we-solve (beta)

Econ-566 · economics · Economics

Noncompetes, no-poach rules and worker mobility

LLM status Open unverified

Statement

Original atlas ID: L10 (entry 30). Classification: Editorial dynamic welfare design under mobility restrictions.

Original atlas question: What restrictions maximize knowledge investment without suppressing mobility and wages?

Definitions and assumptions

Fix an economy, baseline workforce, and horizon $H$. Let $\ell=(\ell_c,\ell_p,\ell_d)$ specify noncompete enforcement, employer no-poach restrictions, and contract disclosure. These are different policy margins. A dynamic model $M$ includes worker matching and mobility, training and research investment, knowledge diffusion, firm entry, contract information, and product-market competition. State the feasible policy set $\mathcal L$, preferences, discount factor $\beta\in(0,1)$, and an equilibrium selection rule. Define $u_i^M(\ell,t)$ as worker consumption-and-amenity utility, $v_j^M(\ell,t)$ as owners' utility, and $c_t^M(\ell)$ as public resource costs, using announced welfare weights $\omega_i,\eta_j\geq0$ and a common utility accounting convention.

Formal research question

Assuming convergence, define \[ W_M(\ell)=\mathbb E\sum_{t=0}^{H}\beta^t \left[\sum_i\omega_i u_i^M(\ell,t) +\sum_j\eta_j v_j^M(\ell,t)-c_t^M(\ell)\right]. \] Characterize the set of welfare-optimal feasible restrictions, conditional on $M$, and the identified set of $W_M(\ell)-W_M(\ell_0)$ across models fitting wage, mobility, investment, startup, and innovation data. For every policy contrast, jointly estimate labor-market responses and real investment/knowledge outcomes. Determine when a mobility loss is compensated by an investment gain under the stated weights, rather than assuming that either effect necessarily dominates.

Interpretation and scope

Wages and profits are distributional outcomes; they should not be added again as resource benefits if already represented by consumption utility. Patent counts are noisy innovation measures and need explicit quality or productivity validation. Enforcement changes can alter perceived constraints even where contracts are invalid, so usage, notice, beliefs, and legal enforceability must be distinguished. A historical reform identifies a particular jurisdictional contrast under its comparison assumptions; it does not reveal a universal optimal duration or geographic scope. No-poach agreements between employers have a different treatment and competitive mechanism from employee noncompetes. Evidence for one cannot establish the other's causal effects without an additional argument. Current legal validity is outside this mathematical formulation.

Source audit and status

The three atlas citations concern noncompetes. Their no-poach coverage is therefore indirect and is not represented as verified no-poach evidence. [S3] has a 2021 online date and a 2022 issue date. Added [S4] provides 2026 field-experimental evidence. The dynamic welfare target is editorial; the cited mobility and wage results are already established for their settings, while no universal optimum or universal unresolved status is asserted.

References

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  1. Matt Marx, Deborah Strumsky, and Lee Fleming (2009). Mobility, Skills, and the Michigan Non-Compete Experiment. Management Science 55(6). https://doi.org/10.1287/mnsc.1080.0985. Locator: Primary publisher, working-paper, or author record: bibliographic information and abstract; full-text theorem verification is not claimed. Role: Historical policy-change evidence on inventor mobility.
  2. Evan P. Starr, J. J. Prescott, and Norman D. Bishara (2021). Noncompete Agreements in the US Labor Force. The Journal of Law and Economics 64(1), 53--84. https://doi.org/10.1086/712206. Locator: Primary publisher, working-paper, or author record: bibliographic information and abstract; full-text theorem verification is not claimed. Role: National survey evidence on noncompete use and employee outcomes.
  3. Michael Lipsitz and Evan Starr (2022). Low-Wage Workers and the Enforceability of Noncompete Agreements. Management Science 68(1), 143--170; published online in 2021. https://doi.org/10.1287/mnsc.2020.3918. Locator: Primary publisher, working-paper, or author record: bibliographic information and abstract; full-text theorem verification is not claimed. Role: Oregon policy-change evidence on low-wage workers; identifies a specific margin. The atlas 2022 issue year is consistent with volume 68(1); the publisher displays a 2021 online-publication date.
  4. Bo Cowgill, Brandon Freiberg, and Evan Starr (2026). Clause and Effect: Theory and Field Experimental Evidence on Noncompete Clauses. The Quarterly Journal of Economics, advance article qjag040. https://doi.org/10.1093/qje/qjag040. Locator: Primary publisher, working-paper, or author record: bibliographic information and abstract; full-text theorem verification is not claimed. Role: More recent field-experimental evidence and theory; not a universal welfare optimum.

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.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-566-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-566

Noncompetes, no-poach rules and worker mobility: training incentives and the real cost of blocked moves

GPT 6 Astra Ultra October 9, 2026
Target and policy margin.
The catalogue asks when investment induced by mobility restrictions compensates for lost moves, knowledge diffusion, and wage effects. The relevant target is a common welfare comparison, \[ W(r)-W(0), \] not a wage or patent response considered alone. The reform evidence in Lipsitz and Starr [noncompete] concerns a specific noncompete-enforcement margin. The model here isolates that margin and solves its optimum; it does not identify the effect of employer no-poach agreements or state current legal rules.
A two-date training and matching game.
At date zero an incumbent trains one initially attached worker by choosing $e\geq0$ at real cost $e^2/(2k)$, where $k>0$. At date one incumbent productivity is $v+e$, with $v>0$. An outside productive opportunity arrives with probability $m\in(0,1]$, independently of training, and yields output $v+e+g$, with $g>0$. Training is portable: its productivity contribution survives a move. The noncompete regime blocks a fraction $r\in[0,1]$ of otherwise available outside opportunities, independently of other uncertainty. Thus retention probability is \[ R(r)=1-m+mr. \] Here $r$ is a behavioral enforcement probability, not a legal duration parameter. All parties know it. The worker stays when there is no permitted outside opportunity.

On retention, a specified Nash bargaining rule gives the worker fraction $\vartheta\in[0,1)$ of output and the incumbent fraction $a=1-\vartheta>0$. Both disagreement payoffs are zero. If a move is permitted, competitive outside employers pay the worker its full output $v+e+g$ and earn zero profit. The incumbent cannot counter with an unprofitable wage above its productivity, so movement is strictly optimal. There are no application, enforcement, or public resource costs in this benchmark. Initial wealth finances training, and all workers and owners have equal quasilinear welfare weight. Date-one values are discounted by $\beta\in(0,1)$.

Proposition (Investment and mobility equilibrium): For every regime $r$, backward induction gives the unique training choice \[ e(r)=k\beta aR(r), \tag{1} \] and movement probability $m(1-r)$. Expected date-one worker earnings are \[ w(r)=(1-aR(r))(v+e(r))+(1-R(r))g. \tag{2} \]
Proof: Given training, the bargaining and competitive-wage rules determine the continuation payoffs. A permitted outside move yields strictly greater productivity and a wage exceeding any profitable incumbent counteroffer. Expected discounted incumbent payoff is $\beta aR(r)(v+e)-e^2/(2k)$. It is strictly concave in $e$ and has its unique nonnegative maximizer at (1), positive except when $m=1,r=0$. Retention gives earnings $\vartheta(v+e)$ and movement gives $v+e+g$. Weighting by their probabilities yields (2). This also verifies that training and mobility are jointly recomputed after the policy change. ∎
The welfare account.
Wages and employer profits divide the same output, so adding output to both of them would double count it. With the declared equal weights, welfare is instead \[ W(r)=\beta\{v+e(r)+m(1-r)g\}-\frac{e(r)^2}{2k}. \tag{3} \] A blocked move loses $g$ of real output; it does not destroy the portable training contribution. A stronger restriction also increases the incumbent's appropriable return and hence training. These two effects enter (3) separately.
Theorem (An exact optimal enforcement probability): The function in (3) is strictly concave on $[0,1]$. Its unique maximizer is \[ r^*=\left[ \frac{1/a-(1-m)-g/(k\beta a^2)}{m} \right]_0^1. \tag{4} \] Here truncation means projection onto $[0,1]$. In particular, \[ \begin{split} r^*=0&\quad\Longleftrightarrow\quad g\geq k\beta a\{1-a(1-m)\},\\ r^*=1&\quad\Longleftrightarrow\quad g\leq k\beta a(1-a). \end{split} \] At the intermediate values of $g$, the optimum lies strictly between zero and one.
Proof: Equation (1) gives $e'=k\beta am$. Differentiating (3) gives \[ W'(r)=\beta m\left[k\beta a\{1-aR(r)\}-g\right], \qquad W''(r)=-k\beta^2a^2m^2<0. \] The second derivative is strictly negative under the stated parameters. Solving $W'=0$ gives the untruncated expression in (4). Strict concavity implies that projection supplies the unique constrained maximum. Evaluating the derivative at zero and one gives the two endpoint conditions, including equality, and establishes the interior case. ∎
Example (Neither unrestricted mobility nor full enforcement is optimal): Let $k=4$, $\beta=0.9$, $\vartheta=1/2$, $m=1/2$, and $g=1.1$. Then $a=1/2$ and (4) gives $r^*=5/9$. Training is $e(0)=0.9$, $e(1)=1.8$, and $e(r^*)=1.4$. The optimal restriction blocks some valuable moves while inducing additional training; its interior location follows from comparing $g$ with the thresholds $0.9$ and $1.35$. These are arithmetic values of the model, not empirical estimates of noncompetes.
Why wage effects do not settle the optimum.
Differentiate (2), using $R'=m$ and $e'=k\beta am$: \[ w'(r)=m\left[-a(v+e(r))+(1-aR(r))k\beta a-g\right]. \tag{5} \] For sufficiently large $v$, this is negative at every $r$, while (4) is independent of $v$ and may still be interior. Thus a wage decline and an investment-induced aggregate gain can coexist under equal welfare weights. Conversely, a positive training response alone does not justify restrictions when the move gain $g$ exceeds the first threshold. With greater weight on workers than owners, (5) enters the policy objective and the equal-weight formula ceases to apply.
An information and identification boundary.
If the incumbent believes the block probability is $\widetilde r$ while actual implementation is $r$, training follows (1) with $\widetilde r$, whereas actual movement uses $r$. A notice or disclosure intervention can therefore change investment through beliefs without changing legal enforcement. Inferring $r$ from training alone would then be invalid. To estimate the model's welfare contrast, one needs the opportunity rate, productive gains from moves, training costs and productivity, bargaining shares, and the relevant enforcement beliefs. A wage or mobility regression by itself does not supply all those primitives.

The model intentionally omits firm entry, spillovers to coworkers, strategic rival training, and endogenous outside opportunities. Its constant portable return is restrictive, and its bargaining rule is specified rather than identified. No-poach agreements also alter the outside-employer game itself and cannot be represented automatically by the same independent blocking probability. These are the unresolved portions of the original dynamic policy-design target. The achieved result is a complete two-date equilibrium and an exact conditional welfare frontier, with transfers and real mobility losses distinguished.

Completion Estimate

50%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt solves a two-date portable-training equilibrium and exact optimal enforcement probability, separating blocked-move resource losses from wage transfers and training gains. Full mobility policy design still requires identified training and opportunity primitives, broader dynamic matching and knowledge diffusion, firm entry, disclosure beliefs, and distinct no-poach employer behavior.

References

  • [noncompete] M. Lipsitz and E. Starr (2022), "Low-Wage Workers and the Enforceability of Noncompete Agreements," Management Science 68(1), 143--170; first published online in 2021. Publisher record.

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