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\documentclass[11pt]{article}

\usepackage{mathprelude}   % geometry, all math/econ macros, theorems, hyperref, cleveref
\usepackage{meconbib}      % biblatex + biber (default). Older journals: \usepackage[natbib]{meconbib}
\bibresource{references}   % -> references.bib in this folder

\title{Lecture Notes: A Topic in Microeconomic Theory}
\author{Gavin Qu}
\date{\today}

\begin{document}
\maketitle
\tableofcontents
\bigskip

\section{Introduction}
These notes show off the shared macros. The reals are \(\R\), an expectation is
\(\E[X]\), and a maximizer is \(\displaystyle x\eqm \in \argmax_{x \in \R} f(x)\).
The classic reference on asymmetric information is \textcite{akerlof1970}.

\section{Definitions and results}

\begin{definition}[Convex set]\label{def:convex}
A set \(C \subseteq \R^{n}\) is \emph{convex} if
\(\lambda x + (1-\lambda) y \in C\) for all \(x,y \in C\) and \(\lambda \in [0,1]\).
\end{definition}

\begin{assumption}\label{as:concave}
The function \(\ph : \R^{n} \to \R\) is concave and integrable.
\end{assumption}

\begin{theorem}[Jensen's inequality]\label{thm:jensen}
Under \Cref{as:concave}, \(\;\ph(\E[X]) \ge \E[\ph(X)]\) for any integrable \(X\).
\end{theorem}

\begin{proof}
Let \(L\) be a supporting line of \(\ph\) at \(\E[X]\). Concavity gives
\(\ph(x) \le L(x)\), so \(\E[\ph(X)] \le \E[L(X)] = L(\E[X]) = \ph(\E[X])\).
\end{proof}

By \Cref{thm:jensen}, the variance \(\Var(X) = \E[X^2] - (\E X)^2 \ge 0\).

\begin{remark}
Set-builder notation with a scaling bar: the upper contour set is
\(\setb{x \in \R^{n} \given \ph(x) \ge c}\), which is convex by \Cref{def:convex}.
\end{remark}

\printrefs
\end{document}