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%	TITLE PAGE
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\title[]{Quantitative Macro-Labor:\\ Inequality in Heterogeneous Agent Models} % The short title appears at the bottom of every slide, the full
% title is only on the title page

\author{Professor Griffy} % Your name
\institute[University at Albany, SUNY] % Your institution as it will appear on the bottom of
% every slide, may be shorthand to save space
{
UAlbany  \ % Your institution for the title page
}
\date{Fall 2024} % Date, can be changed to a custom date

\begin{document}

\begin{frame}
  \titlepage % Print the title page as the first slide
\end{frame}




% ----------------------------------------------------------------------------------------
%	PRESENTATION SLIDES
% ----------------------------------------------------------------------------------------

% ------------------------------------------------
\section{Introduction} % Sections can be created in order to organize your presentation into discrete blocks, all sections and subsections are automatically printed in the table of contents as an overview of the talk
% ------------------------------------------------

\begin{frame}
  \frametitle{Announcements}
  \begin{itemize}
  \item Today: Continue heterogeneous agent models.
  \item Start writing down and solving your model.
  \item Due in 3 weeks!
  \item No office hours today (please email if you have questions!)
  \end{itemize}
\end{frame}

% ------------------------------------------------

\begin{frame}
  \frametitle{Model Project}
  \begin{itemize}
  \item Write down and solve a model that can explain your empirical regularities.
  \item Solving a model is tricky. Try to get a solution for ``calibrated'' parameters.
  \item Try to pick a model somewhat similar to the code I've provided.
  \item Start with a model that we have either talked about or is widely used.
  \item I'll try to help as much as I can.
  \item We start presentations after Thanksgiving!
  \end{itemize}
\end{frame}

% ------------------------------------------------

\section{Inequality in a Heterogeneous Agent Model} % Sections can be created in order to organize your presentation into discrete blocks, all sections and subsections are automatically printed in the table of contents as an overview of the talk

% ------------------------------------------------

\begin{frame}
  \frametitle{Thinking about Inequality}
  \begin{itemize}
  \item We started this semester discussing the role of inequality between individuals in determining outcomes.
  \item What is the source of this inequality?
  \end{itemize}
\end{frame}

% ------------------------------------------------

\begin{frame}
  \frametitle{Huggett, Ventura, and Yaron (2011)}
  \begin{itemize}
  \item Questions:
    \begin{itemize}
    \item How much of life-cycle inequality is ex-ante? i.e., how much is beyond an individual's control?
    \item Which of the initial conditions are important?
    \end{itemize}
  \item Outline:
    \begin{itemize}
    \item Life-cycle production economy with
    \item Risk-aversion, borrowing constraints, and human capital accumulation,
    \item competitively determined prices, social security (GE)
    \end{itemize}
  \item ex-ante heterogeneity in wealth, human capital, learning ability.
  \end{itemize}
\end{frame}

% ------------------------------------------------

\begin{frame}
  \frametitle{Model Environment}\label{environment}
  \begin{itemize}
  \item Life-cycle model: age discrete, indexed by $j$; retirement at age $J_{R}$ die at age $J$
  \item Agents:
    \begin{itemize}
    \item Age-j, born at time t households: $(j,t,k,h,\ell)$
    \item competitive firms who set interest rate and wages.
  \end{itemize}
  \item Technology:
    \begin{itemize}
    \item Endogenous human capital accumulation.
    \item Borrowing constraints. $k \geq \underline{k}$
    \item Uncertainty over rental rate of labor (standard inc. mkts.) \& rate of return to human capital.
    \end{itemize}
  \item Initial heterogeneity:
    \begin{itemize}
    \item Initial wealth ($a_{0}$), human capital ($h_{0}$), and learning ability ($\ell$).
    \end{itemize}
  \item Prices determined by overlapping generations.
  \end{itemize}
\end{frame}

% ------------------------------------------------

\begin{frame}
  \frametitle{Household's Problem}\label{environment}
  \begin{itemize}
  \item Sequential Formulation for agent born at time $t$:
    \begin{align}
      \max_{\{c_{j},l_{j},s_{j},h_{j},k_{j}}_{j=1}^{J} &E[\sum_{j=1}^{J}\beta^{j-1}u(c_{j})]\\
      \text{s.t.  } c_{j} + k_{j + 1} &= (1 + r_{t + j - 1})k_{j} + e_{j} - T_{j,t+j-1}(e_{j},k_{j})\\
      e_{j} &= R_{t + j-1}h_{j}(1 - s_{j}) \forall\;j<J_{R},\;0\;else\\
      h_{j + 1} &= e^{z_{j + 1}}H(h_{j},s_{j},a)
    \end{align}
  \item GE Objects:
    \begin{itemize}
    \item Prices: interest rate \& wages.
    \item Social security income $T$
    \end{itemize}
  \item Multiple generations alive at same time. Hard to solve!
  \item Luckily, no aggregate uncertainty, can solve in stationary equilibrium.
  \end{itemize}
\end{frame}

% ------------------------------------------------

\begin{frame}
  \frametitle{Firm's Problem}\label{environment}
  \begin{itemize}
  \item Firms are competitive.
  \item i.e., they pay the marginal product.
  \item Overlapping generations $\righrarrow$ balanced growth model.
  \item Firm's problem:
    \begin{align}
      \max_{K,L}&\pi F(K,LA) - RL - rK\\
      A_{t + 1} &= (1 + g)A_{t}
    \end{align}
  \item Cross-cohort growth rates of human capital (effectively).
  \end{itemize}
\end{frame}

% ------------------------------------------------

\begin{frame}
  \frametitle{Balanced Growth Competitive Equilibrium}
  \begin{itemize}
  \item A balanced-growth equilibrium is a collection of decisions for each cohort at each age, a set of factor prices, government taxes and expenditures, and an initial distribution $\psi$ such that
    \begin{enumerate}
    \item Agents optimize taking as given the factor prices.
    \item Prices are formed competitively from the firm's problem.
    \item Resource feasibility: expenditures = output in the aggregate.
    \item Balanced government budget.
    \item Balanced growth: $c, k, T, G$ grow over time, while portfolio allocation and interest rate remains constant.
    \item Aggregates are consistent with individual policy rules: $K = \sum_{j}\int k d\psi$, $L = \sum_{j}\int \epsilon d\psi$, $C = \sum_{j}\int c d\psi$, $T = \sum_{j}\int T(.) d\psi$
    \end{enumerate}
  \end{itemize}
\end{frame}

% ------------------------------------------------

\begin{frame}
  \frametitle{Household's Problem}\label{environment}
  \begin{itemize}
  \item Sequential Formulation for agent born at time $t$:
    \begin{align}
      \max_{\{c_{j},l_{j},s_{j},h_{j},k_{j}}_{j=1}^{J} &E[\sum_{j=1}^{J}\beta^{j-1}u(c_{j})]\\
      \text{s.t.  } c_{j} + k_{j + 1} &= (1 + r_{t + j - 1})k_{j} + e_{j} - T_{j,t+j-1}(e_{j},k_{j})\\
      e_{j} &= R_{t + j-1}h_{j}(1 - s_{j}) \forall\;j<J_{R},\;0\;else\\
      h_{j + 1} &= e^{z_{j + 1}}H(h_{j},s_{j},a)
    \end{align}
  \item GE Objects:
    \begin{itemize}
    \item Prices: interest rate \& wages.
    \item Social security income $T$
    \end{itemize}
  \item Multiple generations alive at same time. Hard to solve!
  \item Luckily, no aggregate uncertainty, can solve in stationary equilibrium.
  \end{itemize}
\end{frame}

% ------------------------------------------------

\section{Calibration}
% ------------------------------------------------

\begin{frame}
  \frametitle{Empirical Preliminaries}\label{empiricalpreliminaries}
  \begin{itemize}
  \item Model parameters: ${\only<1>{\color{red}}\sigma = 2, \beta = \frac{1}{1 + r_{F}}}$
  \item Power utility + unemp leisure: ${\only<1>{\color{red}}u(c) = \frac{c^{1 - \sigma} - 1}{1 - \sigma}}$
  \item HC Evolution: $h'=e^{\epsilon}(h+H(h,\ell,s))={\only<1>{\color{red}}e^{\epsilon}(h+\ell\times(h s)^{{\only<2>{\color{blue}}\alpha_{H}}})}$
  \item Initial conditions:
    \begin{itemize}
    \item $(k_{0},h_{0},\ell) \sim LN(\only<2>{\color{blue}}{\psi, \Sigma}$)
    \item Correlations $\only<2>{\color{blue}}{\rho_{KH}, \rho_{KL}, \rho_{HL}}$
    \end{itemize}
  \item Most results: $\sigma_{K} = 0$
  \end{itemize}
\end{frame}

% ------------------------------------------------

\begin{frame}
  \frametitle{Identification}\label{empiricalpreliminaries}
  \begin{itemize}
  \item They preset parameters to non-controversial values ($\sigma = 2$, etc.)
  \item Controversial problem: disciplining human capital accumulation.
  \item Specifically, how to separate depreciation from time allocation.
  \item Their approach:
    \begin{itemize}
    \item Argue that households stop accumulating human capital at some point. Call this age $t$.
    \item Then, any change in human capital from ages $t$ to $t + n$ is due to depreciation.
    \item Use this \& second-moments to identify human capital.
    \end{itemize}
  \end{itemize}
\end{frame}

% ------------------------------------------------

\begin{frame}
  \frametitle{Identification II}\label{empiricalpreliminaries}
  \begin{itemize}
  \item Initial conditions. How to discipline them?
  \item Here, assume that model is correct and pick initial conditions that best fit the data.
  \item Initial human capital: shifts the intercept of the earnings profile.
  \item Learning ability: rotates (change the slope) of the earnings profile.
  \item Match first and second (and third) moments of earnings distributions/profiles for these initial conditions.
  \item Match age-23 wealth for initial wealth.
  \end{itemize}
\end{frame}

% ------------------------------------------------

\begin{frame}
  \frametitle{Empirical Preliminaries}\label{empiricalpreliminaries}
\includegraphics[width=\textwidth]{HVY2.png}
\end{frame}

% ------------------------------------------------

\section{Inequality}
% ------------------------------------------------

\begin{frame}
  \frametitle{Sources of Inequality}\label{environment}
  \begin{itemize}
  \item Sequential Formulation for agent born at time $t$:
    \begin{align}
      \max_{\{c_{j},l_{j},s_{j},h_{j},k_{j}}_{j=1}^{J} &E[\sum_{j=1}^{J}\beta^{j-1}u(c_{j})]\\
      \text{s.t.  } c_{j} + k_{j + 1} &= (1 + r_{t + j - 1})k_{j} + e_{j} - T_{j,t+j-1}(e_{j},k_{j})\\
      e_{j} &= R_{t + j-1}h_{j}(1 - s_{j}) \forall\;j<J_{R},\;0\;else\\
      h_{j + 1} &= e^{z_{j + 1}}H(h_{j},s_{j},a)
    \end{align}
  \item Initially, agents differ in terms of $k_{0}, h_{0}, \ell$
  \item Agents face uncertainty over their future stream of consumption.
  \item Differences in human capital $\rightarrow$ $e_{j}\downarrow\propto\Delta h$
  \end{itemize}
\end{frame}

% ------------------------------------------------

\begin{frame}
  \frametitle{Sources of Inequality}\label{environment}
  \begin{itemize}
  \item Sequential Formulation for agent born at time $t$:
    \begin{align}
      \max_{\{c_{j},l_{j},s_{j},h_{j},k_{j}}_{j=1}^{J} &E[\sum_{j=1}^{J}\beta^{j-1}u(c_{j})]\\
      \text{s.t.  } c_{j} + k_{j + 1} &= (1 + r_{t + j - 1})k_{j} + e_{j} - T_{j,t+j-1}(e_{j},k_{j})\\
      e_{j} &= R_{t + j-1}h_{j}(1 - s_{j}) \forall\;j<J_{R},\;0\;else\\
      h_{j + 1} &= e^{z_{j + 1}}H(h_{j},s_{j},a)
    \end{align}
  \item Agents also make a portfolio allocation decision.
  \item Uncertainty important: human capital is a risky asset; capital is riskless.
  \item What does this mean in terms of initial inequality?
  \end{itemize}
\end{frame}

% ------------------------------------------------

\begin{frame}
  \frametitle{Thinking about Uncertainty in Macroeconomic Models}
  \begin{itemize}
  \item Returning to our Euler Equation.
  \item Wealthy agents:
    \begin{align}
      u'(c_{i,t}) &= \beta E[(1 + \underbrace{r_{t + 1}}_{GE})\underbrace{u'(c_{i,t+1})}_{Closer\;to\;Linear}]
    \end{align}
  \item Poor Agents:
    \begin{align}
      u'(c_{i,t}) &= \beta E[(1 + \underbrace{r_{t + 1}}_{GE})\underbrace{u'(c_{i,t+1})}_{Non-\;Linear}]
    \end{align}
\item What does this mean for portfolio allocation?
\end{itemize}
\end{frame}


% ------------------------------------------------

\begin{frame}
  \frametitle{Some Exploration I}\label{empiricalpreliminaries}
\includegraphics[width=\textwidth]{HVY3.png}
\end{frame}

% ------------------------------------------------

\begin{frame}
  \frametitle{Some Exploration II}\label{empiricalpreliminaries}
\includegraphics[width=\textwidth]{HVY4.png}
\end{frame}

% ------------------------------------------------

\begin{frame}
  \frametitle{Initial Conditions}\label{empiricalpreliminaries}
\includegraphics[width=\textwidth]{HVY5.png}
\end{frame}

% ------------------------------------------------

\begin{frame}
  \frametitle{Initial Conditions vs. Shocks}\label{empiricalpreliminaries}
\includegraphics[width=\textwidth]{HVY6.png}
\end{frame}

% ------------------------------------------------

\begin{frame}
  \frametitle{Initial Conditions and Inequality}\label{empiricalpreliminaries}
\includegraphics[width=\textwidth]{HVY7.png}
\end{frame}

% ------------------------------------------------

\begin{frame}
  \frametitle{What are your thoughts?}
  \begin{itemize}
  \item ?
\end{itemize}
\end{frame}


% ------------------------------------------------

\section{Conclusion}
% ------------------------------------------------


\begin{frame}
  \frametitle{Conclusion}
  \begin{itemize}
  \item Inequality in a heterogeneous agents environment.
  \item Next time: Estimation/linearization?
  \item Start your final projects! Don't wait to last minute.
  \end{itemize}
\end{frame}


\end{document}
