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{\noindent Instructor: \textit{Professor Griffy}\\ Due: \textit{Mar., 6th 2025}\\ AECO 701}

\begin{center} \Large Problem Set 3 \end{center} \vspace{1em}

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\noindent \textbf{Problem 1: The Lucas Critique and Rational Expectations)}
Suppose that a household faces the following problem:

    \begin{align}
      \max_{c_{1},c_{2}} u(c_{1}) &+ \beta u(c_{2})\\
      \text{s.t.  } c_{1} + \frac{c_{2}}{1 + r} &= w_{1} + \frac{w_{2}}{1 + r} + T_{1}
    \end{align}

where $u'(c) > 0$, $u''(c) < 0$, $u'(0) = \infty$, and (for now), $w_{1}$, $w_{2}$, and $T_{1}$ are constants. Assume that the interest rate, $r$, is fixed throughout.

\begin{enumerate}[label=\textbf{\alph*})]
\item Solve for the household's first order conditions and derive the expressions for consumption $c_{1}$ and $c_{2}$.


\item Assume that workers have log-utility $u(c) = ln(c)$, and that the parameters take on the following values: $r = 0.04$, $\beta = \frac{1}{1 + r}$, $T_{1} = 0.0$. Now assume that $w_{1}$ and $w_{2}$ are independently and identically distributed according to $w_{t}\sim LN(0,1)$. Complete the following
\begin{enumerate}
\item Formulate the problem using the appropriate expectations operator and budget constraints, and then derive expressions for $c_{1}$ and $c_{2}$.
\item Simulate the model using the decision rules you found in part (a) and the wage distributions above and plot $c_{1}$ for the 10th, 30th, 50th, 70th, and 90th percentiles of the lifetime income $(w_{1} + \frac{w}_{2}{1 + r})$ distribution.
\item Calculate the slope of the consumption function with respect to $w_{1}$.
\end{enumerate}

\item Now suppose that a policymaker would like to estimate the impact of wages on consumption for future use in stimulating the economy. Run the following regression on your simulated data: $c_{t} = \gamma w_{t} + \epsilon_{t}$. How does $\gamma$ compare to the slope of the consumption function?

\item Now suppose that wages are an AR(1) process. Simulate your model again, but use the following for wages: $w_{1} \sim LN(0, 1)$, $ln(w_{2}) = \rho ln(w_{1}) + \epsilon_{2}$, where $\rho = 0.95$ and $\epsilon\sim N(0, 1)$.
\begin{enumerate}
\item Now, draw a random sample of log-normal wages for period 1. Then draw an equal-sized sample of normally-distributed errors for period 2 and calculate wages in period 2 as $ln(w_{2}) = \rho ln(w_{1}) + \epsilon_{2}$. Now plot $c_{1}$ for the 10th, 30th, 50th, 70th, and 90th percentiles of the lifetime income $(w_{1} + \frac{w}_{2}{1 + r})$ distribution.
\item Calculate the slope of the consumption function with respect to $w_{1}$.
\end{enumerate}

\item Now suppose that a policymaker would like to use a reduced-form estimate to stimulate the economy. Run the following regression on your simulated data: $c_{t} = \gamma w_{t} + \epsilon_{t}$. Now Suppose that the policymaker implements a one-time payment of $T_{1} = 0.5$, which they see as equivalent to $w_{1}$. Simulate the model with this transfer and compare the impact on consumption to the estimate $\hat{\gamma}$.

\end{enumerate}
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