2009-09-08 15:55:19 +02:00
|
|
|
\documentclass[10pt]{article}
|
|
|
|
\usepackage{array,natbib,times}
|
|
|
|
\usepackage{amsmath, amsthm, amssymb}
|
|
|
|
|
|
|
|
%\usepackage[pdftex,colorlinks]{hyperref}
|
|
|
|
|
|
|
|
\begin{document}
|
|
|
|
|
|
|
|
\title{Implementation of Ramsey Optimal Policy in Dynare++, Timeless Perspective}
|
|
|
|
|
|
|
|
\author{Ondra Kamen\'\i k}
|
|
|
|
|
|
|
|
\date{June 2006}
|
|
|
|
\maketitle
|
|
|
|
|
|
|
|
\textbf{Abstract:} This document provides a derivation of Ramsey
|
|
|
|
optimal policy from timeless perspective and describes its
|
|
|
|
implementation in Dynare++.
|
|
|
|
|
|
|
|
\section{Derivation of the First Order Conditions}
|
|
|
|
|
|
|
|
Let us start with an economy populated by agents who take a number of
|
|
|
|
variables exogenously, or given. These may include taxes or interest
|
|
|
|
rates for example. These variables can be understood as decision (or control)
|
|
|
|
variables of the timeless Ramsey policy (or social planner). The agent's
|
|
|
|
information set at time $t$ includes mass-point distributions of these
|
|
|
|
variables for all times after $t$. If $i_t$ denotes an interest rate
|
|
|
|
for example, then the information set $I_t$ includes
|
|
|
|
$i_{t|t},i_{t+1|t},\ldots,i_{t+k|t},\ldots$ as numbers. In addition
|
|
|
|
the information set includes all realizations of past exogenous
|
|
|
|
innovations $u_\tau$ for $\tau=t,t-1,\ldots$ and distibutions
|
|
|
|
$u_\tau\sim N(0,\Sigma)$ for $\tau=t+1,\ldots$. These information sets will be denoted $I_t$.
|
|
|
|
|
|
|
|
An information set including only the information on past realizations
|
|
|
|
of $u_\tau$ and future distributions of $u_\tau\sim N(0\sigma)$ will
|
|
|
|
be denoted $J_t$. We will use the following notation for expectations
|
|
|
|
through these sets:
|
|
|
|
\begin{eqnarray*}
|
|
|
|
E^I_t[X] &=& E(X|I_t)\\
|
|
|
|
E^J_t[X] &=& E(X|J_t)
|
|
|
|
\end{eqnarray*}
|
|
|
|
|
|
|
|
The agents optimize taking the decision variables of the social
|
|
|
|
planner at $t$ and future as given. This means that all expectations
|
|
|
|
they form are conditioned on the set $I_t$. Let $y_t$ denote a vector
|
|
|
|
of all endogenous variables including the planer's decision
|
|
|
|
variables. Let the number of endogenous variables be $n$. The economy
|
|
|
|
can be described by $m$ equations including the first order conditions
|
|
|
|
and transition equations:
|
|
|
|
\begin{equation}\label{constr}
|
|
|
|
E_t^I\left[f(y_{t-1},y_t,y_{t+1},u_t)\right] = 0.
|
|
|
|
\end{equation}
|
|
|
|
This lefts $n-m$
|
|
|
|
the planner's control variables. The solution of this problem is a
|
|
|
|
decision rule of the form:
|
|
|
|
\begin{equation}\label{agent_dr}
|
|
|
|
y_t=g(y_{t-1},u_t,c_{t|t},c_{t+1|t},\ldots,c_{t+k|t},\ldots),
|
|
|
|
\end{equation}
|
|
|
|
where $c$ is a vector of planner's control variables.
|
|
|
|
|
|
|
|
Each period the social planner chooses the vector $c_t$ to maximize
|
|
|
|
his objective such that \eqref{agent_dr} holds for all times following
|
|
|
|
$t$. This would lead to $n-m$ first order conditions with respect to
|
|
|
|
$c_t$. These first order conditions would contain unknown derivatives
|
|
|
|
of endogenous variables with respect to $c$, which would have to be
|
|
|
|
retrieved from the implicit constraints \eqref{constr} since the
|
|
|
|
explicit form \eqref{agent_dr} is not known.
|
|
|
|
|
|
|
|
The other way to proceed is to assume that the planner is so dumb that
|
|
|
|
he is not sure what are his control variables. So he optimizes with
|
|
|
|
respect to all $y_t$ given the constraints \eqref{constr}. If the
|
|
|
|
planner's objective is $b(y_{t-1},y_t,y_{t+1},u_t)$ with a discount rate
|
|
|
|
$\beta$, then the optimization problem looks as follows:
|
|
|
|
\begin{align}
|
|
|
|
\max_{\left\{y_\tau\right\}^\infty_t}&E_t^J
|
|
|
|
\left[\sum_{\tau=t}^\infty\beta^{\tau-t}b(y_{\tau-1},y_\tau,y_{\tau+1},u_\tau)\right]\notag\\
|
|
|
|
&\rm{s.t.}\label{planner_optim}\\
|
|
|
|
&\hskip1cm E^I_\tau\left[f(y_{\tau-1},y_\tau,y_{\tau+1},u_\tau)\right]=0\quad\rm{for\ }
|
|
|
|
\tau=\ldots,t-1,t,t+1,\ldots\notag
|
|
|
|
\end{align}
|
|
|
|
Note two things: First, each constraint \eqref{constr} in
|
|
|
|
\eqref{planner_optim} is conditioned on $I_\tau$ not $I_t$. This is
|
|
|
|
very important, since the behaviour of agents at period $\tau=t+k$ is
|
|
|
|
governed by the constraint using expectations conditioned on $t+k$,
|
|
|
|
not $t$. The social planner knows that at $t+k$ the agents will use
|
|
|
|
all information available at $t+k$. Second, the constraints for the
|
|
|
|
planner's decision made at $t$ include also constraints for agent's
|
|
|
|
behaviour prior to $t$. This is because the agent's decision rules are
|
|
|
|
given in the implicit form \eqref{constr} and not in the explicit form
|
|
|
|
\eqref{agent_dr}.
|
|
|
|
|
|
|
|
Using Lagrange multipliers, this can be rewritten as
|
|
|
|
\begin{align}
|
|
|
|
\max_{y_t}E_t^J&\left[\sum_{\tau=t}^\infty\beta^{\tau-t}b(y_{\tau-1},y_\tau,y_{\tau+1},u_\tau)\right.\notag\\
|
|
|
|
&\left.+\sum_{\tau=-\infty}^{\infty}\beta^{\tau-t}\lambda^T_\tau E_\tau^I\left[f(y_{\tau-1},y_\tau,y_{\tau+1},u_\tau)\right]\right],
|
|
|
|
\label{planner_optim_l}
|
|
|
|
\end{align}
|
|
|
|
where $\lambda_t$ is a vector of Lagrange multipliers corresponding to
|
|
|
|
constraints \eqref{constr}. Note that the multipliers are multiplied
|
|
|
|
by powers of $\beta$ in order to make them stationary. Taking a
|
|
|
|
derivative wrt $y_t$ and putting it to zero yields the first order
|
|
|
|
conditions of the planner's problem:
|
|
|
|
\begin{align}
|
|
|
|
E^J_t\left[\vphantom{\frac{\int^(_)}{\int^(\_)}}\right.&\frac{\partial}{\partial y_t}b(y_{t-1},y_t,y_{t+1},u_t)+
|
|
|
|
\beta L^{+1}\frac{\partial}{\partial y_{t-1}}b(y_{t-1},y_t,y_{t+1},u_t)\notag\\
|
|
|
|
&+\beta^{-1}\lambda_{t-1}^TE^I_{t-1}\left[L^{-1}\frac{\partial}{\partial y_{t+1}}f(y_{t-1},y_t,y_{t+1},u_t)\right]\notag\\
|
|
|
|
&+\lambda_t^TE^I_t\left[\frac{\partial}{\partial y_{t}}f(y_{t-1},y_t,y_{t+1},u_t)\right]\notag\\
|
|
|
|
&+\beta\lambda_{t+1}^TE^I_{t+1}\left[L^{+1}\frac{\partial}{\partial y_{t-1}}f(y_{t-1},y_t,y_{t+1},u_t)\right]
|
|
|
|
\left.\vphantom{\frac{\int^(_)}{\int^(\_)}}\right]
|
|
|
|
= 0,\label{planner_optim_foc}
|
|
|
|
\end{align}
|
|
|
|
where $L^{+1}$ and $L^{-1}$ are one period lead and lag operators respectively.
|
|
|
|
|
|
|
|
Now we have to make a few assertions concerning expectations
|
|
|
|
conditioned on the different information sets to simplify
|
|
|
|
\eqref{planner_optim_foc}. Recall the formula for integration through
|
|
|
|
information on which another expectation is conditioned, this is:
|
|
|
|
$$E\left[E\left[u|v\right]\right] = E[u],$$
|
|
|
|
where the outer expectation integrates through $v$. Since $J_t\subset
|
|
|
|
I_t$, by easy application of the above formula we obtain
|
|
|
|
\begin{eqnarray}
|
|
|
|
E^J_t\left[E^I_t\left[X\right]\right] &=& E^J_t\left[X\right]\quad\rm{and}\notag\\
|
|
|
|
E^J_t\left[E^I_{t-1}\left[X\right]\right] &=& E^J_t\left[X\right]\label{e_iden}\\
|
|
|
|
E^J_t\left[E^I_{t+1}\left[X\right]\right] &=& E^J_{t+1}\left[X\right]\notag
|
|
|
|
\end{eqnarray}
|
|
|
|
Now, the last term of \eqref{planner_optim_foc} needs a special
|
|
|
|
attention. It is equal to
|
|
|
|
$E^J_t\left[\beta\lambda^T_{t+1}E^I_{t+1}[X]\right]$. If we assume
|
|
|
|
that the problem \eqref{planner_optim} has a solution, then there is a
|
|
|
|
deterministic function from $J_{t+1}$ to $\lambda_{t+1}$ and so
|
|
|
|
$\lambda_{t+1}\in J_{t+1}\subset I_{t+1}$. And the last term is equal
|
|
|
|
to $E^J_{t}\left[E^I_{t+1}[\beta\lambda^T_{t+1}X]\right]$, which is
|
|
|
|
$E^J_{t+1}\left[\beta\lambda^T_{t+1}X\right]$. This term can be
|
|
|
|
equivalently written as
|
|
|
|
$E^J_{t}\left[\beta\lambda^T_{t+1}E^J_{t+1}[X]\right]$. The reason why
|
|
|
|
we write the term in this way will be clear later. All in all, we have
|
|
|
|
\begin{align}
|
|
|
|
E^J_t\left[\vphantom{\frac{\int^(_)}{\int^(\_)}}\right.&\frac{\partial}{\partial y_t}b(y_{t-1},y_t,y_{t+1},u_t)+
|
|
|
|
\beta L^{+1}\frac{\partial}{\partial y_{t-1}}b(y_{t-1},y_t,y_{t+1},u_t)\notag\\
|
|
|
|
&+\beta^{-1}\lambda_{t-1}^TL^{-1}\frac{\partial}{\partial y_{t+1}}f(y_{t-1},y_t,y_{t+1},u_t)\notag\\
|
|
|
|
&+\lambda_t^T\frac{\partial}{\partial y_{t}}f(y_{t-1},y_t,y_{t+1},u_t)\notag\\
|
|
|
|
&+\beta\lambda_{t+1}^TE^J_{t+1}\left[L^{+1}\frac{\partial}{\partial y_{t-1}}f(y_{t-1},y_t,y_{t+1},u_t)\right]
|
|
|
|
\left.\vphantom{\frac{\int^(_)}{\int^(\_)}}\right]
|
|
|
|
= 0.\label{planner_optim_foc2}
|
|
|
|
\end{align}
|
|
|
|
Note that we have not proved that \eqref{planner_optim_foc} and
|
|
|
|
\eqref{planner_optim_foc2} are equivalent. We proved only that if
|
|
|
|
\eqref{planner_optim_foc} has a solution, then
|
|
|
|
\eqref{planner_optim_foc2} is equivalent (and has the same solution).
|
|
|
|
|
2011-03-11 23:35:08 +01:00
|
|
|
%%- \section{Implementation}
|
|
|
|
%%-
|
|
|
|
%%- The user inputs $b(y_{t-1},y_t,y_{t+1},u_t)$, $\beta$, and agent's
|
|
|
|
%%- first order conditions \eqref{constr}. The algorithm has to produce
|
|
|
|
%%- \eqref{planner_optim_foc2}.
|
|
|
|
%%-
|
2009-09-08 15:55:19 +02:00
|
|
|
\end{document}
|