1\section{Theoretical framework}\label{sec:theory}23We use a deliberately small model to organise the hypotheses and to be explicit about what posted-price data can identify. Retailer (banner) $r$ sells product $i$ in week $t$ to two consumer types in the spirit of \citet{varian1980} and \citet{salop1977}: a share $1-\lambda_i$ of \emph{loyal} shoppers who buy at their habitual banner if the posted price does not exceed their reservation value $v_i$, and a share $\lambda_i$ of \emph{shoppers} who observe all posted prices (for instance through a comparison service) and buy from the cheapest banner. Banner $r$ faces marginal procurement cost $c_{irt}$, a format-specific cost shifter $\kappa_r$ (service level, store size, location rents) and chooses a posted price $p_{irt}$ and a promotion state $s_{irt}\in\{0,1\}$. Promotions are temporary reductions below a \emph{reference} price $\bar p_{irt}$ that is costly to change (a menu cost $m$ in the sense of \citealp{golosov2007} and \citealp{midrigan2011}) and that loyal shoppers use as their norm.45\paragraph{Equilibrium structure.} With $R$ banners carrying the product, the standard result is a mixed-strategy equilibrium in which each banner posts its reference price $\bar p_{ir}=\bar p_{ir}(\kappa_r, c_{ir}, v_i)$ most of the time and randomises temporary reductions to attract shoppers; the probability of a promotion increases in $\lambda_i$ and in the number of rival banners, and the depth of the reduction is bounded by the gap between the reference price and the shopper-competitive price. Three properties are robust across variants of the model and generate our registered hypotheses.67\begin{enumerate}8\item \textbf{Persistent reference-price dispersion (H1).} Reference prices differ across banners by $\kappa_r$ and $c_{ir}$, which move slowly; hence log gaps between banners on the same product should be large relative to zero and autocorrelated across weeks, whereas a model in which banners price at a common marginal cost predicts gaps near zero and no persistence. Within a company, format differentiation ($\kappa_{\text{discount}}<\kappa_{\text{supermarket}}$) predicts a \emph{stable sign} of the gap (the discount banner is cheaper), which is testable on the Metro/Super~C and Maxi/Provigo pairs.9\item \textbf{Sales double the volatility of posted prices without moving the reference (H2, H3).} Because promotions are temporary deviations, the variance of weekly log price changes decomposes into a reference component and a promotion component; the ratio of total to reference volatility exceeds one by the promotion share and depth. The model with a menu cost on $\bar p$ predicts that the reference price after a sale equals the reference before it (no drift). The alternative ``reference-price inflation'' hypothesis---the retailer raises $\bar p$ before or after a sale so that the displayed discount looks larger---predicts a pre-sale elevation or a post-sale increase of the reference.10\item \textbf{Competitive spillovers are an equilibrium feature, not a causal response (H4).} In the mixed-strategy equilibrium promotions are independent draws conditional on the state; observed co-movement of prices across banners arises from common cost shocks $c_{it}$ and common calendar events (circular weeks), so lead--lag regressions of one banner's changes on another's identify a response only if a rival's change is exogenous. We have no such instrument; the model therefore tells us to treat contemporaneous and lagged associations as descriptive and to use \emph{leads} as a placebo: a response cannot precede its cause.11\end{enumerate}1213\paragraph{Measured inflation and the basket (H5, H6).} If the set of listings changes over time (entry of expensive items, exit of cheap ones), an average of posted prices moves even when every continuing item's price is unchanged; a matched-model index conditions on continuing items and recovers the price change of the fixed set. The difference is an identified object (the composition effect) that does not require quantities. The shopper's problem is a set-covering problem: choose at most $K$ banners and an assignment of items to banners to minimise the cost of a fixed basket; the model predicts decreasing returns to $K$ because the cheapest banner for one item is often the cheapest for others (a low $\kappa_r$ shifts all prices).1415\paragraph{What is not identified.} $\lambda_i$, $v_i$, $m$ and $c_{irt}$ are not identified from posted prices alone; neither are store-level $\kappa$ within a banner, since stores are unobserved---although the store-mixture oscillation of Section~\ref{sec:mixture} reveals that within-banner store dispersion exists and bounds its size. We make no attempt to estimate these primitives.16