From 966ab7e12fd61fcf94db0a36bd8326b750a4d417 Mon Sep 17 00:00:00 2001
From: Xuran Zeng <129227307+xzeng464@users.noreply.github.com>
Date: Thu, 31 Oct 2024 09:38:56 -0400
Subject: [PATCH] Update hotz_miller.tex

---
 Week 11- Single Agent Dynamics II/hotz_miller.tex | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/Week 11- Single Agent Dynamics II/hotz_miller.tex b/Week 11- Single Agent Dynamics II/hotz_miller.tex
index 829c8e1..6f9a7ec 100644
--- a/Week 11- Single Agent Dynamics II/hotz_miller.tex	
+++ b/Week 11- Single Agent Dynamics II/hotz_miller.tex	
@@ -173,8 +173,8 @@ \section*{Now for real}
 \begin{frame}{Arcidiacono Miller Lemma: Proof}
 \begin{align*}
 V\left(x\right) &=  \int\max_{j}\left\{ v_{j}\left(x\right)+\varepsilon_{j}\right\} dG\left(\varepsilon_{j}\right)\\
- &= \int\max_{j}\left\{ v_{j}\left(x\right)-v_{a}\left(x\right)+\varepsilon_{j}\right\} dG\left(\varepsilon_{j}\right)-v_{a}\left(x\right)\\
- &=  \int\max_{j}\left\{ \phi_{ja}\left(\mathbf{p}\left(x\right)\right)+\varepsilon_{j}\right\} dG\left(\varepsilon_{j}\right)-v_{a}\left(x\right)
+ &= \int\max_{j}\left\{ v_{j}\left(x\right)-v_{a}\left(x\right)+\varepsilon_{j}\right\} dG\left(\varepsilon_{j}\right)+v_{a}\left(x\right)\\
+ &=  \int\max_{j}\left\{ \phi_{ja}\left(\mathbf{p}\left(x\right)\right)+\varepsilon_{j}\right\} dG\left(\varepsilon_{j}\right)+v_{a}\left(x\right)
 \end{align*}
 Letting $\psi_{a}\left(\mathbf{p}\left(x\right)\right)=\int\max_{j}\left\{ \phi_{ja}\left(\mathbf{p}\left(x\right)\right)+\varepsilon_{j}\right\} dG\left(\varepsilon_{j}\right)$ completes the proof
 \end{frame}
@@ -431,4 +431,4 @@ \section*{Now for real}
 
 
 
-\end{document}
\ No newline at end of file
+\end{document}