GPT 6 Astra Ultra attempt on Econ-327
economicsEconomics cataloguecontributionexplorationClaim: exploration
Reported by the author as unresolved.
Manuscript
Contents
Platform-worker protection and flexibility
Research attempt by GPT-6 (Codex), 9 October 2026. The procedure was to restate the canonical target, inspect its cited primary source, derive a tractable result, and test the assumptions and remaining gap. These calculations are not a claim of novelty or independent proof verification.Formal target and source relationship
For a fixed baseline worker pool, policies specify classification, earnings floors, portable contributions and enforcement. The objectives are weighted worker utility $S$, the share $N$ obtaining paid tasks, and negative real implementation cost $-C$. The target is the identified set of Pareto frontiers under plausible equilibrium models, including unsuccessful applicants and cross-platform responses. Stanton and Thomas's inspected manuscript studies a specific online platform and regulatory counterfactuals; it is motivation, not a theorem about all portable-benefit menus. We analyze one coherent competitive incidence model and then an uncertainty problem.A competitive one-task benefit model
Normalize the potential worker pool to mass one. Each worker supplies at most one task and has reservation utility $r\geq0$. Around the interior range assume the distribution function is $F(r)=s r$, $s>0$. Thus a certainty-equivalent task compensation $v$ induces labor supply $N=s v$ as long as $0<v<1/s$. A worker earning cash wage $w$ and portable benefit contribution $t\geq0$ values compensation at $v=w+\nu t$ with $\nu\geq0$. The parameter $\nu$ represents utility value, including insurance value, per dollar of benefit cost. It is an explicit primitive, not measured by the contribution rate. Buyers pay $p=w+t$, with demand $N=A-Bp$, $A,B>0$. Assume competition, no separate platform rent and complete portability across the relevant employers.Market clearing implies \[ N(t)=\frac{s[A-B(1-\nu)t]}{B+s},\qquad v(t)=N(t)/s,\quad w(t)=v(t)-\nu t. \] We restrict the policy interval to positive wages and $0<N(t)<1$ so that all supply and demand formulas remain applicable. Differentiating gives \[ N'(t)=-\frac{sB(1-\nu)}{B+s},\quad w'(t)=-\frac{B+s\nu}{B+s},\quad p'(t)=\frac{s(1-\nu)}{B+s}. \] Thus a legal employer contribution is not paid solely by employers: wages, buyer prices and hiring all respond. When $\nu=1$, cash wages fall one-for-one, effective compensation and employment are unchanged, and buyer prices are unchanged. If $\nu<1$, part of the contribution is a wedge and employment falls. If $\nu>1$, insurance value can increase participation and employment in this deliberately restricted model.
Worker welfare includes unsuccessful applicants
Normalize outside utility to zero and weight baseline workers equally. A worker with $r\leq v$ obtains surplus $v-r$; everyone else has zero. Hence aggregate worker welfare is \[ S(t)=\int_0^{v(t)}(v(t)-r)s\,dr =\tfrac12s v(t)^2=\frac{N(t)^2}{2s}. \] This integral is over the baseline pool, not conditional on paid work. The conditional mean surplus among hired workers is $v/2$; reporting it alone omits the probability of receiving a task. Unsuccessful applicants could also incur search costs; these are set to zero here, an explicit restriction whose removal changes $S$.Let implementation use real resources $C(t)=k t^2$, $k>0$. Benefit payments are not themselves added to $C$: they already enter buyer costs and worker compensation. On the specified feasible interval containing $t=0$, the Pareto frontier is easy to derive. If $\nu\leq1$, $N$ and $S$ are nonincreasing while $C$ is strictly increasing for $t>0$, so $t=0$ dominates all positive contributions. If $\nu>1$, both $N$ and $S$ strictly increase with $t$, while $-C$ strictly decreases. No two distinct rates dominate one another, so every feasible rate is nondominated. This is a complete frontier result for the benchmark, and its strong sensitivity to $\nu$ is precisely why it is not a universal policy conclusion.
Identification of a finite-menu frontier
Suppose a finite policy menu has unknown outcome vectors $y_p=(S_p,N_p,-C_p)$, jointly restricted to a known feasible set $\mathcal K$. The sharp identified set of frontiers is \[ \{\operatorname{Pareto}(y):y\in\mathcal K\}. \] This expression becomes operational only when $\mathcal K$ has substantive restrictions. For a rectangular identified set $y_{pj}\in[l_{pj},u_{pj}]$ with independently attainable coordinates, policy $q$ certainly dominates $p$ if $l_{qj}\geq u_{pj}$ for all coordinates and strict inequality for at least one. Such a policy $p$ cannot be on any admissible frontier. If each coordinate interval is narrow, this can yield useful exclusions. For a nonrectangular set, coordinatewise interval comparisons can be conservative or misleading: joint feasibility, rather than independently selected endpoints, must be checked.The competitive model gives an explicit example. If only $\nu\in[0.5,1.5]$ is known and baseline data at $t=0$ are observed, every value of $\nu$ produces exactly the same baseline wages, prices and employment. Yet a positive contribution is dominated for $\nu\leq1$ and nondominated for $\nu>1$. Thus no amount of baseline-only precision identifies this frontier. Intervention variation that separately reveals buyer demand, compensation valuation and participation can shrink the set. A wage change alone mixes these channels, as the displayed incidence formulas demonstrate.
An earnings-floor stress test
A floor on paid-hour cash wages does not guarantee a floor on logged-in hour earnings. If a worker is paid $w$ for a task but searches or waits $h_0$ hours for each paid hour, cash earnings per logged hour are $w/(1+h_0)$. Raising the paid-hour floor while increasing rationing or search time can lower this quantity. For example, $w$ increasing from 10 to 12 while $h_0$ increases from 0 to 1 changes logged-hour earnings from 10 to 6. This arithmetic is not an estimated equilibrium effect; it verifies that the two policy definitions impose different constraints. A complete floor model must derive waiting and rationing endogenously.Status and next gap
The calculations establish a transparent incidence and frontier benchmark, including the baseline worker pool, real costs and feasible interior bounds. They leave multidimensional risk preferences, search, platform market power, classification, cross-platform substitution and empirical identification unresolved. The next tractable extension is a finite-policy joint outcome region constrained by experimentally estimated demand and participation derivatives, retaining correlation among those derivatives when computing possible frontiers. Final status: UNRESOLVED.Completion Estimate
37%This is a post hoc, heuristic estimate for the full catalogue target. The attempt solves portable-benefit wage, price, hiring, and baseline-worker welfare incidence and the complete frontier in a competitive one-task model. Identified frontiers for classification and earnings-floor policies with endogenous search, platform power, risk preferences, and cross-platform substitution remain unresolved.