% paper/sections/calibration.tex — UQO WP 25, calibration and empirical validation
% Every number below comes from docs/parameter_ledger.csv (56 rows, sources read 2026-10-27),
% data/processed/calibration_targets.csv, or the baseline choices recorded in docs/research_log.md (§02–03).
\section{Calibration and empirical validation}\label{sec:calibration}

\subsection{Principle: calibrated versus illustrative parameters}

The three models share one parameter ledger (\texttt{docs/parameter\_ledger.csv}, 56 rows). Each row records
symbol, units, value, admissible range, source, period and geography, the moment matched, the method, an
uncertainty grade and the URL read on 2026-10-27. We keep two classes apart throughout. \emph{Calibrated}
parameters are pinned by a moment computed from downloaded data (Table~\ref{tab:targets}) or read from a verified
publication or regulation (Table~\ref{tab:ledger}). \emph{Illustrative} parameters have no accepted empirical
counterpart: the information precisions $\tau_H,\tau_A$, the delegation-cost distribution $F_\kappa$, the
supervision time $\tau_s$, the inference and energy intensities $\bar q,\bar e$, the task-productivity profile
$\psi_0,\psi_1,\bar\iota$, the return-quality parameters $m_H,\sigma_{rH},\sigma_{rA}$ and all LLM behavioural
parameters. They are labelled wherever a number appears and swept in the robustness section. When a source could not
be obtained the ledger says \textsc{not obtained} rather than carrying a remembered value; this applies to the
Cooley--Prescott quarterly TFP process, the 2008 and 2020 credit-spread peaks, compute depreciation, the ownership
concentration of installed AI compute and the supervision cost of delegation.

The baseline values of the numerical sections are $\beta=0.96$ (Model~A) and $0.95$ (Model~B), $\gamma=2$,
$\alpha=0.33$, $\delta=0.06$, $\rho_z=0.9$ and $\sigma_\varepsilon=0.013$: rounded textbook values placed inside
the measured ranges, with each deviation from the ledger point estimate noted below.

\subsection{Macro block}

\paragraph{Factor shares.} The BLS nonfarm-business labour-share index (FRED \texttt{PRS85006173}, 2017$=$100)
averages 104.5 over 1984--2026 and 98.6 over 2015--2026. The index has no level, so we anchor it to the Penn World
Table labour-compensation share of 2017 (0.5913, FRED \texttt{LABSHPUSA156NRUG}): the rescaled level is 0.618
(1984--2026), 0.583 (2015--2026) and 0.557 in 2026. This \emph{rescaling assumption} is documented in the build
script; the PWT concept is economy-wide and imputes self-employment labour income while the BLS index covers the
nonfarm business sector, so the level is indicative and only the fall of about ten index points since 2000 is
robust. Two cross-checks bracket it: the BEA compensation share of gross domestic income (FRED
\texttt{A4002E1A156NBEA}) averages 54.6\% over 1984--2025 (50.9\% in 2025), and the business-sector capital share
$\alpha$ in Fernald's utilization-adjusted TFP workbook (FRBSF Working Paper 2012-19) averages 0.354 over 1984--2025
and 0.338 over 1948--2025. The baseline $\alpha=0.33$ sits at the low end of the 0.33--0.38 range. Because the
decline of the labour share is an object the model must speak to \citep{acemoglu2018race,acemoglu2022tasks}, we do
not calibrate the task frontier $\bar\iota$ to reproduce it.

\paragraph{Depreciation, growth and expenditure shares.} BEA current-cost depreciation of private fixed assets over
the previous year-end net stock (FRED \texttt{M1TTOTL1ES000}/\texttt{K1TTOTL1ES000}) gives $\delta=0.055$ over
1984--2024 (0.054 in 2024); this includes residential structures, and equipment, software and above all compute
hardware depreciate faster. The baseline $\delta=0.06$ is the usual rounding. Real GDP (FRED \texttt{GDPC1}) grows
2.72\% a year over 1984--2025; investment and consumption average 0.158 and 0.660 of GDP (FRED \texttt{GPDIC1},
\texttt{PCECC96}; ratios of chained aggregates), 0.182 and 0.680 over 2015--2025. Models~A and~B abstract from
trend growth, so these ratios are validation moments for the zero-AI limit, not targets.

\paragraph{TFP process.} We cumulate Fernald's annual utilization-adjusted TFP growth into a log level and fit an
AR(1). Persistence depends on detrending (Table~\ref{tab:targets}): a linear trend gives $\rho_z=0.876$ over
1984--2025 (0.929 over 1948--2025), an HP filter with $\lambda=100$ gives 0.62 (0.47), and $\lambda=6.25$ gives a
value indistinguishable from zero; the quarterly HP(1600) estimate is 0.70 per quarter, 0.24 annualised. The
innovation standard deviation is more stable: 0.0126 (linear trend, 1984--2025), 0.0135 (1948--2025), 0.0127 in
first differences and 0.0074 under HP(6.25). The linear-trend estimates are the headline because the model's
productivity process is trend-stationary; the baseline uses $\rho_z=0.9$, $\sigma_\varepsilon=0.013$, with
$\rho_z$ 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.

\paragraph{Real rate and discounting.} The ex-post real three-month bill rate (FRED \texttt{TB3MS} less
\texttt{CPIAUCSL} inflation) averages 0.57\% over 1984--2025 (standard deviation 2.31 points), the real federal
funds rate 0.83\%, and the ten-year TIPS yield (\texttt{DFII10}) 0.97\% over 2003--2025, 0.63\% over 2015--2025 and
2.66\% on 2026-10-26. A no-growth representative-agent steady state would imply $\beta=1/(1+r)\in[0.990,0.994]$.
We use $\beta=0.96$ in Model~A, which puts the steady-state return on capital near 4\%, between the riskless rate
and the 9.9\% mean real equity return of 1984--2025: the standard compromise in a one-asset model without a risk
premium. In Model~B, $\beta=0.95$ is set below $1/(1+r)$ so that precautionary saving clears the asset market at a
positive rate \citep{aiyagari1994uninsured,huggett1993riskfree}; the equilibrium $r$ is an output.

\subsection{Wealth and earnings block}

The Federal Reserve's Distributional Financial Accounts (DFA) for 2026:Q2 give net-worth shares of 15.0\% for the top
0.1\%, 32.5\% for the top 1\%, 68.9\% for the top 10\%, 28.8\% for the 50th--90th percentiles and 2.3\% for the
bottom half; in 1989:Q3 the top-1\% and bottom-50\% shares were 22.8\% and 3.5\%. A Gini computed from the five
grouped shares under within-group equality is 0.751, a \emph{lower bound}; survey-based values near 0.85 were not
verified here. Model~B's earnings process is a log AR(1) with $\rho_e=0.96$ and $\sigma_e=0.17$
(seven Rouwenhorst states) and a zero borrowing limit $\underline a=0$ (sensitivity to $-1$ times average earnings).
These lie between the PSID estimates of Flodén and Lindé (persistence 0.9136, innovation variance 0.0426, i.e.\
standard deviation 0.206, read from Table~4 of their 1998 working paper) and the 0.98--1.0 persistence of
Storesletten, Telmer and Yaron; they are a choice, not an estimate.

We are explicit about what the Aiyagari block can and cannot deliver. With Gaussian AR(1) earnings and one safe
asset it reproduces the real-rate and capital-stock moments but understates the top tail: the top-1\% share and the
Gini fall well short of 32.5\% and 0.75 \citep{krusell1998income,achdou2022income}. \citet{hubmer2021sources} show
that matching the post-1980 rise in concentration requires return heterogeneity, discount-factor dispersion or
falling tax progressivity. Our return-quality mechanism ($m_H,\sigma_{rH},\sigma_{rA}$) is a form of return
heterogeneity, but we do not tune it to the DFA shares; we report the model's shares beside the DFA values and
treat the gap as a diagnostic. The 50--70\% of the change in the US wage structure that \citet{acemoglu2022tasks}
attribute to task displacement is likewise a validation reference for the displacement channel.

\subsection{Financial block}

From Shiller's composite price, dividend and CPI series the real total return on US equity averages 8.64\% a year
with standard deviation 17.9\% over 1872--2025, and 9.94\% with 15.7\% over 1984--2025. The arithmetic premium over
three-month bills is 9.17\% over 1934--2025 (standard deviation 17.7, Sharpe ratio 0.52), 8.94\% over 1948--2025
and 9.66\% over 1984--2025; geometric premia are one and a half to two points lower and Mehra and Prescott's
1889--1978 value of 6.18\% is the classic lower reference. Model~A's portfolio mapping uses $\sigma_R=0.16$ (the
1984--2025 value rounded) and, in return units, a predictable component of 0.06, human noise 0.10 and model noise
0.04; the three information parameters are illustrative. Model~C uses a dividend with mean 0.05,
$\rho_\delta=0.5$, $\sigma_\delta=0.01$, dividend variance 0.01, private-signal noise 0.02, model-signal noise 0.005
and a safe rate $r_f=0.01$ per period; only the safe rate is anchored in data (the 0.97\% mean TIPS yield), the rest
are illustrative, and the model does not attempt to explain the equity premium.

Credit parameters come from regulation and from a narrow data window. The initial margin $m=0.5$ is the
Regulation~T floor (12 CFR 220.12(a): 50\% of market value for margin equity securities, 150\% for short sales).
The stress path of haircuts follows Gorton and Metrick's repo-haircut index on structured-product collateral, which
rises from zero in early 2007 to nearly 50\% in late 2008 while Treasury haircuts stayed near zero; this sets the
range of the collateral-amplification experiments, in the spirit of \citet{geanakoplos2010leverage} and
\citet{kiyotaki1997credit}. Investment-grade and high-yield option-adjusted spreads (FRED \texttt{BAMLC0A0CM},
\texttt{BAMLH0A0HYM2}) average 0.88\% and 3.08\% over the only window FRED now distributes, 2023-10-30 to
2026-10-26, with maxima of 1.33\% and 4.61\%; these are low-spread years, so the means are weak long-run targets,
and the 2008 and 2020 peaks are recorded only as chart readings from the Federal Reserve's November 2020
\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}/%
\texttt{TLAACBW027SBOG}) is 0.095 on 2026-10-14, 0.106 on average over 2010--2025 and 0.080 over 1984--2025,
i.e.\ leverage of about 10.6; the bank in Model~C starts from this ratio.

\subsection{AI block}

\emph{Compute price trend.} Epoch AI's GPU series (Hobbhahn and Besiroglu, 2022; 470 GPUs, 2006--2021) finds that
FLOP/s per dollar doubles every 2.46 years (95\% interval 2.24--2.72; 2.07 years for ML-specific GPUs). We map this to
the decline of the compute-capital price, $g_Q=1-2^{-1/2.46}=24.5\%$ a year, and derive an illustrative range
$\delta_Q\in[0.20,0.33]$ for compute depreciation from that trend and a three-to-five-year physical life; no direct
estimate was obtained. \emph{Inference prices.} Epoch AI's 2025 data insight finds that the price of a fixed
capability threshold fell 9$\times$ to 900$\times$ a year depending on the threshold, and a16z's ``LLMflation''
estimate is about 10$\times$ a year; 10$\times$ is our central fee path. List prices recorded on 2026-10-27 were
\$10 (input) and \$50 (output) per million tokens for the frontier models of both Anthropic and OpenAI and
\$0.10/\$0.50 for their small models, a hundredfold gap that spans the fee schedule. Model~B's $\phi_0=0.02$,
$\phi_1=0.002$ (units of average annual earnings) and $\bar q=0.02$ are illustrative: model compute units are not
tokens, and $\bar q$ is a scenario parameter (``inference intensity''), not a constant. \emph{Adoption.} The
Census Bureau's Business Trends and Outlook Survey reports that 19.8\% of businesses used AI in the two weeks before
2026-05-03 (37\% of firms with 250 or more employees, 39.7\% in the information sector), and \citet{bick2024rapid}
find that nearly 40\% of US adults aged 18--64 use generative AI, 23\% of the employed at work in the previous week;
these fix the initial delegation shares of the adoption experiments and bracket $F_\kappa$. \emph{Investment and
energy.} The 2025 capital expenditure of Alphabet, Meta, Amazon and Microsoft from SEC XBRL filings and earnings
releases (Appendix~\ref{app:data}) sums to \$376.1 billion on a calendar basis (\$357.5 billion with Microsoft's
fiscal year), about 1.2\% of US GDP---a flow proxy for the AI capital stock, not a measure of ownership
concentration, which is swept as a top-four share of 0.5--0.9. The IEA's \emph{Energy and AI} report puts
data-centre electricity use at 415 TWh in 2024 (1.5\% of world electricity, 45\% of it in the United States, roughly
187 TWh by our arithmetic) and projects about 945 TWh by 2030. The energy price $p_E$ is the EIA industrial retail
price of 9.64 cents per kWh (August 2026, preliminary; 8.98 a year earlier); since $\bar e$ is illustrative, the
energy component of the fee is illustrative in level even though its price is measured. The supervision cost
$\kappa_s$ has no public measure and is swept over 0--30\% of task value; \citet{brynjolfsson2025generative} and
\citet{eisfeldt2023generative} give the closest firm-level evidence and serve for orientation only.

\subsection{Validation}

\paragraph{Internal.} With $K_Q=0$ the production side of Model~A collapses to $Y=e^zK^\alpha L^{1-\alpha}$ and the
delegation share to zero; the solver is tested against the Brock--Mirman closed form ($\delta=1$, $\gamma=1$) and,
at the baseline parameters, against the $I/Y$ and $C/Y$ moments of Table~\ref{tab:targets}. With $d\equiv0$
Model~B is the \citet{aiyagari1994uninsured} economy and must show its known features: an equilibrium rate below
the rate of time preference and a right-skewed but too-thin wealth distribution. Model~C with one rational,
unlevered investor must price the risky asset at the present value of dividends at the safe rate. These checks are
gates, reported in the computation section, not results.

\paragraph{External roadmap and falsification.} The simulations identify effects only within the structural model;
the following data would test each hypothesis directly, subject to access rights, and none of the tests is performed
here. H1 (adoption threshold) predicts adoption rising in the stakes of the delegated decision and in the
frontier-to-small price gap: BTOS firm microdata and provider usage statistics matched to size and sector would
falsify it if adoption were flat in stakes or responded to the small-model price alone. H2 (common-model
correlation) predicts higher trade correlation and lower depth, conditional on public information, among accounts
using the same model: broker order data or exchange audit trails with an automation flag are needed, and no residual
correlation after controlling for common signals would falsify it. H3 (ownership concentration) predicts that AI
rents accrue to the DFA top groups in proportion to their holdings of compute-owning firms: 13F filings and DFA
equity holdings by wealth group can test it, and productivity gains without a rise in the top-0.1\% share would
contradict it. H4 (collateral feedback) predicts positive comovement of agentic leverage, valuations and investment,
observable in margin-debt statistics, repo haircuts and the capital expenditure of compute owners. H5 (diversity and
limits) and H6 (compute scarcity) are policy counterfactuals, falsifiable only indirectly through the response of
spreads and compute prices to documented supply shocks, of which the SEC capex and IEA energy series are the first
elements.

\input{tables/ledger}
\input{tables/targets}