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‎docs/index.html‎

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<a href="./posts/2026-01-26-new-derivation.html" class="no-external">Summary</a>
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<a href="./posts/2026-01-26-new-derivation.html" class="no-external">Formal derivations</a>
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"text": "Summary\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nLLM verification\n\n\n\n\n\n\n\n\n\n\nJan 30, 2026\n\n\nTom Cunningham\n\n\n\n\n\n\n\n\n\n\n\n\nForecasts of AI & Economic Growth\n\n\n\n\n\n\n\n\n\n\nNov 6, 2025\n\n\nTom Cunningham\n\n\n\n\n\n\n\n\n\n\n\n\nEconomics and Transformative AI\n\n\n\n\n\n\n\n\n\n\nOct 2, 2025\n\n\nTom Cunningham\n\n\n\n\n\n\n\n\n\n\n\n\nOn Deriving Things\n\n\n\n\n\n\n\n\n\n\nJan 30, 2025\n\n\nTom Cunningham\n\n\n\n\n\n\n\n\n\n\n\n\nToo Much Good News is Bad News\n\n\n\n\n\n\n\n\n\n\nDec 26, 2024\n\n\nTom Cunningham\n\n\n\n\n\n\n\n\n\n\n\n\nPremature Optimization and the Valley of Confusion\n\n\n\n\n\n\n\n\n\n\nMay 10, 2024\n\n\nTom Cunningham\n\n\n\n\n\n\n\n\n\n\n\n\nPeer Effects, Culture, and Taxes\n\n\n\n\n\n\n\n\n\n\nApr 28, 2024\n\n\nTom Cunningham\n\n\n\n\n\n\n\n\n\n\n\n\nBloodhounds and Bulldogs\n\n\nOn Perception, Judgment, & Decision-Making\n\n\n\n\n\n\n\nApr 27, 2024\n\n\nTom Cunningham\n\n\n\n\n\n\n\n\n\n\n\n\nThe Influence of AI on Content Moderation and Communication\n\n\n\n\n\n\n\n\n\n\nDec 11, 2023\n\n\nTom Cunningham\n\n\n\n\n\n\n\n\n\n\n\n\nThe History of Automated Text Moderation\n\n\n\n\n\n\n\n\n\n\nNov 18, 2023\n\n\nIntegrity Institute collaborators: Alex Rosenblatt, Jeff Allen, Ejona Varangu, Dave Sullivan, Tom Cunningham\n\n\n\n\n\n\n\n\n\n\n\n\nThinking About Tradeoffs? Draw an Ellipse\n\n\n\n\n\n\n\n\n\n\nOct 25, 2023\n\n\nTom Cunningham, OpenAI.\n\n\n\n\n\n\n\n\n\n\n\n\nExperiment Interpretation and Extrapolation\n\n\n\n\n\n\n\n\n\n\nOct 17, 2023\n\n\nTom Cunningham\n\n\n\n\n\n\n\n\n\n\n\n\nAn AI Which Imitates Humans Can Beat Humans\n\n\n\n\n\n\n\n\n\n\nOct 6, 2023\n\n\nTom Cunningham\n\n\n\n\n\n\n\n\n\n\n\n\nSushi-Roll Model of Online Media\n\n\nPreviously: “pizza model”, “salami model”\n\n\n\n\n\n\n\nSep 8, 2023\n\n\nTom Cunningham, Integrity Institute\n\n\n\n\n\n\n\n\n\n\n\n\nHow Much has Social Media affected Polarization?\n\n\n\n\n\n\n\n\n\n\nAug 7, 2023\n\n\nTom Cunningham, Integrity Institute\n\n\n\n\n\n\n\n\n\n\n\n\nThe Paradox of Small Effects\n\n\n\n\n\n\n\n\n\n\nAug 2, 2023\n\n\nTom Cunningham, Integrity Institute\n\n\n\n\n\n\n\n\n\n\n\n\nRanking by Engagement\n\n\n\n\n\n\n\n\n\n\nMay 8, 2023\n\n\nTom Cunningham\n\n\n\n\n\n\n\n\n\n\n\n\nSocial Media Suspensions of Prominent Accounts\n\n\n\n\n\n\n\n\n\n\nJan 31, 2023\n\n\nTom Cunningham\n\n\n\n\n\n\n\n\n\n\n\n\nOptimal Coronavirus Policy Should be Front-Loaded\n\n\n\n\n\n\n\n\n\n\nApr 5, 2020\n\n\n\n\n\n\n\n\n\n\n\n\nOn Unconscious Influences (Part 1)\n\n\n\n\n\n\n\n\n\n\nDec 8, 2017\n\n\n\n\n\n\n\n\n\n\n\n\nThe Work of Art in the Age of Mechanical Production\n\n\n\n\n\n\n\n\n\n\nSep 27, 2017\n\n\n\n\n\n\n\n\n\n\n\n\nRepulsion from the Prior\n\n\n\n\n\n\n\n\n\n\nMay 26, 2017\n\n\n\n\n\n\n\n\n\n\n\n\nThe Repeated Failure of Laws of Behaviour\n\n\n\n\n\n\n\n\n\n\nApr 15, 2017\n\n\nTom Cunningham\n\n\n\n\n\n\n\n\n\n\n\n\nEconomist Explorers\n\n\n\n\n\n\n\n\n\n\nFeb 25, 2017\n\n\n\n\n\n\n\n\n\n\n\n\nSamuelson & Expected Utility\n\n\n\n\n\n\n\n\n\n\nFeb 25, 2017\n\n\n\n\n\n\n\n\n\n\n\n\nWeber’s Law Doesn’t Imply Concave Representations or Concave Judgments\n\n\n\n\n\n\n\n\n\n\nFeb 25, 2017\n\n\n\n\n\n\n\n\n\n\n\n\nRelative Thinking\n\n\n\n\n\n\n\n\n\n\nApr 30, 2016\n\n\nTom Cunningham\n\n\n\n\n\nNo matching items"
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"text": "Formal derivations\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nLLM verification\n\n\n\n\n\n\n\n\n\n\nJan 30, 2026\n\n\nTom Cunningham\n\n\n\n\n\n\n\n\n\n\n\n\nForecasts of AI & Economic Growth\n\n\n\n\n\n\n\n\n\n\nNov 6, 2025\n\n\nTom Cunningham\n\n\n\n\n\n\n\n\n\n\n\n\nEconomics and Transformative AI\n\n\n\n\n\n\n\n\n\n\nOct 2, 2025\n\n\nTom Cunningham\n\n\n\n\n\n\n\n\n\n\n\n\nOn Deriving Things\n\n\n\n\n\n\n\n\n\n\nJan 30, 2025\n\n\nTom Cunningham\n\n\n\n\n\n\n\n\n\n\n\n\nToo Much Good News is Bad News\n\n\n\n\n\n\n\n\n\n\nDec 26, 2024\n\n\nTom Cunningham\n\n\n\n\n\n\n\n\n\n\n\n\nPremature Optimization and the Valley of Confusion\n\n\n\n\n\n\n\n\n\n\nMay 10, 2024\n\n\nTom Cunningham\n\n\n\n\n\n\n\n\n\n\n\n\nPeer Effects, Culture, and Taxes\n\n\n\n\n\n\n\n\n\n\nApr 28, 2024\n\n\nTom Cunningham\n\n\n\n\n\n\n\n\n\n\n\n\nBloodhounds and Bulldogs\n\n\nOn Perception, Judgment, & Decision-Making\n\n\n\n\n\n\n\nApr 27, 2024\n\n\nTom Cunningham\n\n\n\n\n\n\n\n\n\n\n\n\nThe Influence of AI on Content Moderation and Communication\n\n\n\n\n\n\n\n\n\n\nDec 11, 2023\n\n\nTom Cunningham\n\n\n\n\n\n\n\n\n\n\n\n\nThe History of Automated Text Moderation\n\n\n\n\n\n\n\n\n\n\nNov 18, 2023\n\n\nIntegrity Institute collaborators: Alex Rosenblatt, Jeff Allen, Ejona Varangu, Dave Sullivan, Tom Cunningham\n\n\n\n\n\n\n\n\n\n\n\n\nThinking About Tradeoffs? Draw an Ellipse\n\n\n\n\n\n\n\n\n\n\nOct 25, 2023\n\n\nTom Cunningham, OpenAI.\n\n\n\n\n\n\n\n\n\n\n\n\nExperiment Interpretation and Extrapolation\n\n\n\n\n\n\n\n\n\n\nOct 17, 2023\n\n\nTom Cunningham\n\n\n\n\n\n\n\n\n\n\n\n\nAn AI Which Imitates Humans Can Beat Humans\n\n\n\n\n\n\n\n\n\n\nOct 6, 2023\n\n\nTom Cunningham\n\n\n\n\n\n\n\n\n\n\n\n\nSushi-Roll Model of Online Media\n\n\nPreviously: “pizza model”, “salami model”\n\n\n\n\n\n\n\nSep 8, 2023\n\n\nTom Cunningham, Integrity Institute\n\n\n\n\n\n\n\n\n\n\n\n\nHow Much has Social Media affected Polarization?\n\n\n\n\n\n\n\n\n\n\nAug 7, 2023\n\n\nTom Cunningham, Integrity Institute\n\n\n\n\n\n\n\n\n\n\n\n\nThe Paradox of Small Effects\n\n\n\n\n\n\n\n\n\n\nAug 2, 2023\n\n\nTom Cunningham, Integrity Institute\n\n\n\n\n\n\n\n\n\n\n\n\nRanking by Engagement\n\n\n\n\n\n\n\n\n\n\nMay 8, 2023\n\n\nTom Cunningham\n\n\n\n\n\n\n\n\n\n\n\n\nSocial Media Suspensions of Prominent Accounts\n\n\n\n\n\n\n\n\n\n\nJan 31, 2023\n\n\nTom Cunningham\n\n\n\n\n\n\n\n\n\n\n\n\nOptimal Coronavirus Policy Should be Front-Loaded\n\n\n\n\n\n\n\n\n\n\nApr 5, 2020\n\n\n\n\n\n\n\n\n\n\n\n\nOn Unconscious Influences (Part 1)\n\n\n\n\n\n\n\n\n\n\nDec 8, 2017\n\n\n\n\n\n\n\n\n\n\n\n\nThe Work of Art in the Age of Mechanical Production\n\n\n\n\n\n\n\n\n\n\nSep 27, 2017\n\n\n\n\n\n\n\n\n\n\n\n\nRepulsion from the Prior\n\n\n\n\n\n\n\n\n\n\nMay 26, 2017\n\n\n\n\n\n\n\n\n\n\n\n\nThe Repeated Failure of Laws of Behaviour\n\n\n\n\n\n\n\n\n\n\nApr 15, 2017\n\n\nTom Cunningham\n\n\n\n\n\n\n\n\n\n\n\n\nEconomist Explorers\n\n\n\n\n\n\n\n\n\n\nFeb 25, 2017\n\n\n\n\n\n\n\n\n\n\n\n\nSamuelson & Expected Utility\n\n\n\n\n\n\n\n\n\n\nFeb 25, 2017\n\n\n\n\n\n\n\n\n\n\n\n\nWeber’s Law Doesn’t Imply Concave Representations or Concave Judgments\n\n\n\n\n\n\n\n\n\n\nFeb 25, 2017\n\n\n\n\n\n\n\n\n\n\n\n\nRelative Thinking\n\n\n\n\n\n\n\n\n\n\nApr 30, 2016\n\n\nTom Cunningham\n\n\n\n\n\nNo matching items"
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‎posts/2026-01-26-new-derivation.md‎

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- TODO: Discuss the *separable* case.
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- TODO: add discussion of bottlenecks, & decreasing returns to scale.
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- TODO: mention Amdahl's law.
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- TODO: what if you know t and t', and know a single m changed.
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**Suppose we make people more productive at a subset of tasks, and we see how they reallocate their time, what can we infer about their *overall* productivity?**
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# Summary
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We're going to assume (1) they are optimizing some unobserved objective functionm $y(\cdot)$; (2) that function has constant returns to scale (if they doubled their time on every task, they'd double output).
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**In short:** suppose we make someone more productive at *one* task, and we see how they reallocate their time. What can we infer about their overall productivity, assuming that they are reallocating their time efficiently to produce an objective
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The following results are all adaptions of very old standard economic theory about inferring the effects of a *price* change on economic welfare.
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Suppose someone is allocating their time across $N$ different tasks, $t_1,\ldots,t_N$, in order to maximize some function $y(A_1t_1,\ldots,A_Nt_N)$. Each
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**Formally:** suppose someone is allocating their time across $N$ different tasks, $t_1,\ldots,t_N$, in order to maximize some function $y(A_1t_1,\ldots,A_Nt_N)$. Each
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$$\begin{aligned}
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y(A_1t_1,\ldots,A_nt_n) &&& \text{(output)}\\
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V(A)\equiv\max_{t\in\mathbb{R}^n_+} y(A_1 t_1,\dots,A_n t_n) &&& \text{(value)} \\
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\sum_{i=1}^N t_i =1 &&& \text{(time allocation before AI)}\\
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\sum_{i=1}^N t'_i =1 &&& \text{(time allocation after AI)}\\
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m_i \equiv \frac{A_i'}{A_i} &&& \text{(productivity changes)} \\
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\end{aligned}$$
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| What you observe / assume | What we know about uplift ($\frac{V'}{V}$) | Reference |
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|----------------------------------------------------------------------------------------|:-----------------------------------------------------------------------------------------------------------------------------------------:|-----------------------------------|
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| If you know $y(t_1,\ldots,t_n)$ | Exact $V'/V=\frac{P(p)}{P(p')}$ with $p=1/A$, $p'=1/A'$ | Proposition 3 |
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| A fixed time allocation $t$ (not necessarily optimal), and multipliers $m$ | $m_{\min}\le \frac{y(A'\circ t)}{y(A\circ t)}\le m_{\max}$ | Proposition 3.1 |
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| Only multipliers $m$ (no shares) | $m_{\min}\le V'/V \le m_{\max}$ | Corollary 4.1 (part 3) |
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| Baseline shares $t$ and multipliers $m$ (but not $t'$) | $\Bigl(\sum_i \frac{t_i}{m_i}\Bigr)^{-1}\le \frac{V'}{V} \le m_{\max}$ | Corollary 4.1 (part 1) |
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| Post shares $t'$ and multipliers $m$ (but not $t$) | $m_{\min}\le \frac{V'}{V} \le \sum_i t_i' m_i$ | Corollary 4.1 (part 2) |
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| Both $t$ and $t'$ plus multipliers $m$ | $\Bigl(\sum_i \frac{t_i}{m_i}\Bigr)^{-1}\le \frac{V'}{V} \le \sum_i t_i' m_i$ | Proposition 4 |
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| Small changes, know baseline shares $t$ | Approx $\ln\frac{V'}{V} \approx \sum_i t_i \ln m_i$ | Corollary 7.1 |
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| Large changes, know a path $A(\tau)$ and shares along it | Exact $\ln\frac{V'}{V} = \int_0^1 \sum_i t_i(A(\tau))\,\frac{d}{d\tau}\ln A_i(\tau)\,d\tau$ | Proposition 8 |
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| CES (Assumption C1), $n=2$, only task 2 multiplied by $A_2^{(m)}$, know baseline $t_2$ | Exact $\frac{V'}{V}=\left((1-t_2)+t_2 (A_2^{(m)})^{\varepsilon-1}\right)^{\frac{1}{\varepsilon-1}}$ | Proposition 11 |
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| CES (Assumption C1), $n=2$, observe $t_2,t_2',A_2^{(m)}$ | Identify $\varepsilon=1+\frac{\operatorname{logit}(t_2')-\operatorname{logit}(t_2)}{\ln A_2^{(m)}}$ (then use Prop 11 for $\frac{V'}{V}$) | Proposition 12 (+ Proposition 11) |
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m_i \equiv \frac{A_i'}{A_i} &&& \text{(productivity changes due to AI)} \\
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\end{aligned}
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$$
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| What you observe / assume | What we know about uplift ($\frac{V'}{V}$) | Reference |
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|-------------------------------------------------------------------------------------------------|:-----------------------------------------------------------------------------------------------------------------------------:|-----------------------------|
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| You know $y(\cdot)$, $\bm{m}$ and $\bm{t}$ | Exact $V'/V$ | Prop. 3 |
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| You know $m$ and $t$, and the person cannot adjust their time allocations (so $\bm{t}'=\bm{t}$) | $m_{\min}\le \frac{y(A'\circ t)}{y(A\circ t)}\le m_{\max}$ | Prop. 3.1 |
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| You know $\bm{m}$ but not $\bm{t}$ or $\bm{t}'$ | $m_{\min}\le V'/V \le m_{\max}$ | Cor. 4.1 (part 3) |
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| You know $t$ and multipliers $m$ (but not $t'$) | $\Bigl(\sum_i \frac{t_i}{m_i}\Bigr)^{-1}\le \frac{V'}{V} \le m_{\max}$ | Cor. 4.1 (part 1) |
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| You know $t'$ and multipliers $m$ (but not $t$) | $m_{\min}\le \frac{V'}{V} \le \sum_i t_i' m_i$ | Cor. 4.1 (part 2) |
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| You know $t$ and $t'$ and $m$ | $\Bigl(\sum_i \frac{t_i}{m_i}\Bigr)^{-1}\le \frac{V'}{V} \le \sum_i t_i' m_i$ | Prop. 4 |
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| You know $t$ and $m$, and $\bm{m}\simeq 1$ | $\ln\frac{V'}{V} \approx \sum_i t_i \ln m_i$ | Cor. 7.1 |
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| Large changes, know a path $A(\tau)$ and shares along it | $\ln\frac{V'}{V} = \int_0^1 \sum_i t_i(A(\tau))\,\frac{d}{d\tau}\ln A_i(\tau)\,d\tau$ | Prop. 8 |
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| You assume $y(\cdot)$ is CES, $n=2$, only task 2 multiplied by $A_2^{(m)}$, know baseline $t_2$ | $\frac{V'}{V}=\left((1-t_2)+t_2 (A_2^{(m)})^{\varepsilon-1}\right)^{\frac{1}{\varepsilon-1}}$ | Prop. 11 |
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| You assume $y(\cdot)$ is CES, $n=2$, observe $t_2,t_2',A_2^{(m)}$ | $\varepsilon=1+\frac{\operatorname{logit}(t_2')-\operatorname{logit}(t_2)}{\ln A_2^{(m)}}$(\& use Prop 11 for $\frac{V'}{V}$) | Prop. 12 (+ Proposition 11) |
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