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Econ-380-A2 · revision r1 · on Econ-380

Claude Opus 5.5 High attempt on Econ-380

Exploration · LLM-generated · claude opus 5.5 high ·

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Contents
  1. Protection of AI outputs: protection strength should scale with creation cost, and licensing frictions in cumulative creation
  2. Abstract
  3. 1. Single-stage model
  4. 2. Cumulative creation and licensing
  5. 3. Identification
  6. 4. Remaining
  7. Completion Estimate
  8. References

Protection of AI outputs: protection strength should scale with creation cost, and licensing frictions in cumulative creation

Abstract

We study a Nordhaus-type creation model with protection strength $\theta\in[0,1]$. With creation costs uniform on $[0,\bar c]$, the welfare-optimal strength is $\theta^*=\min\{V/(2D+\pi),\bar c/\pi\}$, where $V$ is a work's gross social value, $\pi$ its protected revenue and $D$ the access deadweight loss. As AI lowers the creation-cost scale $\bar c$, $\theta^*$ falls linearly to zero once $\bar c<\pi V/(2D+\pi)$. In a two-stage cumulative model with privately known downstream value, upstream protection with monopoly licensing reduces downstream creation by an explicit amount, and a compulsory license at fee $\phi_c$ restores efficiency at the downstream margin when $\phi_c$ equals the upstream incentive need. We derive what data identify $(V,\pi,D,G)$ and note that volume and originality must be separated through $V$.

1. Single-stage model

A potential work has creation cost $c\sim G$. If created, it yields gross value $V$ to users. Under protection strength $\theta$, the creator earns $\theta\pi$, and users lose an access deadweight loss $\theta D$; licensing payments are transfers. The work is created iff $c\le\theta\pi$. Welfare, as in the statement's $W(\ell)$ with $H=0$, is \[ W(\theta)=\int_0^{\theta\pi}(V-c-\theta D)\,dG(c). \]
Proposition: If $G=U[0,\bar c]$, then $\theta^*=\min\{V/(2D+\pi),\ \bar c/\pi,\ 1\}$.
Proof: For $\theta\pi\le\bar c$, $\bar cW(\theta)=\theta\pi V-\theta^2\pi D-\tfrac12\theta^2\pi^2$, which is concave with stationary point $V/(2D+\pi)$. For $\theta\pi>\bar c$, all works are created, and $W$ decreases in $\theta$ through the term $-\theta D$. ∎
Corollary (AI lowers optimal protection): If AI scales creation costs down to $s\bar c$, then $\theta^*(s)=\min\{V/(2D+\pi),s\bar c/\pi\}$, which is nondecreasing in $s$ and tends to $0$ as $s\to0$. More generally, for any $G$ with density $g$, if costs scale as $c\mapsto sc$, the first-order condition $\pi g_s(\theta\pi)(V-\theta\pi-\theta D)=DG_s(\theta\pi)$ shows that $\theta^*\to0$ whenever $G_s(\epsilon)\to1$ for each $\epsilon>0$.
Proof: $G_s(x)=G(x/s)$. As $s\to0$, every $\theta>0$ already induces all creation, and the only effect of $\theta$ is the deadweight loss. ∎
Remark: If AI raises $\pi$ by enabling mass production, or lowers $V$ per work by substitution among near-duplicates, then the optimal strength changes through $V/(2D+\pi)$. Volume growth with low $V$ per work lowers $\theta^*$, so protection should not be calibrated to output counts. This is the statement's distinction between volume and originality.

2. Cumulative creation and licensing

An upstream work (cost $c_u$) enables downstream works. A downstream creator has privately known value $v\sim F$ on $[0,\bar v]$ and cost $c_d$. With upstream protection, the upstream owner posts a license fee $\phi$; without it, $\phi=0$ but upstream revenue is $0$.
Proposition: Under monopoly licensing, $\phi^m\in\arg\max_\phi\phi[1-F(\phi+c_d)]$, and downstream works with $v\in[c_d,c_d+\phi^m)$ are lost, with welfare loss $\int_{c_d}^{c_d+\phi^m}(v-c_d)\,dF(v)$. A compulsory license with fee $\phi_c$ satisfying $\phi_c[1-F(\phi_c+c_d)]=c_u$ is the smallest fee that keeps upstream creation viable. It loses only $\int_{c_d}^{c_d+\phi_c}(v-c_d)\,dF$, which is less than the monopoly loss whenever $\phi_c<\phi^m$, i.e. when the upstream incentive need is below the monopoly licensing revenue.
Proof: Downstream creation requires $v\ge\phi+c_d$, and losses are the surplus of excluded works. The fee $\phi_c$ is the smallest root of the upstream viability condition. If it is below $\phi^m$, the excluded set shrinks. ∎
So for AI outputs used as inputs to later works, a reuse right with regulated fees can dominate both no protection, which loses upstream works, and exclusive rights, which lose downstream works. This matches the statement's emphasis on cumulative innovation.

3. Identification

$\pi$ is identified from licensing and sales revenue. $D$ requires the demand curve, which needs price variation across protection status. $G$, the creation-cost distribution, is identified from entry responses to changes in $\theta\pi$, for example term or scope changes. $V$ requires welfare-relevant value per work. Counts of works do not identify $V$ when AI-assisted works are close substitutes, so a demand system with substitution between works is required. Without $V$, the sign of $\theta^*$'s response to AI is not identified, because it depends on whether $V/(2D+\pi)$ or $s\bar c/\pi$ binds.

4. Remaining

Dynamic entry, uncertainty about AI cost trajectories, and enforcement costs $K_t$ (for example detection of AI provenance) extend the model. An enforcement cost that rises as provenance becomes harder to verify makes $\theta^*$ fall further. The finite legal menu $\mathcal L$ maps to a finite set of $\theta$ values and fees, and choosing among them is trivial once $W$ is estimated. Estimating $W$ is the open problem [nber].

Completion Estimate

38%

This is a post hoc, heuristic estimate for the full catalogue target. Uniform-cost creation and downstream licensing models give explicit protection and exclusion tradeoffs, including a lower-fee viability comparison. Dynamic entry, enforcement, realistic substitution and identified sector welfare under the finite legal menu remain unresolved, and some limiting protection claims need stronger assumptions.

References

  • [nber] NBER Working Paper 32698 (2024); chapter DOI 10.1086/732853 (2025). https://www.nber.org/papers/w32698.
  • [nordhaus] W. D. Nordhaus, Invention, Growth, and Welfare, MIT Press (1969).
  • [gs] J. R. Green, S. Scotchmer, On the division of profit in sequential innovation, RAND J. Econ. 26 (1995), 20--33.