1% paper/sections/calibration.tex — UQO WP 25, calibration and empirical validation2% Every number below comes from docs/parameter_ledger.csv (56 rows, sources read 2026-10-27),3% data/processed/calibration_targets.csv, or the baseline choices recorded in docs/research_log.md (§02–03).4\section{Calibration and empirical validation}\label{sec:calibration}56\subsection{Principle: calibrated versus illustrative parameters}78The three models share one parameter ledger (\texttt{docs/parameter\_ledger.csv}, 56 rows). Each row records9symbol, units, value, admissible range, source, period and geography, the moment matched, the method, an10uncertainty grade and the URL read on 2026-10-27. We keep two classes apart throughout. \emph{Calibrated}11parameters are pinned by a moment computed from downloaded data (Table~\ref{tab:targets}) or read from a verified12publication or regulation (Table~\ref{tab:ledger}). \emph{Illustrative} parameters have no accepted empirical13counterpart: the information precisions $\tau_H,\tau_A$, the delegation-cost distribution $F_\kappa$, the14supervision time $\tau_s$, the inference and energy intensities $\bar q,\bar e$, the task-productivity profile15$\psi_0,\psi_1,\bar\iota$, the return-quality parameters $m_H,\sigma_{rH},\sigma_{rA}$ and all LLM behavioural16parameters. They are labelled wherever a number appears and swept in the robustness section. When a source could not17be obtained the ledger says \textsc{not obtained} rather than carrying a remembered value; this applies to the18Cooley--Prescott quarterly TFP process, the 2008 and 2020 credit-spread peaks, compute depreciation, the ownership19concentration of installed AI compute and the supervision cost of delegation.2021The baseline values of the numerical sections are $\beta=0.96$ (Model~A) and $0.95$ (Model~B), $\gamma=2$,22$\alpha=0.33$, $\delta=0.06$, $\rho_z=0.9$ and $\sigma_\varepsilon=0.013$: rounded textbook values placed inside23the measured ranges, with each deviation from the ledger point estimate noted below.2425\subsection{Macro block}2627\paragraph{Factor shares.} The BLS nonfarm-business labour-share index (FRED \texttt{PRS85006173}, 2017$=$100)28averages 104.5 over 1984--2026 and 98.6 over 2015--2026. The index has no level, so we anchor it to the Penn World29Table labour-compensation share of 2017 (0.5913, FRED \texttt{LABSHPUSA156NRUG}): the rescaled level is 0.61830(1984--2026), 0.583 (2015--2026) and 0.557 in 2026. This \emph{rescaling assumption} is documented in the build31script; the PWT concept is economy-wide and imputes self-employment labour income while the BLS index covers the32nonfarm business sector, so the level is indicative and only the fall of about ten index points since 2000 is33robust. Two cross-checks bracket it: the BEA compensation share of gross domestic income (FRED34\texttt{A4002E1A156NBEA}) averages 54.6\% over 1984--2025 (50.9\% in 2025), and the business-sector capital share35$\alpha$ in Fernald's utilization-adjusted TFP workbook (FRBSF Working Paper 2012-19) averages 0.354 over 1984--202536and 0.338 over 1948--2025. The baseline $\alpha=0.33$ sits at the low end of the 0.33--0.38 range. Because the37decline of the labour share is an object the model must speak to \citep{acemoglu2018race,acemoglu2022tasks}, we do38not calibrate the task frontier $\bar\iota$ to reproduce it.3940\paragraph{Depreciation, growth and expenditure shares.} BEA current-cost depreciation of private fixed assets over41the previous year-end net stock (FRED \texttt{M1TTOTL1ES000}/\texttt{K1TTOTL1ES000}) gives $\delta=0.055$ over421984--2024 (0.054 in 2024); this includes residential structures, and equipment, software and above all compute43hardware depreciate faster. The baseline $\delta=0.06$ is the usual rounding. Real GDP (FRED \texttt{GDPC1}) grows442.72\% a year over 1984--2025; investment and consumption average 0.158 and 0.660 of GDP (FRED \texttt{GPDIC1},45\texttt{PCECC96}; ratios of chained aggregates), 0.182 and 0.680 over 2015--2025. Models~A and~B abstract from46trend growth, so these ratios are validation moments for the zero-AI limit, not targets.4748\paragraph{TFP process.} We cumulate Fernald's annual utilization-adjusted TFP growth into a log level and fit an49AR(1). Persistence depends on detrending (Table~\ref{tab:targets}): a linear trend gives $\rho_z=0.876$ over501984--2025 (0.929 over 1948--2025), an HP filter with $\lambda=100$ gives 0.62 (0.47), and $\lambda=6.25$ gives a51value indistinguishable from zero; the quarterly HP(1600) estimate is 0.70 per quarter, 0.24 annualised. The52innovation standard deviation is more stable: 0.0126 (linear trend, 1984--2025), 0.0135 (1948--2025), 0.0127 in53first differences and 0.0074 under HP(6.25). The linear-trend estimates are the headline because the model's54productivity process is trend-stationary; the baseline uses $\rho_z=0.9$, $\sigma_\varepsilon=0.013$, with55$\rho_z$ swept from the HP(100) value 0.62 to the long-sample 0.929 in sensitivity. The Cooley--Prescott quarterly values often quoted could not be verified from the chapter and are not used.5657\paragraph{Real rate and discounting.} The ex-post real three-month bill rate (FRED \texttt{TB3MS} less58\texttt{CPIAUCSL} inflation) averages 0.57\% over 1984--2025 (standard deviation 2.31 points), the real federal59funds rate 0.83\%, and the ten-year TIPS yield (\texttt{DFII10}) 0.97\% over 2003--2025, 0.63\% over 2015--2025 and602.66\% on 2026-10-26. A no-growth representative-agent steady state would imply $\beta=1/(1+r)\in[0.990,0.994]$.61We use $\beta=0.96$ in Model~A, which puts the steady-state return on capital near 4\%, between the riskless rate62and the 9.9\% mean real equity return of 1984--2025: the standard compromise in a one-asset model without a risk63premium. In Model~B, $\beta=0.95$ is set below $1/(1+r)$ so that precautionary saving clears the asset market at a64positive rate \citep{aiyagari1994uninsured,huggett1993riskfree}; the equilibrium $r$ is an output.6566\subsection{Wealth and earnings block}6768The Federal Reserve's Distributional Financial Accounts (DFA) for 2026:Q2 give net-worth shares of 15.0\% for the top690.1\%, 32.5\% for the top 1\%, 68.9\% for the top 10\%, 28.8\% for the 50th--90th percentiles and 2.3\% for the70bottom half; in 1989:Q3 the top-1\% and bottom-50\% shares were 22.8\% and 3.5\%. A Gini computed from the five71grouped shares under within-group equality is 0.751, a \emph{lower bound}; survey-based values near 0.85 were not72verified here. Model~B's earnings process is a log AR(1) with $\rho_e=0.96$ and $\sigma_e=0.17$73(seven Rouwenhorst states) and a zero borrowing limit $\underline a=0$ (sensitivity to $-1$ times average earnings).74These lie between the PSID estimates of Flodén and Lindé (persistence 0.9136, innovation variance 0.0426, i.e.\75standard deviation 0.206, read from Table~4 of their 1998 working paper) and the 0.98--1.0 persistence of76Storesletten, Telmer and Yaron; they are a choice, not an estimate.7778We are explicit about what the Aiyagari block can and cannot deliver. With Gaussian AR(1) earnings and one safe79asset it reproduces the real-rate and capital-stock moments but understates the top tail: the top-1\% share and the80Gini fall well short of 32.5\% and 0.75 \citep{krusell1998income,achdou2022income}. \citet{hubmer2021sources} show81that matching the post-1980 rise in concentration requires return heterogeneity, discount-factor dispersion or82falling tax progressivity. Our return-quality mechanism ($m_H,\sigma_{rH},\sigma_{rA}$) is a form of return83heterogeneity, but we do not tune it to the DFA shares; we report the model's shares beside the DFA values and84treat the gap as a diagnostic. The 50--70\% of the change in the US wage structure that \citet{acemoglu2022tasks}85attribute to task displacement is likewise a validation reference for the displacement channel.8687\subsection{Financial block}8889From Shiller's composite price, dividend and CPI series the real total return on US equity averages 8.64\% a year90with standard deviation 17.9\% over 1872--2025, and 9.94\% with 15.7\% over 1984--2025. The arithmetic premium over91three-month bills is 9.17\% over 1934--2025 (standard deviation 17.7, Sharpe ratio 0.52), 8.94\% over 1948--202592and 9.66\% over 1984--2025; geometric premia are one and a half to two points lower and Mehra and Prescott's931889--1978 value of 6.18\% is the classic lower reference. Model~A's portfolio mapping uses $\sigma_R=0.16$ (the941984--2025 value rounded) and, in return units, a predictable component of 0.06, human noise 0.10 and model noise950.04; the three information parameters are illustrative. Model~C uses a dividend with mean 0.05,96$\rho_\delta=0.5$, $\sigma_\delta=0.01$, dividend variance 0.01, private-signal noise 0.02, model-signal noise 0.00597and a safe rate $r_f=0.01$ per period; only the safe rate is anchored in data (the 0.97\% mean TIPS yield), the rest98are illustrative, and the model does not attempt to explain the equity premium.99100Credit parameters come from regulation and from a narrow data window. The initial margin $m=0.5$ is the101Regulation~T floor (12 CFR 220.12(a): 50\% of market value for margin equity securities, 150\% for short sales).102The stress path of haircuts follows Gorton and Metrick's repo-haircut index on structured-product collateral, which103rises from zero in early 2007 to nearly 50\% in late 2008 while Treasury haircuts stayed near zero; this sets the104range of the collateral-amplification experiments, in the spirit of \citet{geanakoplos2010leverage} and105\citet{kiyotaki1997credit}. Investment-grade and high-yield option-adjusted spreads (FRED \texttt{BAMLC0A0CM},106\texttt{BAMLH0A0HYM2}) average 0.88\% and 3.08\% over the only window FRED now distributes, 2023-10-30 to1072026-10-26, with maxima of 1.33\% and 4.61\%; these are low-spread years, so the means are weak long-run targets,108and the 2008 and 2020 peaks are recorded only as chart readings from the Federal Reserve's November 2020109\emph{Financial Stability Report} (high yield near 2,000 and 800 basis points). SLOOS tightening peaks (83.6 in 2008:Q4, 71.2 in 2020:Q3, against a mean of 6.6) date the stress episodes. Bank book equity over assets (H.8 residual over total assets, \texttt{RALACBW027SBOG}/%110\texttt{TLAACBW027SBOG}) is 0.095 on 2026-10-14, 0.106 on average over 2010--2025 and 0.080 over 1984--2025,111i.e.\ leverage of about 10.6; the bank in Model~C starts from this ratio.112113\subsection{AI block}114115\emph{Compute price trend.} Epoch AI's GPU series (Hobbhahn and Besiroglu, 2022; 470 GPUs, 2006--2021) finds that116FLOP/s per dollar doubles every 2.46 years (95\% interval 2.24--2.72; 2.07 years for ML-specific GPUs). We map this to117the decline of the compute-capital price, $g_Q=1-2^{-1/2.46}=24.5\%$ a year, and derive an illustrative range118$\delta_Q\in[0.20,0.33]$ for compute depreciation from that trend and a three-to-five-year physical life; no direct119estimate was obtained. \emph{Inference prices.} Epoch AI's 2025 data insight finds that the price of a fixed120capability threshold fell 9$\times$ to 900$\times$ a year depending on the threshold, and a16z's ``LLMflation''121estimate is about 10$\times$ a year; 10$\times$ is our central fee path. List prices recorded on 2026-10-27 were122\$10 (input) and \$50 (output) per million tokens for the frontier models of both Anthropic and OpenAI and123\$0.10/\$0.50 for their small models, a hundredfold gap that spans the fee schedule. Model~B's $\phi_0=0.02$,124$\phi_1=0.002$ (units of average annual earnings) and $\bar q=0.02$ are illustrative: model compute units are not125tokens, and $\bar q$ is a scenario parameter (``inference intensity''), not a constant. \emph{Adoption.} The126Census Bureau's Business Trends and Outlook Survey reports that 19.8\% of businesses used AI in the two weeks before1272026-05-03 (37\% of firms with 250 or more employees, 39.7\% in the information sector), and \citet{bick2024rapid}128find that nearly 40\% of US adults aged 18--64 use generative AI, 23\% of the employed at work in the previous week;129these fix the initial delegation shares of the adoption experiments and bracket $F_\kappa$. \emph{Investment and130energy.} The 2025 capital expenditure of Alphabet, Meta, Amazon and Microsoft from SEC XBRL filings and earnings131releases (Appendix~\ref{app:data}) sums to \$376.1 billion on a calendar basis (\$357.5 billion with Microsoft's132fiscal year), about 1.2\% of US GDP---a flow proxy for the AI capital stock, not a measure of ownership133concentration, which is swept as a top-four share of 0.5--0.9. The IEA's \emph{Energy and AI} report puts134data-centre electricity use at 415 TWh in 2024 (1.5\% of world electricity, 45\% of it in the United States, roughly135187 TWh by our arithmetic) and projects about 945 TWh by 2030. The energy price $p_E$ is the EIA industrial retail136price of 9.64 cents per kWh (August 2026, preliminary; 8.98 a year earlier); since $\bar e$ is illustrative, the137energy component of the fee is illustrative in level even though its price is measured. The supervision cost138$\kappa_s$ has no public measure and is swept over 0--30\% of task value; \citet{brynjolfsson2025generative} and139\citet{eisfeldt2023generative} give the closest firm-level evidence and serve for orientation only.140141\subsection{Validation}142143\paragraph{Internal.} With $K_Q=0$ the production side of Model~A collapses to $Y=e^zK^\alpha L^{1-\alpha}$ and the144delegation share to zero; the solver is tested against the Brock--Mirman closed form ($\delta=1$, $\gamma=1$) and,145at the baseline parameters, against the $I/Y$ and $C/Y$ moments of Table~\ref{tab:targets}. With $d\equiv0$146Model~B is the \citet{aiyagari1994uninsured} economy and must show its known features: an equilibrium rate below147the rate of time preference and a right-skewed but too-thin wealth distribution. Model~C with one rational,148unlevered investor must price the risky asset at the present value of dividends at the safe rate. These checks are149gates, reported in the computation section, not results.150151\paragraph{External roadmap and falsification.} The simulations identify effects only within the structural model;152the following data would test each hypothesis directly, subject to access rights, and none of the tests is performed153here. H1 (adoption threshold) predicts adoption rising in the stakes of the delegated decision and in the154frontier-to-small price gap: BTOS firm microdata and provider usage statistics matched to size and sector would155falsify it if adoption were flat in stakes or responded to the small-model price alone. H2 (common-model156correlation) predicts higher trade correlation and lower depth, conditional on public information, among accounts157using the same model: broker order data or exchange audit trails with an automation flag are needed, and no residual158correlation after controlling for common signals would falsify it. H3 (ownership concentration) predicts that AI159rents accrue to the DFA top groups in proportion to their holdings of compute-owning firms: 13F filings and DFA160equity holdings by wealth group can test it, and productivity gains without a rise in the top-0.1\% share would161contradict it. H4 (collateral feedback) predicts positive comovement of agentic leverage, valuations and investment,162observable in margin-debt statistics, repo haircuts and the capital expenditure of compute owners. H5 (diversity and163limits) and H6 (compute scarcity) are policy counterfactuals, falsifiable only indirectly through the response of164spreads and compute prices to documented supply shocks, of which the SEC capex and IEA energy series are the first165elements.166167\input{tables/ledger}168\input{tables/targets}169