\documentclass[12pt]{article}
\setlength{\textheight}{23cm}
\setlength{\textwidth}{17cm}
\setlength{\oddsidemargin}{-0.5cm}
\setlength{\evensidemargin}{-0.5cm}
\setlength{\topmargin}{-1.5cm}
%\setlength{\topmargin}{-2cm}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{natbib}
\usepackage{graphicx}
\usepackage{fancyhdr}
\usepackage{sectsty}
\usepackage{color}
\usepackage{ushort}
\usepackage{subfigure}
\usepackage{caption}
\usepackage{sidecap}
\usepackage{ragged2e}
\usepackage{url}
\usepackage{setspace}
\usepackage{multirow}


%\renewcommand{\baselinestretch}{1}
%\setstretch{1.5}
\setstretch{1.35}
%\setstretch{2}

\newcommand{\fignote}[2]{\begin{center}\parbox[c]{#1}{\footnotesize \setstretch{1} #2} \end{center}}
\newcommand{\fignoteapp}[2]{\begin{center}\parbox[c]{#1}{\footnotesize \setstretch{0.9} #2} \end{center}}
\newcommand{\partder}[2]{\frac{\partial #1}{\partial #2}}
\newcommand\independent{\protect\mathpalette{\protect\independenT}{\perp}}
\def\independenT#1#2{\mathrel{\rlap{$#1#2$}\mkern2mu{#1#2}}}
\newcommand{\txt}[1]{\quad \mbox{#1} \quad}
\newcommand{\undertilde}[1]{\underset{\widetilde{}}{#1}}
\newcommand{\diag}{\mbox{diag}}
\newcommand{\In}{\mbox{In}}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{proposition}{Proposition}
\newtheorem{corollary}{Corollary}
\newtheorem{definition}{Definition}
\newtheorem{assumption}{Assumption}
\newtheorem{claim}{Claim}

\newenvironment{packed_enum}{
	\begin{enumerate}
		\setlength{\itemsep}{0pt}
		\setlength{\parskip}{0pt}
		\setlength{\parsep}{0pt}
	}{\end{enumerate}}

\newenvironment{proof}[1][Proof]{\begin{trivlist}
		\item[\hskip \labelsep {\bfseries #1}]}{\end{trivlist}}
\newenvironment{example}[1][Example]{\begin{trivlist}
		\item[\hskip \labelsep {\bfseries #1}]}{\end{trivlist}}
\newenvironment{remark}[1][Remark]{\begin{trivlist}
		\item[\hskip \labelsep {\bfseries #1}]}{\end{trivlist}}
\newcommand{\qed}{\nobreak \ifvmode \relax \else
	\ifdim\lastskip<1.5em \hskip-\lastskip
	\hskip1.5em plus0em minus0.5em \fi \nobreak
	\vrule height0.5em width0.5em depth0.25em\fi}
\def\citeapos#1{\citeauthor{#1}'s (\citeyear{#1})}
\begin{document}
	\sectionfont{\centering\nohang\large\bfseries}
	\subsectionfont{\centering\nohang\normalsize\bfseries}
	\subsubsectionfont{\centering\nohang\normalsize\bfseries}
	
	
	
\section*{Supplement to Lecture 2: Definition of Competitive Equilibrium}
\begin{center} Macroeconomics, EC2B1 \\ Benjamin Moll \end{center}
In the lecture notes we sketched the definition of a competitive equilibrium for a general, abstract economy. This supplement fills in the gaps and provides the equations.

While the definition of a competitive equilibrium presented here is fairly general, it simplifies the notation somewhat and the maths is less precise than in a fully rigorous treatment. For such full generality and precision, see chapter 10.B of MasColell-Whinston-Green ``Microeconomic Theory" (MWG) referenced in the lecture notes. Either way, as stated in the lecture notes: the important thing is not the precise maths but to understand general structure.

\section{Some Notation}

\begin{itemize}
\item $I$ consumers (households) indexed by $i=1,...,I$
\item $J$ producers (firms) indexed by $j=1,...,J$
\item $K$ factors of production (inputs) indexed by $k=1,...,K$
\item $L$ final goods indexed by $\ell = 1,...,L$
\item Quantities
\begin{itemize}
\item $x_{\ell i}$: household $i$'s consumption of good $\ell$
\item $y_{\ell j}$: firm $j$'s production of good $\ell$
\item $e_{\ell i}$: household $i$'s endowment of good $\ell$
\item $\tilde{x}_{k i}$: household $i$'s supply of factor $k$
\item $\tilde{y}_{k j}$: firm $j$'s use of factor $k$
\item $\tilde{e}_{k i}$: household $i$'s endowment of factor $k$
\end{itemize}
\end{itemize}
Note: many books (MasColell-Whinston-Green,...) use more general notation: they denote both goods and inputs by $y_{1 j},...,y_{Lj}$ and use negative numbers $y_{\ell j} \leq 0$ for inputs. I personally find this confusing which is why I chose to use the somewhat less general but simpler notation presented here.



\section{Primitives of the general economy}
\begin{itemize}
\item Preferences: household $i$'s utility
$$u_i(x_{1i}, ...., x_{Li},\tilde{x}_{1i},...,\tilde{x}_{Ki})$$
\item Technology: firm $j$'s production function for producing good $\ell$
$$y_{\ell j} = f_j(\tilde{y}_{1j}, ..., \tilde{y}_{Kj}) $$
\item Resource constraints (feasibility):
\begin{align*}
\mbox{Goods:} \quad \underbrace{\sum_{i=1}^I x_{\ell i}}_{\textrm{total demand of good $\ell$}} &= \underbrace{\sum_{j=1}^J y_{\ell j} + \sum_{i=1}^I e_{\ell i}}_{\textrm{total supply of good $\ell$}}, \quad \mbox{all} \ \ell = 1,...,L\\
\mbox{Factors:} \quad \underbrace{\sum_{j=1}^J \tilde{y}_{k j}}_{\textrm{total demand of factor $k$}} &= \underbrace{\sum_{i=1}^I \tilde{x}_{k i} + \sum_{i=1}^I \tilde{e}_{k i}}_{\textrm{total supply of factor $k$}}, \quad \mbox{all} \ k = 1,...,K
\end{align*}
\end{itemize}

Note: as usual $\sum_{i=1}^I x_{i\ell} = x_{1\ell} + x_{2 \ell} + ... + x_{I\ell}$ and similarly for the other summations

\section{Definition of competitive equilibrium (CE)}
\paragraph{Definition:} a competitive equilibrium are quantities $\{x_{\ell i},y_{\ell j},\tilde{x}_{k i},\tilde{y}_{k j}\}$ and prices $\{p_{\ell},\tilde{p}_k\}$ for $\ell=1,...,L$, $k=1,...,K$, $i=1,...,I$ and $j=1,...,J$ such that:
\begin{enumerate}
\item Utility maximization: taking as given prices $\{p_{\ell},\tilde{p}_k\}$, households maximize utility subject to their budget constraints
\item Profit maximization: taking as given prices $\{p_{\ell},\tilde{p}_k\}$, firms maximize profits
\item Market clearing: demand = supply for each good and each factor
\begin{align*}
\mbox{Goods:} \quad \underbrace{\sum_{i=1}^I x_{\ell i}}_{\textrm{total demand of good $\ell$}} &= \underbrace{\sum_{j=1}^J y_{\ell j} + \sum_{i=1}^I e_{\ell i}}_{\textrm{total supply of good $\ell$}}, \quad \mbox{all} \ \ell = 1,...,L\\
\mbox{Factors:} \quad \underbrace{\sum_{j=1}^J \tilde{y}_{k j}}_{\textrm{total demand of factor $k$}} &= \underbrace{\sum_{i=1}^I \tilde{x}_{k i} + \sum_{i=1}^I \tilde{e}_{k i}}_{\textrm{total supply of factor $k$}}, \quad \mbox{all} \ k = 1,...,K
\end{align*}
\end{enumerate}
As emphasized in the lecture notes, the important thing is the general structure of a competitive equilibrium:
\begin{enumerate}
\item Households maximize taking prices as given
\item Firms maximize taking prices as given
\item All markets clear
\end{enumerate}
This structure will come up over and over again.

	
\end{document}
