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{\noindent Instructor: \textit{Benjamin Griffy}\\ Due: \textit{Apr., 25th 2023}\\ AECO 701}

\begin{center} \Large Short Handout Assignment 1 \end{center} \vspace{1em}

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The assignments this week are meant to review a couple of important concepts for the prelim. They do not need to be turned in, but can hopefully shed some light on a couple sources of confusion.

\noindent \textbf{Thinking about state and choice ariables}
Consider the following stylized neoclassical growth model. A representative agent maximizes utility by choosing consumption, $c$, capital saved, $k'$, and labor supplied, $\ell$. They take prices, $r(K,L)$ and $w(K,L)$ as well as the aggregate law of motion for capital, $K' = G(K)$. They face the following problem:

\begin{align}
  V(k;K) & = \max_{c,l,k'} u(c,1 - \ell) + \beta V(k';K')
  \text{s.t. }\;c + k' \leq& (1 + r(K,L) - \delta)k + w(K,L)\ell\\
  \ell\in (0,1), k_{0}\;given.
\end{align}

where prices are given by a profit maximizing firm:

\begin{align}
  \Pi = \max_{K,L}F(K,L) - wL - rK
\end{align}

The equilibrium in this economy is a recursive competitive equilibrium, where $c,\ell,k'$ solves the workers problem, prices, $w$, and $r$, are determined competitively, aggregation holds, $K = \int k'dF(k)$, $L = \int \ell dF(k)$, and the individual decision rules are consistent with the aggregate laws of motion, i.e., $K' = g(K,K)$, where $g(.)$ is the representative agents decision rule ($k' = g(k,K)$).

\begin{enumerate}[label=\textbf{\alph*})]
\item Why are $k$ and $K$ both state variables? Why are $L$ and $\ell$ not?


\item The function $K' = g(K,K)$ appears innocuous, but it is actually a fairly strict assumption. Describe in your own words, what it means.


\item Why do we ``take prices as given'' when using a representative agent (or any model with atomistic agents)?


Now suppose that labor productivity is determined by the human capital, $h$, of the representative agent. Human capital is an AR1 process that evolves according to the following: $ln(h') = \rho ln(h) + \epsilon_{h}, \epsilon_{h}\sim N(0,\sigma_{h})$. The firms problem is now

  \begin{align}
    \Pi = \max_{K,L} F(k,HL) - rK - wHL
  \end{align}

\item What are the state variables for the workers problem?

\end{enumerate}

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