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Econ-257 · economics · Economics

Collective-bargaining redesign

LLM status Open unverified

Statement

Original subject group: Labor markets and work

Audit classification: Published research agenda. Published metadata and full text checked. Political-policy and coverage questions are explicit; optimization over bargaining systems is editorial.

Definitions and precise research target

Complete editorial problem. Consider a specified economy with workers, firms, and a finite set $\mathcal B$ of bargaining arrangements. An arrangement $b$ defines bargaining coverage, representation rules, agreement extension, contract enforcement, and negotiation level. Let $W_T(b)$, $L_T(b)$, $K_T(b)$, and $R_T(b)$ denote equilibrium compensation, employment, investment, and a preregistered representation index at horizon $T$. Let $U_i(b)$ be worker $i$'s lifetime utility, measured with an explicit wage--amenity preference model, and let $\omega_i\ge0$ be declared welfare weights summing to one. Define $S(b)=E[\sum_i\omega_iU_i(b)]$. The design task is to characterize arrangements maximizing $S(b)$ subject to specified feasibility, fiscal, and employment constraints. Estimate effects separately for represented workers, uncovered workers, entrants, and consumers. A credible analysis must allow bargaining to change firm entry, productivity, wage-setting, political influence, and who becomes represented. State the variation that distinguishes these mechanisms and the equilibrium closure used in counterfactuals. Existing evidence about a particular union's wage premium cannot identify this entire policy ordering. The optimum is conditional on stated objectives and institutional context, rather than a universally best bargaining system.

Definitions and admissible scope. Let the finite arrangement menu specify a separate coverage indicator, membership indicator, negotiation level, extension rule, and dispute procedure. Worker utilities $U_i(b)$ are cardinally normalized using stated consumption, amenity, and risk preferences; wages alone are not utilities. Define normalized weights $\omega_i\ge0$, worker-hours $L_T(b)$, real investment $K_T(b)$, and representation $R_T(b)$, for example the share of eligible workers able to vote for negotiators. Feasibility requires stated inequalities $L_T(b)\ge\underline L$ and $C(b)\le B$. The optimum of a known finite menu is elementary. The substantive task is to identify equilibrium outcomes and welfare rankings, permitting bargaining to affect uncovered workers, consumers, entry, and policy. Define the equilibrium or reduced-form intervention law before interpreting a collective-bargaining counterfactual. The population, objective weights, arrangements, feasibility constraints, utility functions, real resource costs, and treatment of uncertainty are all fixed inputs. Take $\mathcal B$ finite and $E|U_i(b)|<\infty$ for every $i,b$. The requested output is the identified set of feasible welfare rankings and maximizers when equilibrium response laws are not known, not merely enumeration after all utilities have been supplied. The displayed $S$ is explicitly a worker-weighted criterion; a whole-economy variant must add separately weighted consumer, owner, and fiscal payoffs and avoid counting transfers as additional resources.

Variants and assumption changes

  1. Coverage versus membership (source-motivated): vary agreement coverage while holding membership rules fixed, then vary membership conditional on coverage. Estimate separate wage and political-participation effects; overlapping interventions require a factorial design or additional restrictions.
  2. Political feedback (source-motivated extension): let arrangement $b$ change subsequent labor legislation through an explicit voting or lobbying model. Compare the endogenous-policy welfare ranking with a ranking that holds legislation fixed; neither ranking is universally best.

References and source audit

Support and access. Published metadata and full text checked. Political-policy and coverage questions are explicit; optimization over bargaining systems is editorial. Locator: Section 7, pp. 389--390, policy and coverage-versus-membership bullets.
  1. Ethan Kaplan; Suresh Naidu. 2025. Between Government and Market: The Political Economics of Labor Unions. Section 7, Conclusion and Future Research Questions, printed pp. 389–390, bullets on direct policy effects and coverage versus membership. https://econweb.umd.edu/~kaplan/PE_of_Labor.pdf. DOI: https://doi.org/10.1146/annurev-economics-051520-025251.

Common conventions: Source mathematical conventions

These conventions apply to each entry unless explicitly replaced.

Models and identification. A primitive model $m$ specifies all probability laws, feasible choices, information, equilibrium and measurement rules used by its target. The admissible class $\mathcal M$ is fixed independently of outcomes. If $P_O(m)$ is its observable probability law and $\tau(m)$ the target, its identified set at observable law $P$ is \[ \mathcal I_\tau(P)=\{\tau(m):m\in\mathcal M,\ P_O(m)=P\}. \] Point identification means that this set is a singleton. An empty set signals incompatibility of data and assumptions. Coordinate bounds need not describe the joint set. Bounds are sharp only if every retained target is attainable or arbitrarily approachable by compatible models. Finite bounds require explicit restrictions on unobserved quantities; unrestricted confounding, hidden wealth, extrapolation or arbitrary equilibrium selection can yield uninformative sets.

Causal interventions. A potential outcome $Y_i(p)$ is the outcome under a complete policy regime $p$, including timing, financing, information and market spillovers. The target population and units are fixed before treatment. With interference, use the complete assignment vector or a justified exposure mapping; individual no-interference notation alone cannot identify an economy-wide intervention. A difference of mean potential outcomes does not require the cross-world joint distribution. Conditioning on post-treatment migration, survival, enrollment or employment changes the estimand and needs separate justification.

Statistical statements. An estimator, confidence set, forecast or test specifies a sampling law, model class and loss/coverage criterion. Uniform $(1-\alpha)$-coverage means $\inf_{m\in\mathcal M}\Pr_m\{\tau(m)\in C_n\}\geq1-\alpha$, or an explicitly asymptotic version. The whole parameter space trivially has coverage, so informativeness requires a stated width, risk or power objective. A forecasting improvement is relative to a named benchmark, information set and evaluation population.

Equilibrium and optimization. An equilibrium comprises feasible plans, optimality, beliefs where needed, budget constraints and all market/resource clearing. The primitives or a stated benchmark define each correspondence $\mathcal E(p;m)$. Nonemptiness is assumed or proved, never inferred just from compact policy sets. A selected-equilibrium target uses a declared selection rule; a robust target quantifies over all permitted equilibria. Use suprema or infima unless attainment is established. An $\varepsilon$-optimal policy is feasible and within $\varepsilon$ of the finite optimum value. Report an empty feasible set separately.

Welfare and accounting. Normative utility, interpersonal normalization, social weights, numeraire, discounting and the use of public receipts are primitives. Behavioral utility need not coincide with the welfare criterion. Household welfare includes ownership income and rents once; adding profits again to compensating variation generally double-counts them. A monetary equivalent is defined by an explicit transfer and price convention, with monotonicity and range conditions. Average effects, mechanism contributions, transfers and welfare losses are different targets. Infinite sums require convergence and dynamic feasible plans require terminal or no-Ponzi conditions.

Source versions. A working-paper date, revision date and journal publication year are distinguished. A DOI for a journal version is not automatically the DOI of an earlier manuscript. Locators refer to the stated version and printed pages where specified. A metadata-only check does not verify a claim about a full-text conclusion. Every entry's source subsection narrows the provenance claim accordingly.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-257-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-257

Bargaining power, investment, and worker-weighted welfare

Problem ID: J52-253. Status: Unresolved. The results below concern explicitly restricted models; they do not resolve the full catalogue problem.

Question and model

The catalogue asks which collective-bargaining arrangements improve welfare when employment, investment, and other responses are counted. The research agenda in [KN] is considerably broader than any single bargaining model. Here an exact right-to-manage model shows how investment hold-up can make intermediate bargaining power optimal for workers, even when stronger bargaining would benefit them with capital fixed.

There is one firm, a mass $M$ of ex ante identical risk-neutral eligible workers, and an outside wage $w_0$. Employed workers are selected uniformly from this population. Production is \[ F(K,L)=(a+\eta K)L-\frac b2L^2, \qquad a_0=a-w_0>0,\quad b,\kappa>0,\quad\eta\ge0. \] At date zero the firm chooses nonnegative investment $K$ and pays $\kappa K^2/2$. At date one the firm and a union bargain over the wage, taking investment as sunk. At date two the firm chooses employment. The union's bargaining payoff is aggregate wage rent $U=(w-w_0)L$, and the firm's bargaining payoff is operating profit, excluding the sunk investment cost. Nash bargaining maximizes $U^\theta\pi^{1-\theta}$ for $0<\theta<1$, with endpoint arrangements defined by limits. Disagreement yields zero operating payoff and outside wages. Assume \[ b\kappa>\eta^2, \qquad M\ge\frac{a_0}{b(1-\eta^2/(b\kappa))}. \] The second condition ensures that labor supply never binds in the equilibria studied. Average worker welfare is $w_0+U/M$, including eligible workers left unemployed. Owners' profits are not included in this worker-weighted criterion.

Equilibrium and its investment response

Theorem (Unique backward-induction outcome): Write $c=1-\theta/2\in[1/2,1]$ and $\gamma=\eta^2/(b\kappa)<1$. The equilibrium is \[ K_\theta=\frac{\eta c^2a_0}{b\kappa-\eta^2c^2},\qquad z_\theta=a_0+\eta K_\theta=\frac{a_0}{1-\gamma c^2}, \] \[ w_\theta=w_0+(1-c)z_\theta,\qquad L_\theta=\frac{c z_\theta}{b},\qquad U_\theta=\frac{c(1-c)z_\theta^2}{b}. \] Investment and employment weakly decrease with bargaining power $\theta$, strictly when respectively $\eta>0$ and $a_0>0$.
Proof: First relax the employment cap $L\le M$. Given $K$ and a wage with $w-w_0\in[0,z]$, $z=a_0+\eta K$, the strictly concave employment problem gives $L=(z-(w-w_0))/b$ and operating profit $\pi=(z-(w-w_0))^2/(2b)$. Put $x=(w-w_0)/z$. Constants aside, the Nash product is $x^\theta(1-x)^{2-\theta}$. Its log derivative vanishes uniquely at $x=\theta/2$; the second derivative of its log is negative and endpoints give zero when $0<\theta<1$. This proves the relaxed wage and employment expressions conditional on $K$.

The relaxed investment objective is $c^2(a_0+\eta K)^2/(2b)-\kappa K^2/2$. Its second derivative is $c^2\eta^2/b-\kappa<0$, and its first-order condition yields the displayed nonnegative $K_\theta$. We now show that this is also the global optimum with the finite labor supply, including against investment deviations for which the cap binds.

Write $y=w-w_0$. With the cap, employment is $\min\{M,(z-y)_+/b\}$. If $z\le bM/c$, the relaxed bargained outcome is feasible and is still optimal: on the binding-employment branch the Nash product increases up to its boundary whenever that boundary lies below the relaxed optimal wage, and on the other branch it has the unique maximizer just computed. More explicitly, on the binding branch $0\le y\le z-bM$, its interior critical wage is $y=\theta(z-bM/2)$, while on the nonbinding branch it is $y=\theta z/2$. Comparing these critical points with the branch boundary gives the following possibilities when $z>bM/c$: the wage is the kink $y=z-bM$, with operating profit $bM^2/2$, or it is the binding-branch critical point, with operating profit $(1-\theta)M(z-bM/2)$. The latter branch is possible only for $\theta<1$ and $z\ge cbM/(1-\theta)$. The conclusions extend to $\theta=0,1$ by limits.

In all cases the actual bargained operating profit is at most $c^2z^2/(2b)$. Equality holds in the relaxed-feasible region. At the kink this bound follows from $z\ge bM/c$. On the fully binding branch, use \[ M(z-bM/2)\le z^2/(2b),\qquad 1-\theta\le c^2, \] the first inequality being equivalent to $(z-bM)^2\ge0$ and the second to $\theta^2/4\ge0$. Thus actual profit net of investment is globally bounded above by the strictly concave relaxed investment objective. At its unique maximizer $K_\theta$, the assumed labor supply makes $cz_\theta/b\le M$, so that upper bound is attained. No larger capacity-constrained investment can improve it. This proves existence and uniqueness of the actual investment choice and the formula for $z_\theta$.

Both $c^2/(1-\gamma c^2)$ and $c/(1-\gamma c^2)$ increase with $c$, while $c$ decreases with $\theta$. The labor-supply condition follows by evaluating the candidate employment at its maximum $c=1$. Endpoint formulas follow continuously. ∎

The firm anticipates that additional investment raises the bargained wage. This explains why sunk investment must be excluded from its date-one bargaining payoff but included in its date-zero objective. Including it twice would change the question.

The worker-welfare optimum

Theorem (An interior optimum generated by investment): If $\eta=0$, worker welfare is maximized at $\theta=1$. If $\eta>0$, it has a unique maximizer $\theta_*=2(1-c_*)\in(0,1)$, where $c_*\in(1/2,1)$ is the unique root of \[ 1-2c+\gamma(3c^2-2c^3)=0. \] The optimal $\theta_*$ strictly decreases as $\gamma$ increases in $(0,1)$. For fixed investment, in contrast, worker rent increases throughout $\theta\in[0,1]$.
Proof: Factors independent of $c$ aside, worker rent is $f(c)=c(1-c)/(1-\gamma c^2)^2$. Differentiation gives \[ f'(c)=\frac{N(c)}{(1-\gamma c^2)^3},\qquad N(c)=1-2c+\gamma(3c^2-2c^3). \] For $c\in[1/2,1]$, $N'(c)=-2+6\gamma c(1-c)<-1/2$. If $\gamma>0$, then $N(1/2)=\gamma/2>0$ and $N(1)=\gamma-1<0$, proving a unique strict maximum. If $\gamma=0$, $f$ is maximized at $1/2$, giving $\theta=1$. At the interior root, $N_\gamma=c^2(3-2c)>0$ and $N_c<0$, so implicit differentiation gives $dc_*/d\gamma>0$. Consequently $d\theta_*/d\gamma<0$. Finally, with $K$ fixed, $U=\theta(1-\theta/2)z^2/(2b)$ has derivative $(1-\theta)z^2/(2b)\ge0$, strictly positive below one. ∎
Corollary (An employment requirement): Suppose an arrangement must satisfy $L_\theta\ge\underline L$. If $\underline L>L_0$, no arrangement is feasible. Otherwise the feasible set is $[0,\bar\theta]$ for a uniquely defined cap $\bar\theta\in[0,1]$, allowing $\bar\theta=1$. The worker-welfare maximizing feasible arrangement is $\min\{\theta_*,\bar\theta\}$, interpreting $\theta_*=1$ when $\eta=0$.
Proof: Employment is continuous and strictly decreasing in $\theta$, so its upper-level set is exactly the stated interval. The preceding derivative calculation shows that worker welfare increases up to $\theta_*$ and decreases thereafter. Maximizing over the interval proves the claim. ∎

What would connect the calculation to institutions?

The parameter $\theta$ represents bargaining weight in a specified game. It does not by itself describe sectoral coverage, extension rules, contract duration, co-determination, or union political activity. A comparison of those institutions needs a defensible mapping to bargaining timing, outside options, investment technology, and the set of eligible workers. The calculation makes no claim that workers displaced from the firm receive its wage rent: their outside income is included explicitly through $M$.

One potentially testable implication of the restricted model is that an increase in bargaining weight jointly reduces investment and employment while its wage effect is ambiguous once $z_\theta$ responds. Observing that pattern alone does not identify $\theta$ or causal effects, because demand and technology changes also alter these variables. Entry, product-market spillovers, distributional weights among heterogeneous workers, and an empirical identification design remain necessary for the original problem. The interior optimum is a proved property of this model, not an empirical recommendation or a novelty claim.

Completion Estimate

30%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt solves investment, wage and employment equilibrium and a worker-welfare optimum in a right-to-manage bargaining benchmark with an employment constraint. Identified rankings of actual bargaining institutions still require coverage and representation rules, heterogeneous workers, entry, uncovered spillovers and political feedback.

References

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