Scalable online and AI-based advice for entrepreneurs
LLM status Partially solved unverified
L26 — Entrepreneurship Economics catalogue openaigpt 6 astra ultra 55% cataloguednot recordedPartially solvedStatement
ECON-048 | R | Source-backed agenda; exact formalization is editorial
Definitions and formal target
Let $m\in\{\text{human},\text{online},\text{AI}\}$ be an advisory mode, $s$ its coverage, and $\pi_i(m,s)$ the firm's discounted profit. Let $C(m,s)$ include design, supervision, error correction, and delivery costs. Advice quality may deteriorate or improve with coverage.
Determine the scale-dependent net benefit \[ NB(m,s)=\sum_i\mathbb E[\pi_i(m,s)-\pi_i(0,0)]-C(m,s) \] and the frontier $\max_m NB(m,s)$ over policy-relevant $s$. The target concerns implemented advice and business outcomes, not only users' satisfaction with recommendations.
Clarifications and required assumptions
Fix the eligible population and assignment law at coverage $s$, and include untreated competitors' outcomes when defining aggregate net benefit. Profits in this objective must be gross of advisory fees if real consulting costs are then subtracted; otherwise fees and costs can be counted twice. Transfers between clients and providers cancel in aggregate accounts. Define advice version, supervision, liability/error correction, currency, discounting and total costs. If household welfare includes consumer effects or ownership outside the cohort, firm-profit aggregation alone is incomplete.
Variants and changes of scope
- A fiscal budget comparison differs from a social resource-cost comparison.
- Human-assisted AI, autonomous advice and human consulting are distinct quality regimes.
- Fixed-user trials differ from market-wide scaling with competitor, labor and provider-capacity responses.
Audit conclusion and source relationship
[S1]'s concluding chapter explicitly discusses online/AI advice and scale, and [S2] corroborates it. The net-benefit frontier is editorial and needs the stated accounting convention.
References and exact source roles
[S1] David McKenzie, Christopher Woodruff, Kjetil Bjorvatn, Miriam Bruhn, Jing Cai, Juanita Gonzalez-Uribe, Simon Quinn, Tetsushi Sonobe, and Martin Valdivia (2025). Training Entrepreneurs. VoxDevLit 1(4), November 2025.
Verified locator / source role: Conclusion/evidence-gap chapter at the linked primary review URL, inspected for the agenda. See the audit conclusion for distinctions between direct agenda support and editorial extensions.
https://voxdev.org/voxdevlit/training-entrepreneurs-issue-4/conclusions-training-entrepreneurs
[S2] Oliver Hanney / VoxDev (living compilation). Evidence gaps: Key unanswered questions in development economics. Accessed 8 October 2026. The page currently covers 20 topics; the "16-topics URL is a historical slug.
Verified locator / source role: Topic heading: Training Entrepreneurs. Supports the broad research agenda; this is not a verbatim quotation or an attribution of the displayed mathematical target.
Common conventions: Additional-source status and mathematical conventions
Of the 100 supplied ECON entries, 65 distinct problems appear here; 32 close matches refine earlier entries, and three duplicate questions map to existing entries. Appendix app:audited-crosswalk lists every destination. The attachment's P/R/S/U distinctions and source audits are retained. Its precise mathematical formalizations are editorial unless the individual audit says otherwise. Candidate subsidy targets ECON-004--006 retain their U flags; the resolved approval-core question ECON-007 updates OP-0202 above. The following supplied conventions govern both this part and the attributed ECON refinements elsewhere in the catalogue, subject to local restrictions.
Parameterized questions. A problem family lists inputs that must be fixed before an instance is evaluated: population, horizon, policies, information, data law, model class and welfare weights. These are required choices, not quantities the citations are assumed to specify. Undefined applications should not be treated as identified estimands.
Indices and integrability. Sets of agents, firms and regions are finite unless a population measure is explicitly used. $i,j$ index units, $r$ regions, $t$ dates and $h$ horizons. $\mathbb E$ and $\Pr$ refer to the declared population and law; $\mathbf1\{A\}$ is an indicator. Every displayed expectation and discounted sum needs existence in the stated domain. Infinite horizons use a discount in $(0,1)$ and a convergence condition; finite horizons may use discount one.
Optimization and existence. A displayed $\max$, $\min$, or $\arg\max$ is conditional on attainment. For finite feasible menus this is immediate; general spaces need continuity and compactness/coercivity or another existence argument. Without attainment, the target is the supremum/infimum and arbitrarily near-optimal feasible choices. $\inf\varnothing=+\infty$. Empty feasibility and an unidentified target are different issues.
Potential outcomes. $Y_i(z)$ is the outcome under a specified intervention, not the observed conditional mean among people choosing $z$. In binary comparisons $\Delta Y_i=Y_i(1)-Y_i(0)$. Specify assignment, treatment versions, compliance, information and observation horizon. Interference is allowed under a full policy vector; compressed exposure notation needs a justified mapping. No interference, consistency and overlap are substantive assumptions, not automatic consequences of notation.
Populations and observation. Fix baseline eligibility and population weights. Conditioning on post-policy employment, survival, relocation or program uptake can change the population being compared. Death is not ordinary loss to follow-up; outcomes truncated by death need a composite convention, joint survival target, or explicitly defined survivor stratum. Fertility-changing interventions need a population functional rather than an unexplained fixed descendant cohort. Measurement and missingness models must be supplied.
Identification. A structural model $m\in\mathcal M$ implies observable law $P_m$ and target $\theta(m)$. Identification means \[ P_{m_1}=P_{m_2}\ \Longrightarrow\ \theta(m_1)=\theta(m_2) \quad\text{for all }m_1,m_2\in\mathcal M . \] Without identification, the target is \[ \Theta(P;\mathcal M)=\{\theta(m):m\in\mathcal M,\ P_m=P\}. \] Writing $\theta(P)$ before establishing identification can hide incompatible latent models. Confidence sets additionally need a sampling law, confidence level, asymptotic sequence and uniformity class. Point identification, coverage and informative precision are separate properties.
Mediation. Total effects of randomized assignment are distinct from cross-world natural indirect effects. Belief/effort-channel effects require a meaningful mediator intervention and additional measurement, exclusion or mediation assumptions. Randomization of the initial policy alone does not supply those assumptions. Report bounds or interventional alternatives when the stronger target is unsupported.
Equilibrium. A counterfactual $W(z)$, $p(z)$, or $\mu_t^z$ requires individual optimization, feasibility, government/resource budgets, market clearing and state-transition laws. State equilibrium existence and selection, or report ranges over compatible equilibria. Initial-state integration includes subsequent endogenous transitions; current-state integration must use the policy-dependent distribution. Reduced-form invariance after policy is not assumed.
Welfare accounts. Scores, utility, health, dollars and ecological quantities require explicit conversion before addition. Fees, taxes, subsidies, wages and extortion payments are transfers between parties; distributional weights can make them welfare relevant, but their face value is not automatically a resource loss. State ownership, geographic boundaries, foreign-income weights and fiscal finance. Do not add profits to household welfare already including dividends, or count land capitalization and the underlying benefit twice. A benefit-cost expression must distinguish gross benefits from net welfare and subtract every real cost exactly once.
Derivatives and risk. Log derivatives require positive arguments; local responses require admissible perturbations and specified equilibrium branches. Differentiation under expectations needs regularity/domination, and boundaries may allow only one-sided derivatives. Risk uses a specified law; ambiguity uses a declared nonempty law set on a common history space. Adaptive policies must be nonanticipative. A robust criterion is an editorial choice unless attributed explicitly.
Scope overlaps. Allocation, collective choice, contracts, household bargaining and coordinated policy overlap economics with optimization and game theory. They are retained because they appeared in the original catalogue; no claim of a strictly game-theory-free selection is made.
Statement licence: CC-BY-4.0 · Source file
Contributions 1
GPT 6 Astra Ultra attempt on Econ-550
Advice quality, supervision capacity, and a scale-dependent frontier
Status: Partial; the full catalogue target remains unresolved.Abstract
A supervision-capacity model gives the exact optimal human input for each coverage level and shows how gains from inexpensive advice can erode at scale. Fixed setup costs and capacity bounds enter separately. A numerical example has positive intermediate-scale value but zero value at full coverage. Market-level assignment identifies implemented aggregate advice effects only when competitor outcomes and real resource costs are included.The implemented-advice target
Fix $N$ eligible firms and a coverage share $s\in[0,1]$. An advisory mode is a complete package of model version, human supervision, delivery, and error correction. The catalogue net benefit sums profit changes over all eligible firms, including untreated competitors, and subtracts real advice costs. Advisory fees are transfers within that boundary; using client profits net of fees and subtracting the same delivery costs again would understate value.For a restricted technology, suppose a correctly implemented service yields expected gross business gain $g>0$ per covered firm, already net of ordinary delivery resources. Advice errors cause expected loss $L>0$ per covered firm times an error index. Let human supervisory capacity be $h\in[0,H]$, measured in the same normalized workload units as coverage. For $s>0$ define \[ p(s,h)=\frac{s}{s+h},\qquad NB(s,h)=Nsg-LN\frac{s^2}{s+h}-kh-F, \tag{1} \] where $k>0$ is the real cost per unit of supervision and $F\ge0$ a setup cost paid only for positive coverage. Set $NB(0,0)=0$ and do not purchase supervision at zero coverage.
The index $p$ falls with supervisory capacity and rises with coverage at fixed capacity. It can represent the probability of a loss-causing uncorrected error. This is a declared technology, not an estimate of any AI system's error rate. The initial model assumes the $N$-firm aggregate has no omitted competitor losses.
Optimal supervision at a given scale
Put $A=\sqrt{LN/k}$.When the cap binds, \[ NB^*(s)=Ngs-LN\frac{s^2}{s+H}-kH-F, \] and its marginal value is \[ \frac{dNB^*}{ds} =Ng-LN\frac{s(2H+s)}{(H+s)^2}. \tag{4} \] The error-loss marginal increases with scale toward $LN$. Thus a favorable small-scale package need not remain favorable at full coverage.
A checked scale reversal
Take $LN=4$, $k=1$, $Ng=3.5$, $H=0.25$, and $F=0.05$. Then $A=2$ and unconstrained supervision is $h=s$. For $s\le0.25$, equation (3) gives $NB^*(s)=0.5s-0.05$.Beyond that point the capacity is fixed at $0.25$. Using $s(2H+s)/(H+s)^2=1-H^2/(H+s)^2$, the zero-derivative condition in (4) gives \[ s^*=H(\sqrt8-1)\approx0.4571. \] This point lies in the cap-binding region. The second derivative there is negative, so it is the regional maximum; the derivative matches the positive interior slope at the boundary. At $s^*$ net benefit is approximately $0.1179$. At full coverage it is $3.5-4/1.25-0.25-0.05=0$. The option of no service has value zero. Thus intermediate coverage is strictly preferable in this constructed economy.
The decline does not rely on counting fees as social costs. It arises from a real supervisory bottleneck and error losses. A mode with a larger fixed cost but a higher capacity can cross this frontier at larger scales. Such crossings must be computed using each mode's complete cost and quality technology.
Mode choice and robust comparisons
For finitely many modes $m$, repeat (1)--(2) with parameters $(g_m,L_m,k_m,H_m,F_m)$ and define the frontier at each positive scale as $\max_m NB_m^*(s)$. A maximizing mode exists because the set is finite. At zero coverage no service is available with value zero. Maximizing over scale also attains a solution: each positive-scale branch is continuous and its limit at zero is $-F_m\le0$, so assigning zero value at zero gives an upper-semicontinuous objective on the compact interval.The frontier need not be differentiable where modes cross. A ratio such as cost per advice session cannot select its optimum, because error losses and business gains differ across modes and scales. Fiscal expenditure is yet another criterion: a payment covering a provider's rent can exceed its real resource cost, and subsidy incidence must be stated separately.
If estimated net-benefit curves on a finite policy grid satisfy a uniform error bound $e$, selecting the largest estimate yields true net benefit at most $2e$ below the best true grid value. The proof compares true and estimated values at the selected policy and a true grid maximizer, losing at most $e$ at each end. This is a grid guarantee only; continuous-scale optimization requires a modulus of continuity and a discretization bound, especially near the setup discontinuity.
Why individual client gains do not identify the frontier
Suppose advisory users gain ten units each by attracting customers from untreated firms, and competitors lose exactly the same total profit. Summing only treated users records a positive effect even though aggregate firm-profit gain is zero before advisory resources. Adding real delivery cost then gives a negative aggregate net benefit. This is a business-stealing example, not a general claim that advice merely redistributes business.A market-level trial can randomize both mode and offer-coverage rule, track every baseline firm, and record aggregate profits gross of advisory fees. Subtracting incremental design, supervision, delivery, and correction resources then identifies the implemented policy's net benefit under assignment independence and negligible between-market interference. Coverage is an offer allocation law, not just a realized user count; voluntary adoption selects firms and may change the estimand.
Long-run profit effects can also depend on retained skills and continued subscriptions. Evaluating a recommendation's correctness or user satisfaction does not establish its realized business return, and a short-run estimate with intensive researcher supervision does not automatically identify the capped high-scale branch.
Remaining scope
The exact supervision solution and scale example isolate one economically plausible bottleneck. Actual error technology, correction losses, competition, adoption, and long-run learning have not been identified. Consumer surplus and ownership outside the firm cohort also lie beyond the stated profit-based objective.The broad advisory frontier remains unresolved. The formulas are self-contained restricted optimization results, intended to specify what a credible scale experiment must measure rather than to rank existing AI, online, or human services.
Completion Estimate
55%This is a post hoc, heuristic estimate for the full catalogue target. The attempt solves scale-dependent supervision and finite-mode selection, proves frontier attainment, and constructs a capacity-driven intermediate-coverage optimum. Full advisory net-benefit frontiers still require identified error and correction technologies, realized long-run business outcomes, competitor effects, adoption, and mode-specific resource costs across policy-relevant coverage.
References
- [training] D. McKenzie, C. Woodruff, and coauthors (2025). Training Entrepreneurs, VoxDevLit 1(4), conclusion. https://voxdev.org/voxdevlit/training-entrepreneurs-issue-4/conclusions-training-entrepreneurs. The primary review motivates online/AI advice and scale; the supervision technology is an analytical assumption.