From 7be609ad1583d50b81d41c64f47add448c6765a0 Mon Sep 17 00:00:00 2001 From: Vahid Ahmadi Date: Mon, 29 Jun 2026 17:12:43 +0100 Subject: [PATCH 1/4] Add conference slide deck (IMA World Congress 2026) for the VAT paper A 29-frame beamer (metropolis) talk adapting the Chicago template to the main-branch VAT microsimulation paper: - PolicyEngine branding: light-teal header banner + deep-teal text matching the poster, official transparent PE mark in every header corner, centered title slide with the teal wordmark, light-teal section dividers with a slim step-progress bar. - Paper-section order: motivation (ONS/HMRC/OBR figures) -> data & calibration -> static costing -> bunching/dominated region -> behavioural -> conclusion, plus appendix backup slides with working hyperlink nav buttons. - Key equations on the data, bunching, dominated-region and behavioural slides. - Math-safe \pounds everywhere (no stray dollar signs); self-contained author-year citations; figures under Figures/, logos under logo/. - Closing slide with QR to the project site and contact icons. Co-Authored-By: Claude Opus 4.8 (1M context) --- paper/slides/.gitignore | 11 + .../slides/Figures/bunching_analysis_85k.png | Bin 0 -> 266426 bytes .../Figures/dynamic_cost_vs_elasticity.png | Bin 0 -> 231725 bytes .../slides/Figures/dynamic_notch_fit_e017.png | Bin 0 -> 130913 bytes .../Figures/firms_by_turnover_band_85k.png | Bin 0 -> 153794 bytes paper/slides/Figures/firms_impact_2025_26.png | Bin 0 -> 174407 bytes paper/slides/Figures/obr_vat_bunching_85k.png | Bin 0 -> 227330 bytes .../slides/Figures/revenue_impact_2025_26.png | Bin 0 -> 166845 bytes .../Figures/taper_reform_distribution.png | Bin 0 -> 117688 bytes .../Figures/turnover_distribution_85k.png | Bin 0 -> 119243 bytes .../vat_firms_by_turnover_band_85k.png | Bin 0 -> 183330 bytes .../Figures/vat_threshold_revenue_impact.png | Bin 0 -> 156013 bytes paper/slides/logo/policyengine-mark.png | Bin 0 -> 2434 bytes paper/slides/logo/policyengine-teal.png | Bin 0 -> 13907 bytes paper/slides/logo/policyengine-white.png | Bin 0 -> 7901 bytes paper/slides/slides.pdf | Bin 0 -> 1785695 bytes paper/slides/slides.tex | 692 ++++++++++++++++++ 17 files changed, 703 insertions(+) create mode 100644 paper/slides/.gitignore create mode 100644 paper/slides/Figures/bunching_analysis_85k.png create mode 100644 paper/slides/Figures/dynamic_cost_vs_elasticity.png create mode 100644 paper/slides/Figures/dynamic_notch_fit_e017.png create mode 100644 paper/slides/Figures/firms_by_turnover_band_85k.png create mode 100644 paper/slides/Figures/firms_impact_2025_26.png create mode 100644 paper/slides/Figures/obr_vat_bunching_85k.png create mode 100644 paper/slides/Figures/revenue_impact_2025_26.png create mode 100644 paper/slides/Figures/taper_reform_distribution.png create mode 100644 paper/slides/Figures/turnover_distribution_85k.png create mode 100644 paper/slides/Figures/vat_firms_by_turnover_band_85k.png create mode 100644 paper/slides/Figures/vat_threshold_revenue_impact.png create mode 100644 paper/slides/logo/policyengine-mark.png create mode 100644 paper/slides/logo/policyengine-teal.png create mode 100644 paper/slides/logo/policyengine-white.png create mode 100644 paper/slides/slides.pdf create mode 100644 paper/slides/slides.tex diff --git a/paper/slides/.gitignore b/paper/slides/.gitignore new file mode 100644 index 0000000..2c0acc1 --- /dev/null +++ b/paper/slides/.gitignore @@ -0,0 +1,11 @@ +*.aux +*.log +*.nav +*.snm +*.out +*.toc +*.vrb +b1.log +b2.log +build.log +build2.log diff --git a/paper/slides/Figures/bunching_analysis_85k.png b/paper/slides/Figures/bunching_analysis_85k.png new file mode 100644 index 0000000000000000000000000000000000000000..59df44f30798624807274e6313cb443be24df88f GIT binary patch literal 266426 zcmeFZWmHw`-!_VhAl=<5lG4&jccXMmcQ*=1r%0ESbV;`~N_Tfjch`F__ulUPkLMZh z`EbTKAI==Gm?CSgHShbE*LD3?ki4uI5|FUf z@PpeyMAbpb+Q`9K&+aXhjGlvyg|&l)sXnRGTRVGGYb#bp=I4yh=}Ao-9Bk~lnV2kp z{{^G9oiWoVJ5@Vy6*wDlHG3#1MheLH1D9+8C#VNdP!htglwFc{=A2!W-%i6ExU0KS 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@@ -0,0 +1,692 @@ +\documentclass[aspectratio=169]{beamer} + +% ----------------------------------------------------------- +% Theme & Fonts (metropolis, rebranded to PolicyEngine teal) +% ----------------------------------------------------------- +\usetheme{metropolis} +\metroset{sectionpage=none, numbering=fraction, progressbar=none} + +% ----------------------------------------------------------- +% Core packages +% ----------------------------------------------------------- +\usepackage{graphicx} +\usepackage{booktabs} +\usepackage{threeparttable} +\usepackage{subcaption} +\usepackage{ragged2e} +\usepackage{xcolor} +\usepackage{colortbl} +\usepackage{amsmath,amssymb} +\usepackage{tikz} +\usetikzlibrary{decorations.pathreplacing} +\usepackage{makecell,tabularx} +\usepackage[absolute,overlay]{textpos} +\usepackage{tcolorbox} +\tcbuselibrary{skins,breakable} +\usepackage{fontawesome5} +\usepackage{qrcode} +\usepackage{hyperref} + +\graphicspath{{Figures/}} + +% ----------------------------------------------------------- +% PolicyEngine brand palette +% ----------------------------------------------------------- +\definecolor{peteal}{RGB}{44,122,123} % primary teal #2C7A7B +\definecolor{pedarkteal}{RGB}{26,77,77} % deep teal +\definecolor{pegray}{RGB}{102,102,102} +\definecolor{pelight}{RGB}{220,237,236} +\definecolor{darkred}{RGB}{139,0,0} + +% Header colour template matched to the PolicyEngine poster: +% light-teal banner with deep-teal text, teal progress/separator rules. +\setbeamercolor{normal text}{fg=black!88,bg=white} +\setbeamercolor{alerted text}{fg=peteal} +\setbeamercolor{frametitle}{bg=pelight,fg=pedarkteal} +\setbeamercolor{progress bar}{fg=peteal,bg=pelight} +\setbeamercolor{title separator}{fg=peteal} +\setbeamercolor{progress bar in head/foot}{fg=peteal,bg=pelight} +\setbeamercolor{progress bar in section page}{fg=peteal,bg=pelight} +\setbeamercolor{section title}{fg=pedarkteal} +\setbeamercolor{itemize item}{fg=peteal} +\setbeamercolor{itemize subitem}{fg=peteal} +\setbeamercolor{block title}{fg=white,bg=peteal} +\setbeamercolor{block body}{bg=pelight!40} +\setbeamercolor{standout}{fg=white,bg=pedarkteal} + +\hypersetup{colorlinks=true, linkcolor=pedarkteal, citecolor=pedarkteal, urlcolor=peteal} + +% Compact bibliography for the References slide +\setbeamertemplate{bibliography item}{} +\setbeamerfont{bibliography item}{size=\tiny} +\setbeamerfont{bibliography entry author}{size=\tiny} +\setbeamerfont{bibliography entry title}{size=\tiny} +\setbeamerfont{bibliography entry location}{size=\tiny} +\setbeamerfont{bibliography entry note}{size=\tiny} + +% ----------------------------------------------------------- +% Self-contained author-year citations (no natbib/bibtex) +% ----------------------------------------------------------- +\makeatletter +\@namedef{c@au@saez2010}{Saez} \@namedef{c@yr@saez2010}{2010} +\@namedef{c@au@klevenwaseem2013}{Kleven and Waseem} \@namedef{c@yr@klevenwaseem2013}{2013} +\@namedef{c@au@kleven2016}{Kleven} \@namedef{c@yr@kleven2016}{2016} +\@namedef{c@au@liuetal2021}{Liu et al.} \@namedef{c@yr@liuetal2021}{2021} +\@namedef{c@au@liulockwoodtam2024}{Liu, Lockwood and Tam} \@namedef{c@yr@liulockwoodtam2024}{2024} +\@namedef{c@au@blomquistetal2021}{Blomquist et al.} \@namedef{c@yr@blomquistetal2021}{2021} +\@namedef{c@au@bertanhaetal2023}{Bertanha et al.} \@namedef{c@yr@bertanhaetal2023}{2023} +\@namedef{c@au@garicano2016}{Garicano et al.} \@namedef{c@yr@garicano2016}{2016} +\@namedef{c@au@gourioroys2014}{Gourio and Roys} \@namedef{c@yr@gourioroys2014}{2014} +\@namedef{c@au@onji2009}{Onji} \@namedef{c@yr@onji2009}{2009} +\@namedef{c@au@asatryanpeichl2017}{Asatryan and Peichl} \@namedef{c@yr@asatryanpeichl2017}{2017} +\@namedef{c@au@waseem2024}{Waseem} \@namedef{c@yr@waseem2024}{2024} +\@namedef{c@au@keenmintz2004}{Keen and Mintz} \@namedef{c@yr@keenmintz2004}{2004} +\@namedef{c@au@obr2023vat}{OBR} \@namedef{c@yr@obr2023vat}{2023} +\@namedef{c@au@hmt_spring_budget_2024}{HM Treasury} \@namedef{c@yr@hmt_spring_budget_2024}{2024} +\@namedef{c@au@policyengine}{Ghenis et al.} \@namedef{c@yr@policyengine}{2022} +\newcommand{\citet}[1]{\@ifundefined{c@au@#1}{\textbf{??#1??}}% + {\csname c@au@#1\endcsname\ (\csname c@yr@#1\endcsname)}} +\newcommand{\citep}[1]{\@ifundefined{c@au@#1}{\textbf{??#1??}}% + {(\csname c@au@#1\endcsname, \csname c@yr@#1\endcsname)}} +\makeatother + +% ----------------------------------------------------------- +% Helpers +% ----------------------------------------------------------- +\newcommand{\back}[1]{\hfill{\footnotesize\hyperlink{#1}{\beamergotobutton{back}}}} +\newcommand{\E}{\mathbb{E}} +\setbeamertemplate{itemize item}{\textbullet} +\setbeamertemplate{itemize subitem}{\textemdash} + +% Math-safe pound sign: \pounds inside math mode otherwise renders as "$". +\renewcommand{\pounds}{\ifmmode\text{\textsterling}\else\textsterling\fi} + +% PolicyEngine mark in the top-right corner of every frame header. +\addtobeamertemplate{frametitle}{}{% + \begin{tikzpicture}[remember picture,overlay] + \node[anchor=north east,xshift=-0.28cm,yshift=-0.13cm] + at (current page.north east) + {\includegraphics[height=0.62cm]{logo/policyengine-mark.png}}; + \end{tikzpicture}% +} + +% Section divider pages: light-teal background matching the header banner, +% with a row of step boxes that fill in up to the current section. +\newcommand{\totalsections}{6} +\AtBeginSection[]{% + \begingroup + \setbeamercolor{background canvas}{bg=pelight} + \begin{frame}[plain,noframenumbering] + \centering + \vfill + {\usebeamerfont{section title}\Large\bfseries\color{pedarkteal}\insertsectionhead\par} + \vspace{0.7cm} + \edef\curstep{\the\value{section}}% + \begin{tikzpicture}[x=1cm] + \foreach \i in {1,...,\totalsections}{% + \ifnum\i>\curstep\relax + \node[draw=peteal!55,line width=0.7pt,rounded corners=1pt, + minimum width=0.62cm,minimum height=0.2cm,inner sep=0] + at (\i*0.7,0) {}; + \else + \node[fill=peteal,draw=peteal,line width=0.7pt,rounded corners=1pt, + minimum width=0.62cm,minimum height=0.2cm,inner sep=0] + at (\i*0.7,0) {}; + \fi + } + \end{tikzpicture} + \vfill + \end{frame} + \endgroup +} + +% ----------------------------------------------------------- +% Title Information +% ----------------------------------------------------------- +\title{A Firm-Level Microsimulation for VAT Policy Analysis} +\author{Vahid Ahmadi} +\institute{PolicyEngine} +\date{2 July 2026} + +\begin{document} + +% ================================================================= +% MAIN DECK +% ================================================================= + +% --- Centered, branded title slide --- +\begin{frame}[plain] + \centering + \vfill + \vspace{0.6cm} + + % PolicyEngine teal wordmark, centered at top + \includegraphics[height=0.9cm]{logo/policyengine-teal.png} + + \vspace{0.7cm} + + {\usebeamerfont{title}\Large\bfseries\color{pedarkteal}% + A Firm-Level Microsimulation for VAT Policy Analysis\par} + + \vspace{0.55cm} + + {\color{peteal}\rule{0.45\textwidth}{0.8pt}}\par + + \vspace{0.55cm} + + {\large\color{pedarkteal}\bfseries Vahid Ahmadi\par} + \vspace{0.15cm} + {\color{peteal}PolicyEngine\par} + + \vspace{0.5cm} + + {\small\color{pegray}IMA World Congress 2026 \,\textbar\, Brussels \,\textbar\, Parallel Session 4A\par} + + \vfill +\end{frame} + +\section{Motivation} + +% --- Question --- +\begin{frame}{The question} + \begin{center} + \Large + How should the UK \alert{cost reforms} to its VAT registration + threshold --- changes to its \alert{level}, \alert{shape}, or + \alert{rate} --- on a \alert{single, common, reproducible basis}, + + \vspace{0.4cm} + + when the firm-level data needed are not openly available? + \end{center} +\end{frame} + +% --- Motivation --- +\begin{frame}{Why the VAT threshold matters} + \hypertarget{slide:motivation}{} + \begin{itemize}\itemsep0.35cm + \item The UK VAT threshold is a \alert{notch, not a kink}: cross it and + 20\% falls due on the firm's \emph{entire} turnover --- not just the slice above. + \item A discrete liability jump of \alert{$\tau T^{*}=\pounds17{,}000$} + ($=20\%\times\pounds85{,}000$) at the threshold. + \item Threshold was \alert{\pounds85{,}000} (2023--24), raised to + \alert{\pounds90{,}000} on 1 April 2024 --- after a seven-year nominal + freeze, the longest in the tax's history. + \item \alert{Among the highest thresholds in the OECD}, and it sits in a + \emph{dense} part of the firm-size distribution. + \item Firms respond by \alert{bunching just below} it; the cliff-edge + keeps the threshold under recurrent pressure for reform. + \end{itemize} + \begin{textblock*}{3cm}(12.6cm,8.4cm) + \hyperlink{slide:notch_def}{\beamergotobutton{notch vs.\ kink}} + \end{textblock*} +\end{frame} + +% --- Motivation: figures from the paper (ONS / HMRC / OBR) --- +\begin{frame}{Firms cluster just below the threshold} + \hypertarget{slide:motivation_figs}{} + \centering + {\small Cross \alert{\pounds85{,}000} and 20\% VAT falls due on the firm's + \emph{entire} turnover --- a \alert{\pounds17{,}000 jump} --- so firms cluster + \alert{right below it}.} + + \vspace{0.6em} + + \begin{minipage}[t]{0.32\textwidth} + \centering + \includegraphics[height=0.50\textheight,width=\linewidth,keepaspectratio]{firms_by_turnover_band_85k.png}\\[0.3em] + {\scriptsize All UK firms by turnover band (ONS)} + \end{minipage}\hfill + \begin{minipage}[t]{0.32\textwidth} + \centering + \includegraphics[height=0.50\textheight,width=\linewidth,keepaspectratio]{vat_firms_by_turnover_band_85k.png}\\[0.3em] + {\scriptsize VAT-registered firms by band (HMRC)} + \end{minipage}\hfill + \begin{minipage}[t]{0.32\textwidth} + \centering + \includegraphics[height=0.50\textheight,width=\linewidth,keepaspectratio]{obr_vat_bunching_85k.png}\\[0.3em] + {\scriptsize Bunching below \pounds85k (OBR, 2023)} + \end{minipage} +\end{frame} + +% --- This paper --- +\begin{frame}{This paper} + \begin{itemize}\itemsep0.25cm + \item An \alert{open, reproducible firm-level microsimulation} that costs + UK VAT threshold reforms on a common static basis. + \item Built on \alert{synthetic firm microdata} calibrated to published + HMRC + ONS aggregates --- \alert{released openly} and regenerable under + any counterfactual schedule. + \item Prices three reform families --- \alert{level} (move the threshold), + \alert{shape} (a graduated taper), \alert{rate} (a reduced-rate band) --- + \alert{three ways}: + \begin{itemize}\itemsep0.1cm + \item[1.] \textbf{statically} (mechanical reclassification); + \item[2.] by the exact, elasticity-free \textbf{dominated-region} distortion each removes; + \item[3.] \textbf{behaviourally}, conditional on an assumed turnover elasticity $e$. + \end{itemize} + \item \alert{Headline:} only the graduated taper removes the notch's + distortion; raising the threshold is the dearest, least efficient option. + \end{itemize} +\end{frame} + +% --- Literature --- +\begin{frame}{Related literature} + \begin{itemize}\itemsep0.3cm + \item \textbf{Bunching / notch methodology} --- source of the + ``dominated region'' + \begin{itemize} + \item \citet{saez2010}; \citet{klevenwaseem2013}; \citet{kleven2016} + \end{itemize} + \item \textbf{VAT registration threshold (empirical)} + \begin{itemize} + \item \citet{liuetal2021} (UK notch, $\sim$43\% voluntary registration); + \citet{liulockwoodtam2024}; \citet{onji2009}; \citet{asatryanpeichl2017}; \citet{waseem2024} + \end{itemize} + \item \textbf{Does bunching identify a structural elasticity?} + \begin{itemize} + \item \citet{blomquistetal2021}; \citet{bertanhaetal2023} --- \emph{not} non-parametrically identified + \end{itemize} + \item \textbf{Structural models of size-based thresholds} + \begin{itemize} + \item \citet{garicano2016}; \citet{gourioroys2014} + \end{itemize} + \end{itemize} +\end{frame} + +\section{Data and calibration} + +% --- Data sources --- +\begin{frame}{Data: synthetic firms calibrated to public aggregates} + \hypertarget{slide:data}{} + \begin{itemize}\itemsep0.3cm + \item \textbf{No open firm-level VAT microdata exist} $\Rightarrow$ build a + synthetic population calibrated to published aggregates (data year 2023--24). + \item \textbf{Calibration targets:} + \begin{itemize} + \item ONS \emph{UK Business: Activity, Size and Location} --- firm counts by sector $\times$ turnover band + \item HMRC \emph{VAT Annual Statistics} --- registered counts and net VAT liability by band / sector + \end{itemize} + \item \textbf{Scale:} $\approx$\alert{2.94m} firm records, weighted to + represent $\approx$\alert{2.5m} UK firms. + \item \textbf{Accuracy:} $\approx$\alert{90\%} across five calibrated dimensions. + \item Approach mirrors open household tax-benefit microsimulation \citep{policyengine}. + \end{itemize} + \begin{textblock*}{8.6cm}(5.4cm,7.95cm) + \raggedleft + \hyperlink{slide:generation}{\beamergotobutton{how firms are built}}\quad + \hyperlink{slide:turnover_fig}{\beamergotobutton{distribution}} + \end{textblock*} +\end{frame} + +% --- Turnover distribution figure --- +\begin{frame}{The calibrated turnover distribution} + \hypertarget{slide:turnover_fig}{} + \begin{figure} + \centering + \includegraphics[height=0.62\textheight]{turnover_distribution_85k.png} + \end{figure} + \vspace{-0.2cm} + {\footnotesize Density steps down at the \pounds85{,}000 registration + threshold and again at the \pounds150{,}000 Flat Rate Scheme ceiling. + \alert{The step is inherited from coarse HMRC band targets} --- quantified later by a placebo test.} +\end{frame} + +\section{Static costing} + +% --- Static formula + reform families --- +\begin{frame}{Static costing: mechanical reclassification} + \begin{block}{Static revenue at threshold $T$} + \[ R(T) = \sum_i w_i\, v_i\, \mathbf{1}\{y_i \ge T\}. \] + \end{block} + \begin{itemize}\itemsep0.2cm + \item Turnover held fixed $\Rightarrow$ revenue moves \alert{only through + firms whose registration status flips} (those between the old and new cutoff). + \item Three reform \alert{levers}, four schedules, all on one base: + \begin{itemize} + \item \textbf{Level} --- move the threshold up or down + \item \textbf{Shape} --- graduated taper, 0\% at \pounds85k $\to$ 20\% at \pounds105k + \item \textbf{Rate} --- reduced-rate band (10\% or 15\%) in \pounds85k--\pounds105k + \end{itemize} + \item For each, recompute $R=\sum_i w_i v_i$ under the new schedule and difference. + \end{itemize} +\end{frame} + +% --- Validation vs HMRC --- +\begin{frame}{Validation: the April-2024 anchor reform} + \hypertarget{slide:validation}{} + \begin{columns}[T] + \begin{column}{0.56\textwidth} + \begin{figure} + \centering + \includegraphics[width=\linewidth]{vat_threshold_revenue_impact.png} + \end{figure} + \end{column} + \begin{column}{0.44\textwidth} + \footnotesize + \begin{itemize}\itemsep0.25cm + \item Cost the \alert{\pounds85k$\to$\pounds90k} April-2024 rise, my + estimate vs HMRC's published costing. + \item \alert{2025--26: \pounds175m vs \pounds185m} --- agree within + \pounds7--21m per year across the horizon. + \item Both turn to a small \emph{gain} by 2028--29 as fiscal drag lifts + the frozen baseline. + \item Microdata are calibrated to HMRC aggregates $\Rightarrow$ a + \alert{consistency check}, not out-of-sample validation. + \end{itemize} + \end{column} + \end{columns} +\end{frame} + +% --- Threshold sweep --- +\begin{frame}{Static sweep: cost of moving the threshold (2025--26)} + {\footnotesize Validated on the one move HMRC published, \alert{now sweep the + whole schedule} --- the same mechanical reclassification at every threshold.} + \vspace{0.15cm} + \begin{figure} + \centering + \includegraphics[width=0.46\linewidth]{revenue_impact_2025_26.png}\hfill + \includegraphics[width=0.46\linewidth]{firms_impact_2025_26.png} + \end{figure} + \vspace{-0.1cm} + {\footnotesize From a \pounds90k base (\pounds203.8bn), the schedule is + roughly linear, \alert{$\approx$\pounds150--220m per \pounds5k step}: e.g.\ + \alert{$-$\pounds374m} to raise to \pounds100k, \alert{$+$\pounds665m} to + lower to \pounds70k. Even a \pounds30k rise to \pounds120k costs $\sim$\pounds1.2bn --- under 0.6\% of the base.} +\end{frame} + +% --- Reform menu --- +\begin{frame}{The reform menu, one common base} + \hypertarget{slide:menu}{} + \begin{center} + \renewcommand{\arraystretch}{1.35} + \begin{tabular}{@{}l l r l@{}} + \toprule + \textbf{Reform} & \textbf{Lever} & \textbf{Static} & \textbf{Dominated region} \\ + \midrule + Raise to \pounds100k & Level & $-$\pounds508m & relocated \\ + Taper \pounds85--105k & Shape & $-$\pounds336m & \alert{removed} \\ + Reduced 10\% & Rate & $-$\pounds343m & split in two \\ + Reduced 15\% & Rate & $-$\pounds171m & split in two \\ + \bottomrule + \end{tabular} + \end{center} + \vspace{0.2cm} + {\footnotesize Common \pounds85{,}000-notch / \pounds183.6bn 2023--24 base. + \alert{The level move is the most expensive yet removes no distortion}; the + taper removes it entirely for less.\\[0.2em] + \emph{``Split in two''} $=$ a second notch appears where the reduced-rate band + ends (\pounds105k reverts to 20\%), so the empty band is fragmented, not shrunk.} + \begin{textblock*}{3cm}(12.6cm,8.4cm) + \hyperlink{slide:behav_table}{\beamergotobutton{behavioural}} + \end{textblock*} +\end{frame} + +\section{The notch's distortion} + +% --- Bunching is mechanical --- +\begin{frame}{Bunching in the synthetic data is mechanical} + \hypertarget{slide:bunching}{} + \begin{columns}[T] + \begin{column}{0.55\textwidth} + \begin{figure} + \centering + \includegraphics[width=\linewidth,height=0.46\textheight,keepaspectratio]{bunching_analysis_85k.png} + \end{figure} + \vspace{-0.2cm} + {\footnotesize Excess mass over a polynomial no-bunching counterfactual $g$:} + \[ b=\frac{B(y^{*})}{g(y^{*})}\approx\frac{\Delta y^{*}}{y^{*}}, + \qquad E=\!\!\sum_{\text{window}}\!\big(n_j-c_j\big). \] + \end{column} + \begin{column}{0.45\textwidth} + \footnotesize + \begin{itemize}\itemsep0.25cm + \item Reproduced density step: ratio \alert{$b=0.060$} (SE 0.005), + excess mass \alert{$E=8{,}712$ firms} (SE 662). + \item \alert{Placebo:} remove the step from the calibration target $\Rightarrow$ + $E$ collapses to \alert{0--99 firms}. + \item So the step is \alert{inherited from coarse HMRC bands}, not firm + behaviour --- the generator has no location choice. + \item Genuine threshold bunching remains the administrative fact of + \citet{liuetal2021}. + \end{itemize} + \end{column} + \end{columns} +\end{frame} + +% --- Dominated region --- +\begin{frame}{The dominated region: an exact, elasticity-free distortion} + \hypertarget{slide:dominated}{} + \begin{columns}[T] + \begin{column}{0.52\textwidth} + \begin{figure} + \centering + \includegraphics[width=\linewidth,height=0.44\textheight,keepaspectratio]{dynamic_notch_fit_e017.png} + \end{figure} + \vspace{-0.15cm} + {\footnotesize The notch in the firm's profit:} + \[ \pi(y)=\begin{cases} y-c(y), & y Date: Wed, 1 Jul 2026 01:44:17 +0200 Subject: [PATCH 2/4] Update slide deck for the value-added (formulation A) paper Sync onto the new main (PRs #19, #20 merged) and reflect Section 7's value-added recast of the behavioural simulator. - New slide "The firm's problem: VAT on value added (formulation A)": pi(y)=(1-delta)(1-tau f(y))y - C(y;n,e), with delta the deductible-input share (value added (1-delta)y), C the own-factor cost (inputs already netted, so reading C as total cost would double-count), optimum n[(1-delta)(1-tau)]^e, and the new formulation_a_optima figure (delta in {0,0.4,0.8} across notch and taper). - Behavioural slide: note the simulator is a value-added base (formulation A). The e-sensitivity numbers (GBP508m->GBP292m, etc.) are unchanged. - Limitations: the "turnover-tax approximation" bullet now states the simulator taxes value added with a parametric delta (ONS Supply-Use to come), still abstracting from voluntary registration. Rebuilt slides.pdf (30 pages). 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