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\title{\LARGE{\textbf{Spending Allocation under Nominal Uncertainty:}\vspace{3mm}} \\ \small{\LARGE{\textbf{A Model of Effective Price Rigidity}}}\vspace{5mm}}

\author[shortname]{\vspace{-2mm} \hspace{-3mm} Gaetano GABALLO \hspace{28mm}   \and Luigi PACIELLO }
\institute{\vspace{10mm} \hspace{-3mm} HEC Paris and CEPR  \hspace{37mm} EIEF and CEPR}

\date{JEEA Issue 23-6, December 2025}



\begin{document}

\begin{frame}
\titlepage

\end{frame}

\begin{frame}
\frametitle{The Paper in a Nutshell}

\onslide<1-11> $\star$ Countercyclical markups in aggregate key to get $ \pi \uparrow$ and $c \uparrow$
\bigskip

\onslide<2-11>\hspace{1cm} workhorse NK : ~~~~~$~~{\color{red}p_j^{post}/w~\downarrow}~~~~~$~~~~~~$~~\onslide<3-11>\pi^{paid}/\pi^{post} \approx 1~~$
\bigskip

\onslide<4-11>\hspace{1cm} this paper :~~~~~~~~~~~$~~p_j^{post}/w=\bar \mu_j ~~$~~~~~~ $~~{\color{red}\pi^{paid}/\pi^{post} ~\downarrow}$

\medskip
\onslide<3-11>\hspace{1cm}{\color{white} this paper :}~~~~~~~~~{\footnotesize{$\underset{\hyperlink{evidence2}{\beamerbutton{facts $\rightarrow$}}}{\text{markups distribution ${\color{white} \int}$}}$} }~~~~~~ {\footnotesize{demand allocation} }

\bigskip 
\onslide<5-11>$\Rightarrow$ Shopping is frictional +  Inflation triggers Hs' price hunting. 

\medskip

\bigskip

\onslide<6-11>$\star$  Search incentives based on Hs' nominal confusion ($\sim$ Lucas-72): $ \hyperlink{evidence1}{\beamerbutton{facts $\rightarrow$}}$ \medskip

\onslide<7-11>\hspace{0.5cm} $ - $ Hs' perceptions of the nominal level based on local prices \onslide<8-11>  ($\not =$ NK)  \medskip
\label{back_v2}

\onslide<9-11>\hspace{0.5cm} $ - $ Aggregate nominal shocks correlate mistakes on hunting gains  

\bigskip \bigskip

\onslide<10-11>$\star$ A {\color{red}demand-driven} mechanism of \textbf{aggregate} countercyclical markups...
\medskip 

\onslide<11>\hspace{1cm}...with fixed (but {\color{blue}increasing with Hs' unc.}) firm-level markups.

\end{frame}

\begin{frame}
\frametitle{Related Literature}

\bigskip

\begin{itemize}
\item \textbf{Consumers' search in GE}: {\footnotesize Coibion,  Gorodnichenko, and Hong (AER, 2015); Kaplan and Menzio (JPE,2016)}
\begin{itemize}  \small {\color{blue}
\item[-] Them: real determinants (unemployment) of search cost; 
\item[-] Us: real search cost are fixed, nominal shocks influence exp. payoffs;}
\end{itemize}
\smallskip

\item \textbf{Extensive margins}: {\footnotesize Phelps and Winter (1970), Rotemberg and Woodford (1999), Paciello, Pozzi, and Trachter (IER, 2019), Michelacci, Paciello, Pozzi (2020)} - {\color{blue}No nominal uncertainty;}

\bigskip

\item \textbf{Learning from Prices (IO)}: {\footnotesize Fishman (QJE, 1996), Benabou and Gertner (REStud,1993), Janssen and Shelegia (AER, 2015)}   - {\color{blue} No heterogeneous costs of search, infinite demand elasticity;}

\bigskip

\item \textbf{Learning from Prices - Uncertainty (Macro)}:{\footnotesize Lucas (AER, 1972), Amador and Weill (JPE, 2010), Gaballo (REStud, 2018), Chahrour and Gaballo (REStud, forth.)}  - {\color{blue}Learning from \text{competitive} prices: no signaling power;}

\bigskip

\item \textbf{Households' uncertainty}: {\footnotesize Angeletos  and  La’  O (JPE, 2020);  Farhi  and  Werning (AER, 2019); Gabaix (AER, 2020); McKay, Nakamura and Steinsson (AER, 2016) etc... + Mackowiak and Wiederholt (ReStud, 2015)}   - {\color{blue}Non-neutrality relies anyway on firms' rigidity.}

\end{itemize}
\end{frame}


\begin{frame}{Outline}

\begin{itemize}

\item Model

\bigskip

\item Demand-Driven Business Cycles  \smallskip
\begin{itemize}
\item Aggregate Counter-Cyclical Markup \smallskip
\item Firm-level Markups: A-cyclical but Endogenous to Hs' Uncertainty \smallskip
\item Welfare 
\end{itemize}

\bigskip

\item Empirics: a test of inflation-driven shopping reallocation

\bigskip

\item Extensions \smallskip
\begin{itemize} 
\item Households' vs Local Firms' Uncertainty \smallskip
\item Targeted Communication \smallskip
\item Local Varieties \smallskip
\item Firms' Common Uncertainty (sticky prices) 
\end{itemize}
\end{itemize}
\end{frame}

\begin{frame}{Outline}

\begin{itemize}

\item Model

\bigskip {\color{gray}

\item Demand-Driven Business Cycles  \smallskip
\begin{itemize}
\item \color{gray} Aggregate Counter-Cyclical Markup \smallskip
\item Firm-level Markups: A-cyclical but Endogenous to Hs' Uncertainty \smallskip
\item Welfare 
\end{itemize}

\bigskip

\item Empirics: a test of inflation-driven shopping reallocation

\bigskip

\item Extensions \smallskip
\begin{itemize} 
\item \color{gray} Households' vs Local Firms' Uncertainty \smallskip
\item Targeted Communication \smallskip
\item Local Varieties \smallskip
\item Firms' Common Uncertainty (sticky prices) 
\end{itemize}}
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Elements of the model}

\begin{itemize}
\item Two markets for goods: local (markups $>0$) and competitive (no markups)

\bigskip

\item Hs exert shopping effort to buy in competitive markets

\begin{itemize}
\item[-] drive to distant locations (Walmart  vs  local Safeway)\smallskip
\item[-] search info about local markets on temporary sale \smallskip
\item[-]  search info to arbitrage across shops for different varieties \smallskip
\item[-] behavioral/psycological, shopping habits 
\end{itemize}

\bigskip
\item Local productivity blurs the correlation of local prices with aggregate nominal shocks

\bigskip
\item When deciding on shopping effort Hs forecast inflation with local prices 

\begin{itemize}
\item[$\rightarrow$] firms set prices without any friction 
\end{itemize}
 
\end{itemize}

\end{frame}

\begin{frame}
\frametitle{Model: Households}

\bigskip

Household $i \in [0,1]$ on island $j \in [0,1]$:
\begin{eqnarray} \label{eq-HH-utility}
& &E \left[\sum_{\tau=t}^{\infty}\beta^{\tau-t} \left(\frac{c_{ij\tau} ^{1-\frac{1}{\gamma}}}{1-\frac{1}{\gamma}}\,- \, \varphi \ell_{ij\tau} - {\color{red}  \,\psi_{ij}\, s_{ijt}} \right) \big | \Omega^u_{i,j,t} \right]
\end{eqnarray}
{\small with \alert{shopping effort cost $ \, \psi_{ij}$}, with $\color{red}s_{ijt} \in \{0,1 \}$. \pause Her budget set is:} 
\begin{equation} \label{eq-HH-budget}
 c_{ijt}\, \mathcal{P}({\color{red}s_{ijt}}) + R^{-1}_{t} b_{ijt} = W_t \ell_{ijt} + b_{ijt-1}-T_t
\end{equation}\pause 

\vspace{-0.5cm}
\begin{itemize}
\item[ ] {\small $\mathcal{P}({\color{red}0})=p_{jt}$ price of the monopolistic firm in island $j$} \smallskip
\item[ ] {\small$\mathcal{P}({\color{red}1})=P_t$ price in a competitive market }
\end{itemize}  \medskip \pause
where \alert{$\psi_{ij}$} is distributed as in a generalized Pareto\[{\color{red}G(\psi)=1-\kappa \left(1-\frac{\psi}{\Psi} \right)^{\frac{\lambda}{\gamma-1}},} \]
with $\kappa>0, {\color{blue}\lambda>0}$, and $\Psi=\frac{\varphi^{\gamma-1}}{\gamma-1}$  with support 
\begin{align*}
 [\underline{\psi}, +\infty] \text{ with } \gamma \leq 1\hspace{1.5cm} [\underline \psi , \Psi+\underline \psi] \text{ with } \gamma>1
\end{align*}
where $\underline{\psi}$ such that $G(\underline{\psi})=0$.
\smallskip \pause
\end{frame}

\begin{frame}
\frametitle{Model: Timing}


\begin{figure}[htb!] 
\begin{center}

\only<2-5>{{\color{red}Stage 1,}}  \only<4-5>{{\color{blue}Stage 2,}} \only<5>{{\color{teal}Stage 3}}

\vspace{1cm}

{\footnotesize
\begin{tikzpicture}[x=140] 
        \draw (-1,0) -- (0.25,0);   % draw line
          \draw[] (-0.7,-0.1)  --  (-0.7,0.1);  % draw tick
         \draw (-0.7, 0) node[below=10pt] {Shocks and prices};
         \draw (-0.7, 0) node[below=20pt] { $P_t$ and $p_{jt}$ realize}; \pause
        \draw[] (-0.1,-0.1) -- (-0.1,0.1); % draw tick
         \draw (-0.1, 0) node[below=10pt] {Observe};
         \draw (-0.1, 0) node[below=20pt] { \color{red}$\Omega^s_{ijt}=\{p_{jt}\}\cup \Omega_{t-1}$ }; 
         \draw[] (0.25,-0.1) -- (0.25,0.1);  \pause % draw tick 
         \draw[] (0.25,0.0) -- (0.65,1.5) ; % draw tick
          \draw (0.4, 1.0) node[above=0pt] {${\color{red}s_{ijt}}=0$};
          \draw[] (0.25,0.0) -- (0.65,-1.5) ; % draw tick
          \draw (0.4, -1.4) node[above=0pt] {${\color{red}s_{ijt}}=1$}; \pause
         \draw (0.95, 2.0) node[below=0.0pt] {{\color{blue}$c_{ijt}$} given {\color{blue}$\Omega^c_{ijt}=\Omega^s_{ijt}$}};
            \draw (0.84, -1.5) node[above=0.0pt] {Incur cost $\psi_{ij}$};
        \draw (1.0, -1.5) node[below=0.0pt] {{\color{blue}$c_{ijt}$} given {\color{blue}$\Omega^c_{ijt}=\{ P_{t}, \Omega_{ijt}^s \} $}} ; \pause
\end{tikzpicture}
}
\end{center}
\end{figure}

\vspace{1cm}

\only<5>{ \center {\color{teal}$\{ \ell_{ijt}, b_{ijt}$\}} in centralized markets $\Rightarrow$ full info {\color{teal}$\Omega_t=\{ P_{t}, \Omega_{t-1} \}$}.} 

\end{frame}

\begin{frame}
\frametitle{Model: Firms}

\bigskip 

\hspace{0.2cm} - {\color{blue}\textit{Distant} competitive firms :} linear technology in labor, thus 

\begin{equation} 
\color{blue} P_t=W_t \label{eq-P}
\end{equation}

\bigskip \medskip \pause


\hspace{0.2cm} - {\color{red}\textit{Local} heterogeneous monopolist :} maximize real profits under full info
\begin{equation}  
\max_{{\color{black}p_{jt}}} \  \left\{ \, \underbrace{{\color{red}\mathcal{N}({\color{red}p_{jt}})}\, \mathcal{C}({\color{black}p_{jt}})}_{D({\color{black}p_{jt}})} \left(\frac{{\color{black}p_{jt}}}{{\color{black}W_t}}-z_{jt}\right)  \right\} \label{eq-firm-problem}
\end{equation} 
where : \smallskip

\begin{itemize}
\item[-] \small $\ln z_{jt} \sim N(z,\sigma_z)$ are i.i.d. productivity shocks
\item[-] {\color{red}$\mathcal{N}(p_{jt}) \subset (0,1)$ is the mass of agents buying local}
\item[-] $\mathcal{C}(p_{jt})$ denotes individual local consumption 
\end{itemize} \medskip

\end{frame}

\begin{frame}
\frametitle{Model: Aggrgeate Shocks and Monetary Policy}
\begin{itemize}


\item $\Pi_t\equiv P_{t}/P_{t-1} = W_{t}/W_{t-1}$,  because of \eqref{eq-P}. 

\bigskip \medskip \pause

\item Monetary Policy implements a standard Taylor rule
\begin{equation} \label{eq-inflation}
R_t=\beta^{-1}\left(\Pi_{t}\,e^{-\pi^w_t}\right)^\phi
\end{equation}
setting $T_t$ accordingly, with $\phi>1$, where $\pi^w_t \sim N(0,\sigma_{\pi^w}^2)$ represents a stochastic inflation target.
 \end{itemize}



\end{frame}

\begin{frame}
\frametitle{Equilibrium Definition}


\begin{definition}

\vspace{0.2cm}
\footnotesize{Given the past price level and inflation $\{P_{t-1},\Pi_{t-1}\}$ and a distribution of bond holdings $\{b_{ijt-1}\}_{i,j \in [0,1]\times[0, 1]}$, the realizations of aggregate and idiosyncratic shocks, $\pi_{t}$ and $\{z_{jt}\}_{j \in [0,1]}$ respectively, a \textit{log-normal equilibrium} is a collection of log-normally distributed prices $\{P_t, W_t, R_t, \{p_{jt}\}_{j \in [0,1]}\}$, and quantities $\{s_{ijt}, c_{ijt},b_{ijt},\ell_{ijt}\}_{i,j \in [0,1]\times[0, 1]}$ at time $t$ such that: \medskip
\begin{itemize}
\item[-] in each island $j$, each household $i$ chooses $\{s_{ijt}, c_{ijt},b_{ijt},\ell_{ijt}\}$ to maximize the expected utility \eqref{eq-HH-utility} subject to the sequence of budget constraints in \eqref{eq-HH-budget}; 
\item[-] $s_{ijt}$ and $c_{ijt}$ are chosen first, according to the infromation sets $\Omega^s_{ijt}$ and $\Omega^c_{ijt}$ respectively, $b_{ijt}$ and $\ell_{ijt}$ are chosen at the end of the period under perfect information; 
\item[-]  in each island $j$, $p_{jt}$ solves the firm problem in (\ref{eq-firm-problem}); 
\item[-] $W_t$ and $P_t$ are determined, respectively,  by equations \eqref{eq-P}  and (\ref{eq-inflation});
\item[-] $R_t$ and $T_t$ guarantee the equilibrium in the bond market, consistently with the monetary policy rule in \eqref{eq-inflation};
\item [-] $\int s_{ijt} di > 0$ holds almost surely in all islands.
\end{itemize} }
\end{definition}

\bigskip

\textbf{A non-trivial fixed point}: demand elasticity along extensive margins influence pricing, in turn pricing influences  shopping choices by affecting Hs' expectations on $p_{jt}/P_t$. 

\end{frame}

\begin{frame}
\frametitle{Markets open sequentially. at time $t$, H chooses:} 

\begin{itemize}
\item[stage 1] ${\color{red}s_{ijt}=0}$ iff:
\begin{equation*}
  E\left[V_{jt}(\psi_{ij}; 0)-V_{jt}(\psi_{ij}; 1) \,| \,  \left\{ {\color{blue}p_{jt}} \right\}  \right]\geq 0. \label{eq-psi}
\end{equation*}
\end{itemize}
i.e. iff $\psi_{ij}> \hat \psi_{jt}$, with $\hat \psi_{jt}$ being an equilibrium object. \pause

\medskip 

\begin{itemize}
\item[stage 2]  Consumption $c_{ijt}$:
\begin{equation*}
c_{ijt}^{-\frac{1}{\gamma}}=\varphi\,E\left[\frac{\mathcal{P}({\color{red}s_{ijt}})}{W_t} \, \Big | \,\left\{\mathcal{P}({\color{red}s_{ijt}}), {\color{blue}p_{jt}}  \right\}   \right]. 
\end{equation*}
\end{itemize}
  \pause

\begin{itemize}
\item[stage 3]  Labor $\ell_{ijt}$ and Bonds $b_{ijt}$:
\begin{equation*}
1 = \beta\,R_t \, E_t\left[\frac{W_t}{W_{t+1}}\, \Big | \{ \mathcal{P}({\color{red}s_{ijt}}), {\color{blue}p_{jt}} ,W_t, \, R_t\}\right].
\end{equation*}
\end{itemize} \pause \medskip
Monetary Policy implies  
 \[\pi_t=\frac{1}{\phi} E_t\left[\pi_{t+1}| \{ \mathcal{P}({\color{red}s_{ijt}}), {\color{blue}p_{jt}} ,W_t, \, R_t\}\right]+\pi^w_t \]
and finally 
 \[\pi_t=\pi^w_t \]
 as $\left[ \{ \mathcal{P}({\color{red}s_{ijt}}), {\color{blue}p_{jt}} ,W_t, \, R_t\}\right]$ does not contain info about $\pi^w_{t+1} $.

\end{frame}

\begin{frame}
\frametitle{Equilibrium Proposition}

A unique log-normal equilibrium at time $t$ exists such that:

\bigskip \pause

$\star$ the mass of households purchasing in island $j$ is given by
\begin{align*} \label{eq-N-opt}
\mathcal{N}_{jt}(p_{jt})&= \kappa^\frac{\lambda}{\gamma-1}\,e^{{\color{blue}-\lambda}\, \left( {\color{red}\ln p_{jt}- E\left[\ln P_t|\, \Omega^s_{jt}\right]+ \frac{1}{2}\mathcal{S}} \right)},
\end{align*}
with $\hat \psi_{jt}=\frac{\Psi}{\kappa}[1 -  e^{(1-\gamma)\, \left( \ln p_{jt}- E\left[\ln P_t|\, \Omega^s_{jt}\right]+ \frac{1}{2}\mathcal{S}\right)}] \geq 0$ for any $j$ almost surely;

\bigskip \pause

 $\star$ the consumption of household $i$ initially matched to  a island $j$ is
\begin{equation*}
c_{ijt}= \begin{cases} \mathcal{C}_{jt}(p_{jt}) \equiv C^*\,e^{{\color{blue}-\gamma}\, \left( {\color{red}\ln p_{jt}- E\left[\ln W_t|\, \Omega^s_{jt}\right]+ \frac{1}{2}\mathcal{S}}\right)} \label{eq-c-opt} & \text{if} \ \ \psi_{ij}> \hat \psi_{jt} \\
C^*\equiv \varphi^{-\gamma} & \text{otherwise} \end{cases};
\end{equation*}

\bigskip \pause

$\star$ the optimal price posted by the monopolistic firm in island $j$  is
\begin{equation*} \label{eq-p-opt}
p_{jt} =  \underbrace{\frac{(\gamma+\lambda)(1-{\color{blue}\omega})}{(\gamma+\lambda)(1-{\color{blue}\omega})-1}}_{\mu}\, \times \,\frac{W_t}{z_{jt} },
\end{equation*}
provided $z$ is suff. large relative to $\sigma_z$, and $(\gamma+\lambda)(1-\omega)>1$.
\end{frame}

\begin{frame}
\frametitle{Cartoon Summary} 
\only<2>{\includegraphics[width=\textwidth]{./pwp/presentation1-9.pdf}}
\only<3>{\includegraphics[width=\textwidth]{./pwp/presentation1-10.pdf}}
\only<4>{\includegraphics[width=\textwidth]{./pwp/presentation1-11.pdf}}
\only<5>{\includegraphics[width=\textwidth]{./pwp/presentation1-12.pdf}}
\only<6>{\includegraphics[width=\textwidth]{./pwp/presentation1-13bis.pdf}}

\only<2>{\color{white}Two types of firms: local (high markup) and competitive (low/no-markup) \medskip}

\only<3-5>{Two types of firms: local (high markup) and competitive (low/no-markup) \medskip} 

\only<2-3>{\color{white}\textbf{Real rigidity:} Shopping costs individual-specific effort.}

\only<4>{\textbf{Real rigidity:} Shopping costs individual-specific effort.}

\only<5>{\textbf{Info rigidity:} Shopping choice conditional to local prices only. }

\only<6>{Conditional to an aggregate nominal shock: 
\begin{itemize}
\item swimmers: $\uparrow$  consumption due to $\downarrow$ markup  \pause  $\Rightarrow$ {\color{red}\textbf{\textit{effective} price rigidity}}
\end{itemize}}

\end{frame}




\begin{frame}{Outline}

\begin{itemize}

\item Model

\bigskip 

\item Demand-Driven Business Cycles  \smallskip
\begin{itemize}
\item  Aggregate Counter-Cyclical Markup \smallskip
\item Firm-level Markups: A-cyclical but Endogenous to Hs' Uncertainty \smallskip
\item Welfare 
\end{itemize}

\bigskip

\item {\color{gray}Empirics: a test of inflation-driven shopping reallocation

\bigskip

\item Extensions \smallskip
\begin{itemize} 
\item \color{gray} Households' vs Local Firms' Uncertainty \smallskip
\item Targeted Communication \smallskip
\item Local Varieties \smallskip
\item Firms' Common Uncertainty (sticky prices) 
\end{itemize}}
\end{itemize}
\end{frame}


\begin{frame}
\frametitle{Demand-Driven Business Cycle Fluctuations}


Agg. consumption (switchers + non-switchers) of agents type $j$: \medskip
 \[ C_{jt}-C^* =  \underbrace{ \mathcal{N}(p_{jt})}_{\substack{\text{mass of local buyers} \\ \downarrow \text{ with } P_{t} \\ \text{ {\color{blue} less positive}}}}  \quad \times \quad \underbrace{\left(\mathcal{C}(p_{jt})-C^* \right) }_{\substack{\text{local consumption loss} \\ \downarrow \text{ with } P_{t}  \\ \text{ {\color{red} more negative}}}} \]
 
 \bigskip  \bigskip \pause
 
 In particular:   
 
\begin{align*}
\ln C_t - \ln \bar C \, \approx \, \left[{\color{blue}\lambda\,\mu(\omega)^\gamma-\lambda}\, {\color{red}- \, \gamma}\right]\,\bar\alpha\,(1-\omega) \, \pi_t, \label{eq-c-eff}
\end{align*}
with $\bar\alpha$ denoting the steady state share of expenditure in the local market, and $\bar C$  steady state aggregate consumption.

\bigskip \medskip \pause

$\star $  N.B.: {\color{blue}With $\lambda=0$ the comovement would be negative!} \medskip \pause

- This is why firms {\color{red} must} be bad compared to Hs in the NK framework.


\end{frame}

 

\begin{frame}
\frametitle{Aggregate Countercyclical Markup}

 \bigskip

The effective average markup:
\begin{align*}
\mathcal{M}^{eff}_t\equiv \frac{P_t^{eff}}{W_t} =  1 + {\color{red}\alpha_t} \, \left(\mu-1 \right),
\end{align*}
where $\alpha_t$ denotes the market share of local sellers and is given by 
\begin{equation*}
{\color{red}\alpha_t} = \frac{\bar{\mathcal{N}}\,\bar{\mathcal{C}}\,e^{-(\lambda+\gamma)\,(1-\omega) \pi_t}}{1-\bar{\mathcal{N}}\,e^{-\lambda\,(1-\omega) \pi_t}+\bar{\mathcal{N}}\,\bar{\mathcal{C}}\,e^{-(\lambda+\gamma)\,(1-\omega) \pi_t}}. 
\end{equation*}
where $\bar{\mathcal{C}}$, $\bar{\mathcal{N}}$ denote steady states values. 

 \bigskip\pause

${\color{red}\alpha_t}$ co-moves negatively with $\pi_t$, in fact: 
\begin{equation*}
\ln \alpha_t-\ln\bar\alpha \approx  -\left[\bar\alpha\,\lambda\,\mu^\gamma+(1-\bar\alpha)\,(\gamma+\lambda)\right]\, (1-\omega) \pi_t
\end{equation*}
where $\bar\alpha$ is the value of $\alpha_t$ at $\pi_t=0$.

\bigskip \pause

\begin{center}{\color{red}The aggregate markup is countercyclical.}\end{center}

\pause

\begin{center}{\color{blue}Let us now turn to how $\omega$ shapes $\mu$ }\end{center}

\end{frame}

\begin{frame}
\frametitle{Learning from Prices and Local Markups}

\medskip

\, \,  $\rightarrow \, \, {\color{blue}\omega}$ denotes the elasticity of Hs' inflation exp. w.r.t. local prices.
\begin{align*}
E[\pi_{t}\, | \, \Omega^{s}_{t}]  &=\,\underbrace{\frac{\sigma_z^{-2}}{\sigma_{\pi}^{-2}+\sigma_{z}^{-2}}}_{{\color{blue}\omega}}\,(\,\ln p_{jt} \,- \, E[\ln p_{jt}|\Omega_{t-1}]\, ).
\end{align*}


\medskip \pause

\begin{figure}
\includegraphics[width=0.5\textwidth,trim={0 0cm  0 14cm}]{figures/slides_markupGG}
\end{figure}

\smallskip  

The optimal markup set by firm $j$ is endogenous but {\color{blue}time/state-invariant}: \medskip
\begin{equation*} 
\mu({\color{blue}\omega})= \begin{cases} \frac{(\gamma+\lambda)(1-{\color{blue}\omega})}{(\gamma+\lambda)(1-{\color{blue}\omega})-1}~~~~~~~~~\text{with}~~~~~~\omega<\frac{\gamma+\lambda-1}{\gamma+\lambda},\\
+\infty \hspace{2.22cm}\text{with}~~~~~~\omega \geq \frac{\gamma+\lambda-1}{\gamma+\lambda}, \end{cases}
\end{equation*}
\end{frame}

\begin{frame}
\frametitle{Confusion and discretion in pricing}


\only<1>{\begin{figure}
\includegraphics[width=\textwidth]{./figures/figure_markup_GG_bis1.eps}
\end{figure}

\bigskip}


\only<1>{\color{blue}Without any signaling power we have the standard firm problem.}

\only<2>{\begin{figure} 
\includegraphics[width=\textwidth]{./figures/figure_markup_GG_bis2.eps}
\end{figure}

\bigskip}


\only<2>{\color{blue}Signalling power induces firms to increase their markups.}

\only<3-4>{\begin{figure}
\includegraphics[width=\textwidth]{./figures/figure_markup_GG_bis3_old.eps}
\end{figure}

\bigskip}


\only<3>{\color{blue}However, as Hs anticipate $\mu^*$, their demand shifts down!}

\only<4>{\color{red} Signaling power creates a commitment problem, hurting firms' profits.}


\end{frame}



\begin{frame}
\frametitle{Demand-Driven Business Cycle Fluctuations}

\begin{figure}[htb!] 
\center
\caption{The consumption-inflation comovement}\label{fig-C}

\only<1-2>{\includegraphics[width=7cm, height=5cm]{./figures/figure_theory1_slides_v1.pdf}}

\only<3>{\includegraphics[width=7cm, height=5cm]{./figures/figure_theory1_slides_v2.pdf}} 

\only<4>{\includegraphics[width=7cm, height=5cm]{./figures/figure_theory1_slides_v3.pdf}}


\end{figure}

\medskip

\only<1>{\center{{\color{red}dashed :} ${\color{red}(\gamma=1, \omega=0)} \Rightarrow \left[\lambda\,\mu(\omega)^\gamma-\lambda\, - \, \gamma\right] {\color{red}= 0}$}}

\only<2>{\center{{\color{blue}solid :} $({\color{blue}\gamma=1, \omega>0}) \Rightarrow \left[\lambda\,\mu(\omega)^\gamma-\lambda\, - \, \gamma\right] $ {$\color{blue}> 0$}}}



\only<3>{\center{dotted :  $(\gamma=1.5, \omega>\bar \omega) \Rightarrow \left[\lambda\,\mu(\omega)^\gamma-\lambda\, - \, \gamma\right] > 0$}}

\only<4>{\center{dotted : $(\gamma=0.5, \omega \geq 0) \Rightarrow \left[\lambda\,\mu(\omega)^\gamma-\lambda\, - \, \gamma\right] > 0$}}

\end{frame}


\begin{frame}{Welfare and Policies of Uncertainty Reduction}
\medskip
\begin{center}
$\uparrow$ inflation volatility~~~$\Rightarrow$~~~$\uparrow$ $\omega=\frac{1}{1+\frac{\sigma^2_z}{\sigma^2_\pi}}$~~~$\Rightarrow$~~~$\uparrow$ $\mu$
 \end{center}
 
 \bigskip
 
\begin{figure}[!hbtp] 
\begin{center}
\includegraphics[width=\textwidth]{welfareGG_march11_final.eps}
\begin{minipage}{\textwidth}
\footnotesize{Note: The figure shows how welfare varies with $\sigma^2_\pi$. The vertical dotted line corresponds to the baseline value of $\sigma_\pi=0.35\%$. Welfare is calculated in equivalent consumption growth with respect to baseline consumption. We fixed $\gamma=.5, \lambda=4$ and $\varphi=1$. Calibration is such that at baseline, $\rho=0.1$ and $\omega=0.29$ for the dashed line and $\omega=0.19$ for the solid line.}
\end{minipage}
\end{center}
\end{figure}
\end{frame}





\begin{frame}{Outline}

\begin{itemize}

\item Model

\bigskip 

\item Demand-Driven Business Cycles  \smallskip
\begin{itemize}
\item  Aggregate Counter-Cyclical Markup \smallskip
\item Firm-level Markups: A-cyclical but Endogenous to Hs' Uncertainty \smallskip
\item Welfare 
\end{itemize}

\bigskip

\item Empirics: a test of inflation-driven shopping reallocation

\bigskip

\item {\color{gray}Extensions \smallskip
\begin{itemize} 
\item \color{gray} Households' vs Local Firms' Uncertainty \smallskip
\item Targeted Communication \smallskip
\item Local Varieties \smallskip
\item Firms' Common Uncertainty (sticky prices) 
\end{itemize}}
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Empirics: Effective vs Posted Inflation} \label{back}

\smallskip

- \textbf{effective inflation}: $\pi_{t}^{eff}$ effective per unit price \smallskip

- \textbf{posted inflation}: $\pi_{t}^{pos}$ counterfactual with fixed spending shares \smallskip \pause
\bigskip

-\textbf{ inflation gap}  $\pi_{t}^{eff} - \pi_{t}^{pos}$ from spending reallocation \smallskip

\pause
\begin{itemize}
\item covariance with inflation: $0$ in NK $\, \, \hyperlink{figNK}{\beamerbutton{fig NK}}$; \pause
\item  $\not=0$ in our model 

$$
\pi_t^{eff}-\pi^{pos}_t \approx  -\bar\alpha\,(\mu-1)\,\underbrace{\left[\bar\alpha\,\lambda\,\mu^\gamma +(1-\bar\alpha)\,(\lambda+\gamma) \right]}_{\log \alpha_t - \log \bar \alpha}\,(1-\omega)\,\pi_t.
$$

\bigskip

\item $\color{red}<0$ if households less informed than firms

\end{itemize} 
\end{frame}

\begin{frame}
\frametitle{Data} 

\begin{itemize}

\item Information Resources Inc. (``IRI"), weekly price and quantity information 2001-2011 on items (UPC $\times$ Store $\times$ Market). 

\bigskip 
\item Method: Coibion et al. (AER15), Gagnon et al. (AER17)
\medskip 
\item \textcolor{blue}{Effective} vs \alert{Posted} price and inflation for UPC $j$ in market $m$:
\begin{align*}
\color{blue} p^{eff}_{mj,t} &= \color{blue} \frac{\sum_{s\in S_m} TR_{mjs,t}}{\sum_{s\in S_m} TQ_{mjs,t}}\implies \color{blue}  \pi^{eff}_{mj,t}=\ln{p^{eff}_{mj,t}}-\ln{p^{eff}_{mj,t-1}}\\
\color{red}  p^{pos}_{mj,t} &= \color{red}  \sum_{s\in S_m} \bar w_{msj} \frac{TR_{mjs,t}}{TQ_{mjs,t}}\implies \color{red} \pi^{pos}_{mj,t}=\ln{p^{pos}_{mj,t}}-\ln{p^{pos}_{mj,t-1}}
\end{align*}


\item Average over $j$ within category $c$ and market $m$, monthly: $$\pi^{eff}_{mc,t}=\frac{1}{J}\sum_{j\in c}\pi^{eff}_{mj,t} \ \ \text{and} \ \ \  \pi^{pos}_{mc,t}=\frac{1}{J}\sum_{j\in c}\pi^{pos}_{mj,t}$$ 

\item Average over $c$ within market $m$, monthly: $\pi_{m,t}=\frac{1}{N_c}\sum_{c \in S_m}\,\pi^{pos}_{mc,t}$.

\end{itemize}
\end{frame}



\begin{frame}{Effective vs Posted Inflation}
\small

\begin{table}[h!]

{
\begin{tabular}{l|ccccc}
\hline\hline
& & &  &    \\
$\pi^{eff}_{m,t}-\pi^{pos}_{m,t} $ & (1) & (2) &  (3) & (4) & (5) \\
 & & &  & \\
 \hline
 & & &  && \\
 &\multicolumn{5}{c}{\textbf{CGH's sample}}\\
Inflation, $ \pi_{m,t}$   &$-0.22^{***}$ & &$-0.24^{***}$  \\
  & (0.060)&  &  (0.06)    \\
Unemployment, $u_{m,t}$   & &$-0.12^{***}$   &$-0.14^{***}$ \\
 & & (0.034) &  (0.035) \\
FFR shock, $\epsilon_{t}$      & &  & & $0.04^{***}$ & $0.04$ \\
 & &  &   &  (0.007) &  (0.064)\\
\hline
& & &  &&  \\ 
 &\multicolumn{5}{c}{\textbf{GLSS's sample}}\\
Inflation, $\pi_{m,t}$   &$-0.09^{***}$ & &$-0.09^{***}$  \\
  &(0.028) & &  (0.029)    \\
Unemployment, $u_{m,t}$   &&$0.03$   &0.015 \\
 && (0.032) &  (0.033) \\
FFR shock, $\epsilon_{t}$     & &  & & $0.03^{***}$&$0.08^{**}$   \\
 & &  &   &   (0.004)&  (0.032)\\

 \hline
Robust S.E.      & Yes& Yes & Yes& Yes& Yes\\
   Stratum F.E.  & Yes& Yes & Yes& Yes& Yes \\
Month F.E.   & Yes & Yes  & Yes& No& No\\
\hline 
\hline 
\end{tabular}%
}
\begin{minipage}{11cm}
\vspace{0.2cm}
\footnotesize{Note: We regress the gap between paid and posted price inflation in each stratum on the market specific inflation rate in columns (1) and (3), on the market specific unemployment rate in columns (2) an (3), on the FFR shocks obtained from Agrippino et al. (2015, 2016, 2018) with the narrative method in column (4) and high frequency method in column (5).}
\end{minipage}
\end{table}


\end{frame}

\begin{frame}{Outline}

\begin{itemize}

\item Model

\bigskip 

\item Demand-Driven Business Cycles  \smallskip
\begin{itemize}
\item  Aggregate Counter-Cyclical Markup \smallskip
\item Firm-level Markups: A-cyclical but Endogenous to Hs' Uncertainty \smallskip
\item Welfare 
\end{itemize}

\bigskip

\item Empirics: a test of inflation-driven shopping reallocation

\bigskip

\item Extensions \smallskip
\begin{itemize} 
\item  Households' vs Local Firms' Uncertainty \smallskip
\item Targeted Communication \smallskip
\item Local Varieties \smallskip
\item Firms' Common Uncertainty (sticky prices) 
\end{itemize}
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{An extended info structure}

$\star$ Local firms' uncertainty about inflation innovations:
$$E[\pi_{t}\, | \, \Omega^{f}_{jt}]={\color{red}\delta} \left(\pi_t + u_{jt}\right)$$
so that $p_{jt} =  \mu\,  e^{E\left[\ln W_t|\, \Omega_{jt}^f\right]+\frac{1}{2}\mathcal{V} - \ln z_{jt} }$;

\bigskip \bigskip \pause

$\star$ Households' expectations are expressed as
\begin{align*}
E[\pi_{t}\, | \, \Omega^{s}_{jt}]  &={\color{brown}\rho}\left(\pi_t + \nu_{jt}\right)+{\color{blue}\omega}\,(\ln p_{jt} -E[\ln p_{jt}|\Omega_{t-1}] ),
\end{align*}
where $\nu_{jt}\sim N(0,\sigma^2_\nu)$ and $u_{jt}\sim N(0,\sigma_u^2)$ i.i.d.

\bigskip \bigskip \pause

$\star$ Households perceived real local price is 
$$
\int_0^1\ln p_{jt}- E\left[\ln P_t|\, \Omega^s_{jt}\right] dj = \ln \mu + \underbrace{(\delta-\zeta)}_{\text{info gap}}\,\pi_t.
$$
where $\zeta \equiv  {\color{blue}\omega} {\color{red}\delta} +{\color{brown}\rho}$.
\end{frame}

\begin{frame}
\frametitle{Information gap}

\begin{figure}[htb!] 
\center
\caption{The consumption-inflation comovement and the information gap}\label{fig-C-gap}

\includegraphics[width=8cm, height=5cm]{./figure_theory2.eps} 

\begin{minipage}{10cm}
\vspace{0.5cm}
\footnotesize{Note: We set $\sigma_z/\sigma_{\pi}$ so that $\omega=0.25$ and $\kappa$ so that $\bar\alpha=0.66$ in all simulations. We report the elasticity of consumption to an inflation shock at different combinations of $\lambda$ and $\gamma$, and for different values of $\delta-\zeta$ on the horizontal axis obtained either by varying $\delta$ and/or $\rho$. In the baseline specification we use $\lambda=4$ and $\gamma=1$ (solid blue line). The red dashed line plots a counterfactual where $\mu=(\lambda+\gamma)/(\lambda+\gamma-1)$. The black dotted line also sets $\lambda=0$.}
\end{minipage}
\end{figure}

\end{frame}

\begin{frame}
\frametitle{Local Varieties}

$c_{ijt}$ is an aggregate over the $X$ varieties,
\begin{equation*}\label{eq-c-j2}
c_{ijt} = X\,\left(\frac{1}{X}\sum_{x=0}^X c_{xijt}^\frac{\epsilon-1}{\epsilon}\right)^\frac{\epsilon}{\epsilon-1}
\end{equation*}
with $\epsilon>1$ being the elasticity of substitution across varieties, whose price is:
\begin{equation*}\label{eq-p2}
p_{jt} = \left(\frac{1}{X}\sum_{n=0}^N p_{xjt}^{1-\epsilon}\right)^\frac{1}{1-\epsilon}.
\end{equation*}

\medskip

Each local firm $x\in\{1, 2, ...., X\}$ chooses the price $p_{xjt}$ that maximizes profits 
\begin{equation*}  \label{eq-firm-problem2}
p_{xjt} = \textrm{argmax}_ {p} \,  \mathcal{N}_{jt}(p)\, \mathcal{D}_{xjt}(p)  \,\left(\frac{p}{W_t}-{z_{jt}}\right)
\end{equation*}
with $\mathcal{D}_{xjt}(p)=\int c_{xijt}\,di = c_{ijt}\left(\frac{p}{p_{jt}}\right)^{-\epsilon}di$. 

\medskip


In equilibrium, $\mu=\frac{\xi}{\xi-1}$ where
\begin{equation*} \label{eq-mu_X}
\xi=\left(1-\frac{1}{X}\right)\,\epsilon+\frac{1}{X}\,(\gamma+\lambda)(1-\omega).
\end{equation*}

\end{frame}


\begin{frame}
\center
\Huge Conclusions
\end{frame}


\begin{frame}
\frametitle{Conclusions}

\medskip

We have built on the idea that {\color{red} inflation provides incentives to more aggressive shopping.}  \bigskip \bigskip \pause

\textit{We turn up-down NK logic:} firms price frictionless but households act frictionally (shopping effort + learning from prices). 

\bigskip \bigskip  \pause

A \textit{pencil-and-hand} model where: \smallskip \pause

{\center {\color{blue} \textbf{Demand-driven Countercyclical Aggregate Markup +} }} \medskip \pause

\hfill {\color{red} \textbf{...but fixed firm-level markups!}}

\end{frame}

\begin{frame}
\center
\Huge Thanks
\end{frame}

\appendix
\backupbegin
\begin{frame}
\center
\Huge Appendix
\end{frame}

\begin{frame}
\frametitle{R.3. The response of the inflation gap} \label{figNK}
\begin{figure}
\center
\includegraphics[width=0.7\textwidth]{./figure_NK.eps} 
\end{figure}

\bigskip

With deflationary shocks consumers reallocate towards local markets to save shopping effort.  

\hspace{9cm} $\, \, \hyperlink{back}{\beamerbutton{back}}$
\end{frame}

\begin{frame}
\frametitle{Micro-evidence on Markup Cyclicality}\label{evidence2}
\begin{itemize}


\item \textbf{Macro models call for countercyclical markups}

\medskip

{\footnotesize{Bils, Klenow and Malin (2018): ``Thus, countercyclical price markups deserve a central place in business-cycle research."}}

\bigskip 

\item \textbf{Micro evidence seems pointing towards a-cyclical markups} 

\medskip

{\footnotesize{Anderson, Rebelo and Wong (2018): ``markups are relatively stable over time and mildly procyclical";


\medskip
 
Burstein, Carvalho and Grassi (2020): ``exploiting different reduced-form measures of markup cyclicality, two researchers may arrive at opposing conclusions even within a single dataset."} }

\bigskip  \bigskip  


\item[] \textbf{\color{red}Q: How to reconcile Macro call with Micro evidence?}

\end{itemize}
\bigskip  \bigskip

\hspace{9cm} $\, \, \hyperlink{back_v2}{\beamerbutton{back}}$

\end{frame}



\begin{frame}
\frametitle{How well households predict inflation?}\label{evidence1}
\hypertarget{Table}{}
\begin{figure}[!hbtp]
\begin{center} 
\includegraphics[width=\textwidth]{Michigan.pdf}
\end{center}
\end{figure}
\bigskip \bigskip
\tiny{{\color{gray}Source: D'Acunto et al. (2019)}}
\end{frame}



\begin{frame}
\frametitle{Three Facts on Expectations and Shopping}

\linespread{1}
\bigskip

\begin{itemize}

\item \textbf{Shopping prices influence expectations}

\smallskip

{\footnotesize{
Expectation Survey Bank of England: first source of information 

\smallskip

D'Acunto, Malmendier, Ospina and Weber (2021): CPI Household $\sim 0.2$;
CPI Frequency  $\sim 0.3$}}

\bigskip 

\item \textbf{Arbitrage across shops: valuable but limited}

\smallskip

{\footnotesize{Menzio and Kaplan (2016):
visiting one additional store: $\sim -0.6\%$ Indiv-CPI; shopping concentrated:  $~2.3$ shops visited on average per quarter}}

\bigskip 

\item \textbf{Expected$^2$ inflation less dispersed than experienced$^1$} 

\smallskip

{\footnotesize{
$^1$ Michigan Survey: inter. range $3.8\%$

\smallskip
 
$^2$ Kaplan and Schulhofer-Wohl (2017): inter. range $7.3\%$} }

\bigskip  


\item[] \textbf{\color{red}Q: how to reconcile the frictional household (micro) with output-inflation comovement (macro)?}

\end{itemize}

\hspace{9cm} $\, \, \hyperlink{back_v2}{\beamerbutton{back}}$

\end{frame}

\backupend

\end{document}
