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Econ-616 · economics · Economics

Informality and firm growth

LLM status Open unverified

Statement

Original atlas ID: D4 (entry 34). Classification: Editorial registration-versus-policy-bundle identification problem.

Original atlas question: Does formalization cause firms to become more productive, or do productive firms simply formalize?

Definitions and assumptions

Fix baseline enterprises, workers, and horizon $H$. Distinguish active-firm registration $F_{jt}\in\{0,1\}$ from compliant-job status $G_{ijt}\in\{0,1\}$; a registered firm can employ unregistered workers. Define $F_{jt}=0$ after closure, and $G_{ijt}=0$ whenever worker $i$ has no job at firm $j$, including after closure or during nonemployment. Otherwise $G_{ijt}=1$ precisely when that job satisfies the specified compliance rules. A policy vector $\pi$ specifies registration costs, tax schedules, labor requirements, enforcement, and access to services. Let $V_{jt}(\pi)$ be real value added, $L_{jt}(\pi)$ total labor hours, and $S_{jt}(\pi)$ enterprise survival, tracking baseline firms after exit and entry changes. Set output and hours to zero after genuine closure, and report survival separately. A structural model $M$ specifies demand, production, selection into formality, evasion, and market adjustment.

Formal research question

For announced policy contrasts $\pi_1,\pi_0$, identify \[ \Theta_H=\left( \mathbb E[V_{jt}(\pi_1)-V_{jt}(\pi_0)],\; \mathbb E[S_{jt}(\pi_1)-S_{jt}(\pi_0)],\; \Delta F_t,\Delta G_t,\Delta L_t\right)_{t=0}^H, \] where $\Delta F_t,\Delta G_t,\Delta L_t$ use specified baseline firm and worker populations. Separately, in a model defining an intervention that changes registration while holding the other policy components fixed, identify or bound the registration effect on investment and productivity. State when that intervention is feasible and when the data only identify a bundled reform. For aggregate productivity, use a defined output-hours or production-function measure with positive denominators and explicit price adjustment.

Interpretation and scope

Productive firms can select into registration; the observed registered-unregistered gap therefore does not identify formalization's causal effect. A registration-assistance instrument requires exclusion from productivity other than through registration, which can fail if assistance also supplies training or information. Compliance costs, access to finance, and enforcement changes are distinct policy channels. Conditioning on surviving firms can conceal induced exit or entry, and revenue productivity can change merely because prices change. The formal sector's output share, the fraction of registered firms, and the fraction of compliant jobs need not move together. Employment composition and workers' outside options belong to the equilibrium response and cannot be inferred from registry counts alone.

Source audit and status

[S1] reviews informal firms and [S2] explicitly distinguishes informality margins. [S3], verified as NBER Working Paper 34445, was issued in 2025 and revised in July 2026. It studies minimum wages and informality; it is adjacent evidence, not a direct formalization experiment. The registration-versus-bundled-policy identification problem is editorial, and this audit does not certify universal unresolved status.

References

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  1. Rafael La Porta and Andrei Shleifer (2014). Informality and Development. Journal of Economic Perspectives 28(3), 109--126. https://doi.org/10.1257/jep.28.3.109. Locator: Primary publisher, working-paper, or author record: bibliographic information and abstract; full-text theorem verification is not claimed. Role: Evidence and conceptual distinctions among informal firms.
  2. Gabriel Ulyssea (2018). Firms, Informality, and Development: Theory and Evidence from Brazil. American Economic Review 108(8). https://doi.org/10.1257/aer.20141745. Locator: Primary publisher, working-paper, or author record: bibliographic information and abstract; full-text theorem verification is not claimed. Role: Model distinguishing firm and labor informality; supports separate extensive/intensive margins.
  3. Ellora Derenoncourt, François Gerard, Lorenzo Lagos, and Claire Montialoux (2025). Minimum Wages and Informality. NBER Working Paper 34445, issued November 2025; revised July 2026. https://www.nber.org/papers/w34445. Locator: Primary publisher, working-paper, or author record: bibliographic information and abstract; full-text theorem verification is not claimed. Role: Minimum-wage policy and informal-sector wage spillovers; adjacent evidence rather than direct causal evidence on formalization. Primary record verifies source identity and 2026 revision. The mismatch between wage-policy evidence and formalization effects is flagged in the entry.

Integrated refinement: ECON-071

Informal firms as stepping stones to formal growth

This refinement adopts the additional source conventions (audited:conventions), subject to its local restrictions. Initially informal firms' productive transition paths. Track an initially informal cohort through policy-induced registration, exit and growth. Let $F_{i,t}(z)$ denote status, with an explicit post-closure rule, and $\Pi_{i,t}(z)$ net profits including exit outcomes. Identify \[ P_z(h)=\Pr(F_{i,h}(z)=1\mid F_{i,0}=0),\qquad V_z=\mathbb E\sum_{t=0}^{T}\delta^t\Pi_{i,t}(z). \] Declare registration costs, taxes, enforcement and formal-sector benefits as components of the full regime. Use pretreatment cohort definitions and consistent real-cost/profit accounting. Distinguish genuinely productive transitions from relabeling, survival selection and displacement of other firms. Registration, compliant jobs, sustained productivity and household welfare are separate outcomes, so administrative counts alone do not establish growth.

Supplied source audit for ECON-071

Local [S1], [S2], etc. in this audit and the references immediately below belong to ECON-071, independently of the original entry's reference numbering. [S1] and [S2] directly support informal firms as potential stepping stones. The fixed-cohort transition/value formulation is editorial; the author's name is Gabriel Ulyssea, despite a spelling inconsistency on a review citation line.

References for integrated refinement ECON-071

[S1] Gabriel Ulyssea, Matteo Bobba, Lucie Gadenne, and Mariaflavia Harari (2025). Informality. VoxDevLit 6(2), April 2025.

Verified locator / source role: Primary publisher page checked for review identity, authors, version and background. The specific published agenda is corroborated in [S2]; the exact formal target is editorial.

https://voxdev.org/voxdevlit/informality

[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: Informality. Supports the broad research agenda; this is not a verbatim quotation or an attribution of the displayed mathematical target.

https://voxdev.org/topic/evidence-gaps-key-unanswered-questions-across-16-topics-development-economics

Common conventions: Source mathematical conventions

Unless an entry states otherwise, agent and firm sets are finite. State and action spaces are measurable, strategies and outcome maps are measurable, probability laws are specified, and expectations exist for the targets under discussion. Infinite-horizon discount factors lie in $(0,1)$ and discounted objectives are finite. Norms, tolerances and complexity input sizes must be specified for computational comparisons. Constants or primitives that appear locally are inputs, not empirical facts asserted by this catalogue.

Identification. A statistical model $m\in\mathcal M$ implies an observable law $P_m$ and target $\theta(m)$. At law $P$, its identified set is \[ \Theta(P;\mathcal M)=\{\theta(m):m\in\mathcal M,\ P_m=P\}. \] Point identification holds when this set is a singleton, or equivalently when $P_{m_1}=P_{m_2}$ implies $\theta(m_1)=\theta(m_2)$ for all compatible models. Sharp bounds mean bounds attained, or approached when the boundary is not attained, by compatible models. Writing this set is a definition, not a solution: the substantive task is to establish informative restrictions and derive or estimate the set. An unrestricted-confounding model may give only trivial bounds.

Inference. Identification, estimability and valid uncertainty quantification are separate questions. Fix $\alpha\in(0,1)$. A confidence procedure $C_n$ over a declared law class $\mathcal P$ has asymptotic uniform coverage at level $1-\alpha$ when \[ \liminf_{n\to\infty}\inf_{P\in\mathcal P}\Pr_P\{\theta(P)\in C_n\}\geq1-\alpha. \] For set-identified targets the coverage event must be defined separately, for example coverage of the full identified set. If an entry uses identification-indexed models instead of uniquely indexed laws, the same coverage convention is applied over those models. Pointwise limits do not establish uniform validity, and asymptotic validity does not guarantee finite-sample precision. Sample sizes, dependence structures and weak-identification sequences are part of each application.

Counterfactuals. $Y_i(a)$ is a potential outcome under a specified intervention, including its timing, implementation and versions. It is not $Y_i$ conditional on voluntarily choosing $a$. A full intervention vector permits interference; shorthand scalar treatment notation does not silently impose no interference. Assignment, selection, attrition, anticipation and measurement must be specified. A claim about an instrument's local effect is not automatically a population policy effect.

Economic equilibrium. A counterfactual model includes best responses, constraints, feasibility, market clearing, state transitions and expectations consistent with the stated information. When several equilibria exist, either a declared selector is an input or the target is set-valued. Comparisons across policy regimes must use the stated selection convention. Reduced-form relationships are not presumed invariant after an equilibrium-changing intervention.

Optimization and welfare. Optimization is over the stated feasible set. If attainment is not established, $\sup$ or $\inf$ and $\varepsilon$-optimal choices replace an asserted maximizer or minimizer. For example, with value $V=\sup_{a\in\mathcal A}W(a)<\infty$, an $\varepsilon$-optimal set is $\{a\in\mathcal A:W(a)\geq V-\varepsilon\}$ for $\varepsilon>0$. Benefits, costs, risk and health or environmental outcomes can be aggregated only using declared units and conversion weights. Interpersonal utility weights, the treatment of fiscal transfers and foreign profits, discounting, and the behavioral welfare criterion are explicit inputs. A comparative-static derivative additionally requires existence and appropriate differentiability of the selected counterfactual.

Sources. Local labels [S1], [S2], and so forth restart in every question. Locators identify the relevant abstract, model, section or result at the level actually checked; a bibliographic or abstract-level verification is not represented as inspection of an entire proof. Working papers are distinguished from later publications and changing versions. Source summaries are paraphrased. All URL checks and editorial formulations refer to the audit date stated above.

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

Econ-616-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-616

Registration timing, firm scale and compliant jobs: a dynamic threshold benchmark

Catalogue problem: O17-612. Status: Unresolved. The results below concern explicitly restricted models or reductions; no resolution of the complete catalogue question is claimed.

Question and scope

The catalogue distinguishes registration, compliant jobs, productive growth and survival. Ulyssea [ulyssea] provides a model and evidence with distinct firm and labor informality margins. We solve a transparent small-open-economy benchmark with growing productivity, a sunk registration cost and a separate job-compliance choice. It shows both how productive firms select into registration and why registration counts need not measure technology gains or compliant employment.

Operating decisions and accounting

A baseline firm with productivity $a>0$ produces real value added $V=a\sqrt{\ell}$ from $\ell\geq0$ labor hours, with no intermediate inputs. Output price is one. An outside sector fixes the base wage $w>0$ and supplies enough labor elastically that the firm does not affect it. Financing and enforcement payments can make the firm's effective cost per labor hour differ from $w$. These are private cost wedges; they are not all real social resource losses.

An informal firm pays effective hourly cost $c_I>0$, pays no modeled output tax, and has only noncompliant jobs. A registered firm pays output tax $\tau\in[0,1)$ and chooses compliant jobs at additional hourly charge $\gamma\geq0$ or noncompliant jobs subject to expected hourly penalty $\phi\geq0$. With base formal financing cost $\bar c_F>0$, set \[ c_F=\bar c_F+\min\{\gamma,\phi\}. \] The firm uses a common compliance status for its workforce; tie-breaking at $\gamma=\phi$ selects compliance. Registered operation also incurs fixed per-period private cost $k\geq0$. Registering costs a one-time amount $F\geq0$, and registration is irreversible. All receipts, taxes and financing payments enter private profits consistently; no aggregate welfare result is inferred by treating every payment as a resource cost.

Lemma (Scale, profits and the two informality margins): The unique optimal hours, value added and operating profits are \[ \ell_I(a)=\left(\frac{a}{2c_I}\right)^2, \quad V_I(a)=\frac{a^2}{2c_I}, \quad \pi_I(a)=\frac{a^2}{4c_I}, \] \[ \ell_F(a)=\left(\frac{(1-\tau)a}{2c_F}\right)^2, \quad V_F(a)=\frac{(1-\tau)a^2}{2c_F}, \quad \pi_F(a)=\frac{(1-\tau)^2a^2}{4c_F}-k. \] Formal jobs are compliant exactly when $\gamma\leq\phi$, under the tie rule. In particular, registered employment may be entirely noncompliant.
Proof: For a regime with after-tax revenue share $b>0$ and effective hourly cost $c>0$, set $x=\sqrt\ell\geq0$. Variable profit is $bax-cx^2$, a strictly concave quadratic with unique optimum $x=ba/(2c)$. Substituting yields hours, gross value added and variable profit $b^2a^2/(4c)$. Use $b=1,c=c_I$ for informal operation and $b=1-\tau,c=c_F$ for registered operation, subtracting its fixed cost. Conditional on registration, the labor-compliance choice minimizes its per-hour payment, so the lower of $\gamma$ and $\phi$ is chosen. ∎

An exact dynamic registration rule

At dates $t=0,1,\ldots$, productivity follows a known path $a_t=a_0g^t$, $a_0>0$, $g>1$. The private discount factor is $\delta\in(0,1)$ with $\delta g^2<1$, ensuring all profit sums converge. The initially informal firm chooses a registration date $T\in\{0,1,\ldots\}\cup\{\infty\}$; $T=\infty$ means never register. The sunk cost is paid at the beginning of date $T$, and registered operation starts immediately. There are no liquidity constraints on paying $F$.

Define \[ d=\frac{(1-\tau)^2}{4c_F}-\frac1{4c_I}. \] Assume $d>0$: at sufficiently large scale the registered operating technology and financing bundle has a private advantage. This is a substantive parameter restriction, not a universal property of formality.

Theorem (Optimal registration time): Relative to remaining informal forever, the value of registration at finite date $T$ is \[ \Delta\mathcal V(T)=\delta^T\left\{ \frac{d a_0^2g^{2T}}{1-\delta g^2} -\frac{k}{1-\delta}-F\right\}. \] An optimal registration date exists and is finite. With the earliest-date rule among ties, it is \[ T^*=\min\{t\geq0:d a_t^2\geq k+(1-\delta)F\}. \] Reducing the registration cost $F$ weakly advances registration, while increasing the recurrent cost $k$ weakly delays it. The rule compares the current operating advantage with the opportunity cost of paying the sunk fee one period earlier.
Proof: The registered-minus-informal operating profit at date $t$ is $d a_t^2-k$. Discounting from $T$ onward and subtracting the sunk fee gives two convergent geometric sums, proving the value formula. Its successive difference is \begin{align*} \Delta\mathcal V(T+1)-\Delta\mathcal V(T) &=\delta^T\{-d a_T^2+k+(1-\delta)F\}. \end{align*} Since $a_T$ grows strictly, this difference is positive before the displayed threshold, nonpositive at it, and strictly negative thereafter except possibly at the single equality date. Thus the value sequence increases up to the threshold and decreases after it, with a tie between adjacent dates only at equality. The threshold is reached in finite time because $d>0$ and $a_T\to\infty$.

Never registering has incremental value zero. The finite-date value is positive for all sufficiently large $T$, since $d a_T^2/(1-\delta g^2)$ eventually exceeds the finite constants in braces. Hence the maximizer identified above weakly exceeds a strictly positive finite-date value and is preferable to never registering. The threshold comparisons in $F$ and $k$ prove the comparative statics. ∎

For a distribution of initial productivities, the registered share by date $h$ is therefore identified within this structural model from that distribution and the primitives: \[ P(h)={\mathbb{P}}\left(d a_0^2g^{2h}\geq k+(1-\delta)F\right). \] Every initially informal firm is included, not just survivors. In this benchmark firms never close: informal profits are positive, and at optimal registration the operating advantage is nonnegative and remains so. This simplifies survival rather than estimating an exit effect.

Proposition (A finite planning horizon): If the firm instead ends operation at a fixed date $H\geq0$, write $G_m(x)=\sum_{j=0}^{m-1}x^j$. The incremental value of registering at $T\leq H$ is \[ \Delta\mathcal V_H(T)=\delta^T\{d a_T^2G_{H-T+1}(\delta g^2) -kG_{H-T+1}(\delta)-F\}. \] Let $T^*$ be the threshold date in Theorem Theorem (Optimal registration time) and set $\bar T=\min\{T^*,H\}$. Among finite dates, $\bar T$ is an earliest maximizer. If $\Delta\mathcal V_H(\bar T)>0$, registration at $\bar T$ is optimal; if it is negative, never registering is optimal; at zero both are optimal. Thus crossing the one-period timing threshold does not by itself justify paying the fee near a terminal date.
Proof: Summing operating differences only through $H$ gives the displayed formula. For $T<H$, all operating differences from $T+1$ through $H$ cancel in the comparison of adjacent registration dates, leaving \[ \Delta\mathcal V_H(T+1)-\Delta\mathcal V_H(T) =\delta^T\{-d a_T^2+k+(1-\delta)F\}. \] Its sign changes at the same threshold as in the infinite-horizon proof. Capping that threshold at $H$ therefore maximizes over the available finite dates. The outside option of never registering has incremental value zero, which gives the final comparison. ∎

Growth, measured productivity and selection are different

Proposition (Scale expansion without a technology effect): Under $d>0$, registration at a fixed productivity $a$ strictly raises real value added and labor hours. It leaves the production-function parameter $a$ unchanged and strictly lowers average value added per labor hour. Specifically, \[ \frac{V_F}{V_I}=R:=\frac{(1-\tau)c_I}{c_F}>1, \qquad \frac{\ell_F}{\ell_I}=R^2, \qquad \frac{V_F/\ell_F}{V_I/\ell_I}=\frac1R<1. \] If $\phi<\gamma$, the same scale expansion creates no compliant jobs in the model.
Proof: The inequality $d>0$ is equivalent to $c_F<(1-\tau)^2c_I$. Hence $R>(1-\tau)^{-1}\geq1$, with strict inequality. The ratios follow by substitution from Lemma Lemma (Scale, profits and the two informality margins). Both potential production functions have exactly the same parameter $a$; larger hours operate farther down its decreasing marginal product. The compliance claim follows from the separately optimized per-hour status choice. ∎

The fall in output per hour is a well-defined ratio, not a deterioration of the firm's technology. A financing or cost improvement raises scale on a diminishing-returns production function, which can lower average labor productivity. Conversely a registered--unregistered difference in an estimated technology parameter can arise entirely from selection.

Proposition (A selected technology gap with zero causal technology change): Suppose at a fixed date the cross-sectional productivity distribution has positive probability on each side of the registration threshold and finite mean. Under the earliest-date rule, every registered firm's productivity is at least the threshold and every unregistered firm's productivity is strictly below it. Therefore \[ {\mathbb{E}}[a\mid\text{registered}]>{\mathbb{E}}[a\mid\text{unregistered}], \] although the causal effect of changing registration on the production-function parameter $a$ is zero for every firm.
Proof: The monotone productivity path means a firm has registered by the current date exactly when its current productivity meets the threshold in Theorem Theorem (Optimal registration time). On the unregistered event, the nonnegative distance below the threshold is strictly positive almost surely, so its conditional expectation is positive; on the registered event productivity is weakly above the threshold. Their conditional means are thus strictly ordered. In the model the entire productivity path is specified before registration and does not depend on its timing, so both registration interventions have the same $a$ path. ∎

The theorem does not say registration never affects productivity in reality. It demonstrates that a positive registered-firm technology gap is consistent with zero causal technology effect even when formalization genuinely raises output and profits.

What is the intervention, and what remains?

The modeled transition switches taxes, access to a lower financing-cost regime, recurrent obligations and possibly job compliance together. It is a bundle attached to registration. A pure registry-label intervention that holds the operating revenue share, hourly costs, feasible choices and all fees fixed changes none of the objective functions above and therefore changes neither optimal hours nor output. That zero result follows from identical choice problems; it should not be mislabeled as the effect of the actual bundle.

A subsidy to $F$ can identify a timing response under the model only if it does not also supply management training, relax a binding liquidity constraint or alter $c_F$. Registry counts alone do not identify these channels. Estimating the finite-horizon catalogue vector requires baseline-firm output and hours, separate worker-compliance observations, and survival measurement. The present model supplies exact transition shares and private discounted values but deliberately has no entry, exit, wage equilibrium adjustment or displacement of other firms. It is not an aggregate welfare calculation: tax and financing receipts accrue to counterparties and would have to be included once in a full closure.

A stochastic productivity process or a temporary formal-status option would introduce an option value absent from the deterministic timing comparison. Credit rationing could prevent payment of a profitable entry fee. Endogenous labor supply would change both scale and the wage comparison. These are substantive remaining gaps in the original policy and identification problem. The completed restricted results isolate registration timing, scale, selection and job compliance rather than treating them as a single outcome.

Completion Estimate

44%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt solves registration timing and operating scale with separate job compliance, and proves selection can create a technology gap despite zero causal technology change. The full formalization policy vector still requires causal output, survival and employment evidence, feasible separation of registry labels from bundles, and entry, exit, credit and wage-equilibrium responses.

References

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