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\title{Chahrour Exam Questions}
\author{Gabe Sekeres}
\course{ECON 6130}
\term{Fall 2024}

\singlespacing

\begin{document}
\maketitle


\tableofcontents

\newpage

\section{Qual 2024}

\subsection{Questions}

Consider an economy populated by identical consumer-worker households with preferences for consumption and labor effort given by
\[
\expect_0 \sum_{t=0}^\infty \beta^t \curll \frac{C_t^{1-\sigma} - 1}{1-\sigma} - \gamma_s S_t - \gamma_n N_t \curlr
\]
Notice that now the household supplies search effort $S_t$ and work effort $N_t$ at constant marginal disutilities $\gamma_s$ and $\gamma_n$ respectively. The household earns income from the wages, $W_t$, paid to currently-employed workers $N_t$ and any profits $\pi_t$ earned by the firm, which it owns. In making their decisions, both the household and the firm take the wage as exogenous.

Unlike in class, assume that workers who match with an employer today start productive work in the following period. From the household perspective, this means that $N_t$ evolves according to
\[
N_{t+1} = (1-\delta_n)N_t+p_tS_t
\]
where $\delta_n$ is the worker separation rate and $p_t$ is the household probability that a unit of search effort results in a job. 

Firms hire workers to produce output, according to the production function
\[
Y_t = A_tN_t
\]
In order to hire a worker, firms must post a vacancy at cost $\phi$. A vacancy results in a match (and a future employed worker) with probability $q_t$, so that from the firm's perspective
\[
N_{t+1} = (1-\delta_n) N_t + q_tV_t
\]
The firm's objective is to maximize the discounted present value of profits, given by
\[
V_0 = \expect_0 \sum_{t=0}^\infty \beta^t \curll \frac{\lambda_{1t}}{\lambda_{10}}\pi_t \curlr 
\]
where $\lambda_t$ is the marginal utility of consumption in period $t$. 

Equilibrium matches are determined by an aggregate matching function $M(V_t,S_t) = \chi V_t^\varepsilon S_t^{1-\varepsilon}$. Each vacancy has an equal chance of being matched, so that in equilibrium
\[
q_t = M(V_t,S_t) / V_t = \chi (V_t / S_t)^{\varepsilon - 1}
\]
Conversely,
\[
p_t = M(V_t,S_t) / S_t = \chi (V_t / S_t)^{\varepsilon}
\]

Finally, technology is purely exogenous and evolves according to an AR(1) process in logs,
\[
\log(A_t) = \rho \log(A_{t-1}) + \varepsilon_t
\]



\begin{enumerate}
	\item In the language of the course, list separately the endogenous jump variables, the endogenous state variables, and the exogenous state variables in this model. Finally, make a list of all of the exogenous parameters of this economy. (5 points)
	\item Write the \underline{household}'s Lagrangian optimization problem and find the first order necessary conditions for optimality of the household. Denote the multipliers on the household budget constraint and labor evolution constraint with $\lambda_{1t}$ and $\lambda_{2t}$ respectively. (10 points)
	\item Write the \underline{firm}'s Lagrangian optimization problem and find the first order necessary conditions for optimality. Denote the multipliers on constraints (2) and (3) with $\theta_{1,t}$ and $\theta_{2,t}$ respectively. (10 points)
	\item Now write the Bellman equation that corresponds to the \underline{social planner}'s optimization problem in this economy and find the first order necessary conditions for optimality using the envelope theorem. (10 points)
	\item Log-linearize the \underline{vacancy posting condition} (first order condition for $V_t$) from the firm problem. You should proceed from first principals (\ie replace $V$ with $\exp(v)$, etc and compute a first-order Taylor approximation). You should log-linearize the equations around the steady-state, and you may treat the steady-state values of endogenous variables, like $N$, $C$, $V$, etc, as parameters. (5 points)
	\item Suppose I wanted to see what would happen over time to employment in the economy in response to temporary increase in the level of risk-aversion ($\sigma$) in the economy. Decide which of the four solution algorithms (linearization, shooting, value function, or projection) that we studied you would use to answer this question, and explain why you chose that method. Your response should include a brief description of the method you choose, and an explanation of why other methods are not as well-suited to the question. (20 points)
\end{enumerate}
\subsection{Solutions}

(Ryan's Solutions)

\begin{enumerate}
	\item We have that the one exogenous state variable is $A_t$, the one endogenous state variable is $N_t$, and the jump variables are $C_t$, $V_t$, $Y_t$, $q_t$, $p_t$, and $S_t$. The parameters are $W_t$, $\beta$, $\sigma$, $\gamma_s$, $\gamma_n$, $\delta_n$, $\chi$, $\varepsilon$, $\rho$, and $\varepsilon_t$.
	\item The household's problem is
	\[
	\max_{\{C_t,S_t,N_{t+1}\}} \expect_0 \sum_{t=0}^\infty \beta^t \Bigg\{ \frac{C_t^{1-\sigma} - 1}{1-\sigma} - \gamma_s S_t - \gamma_n N_t + \lambda_{1,t}\parl \pi_t + W_tN_t - C_t\parr + \lambda_{2,t}\parl (1-\delta_n) N_t + p_tS_t - N_{t+1}\parr\Bigg\}
	\]
	Which admits the first order conditions
	\begin{align*}
		C_t^{-\sigma} &= \lambda_{1,t} &&(C_t) \\
		\gamma_s &= p_t\lambda_{2,t} &&(S_t) \\
		\lambda_{2,t} &= \beta \expect_{t}\Big[ \lambda_{1,t+1} W_{t+1} - \gamma_n + \lambda_{2,t+1}(1-\delta_n)  \Big] &&(N_{t+1})
	\end{align*}
	\item The firm's problem is 
	\[
	\max_{\{Y_t,V_t,N_{t+1}\}} \expect_0 \sum_{t=0}^\infty \beta^t \curll \frac{\lambda_{1,t}}{\lambda_{1,0}} (\underbrace{Y_t - W_tN_t - \phi V_t}_{\pi_t}) + \theta_{1,t}(A_tN_t - Y_t) + \theta_{2,t} ((1-\delta_n)N_t + q_tV_t - N_{t+1})\curlr 
	\]
	Which admits the first order conditions
	\begin{align*}
		\frac{\lambda_{1,t}}{\lambda_{1,0}} &= \theta_{1,t} &&(Y_t) \\
		\phi \frac{\lambda_{1,t}}{\lambda_{1,0}} &= \theta_{2,t}q_t &&(V_t) \\
		\theta_{2,t} &= \beta \expect \barl \theta_{1,t+1} A_{t+1} - \frac{\lambda_{1,t+1}}{\lambda_{1,0}}W_{t+1} - \theta_{2,t+1}(1-\delta_n)\barr &&(N_{t+1}) 
	\end{align*}
	Combining, we get that
	\[
	\frac{\phi}{q_t}\lambda_{1,t} = \beta \expect \barl \lambda_{1,t+1}(A_{t+1}-W_{t+1}) - (1-\delta_n) \frac{\phi}{q_{t+1}}\lambda_{1,t+1} \barr
	\]
	\item The planner's value function is
	\[
	V(N_t,A_t) = \max_{\{C_t,S_t,V_t,N_{t+1}\}} \frac{C^{1-\sigma}-1}{1-\sigma} - \gamma_sS_t - \gamma_n N_t + \beta \expect \barl V(N_{t+1},A_{t+1})\barr
	\]
	subject to
	\[
	N_{t+1} = (1-\delta_n)N_t + \chi V_t^\varepsilon S_t^{1-\varepsilon} \quad \text{ and } \quad C_t = A_tN_t - \phi V_t
	\]
	Since 
	\[
	V_t = \parl \frac{N_{t+1}-(1-\delta_n)N_t}{\chi S_t^{1-\varepsilon}}\parr^\frac{1}{\varepsilon}
	\]
	we have that
	\[
	V(N_t,A_t) = \max_{\{S_t,N_{t+1}\}} u\parl A_t N_t - \phi  \parl \frac{N_{t+1}-(1-\delta_n)N_t}{\chi S_t^{1-\varepsilon}}\parr^\frac{1}{\varepsilon}\parr - \gamma_sS_t - \gamma_nN_t + \beta \expect \barl V(N_{t+1},A_{t+1})\barr
	\]
	The first order conditions are
	\begin{align*}
		-\gamma_s + u'(\cdot) \phi \frac{1}{\varepsilon} \parl \frac{N_{t+1}-(1-\delta_n)N_t}{\chi S_t^{1-\varepsilon}}\parr^{\frac{1}{\varepsilon}-1} (\varepsilon - 1) \frac{N_{t+1}-(1-\delta_n)N_t}{\chi}S_t^{\varepsilon-2} &= 0 &&(S_t) \\
		-u'(\cdot)\phi \frac{1}{\varepsilon} \parl \frac{N_{t+1}-(1-\delta_n)N_t}{\chi S_t^{1-\varepsilon}}\parr^{\frac{1}{\varepsilon}-1} \frac{1}{\chi S_t^{1-\varepsilon}} + \beta \expect\barl V_N(N_{t+1},A_{t+1})\barr &= 0 &&(N_{t+1})
	\end{align*}
	By the Envelope condition, we have that
	\[
	V_N(N_{t},A_{t}) = u'(\cdot) \phi \frac{1}{\varepsilon} \parl \frac{N_{t+1}-(1-\delta_n)N_t}{\chi S_t^{1-\varepsilon}}\parr^{\frac{1}{\varepsilon}-1} \frac{1}{\chi S_t^{1-\varepsilon}} (1-\delta_n) + A_tu'(\cdot)
	\]
	Combining, we get the Bellman Equation
	\[
	u'(\cdot)\frac{\phi}{\varepsilon} \parl \frac{N_{t+1}-(1-\delta_n)N_t}{\chi S_t^{1-\varepsilon}}\parr^{\frac{1}{\varepsilon}-1} \frac{1}{\chi S_t^{1-\varepsilon}} = \beta \expect\barl u'(\cdot) \curll A_{t+1} + \frac{(1-\delta_n)\phi}{\varepsilon \chi S_{t+1}^{1-\varepsilon}} \parl \frac{N_{t+2}-(1-\delta_n)N_{t+1}}{\chi S_{t+1}^{1-\varepsilon}}\parr^{\frac{1}{\varepsilon}-1}\curlr \barr
	\]
	\item We will log-linearize the combined posting condition from above:
	\[
	\frac{\phi}{q_t}\lambda_{1,t} = \beta \expect \barl \lambda_{1,t+1}(A_{t+1}-W_{t+1}) - (1-\delta_n) \frac{\phi}{q_{t+1}}\lambda_{1,t+1} \barr
	\]
	For each variable, denote $\tilde{x} = \log(x)$. We have that this becomes
	\[
	\frac{\phi}{\exp(\tilde{q}_t)} = \beta \expect \barl \frac{\exp(\tilde{\lambda}_{1,t+1})}{\exp(\tilde{\lambda}_{1,t})}\parl \exp(\tilde{a}_{t+1}) - \exp(\tilde{w}_{t+1}) - (1-\delta_n)\frac{\phi}{\exp(\tilde{q}_{t+1})}\parr\barr
	\]
	Using the first order Taylor expansion about the steady state, where the steady state variables are denoted by subscripts, the left side is approximated by
	\[
	\frac{\phi}{\exp(q_{ss})(\hat{q}_t)} \equiv \frac{-\phi \hat{q}_t}{\exp(q_{ss})}
	\]
	where $\hat{q}_t = \tilde{q}_t - q_{ss}$. The right hand side is approximated by
	\[
	\beta \expect \barl \exp(a_{ss}) (\hat{a}_t) - \exp(w_{ss}) (\hat{w}_t) + (1-\delta_n) \frac{\phi \hat{q}_{t+1}}{\exp(q_{ss})}\barr \equiv \beta \expect \barl A_{ss}\hat{a}_t - W_{ss} \hat{w}_t + (1-\delta_n) \frac{\phi \hat{q}_{t+1}}{\exp(q_{ss})}\barr
	\]
	\item We should use projection, which captures both uncertainty and non-linearities. We would approximate policy functions with weights on basis functions, so for policy $h(x)$, where we know $\{x_i\}_{i=1}^I$, we would approximate
	\[
	\hat{h}(x) = \sum_{i=1}^I a_i b_i(x)
	\]
	where $b_i$ are basis functions and $a_i$ are weights. 
	
	The other methods each have issues. Log-Linearization would not capture the effects of risk aversion, since it assumes agents are implicitly risk-neutral. Shooting requires perfect foresight, so we would not see the effects of uncertainty, and value function iteration is slow, since we have smooth choice sets.
\end{enumerate}


\newpage
\section{Final 2023}

\subsection{Questions}

\begin{enumerate}
	\item \textbf{Model with production externality}. Consider an economy populated by identical producer-consumer households with preferences for consumption given by
	\[
	\expect_0 \sum_{t=0}^\infty \beta^t \log(C_t)
	\]
	Households are endowed with an initial stock of capital $K_0$. Household-level capital evolves according to the equation
	\[
	K_{t+1} = (1-\delta) K_t + I_t
	\]
	Household-level consumption is equal to output minus investment
	\[
	C_t = A_tK_t^\alpha - I_t
	\]
	where $Y_t = A_tK_t^{\alpha}$ is the production function and $0 < \alpha < 1$.
	
	Finally, assume that total factor productivity, $A_t$, is taken as exogenous by each household, but depends on the aggregate level of current capital according to
	\[
	\log(A_t) = a_0 \log(\bar{K}_t)
	\]
	with $a_0 \ge 0$ and $a_0 + \alpha < 1$. In equilibrium, $K_t = \bar{K}_t$.
	\begin{enumerate}
		\item In the language of the course, list separately the endogenous jump variables, the endogenous state variables, and the exogenous state variables in this model. Finally, make a list of all of the exogenous parameters of this economy. (5 points)
		\item Write the household’s Lagrangian optimization problem and find the first order necessary conditions for optimality of the household. Denote the multipliers on constraints (1) and (2) with $\lambda_{1,t}$ and 
	\end{enumerate}
\end{enumerate}























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