%slides_soe_rbc.tex
%Slides for Chapter 4 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 4   

the open economy 

real-business-cycle model}

\vspace{1in}

\end{minipage}
\end{slide}

\topleft{\textit{Open Economy Macroeconomics, Chapter 4} }
\topright{M.\ Uribe and S.\ Schmitt-Groh\'e}


\begin{slide}
{\bf Motivation:}

In the previous chapter, we built a model of the open economy driven by productivity shocks and argued that it  
 can capture the observed
countercyclicality of the trade balance.  We also established that
two features of the model are important for  making this prediction
possible. First, productivity shocks must be sufficiently
persistent. Second, capital adjustment costs must not be too strong.
In this chapter, we
ask more questions about the ability of that model to explain observed business cycles. In particular, we 
ask whether it can explain
the sign and magnitude of  business-cycle indicators, such as the standard deviation, serial correlation, and correlation with output of output, consumption, investment, the trade balance, and the current account. 


\end{slide}

\begin{slide}
{\bf The Small Open Economy RBC Model}

To make the models studied in chapters 2 and 3  more empirically realistic and to give them a better chance to account for observed business-cycle regularities add:

\begin{enumerate}
  \item  endogenous labor supply and demand
  \item  uncertainty in the technology shock process
  \item  capital depreciation.
\end{enumerate}

 The resulting theoretical framework is known as the Small Open Economy Real-Business-Cycle
model, or, succinctly, the SOE-RBC model.



\end{slide}

\begin{slide}

{\bf The Household's Maximization Problem}
\begin{equation}
\label{eq:soe-rbc-utility-edeir}
\tag{4.1}
E_0 \sum_{t=0}^{\infty} \beta^t U(c_t,h_t)
\end{equation}
subject to
\begin{equation}
\label{eq:soe-rbc-bc-edeir}
\tag{4.2}
c_t + i_t +  \Phi(k_{t+1}-k_t)+ (1+r_{t-1})  d_{t-1} = 
y_t + d_t
\end{equation}
\begin{equation}
\label{eq:soe-rbc-y-edeir}
\tag{4.3}
y_t = A_tF(k_t,h_t)
\end{equation}
\begin{equation}
\label{eq:soe-rbc-i-edeir}
\tag{4.4}
k_{t+1} = (1-\delta)k_t + i_t
\end{equation}
\begin{equation}
\label{eq:soe-rbc-no-Ponzi-edeir}
\tag{4.5}
\lim_{j \rightarrow \infty}
E_t
\frac{d_{t+j}}{\prod_{s=0}^j(1+r_s)} \le 0
\end{equation}

Capital adjustment cost, $\Phi(0)=\Phi'(0) =0$; $\Phi''(0)>0 $
\end{slide}

\begin{slide}
Additions/differences to the model analyzed in Chapter 3
\\
$\bullet$ endogenous labor supply, $U(c_t, h_t)$
\\
$\bullet$ endogenous labor demand, $F(k_t, h_t)$
\\
$\bullet$ uncertainty, $A_t$ is stochastic
\\
$\bullet$ the interest rate is no longer constant, $r_{t}\neq r$
\\
$\bullet$ depreciation,  $\delta$ no longer $ 0$
\end{slide}

\begin{slide}
{\small
{\bf Household's Optimality Conditions}
\begin{equation}
\label{eq:soe-rbc-bc-combined-edeir}
\tag{4.6}
c_t + k_{t+1}-(1-\delta)k_t+ \Phi(k_{t+1}-k_t)+ (1+r_{t-1})  d_{t-1} 
= A_tF(k_t,h_t)+ d_t
\end{equation}
\begin{equation}
\label{eq:soe-rbc-foc-d-edeir}
\tag{4.7}
\lambda_t = \beta (1+r_t)E_t\lambda_{t+1}
\end{equation}
\begin{equation}
\label{eq:soe-rbc-foc-c-edeir}
\tag{4.8}
 U_c(c_t,h_t)
= \lambda_t 
\end{equation}
\begin{equation}
\label{eq:soe-rbc-foc-h-edeir}
\tag{4.9}
-U_h(c_t,h_t) = \lambda_tA_t F_h(k_t,h_t)
\end{equation}
\begin{equation}
\tag{4.10}
\label{eq:soe-rbc-foc-k-edeir}
1+\Phi'(k_{t+1}-k_t)
= \beta
E_t \frac{\lambda_{t+1}}{\lambda_t }
\left[
A_{t+1}F_k(k_{t+1},h_{t+1})
+1-\delta + \Phi'(k_{t+2}-k_{t+1})
\right]
\end{equation}
}
\end{slide}

\begin{slide}

{\bf Inducing Stationarity: External debt-Elastic Interest Rate (EDEIR)}
\begin{equation}
\label{eq:soe-rbc-r-edeir}
\tag{4.14}
r_t = r^* + p(\widetilde{d}_t)
\end{equation}
$r^*=$  constant world interest rate\\
$p(\tilde{d_t})=$  country interest-rate premium
\\
$\tilde{d}_t=$ cross-sectional average of debt

In equilibrium cross-sectional average of debt must equal individual debt 
\begin{equation}
\tag{4.15}
\label{eq:edeir-dtilde=d}
\widetilde{d}_t = d_t
\end{equation}



{\bf Evolution of Total Factor Productivity, AR(1) process}
\begin{equation}
\tag{4.12}
\label{eq:soe-rbc-productivity-shock}
\ln A_{t+1} = \rho \ln A_t + \widetilde{\eta}  \epsilon_{t+1}
\end{equation}

\end{slide}

\begin{slide}


{\bf The Trade Balance}
\begin{equation}
\tag{4.20}
\label{eq:soe-rbc-tb-edeir}
tb_t =y_t-c_t - i_t - \Phi(k_{t+1}-k_t)
\end{equation}

{\bf The Current Account}
\begin{equation}
\tag{4.21}
\label{eq:soe-rbc-ca=tb-rd-edeir}
ca_t = tb_t - r_{t-1}d_{t-1}
\end{equation}



\end{slide}



\begin{slide}
{\bf \large Equilibrium Conditions}
{\small 

\begin{equation}
\tag{4.11}
\label{eq:soe-rbc-Uh/Uc-edeir}
-\frac{U_h(c_t,h_t)}{U_c(c_t,h_t)}
= A_t F_h(k_t,h_t)
\end{equation}
\begin{equation}
\tag{4.16}
\label{eq:soe-rbc-bc-combined-eqm-edeir}
c_t+ k_{t+1}-(1-\delta) k_t + \Phi(k_{t+1}-k_t)+[1+r^*+p(d_{t-1})]  d_{t-1} 
=A_tF(k_t,h_t) + d_t
\end{equation}
\begin{equation}
\label{eq:soe-rbc-foc-d-eqm-edeir}
\tag{4.17}
 U_c(c_t,h_t) = \beta (1+r^* + p(d_t))E_t  U_c(c_{t+1},h_{t+1})
\end{equation}
\begin{equation}
\label{eq:soe-rbc-foc-k-eqm-edeir}
\tag{4.18}
1= \beta
E_t \left\{ \frac{ U_c(c_{t+1},h_{t+1})}{ U_c(c_t,h_t)}
\frac{\left[
A_{t+1}F_k(k_{t+1},h_{t+1})
+1-\delta + \Phi'(k_{t+2}-k_{t+1})
\right]}{1+\Phi'(k_{t+1}-k_t)
}
\right\}
\end{equation}
This is a system of non-linear stochastic difference equations. It  does not have a closed form solution. We will use numerical  techniques to find a first-order accurate approximate solution around the nonstochastic steady state. This is a local approximation. (Later we will also consider a global solution method.)


[For capital, the system is a second-order difference equation as it features $k_t$, $k_{t+1}$ and $k_{t+2}$. We would like to have a system of first-order difference equations. To this end, introduce the auxiliary variable $k^f_t$ and impose
\[
k^f_t = k_{t+1}
\] 
Note that $k^f_t$ is in the information set of period $t$. This equation together with the four above equations forms a system of  stochastic first-order difference equations in the 5 unknowns: $c_t$, $h_t$, $d_{t-1}$, $k_t$, and $k^f_t$.]

}
\end{slide}


\begin{slide}
{\bf First-order accurate approximation of equilibrium dynamics around the non-stochastic steady state}



\verb7edeir_ss.m7\\
\verb7edeir_model.m7\\
\verb7edeir_run.m7\\


\end{slide}

\begin{slide}
{\bf Derivation of a first-order accurate approximation}

{\tiny Based on Schmitt-Groh\'e and Uribe JEDC 2004. }


The EDEIR model developed above gives rise to equilibrium conditions of the form
\begin{equation}
\label{eq:eqm-conditions}
E_t f(y_{t+1}, y_t, x_{t+1}, x_t) = 0
\end{equation}
where \\
$x_t=$ $n_x\times 1 $ vector of predetermined (or state) variables\\
$y_t=$ $n_y\times 1 $ vector of nonpredetermined (or control) variables\\
$x_0$ is an $n_x\times 1 $ vector of initial conditions\\
Terminal condition: $\lim_{j\rightarrow \infty} E_t 
\left[
\begin{array}{cc}
x_{t+j}& y_{t+j}
\end{array}
\right]' 
\rightarrow  
\left[
\begin{array}{cc}
\bar{x} &\bar{y}
\end{array}
\right]' 
$
\\
Let
\[
n =n_x+n_y
\]
Then we have that 
\[
f: R^{n_y}\times R^{n_y}\times R^{n_x}\times R^{n_x}\rightarrow R^n
\]
\end{slide}
\begin{slide}

A large class of dynamic stochastic general equilibrium models can be written in the form given in \eqref{eq:eqm-conditions}. And most studies in real and monetary business cycle analysis use models belonging to this class. Of course, there are also many types of  models that do not fit into that class. For example, models with occasionally binding constraints.

Partition state vector $x_t$ 
\[
x_t = \left[\begin{array}{c}
x^1_t\\
x^2_t\end{array}\right]
\]
$x^1_t=$ vector of endogenous predetermined state variables\\
$x^2_t=$ vector of exogenous state variables

We assume that the exogenous state evolves as: 
\begin{equation}
\label{eq:evolution_x2}
x^2_{t+1} = \tilde{h}( x^2_t, \sigma) + \sigma  \tilde{\eta} \epsilon_{t+1},
\end{equation}
$\sigma = $ parameter scaling the amount of uncertainty. ($\sigma = 0$ is perfect foresight.)

\end{slide}
\begin{slide}

Solution to models that are described by  \eqref{eq:eqm-conditions}
 and \eqref{eq:evolution_x2} can then be expressed as:
\begin{equation}
\label{eq:solutiony}
y_t = \hat{g}(x_t)
\end{equation}
\begin{equation}
\label{eq:solutionx}
x_{t+1} = \hat{h}(x_t) + \sigma \eta  \epsilon_{t+1}
\end{equation}
where
\[
\eta = 
\left[
\begin{array}{c}
\emptyset
\\
\tilde{\eta}
\end{array}
\right].
\]



The shape of the functions $\hat{h}$ and $\hat{g}$ will in general depend on the amount of uncertainty in the economy. 
\end{slide}
\begin{slide}
Key idea of perturbation: parameterize the amount of uncertainty as follows
\begin{equation*}
\label{eq:solutiony}
\hat{g}(x_t) = g(x_t,\sigma)
\quad \mbox{where } \quad g:R^{n_x}\times R^+ \rightarrow R^{n_y}
\end{equation*}
\begin{equation*}
\hat{h}(x_t) =h(x_t,\sigma) \quad \mbox{where } 
\quad h:R^{n_x}\times R^+ \rightarrow R^{n_x}
\end{equation*}

Then we can write the solution to the model described by \eqref{eq:eqm-conditions}
 and \eqref{eq:evolution_x2} as 
\begin{eqnarray}
\label{eq:solutiony_sigma}
y_t &=& g(x_t,\sigma)\\
\label{eq:solutionx_sigma}
x_{t+1} &=& h(x_t,\sigma) + \sigma \eta  \epsilon_{t+1}
\end{eqnarray}

Perturbation methods perform a  \textit{local} approximation of  $g(x,\sigma)$ and $h(x,\sigma)$ around a particular point $(\bar{x}, \bar{\sigma})$
\end{slide}
\begin{slide}
First-order  Taylor series expansion of $g$ and $h$ around $(x,\sigma)=(\bar{x},\bar{\sigma})$  
\begin{eqnarray*}
g(x,\sigma) 
&=& 
g(\bar{x},\bar{\sigma})+ g_x(\bar{x},\bar{\sigma})(x-\bar{x}) + g_{\sigma}(\bar{x},\bar{\sigma}) (\sigma-\bar{\sigma}) + h.o.t.
\\  
h(x,\sigma)
&=& 
h(\bar{x},\bar{\sigma}) +  h_x(\bar{x},\bar{\sigma}) (x-\bar{x}) + h_{\sigma}(\bar{x},\bar{\sigma})(\sigma-\bar{\sigma}) + h.o.t.
\end{eqnarray*}
$h.o.t.=$ higher order terms\\
Unknowns: $g(\bar{x},\bar{\sigma}),  g_x(\bar{x},\bar{\sigma}), g_{\sigma}(\bar{x},\bar{\sigma}), h(\bar{x},\bar{\sigma}),   h_x(\bar{x},\bar{\sigma}) ,  h_{\sigma}(\bar{x},\bar{\sigma})$
\\
To identify these terms, substitute the proposed solution  given by equations~(\ref{eq:solutiony_sigma}) and (\ref{eq:solutionx_sigma}) into  equation~(\ref{eq:eqm-conditions}), and define 
\begin{eqnarray}
\label{eq:soe_rbc_slides_F(x,sig)}
F(x,\sigma) 
&\equiv&
E_t f( g(h(x,\sigma)+\eta \sigma \epsilon',\sigma),g(x,\sigma),h(x,\sigma)+\eta \sigma\epsilon',x)
\label{eq:sol_sub}\\
&=& 
0.
\nonumber
\end{eqnarray}
Here we are dropping time subscripts. We use a prime to indicate variables dated in period $t+1$. 

Because $F(x,\sigma)$ must be equal to zero for any possible values of $x$ and $\sigma$, it must be the case that the derivatives of any order of $F$ must also be equal to zero. Formally, 
\begin{equation}
\label{eq:dF=0} F_{x^k{\sigma}^j}(x,\sigma)=0 \quad \forall x,\sigma,j,k,
\end{equation}
where $F_{x^k{\sigma}^j}(x,\sigma)$ denotes the derivative of $F$ with respect to  $x$ taken  $k$ times and with respect to $\sigma$ taken  $j$ times. 

\end{slide}
\begin{slide}
{\bf What point to approximate around?}\\
We need to evaluate the derivatives of $F(x,\sigma)$,  $F_{x^k{\sigma}^j}(x,\sigma)$, at the point we are approximating the equilibrium around. In general this is difficult if not impossible. But there are some points for which evaluation of those derivatives is possible. 

One such point is the non-stochastic steady state, $(x,\sigma)= (\bar{x}, 0)$, where $\bar{x}$ denotes the non-stochastic steady state value of $x_t$. For this point we know: $y_t=\bar{y}$, $y_{t+1}=\bar{y}$, and $x_{t+1}=\bar{x}$, where $\bar{y}$ denotes the non-stochastic steady state of $y_t$. 


Another point one can evaluate the derivatives of $F(x,\sigma)$ at is  $x_t\neq \bar{x}$ and $\sigma=0$. This works in cases in which one can find the exact deterministic solution of a model.  In that case one can find $y_t$, $y_{t+1}$ and $x_{t+1}$ for $(x_t, \sigma)=(x_t, 0)$ but needs to resort to approximation techniques to characterize the solution to the stochastic version of the economy. 

For the remainder of this chapter we will focus on approximation around the non-stochastic steady state  $(x,\sigma) = (\bar{x}, 0)$.

\end{slide}


\begin{slide}
Let's write again the first-order  Taylor series expansion of $g$ and $h$ but this time around the non-stochastic steady state, $(x,\sigma)=(\bar{x},0)$  
\[
g(x,\sigma) 
= 
g(\bar{x},0)+ g_x(\bar{x},0)(x-\bar{x}) + g_{\sigma}(\bar{x},0) (\sigma-0)
\]
\[
h(x,\sigma)
= 
h(\bar{x},0) +  h_x(\bar{x},0) (x-\bar{x}) + h_{\sigma}(\bar{x},0)(\sigma-0) 
\]

We wish to find:

$g(\bar{x},0)$

$g_x(\bar{x},0)$

$g_{\sigma}(\bar{x},0)$

$h(\bar{x},0)$

$h_x(\bar{x},0)$

$h_{\sigma}(\bar{x},0)$

\end{slide}


\begin{slide}
{\bf Find $g(\bar{x},0)$ and $h(\bar{x},0)$}

From  \eqref{eq:solutiony_sigma}
\[
g(\bar{x},0) = \bar{y}
\]

From  \eqref{eq:solutionx_sigma} 
\[
h(\bar{x},0) = \bar{x}
\]


\end{slide}


\begin{slide}
{\bf Find $h_{\sigma}$ and $g_{\sigma}$}

Recall \eqref{eq:soe_rbc_slides_F(x,sig)} 
\begin{eqnarray*}
0&=& F(x,\sigma) \\
&=&
E_t f( g(h(x,\sigma)+\eta \sigma \epsilon',\sigma),g(x,\sigma),h(x,\sigma)+\eta \sigma\epsilon',x)
\end{eqnarray*}
The first derivative of $F(x,\sigma)$ with respect to $\sigma$ evaluated at $(x,\sigma)=(\bar{x}, 0)$
\begin{eqnarray*}
0
&=&
F_{\sigma}(\bar{x},0) 
\\
&=&
 f_{y'}(\bar{y}, \bar{y},\bar{x},\bar{x}) \left[
g_x(\bar{x},0)  h_{\sigma}(\bar{x},0) + g_{\sigma}(\bar{x},0)
\right]
\\&&+ 
f_{y}(\bar{y}, \bar{y},\bar{x},\bar{x}) 
 g_{\sigma}(\bar{x},0)
\\&&+
f_{x'}(\bar{y}, \bar{y},\bar{x},\bar{x}) 
  h_{\sigma}(\bar{x},0) 
\end{eqnarray*}

Let $f_{i} \equiv  f_{i}(\bar{y}, \bar{y},\bar{x},\bar{x}) $ for $i=y','y,x',x$ 

Note that we can evaluate $f_i$ because we know the function $f$ and we know the steady state $(\bar{y},\bar{x})$
\end{slide}
\begin{slide}

Rearrange to obtain
\[
\left[
\begin{array}{cc}f_{y'}g_x + f_{x'}& f_{y'}+f_{y}\end{array}
\right]
\left[
\begin{array}{c}
h_{\sigma}\\
g_{\sigma}
\end{array}
\right]
=
0
\]
This is a linear homogenous equation in n unknowns. For it to have a unique solution it must be that 
\begin{equation}
\left[
\begin{array}{c}
h_{\sigma}\\
g_{\sigma}
\end{array}
\right]
=
\left[
\begin{array}{c}
\emptyset\\
\emptyset
\end{array}
\right]
\end{equation}
This is an important result. It says that up to first-order accuracy one need not correct the constant term or the slope term of the approximation for the presence of uncertainty. The policy function is the same as under perfect foresight but for the additive stochastic error term. (the solution displays the certainty equivalence principle)

Up to first order accuracy the solution is: 
\begin{eqnarray*}
y_t &=& \bar{y}+ g_x(\bar{x},0)(x-\bar{x}) 
\\  
x_{t+1} &=& \bar{x} +  h_x(\bar{x},0) (x-\bar{x})+ \sigma \eta \epsilon_{t+1}
\end{eqnarray*}


\end{slide}
\begin{slide}


Consider the unconditional expectations of $x_t$ of the first-order accurate approximation: 
\begin{eqnarray*}
E(x_t)
& = & 
E
\left\{ 
\bar{x} + h_x(\bar{x},0) (x_t-\bar{x}) 
 + h_{\sigma}(\bar{x}, 0) (\sigma -0)
\right\}
\\
&=&
\bar{x} + h_x(\bar{x},0) ( E(x_t)-\bar{x})  + 0
\end{eqnarray*}

It follow that up to first order accuracy: 
\[
E x_t = \bar{x} \quad \mbox{and} \quad E y_t = \bar{y}
\]
or in words the unconditional expectation is the same as the mean. Hence first-order accurate approximations will not be helpful to approximate average risk premia (they would all be zero) or the average welfare associated with different monetary or fiscal policy that all give rise to the same nonstochastic steady state (all policies give the same welfare in the steady state). 
\end{slide}

\begin{slide}
{\bf Find $h_x(\bar{x},0)$ and $g_x(\bar{x},0)$}

Start again from \eqref{eq:soe_rbc_slides_F(x,sig)}
\begin{eqnarray*}
0&=& F(x,\sigma) \\
&=&
E_t f( g(h(x,\sigma)+\eta \sigma \epsilon',\sigma),g(x,\sigma),h(x,\sigma)+\eta \sigma\epsilon',x)
\end{eqnarray*}
The first derivative of $F(x,\sigma)$ with respect to $x$ evaluated at $(x,\sigma)=(\bar{x}, 0)$
\begin{eqnarray*}
0
&=&
F_{x}(\bar{x},0) 
\\
&=&
f_{y'} g_x h_{x}
+
f_{y} g_{x}
+
f_{x'}h_{x}
+ 
f_{x}
\end{eqnarray*}
Rearrange to
\[
\left[
\begin{array}{cc}f_{x'} & f_{y'}\end{array}
\right]
\left[
\begin{array}{c}
I\\
g_{x}
\end{array}
\right]
h_x = -
\left[
\begin{array}{cc}f_{x} & f_{y}\end{array}
\right]
\left[
\begin{array}{c}
I\\
g_{x}
\end{array}
\right]
\]
To solve this expression for $h_x$ and $g_x$ use a Schur decomposition. We describe this in detail in Appendix 4.14 of the Chapter. 
The Matlab program \verb7gx_hx.m7 posted on our website with the materials for Chapter 4 performs this step. 

\end{slide}

\begin{slide}
{\small Taking stock: 

Thus far we have presented a first-order accurate approximation technique. 

Next we will discuss how to implement this technique in the case of the EDEIR model. 

It should be clear that an important element of the implementation is 
finding numerical values for the derivatives of the function $f(\cdot)$ at the  non-stochastic steady state. 

Our approach is to use the Symbolic Math Toolbox of Matlab to do most of the work. This has several advantages. One is that the room for error is much smaller and the other is that it eliminates any tedious linearization by hand. 

To allow a Symbolic Math toolbox to implement the linearization it is convenient to specify functional forms for the utility, production, country premium, and adjustment cost functions. What will matter, given that we perform a first-order approximation to the equilibrium conditions, is at most the first and second derivatives of those functions. 


}
\end{slide}

\begin{slide}
{\bf Functional Forms}


{\bf Period utility function } 
\[
U(c,h) = 
\frac{
\left(
c-\omega^{-1} h^{\omega}
\right)^{1-\sigma}
-1
}
{1-\sigma};
\quad \omega>1; \sigma>0
\]
{\bf Debt-elastic interest rate} 
\[
p(d) = \psi \left(e^{d-\bar{d}} -1 \right); \quad \psi >0 
\]
{\bf Production function}
\[
F(k,h) = k^{\alpha}h^{1-\alpha}; \quad \alpha \in (0,1)
\]
{\bf Adjustment cost function}
\[
\Phi(x) = \frac{\phi}{2} x^2; \quad \phi>0
\]
6 structural parameters: $\sigma$, $\omega$, $\psi$, $\bar{d}$, $\alpha$, $\phi$
\end{slide}

\begin{slide}



{\bf Characterizing the Deterministic Steady State}

The steady state is the quadruple $(d, k, c , h)$ satisfying 

\begin{equation}
\tag{4.11'}
\label{eq:soe-rbc-Uh/Uc-edeir_ss}
-\frac{U_h(c,h)}{U_c(c,h)}
= A F_h(k,h)
\end{equation}
\begin{equation}
\tag{4.16'}
\label{eq:soe-rbc-bc-combined-eqm-edeir_ss}
c+ \delta k + (r^*+p(d))  d
=A F(k,h ) 
\end{equation}
\begin{equation}
\label{eq:soe-rbc-foc-d-eqm-edeir_ss}
\tag{4.17'}
1 = \beta (1+r^* + p(d))
\end{equation}
\begin{equation}
\label{eq:soe-rbc-foc-k-eqm-edeir_ss}
\tag{4.18'}
1= \beta
\left[
A F_k(k,h)
+1-\delta  
\right]
\end{equation}
\end{slide}

\begin{slide}
Using the assumed functional forms the steady state becomes
\begin{equation}
\tag{4.11''}
\label{eq:soe-rbc-Uh/Uc-edeir_ss'}
h^{\omega -1}
= A (1-\alpha) (k/h)^{\alpha}
\end{equation}
\begin{equation}
\tag{4.16''}
\label{eq:soe-rbc-bc-combined-eqm-edeir_ss'}
c+ \delta k + (r^*+ \psi(e^{d-\bar{d}}))  d
=A (k/h)^{\alpha } h 
\end{equation}
\begin{equation}
\label{eq:soe-rbc-foc-d-eqm-edeir_ss'}
\tag{4.17''}
1 = \beta (1+r^* + \psi(e^{d - \bar{d}}-1))
\end{equation}
\begin{equation}
\label{eq:soe-rbc-foc-k-eqm-edeir_ss}
\tag{4.18''}
1= \beta
\left[
A \alpha (k/h)^{\alpha -1}
+1-\delta  
\right]
\end{equation}
This is a system of 4 equations in 4 unknown endogenous variables, $(c, d, h, k)$ and 7 unknown parameters, $\omega, \alpha, \delta, r^*, \psi,  \bar{d}, \beta$. 
 (From \eqref{eq:soe-rbc-productivity-shock}, we know that in steady state $A=1$). The model has 4 additional structural parameters, $\sigma$, $\phi$, $\rho$, $\tilde{\eta}$, which do not enter the steady state but which also need to be assigned values to. In sum, there are 11 structural parameters to be calibrated. They are: 
\[
\left[
\begin{array}{ccccccccccc}
\omega& \alpha& \delta& r^*& \beta& 
\sigma& \phi& \rho& \tilde{\eta}&    \bar{d}& \psi
\end{array}
\right]
\]
\end{slide}

\begin{slide}
We assume that the time unit is one year and calibrate the model to the Canadian economy. We adopt (almost) the same calibration as  Mendoza (1991).

\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
$ \sigma$&
$1+r^*=1/\beta$&
$\delta$&
$\alpha$&
$ \omega$&
$\phi$&  
$\rho$& 
$\sigma_{\epsilon}$&$\bar{d}$\\
\hline
  2 & 
1.04&
0.1&
0.32&
1.455 &
0.028&
0.42&
0.0129
&
0.7442  
\\
\hline
\end{tabular}

Comment:  Mendoza's model uses a different stationarity inducing device (an internal discount factor (IDF) model, which we discuss in detail in section 4.10.4) and hence that calibration does not assign a value to $\psi$. As in Schmitt-Groh\'e and Uribe (2003), we set $\psi$ to ensure that the EDEIR model predicts the same volatility of the current-account-to-output ratio as the IDF model. The value that achieves that is
\[
\psi = 0.000742
\]
Given values for the structural parameters,  the steady state can be computed using the  Matlab program \verb7edeir_ss.m7 available on the book's Website. This yields

\centerline{\begin{tabular}{|c|c|c|c|}
 \hline 
$c$&$d$&$h$&$k$\\
\hline 
    1.1170
&   0.7442
&   1.0074
&    3.3977\\
\hline 
\end{tabular}
}



\end{slide}


\begin{slide}
{\bf The Calibration Strategy}

To obtain the values of the structural parameters shown in the previous slide (and Table 4.1 in the book), three types of  restrictions were imposed: \\
{\bf Category a:} restrictions using  sources unrelated to the data that the model aims to explain,  {\bf 4} parameters: $\sigma=2$, $\delta=0.1$, $r^*=0.04$, $\beta=1/(1+r^*)$. 
\\
{\bf Category b:} restrictions to match first moments of the data that the model aims to explain, {\bf 2} parameters:  $\alpha$, $\overline{d}$
\begin{align*}
\mbox{labor share } =& 0.68\\
\mbox {trade-balance-to-output ratio } = & 0.02
\end{align*}
{\bf Category c:} restrictions to match second moments of the data that the model aims to explain, {\bf 5} parameters:  $\omega$, $\phi$,  $\psi$, $\rho$,  $\tilde{\eta}$. The second moments to be matched are: 
$\sigma_y$, $\sigma_h$, $\sigma_i $, $\sigma_{tb/y}$, corr$(\ln y_t, \ln y_{t-1})$ 
\end{slide}
\begin{slide}
How to implement this calibration strategy? The restrictions in category a translate immediately into values for structural parameters. To go from the restrictions in categories b and c to the values of the structural parameters shown in Table 4.1, one proceeds as follows: 

The labor share, $s_h$,  is defined as
\[
s_h = \frac{w h }{y}
\]
In the decentralized economy we have
\[
A_tF_2(k_t, h_t) = w_t
\]
Thus, in the steady state: 
\[
s_h = \frac{A F_2(k,h) h}{A F(k,h)}
\]
Using the assumed functional form for $F(\cdot)$ yields
\[
s_h = (1-\alpha)
\]
Hence we have that $\alpha = 1-s_h = 1-0.68 $, that is, 
\[
\alpha = 0.32
\]

\end{slide}
\begin{slide}
Let $\theta$ denote the vector of structural parameters we still need to assign numerical values to, that is, let
\[
\theta \equiv \left[
\begin{array}{cccccc} 
\omega &
\bar{d}&
\phi &
\psi &
\rho &
\tilde{\eta}
\end{array}
\right]
\]
The calibration strategy described on the previous slide consists of the following steps: 

Step 1: guess a value for each element of $\theta$

Step 2: Given the guess for $\omega$ find $h$ using~\eqref{eq:soe-rbc-foc-k-eqm-edeir_ss}
\[
\frac{k}{h} = \left(\frac{r^* + \delta}{A \alpha}\right)^{\frac1{\alpha-1}}
\]
in ~\eqref{eq:soe-rbc-Uh/Uc-edeir_ss'}
\[
h = \left((1-\alpha) A (k/h)^{\alpha}\right)^{1/(\omega-1)}
\]
With $h$ in hand,  find $k$ and $y$, as $k= (k/h) h$ and $y = A (k/h)^{\alpha} h$, respectively. 
\end{slide}
\begin{slide}

Step 3: Let $s_{tb}$ denote the trade-balance-to-output ratio. In the steady state, 
\[
s_{tb} = \frac{r^* d}{y}
\]
Solve this expression for $d$
\[
d = \frac{s_{tb} y }{r^*}
\]
Then use \eqref{eq:soe-rbc-foc-d-eqm-edeir_ss'}
 and the restriction that $\beta(1+r^*) =1$ to obtain
\[
\bar{d} = d
\]
Step 4: Find $c$ from ~\eqref{eq:soe-rbc-bc-combined-eqm-edeir_ss'}
\[
c = y - \delta k - r^* d
\]
Step 5: With the steady state values of $(c,k,d,h)$ and all structural parameters in hand compute the model's predictions for 
\[
x(\theta) \equiv
\left[
\begin{array}{ccccc}
\sigma_y & \sigma_h& \sigma_i& \sigma_{tb/y}&corr(\ln y_t, \ln y_{t-1})
\end{array}
\right]
\]

\end{slide}
\begin{slide}


Step 6: Find the distance
\[
D = |x(\theta) - x^*|
\]
where $x^*$ denotes the vector of targeted  moments observed in Canadian data

Step 7: Keep adjusting $\theta$ until $D$ is less than some threshold $D^*$.

 Comment:  In general there does not exist a $\theta$ that makes the distance $D$ exactly equal to zero. Hence one has to pick some threshold for the distance,  $D^*$.
 
\end{slide}
\begin{slide}
Before analyzing to which extend the SOE-RBC model can account for the observed Canadian business cycle, let's first study the predictions of this model regarding the prediction for which we build intuition in Chapters 2 and 3. 

In particular, there we showed  that


$\bullet$  the more persistent productivity shocks are, the more
likely an initial deterioration of the trade balance will be.

$\bullet$ the more pronounced are capital adjustment costs,
the smaller will be the initial trade balance deterioration in response
to a positive and persistent productivity shock.

$\bullet$ the more persistent the technology shock is, the higher the  volatility of consumption relative to output will be. 


The next three figures show that these analytical results do indeed hold  in the fully-fledged stochastic dynamic  open economy real-business-cycle model. 

\end{slide}

\begin{slide}
\centerline{\tiny \bf Impact response of the trade balance as a function of the persistence of the technology shock}

\centerline{\includegraphics[width = 6in]{soe_rbc_irtbys_rho_slides.eps}}
%Produced with: Z:\uribe\book\soe_rbc\edeir\edeir_sensitivity\edeir_run.m

{\tiny Notes. The figure shows the impact response of the trade balance in response to a one percent positive innovation in productivity predicted by the EDEIR model presented in  Chapter 4. The response of the trade balance  is measured in units of steady-state output.  All parameters other than $\rho$ take the values shown  in Table 4.1. The open circle indicates the baseline value of $\rho$. 



Comments: The figure shows that the more persistent  the productivity shock is  the smaller the impact response of the trade balance will be. For $\rho>0.3$, the response of the trade balance is negative, confirming the analytical results of chapters 2 and 3.

}

\end{slide}

\begin{slide}
\centerline{\tiny \bf Impact response of the trade balance as a function of capital adjustment costs}

\centerline{\includegraphics[width = 6in]{soe_rbc_irtbys_phi_slides.eps}}
%Produced with: Z:\uribe\book\soe_rbc\edeir\edeir_sensitivity\edeir_run.m

{\tiny Notes. The figure shows the impact response of the trade balance in response to a one percent positive innovation in productivity as  a function of the size of capital adjustment costs, $\phi$, predicted by the EDEIR model presented in  Chapter 4. The response of the trade balance  is measured in units of steady-state output.  All parameters other than $\rho$ take the values shown  in Table 4.1. The open circle indicates the baseline $\phi$ value. 


Comments: The figure shows that the higher capital adjustment costs are  the larger the impact response of the trade balance will be. For $\phi>0.06$, the response of the trade balance turns positive, confirming the analytical results of chapters 2 and 3.

}

\end{slide}









\begin{slide}
{\tiny \bf Relative volatility of consumption as a function of the persistence of the stationary technology shock}

\centerline{\includegraphics[width = 6in]{soe_rbc_emerging_sigmac_sigmay_slides.eps}}
%Produced with: 
%Z:\uribe\book\soe_rbc\edeir\edeir_sensitivity\edeir_run.m
{\tiny Notes. The relative standard deviation shown is that implied by the EDEIR model presented in Chapter 4. All parameters other than $\rho$ take the values shown  in Table 4.1.  The open circle indicates the baseline value of $\rho$.



Comments: The figure shows that the more persistent stationary productivity shocks are, the higher the standard deviation of consumption relative to the standard deviation of output will be, just as derived analytically in the permanent income model of Chapter 2. 

}
\end{slide}

\begin{slide}
We now turn an analysis of second moments predicted by the SOE-RBC model and compare them to the Canadian data. 

\end{slide}

\begin{slide}
\centerline{\bf Some Empirical Regularities of the Canadian Economy}
Why Canada? Because it is a small open economy and it is the economy studied in Mendoza (1991). 

\centerline{\small
\begin{tabular}{|c|c|c|c|}
\hline
Variable&\multicolumn{3}{|c|}{Canadian Data}\\
\cline{2-4}
&$\sigma_{x_t}$
&$\rho_{x_t,x_{t-1}}$
&$\rho_{x_t,GDP_t}$
\\
\hline
$y$           &  2.8&  0.61&      1\\
$c$           &  2.5&   0.7&   0.59\\
$i$           &  9.8&  0.31&   0.64\\
$h$           &    2&  0.54&    0.8\\
$\frac{tb}{y}$&  1.9&  0.66&  -0.13\\
\hline
\end{tabular}}

\centerline{\tiny Source: Mendoza AER, 1991. Annual data. Log-quadratically detrended.}

{\small

{\bf Comments}
\begin{itemize}
\item Volatility ranking:
$\sigma_{tb/y}<$$\sigma_c<$$\sigma_y<$$\sigma_i$. 
\item Consumption, investment, and hours are procyclical. 
\item The trade-balance-to-output ratios is countercyclical. 
\item All variables considered are positively serially correlated. 
\item Similar stylized facts emerge from other small developed countries (see, e.g., chapter 1). 
\end{itemize}

}

\end{slide}



\begin{slide}
\centerline{\bf Empirical and Theoretical Second Moments}


{\small
\begin{center}
\begin{tabular}{|c|ccc|ccc|ccc|}
\hline
&\multicolumn{6}{|c|}{Canadian Data}&&&\\
&\multicolumn{3}{c|}{1946 to 1985}&\multicolumn{3}{c|}{1960 to 2011}&\multicolumn{3}{|c|}{Model}\\
&$\sigma_{x_t}$ &$\rho_{x_t,x_{t-1}}$ &$\rho_{x_t,y_t}$ 
&$\sigma_{x_t}$ &$\rho_{x_t,x_{t-1}}$ &$\rho_{x_t,y_t}$
&$\sigma_{x_t}$  &$\rho_{x_t,x_{t-1}}$ &$\rho_{x_t,y_t}$\\
 \hline
$y$  &  2.8&  0.6& 1 
&   3.7&  0.9&  1
&  3.1 &  0.6&  1
\\
$c$
&  2.5&   0.7&   0.6  
&   2.2&  0.7&  0.6
&  2.7&  0.8&  0.8
\\
$i$ &  9.8&  0.3&   0.6   
&  10.3&  0.7&  0.8
&  9.0&  0.1&  0.7
\\
$h$&    2.0&  0.5&    0.8   
&   3.6&  0.7&  0.8
&  2.1&  0.6&  1
\\
$\frac{tb}{y}$
&  1.9&  0.7&  -0.1  
&   1.7&  0.8&  0.1
&  1.8&  0.5&  -0.04
\\
$\frac{ca}{y}$&    &     &        &&&&  1.4&  0.3&  0.05\\
\hline
\end{tabular}
\end{center}

}

{\small
{\bf Comments:}

$\bullet$  $\sigma_h$, $\sigma_i$, $\sigma_y$,
$\sigma_{tb/y}$, and $\rho_{y_t,y_{t-1}}$  were targeted by calibration, so no real test  here.  

$\bullet$ model correctly places $\sigma_c$ 
below $\sigma_y$ and $\sigma_i$ and above $\sigma_h$ and $\sigma_{tb/y}$.  

$\bullet$ model correctly makes $tb/y$ countercyclical. 

$\bullet$  model overestimates the correlations of hours and consumption with output. 


}
\end{slide}



\begin{slide}
\centerline{\bf Response to a Positive Technology Shock} 

%\label{fig:mendoza91} 


\centerline{\includegraphics{soe_rbc_edeir.eps}  }

\centerline{\tiny Source: Schmitt-Groh\'e and Uribe (JIE, 2003)}

\end{slide}

\begin{slide}


{\bf Comments:}
\begin{itemize}
\item Output, consumption, investment, and hours expand. 
\item The trade balance deteriorates. 
\end{itemize}
\end{slide}




\begin{slide}
{\bf 4.9 The Complete Asset Markets (CAM) Model}
\[
%\label{eq:cam-bc}
E_t r_{t+1}b_{t+1}  = b_t + y_t-c_t- i_t - \Phi(k_{t+1}-k_t),
\]
\[
%\label{eq:cam-no-Ponzi}
\lim_{j\rightarrow \infty} E_t 
q_{t+j} b_{t+j} \ge 0,
\]
\[
q_t = r_1 r_2 \dots r_t,
\]
\[
%\label{eq:cam-foc-d}
\lambda_t r_{t+1} = \beta \lambda_{t+1}.
\]
\[
%\label{eq:cam-foc-d-foreign}
\lambda^*_t r_{t+1} = \beta \lambda^*_{t+1}.
\]
\[
\frac{\lambda_{t+1}}{\lambda_t}
=
\frac{\lambda^*_{t+1}}{\lambda^*_t}.
\]
\[
\lambda_t= \xi \lambda^*_t,
\]
\[
%\label{eq:cam-risk-sharing}
\lambda_t= \psi_4,
\]

{\bf Calibration:} Set $\psi_4$ so that steady-state consumption equals steady-state consumption in the model with Uzawa preferences. 


\end{slide}

\begin{slide}
\begin{center}
{\bf \large The SOE-RBC Model With Complete Asset Markets: Predicted Second Moments}
%created with c:\uribe\teaching\econ366\lecture_notes\cam\cam_run.m
\medskip
  
\begin{tabular}{ccccccc}
\hline\hline
&\multicolumn{2}{c}{$\sigma_{x_t}$}  &\multicolumn{2}{c}{$\rho_{x_t,x_{t-1}}$} &\multicolumn{2}{c}{$\rho_{x_t,GDP_t}$}\\
variable&CAM&EDEIR&CAM&EDEIR&CAM&EDEIR\\
 \hline
$y$ &  3.1&3.1&  0.61&0.62&  1.00&1.00\\
$c$&  1.9&2.71&  0.61&0.78&  1.00& 0.84\\
$i$ &  9.1& 9.0& 0.07& 0.07& 0.66&0.67\\
$h$&  2.1&2.1&  0.61&0.62&  1.00&1.00\\
$\frac{tb}{y}$&  1.6&1.78&  0.39& 0.51& 0.13&-0.04\\
$\frac{ca}{y}$&  3.1&1.45&  -0.07&0.32&  -0.49&0.05\\
\hline\hline
\end{tabular}
\begin{quote}
{\tiny Note. Standard deviations are measured in percentage points. The columns labeled CAM are produced with the Matlab program \verb!cam_run.m! available at \\
\verb!http://www.columbia.edu/~mu2166/closing.htm!.}
\end{quote}
\end{center}
\end{slide}



\begin{slide}
{\bf \small Impulse Response to a Unit Technology Shock\\
One-Bond Versus Complete Asset Market Models}\\
%\label{figureedeir_cam} 
\centerline{\includegraphics[width=6in]{edeir_cam.eps} }
{\tiny  Dash-diamond, EDEIR model. Dash-dotted, complete-asset-market model.}

\end{slide}





\begin{slide}
{\bf \large 4.10 Alternative Ways to Induce Stationarity}

\end{slide}


\begin{slide}
{\bf 4.10.1 The Internal Debt-Elastic Interest  Rate (IDEIR) Model}
\[
%\label{eq:deir-r}
r_t = r + p(d_t),
\]
The Euler equation becomes
\[
%\label{eq:deir-foc-d}
\lambda_t = \beta [1+r + p(d_t) + 
p'(d_t)d_t]
E_t\lambda_{t+1}
\]
\[p(d) = \psi_2 \left(
e^{d-\bar{d}} -1
\right),
\]
{\bf Calibration:} Same as in the external case. Note that the steady-state value of debt is no longer equal to $\bar{d}$. Instead, $d$ solves
\[(1+d)e^{d-\bar{d}}=1 \Rightarrow d=0.4045212.\]
 

\end{slide}


\begin{slide}
\centering

{\bf Internal Debt-Elastic Interest-Rate}

\begin{tabular}{cccc}
\hline\hline
Variable& $\sigma_{x_t}$&$\rho_{x_t,x_{t-1}}$&$\rho_{x_t,GDP_t}$\\
\hline
$y$&  3.1&   0.62&       1\\
$c$&  2.5&   0.76&    0.89\\
$i$&    9&  0.068&    0.68\\
$h$&  2.1&   0.62&       1\\
$tb/y$&  1.6&   0.43&  -0.036\\
$ca/y$&  1.4&   0.31&   0.041\\
\hline\hline
\end{tabular}

\end{slide}


\begin{slide}
\centerline{\bf Internal Debt-Elastic Interest Rate Premium}
\centerline{\bf Response to a Positive Technology Shock} 
\centerline{\includegraphics{deire.eps}  }

{\small Comment:
The economy with internal debt-elastic interest rate premium behaves very simularly to the economies featuring other stationarity inducing devices. 
}
\end{slide}

\begin{slide}
{\bf 4.10.2 The Portfolio Adjustment Cost (PAC) Model}
\[
%\label{eq:bac-bc}
d_t=(1+r_{t-1}) d_{t-1} -
y_t+c_t+ i_t + \Phi(k_{t+1}-k_t)
+ \frac{\psi_3}2 (d_t-\bar{d})^2
\]

\[
%\label{eq:bac-foc-d}
\lambda_t [1-\psi_3(d_t-\bar{d})] = \beta (1+r_t)E_t\lambda_{t+1}
\]


{\bf Calibration}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
$ \beta$&
$ \bar{d}$&
$ \psi_3$&
$r$\\
\hline
0.96 & 
0.7442 &
0.00074&
$\beta^{-1}-1$\\
\hline
\end{tabular}
\end{center}


\end{slide}

\begin{slide}
{\bf 4.10.3 The External Discount Factor (EDF) Model}
\[
%%\label{eq:simple-theta_{t+1}}
\theta_{t+1} =   \beta(\tilde{c}_t,\tilde{h}_t) \theta_t  \quad \quad t\ge 0,
\]
\[\theta_0=1,\]
where $\tilde{c}_t$ and $\tilde{h}_t$ denote  per capita consumption and hours worked. 
\[
%%\label{eq:simple-foc-d}
\lambda_t = \beta(\tilde{c}_t,\tilde{h}_t)(1+r_t)E_t\lambda_{t+1}
\]
\[
%%\label{eq:simple-foc-c}
\lambda_t = U_c(c_t,h_t)
\]
\[
%%\label{eq:simple-foc-h}
-U_h(c_t,h_t) = \lambda_tA_t F_h(k_t,h_t)
\]
\begin{eqnarray*}
\lambda_t [1+{\Phi'}_t]
&=& \beta(\tilde{c}_t,\tilde{h}_t)
E_t \lambda_{t+1} 
[
A_{t+1}F_k(k_{t+1},h_{t+1})
\\
&+&
1-\delta + {\Phi'}_{t+1}
]
\end{eqnarray*}
{In Equilibrium}
\[
%\label{eq:c=tildec}
\mbox{$c_t=\tilde{c}_t$ and $h_t=\tilde{h}_t$}
\]
\end{slide}



\begin{slide}
To be added: 






4.10.4 The Internal Discount Factor (IDF) Model

4.10.5 The Model with No Stationarity-Inducing Features (NSIF) 

\end{slide}






\begin{slide}
{\bf \large 4.11 The Perpetual-Youth (PY) Model }

to be added 
\end{slide}

\begin{slide}
{\bf \large 4.12 Inducing Stationarity: Quantitative Comparison of Alternative Methods}

\end{slide}



\begin{slide}
\centerline{\bf \small Impulse Response to a Unit Technology Shock in Models 1 Through 5 }
%\label{figure:mainfigure} 
\centerline{\includegraphics{soe_rbc_mainfigure_book.eps}  }
{\tiny Source: Schmitt-Groh\'e and Uribe (JIE, 2003).
 Note. Solid line, endogenous discount factor.  Squares, endogenous discount factor without internalization.  Dashed line, Debt-elastic interest rate. Dash-dotted line, Portfolio adjustment cost. 
Dotted line, complete asset markets.  Circles, No stationarity inducing elements.


}
\end{slide}


\begin{slide}


{\tiny

%\onehalfspacing
{\bf Observed and Implied Second Moments}
%\label{table:maintable}
%created with c:\joint\closing\maintable.m
\medskip

\begin{tabular}{lcccccc}
\hline\hline
& Data & Model 1& Model 1a & Model 2&Model 3 & Model 4\\
\hline
\multicolumn{5}{l}{\underline{Standard Deviations}}\\
$y$                                       &    2.8&     3.1&     3.1&     3.1&     3.1&   3.1\\
$c$                                       &    2.5&     2.3&     2.3&     2.7&     2.7&   1.9\\
$i$  &    9.8&     9.1&     9.1&       9&       9&   9.1\\
$h$                                       &      2&     2.1&     2.1&     2.1&     2.1&   2.1\\
$tb/y$                          &    1.9&     1.5&     1.5&     1.8&     1.8&   1.6\\
$ca/y$                          &       &     1.5&     1.5&     1.5&     1.5&     \\
\multicolumn{5}{l}{\underline{Serial Correlations}}\\
$y$                              &   0.61&    0.61&    0.61&    0.62&    0.62&  0.61\\
$c$                              &    0.7&     0.7&     0.7&    0.78&    0.78&  0.61\\
$i$                              &   0.31&    0.07&    0.07&   0.069&   0.069&  0.07\\
$h$                              &   0.54&    0.61&    0.61&    0.62&    0.62&  0.61\\
$tb/y$ &   0.66&    0.33&    0.32&    0.51&     0.5&  0.39\\
$ca/y$ &       &     0.3&     0.3&    0.32&    0.32&    \\
\multicolumn{5}{l}{\underline{Correlations with Output}}\\
$c$                                &   0.59&    0.94&    0.94&    0.84&    0.85&     1\\
$i$                                &   0.64&    0.66&    0.66&    0.67&    0.67&  0.66\\
$h$                                &    0.8&       1&       1&       1&       1&     1\\
$tb/y$                   &  -0.13&  -0.012&  -0.013&  -0.044&  -0.043&  0.13\\
$ca/y$                   &       &   0.026&   0.025&    0.05&   0.051  \\
\hline\hline
\end{tabular}

\centerline{\tiny Source: Schmitt-Groh\'e and Uribe (JIE, 2003)}
\begin{quote}
Note.  Standard deviations are measured in percent per year.  
\end{quote}

%%%%%%%%%%%%%%%%%

}

\end{slide}


\begin{slide}
{\bf \large 4.13 Global Solution Methods}

To be added

\end{slide}


\end{document}
