_2_3_excess_supply.eps}}
%This figure was produced by running figure_9_2_3_excess_supply.m in Z:\uribe\book\dnwr\graphical_model 
\end{minipage}\hfill \begin{minipage}{4.5in}
$\Rightarrow$ to clear the labor market the real wage must fall, but nominal wages cannot fall due to DNWR (assume $\gamma=1$). Thus, unless the monetary authority devalues, $\mathcal{E}\uparrow$, the labor market will fail to clear. 


\end{minipage}

\bigskip 

\centerline{In eqm, absent a devaluation there is unemployment, $h_t < \bar{h}$}


In the following sections, we will use the graphical framework (as well as quantitative methods) to analyze boom and bust cycles. 

\end{slide}



\begin{slide}

\centering 

{\bf \Large 9.2 Currency Pegs}

\end{slide}

\begin{slide}
The exchange rate policy is a peg
\[
\mathcal{E}_t=\mathcal{E}_0; \quad  \forall t\ge0. 
\]
Use the graphical apparatus just developed to show that a {\bf boom-bust cycle} leads---as documented in Figure 9.1---to

-- nominal wage growth and real appreciation during the boom phase

-- involuntary unemployment and insufficient real depreciation during the bust phase

\end{slide}


\begin{slide}
\centerline{\bf Figure 9.4 Adjustment to a Boom-Bust Cycle under a Currency Peg}

\begin{minipage}[t]{7in}
\rightline{\includegraphics[width=7in]{dnwr_mc_p.eps}}
%produced November 5, 2014 with 
%c:\data\uribe\book\dnwr\supply_prudential.m
\end{minipage}
\begin{minipage}[b]{3in}

\vfill

$c^T_1<c^T_0$

\rightline{negative external  shock} 
\rightline{possibly caused by $r_t\uparrow$}\end{minipage}
\end{slide}

\begin{slide}


{\small Observations on Figure 9.4: Adjustment in a Boom-Bust Episode
 
The initial situation is point $A$. At point $A$ there is full employment, $h=\bar{h}$. 

Now a boom starts. We capture this by an increase in $c^T$ (perhaps because $r$ falls). Given nominal wages the economy moves to point B. But at point B, there is excess demand for labor. Thus nominal wages will rise. By how much? Until the excess demand for labor has disappeared. That will be at point C. Thus the boom leads to an increase in nominal wages ($W\uparrow$) and a real appreciation ($p\uparrow$). The economy continues to operate at full employment. 

Next the boom is over and the bust comes. We capture this by assuming that $c^T$ falls back to its original level, $c^T_0$. 
 
This shifts the demand for nontradables back to its original position. The new intersection between supply and demand is at point $D$. At D, labor supply exceeds labor demand. However, because nominal wages are downwardly rigid and the nominal exchange rate is fixed, the supply schedule does not shift, (for simplicity, in the figure,  we assume $\gamma =1$). Thus the economy is stuck at point D. At point D, there is involuntary unemployment ($\bar{h}-h^{bust}$) and there is insufficient real depreciation, i.e., $p_t$ does not fall enough (that is, does not fall to $p_0$) to restore full employment. 

}\end{slide}



\begin{slide}
{\bf The Link between Volatility and Average Unemployment }

The present model predicts that aggregate volatility increases the mean level of unemployment.

This prediction gives rise to large welfare benefits of stabilization policy. 

--- This prediction is \underline{not} due to the assumption of downward nominal wage rigidity, but due to the assumption that employment is determined by the minimum of labor demand and labor supply. (Note: key difference with Calvo-style sticky wage models in which employment is always demand determined.)

--- Downward nominal wage rigidity \underline{amplifies}  the connection between aggregate volatility and mean unemployment. 

\end{slide}




\begin{slide}
To see this consider the following example: 

$U(A(c^T_t,c^N_t))=\ln c^T_t + \ln c^N_t$ 

$d_t=0 $ (no access to international financial markets)

$\Rightarrow c^T_t=y^T_t$. 

$y^T_t=\left\{\begin{array}{ll}
1+\sigma  & \mbox{prob    } \frac12\\
1-\sigma  & \mbox{prob    } \frac12
\end{array}\right.$

$E(y^T_t)=1$ and var$(y^T_t)=\sigma^2$. 

$F(h_t) = h_t^{\alpha}$ 

$\bar{h}=1$  
 
$\mathcal{E}_t=\mathcal{E} $ (currency peg)

$W_{-1} = \alpha \mathcal{E}$
\end{slide}
\begin{slide}

The equilibrium conditions associated with this economy are (we list all except wage adjustment, for which we will consider two cases):
\[
\frac{c^T_t}{c^N_t} = p_t
\]
\[
\alpha p_t (h_t)^{\alpha-1} = W_t/\mathcal{E}
\]
\[
c^T_t = y^T_t
\]
\[
c^N_t = h_t^{\alpha}
\]
Step 1: Find labor demand: $h^d_t =  \frac{\alpha y^T_t}{ W_t /\mathcal{E}}$

Step 2: Find equilibrium labor as $h_t = \min\{\bar{h}, h^d_t\}$

\end{slide}
\begin{slide}
Case 1: Assume bi-directional nominal wage rigidity. 
\[W_t = \alpha \mathcal{E}\]
Then,  $h^d_t=y^T_t$, and  the equilibrium level of employment is
\[
h_t 
=
 \left\{
\begin{array}{cl}
1-\sigma&\mbox{ if $y^T_t =1-\sigma$}\\
1&\mbox{ if $y^T_t =1+\sigma$}
\end{array}
\right.
\]

Let $u_t\equiv \bar{h}-h_t$ denote the unemployment rate. It  follows that the equilibrium  distribution of $u_t$ is given by 
\[
u_t 
=
 \left\{
\begin{array}{cl}
\sigma&\mbox{with probability $\frac12$}\\
0&\mbox{with probability $\frac12$}
\end{array}
\right..
\]
The unconditional mean of the unemployment rate is  then given by
\[
E(u_t) = \frac{\sigma}2.
\]
Average level of unemployment increases linearly with the volatility of tradable endowment, in spite of the fact that wage rigidity is symmetric!  
\end{slide}




\begin{slide}
Case 2: assume `only' downward nominal wage rigidity, $W_t\ge W_{t-1}$ 


Then, $W_t = \alpha (1+\sigma) > \alpha$ and 
\[
h_t 
=
 \left\{
\begin{array}{cl}
\frac{1-\sigma}{1+\sigma}&\mbox{ if $y^T_t =1-\sigma$}\\
1&\mbox{ if $y^T_t =1+\sigma$}
\end{array}
\right.
\]
$E(u_t) = \sigma/(1+\sigma)>\sigma/2$ (recall that $\sigma$ must be less than 1). 


Thus uni-directional wage rigidity exacerbates the link between mean unemployment and volatility. 
\end{slide}

 
\begin{slide}
\centerline{\bf The Peg-Induced Externality}

Under a currency peg and downward nominal wage rigidity, a good shock, in this example that follows a fall in the country interest rate $r_t$, can be the prelude to  bad things to happen later.  The reason is that individual agents do not internalize that during the boom nominal wages increase too much, putting the economy in a vulnerable position once the good shock fades away. 

Consider the following environment:
\[U(A(c^T_t,c^N_t)) = \ln c^T_t + \ln c^N_t\]
\[F(h_t) = h_t^{\alpha}; \quad 0<\alpha<1\]
\[\bar{h}=1; \quad 
y^T_t=y^T>0; \quad
\gamma=1; \quad
\beta(1+r)=1; \quad
d_0=0; \quad 
w_{-1} = \alpha y^T\]
\[
r_t  = \left\{
\begin{array}{ll}
r&t>0\\
\underline{r}<r& t=0
\end{array}
\right.
\]
\end{slide}

\begin{slide}
\begin{center}
 {\bf The Peg-Induced Externality} (Continued)
\end{center}

In a couple of slides, you'll find  the solution of the equilibrium  in algebraic and graphical form. To help the interpretation of those slides,  it is of use to discuss intuitively what is going on in this economy:

--- The fall in the interest rate in period 0 induces an expansion in the desired demand for consumption goods, of all types, tradables and nontradable. 

--- The increased demand for tradables causes a trade balance deficit, a deficit in the current account, and an increase in external debt in period 0. 

--- The increased demand for nontradables cause a rise in wages and a rise in the relative price of nontradables (i.e., an appreciation of the real exchange rate). 
\end{slide}

\begin{slide}
 {\bf The Peg-Induced Externality} (Continued)

--- In period 1, the interest rate goes back up to its permanent value $r$, causing a contraction in the demand for consumption goods (both tradables and nontradables), and a reversal in the trade balance and the current account. 

--- The contraction in the demand for nontradables causes a derived contraction in the demand for labor. However, because nominal wages are downwardly rigid and the nominal exchange rate is fixed, the real wage fails to fall, causing involuntary unemployment. 

--- Involuntary unemployment is highly persistent. (In fact, in this example,  because $\gamma=1$, it never disappears.)


\end{slide}

\begin{slide}
\begin{center}
 {\bf The Peg-Induced Externality} (Continued)
\end{center}
The equilibrium has the following closed-form solution: 
\[
c^T_0 = y^T \left[\frac{1}{1+\underline{r}}+\frac{r}{1+r}
\right]>y^T
\]
\[
c^T_t = y^T 
\left[
\frac1{1+r} + \frac{r}{1+r} \frac{1+\underline{r}}{1+r}
\right]<y^T,  
\]
\[
d_t = y^T \left[
1- \frac{1+\underline{r}}{1+r}
\right]>0, 
\]
\[
h_0 = 1;
\]
\[
h_1=h_2\dots = \frac{1+\underline{r}}{1+r}<1
\]
The following slide displays the same information graphically.
\end{slide}

\begin{slide}
\centerline{\bf Figure 9.5 A Temporary Decline in the Country Interest Rate}
%\label{fig:dnwr_dnwr_temp_decline_in_r}

\centerline{\includegraphics[width=6in]{dnwr_temp_decline_in_r.eps}}

{\tiny 
\rule{1cm}{0.2mm} currency peg \hfill  \_ \_ \_ \_ flexible wage economy or optimal exchange rate economy

}

%produced with 
%c:\data\uribe\book\dnwr\temp_decline_in_r.m
\end{slide}


 





\begin{slide}

\begin{center}
{\bf \Large Section 9.3

 Optimal Exchange Rate Policy}
\end{center}
\end{slide}
\begin{slide}

{\bf Motivation}

$\bullet$ We have just seen that under an exchange rate peg a negative external shock  may lead to involuntary unemployment. How would optimal exchange rate policy look like?

$\bullet$ In this section we show that under optimal policy there is (1) full employment and (2) negative external shocks call for devaluations.

$\bullet$ We begin with a graphical explanation. 

\end{slide}

\begin{slide}
{\bf The graph on the next slide illustrates how the optimal exchange rate policy works}\\ {\small The initial situation is point $A$. Suppose a negative shock (possibly an increase in the country interest rate),  shifts the demand schedule down and to the left. Without government intervention, the supply schedule does not move, because $W$ is downwardly rigid.  The equilibrium would then be  point $B$, with involuntary unemployment equal to $\bar{h}-h^{PEG}$. 

Suppose now that the government devalues the domestic currency from $E_0$ to $E_1>E_0$. The devaluation lowers the real wage from $W_0/E_0$ to $W_0/E_1$, causing the supply schedule to shift down and to the right. 

If the devaluation is just right, the new supply schedule will cross the new demand schedule at point $C$, preserving full employment (we say preserving and not restoring full employment because the economy jumps from $A$ to $C$, without visiting $B$). 

Note: the fall in labor cost caused by the drop in the real wage allows firms to cut prices from $p_0$ to $p^{OPT}$ and induces households to switch expenditure  away from tradables and toward nontradables. 

Take another look at the graph for the analytical example in of a temporary fall in $r_0$. The broken lines display the equilibrium under optimal exchange-rate policy.


}


\end{slide}

\begin{slide}
\begin{center}
{\bf \large  Optimal Exchange-Rate Policy  }\\(again, assume  $\gamma =1$)
\\
\includegraphics[width=6in]{dnwr_mc_p_essim_slides.eps}
%produced with z:\uribe\default\mc_p_peg_only.m
\\
$c^T_1<c^T_0$ (negative  shock, possibly $r_t\uparrow$) \\ $\mathcal{E}_1>\mathcal{E}_0$
(optimal devaluation)

\end{center}
\end{slide}


\begin{slide}
{\bf \large The Ramsey Optimal Exchange-Rate Policy}

The Ramsey government solves the problem:
\[
\max \mathbb{E}_0 \sum_{t=0}^{\infty} \beta^t U(A(c^T_t, F(h_t)))
\]
subject to (9.9) and (9.13)-(9.19). 

Strategy: solve a less constrained problem and then show that its solution satisfies  (9.9) and (9.13)-(9.19). 

\end{slide}


\begin{slide}
Consider the less restricted problem of choosing $\{c^T_t,h_t,d_{t+1}\}_{t=0}^{\infty}$ to 
\[
\max \mathbb{E}_0 \sum_{t=0}^{\infty} \beta^t U(A(c^T_t, F(h_t)))
\]
subject to the following subsect of the equilibrium coditions:
\[
\tag{9.9}%\label{eq:dnwr_market-clearing-traded-competitive}
c^T_t + d_t 
=
y^T_t + \frac{d_{t+1}}{1+r_t}
\]
\[
\tag{9.18}
%\label{eq:dnwr_labor-supply-competitive}
h_t 
\le
\bar{h}
\]
{\small Clearly,  the solution for labor is $h_t=\bar{h}$ for all $t$ or  full employment
at all times. 
  Intuition: One nominal friction and one instrument that can fully offset it. Hence possible to obtain the flexible wage allocation (which here coincides with the first best). 
But to show that this is indeed 
the 
allocation under the optimal exchange-rate policy, we must show that 
 the solution to the above  social planner's problem satisfies the all of the competitive equilibrium  conditions, that is, conditions  (9.9) and (9.13)-(9.19).  
  To see this, proceed by construction: set $\lambda_t$ to satisfy (9.13), $p_t$ to satisfy (9.15), $w_t$ to satisfy (9.16), $\epsilon_t$ to satisfy (9.17), (9.18) is a constraint of the social planner's problem, and so is (9.9). Because $h_t=\bar{h}$, (9.19) holds. That (9.14) holds follows from the definition of $\lambda_t$ and the first-order condition of the social planner.  }
\end{slide}


\begin{slide}
An important reference point: The {\bf full-employment real wage}, denoted  $\omega(c^T_t)$, 
 defined as  the real wage that clears the labor market,
\[
\omega(c^T_t) \equiv
\frac{A_2(c^T_t,F(\bar{h}))}{A_1(c^T_t,F(\bar{h}))}
F'(\bar{h})
;\quad 
\omega'(c^T_t)>0
\]

Set the (gross) devaluation rate, $\epsilon_t = \mathcal{E}_t/\mathcal{E}_{t-1}$, to eliminate unemployment:
\[
\epsilon_t 
\ge
 \frac{\gamma W_{t-1}/\mathcal{E}_{t-1}}{\omega(c^T_t)}
\]
Note: There is a whole family of optimal exchange-rate policies. Under any member of this policy, $h_t=\bar{h}$ and $w_t=\omega(c^T_t)$ for all $t$. 
\end{slide}



\begin{slide}
{\bf When is it inevitable to devalue?}

Optimal exchange rate policy is: 
\[
\epsilon_t 
\ge
 \frac{\gamma W_{t-1}/\mathcal{E}_{t-1}}{\omega(c^T_t)}
\]

Because $\omega'(c^T_t)>0$, optimal devaluations occur in periods of contraction of aggregate demand. It follows that contractions are devaluatory as opposed to devaluations being contractionary. 

\end{slide}



\begin{slide}
Under optimal exchange rate policy external debt and tradable consumption are determined by the solution to
\[
v^{OPT}(y^T_t,r_t,d_t)
=
\max_{\{d_{t+1}, c^T_t\}} \left\{
 U(A(c^T_t,F(\bar{h}))
+\beta \mathbb{E}_t
v^{OPT}(y^T_{t+1},r_{t+1},d_{t+1})
\right\}
\]
subject to
\[
y^T_t + \frac{d_{t+1}}{1+r_t} = d_t +c^T_t
\]

\end{slide}

 
\begin{slide}
\centering
{\bf \Large 9.4 Empirical  Evidence On


 Downward Nominal Wage Rigidity}



\end{slide}


 
\begin{slide}

$\bullet$ Downward nominal wage rigidity is the central friction in the present model $\Rightarrow$ natural to ask if it is  empirically relevant. 

$\bullet$  Downward nominal wage rigidity  has been studied empirically from a number of perspectives:

--- Evidence from micro and macro data. 

--- Studies focusing on rich, emerging, and poor countries. 

---  Studies focusing on formal and informal labor markets. 

$\bullet$ {\bf By product:} Will obtain an estimate of the parameter $\gamma$ governing wage stickiness in the model (useful for quantitative analysis). 

\end{slide}


\begin{slide}
\begin{center}
{\bf Downward Nominal Wage Rigidity

A.)  Evidence From Micro Data from Developed Countries}

\end{center}

\begin{enumerate}
\item United States, 1986-1993, SIPP panel data
  \item United States, 1996-1999, SIPP panel data
  \item United States, 1997-2016, CPS panel data
  \item Other Developed Countries
\end{enumerate}
\end{slide}


\begin{slide}
\begin{center}
{\bf \large 1.) United States, 1986-1993, SIPP panel data}

{\bf Probability of Decline, Increase, or No Change in   Wages }

\begin{tabular}{lrr}
\hline \hline
&\multicolumn{2}{c}{Interviews One Year apart}
\\
&Males & Females\\\hline 
{Decline}&  5.1\%& 4.3\%\\
{\bf Constant} & 53.7\%& 49.2\%\\
Increase& 41.2\%& 46.5\%\\
\hline\hline
\end{tabular}
\end{center}

{\tiny Source: Gottschalk (2005). Note: Male and female hourly workers not in school,  18 to 55 at some point during the panel.  All nominal-wage changes are within-job wage changes, defined as changes while working for the same employer. SIPP panel data.\par}


$\bullet$ Large mass at `Constant'  suggests nominal wage rigidity. 

$\bullet$ Small mass at `Decline' suggests downward nominal wage rigidity.
\end{slide}

\begin{slide}
\centering
\begin{center}
{\bf \large 2.) United States 1996-1999, SIPP panel data}

{\bf Distribution of Non-Zero Nominal Wage Changes}\\
\includegraphics[width=6in]{dnwr_Barattieri_Basu_Gottschalk.eps}

{\small %Quarterly, Within-job and job changers  1996-1999.
 Source:  Barattieri, Basu,  and Gottschalk (2012). SIPP panel data.}

\end{center}
\end{slide}




\begin{note}
\begin{center}
{\bf \large 3.a) United States 2011, CPS data.}

Distribution of  Nominal Wage Changes, U.S.\ 2011\\
\includegraphics[width=6in]{dnwr_Daly_Hobijn_Lucking.eps}  

{\tiny Source: Daly, Hobijn, and Lucking (2012). }

\end{center}
\end{note}

\begin{slide}
{\bf \large 3.b) The next  slide shows the distributions of Nominal Wage Changes in the   United States for each year from   1997 to 2016,  from CPS panel data}

{\bf Nominal wage change:} Year-over-year log changes in nominal hourly wages of hourly-paid job stayers.

$\bullet$ In each panel, the horizontal axis shows bins of  year-over-year percent change in the nominal hourly wage of an hourly-paid jobstayer.  The bin size is two percent, with the exception of a wage-freeze bin, which is defined as an exact zero change. 

$\bullet$ The vertical axis measures the share of workers in each bin. 

$\bullet$ Each wage change distribution is based on about 5,000 workers. 

\end{slide}



\begin{slide}
\centerline{\includegraphics[width=8in]{z:/joint/jobstayers/data/plot_wage_change_densities.eps}  }
%produced with  plot_wage_change_densities.m in the directory z:\joint\jobstayers\data
\centerline{\tiny Source:  Jo, Schmitt-Groh\'e, and Uribe (2017). }


\end{slide}

\begin{slide}
{\bf \large Observations on the figure}

$\bullet$ Large spike at zero wage changes. 

$\bullet$ Many more wage increases than wage cuts. 

$\bullet$ Fraction of wage freezes is cyclical, rises from 15 percent in 2007 to 20 percent in 2009. 

$\bullet$ Much smaller cyclical increase in wage cuts. 



\end{slide}

\begin{slide}
{\bf \large 4.) Micro Evidence On Downward Nominal Wage Rigidity From Other Developed Countries}

$\bullet$ Canada: Fortin (1996).

$\bullet$ Japan:  Kuroda and  Yamamoto (2003). 

$\bullet$ Switzerland: Fehr and Goette (2005). 

$\bullet$ Industry-Level Data: Holden and Wulfsberg (2008), 19 OECD countries from 1973 to 1999. 

\end{slide}

\begin{slide}
\begin{center}
{\bf \large B.) Evidence From Informal Labor Markets}
\end{center}

$\bullet$ Are nominal wages
downwardly flexible in 
  in informal labor markets, where labor unions, wage legislation, or regulation  play, if any,  a small role?  

$\bullet$ Kaur (2012) 
addresses this issue by 
examining  the behavior of nominal wages, employment, and rainfall in casual daily agricultural labor markets in rural India (500  districts from 1956 to 2008). 

$\bullet$ Finds asymmetric nominal wage adjustment: 

--- $W_t$ increases in response to positive rainfall shocks

--- $W_t$ failure to fall,  labor rationing, and unemployment are observed  in response to negative rain shocks. 

$\bullet$  Inflation (uncorrelated with local rain shocks) tends to moderate rationing and unemployment during negative rain shocks, suggesting  downward rigidity  in nominal rather than real wages. 

\end{slide}

\begin{slide}
\centerline{\bf \large C.) Evidence From the Great Depression in the U.S.}

$\bullet$  How do nominal wages behave during extraordinary contractions? 

$\bullet$ The next slide shows  the nominal wage rate and the consumer price index in the  United States from 1923:1-1935:7. 

$\bullet$ Between 1929 and 1931
 the U.S. economy experienced an enormous contraction in employment of  31\%. 

$\bullet$ Nonetheless, during this period nominal hourly wages fell by 0.6\% per year, while  consumer prices fell by 6.6\% per year. See the figure on the next slide. 

$\bullet$ A similar pattern is observed during the second half of the Depression. By 1933, real wages were 26\% higher than in 1929, in spite of a highly  distressed labor market.
\end{slide}


\begin{slide}
\begin{center}
{\bf Figure 9.8 Nominal Wage Rate and Consumer Prices, United States 1923:1-1935:7}

\bigskip

\includegraphics[width=5.5in]{dnwr_Great_Depression_US.eps}

\end{center}
{\tiny Solid line: natural logarithm of  an index of 
manufacturing money wage rates. 
Broken line: logarithm of the consumer price index. \par }
%Produced with
%great_depression_us.m 
%located at
%c:\data\uribe\book\dnwr\data
\end{slide}


\begin{slide}
{\bf \large D.) Evidence From the Great Depression In Europe}

$\bullet$ Countries that left the gold standard  earlier recovered faster than countries that remained on gold. 

--- {Left Gold Early (sterling bloc):} 
 United Kingdom, Sweden, Finland, Norway, and Denmark.

--- {\bf Countries That Stuck To Gold (gold bloc):}
France, Belgium, the Netherlands, and Italy. 

$\bullet$ The gold standard  is  akin to a  currency peg. A peg not to another  currency, but to gold. 

$\bullet$ When the sterling-bloc left gold, they effectively devalued, as their currencies lost value against gold.  

$\bullet$ The figure on the next  slide shows that  between 1929 and 1935 the sterling-bloc  experienced less real wage growth and a larger increase in industrial production than  the gold bloc. 
\end{slide}


\begin{slide}
\centerline{\bf Changes In Real Wages and Industrial Production, 1929-1935}
\centerline{\includegraphics[width=4.5in]{dnwr_EichengreenSachsFigure2.eps}}

\end{slide}

\begin{slide}
\begin{center}
{\bf \large E.)  Evidence From Emerging Countries}
\end{center}


$\bullet$ {Argentina} pegged the peso at a 1-to-1 rate to the dollar between 1991 and 2001.  

$\bullet$ Starting in 1998, the economy was buffeted by a number of large negative shocks (weak commodity prices, large devaluation in Brazil, large increase in country premium). 

$\bullet$ Not surprisingly, between 1998 and 2001, unemployment rose sharply. See the figure on the next slide. 

$\bullet$ Nonetheless, nominal wages remained  remarkably flat. 

$\bullet$ This evidence 
is consistent with 
downward nominal wage rigidity, and  suggests that   $\gamma$, the parameter governing downward wage rigidity in the model,  is about 1.

$\bullet${\small Why $\gamma\approx 1$? The slackness condition $(\bar{h}-h_t)(W_t-\gamma W_{t-1})$ (recall $\epsilon_t=1$ between 1991 and 2001), implies that if unemployment is growing, then wages must grow at the gross rate $\gamma$.   Argentine wages were  flat $\Rightarrow \gamma \approx 1$.}
\end{slide}
\begin{slide}
\begin{center}
{\bf \large  Argentina 1996-2006 }\\
\includegraphics[width=6.5in,angle=-0]{dnwr_arg_unemployment_wage_slides.eps}\\

%{{\bf Memo:} Average annual CPI inflation 1998-2001: -0.86\% }\\
{\bf Implied Value of $\gamma$:} Around unity.



%produced with c:\data\uribe\default\arg_unemployment_wage.m

\end{center}
 
\end{slide}


\begin{slide}
\begin{center}
{\bf \large Evidence From Peripheral Europe (2008-2011)}
\end{center}

$\bullet$ The next slide shows the unemployment rate and  nominal wage growth between 2008:Q1 and 2011:Q2  in 12 European countries that were either in the eurozone or pegging to the euro. 

$\bullet$ Between 2008 and 2011, all countries in the periphery of Europe experienced increases in unemployment;  Some very large increases. 

$\bullet$ In spite of extreme duress in the labor market, nominal hourly wages experienced  increases in most countries and modest declines in only a few. 

$\bullet$ The slide following the table explains how to use the information in the table to  infer a range for $\gamma$. 

\end{slide}

\begin{slide}
\begin{center}
{\bf \large Table 9.2 Unemployment, Nominal Wages, and  $\boldsymbol{\gamma}$\\ Evidence from the Eurozone}

{\small
\begin{tabular}{lcccc}
\hline \hline
&\multicolumn{2}{c}{Unemployment Rate}& Wage Growth&Implied\\
& 2008Q1
& 2011Q2
& $\frac{W_{2011Q2}}{W_{2008Q1}}$
& Value of 
\\Country
& {\tiny (in percent)} 
& {\tiny (in percent)} 
& {\tiny (in percent)}
&$\gamma$\\
\hline
Bulgaria&  6.1&  11.3&  43.3&  1.028\\
Cyprus&  3.8&  6.9&  10.7&  1.008\\
Estonia&  4.1&  12.8&  2.5&  1.002\\
Greece&  7.8&  16.7&  -2.3&  0.9982\\
Ireland&  4.9&   14.3&   0.5&  1.0004\\
Italy     &  6.4&      8.2&  10.0& 1.007\\ 
Lithuania&  4.1&  15.6&  -5.1&  0.996\\
Latvia&  6.1&  16.2&  -0.6&  0.9995\\
Portugal&  8.3&  12.5&  1.91&  1.001\\
Spain&  9.2&  20.8&  8.0&  1.006\\
Slovenia&  4.7&  7.9&  12.5&  1.009\\
Slovakia&  10.2&  13.3&  13.4&  1.010\\
\hline \hline 
 \end{tabular}}
\end{center}
{\tiny 
Note.  $W$ is an index of nominal average hourly labor cost in manufacturing, construction, and services, including the public sector (except for Spain). 
Source: Schmitt-Groh\'e and Uribe (JPE, 2016)


}
\end{slide}

\begin{slide}
{\bf How To Infer $\gamma$  From European Data}

As explained in the analysis of the Argentine Convertibility Plan,   the slackness condition of the model, $(W_t-\gamma W_{t-1})(\bar{h}-h_t)$,  implies that if unemployment increases form one period to the next, then nominal wages must be growing at the rate $\gamma$: $\frac{W_t}{W_{t-1}}=\gamma$. 

How to calculate $\gamma$: 
\[
\gamma = \left(
\frac{W_{2011:Q2}}{W_{2008:Q1}}
\right)^{\frac1{13}}
\]
Subtract 0.6\% per quarter to adjust for foreign inflation and long-run growth (because they are not explicitly incorporated in the model) to obtain the estimate:
\[
\mathbf{\gamma \in [ 0.99,1.022]}
\]
\end{slide}







\begin{slide}
\centerline{\bf \Large Quantitative Analysis}
\bigskip
\centerline{(Sections 9.5-9.8 and 9.10)}

\vspace{2in}

Replication files: \verb7 usg_dnwr.zip 7 available online with the materials for this chapter.

\end{slide}








\begin{slide}
{\bf \large Functional Forms}

Assume a CRRA form for preferences, a CES form for the aggregator  of tradables and nontradables, and an isoelastic form for the production function of nontradables: 
\[
U(c) = \frac{c^{1-\sigma}-1}{1-\sigma}
\]
\[
A(c^T,c^N) = 
\left[
a (c^T)^{1-\frac{1}{\xi}}
+ (1-a) (c^N)^{1-\frac{1}{\xi}}
\right]^{\frac1{1-\frac1{\xi}}}
\]
\[
F(h) = h^{\alpha},
\]
with $\sigma, \xi, a, \alpha>0$. 


\end{slide}

\begin{slide}
{\bf The case of Equal Intra- and Intertemporal Elasticities of Substitution}

Consider the case
\[\xi = \frac1{\sigma}\]
Why this case is of interest:

$\bullet$ It makes the determination of the equilibrium levels of debt, $d_t$, and consumption of tradables, $c^T_t$, independent of the level of activity in the nontraded sector (see the next slide). 
 As a result, the  welfare consequences of exchange-rate policy or nominal wage rigidity are fully attributable to their effect on unemployment, and not on their effect on the accumulation of external debt.

$\bullet$ It  facilitates the computation of equilibrium, as the equilibrium dynamics of $d_t$ and $c^T_t$ can be computed  separately from the equilibrium dynamics of $h_t$, $w_t$, $c^N_t$, and $p_t$. 

$\bullet$ As we will argue shortly, $\sigma=1/\xi=2$ is  empirically plausible. 
\end{slide}

\begin{slide}
{\bf \large Debt and  Tradable Consumption when $\xi=\frac1{\sigma}$}

In this case,
\[
U(A(c^T_t,c^N_t)) = 
\frac{a{c^T_t}^{1-\sigma}
+
(1-a) {c^N_t}^{1-\sigma} -1}
{1-\sigma},
\]
which is separable in $c^T_t$ and $c^N_t$. Then  $d_t$ and $c^T_t$  solve
\[
c^T_t + d_t = y^T_t + \frac{d_{t+1}}{1+r_t}
\]
\[
(c^T_t)^{-\sigma} = \beta (1+r_t) \mathbb{E}_t
(c^T_{t+1})^{-\sigma} 
\]
This subsystem is independent of $h_t$, $w_t$, $p_t$, and $c^N_t$. 

It can be cast as a Bellman equation problem, which facilitates the quantitative analysis (more on the next slide). 

\end{slide}


\begin{slide}
{\bf \large Approximating Equilibrium Dynamics Under Optimal Exchange-Rate Policy when $\xi=1/\sigma$ }\\
 Equilibrium processes $\{c^T_t,d_{t+1}\}$ solve the Bellman equation problem
\begin{equation}
\label{eq:dnwr_Bellman_optimal_exchange_rate_policy}
v^{OPT}(y^T_t, r_t,d_t) = \max_{\{d_{t+1},c^T_t\}} 
\left\{
U(A(c^T_t,F(\bar{h}))) + \beta E_tv^{OPT}(y^T_{t+1},r_{t+1},d_{t+1})
\right\}
\end{equation}
\begin{equation}
\label{eq:dnwr_Bellman_optimal_exchange_rate_policy_constraint}
\mbox{ subject to }
c^T_t + d_t = y^T_t+\frac{d_{t+1}}{1+r_t}; \quad \mbox{ and } d_{t+1}\le\bar{d}.
\end{equation}
Approximate by value function iteration over a discretized state space $(y^T_t,r_t,d_t)$. 
Use 21 values for $y^T_t$ and 11 for $r_t$ (the estimated joint process $(y^T_t,r_t)$ is given below). Use 501 equally spaced points for $d_t$ between 1 and 8.34.

Given approximated solutions to $d_{t+1}$ and $c^T_t$, all other variables of the model can be  backed out:\\
\ $h_t=\bar{h}$,\\
$p_t = \frac{A_2(c^T_t,F(\bar{h}))}{ 
A_1(c^T_t,F(\bar{h}))}$,  \\
 $w_t = p_tF'(\bar{h})$, \\
and a chosen member of the family $\epsilon_t\ge \gamma w_{t-1}/w_t.$

\end{slide}

\begin{slide}
{\bf Approximating Equilibrium Under  Currency Pegs with $\xi=1/\sigma$}

When  $\xi=1/\sigma$, the solution for  $d_{t+1}$ and $c^T_t$ is as before, since it does not depend on the exchange-rate policy. 
 for the  optimal exchange-rate policy 

The determination of $w_t$ requires knowledge of past real wages. So, $w_{t-1}$ is a relevant endogenous state. This is different from the equilibrium under optimal exchange-rate policy. 

Given 
$w_{t-1}$ and $c^T_t$, the solution   for $w_t$ is static (simple): First, conjecture that $h_t=\bar{h}$, and obtain $w_t = \frac{A_2(c^T_t,F(\bar{h}))}{ 
A_1(c^T_t,F(\bar{h}))}F'(\bar{h})$. If $w_t \ge \gamma w_{t-1}$, this is the  solution.  Otherwise,  $w_t=\gamma w_{t-1}$, and   $h_t$  solves 
$w_t =\frac{A_2(c^T_t,F(h_t))}{ 
A_1(c^T_t,F(h_t))}F'(h_t)$.

Discretization of $w_{t-1}$ grid: 500 points between  0.25 and 6, equally spaced in logs. 

\end{slide}

\begin{slide}
{\bf Approximating Equilibrium when $\xi\neq 1/\sigma$}

{\bf Optimal Exchange-Rate Policy:}  As before, the solution is separable.  
First, the processes $d_{t+1}$ and $c^T_t$  solve the Bellman equation problem \eqref{eq:dnwr_Bellman_optimal_exchange_rate_policy}-\eqref{eq:dnwr_Bellman_optimal_exchange_rate_policy_constraint}. Then, all other variables ($w_t$, $p_t$, $c^N_t$) are then easily backed out. 

{\bf Currency Peg:} 
The  the solution  is 
no longer separable. All equilibrium conditions must be solved jointly and becomes  
more complicated.  Schmitt-Groh\'e and Uribe (JPE, 2016) develop an algorithm to handle this case. 

\end{slide}




\begin{slide}
{\bf \large The Driving Process:}

{\bf Data:}  Argentine data over the period 1983:Q1---2001:Q3.   Exclude the period 2001:Q4 to present, because of the default episode in 2002 (no default in the model).  

{\bf  Empirical Measure of $y^T_t$:}  sum of GDP in agriculture, manufacturing, fishing, forestry, and mining. Quadratically detrended. 

{\bf Empirical Measure of $r_t$:} Sum of Argentine EMBI+ plus 90-day Treasury-Bill rate minus a measure of U.S.\ expected inflation. 


The following slide displays the two time series. 
\end{slide}

\begin{slide}

\begin{center}
{\bf Tradable  Output and Country Interest Rate\\ Argentina 1983:Q1 to 2001:Q3}

\begin{tabular}{cc}
{\tiny \bf   Traded Output}&
{\tiny \bf Country Interest Rate}
\\
\includegraphics[width=4in]{dnwr_lgdp_traded_data_slides.eps}
&
\includegraphics[width=4in]{dnwr_arg_int_rate_data_slides.eps}
\\
\end{tabular}
\end{center}
\end{slide}

\begin{slide}
Estimate the  AR(1) system
\begin{equation*}
\label{eq:AR(1)_yT_and_r}
\left[
\begin{array}{c}
\ln  y^T_t\\
\ln \frac{1+r_t}{1+r}
\end{array}
\right]
=
A 
\left[
\begin{array}{c}
\ln  y^T_{t-1}\\
\ln \frac{1+r_{t-1}}{1+r}\end{array}
\right]
+ 
\epsilon_t,
\end{equation*}

{\bf  OLS Estimate of the Driving Process}
\[
A = \left[
\begin{array}{cc}
0.79&-1.36\\
-0.01&0.86
\end{array}
\right] ;\quad \quad 
\Sigma_{\epsilon} = \left[
\begin{array}{cc}
0.00123&-0.00008\\
-0.00008&0.00004
\end{array}
\right]; 
\]
\[  r=0.0316 \mbox{ (3.16\% per quarter).}\]
\end{slide}


\begin{slide}
\begin{center}
{\bf Some Unconditional Summary Statistics}\\
\begin{tabular}{lcc}
\hline\hline
Statistic&$y^T$&$r$\\
\hline
Std. Dev.&12\%&6\%yr\\
Serial Corr.&0.95&0.93\\
Corr($y^T_t,r_t$)&\multicolumn{2}{c}{-0.86}\\
Mean&1&12\%yr\\
\hline\hline
\end{tabular}
\end{center}

Comments: (1) High volatility of both  $y^T_t$ and $r_t$; (2) negative  correlation between $y^T_t$ and $r_t$ (when it rains it pours); (3) High mean country interest rate.  

\end{slide}

\begin{slide}
{\bf \large Calibration}
%\label{table:dnwr_calibration}

\medskip

\begin{tabular}{ccl}
\hline\hline
Parameter&Value&Description\\
\hline
$\gamma$&0.99&Degree of downward nominal wage rigidity\\
$\sigma$&2&Inverse Intertemp. elast. of  subst.\\
$y^T$&1&Steady-state tradable output\\
 $\bar{h}$&1&Labor endowment\\
$a$&0.26&Share of tradables\\
 $\xi$&0.5&Intratemp. elast. of subst.\\ 
 $\alpha$&0.75&Labor share in nontraded sector\\
$\beta$&0.9635&Quarterly subjective discount factor\\
\hline\hline
\end{tabular}

Note: $\sigma=2$ is widely used in business cycle analysis, and  $\xi=0.5$ is within the range of values estimated for emerging countries (see the survey by Akinci, 2011).  Consequently, the restriction $\xi=1/\sigma$ is  quite compelling on empirical and computational grounds. 

\end{slide}

\begin{slide}

\begin{center}
{\bf Crisis Dynamics Under A Currency Peg \\and Under Optimal Exchange-Rate Policy}
\end{center}
 We are interested in 
characterizing quantitatively the response  of the model economy to 
large contractions like the ones observed in Argentina in 2001 and in the periphery of Europe in 2008. In Argentina, for instance, traded output fell by 2 standard deviations in a period of two and a half years (10 quarters).  Accordingly, we use the following operational definition of  an  external crisis. 


{\bf Definition of an External Crisis.} A crisis is  a situation in which in period $t$ 
tradable output, $y^T_t$,  is at or above average, and 10 quarters later, in period $t+10$,  it is at least two standard deviations below trend. 

{\bf The Typical  External Crisis:} Simulate the model  for 20 million periods. Extract all windows of time in which $y^T_t$ conforms to the definition of a crisis. For each variable of interest, average all windows and subtract its unconditional mean (i.e., the mean taken over the 20 million observations). 
\end{slide}

\begin{slide}
\begin{center}
{\bf Figure 9.11 Sources of an External Crisis}

\bigskip

% \label{fig:dnwr_plot_yT_r}
\includegraphics[width=6in]{dnwr_plot_yT_r.eps}

{\tiny Note. Replication file plot\_ir.m in usg\_dnwr.zip.}

%produced with 
%plot_ir.m
%in c:\data\uribe\book\dnwr
%figure(3)
\end{center}

Comments: (1) Because $y^T_t$ and $r_t$ are negatively correlated, the collapse in $y^T_t$ coincides with a sharp increase in the country interest rate.  (2) The response of $y^T_t$ and $r_t$ are exogenous to the model, so this plot is independent of the exchange-rate policy.  The next slide displays the response of the endogenous variables. 
\end{slide}

\begin{slide}
{\bf \large Assumed Optimal Exchange-Rate Policy}


From the family of optimal exchange-rate policies, we pick
\[
\epsilon_t = \frac{w_{t-1}}{\omega(c^T_t)}
\]
Properties of this policy: 

{\bf (1)} It implies that the nominal wage rate, $W_t$, and the nominal price of nontradables, $P^N_t$, are constant at all times. Note: It fully stabilizes the  (factor) price that suffers from nominal rigidity. 


{\bf (2)} It induces zero inflation and zero devaluation on average. \\

{\bf (3)} From (2), we have that the assumed optimal exchange-rate policy 
delivers devaluations ($\epsilon>1$) and revaluations ($\epsilon_t<1$) over the business cycle. Specifically, the government devalues during downturns and revalues during booms.  
After  presenting  the 
dynamics of extrnal crises 
predictied by  the DNWR model,  we will   provide some empirical support for property (3). 

\end{slide}


\begin{slide}
\begin{center}
{\bf \large External Crisis  under \\
Alternative Exchange-rate Policies  with  \\
Downward Niminal Wage Rigidity}
\end{center}

We are now   ready to present the predicted equilibrium dynamics
during a typical crisis 
  under a currency peg and under the optimal exchange rate policy we selected. 

The next slide shows with  solid lines the response of the economy under a currency peg and with broken lines the response under the assumed optimal exchange-rate policy. 

\end{slide}






\begin{slide}
\centering
{\bf \large Pegs Amplify Negative External Shocks}\\
\includegraphics[width=5in,angle=-90 ]{dnwr_plot_ir_slides.eps}\\
%produced with 
%c:\data\uribe\book\dnwr
%plot_ir_slides.m
\rule{2cm}{0.7mm} Currency Peg \hfill
\_ \_ \_ Optimal Exchange-Rate Policy 

{\tiny Note. Replication file plot\_ir.m in usg\_dnwr.zip.}

\end{slide}

\begin{slide}
{\bf Observations}\\
$\bullet$ Large contraction in $c^T$, driven primarily by the hike in the country interest rate. The the trade balance, $y^T_t-c^T_t$,  actually improves  in  spite of the fact that $y^T$ falls sharply. The response of $c^T$ is independent of exchange-rate policy, because $\xi=1/\sigma$.  \\
%
$\bullet$ {\bf Currency Pegs: } large increase in unemployment (25\%),  because the real wage does not fall sufficiently (stays 40\% above the full-employment real wage). Firms don't cut prices because labor cost remains high. As a result, consumers don't switch spending away from tradables and toward nontradables. \\
%
$\bullet$ {\bf Optimal Exchange-Rate Policy:} It cannot 
avoid the contraction in the tradable sector. But it can prevent the external crisis to spreead to the nontradable sector. 
In fact, it preserves full employment throughout the crisis (this result was also established analytically earlier in this chapter). Large devaluations of around 30\% per year for 2.5 years. Consistent with devaluations post Convertibility in Argentina. Devaluations bring the real wage down ($\mathcal{E}\uparrow \Rightarrow w=W/\mathcal{E}\downarrow$), fostering employment and allowing the real exchange  to depreciate ($p=P^N/\mathcal{E} \downarrow$). Real depreciation facilitates expenditure switch toward nontradables. 

\end{slide}


\begin{slide}
{\bf Devaluations and Revaluations in Reality}\\

Do countries devalue during crises and revalue 
when the crisis is over, and why? Look at the next graph. It displays the devaluation rate and the inflation rate for two sets of Latin American countries during the global crisis of 2008. One set is Argentina, and the other includes Chile, Colombia, Mexico, Peru, and Uruguay. 

During the crisis, all countries devalued significantly. However, during the recovery, all countries but Argentina revalued their currencies. The countries that revalued experienced lower inflation than Argentina. 

\end{slide}

\begin{slide}
\begin{center}
{\bf Devaluation and Inflation In Latin America: 2006-2011}
%\label{fig:dnwr_revaluations}

\includegraphics[height=3.9in]{dnwr_revaluations.eps}

\rule{2cm}{1mm} Argentina
  \, \, \,
\rule{0.22cm}{1mm} 
\rule{0.22cm}{1mm}
\rule{0.22cm}{1mm}
Others 
\, \,
\rule{2cm}{0.1mm}  NBER reference dates

\end{center}
%produced with 
%dnwr_revaluations.m in
%c:\data\uribe\foco_economico\fe3
\end{slide}


\begin{slide}
{\bf \large  Are devaluations expansionary as  predicted by the DNWR model?}

Two address this issue, we take another look at two episodes of exiting a currency peg:

$\bullet$ Ending Convertibility: Argentina 1996-2006

$\bullet$ Exiting the Gold Standard: Europe 1929-1935

\end{slide}

\begin{slide}
\centerline{\bf  Argentina Post Convertibility 1996-2006}
\begin{minipage}{5in}
\centerline{\includegraphics[width=5in,angle=-0]{dnwr_arg_unemployment_wage_slides.eps}}
{\tiny {Memo:} Average annual CPI inflation 1998-2001: -0.86\% 

}
%produced with c:\data\uribe\default\arg_unemployment_wage.m
\end{minipage}\begin{minipage}{5in}
{\tiny Observations:$\bullet$ Argentine peso pegged to the U.S.\ dollar from April 1991 to December 2001. (Convertibility Plan)
$\bullet$ Since 1998  severe recession,  subemployment rate reaches  35 percent. 
$\bullet$ but no nominal wage decline in 1998-2001; 
$\bullet$ December 2001  Argentina  devalues   by 250\%; 
$\bullet$ leads to large decline in the real wage;
$\bullet$ labor market conditions then improved quickly. 
$\bullet$ sizable fall in real wages right  after the Dec 2001 devaluation suggests that the 1998-2001 period was one of censored wage deflation. 
$\bullet$ nominal wage growth after devaluation  suggests upward flexibility of  nominal wages;  

Overall, predicted expansionary effects of devaluation are consistent with dynamics in Argentina post Convertibility


}
 \end{minipage}
\end{slide}
\begin{slide}
{\bf Exiting the Gold Standard: Europe 1929 to 1935}

$\bullet$ Friedman and Schwartz (1963) observe that  countries that left  gold   early (the sterling bloc) enjoyed  more rapid recoveries than countries that stayed on gold longer (the gold bloc). 

$\bullet$ Sterling bloc: United Kingdom, Sweden, Finland, Norway,  Denmark.

$\bullet$ Gold bloc:  France, Belgium, the Netherlands, and Italy. 


$\bullet$ Eichengreen and Sachs (1986) observe that  real wages behaved differently in countries that left gold early (ie devalued) and in countries  that stayed on gold longer (ie stayed on the peg). Take a look at the next figure, which is redrawn from Eichengreen and Sachs. 


\end{slide}
\begin{slide}{\bf Changes In Real Wages and Industrial Production, 1929-1935}

\begin{minipage}{4.5in}\leftline{\bf Figure 9.15}
\rightline{\includegraphics[width=4.5in]{dnwr_EichengreenSachsFigure2.eps}}
%produced with fig2_Eichengreen_Sachs.m in z:\uribe\book\dnwr
%{\tiny Redrawn from Eichengreen and Sachs (1985). }
\end{minipage}
\begin{minipage}{5in} {\tiny 
$\bullet$ The horizontal axis measures the 1935 real wage, defined as the nominal wage deflated by the wholesale price index; $\bullet$ The vertical axis measures industrial production in 1935; $\bullet$ Real wages and industrial production are normalized to 100 in 1929; $\bullet$     The solid line is the fitted regression line: $IP = 170.0  - 0.561 \times \mbox{Real Wage}$.

$\bullet$ relative to their respective 1929 levels real wages in 1935 in the sterling bloc countries were lower than real wages in the gold bloc countries. $\bullet$ And industrial production in the sterling bloc countries in 1935 exceeded their respective 1929 levels whereas industrial production in the gold bloc countries was below their respective 1929 levels. 


$\bullet$ Taken together, these two  facts  suggest that  in the great depression years nominal wages were downwardly rigid in Europe and that abandoning a peg during a recession can be expansionary. 

}
\end{minipage}
\end{slide}
\begin{slide}
\begin{center}
{\bf \large The Welfare Costs of  Currency Pegs}\\
\end{center}
{\small Find the compensation, measured as percent increase in the stream of consumption in the peg economy,  denoted  $\Lambda(s_t)$, that makes agents indifferent between living under a peg or under the optimal exchange-rate policy, given the current state $s_t=(y^T_t,r_t,d_t,w_{t-1})$.  This compensation is implicitly given by
\[
 \mathbb{E}
\left\{\left.
\sum_{j=0}^{\infty}
\beta^j U\left(c^{\mbox{PEG}}_{t+j}(1+\Lambda(s_t))/100\right) \right|
s_t
\right\}
=
 \mathbb{E}
\left\{\left.
\sum_{j=0}^{\infty}
\beta^j U\left(c^{\mbox{OPT}}_{t+j}\right) \right|
s_t
\right\},
\]
Solve for $\Lambda(s_t)$ to obtain
\[
\Lambda(s_t)
=100 
\left\{
\left[
\frac{v^{OPT}(y^T_t,r_t,d_t)(1-\sigma)+(1-\beta)^{-1}}
{v^{PEG}(y^T_t,r_t,d_t,w_{t-1})(1-\sigma)+(1-\beta)^{-1}} 
\right]^{1/(1-\sigma)} - 1
\right\},
\]
where $
v^{OPT}(y^T_t,r_t,d_t) \equiv 
 \mathbb{E}
\left\{\left.
\sum_{j=0}^{\infty}
\beta^j U\left(c^{\mbox{OPT}}_{t+j}\right) \right|
s_t
\right\}
$\\
and 
$v^{PEG}(y^T_t,r_t,d_t,w_{t-1}) \equiv 
 \mathbb{E}
\left\{\left.
\sum_{j=0}^{\infty}
\beta^j U\left(c^{\mbox{PEG}}_{t+j}\right) \right|
s_t
\right\}
$,\\ 
with the expectation taken over  the   distribution of $s_t$ in the peg economy. 
The welfare cost of a peg,  $\Lambda(s_t)$, is a random variable as it is a function of the  state in period $t$, $s_t$. When $\sigma=1/\xi$ only state variable that is policy dependent is $w_{t-1}$ and $c^T_t$ is policy independent. Thus, when 
$\sigma=1/\xi$
the only source of welfare loss of suboptimal exchange rate policy stems from the dynamics of $c^N_t$.  


}

\end{slide}


\begin{slide}
\begin{center}

{\bf \large The Welfare Costs of Currency Pegs}

 %\label{table:dnwr_sensitivity}}

\begin{tabular}{lccc}
\hline 
\hline 
 & \multicolumn{2}{c}{Welfare Cost  }& Unemployment \\
\cline{2-3}
Model&Mean&Median&Mean Rate \\
\hline
Baseline ($\gamma=0.99$) & 7.8&7.2&11.7\\
%welfare numbers are produced by running 
%simu_welf.m in 
%c:\data\uribe\book\dnwr
%Average unemployment was produced by running simu.m, using as input pfi_peg.mat. In turn, simu_pfi_peg.mat was produced by running pfi_peg.m. All of these scripts are located at
%c:\data\uribe\book\dnwr
\hline 
\hline 
\end{tabular}
\end{center}

{\tiny Note. The welfare cost of a currency peg is expressed in percent of consumption. Welfare costs are computed over the distribution of the state $(y^T_t,r_t,d_t,w_{t-1})$ induced by  the peg economy. 
Replication files: simu\_welf.m (welfare cost) and simu.m (unemployment) in usg\_dnwr.zip.

}

{\bf Observation:} Large welfare costs of currency pegs. All of the cost is explained by lost consumption of nontradables due to unemployment in that sector. 
\end{slide}

\begin{slide}
\centerline{\bf Probability Density Function of the Welfare Cost of  Currency Pegs}
%\label{fig:dnwr_welfare_cost}

\centerline{\includegraphics[width=4.5in]{dnwr_welfare_cost.eps} }
%produced by running
%plot_welf.m in
%c:\data\uribe\book\dnwr
%\begin{quote}
%{\small Note. The welfare cost of a currency peg is 
%defined in 
%equation~(\ref{eq:dnwr_welfare_cost_of_peg}). 
%The density function of welfare costs  is computed over the distribution of the state $(y^T_t,r_t,d_t,w_{t-1})$ induced by  the peg economy. }
%\end{quote}

{\tiny Replication  file plot\_welf.m in usg\_dnwr.zip.}



{\bf Observation:} The distribution of welfare costs  of pegs is highly skewed to the right, suggesting the existence of initial states, $(y^T_t,$ $r_t,d_t,$ $w_{t-1})$, 
in which pegs are highly costly in terms of unemployment.  The next slide identifies such states. 
\end{slide} 


\begin{slide}
\begin{center}
{\bf Welfare Cost of  Currency Pegs and  the  Initial State}
%\label{fig:dnwr_welfare_cost_wb_d}

\includegraphics[width=5in]{dnwr_welfare_cost_wb_d.eps}
\end{center}

{\tiny 
Note. In each plot, all states except the one shown on the horizontal axis are fixed at their unconditional mean values. The dashed vertical lines indicate the unconditional mean of the state  displayed on the horizontal axis (under a currency peg if the state is endogenous).  Replication  file plot\_welf.m in usg\_dnwr.zip. \par }


{\small {\bf Observation: }Currency pegs are more costly the higher the initial past wage, the higher the initial stock of external debt, the lower the initial endowment of tradables, and the higher the initial country interest rate.}
%{\small Note. In each plot, all states except the one shown on the horizontal axis are fixed at their unconditional mean values. The dashed vertical lines indicate the unconditional mean of the state  displayed on the horizontal axis (under a currency peg if the state is endogenous).}
%produced with 
%plot_welf.m in
%c:\data\uribe\book\dnwr
%call figure(2)
\end{slide} 


\begin{slide}

{\bf \large Alternative Parameterizations and Model Specifications}

\bigskip

9.10 Varying the Degree of Wage Rigidity\\
9.11 Symmetric Wage Rigidity\\
9.13 Endogenous Labor Supply\\
9.14 Production in the Traded Sector\\
9.15 Product Price Rigidity\\


\end{slide} 


\begin{slide}
\begin{center}
{\bf \large 
Varying the Degree of Downward Wage Rigidity}

 %\label{table:dnwr_sensitivity}}

\begin{tabular}{lccc}
\hline 
\hline 
 & \multicolumn{2}{c}{Welfare Cost  }& Unempl. \\
\cline{2-3}
Model&Mean&Median&Rate \\
\hline
Baseline ($\gamma=0.99$) & 7.8&7.2&11.7\\
%the entries for low and high wage rigidity were produced by running
%plot_welf_gama.m  
%in
%c:\data\uribe\book\dnwr
%look at the variables [GAMA WC U]
Lower Downward Wage Rigidity\\
$\gamma= 0.98$&5.7&5.3&8.9\\
$\gamma= 0.97$&3.5&3.3&5.6\\
$\gamma= 0.96$&2.8&2.7&4.6\\
Higher Downward Wage Rigidity\\ $\gamma=0.995$&
14.3&   13.0&19.5\\
%welfare numbers are produced by running 
%simu_welf.m in 
%c:\data\uribe\book\dnwr
%Average unemployment was produced by running simu.m, using as input pfi_peg.mat. In turn, simu_pfi_peg.mat was produced by running pfi_peg.m. All of these scripts are located at
%c:\data\uribe\book\dnwr
\hline 
\hline 
\end{tabular}
%\begin{quote}
%{\small Note. The welfare cost of a currency peg is expressed in percent of consumption per quarter, see equation~(\ref{eq:dnwr_welfare_cost_of_peg}). 
%The mean and median of the welfare cost of a peg is computed over the distribution of the state $(y^T_t,r_t,d_t,w_{t-1})$ induced by  the peg economy. }
%\end{quote}
\end{center}
{\bf Observation:}  Sizable welfare costs and unemployment even for highly flexible wages, e.g., $\gamma=0.96$. Recall, $\gamma=0.96$ means that wages can fall frictionlessly by 16\% per year. 
\end{slide}


\begin{slide}
\centerline{\bf  Symmetric Wage Rigidity}

{Is more wage flexibility always welfare increasing?}

Not always. We have just seen that
 the welfare costs of a currency peg increase as
the degree of downward wage rigidity, 
 $\gamma$, increases. So the answer here is Yes. 


We now consider a 
different way of increasing wage rigidity, namely,   bi-directional wage rigidity:
\[
\frac{1}{\gamma}\ge \frac{W_t}{W_{t-1}}\ge \gamma
\]
We will see that this increase in wage rigidity is welfare enhancing. 
\end{slide}


\begin{slide}


\centerline{\bf \large The Welfare Costs of Pegs:  Symmetric Wage Rigidity}
\centerline{\small  ($\gamma=0.99$)}

 %\label{table:dnwr_sensitivity}}
\begin{tabular}{lcc}
\hline 
\hline 
 & \multicolumn{1}{c}{Welfare Cost  }& Unempl. \\
&Mean&Rate \\
\hline
Downward only: $\frac{W_t}{W_{t-1}}\ge \gamma $ & 7.8&11.7\\
%welfare numbers are produced by running 
%simu_welf.m in 
%c:\data\uribe\book\dnwr
%Average unemployment was produced by running simu.m, using as input pfi_peg.mat. In turn, simu_pfi_peg.mat was produced by running pfi_peg.m. All of these scripts are located at
%c:\data\uribe\book\dnwr
Upward and downward:
$\frac1{\gamma}\ge\frac{W_t}{W_{t-1}}\ge \gamma$
&3.3&5.2\\
%produced by running simu_welf.m (for welfare numbers) and simu.m  (for unemployment), using as input file pfi_peg_sym.mat, which in turn is produced by running pfi_peg_sym.m all of these files are in
%c:\data\uribe\book\dnwr
\hline 
\hline 
\end{tabular}
%\begin{quote}
%{\small Note. The welfare cost of a currency peg is expressed in percent of consumption per quarter, see equation~(\ref{eq:dnwr_welfare_cost_of_peg}). 
%The mean and median of the welfare cost of a peg is computed over the distribution of the state $(y^T_t,r_t,d_t,w_{t-1})$ induced by  the peg economy. }
%\end{quote}

{ 
$\bullet$ Welfare costs under symmetric rigidity, while still large, are half that under downward wage rigidity. 
 { Thus  greater wage flexibility is welfare decreasing.}  Why? Symmetric wage rigidity alleviates the peg-induced externality (we saw this theoretically).

 
$\bullet$ To the extent that downward  wage rigidity is the case of greatest empirical relevance, this result suggests that models with upward and downward wage rigidity underestimate the welfare costs of currency pegs.

}
\end{slide}

\begin{slide}

\centerline{\bf   Endogenous Labor Supply}

Thus far, we have assumed that households supply $\bar{h}$  units of labor inelastically. How should  an endogenous labor supply  affect the main predictions of the model?
 
$\bullet$ Now negative income shocks (e.g., $y^T\downarrow$),   may cause an  increase in labor supply, elevating the  level of involuntary unemployment. 

$\bullet$ 
Thus far, the model included only one type of non-work activity, namely involuntary unemployment. Now there will be two, involuntary unemployment
(or involuntary lesure) 
 and  voluntary leisure (or voluntary unemployment).   How do voluntary and involuntary unemployment enter in the utility function? Are they substitutes or complements? How substitutable are they? This margin will make a difference in determining the welfare costs of currency pegs. 

With these questions in mind, 
 let's now  introduce a labor choice more formally. 

\end{slide}

\begin{slide}

Assume that the period utility function  is increasing in consumption, $c_t$, and leisure, $\ell_t$, 
\[
U(c_t, \ell_t ) = 
\frac{c_t^{1-\sigma}-1}{1-\sigma}
+ \varphi \frac{{\ell_t}^{1-\theta}-1}{1-\theta}
\]
 The desired  demand for leisure,  (or desired supply of labor)  is a notional object  that results from assuming that the household can work as many hours as it wishes at the going real wage:
\begin{equation*}
%\label{eq:dnwr_demand _leisure}
\varphi (\ell_t^v)^{-\theta} = w_t \lambda_t,
\end{equation*}
where $\ell^v_t$ denotes voluntary leisure. 
Let $\bar{h}$ denote the total endowment of hours per period. The desired (or voluntary) labor supply is given by
\[
h^v_t = \bar{h}-\ell^v_t.
\]
Let $h_t$ denote actual hours worked. 
\end{slide}
\begin{slide}
A no-slavery condition: 
\begin{equation*}
%\label{eq:dnwr_hv>=h}
h^v_t \ge h_t,
\end{equation*}
Closing the labor market
\begin{equation*}
%\label{eq:dnwr_hs_slackness}
(h^v_t-h_t)\left(w_t -  \gamma \frac{w_{t-1}}{\epsilon_t}\right) = 0.
\end{equation*}

The above four  equations replace the conditions $h_t \le \bar{h}$ and  $(\bar{h}-h_t)(w_t-\gamma w_{t-1}/\epsilon_t)=0$ of the  economy with inelastic labor supply. The rest of the equilibrium conditions are unchanged. 

Involuntary unemployment, or, synonymously, involuntary leisure, denoted $u_t$, is given by
\[
u_t = h^v_t - h_t
\]
\end{slide}

\begin{slide}

Policy evaluation requires addressing an important question:

 {\bf How should voluntary and involuntary leisure enter in the period utility function?} 

One possibility is to assume that $\ell^v_t$ and $u_t$ are perfect substitutes. In this case, the second argument of the utility function becomes 
$\ell_t = \ell^v_t+u_t$. Is this assumption empirically realistic? 
\end{slide}

\begin{slide}


 The existing empirical literature  seems to   reject this assumption:

{ $\bullet$ Krueger and Mueller (2012): 
the unemployed enjoy leisure activities to a lesser degree than the employed and  on a typical day report higher levels of sadness than the employed.

$\bullet$ Winkelmann and 
Winkelmann (1998): Unemployment has a large 
non-pecuniary 
detrimental effect on life satisfaction.

$\bullet$ Krueger and Mueller (2012):
Unemployed  spend 101 minutes more per day on job search than employed (not surprising). However, job search generates the highest feeling of sadness after personal care out of 13 time-use categories. }

$\Rightarrow$ Better specification appears to be : $
\ell_t = \ell^v_t + \delta u_t
$

with $\delta<1$. Will consider three values, 1, 0.75, and 0.5.
\end{slide}
\begin{slide}

\centerline{\bf Endogenous Labor Supply And}

\centerline{\bf  The Welfare Costs of Currency Pegs}

% \label{table:dnwr_endogenous_labor}
\centerline{\begin{tabular}{lccc}
\hline 
\hline 
 & \multicolumn{2}{c}{Welfare Cost }&Unemployment  \\
\cline{2-3}
Parameterization&Mean&Median&Rate \\
\hline
Baseline (inelastic labor supply) & 7.8&7.2&11.7\\
%the entries for low and high wage rigidity were produced by running
%plot_welf_gama.m  
%in
%c:\data\uribe\book\dnwr
%look at the variables [GAMA WC U]
\multicolumn{3}{l}{Endogenous Labor Supply $\ell_t = \ell^v_t + \delta u_t$}\\
$\delta = 0.5$&16.5&15.2&30.9\\
$\delta=0.75$&8.2&7.5&30.9\\
$\delta=1$&1.7&1.5&30.9\\
%for all values of delta, produced by running
%simu_welf.m in
%c:\data\uribe\book\dnwr\endogenous_labor
\hline 
\hline 
\end{tabular}
%\begin{quote}
%{\small Note. The welfare cost of a currency peg is expressed in percent of consumption per quarter.  Unemployment rates are expressed in percent.}
%\end{quote}
}
{\bf Observations:} (1) Unemployment larger under endogenous labor supply specification.  (2) Welfare cost  of peg larger or smaller depending on preferences about involuntary leisure, $\delta$. 
\end{slide}





\begin{slide}
\centerline{\bf  Product Price Rigidity}

Assume now that nominal wages are fully flexible and that instead prices are sticky. Consider downward nominal price ridigity and symmetric price rigidity. 
\[
\mbox{downward price rigidity: } \quad \quad \frac{P^N_t}{P^N_{t-1}} \ge \gamma_{p} 
\]
\[
\mbox{symmetric price rigidity: } \quad \quad \frac1{\gamma_{p} }\ge \frac{P^N_t}{P^N_{t-1}} \ge \gamma_{p} 
\]

Calibrate models as before, but set $\gamma=0$ and $\gamma_p = 0.99$. 

\end{slide}





\begin{slide}

{\bf Adjustment to a Negative External Shock with Downward Price Rigidity Under A Currency Peg}

\centerline{\includegraphics[width=6in]{dnwr_mc_p_dnpr.eps}}

\centerline{\tiny  $c^T_1<c^T_0; \quad  \quad \gamma_p=1$}
%produced with 
%c:\data\uribe\book\dnwr\dnpr\mc_p.m

\end{slide}
\begin{slide}
{\bf Observations:} When $c^T$ falls from $c^T_0$ to $c^T_1$, full employment would occur if prices could decline to $\rho(c^T_1)$, point C in the figure. But because of downward nominal price rigidity, $P^N_t/\mathcal{E}$ remains at $\rho(c^T_0)$. At this price households only demand $F(h^{bust})$ units of nontradables and the economy suffers of unemployment due to week demand.

Optimal policy calls for a  devaluation that lowers $P^N_t/\mathcal{E}_t$ down to $\rho(c^T_1)$ and restores full employment. Hence contractions continue to be devaluatory!

The economy continues to suffer from a peg induced externality. Increases in $P^N_t$ during booms should be limited to avoid unemployment during the recession phase of the cycle. 


\end{slide}





\begin{slide}
\centerline{\bf Price Rigidity And The Welfare Costs of Currency Pegs \label{table:dnwr_price_rigidity}}


\begin{tabular}{lcc}
\hline 
\hline 
 & \multicolumn{1}{c}{Welf}&  Unempl \\
Parameterization&Cost&Rate\\
\hline
Baseline (wage rigidity, $\gamma=0.99$ and $\gamma_p=0$.) & 7.8&11.7\\
%produced  by running
%simu_welf.m (for welfare costs)  and simu.m (for unemployment)
 %in
%c:\data\uribe\book\dnwr
\multicolumn{3}{l}{Nominal Price Rigidity ($\gamma=0$, $\gamma_p=0.99)$}\\
\,\,\,Downward Price Rigidity, $P^N_t/P^N_{t-1} \ge \gamma_p $ &9.9&14.1\\
%produced  by running
%simu_welf.m (welfare) and simu.m (unemployment)
%c:\data\uribe\book\dnwr\dnpr
\,\,\,Symmetric Price Rigidity, $1/\gamma_p \ge P^N_t/P^N_{t-1}\ge \gamma_p $ &4.4&6.6\\
%produced  by running
%simu_welf.m (welfare) and simu.m  in
%c:\data\uribe\book\dnwr\dnpr
%use pfi_peg_sym.mat
Calvo Price Rigidity, $\theta =0.7$& 3.6&N/A\\ 
\hline 
\hline 
\end{tabular}


{\tiny Note. The welfare cost of a currency peg is expressed in percent of consumption per quarter.  Unemployment rates are expressed in percent.


}
%
\end{slide}


\begin{slide}
{\bf Observations.} The welfare costs of pegs under downward price rigidity are as large as under downward wage rigidity. 
The welfare costs under symmetric price rigidity  are only about half as large as under downward price rigidity. It follows that adding upward price rigidity ameliorates the peg induced externality, just like adding upward wage rigidity amerliorates it in the model model with wage rigidity.

The last line, labeled Calvo pricing, pertains to an economy with Calvo-type price rigidity in the nontraded sector. The probability of not being able to change the nominal price is $\theta=0.7$ per period. The calibration of the shocks is the same as in the baseline model. The Calvo model is one of symmetric price rigidity and so it is not surprising that the welfare costs of pegs are most similar to those associated with the economy with the bi-directional price rigidity studied earlier. 


\end{slide}

\begin{slide}
{\bf  Summary of  Theoretical results:}

$\bullet$ {\bf Analytical results}
{\small 
--- The combination of a currency peg and downward nominal wage rigidity creates an externality that amplifies the severity of contractions. 
\\
--- The model predicts that the average rate of unemployment is increasing in the degree of aggregate volatility.  
\\
--- The DNWR model  predicts  significant amplification of booms-bust cycles under suboptimal monetary policy (and in particular under currency pegs.). 
}

$\bullet$ {\bf Quantitative Results: }

{\small 

--- The  costs of currency pegs  due to downward nominal wage rigidity are  large, \\
$\cdot$  in terms of welfare, 4 to 10\% of consumption per period.\\
$\cdot$  and in terms of unemployment, 10 to 30\%.\\
--- These results are robust to a variety of changes parameter 
values and model specifications, including endogenous labor supply, bidirectional nominal rigidity,  and product price rigidity.  \\
--- The welfare costs of currency pegs are higher the higher the past real wage, the higher the level of external debt, the lower the level of tradable output, and the higher the country interest rate. 
}

\end{slide}
\begin{slide}
\centerline{\bf \large Section 9.12 }
\centerline{\bf \large  The Mussa Puzzle}
\end{slide}
\begin{slide}
Mussa (1986) is an empirical paper that provides evidence against the ``nominal exchange rate regime neutrality'' hypothesis. 

Methodology: compare variations in nominal and real exchange rates in quarterly data from 13 industrialized countries during the fixed exchange rate period (1957-1970) with variations in the floating exchange rate period (1973-1984)

Observables: 
\\
$
\mathcal{E}_t = \mbox{nominal exchange rate vis-\'a-vis the U.S. dollar}
$
\\
$
P^*_t = \mbox{U.S.\ Consumer Price Index }
$
\\
$P_t = \mbox{Domestic Consumer Price Index }
$
\\
$
RER_t = \frac{\mathcal{E}_t P^*_t}{P_t}
$

Take a look at the next figure. It shows 3 series: (1) the natural logarithms of the dollar French franc nominal and real exchange rates as well as the log of the relative CPIs. 

\end{slide}
\begin{slide}
\centerline{\includegraphics[width=8in]{dnwr_Mussa_Figure1_France.eps}}
{\tiny Source: This is figure 1 of Mussa 1986. Solid line: nominal exchange rate, $\mathcal{E}_t$; plus-line:  ratio of CPIs, $P^*_t/P_t$; diamond line: real exchange rate, $RER_t=\mathcal{E}_t P^*_t / P_t$; variables are shown in natural logarithms. 

}
\end{slide}
\begin{slide}
{\bf Comments on the figure: }

Quarterly data, 1957:Q2 to 1984:Q3. 

Fixed exchange rate: Bretton Woods, 1957:Q2-1970:Q4.\\
 French franc was pegged to the U.S.\ dollar. 

Floating exchange rate: Post-Bretton Woods, 1973:Q1-1984:Q3.


Selection of sample period:  Why a gap in the sample period? Mussa argues that 1970Q4 to 1973Q1 is a transition period. The float starts in March 1973. 
Why does Mussa start his sample in  1957, this was afterall not the beginning of the Bretton Woods agreement? Mussa justifies the start date by saying that this is when  the IFS tapes start having the raw data. 
Why does Mussa end the sample 1983Q3? This must be the most recent observation when he wrote the paper.
\end{slide}
\begin{slide}

$\bullet$ The graph shows that during the Bretton Woods period (1957-1970) the nominal exchange rate was basically constant. In the post-Bretton Woods period (1973-1984) it fluctuates quite a bit. This confirms treating these two periods as having a different nominal exchange rate regime. 

$\bullet$ The striking feature of the graph is that the real exchange rate mimics the behavior of the nominal exchange rate throughout.

$\bullet$ The flipside of this observation is that the times series properties  of relative prices, $P^*_t/P_t$, remained remarkable constant 
across the two periods. 

$\bullet$ Mussa argues that under the  `nominal exchange rate regime neutrality' hypothesis one should not observe such a high correlation between nominal and real exchange rates. 



\end{slide}
\begin{slide}
{\small Mussa documents three facts: \\
1.) The standard deviation of the real deprecation rate is smaller during the peg era. 
\\
2.) The standard deviations and serial correlations of the nominal and real depreciation rates are similar to each other in each exchange rate regime. 
\\
3.) The volatility of national  CPI inflation  is about the same during the fixed and the floating regime sample period. 
\\
Let
\[
\epsilon^{RER}_t \equiv \ln RER_t - \ln RER_{t-1}
\]
\[
\epsilon_t \equiv \ln \mathcal{E}_t - \ln \mathcal{E}_{t-1}
\]
\[
\epsilon^p_t \equiv \ln P^*_t/P_t - \ln P^*_{t-1}/P_{t-1}
\]


Illustrate these 3 facts for  France

\centerline{
\begin{tabular}{lcc}
\hline
\hline
France& 1957Q2-1970Q4&1973Q1-1984Q3\\
\hline
var$(\epsilon^{RER}_t)$ & 5.258&23.590\\
var$(\epsilon_t)$ & 8.197&24.275\\
corr$(\epsilon^{RER}_t,\epsilon^{RER}_{t-1})$&0.1150&0.3376\\
corr$(\epsilon_t,\epsilon_{t-1})$&0.1776&0.4317\\
var$(\epsilon^p_t)$ & 1.543&0.540\\
\hline
\hline
\end{tabular}}

Taken from Table 1.4 of Mussa (1986)


 }
\end{slide}

\begin{slide}
{\bf \large Theoretical Implications of the Mussa Puzzle}

$\bullet$ The Mussa facts are referred to as a puzzle because  they suggest that relative prices depend on the behavior of nominal prices. This is inconsistent with flexible-price models in the
RBC tradition,
possibly augmented with a demand for money,  
 which represented the predominant paradigm around the time of Mussa's writing (e.g., Stockman, 1988). 

$\bullet$ 
The Mussa puzzle suggests a role for nominal rigidities, which gained renewed 
popularity since the mid 1990s. 
An early analysis of the Mussa puzzle through the lens  of a sticky-price model is Monacelli (2004).

$\bullet$ Here, we wish to address two questions: \\
{\bf (a)} Are  the predictions of the DNWR   model consistent with the
Mussa facts.\\
{\bf (b)}  When seen through the lens of the DNWR model, do the Mussa facts tell us anything about the optimality or not of the monetary/exchange-rate regime post Bretton Woods? 
\end{slide}


\begin{slide}
{\bf Real and Nominal Exchange Rates Under Fixed And Floating Exchange-Rate Regimes as Predicted by the DNWR Model}
%\label{table:dnwr_Mussa}}

\medskip

\begin{center}

\begin{tabular}{lccc}
\hline
\hline
& Peg&\multicolumn{2}{c}{Float}\\
\cline{3-4}
&&Optimal & Anti optimal\\
\hline
std$(\epsilon^{RER}_t)$ & 12.0&32.5&5.2\\
std$(\epsilon_t)$ & 0&45.2&44.0\\
corr$(\epsilon^{RER}_t,\epsilon^{RER}_{t-1})$&0.18&-0.04&-0.04\\
corr$(\epsilon_t,\epsilon_{t-1})$&--&-0.04&0.95\\
corr$(\epsilon^{RER}_t,\epsilon_t)$&--&0.99&-0.15\\
std$(\pi_t)$ & 13.2&13.2&44.3\\
\hline
\hline
\end{tabular}

\end{center}
{\tiny 
Note. Standard deviations are expressed in percent per year. The optimal floating exchange-rate policy is given by 
$\epsilon_t=w_{t-1}/\omega(c^T_t)$, and the suboptimal floating exchange-rate policy is given by $\epsilon_t = \omega(c^T_t)/w_{t-1}$.
\par 
}

%Produced with simu.m simu_opt.m and simu_anti_opt.m. The mat files used in these programs, in turn, were produced with pfi_peg.m, vfi_opt.m, and anti_opt.m, respectively. All programs are in
%c:\data\uribe\book\dnwr
\end{slide}


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

$\bullet$ 
In line with Mussa's first fact, the predicted standard deviation of the real depreciation rate is  much larger under the optimal floating exchange rate policy  than under the peg (first line of the table). 

$\bullet$ In line with Mussa's second fact, under the optimal flexible exchange  rate regime, the nominal and real exchange rates have similar standard deviation and first-order  serial correlations, and are highly positively  contemporaneously correlated (lines 2-5 of the table). 

$\bullet$ In line with Mussa's third fact, the predicted volatility of CPI inflation
 is the same under the peg and the optimal floating regime (last line of the table). 
\end{slide}

\begin{slide}
{\bf Was Post-Bretton-Woods Exchange-Rate Policy Optimal?}\\
$\bullet$ A common practice that needs clarification: 
Often, empirical studies classify exchange-rate regimes into fixed or floating, and then derive stylized facts associated with each regime. This practice is problematic because in reality there is  an infinite family of floating exchange-rate rebimes, not just one, which  can induce different  dynamics of nominal and real variables. \\
$\bullet$ As an  illustration,  consider the `anti optimal'  exchange rate policy:
\centerline{$
\epsilon_t = \frac{\omega(c^T_t)}{w_{t-1}}.
$}
which is the inverse of the optimal policy studied earlier in the quantitative analysis. \\
$\bullet$ The last column of the previous table 
shows that under
the anti optimal 
  floating exchange-rate policy, the model fails to capture all three of the Mussa facts.\\
$\bullet$ 
  Seen through the lens of the 
DNWR model,  it follows that Mussa's facts can be interpreted as suggesting that during the early post-Bretton-Woods period, overall,
the dynamics of inflation and nominal and real exchange rates were consistent with the optimal exchange rate policy. 
\end{slide}




\begin{slide}
{\bf Empirical Evidence on Real Effects of Monetary Instability}

\begin{enumerate}
  \item Industrialized countries, Mussa 1986, method, compare fixed and float.
  \item United States, Christiano, Eichenbaum, Evans  2005, method, VAR identification. 
  \item small open economies, to be added, method, VAR identification.
  \end{enumerate}  
\end{slide}
\begin{slide}
{Real Effects of Monetary Instability: Empirical Evidence from VARs}


The dynamic effects of a shock to monetary policy
$\bullet$
\[
R_t = f(\Omega_t) + \epsilon_t
\]
\begin{description}
  \item $R_t$ the federal funds rate
  \item $f(.)$ is a linear function
  \item $\epsilon_t$ is a monetary policy shock 
\end{description}  
$\bullet$
 Identification assumption: $\epsilon_t \perp \Omega_t$
\[Y_t = \left[ \begin{array}{c} Y_{1t}\\R_t\\Y_{2t} \end{array} \right] \]

\end{slide}
\begin{slide}

$ Y_{1t}$  is part of $\Omega_t$, which does not respond contemporaneously to monetary policy shocks

\[{Y_{1t}}_{6\times 1} = \left[ \begin{array}{c} 
\mbox{log of real GDP$_t$}\\
\mbox{log of real $C_t$}\\
\mbox{log of GDP deflator}\\
\mbox{log of real investment}\\
\mbox{log of real wage}\\
\mbox{log of labor productivity}\\
\end{array} \right] 
\]

\[{Y_{2t}}_{2\times 1} = \left[ \begin{array}{c} 
\mbox{log of real profits}\\
\mbox{growth rate of M2}
\end{array} \right] 
\]

\[
\Omega_t = \begin{array}{rrrrr} Y_{1t},&Y_{1t-1},&Y_{1t-2},&Y_{1t-3},&Y_{1t-4}\\
&R_{t-1},&R_{t-2},&R_{t-3},&R_{t-4}\\
&Y_{2t-1},&Y_{1t-2},&Y_{2t-3},&Y_{2t-4}\\
\end{array}\]

\end{slide}

\begin{slide}


$\bullet$ Sample: 1965:3-1995:3

$\bullet$ Ignoring a constant the VAR is 
\[
Y_t = A_1 Y_{t-1} + A_2 Y_{t-2} +A_3 Y_{t-3} + A_4 Y_{t-4} + C \eta_t
\]
Let $C\eta_t = u_t$

$\bullet$  $C$ is 9 by 9 and lower triangular (zeros above the diagonal)  
\\ $\bullet$ $C$ has ones on the diagonal
\\ $\bullet$ ${\eta_t}_{9\times 1}\sim$ mean 0, serially uncorrelated, diagonal covariance matrix. 
\\$\bullet$  Element 7 of $\eta_t$ is $\epsilon_t$, the monetary policy shock. 
\\$\bullet$  $\epsilon_t>0$ means a contractionary policy shock
\\$\bullet$  Estimate $A_{i}$ for $i=1,2,3,4$
\\$\bullet$  Estimate $C$, how? var($u_t) = C \Sigma_{\eta} C'$
\\$\bullet$  Shaded area is the 95 percent confidence interval computed using the methodology of Sims and Zha 
\end{slide}

\begin{slide}


\begin{tabular}{cc}
\includegraphics[width=4in]{dnwr_cee_ir_i.eps}
&
\includegraphics[width=4in]{dnwr_cee_ir_output.eps}
\\
\includegraphics[width=4in]{dnwr_cee_ir_c.eps}
&
\includegraphics[width=4in]{dnwr_cee_ir_ivv.eps}
\end{tabular}

{\tiny These plots are taken from Figure 1 of Christiano, Eichenbaum, Evans, JPE 2005.}

\end{slide}

\begin{slide}
{Real Effects of an expansionary monetary policy shock from CEE}

1.)   Y, C, I
 
  $\bullet$ Y, C, I respond in a hump-shaped fashion
 \\
 $\bullet$  Y peaks 4-6 quarters after the shock
 \\
 $\bullet$  C peaks 4 quarters after the monetary policy shock
 \\
 $\bullet$ I peaks 5 quarters after the monetary policy shock
 \\
 $\bullet$ The impulse response is significant until $t=8$. 

2.) R

  $\bullet$ increases by 60 basis points
  \\
$\bullet$ peak response on impact
  \\
$\bullet$ is over after 4-5 quarters [long before peak output]

While there is quite some debate about what are the correct identification assumptions, there is considerable agreement about the qualitative effects of  a monetary policy shock in the sense that inference is robust across a large subset of the identification schemes that have been considered in the literature. 

\end{slide}

\begin{slide}
\centerline{ \bf Section 9.16}
\centerline{\bf \;large   Staggered  Price Setting:}
\centerline{\bf \large The Calvo Model}

\end{slide}

\begin{slide}
The Calvo model:

--- is perhaps  {\it the} canonical models used in monetary economics

--- assumes staggered price setting


--- proposed by   Calvo (1983) and later refined by  Woodford (1996) and   Yun (1996)
 
--- price rigidity is bidirectional, that is, the upward and downward adjustment of nominal prices is sluggish 

\end{slide}

\begin{slide}


Differences to the models of nominal rigidities studied earlier in the chapter: 

(1) firms are forced  to satisfy demand even if the price is below marginal cost. An implication of this difference is that in  the Calvo model the labor supply must be wage elastic for price stickiness to have first-order effects

(2) Calvo model 
assumes imperfect competition in product markets. This 
assumption allows firms be price setters and to have nonzero finite demand  
even when their prices differ from those of their closest competitors. 


(3) wages are flexible and there is no involuntary unemployment.
\end{slide}

\begin{slide}
{\bf Households}

\begin{equation}
\label{eq:dnwr_cy_utility}
\tag{9.41}
\max  E_0 \sum_{t=0}^{\infty} \beta^t
[U(c_t) - V(h_t)]
\end{equation}

subject to 
\begin{equation}
\label{eq:dnwr_cy_aggregator}
\tag{9.42}
c_t = A(c^T_t,c^N_t)
\end{equation}
\begin{equation}
\label{eq:dnwr_cy_budget_constraint}
\tag{9.43}
P^T_t c^T_t + P^N_t c^N_t +
\mathcal{E}_t d_t =
P^T_t y^T_t +W_th_t+\Phi_t
+ T_t +  \frac{\mathcal{E}_t d_{t+1}}{1+r_t} 
\end{equation}
\begin{equation}
\label{eq:dnwr_cy_no_Ponzi}
\tag{9.44}
d_{t+1} \le \bar{d}
\end{equation}
\end{slide}

\begin{slide}

Assume law of one price holds for tradables 
$$
P^T_t = \mathcal{E}_t
$$ 
 
Optimality conditions of the household's problem are: \eqref{eq:dnwr_cy_aggregator}-\eqref{eq:dnwr_cy_no_Ponzi}
\[
\frac{A_2(c^T_t,c^N_t)}{A_1(c^T_t,c^N_t)}
=p_t
\]
\[
\lambda_t = U'(c_t) A_1(c^T_t,c^N_t)
\]
\[
\frac{\lambda_t}{1+r_t} = \beta E_t
\lambda_{t+1} + \mu_t
\]
\[
 \mu_t \ge 0
\]
\[
\mu_t(d_{t+1}-\bar{d})=0
\]
\[
V'(h_t) = \lambda_t \frac{W_t}{P^T_t}
\]
\end{slide}

\begin{slide}
{\bf 9.16.2 Firms Producing Final Nontraded Goods}

Production technology:
\begin{equation}
\label{eq:dnwr_cy_dixit_stiglitz}
\tag{9.45}
y^N_t  = 
\left[
\int_0^1
\left(a^N_{it}\right)^{1-\frac1{\mu}} di
\right]^{\frac1{1-\frac1{\mu}}},
\end{equation}
$y^N_t=$ output of the  final nontraded good\\
$a^N_{it}=$ quantity of intermediate goods of type $i\in [0,1]$ used in the 
production of the final nontraded good\\
$\mu>1$  elasticity of substitution across varieties\\
 

Environment:  perfect competition
\end{slide}

\begin{slide}

Firm profits: 
\begin{equation*}
%\label{eq:dnwr_cy_profits_final}
P^N_t y^N_t -
\int_0^1
P^N_{it} a^N_{it}di,
\end{equation*}
Profit maximization implies intermediate input demand of the form
\begin{equation}
\label{eq:dnwr_cy_demand_aNit}
\tag{9.46}
a^N_{it} = y^N_t 
\left(
\frac{P^N_{it}}
{P^N_t}
\right)^{-\mu}
\end{equation}

$\bullet$ demand for the  intermediate good of variety $i$ 
is increasing in the level of final output and decreasing in the relative price of the variety in terms of the final good, with a price  elasticity of $-\mu$. 

Using this expression to eliminate $a^N_{it}$ from the 
Dixit-Stiglitz aggregator \eqref{eq:dnwr_cy_dixit_stiglitz}
 yields 
\begin{equation}
\label{eq:dnwr_cy_PN}
\tag{9.47}
P^N_t = 
\left[
\int_0^1
(P^N_{it})^{1-\mu}di
\right]^{\frac1{1-\mu}}
\end{equation}
$\bullet$  price of the final nontraded good is increasing and homogeneous of degree one in the price of the intermediate nontraded goods 
\end{slide}
\begin{slide}
{\bf  9.16.2 Firms Producing Nontraded Intermediate Goods}

Production technology for variety $i$
\begin{equation}
\label{eq:dnwr_cy_production_function}
\tag{9.48}
y^N_{it} = h_{it}^{\alpha}; \quad \alpha\in(0,1]
\end{equation}

Environment: Monopolistic competition, firms are price setters

Production is demand determined. This means that, given the posted price $P^N_{it}$, firms must set production to ensure that all customers are served, that is, 
\begin{equation}
\label{eq:dnwr_cy_output_demand_determined}
\tag{9.49}
y^N_{it}=a^N_{it}
\end{equation}
\end{slide}
\begin{slide}

Profits
\[
\Phi_{it} = P^N_{it}a^N_{it}-(1-\tau)W_th_{it}
\]

$\tau = $ a labor  subsidy to offset distortions  introduced by imperfect competition. Its presence facilitates the characterization of optimal monetary policy, as it  results in a model with a single distortion, namely the one stemming from price rigidity. That is, monetary policy here will not be called up to correct distortions stemming from imperfect competition. 

\end{slide}
\begin{slide}
{\bf Price setting problem of producer of variety $i$}

Use \eqref{eq:dnwr_cy_demand_aNit}, 
\eqref{eq:dnwr_cy_production_function}, and 
\eqref{eq:dnwr_cy_output_demand_determined}
to eliminate  $a^N_{it}$,  $h_{it}$, and $y^N_{it}$, 
respectively, 
from  the expression for profits in period $t$ :
\[
 P^N_{it}
y^N_{t}
\left(
\frac{P^N_{it}}
{P^N_t}
\right)^{-\mu}
-
(1-\tau) W_t {y^N_t}^{\frac1{\alpha}} \left(
\frac{P^N_{it}}
{P^N_t}
\right)^{-\frac{\mu}{\alpha}}.
\]


Under price flexibility, the optimal pricing decision consists in choosing $P^N_{it}$ to maximize the above expression.


Under price stickiness, the pricing problem  is  different because 
 firms, by assumption,  cannot reoptimize prices every period. 


With probability $\theta \in (0,1)$ a firm cannot reset its price in period $t$ and must charge the same price as in the previous period, 
and  with probability $1-\theta$ it can adjust the price freely. 
Consider the pricing decision of a firm that can reoptimize its price in period $t$.

\end{slide}
\begin{slide}

Let $\tilde{P}^N_{it}$ denote the price 
chosen in $t$. 
Then, with probability $\theta$, the price will continue to be $\tilde{P}^N_{it}$ in period $t+1$. 
With probability $\theta^2$, the price will continue to be $\tilde{P}^N_{it}$ in period $t+2$, and so on. In general, 
with probability $\theta^s$ the price will continue to be $\tilde{P}^N_{it}$ in period $t+s$. 

 The original Calvo (1983) formulation assumes that $\tilde{P}^N_{it}$ is set following an ad hoc rule of thumb. The innovation introduced by Yun (1996) is to assume that the firm picks 
$\tilde{P}^N_{it}$ in a profit-maximizing fashion. 

As in much of the modern new-Keynesian literature (e.g., Woodford, 2003), we adopt Yun's approach.   Specifically, the present discounted value of profits associated with $\tilde{P}^N_{it}$ is given by
\[
E_t 
\sum_{s=0}^{\infty}
Q_{t,t+s} \theta^s
\left[
 \tilde{P}^N_{it}
y^N_{t+s}
\left(
\frac{\tilde{P}^N_{it}}
{P^N_{t+s}}
\right)^{-\mu}
-
(1-\tau)W_{t+s} {y^N_{t+s}}^{\frac1{\alpha}} \left(
\frac{\tilde{P}^N_{it}}
{P^N_{t+s}}
\right)^{-\frac{\mu}{\alpha}}
\right],
\]
$Q_{t,t+s}=$ state-contingent nominal discount factor that converts 
nominal payments in period $t+s$ into a nominal payment in period $t$. 




\end{slide}
\begin{slide}
{\small 

The firm picks $\tilde{P}^N_{it}$ to maximize the pdv of profits. The FOC is: 
\begin{equation}
\label{eq:dnwr_cy_foc_PNTILDE}
\tag{9.50}
E_t 
\sum_{s=0}^{\infty}
Q_{t,t+s} \theta^s
y^N_{t+s}
\left(
\frac{\tilde{P}^N_{it}}
{P^N_{t+s}}
\right)^{-\mu}
\left\{ \frac{\mu-1}{\mu}
 \tilde{P}^N_{it}
-\frac1{\alpha}
(1-\tau) W_{t+s} 
\left[
{y^N_{t+s}} \left(
\frac{\tilde{P}^N_{it}}
{P^N_{t+s}}
\right)^{-\mu}
\right]^{\frac{1-\alpha}{\alpha}}
\right\} = 0.
\end{equation}

$ \frac{\mu-1}{\mu}
 \tilde{P}^N_{it}=$  marginal revenue in period $t+s$\\
$\frac1{\alpha}(1-\tau) W_{t+s} 
\left[
{y^N_{t+s}} \left(
\frac{\tilde{P}^N_{it}}
{P^N_{t+s}}
\right)^{-\mu}
\right]^{\frac{1-\alpha}{\alpha}}
=
\frac{(1-\tau)W_{t+s} }{\alpha (h^N_{it+s})^{\alpha-1}}$  marginal cost in period $t+s$.
  
$y^N_{t+s}
\left(
\frac{\tilde{P}^N_{it}}
{P^N_{t+s}}
\right)^{-\mu}=$  quantity sold in period $t+s$

$\bullet$ price set in period $t$  equates the present discounted values of marginal revenues and marginal costs weighted by the level of production 



}

\end{slide}
\begin{slide}
{\bf \small Crisis Dynamics in the Calvo Model: The Role Of Exchange-Rate Policy}

\includegraphics[width=9.5in]{dnwr_cy_plot_ir_slides.eps}

{\tiny \rule{2cm}{1mm} Currency Peg
  \, \, \,
\rule{0.22cm}{1mm} 
\rule{0.22cm}{1mm}
\rule{0.22cm}{1mm}
Optimal Exchange Rate Policy 


{\tiny Note.
Replication file typical\_crisis.m  in  calvo.zip available online  with the materials for this chapter. }


}

%produced with 
%z:\data\uribe\dnwr\calvo\typical_crisis_slides.m
\end{slide}

\end{document}

\subsection{Aggregation and Equilibrium}
The firm's  first-order condition~\eqref{eq:dnwr_cy_foc_PNTILDE}
has only one firm-specific variable, namely, $\tilde{P}^N_{it}$. 
Therefore, 
 all firms that get to reoptimze 
their
price in period $t$ will choose the same price, regardless of idiosyncratic price histories. 
This means that we can drop the subscript $i$ from $\tilde{P}^N_{it}$, which greatly facilitates aggregation. 
 Using this fact and equation~\eqref{eq:dnwr_cy_PN}, we can write the price of the final nontradable good  as
\begin{eqnarray*}
(P^N_t)^{1-\mu} &=&
\int_0^1
(P^N_{it})^{1-\mu}di
\\
&=&
\theta {P^N_{t-1}}^{1-\mu} + (1-\theta) \left(\tilde{P}^N_t\right)^{1-\mu}.
\end{eqnarray*}
The first term in the second equality uses the fact that, because the
 firms that do not get to change 
their price 
in period $t$ 
 are picked randomly, the average
of the 
 prices charged by these  $\theta$ firms equals the price
of the final good 
 in period $t-1$. Dividing both sides
of the above expression 
 by $(P^N_t)^{1-\mu}$ yields
\[
1 = \theta (\pi^N_t)^{\mu-1} + (1-\theta)
(\tilde{p}^N_t)^{1-\mu},
\]
where $\pi^N_t \equiv P^N_t/P^N_{t-1}$ denotes the gross rate of inflation  of  nontradables, and $\tilde{p}^N_t\equiv 
\tilde{P}^N_t/P^N_t$ denotes the relative price of reoptimized prices in terms of final nontraded goods. 


We assume that firms are owned by domestic households. This means that  the nominal discount factor used by firms, $Q_{t,t+s}$, must coincide with that of private households,  
\[
Q_{t,t+s} = \beta^s \frac{\lambda_{t+s}P^T_t}{\lambda_tP^T_{t+s}}.
\]

Let  $p_t\equiv P^N_t /P^T_t$ denote the relative price of nontradables in terms of tradables, and use the above expression for $Q_{t,t+s}$, to write  the first-order condition~\eqref{eq:dnwr_cy_foc_PNTILDE} as
\begin{eqnarray}
\label{eq:dnwr_cy_foc_PNTILDE_rec}
E_t 
\sum_{s=0}^{\infty}
\beta^s \theta^s \frac{\lambda_{t+s}}{\lambda_t}
y^N_{t+s}
\left(
\tilde{p}^N_t \prod_{k=1}^s \frac1{\pi^N_{t+k}}
\right)^{-\mu}
\\
\left\{ 
\frac{\mu-1}{\mu}
\left(
\tilde{p}^N_t \prod_{k=1}^s \frac1{\pi^N_{t+k}}
\right)
p_{t+s}
-\frac{1-\tau}{\alpha}
 w_{t+s} 
\left[
{y^N_{t+s}} 
\left(
\tilde{p}^N_t \prod_{k=1}^s \frac1{\pi^N_{t+k}}
\right)^{-\mu}
\right]^{\frac{1-\alpha}{\alpha}}
\right\} = 0,\nonumber
\end{eqnarray}
where $w_t\equiv W_t/\mathcal{E}_t$ denotes the real wage expressed in terms of tradable goods. 
For the characterization of equilibrium
it is convenient to write this equilibrium condition   in recursive form. To this end, first define  the present discounted values of marginal revenue and marginal cost as follows
\begin{equation*}
pvmc_t = 
\frac{1-\tau}{\alpha}
E_t 
\sum_{s=0}^{\infty}
\beta^s \theta^s \frac{\lambda_{t+s}}{\lambda_t}
y^N_{t+s}
\left(
\tilde{p}^N_t \prod_{k=1}^s \frac1{\pi^N_{t+k}}
\right)^{-\mu}
 w_{t+s} 
\left[
{y^N_{t+s}} 
\left(
\tilde{p}^N_t \prod_{k=1}^s \frac1{\pi^N_{t+k}}
\right)^{-\mu}
\right]^{\frac{1-\alpha}{\alpha}}
\end{equation*}
and 
\begin{equation*}
pvmr_t = 
\frac{\mu-1}{\mu}
E_t 
\sum_{s=0}^{\infty}
\beta^s \theta^s \frac{\lambda_{t+s}}{\lambda_t}
\left(
\tilde{p}^N_t 
\prod_{k=1}^s \frac1{\pi^N_{t+k}}
\right)^{1-\mu}
p_{t+s}y^N_{t+s}.
\end{equation*}
Then we can rewrite  equation~\eqref{eq:dnwr_cy_foc_PNTILDE_rec} as 
\begin{equation*}
%\label{eq:dnwr_cy_pvmr=pvmc}
pvmr_t = pvmc_t.  
\end{equation*}
Now, applying some algebra,  express $pvmc_t$ and $pvmr_t$ in recursive form as
\begin{equation*}
pvmc_t = 
\frac{1-\tau}{\alpha} 
(y^N_t)^{\frac1{\alpha}}  w_t 
(\tilde{p}^N_t)^{-\frac{\mu}{\alpha}}
+
\beta \theta
E_t 
\frac{\lambda_{t+1}}{\lambda_t}
\left(
\frac{\tilde{p}^N_t}
{\tilde{p}^N_{t+1}}
\frac1{\pi^N_{t+1}}
\right)^{-\frac{\mu}{\alpha}}
pvmc_{t+1}
\end{equation*}
and
\begin{equation*}
pvmr_{t}=
\frac{\mu-1}{\mu} 
y^N_t  p_t 
(\tilde{p}^N_t)^{1-\mu}
+
\beta \theta
E_t 
\frac{\lambda_{t+1}}{\lambda_t}
\left(
\frac{\tilde{p}^N_t}
{\tilde{p}^N_{t+1}}
\frac1{\pi^N_{t+1}}
\right)^{1-\mu}
pvmr_{t+1}.
\end{equation*}

Total hours worked, denoted $h_t$, is the sum of hours worked across all intermediate goods producing firms, that is, 
\[
h_t = \int_0^1 h_{it} di. 
\]
 To obtain a relationship between total hours worked and aggregate output of nontradables,  use~\eqref{eq:dnwr_cy_production_function} to eliminate $h_{it}$ from the above expression. 
Then we have that  
$
h_t = \int_0^1 y_{it}^{1/\alpha}  di
= \int_0^1 
\left[
y^N_{t}
\left(
\frac{P^N_{it}}
{P^N_t}
\right)^{-\mu}
\right]^{1/\alpha}  di
= {y^N_{t}}^{1/\alpha} 
\int_0^1 
\left(
\frac{P^N_{it}}
{P^N_t}
\right)^{-\mu/\alpha}  di
. 
$
Let 
\[
s_t \equiv 
\int_0^1 
\left(
\frac{P^N_{it}}
{P^N_t}
\right)^{-\mu/\alpha}  di.
\] 
The variable $s_t$ measures price dispersion. If all $i$ varieties sell for the same price, then $s_t=1$, otherwise, as will become clear  shortly,  $s_t\ge 1$. 
Using the definition of $s_t$ in the above expression 
yields
\[
y^N_t  = s_t^{-\alpha}h_t^{\alpha} .
\]
This expression shows that price dispersion acts as a negative productivity shock.
The higher is price dispersion, the lower will be the aggregate level of output associated with a given level of employment. When prices are fully flexible, all firms charge the same price ($P^N_{it}=P^N_t$), so $s_t=1$ and $y^N_t = h_t^{\alpha}$. 
 It follows that  $s_t$ represents a measure of output loss due to the presence of price rigidity.  

We now show that  $s_t\ge 1$. To this end, let $v_{it} \equiv
\left(
\frac{P^N_{it}}
{P^N_t}
\right)^{1-\mu} $, so that 
\[
s_t = \int_0^1 v_{it}^{\frac{\mu}{(\mu-1)\alpha}}  di. 
\]
The exponent on $v_{it}$, $\frac{\mu}{(\mu-1)\alpha}$,  is greater than 1 because $\mu>1$ and $0<\alpha\le 1$, making the right hand side a convex function of $v_{it}$. Then, by Jensen's inequality, we have
\[
\int_0^1 v_{it}^{\frac{\mu}{(\mu-1)\alpha}}  di\ge \left( \int_0^1 v_{it} di \right)^{\frac{\mu}{(\mu-1)\alpha}}. 
\]
By equation~\eqref{eq:dnwr_cy_PN}, the right hand side of this equation is 1, and the left-hand side is $s_t$. Therefore, we have shown that 
\[
s_t\ge 1.
\]
Using the fact that all firms that cannot change their price in a given period, charge last period's price and that all firms that can change their price choose the same price, namely, $\tilde{P}^N_t$, it follows that  the measure of price dispersion,  $s_t$,  evolves over time as 
\begin{eqnarray*}
s_t &=& \int_0^1 
\left(
\frac{P^N_{it}}
{P^N_t}
\right)^{-\mu/\alpha}  di\\
&=&\theta
\int_0^{1} 
\left(
\frac{{P}^N_{it-1}}
{P^N_t}
\right)^{-\mu/\alpha}  di 
+
(1-\theta)
\left(
\frac{\tilde{P}^N_{t}}
{P^N_t}
\right)^{-\mu/\alpha}  \\
&=&
\theta s_{t-1} (\pi_t^N)^{\mu/\alpha}
+
(1-\theta) (\tilde{p}^N_t)^{-\mu/\alpha}.
\end{eqnarray*}
In equilibrium the market for nontradable goods clears,
\[
y^N_t = c^N_t. 
\]
Aggregate profit income is
\begin{eqnarray*}
\Phi_t &=& \int_0^1 \Phi_{it} di
\\
& =& \int_0^1  \left(P^N_{it} y^N_{it} - (1-\tau)W_t h_{it}\right) di 
\\
& =& P^N_{t} y^N_{t} - (1-\tau)W_t h_t,
\end{eqnarray*}
where the last equality follows from the facts that the producers of final nontraded goods make zero profits and from the definition of $h_t$. 
The government finances labor subsidies with lump-sum taxes. This implies that the period-by-period budget constraint of the government is 
\[
\tau W_t h_t = T_t.
\]
Combining the above three expressions
with the household's sequential budget constraint \eqref{eq:dnwr_cy_budget_constraint} 
yields the familiar resource constraint  
\[
c^T_t + d_t = y^T_t + \frac{d_{t+1}}{1+r_t}. 
\]

Finally, we can write the gross growth rate of the 
relative price of nontradables in terms of tradables as the ratio of the inflation of nontradables to the depreciation rate, 
\[
\frac{p_t}{p_{t-1}} = \frac{\pi^N_t}{\epsilon_t},
\]
where, as before, $\epsilon_t \equiv \mathcal{E}_t/\mathcal{E}_{t-1}$ denotes the gross rate of depreciation of the domestic currency.




Then a competitive equilibrium
is  a set of processes $c^T_t$, $\pi^N_t$, $p_t$, $h_t$, $\lambda_t$, $w_t$, $\tilde{p}^N_t$, $y^N_t$, $s_t$, $d_{t+1}$,  $\mu_t$, $pvmc_t$, and $pvmr_t$ satisfying
\begin{equation}
\label{eq:dnwr_cy_resource_constraint_eqm}
c^T_t + d_t = y^T_t + \frac{d_{t+1}}{1+r_t}, 
\end{equation}
\begin{equation}
\label{eq:dnwr_cy_lambda_eqm}
\lambda_t = U'(A(c^T_t,y^N_t)) A_1(c^T_t,y^N_t),
\end{equation}
\begin{equation}
\label{eq:dnwr_cy_Euler_eqm}
\frac{\lambda_t}{1+r_t} 
= \beta E_t \lambda_{t+1} + \mu_t,
\end{equation} 
\begin{equation}
\label{eq:dnwr_cy_d<=dbar_eqm}
d_{t+1} \le \overline{d},
\end{equation}
\[ \mu_t \ge 0,\]
\[
%{eq:dnwr_cy_slackness_eqm}
\mu_t(d_{t+1}-\bar{d})=0,
\]
\begin{equation}
\label{eq:dnwr_cy_demand_for_nontradables_eqm}
\frac{A_2(c^T_t,y^N_t)}{A_1(c^T_t,y^N_t)}
=p_t,
\end{equation}
\begin{equation}
\label{eq:dnwr_cy_supply_of_labor_eqm}
V'(h_t) = \lambda_t w_t,
\end{equation}
\begin{equation}
\label{eq:dnwr_cy_h = s yN_eqm}
y^N_t  = s_t^{-\alpha}h_t^{\alpha}, 
\end{equation}
\begin{equation}
\label{eq:dnwr_cy_s_evolution_eqm}
s_t = 
\theta s_{t-1} (\pi_t^N)^{\mu/\alpha}
+
(1-\theta) (\tilde{p}^N_t)^{-\mu/\alpha},
\end{equation}
\begin{equation}
\label{eq:dnwr_cy_piN_eqm}
1 = \theta (\pi^N_t)^{\mu-1} + (1-\theta)
(\tilde{p}^N_t)^{1-\mu},
\end{equation}
\begin{equation}
\label{eq:dnwr_cy_pvmr=pvmc_eqm}
pvmr_t = pvmc_t,
\end{equation}
  \begin{equation}
\label{eq:dnwr_cy_pvmc_recursive_eqm}
pvmc_t = 
\frac{1-\tau}{\alpha} 
(y^N_t)^{\frac1{\alpha}}  w_t 
(\tilde{p}^N_t)^{-\frac{\mu}{\alpha}}
+
\beta \theta
E_t 
\frac{\lambda_{t+1}}{\lambda_t}
\left(
\frac{\tilde{p}^N_t}
{\tilde{p}^N_{t+1}}
\frac1{\pi^N_{t+1}}
\right)^{-\frac{\mu}{\alpha}}
pvmc_{t+1},
\end{equation}
\begin{equation}
\label{eq:dnwr_cy_pvmr_recursive_eqm}
pvmr_{t}=
\frac{\mu-1}{\mu} 
y^N_t  p_t 
(\tilde{p}^N_t)^{1-\mu}
+
\beta \theta
E_t 
\frac{\lambda_{t+1}}{\lambda_t}
\left(
\frac{\tilde{p}^N_t}
{\tilde{p}^N_{t+1}}
\frac1{\pi^N_{t+1}}
\right)^{1-\mu}
pvmr_{t+1},
\end{equation}
\begin{equation}
\label{eq:dnwr_cy_epsilon_eqm}
p_t = p_{t-1} \frac{\pi^N_t}{\epsilon_t},
\end{equation}
given exogenous processes $y^T_t$ and $r_t$, 
 initial conditions $d_0$ and $s_{-1}$, 
an exchange-rate policy  $\epsilon_t$,  and a subsidy policy $\tau$. 

We  assume that the government sets the labor subsidy so as to offset the distortion in the labor market created by 
the presence of imperfect competition in product markets, that is, 
\begin{equation}
\label{eq:dnwr_cy_tau_optimal}
\tau = \frac1{\mu}.
\end{equation}

\subsection{The Flexible-Price Equilibrium}
Suppose all firms get the opportunity to set their price optimally every period, that is, $\theta=0$. This is an environment in which all prices are fully flexible. In this environment, equilibrium condition~\eqref{eq:dnwr_cy_piN_eqm} collapses to
\[
\tilde{p}^N_t = 1.
\]
It then follows from equilibrium condition~\eqref{eq:dnwr_cy_s_evolution_eqm}   that 
\[
s_t = 1,
\]
which says that there is no efficiency loss due to price dispersion. This means that a given level of employment yields the maximum level of final nontradable goods. Formally, equilibrium condition~\eqref{eq:dnwr_cy_h = s yN_eqm} now becomes
\[
y^N_t = h_t^{\alpha}. 
\]
Finally, 
\eqref{eq:dnwr_cy_pvmr=pvmc_eqm}-\eqref{eq:dnwr_cy_pvmr_recursive_eqm}
collapse to the static condition
\[
p_t = \frac{w_t}{\alpha h_t^{\alpha-1}},
\]
which says that 
in equilibrium 
firms
behave 
as if they operated in a perfectly competitive environment, in the sense that they 
 equate  price  to marginal cost every period.  

A flexible-price equilibrium is then a set of processes $c^T_t$, 
$h_t$, $d_{t+1}$,  $\lambda_t$ and $\mu_t$ satisfying 
\begin{equation}
\label{eq:dnwr_cy_resource_constraint_fpeqm}
c^T_t + d_t = y^T_t + \frac{d_{t+1}}{1+r_t}, 
\end{equation}
\begin{equation}
\label{eq:dnwr_cy_lambda_fpeqm}
\lambda_t = U'(A(c^T_t,h_t^{\alpha})) A_1(c^T_t,h_t^{\alpha}),
\end{equation}
\begin{equation}
\label{eq:dnwr_cy_Euler_fpeqm}
\frac{\lambda_t}{1+r_t} 
= \beta E_t \lambda_{t+1} + \mu_t,
\end{equation}
\begin{equation}
\label{eq:dnwr_cy_d<=dbar_fpeqm}
d_{t+1} \le \overline{d},
\end{equation}
\begin{equation}
\label{eq:dnwr_cy_mu>0_fpeqm}
 \mu_t \ge 0,
\end{equation}
\begin{equation}
\label{eq:dnwr_cy_slackness_fpeqm}
\mu_t(d_{t+1}-\bar{d})=0,
\end{equation}
\begin{equation}
\label{eq:dnwr_cy_supply_of_labor_fpeqm}
V'(h_t) = \alpha \lambda_t 
\frac{A_2(c^T_t,h_t^{\alpha})}{A_1(c^T_t,h_t^{\alpha})}
 h_t^{\alpha-1}, 
\end{equation}
given exogenous processes $y^T_t$ and $r_t$ and the initial condition $d_0$. 
This is indeed the best possible equilibrium, in the sense that its associated allocation is the solution to the social planner's problem
\[
\max_{\{c^T_t,h_t,d_{t+1}\}} E_0 \sum_{t=0}^{\infty} \beta^t
[U(A(c^T_t,h_t^{\alpha})) - V(h_t)]
\]
subject to
\begin{equation}
\tag{\ref{eq:dnwr_cy_resource_constraint_fpeqm}}
c^T_t + d_t = y^T_t + \frac{d_{t+1}}{1+r_t}, 
\end{equation}
and
\begin{equation}
\tag{\ref{eq:dnwr_cy_d<=dbar_fpeqm}}
d_{t+1} \le \overline{d}.
\end{equation}
As the following analysis shows, the flexible-price equilibrium provides a useful benchmark for the characterization of optimal exchange-rate policy. 

\subsection{Optimal Exchange-Rate Policy}
Is there an exchange-rate policy that can support the flexible-price equilibrium? The answer to this question turns out to be yes. To see this,
suppose  for simplicity that $s_{-1}=1$. (Exercise~\ref{exercise:Yun_AER} considers the general case $s_{-1}>1$.) 
Now imagine  an exchange-rate policy that fully stabilizes the nominal price of nontradables, so that 
\[
\pi^N_t = 1,
\]
for all $t$. We will show shortly that such an exchange-rate policy does exist. 
Combining the above equation with equilibrium condition~\eqref{eq:dnwr_cy_piN_eqm}, we obtain
$
\tilde{p}^N_t = 1$, for all $t$. 
This expression implies that firms that get to change their price in the current period choose 
  the average price level $P^N_t$. 
The fact that $\pi^N_t=\tilde{p}^N_t=1$ then implies, from 
 equilibrium condition~\eqref{eq:dnwr_cy_s_evolution_eqm},  that 
$s_t=1$ for all $t$.  That is, there is no output loss due to price dispersion, or $y^N_t = h_t^{\alpha}$. Finally, using the above results we have that 
\eqref{eq:dnwr_cy_pvmr=pvmc_eqm}-\eqref{eq:dnwr_cy_pvmr_recursive_eqm}
become 
\begin{equation}
\label{eq:dnwr_cy_p_mc_optimal_eqm}
p_t = \frac{w_t}{\alpha h_t^{\alpha-1}},
\end{equation}
for all $t$.  This completes the proof that a policy that fully stabilizes the price of nontradables achieves the flexible-price equilibrium. 

To obtain the optimal devaluation rate, use the flexible-price processes for $c^T_t$ and $y^N_t$ to construct the flexible-price real exchange rate, $p_t$, from equilibrium condition~\eqref{eq:dnwr_cy_demand_for_nontradables_eqm}. Then use this process for $p_t$ to evaluate equilibrium condition~\eqref{eq:dnwr_cy_epsilon_eqm}, which gives the optimal
 process $\epsilon_t$, 
\[
\epsilon_t = \frac{p_{t-1}}{p_t}.
\]
It follows that any shock that causes a real exchange rate depreciation
(a fall in $p_t$)
 in the flexible-price equilibrium, such as a contraction in tradable output $y^T_t$ or an increase in the interest rate $r_t$, must be accompanied  by a devaluation. Thus, the present model shares two important predictions with the models of downward nominal wage rigidity or downward nominal price rigidity studied earlier in this chapter. First, stabilization of the nominal price of nontradables is optimal. And second, contractions are devaluatory. 

\subsection{The Open Economy New-Keynesian Phillips Curve}

At this point it is of interest to digress and derive a linearized version of 
the equilibrium price setting conditions 
\eqref{eq:dnwr_cy_pvmr=pvmc_eqm}-\eqref{eq:dnwr_cy_pvmr_recursive_eqm}, which gives rise to an open-economy version of the familiar  new-Keynesian Phillips curve. 


 Define $
\widehat{x}_t = \ln (x_t/x),
$
 where $x$ denotes the deterministic steady-state value of $x_t$. We perform the log-linearization around the deterministic steady state with $\pi^N=1$. The log-linear version of \eqref{eq:dnwr_cy_pvmc_recursive_eqm}
 is
\[
\widehat{pvmc}_t 
= -\frac{\mu}{\alpha} \widehat{\tilde{p}}^N_t
+(1-\beta\theta)
\left(
\frac1{\alpha}\widehat{y}^N_t
+\widehat{w}_t
\right)
+ \beta\theta
E_t
\left[
\widehat{\lambda}_{t+1}
-\widehat{\lambda}_t 
+ \frac{\mu}{\alpha}
\left(
\widehat{\tilde{p}}^N_{t+1}
+\widehat{\pi}^N_{t+1}
\right)
+ \widehat{pvmc}_{t+1}
\right].
\]
The  log-linear version of~\eqref{eq:dnwr_cy_piN_eqm} is
\[
\widehat{\tilde{p}}^N_t
= \frac{\theta}{1-\theta} \widehat{\pi}^N_t.
\]
Combine the above two expressions to eliminate  $\widehat{\tilde{p}}^N_t
$, which yields
\[
\widehat{pvmc}_t 
= -\frac{\mu}{\alpha}
\frac{\theta}{1-\theta}
 \widehat{\pi}^N_t
+(1-\beta\theta)
\left(
\frac1{\alpha}\widehat{y}^N_t
+\widehat{w}_t
\right)
+ \beta\theta
E_t
\left[
\widehat{\lambda}_{t+1}
-\widehat{\lambda}_t 
+ \frac{\mu}{\alpha}
\frac1{1-\theta}
\widehat{\pi}^N_{t+1}
+ \widehat{pvmc}_{t+1}
\right].
\]
The log-linearization of \eqref{eq:dnwr_cy_pvmr_recursive_eqm} is 
\[
\widehat{pvmr}_t 
= (1-\mu)
\frac{\theta}{1-\theta}
 \widehat{\pi}^N_t
+(1-\beta\theta)
\left(
\widehat{y}^N_t
+\widehat{p}_t
\right)
+ \beta\theta
E_t
\left[
\widehat{\lambda}_{t+1}
-\widehat{\lambda}_t 
- (1-\mu)\frac1{1-\theta}
\widehat{\pi}^N_{t+1}
+ \widehat{pvmr}_{t+1}
\right].
\]
And the log-linearization of 
\eqref{eq:dnwr_cy_pvmr=pvmc_eqm} is 
\[
\widehat{pvmr}_t = \widehat{pvmc}_t.
\]
Using this expression in the
linearized versions of 
\eqref{eq:dnwr_cy_pvmr_recursive_eqm} and \eqref{eq:dnwr_cy_pvmc_recursive_eqm}
yields
\[
0
=
\left(-\frac{\mu}{\alpha}-1+\mu\right)
\frac{\theta}{1-\theta}
 \widehat{\pi}^N_t
+(1-\beta\theta)
\left[
\left( \frac1{\alpha}-1\right)
\widehat{y}^N_t
+ \widehat{w}_t
-\widehat{p}_t
\right]
+ \beta
E_t
\left[
\left(
\frac{\mu}{\alpha}+1-\mu
\right)\frac\theta{1-\theta}
\widehat{\pi}^N_{t+1}.
\right]
\]
Now let 
\[
\widehat{mc}_t \equiv 
\left( \frac1{\alpha}-1\right)
\widehat{y}^N_t
+ \widehat{w}_t
-\widehat{p}_t
\]
 as
the log-deviation of aggregate marginal cost in the nontraded sector from  steady-state. 
Combining this definition with the above expression yields
\begin{equation}
\label{eq:dnwr_cy_new_keynesian_phillips_curve}
\widehat{\pi}^N_t
=
 \beta
E_t
\widehat{\pi}^N_{t+1} 
+
\kappa
\widehat{mc}_t
,\end{equation}
with 
$
\kappa = \left[\frac{(1-\theta)(1-\beta\theta)}
{ \theta \left(
\frac{\mu}{\alpha}+1-\mu
\right)
}
\right]>0.
$
This expression is known as the new Keynesian Phillips curve. 
It relates current inflation to  expected future inflation and marginal costs. 

\subsection{Crisis Dynamics in the Calvo Model}

In this section, we numerically characterize the equilibrium dynamics triggered by an external crisis. The exercise is identical to the one conducted in  section~\ref{sec:dnwr_crises}. As in that section, we compare two alternative exchange-rate policies, the optimal one and a currency peg. 

To simulate equilibrium dynamics, we follow much of the new-Keynesian literature and employ perturbation methods.\footnote{Alternatively, one could apply global methods as we did earlier in this chapter. The present setup, however, is somewhat more complex, as it includes two forward-looking equations in addition to the consumption Euler equation,  namely equations \eqref{eq:dnwr_cy_pvmc_recursive_eqm} and  \eqref{eq:dnwr_cy_pvmr_recursive_eqm}. 
Both models have the same number of state variables. In the Calvo model the  state variables are $d_t$ and $s_{t-1}$, and in the model of the previous sections they are $d_t$ and $w_{t-1}$.}
Since the perturbation method cannot handle occasionally binding constraints, we must eliminate the borrowing limit,  \eqref{eq:dnwr_cy_d<=dbar_eqm}.
In its place, we assume an externally debt-elastic interest rate, following   Schmitt-Groh\'e and Uribe (JIE, 2003) (see also the exposition in  section~\ref{sec:soe-rbc-edeir} of chapter~\ref{chapter:soe-rbc}). 
Specifically, we assume that 
\begin{equation}
\label{eq:dnwr_cy_edeir}
r_t = r^*_t + \psi
\left[
e^{d_{t+1}-\overline{d}}-1
\right],
\end{equation}
where $r_t$ is now an endogenous variable and 
 $r^*_t$ is an exogenous variable. 

A competitive equilibrium is then 
 a set of processes $c^T_t$, $r_t$, $p_t$, $h_t$ , $\lambda_t$, $w_t$, $\tilde{p}^N_t$, $y^N_t$, $s_t$,  $d_{t+1}$, $\pi^N_t$, $pvmc_t$, and $pcmr_t$ satisfying 
\eqref{eq:dnwr_cy_resource_constraint_eqm},
\eqref{eq:dnwr_cy_lambda_eqm}, 
\eqref{eq:dnwr_cy_demand_for_nontradables_eqm}-%\eqref{eq:dnwr_cy_piN_eqm}, 
\eqref{eq:dnwr_cy_epsilon_eqm}, 
\eqref{eq:dnwr_cy_edeir}, 
and 
\begin{equation}
\label{eq:dnwr_cy_Euler_edeir_eqm}
\frac{\lambda_t}{1+r_t} 
= \beta E_t \lambda_{t+1},
\end{equation}
given exogenous processes $y^T_t$ and $r^*_t$, 
 initial conditions $d_{0}$ and $s_{-1}$, 
an exchange-rate policy,  $\epsilon_t$, and the  labor subsidy policy~\eqref{eq:dnwr_cy_tau_optimal}. 


\begin{table}
\caption{Calibration of the Calvo Model}
\label{table:dnwr_cy_calibration}

\medskip

\begin{center}
\begin{tabular}{ccl}
\hline\hline
Parameter&Value&Description\\
\hline
$\theta$&0.7&Probability of no price change in nontraded sector\\
$\mu$&6&Elasticity of subst. across intermediate nontradables\\
$\sigma$&2&Inverse of intertemporal elasticity of consumption\\
$\beta$&$1.0316^{-1}$&Quarterly subjective discount factor\\
$\varphi$&1.11&Preference parameter\\
$\chi$&1&Preference parameter\\
 $\bar{h}$&3&Labor endowment\\
$a$&0.26&Share of tradables\\
 $\xi$&0.5&Elasticity of substitution between tradables and nontradables\\
 $\alpha$&0.75&Labor share in nontraded sector\\
$\psi$&0.0000335&Parameter of debt-elastic interest rate\\
$\overline{d}$&   2.9014&Parameter of debt-elastic interest rate\\
$y^T$&1&Steady-state tradable output\\
$r^*$&0.0316&Steady-state interest rate (quarterly)\\
\hline\hline
\end{tabular}
\end{center}
\begin{quote}
{\small Note. The time unit is a quarter. 
The values assigned to $\sigma$, $a$, $\xi$, $\alpha$, $y^T$, and $r^*$ are taken from table~\ref{table:dnwr_calibration}. 
The values assigned to $\varphi$, $\chi$, and $\bar{h}$ are taken from section~\ref{sec:dnwr_endogenous_labor}}.
\end{quote}
\end{table}
Table~\ref{table:dnwr_cy_calibration} summarizes the calibration of the  model. 
The time unit is one quarter. 
As in previous sections, we adopt the functional forms 
$$
U(c) = \frac{c^{1-\sigma}-1}{1-\sigma},
$$
$$
A(c^T,c^N) = 
\left[
a (c^T)^{1-\frac{1}{\xi}}
+ (1-a) (c^N)^{1-\frac{1}{\xi}}
\right]^{\frac1{1-\frac1{\xi}}},
$$
and 
$$
F(h) = h^{\alpha},
$$
with $\sigma=1/\xi=2$, $a = 0.26$, and $\alpha=0.75$. 
As in section~\ref{sec:dnwr_endogenous_labor}, we assume that the marginal disutility of labor takes the form 
\[
V'(h) = \varphi (\bar{h}-h)^{-\chi},
\]
with $\varphi=1.11$,  $\bar{h}=3$, and $\chi=1$. 
These parameter values ensure 
a Frisch wage elasticity of labor supply of 2, 
a share  of time spent working of one third,   and a steady-state value of 
hours worked  of unity.  
The bivariate driving forces $(y^T_t,r^*_t)$ is assumed to follow the  AR(1) process
\begin{equation}
\label{eq:dnwr_cy_AR(1)_yT_and_r}
\left[
\begin{array}{c}
\ln  y^T_t\\
\ln \frac{1+r^*_t}{1+r^*}
\end{array}
\right]
=
\left[
\begin{array}{cc}
0.79&-1.36\\
-0.01&0.86
\end{array}
\right] 
\left[
\begin{array}{c}
\ln  y^T_{t-1}\\
\ln \frac{1+r^*_{t-1}}{1+r^*}\end{array}
\right]
+ 
\sigma_{\epsilon}
\Gamma\,
\epsilon_t,
\end{equation}
with 
\[
\sigma_{\epsilon} = 1;
\quad \quad 
\Gamma\Gamma' = \left[
\begin{array}{cc}
0.00123&-0.00008\\
-0.00008&0.00004
\end{array}
\right]; \quad \quad  r=0.0316,
\]
which is identical to the one   given in equation~\eqref{eq:dnwr_AR(1)_yT_and_r}, with $r^*_t$ and $r^*$ taking the places of $r_t$ and $r$, respectively. 
As earlier in this chapter, the primary concern here is to characterize the 
the cyclical behavior of the economy under alternative exchange-rate policies and not to ascertain the contribution of particular shocks to short-run fluctuations.\footnote{This issue is taken up in exercise~\ref{exercise:dnwr_cy_var_decomp}.} For this reason, 
the results presented below obtain under 
any matrix $\Gamma$ satisfying the above restriction.  We introduce the scalar $\sigma_{\epsilon}$, because it facilitates the Taylor expansion of the equilibrium conditions around the deterministic steady state $\sigma_{\epsilon}=0$. This will turn out to be  particularly useful later in the section, when we approximate the model up to second order  to compute  welfare. 


There are two new parameters in the present model, $\theta$, the probability  of not being able to change the price in a given period, and $\psi$, the parameter governing the debt elasticity of the interest rate. 
There is scanty empirical evidence on the frequency of nominal price changes in nontraded goods in emerging countries. A reason for this scarcity is likely given by the fact that for much of the postwar period  emerging countries experienced relatively high levels of inflation. An exception is Gagnon (2009), who 
using Mexican micro consumer-price data 
reports a monthly frequency of nominal price changes in nonregulated services in Mexico 
slightly below  10 percent 
over two low-inflation periods,   one preceding and the other following the 1994 `Tequila' crisis. This evidence suggests a quarterly probability of no price change  of nontradables of 70 percent. Accordingly, we set $\theta$ equal to 0.7. 
We calibrate $\psi$ to match the 
unconditional standard deviation of (percentage deviations from trend of) tradable consumption of 18.5 percent  implied by all of the models studied 
earlier in this chapter.\footnote{Those models do not incorporate a debt-elastic interest rate. Instead, stationarity is induced by setting $\beta(1+r)<1$ and approximating  the equilibrium with global methods  
 (see section~\ref{sec:dnwr_calibration}).} The implied value of $\psi$ is 0.0000335. 

We
are now ready to quantitatively characterize the behavior of the model during a large external crisis.  The  definition of a large crisis follows the one given in section~\ref{sec:dnwr_definition_crisis}. Specifically, 
an external crisis that starts in  period $t$ is   a situation in which tradable output is at or above trend in quarter $t$ and at least two standard deviations below trend in quarter $t+10$.  We   simulate the model for three million quarters and identify windows $(t-10,t+30)$ in which  movements in traded output  conform to the definition of an external crisis.  Then, for each variable of interest
we average all  windows
 and subtract the  respective mean taken over the entire sample of three million periods. 
The beginning of  the typical crisis  is normalized at $t=0$.  The behavior of the exogenous variables $y^T_t$ and $r^*_t$ are displayed in figure~\ref{fig:dnwr_plot_yT_r}.  The large decline in tradable output in the typical crisis is coupled with a large increase in the country interest rate due to the strong negative correlation  between  $y^T_t$ and $r^*_t$ present in Argentine data. 

\begin{figure}

\begin{center}

\caption{Crisis Dynamics in the Calvo Model: The Role Of Exchange-Rate Policy}

\label{fig:dnwr_cy_plot_ir}

\includegraphics[width=5.5in]{dnwr_cy_plot_ir.eps}

\rule{2cm}{1mm} Currency Peg
  \, \, \,
\rule{0.22cm}{1mm} 
\rule{0.22cm}{1mm}
\rule{0.22cm}{1mm}
Optimal Exchange Rate Policy 

\end{center}

{\small Note.
This graph is produced by the Matlab programs in  calvo.zip available online  with the materials for this chapter. 
}

%produced with 
%c:\data\uribe\dnwr\calvo\typical_crisis.m
\end{figure}
Figure~\ref{fig:dnwr_cy_plot_ir} displays the behavior of the endogenous variables of the model during the typical external crisis under two alternative exchange-rate policies, a currency peg, shown with solid lines, and the optimal exchange rate policy, shown with dashed lines.
 Comparing figures~\ref{fig:dnwr_plot_ir} and \ref{fig:dnwr_cy_plot_ir}
suggests that the dynamic response of the Calvo model is similar to that implied by the model with downward nominal wage rigidity. 
 Because preferences are separable in tradable and nontradable consumption ($\sigma=1/\xi$), the  responses of consumption of tradables  
(row 3 column 2), 
 the level of external debt (not shown), and the level of the trade balance (not shown) are identical under the peg and the optimal exchange-rate policy. The crisis causes a contraction in tradable consumption of about 25 percent. 
 

The primary role of the optimal exchange rate policy is to prevent the external crisis from spreading to the nontraded sector. In fact, employment is unaffected by the crisis as shown in the top left panel of the figure. To see this analytically, combine equilibrium conditions
~\eqref{eq:dnwr_cy_lambda_eqm},
~\eqref{eq:dnwr_cy_demand_for_nontradables_eqm},
\eqref{eq:dnwr_cy_supply_of_labor_eqm},
and ~\eqref{eq:dnwr_cy_p_mc_optimal_eqm}, and use the assumption  $\sigma=1/\xi$ to obtain
\[
\alpha(1-a) (\overline{h}-h_t) = \varphi h_t^{1-\alpha+\alpha/\xi} ,
\]
which implies that $h_t$ is constant over time under the optimal exchange-rate policy. This prediction is akin to the full-employment result obtained under optimal exchange-rate policy in the model with downward nominal wage rigidity expounded earlier in this chapter.  The monetary authority manages to maintain full  employment by implementing large devaluations throughout the crisis in the order of 30 to 40 percent per year. The  devaluations allow the price of nontradables, which is rigid in nominal terms, to decline in real terms, that is,  they allow the real exchange rate to depreciate. In turn, the depreciation induces an expenditure switch away from tradables and toward nontradables, which is the optimal adjustment under flexible prices.  
By contrast, under the currency peg, employment falls by about 15 percent.  The reason is that 
due to price stickiness, 
throughout the crisis the relative price of nontradables stays too high to be consistent with firms operating at full capacity. 

Finally, like in the model with nominal price rigidity of section~\ref{sec:dnwr_price_rigidity_nontraded_sector}, the real wage displays a significant downward adjustment during the crisis both under the optimal exchange-rate policy and under the peg. In fact, under the peg, the real wage falls by more than under the optimal policy. The reason is that wages are fully flexible, whereas firms are rationed by weak demand, 
which 
causes the market clearing real wage to plummet. 
This prediction of the sticky-price model is counterfactual. 
The lack of sizable downward 
wage 
adjustment in the private sector observed during crises in fixed-exchange-rate economies or in monetary unions (see, for instance, the cases of 
Argentina 1998-2001 and 
the periphery of Europe 2008-2011 shown in figures~\ref{fig:dnwr_europe_data} and \ref{fig:dnwr_arg_unemployment_wage_2001}) suggests that downward wage rigidity is a more plausible source of nominal rigidity for  understanding the adjustment of emerging economies  with fixed exchange rates. 
