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Econ-189 · economics · Economics

Measuring employer wage-setting power

LLM status Open unverified

Statement

Original subject group: Labor markets and work

Audit classification: Published research agenda. Metadata and full chapter checked. Passthrough alone is insufficient under the stated model; the counterfactual wage-component estimand is editorial.

Definitions and precise research target

Complete editorial problem. Fix a labor market, a time interval, and a population of employer--worker matches. Let $w_{ij}>0$ be worker $i$'s hourly compensation at firm $j$, and let $p_{ij}>0$ be the marginal revenue product of that worker's labor, holding the firm's other inputs fixed. Define the latent or model-measured wedge $d_{ij}=\log p_{ij}-\log w_{ij}$. Let $a_{ij}$ denote job amenities, $s_i$ worker skills, and $c_{ij}$ contract incentives. In a structural model $M$, let $w^{C}_{i}(M)>0$ be compensation when the firm's ability to restrict labor supply or bargain over rents is removed while preferences, production, and aggregate resources remain fixed. Define the competition intervention and selected counterfactual job explicitly. Compare the same baseline worker's selected counterfactual wage, even if the employer changes; report nonemployment and nonexistent positive wages separately. The market-power component is $m_i(M)=\log w^{C}_{i}(M)-\log w_{i,j(i)}$. Given matched records, production information, and specified exogenous shocks, identify or sharply bound $E[m_i(M)]$ over models consistent with the data. Establish which restrictions distinguish this component from amenity compensation, sorting, productivity measurement error, and incentive pay. Report sensitivity to the definition of competition and to equilibrium reallocation. The task concerns the identified set in a declared market; neither a wage wedge nor a passthrough coefficient alone constitutes the answer.

Definitions and admissible scope. Use a probability measure $\mu$ on baseline workers, including a separately coded outside option $j=0$. Production revenue $R_j(\ell,k)$ is differentiable and $p_{ij}=\partial R_j/\partial\ell_i$ is a latent technological marginal product, not automatically an observed variable. A structural model specifies preferences, technologies, offer sets, bargaining or wage-posting rules, resource constraints, and an equilibrium selection rule. A competition intervention $C$ must replace those wage-setting rules with a stated price-taking or other explicit benchmark. Let $J_i^C$ be the selected counterfactual job, and restrict the log-wage target to baseline employed workers with positive baseline and counterfactual earnings; report the mass outside this domain. For observable law $P_O$, define $\mathcal M(P_O)=\{M\in\mathcal M_0:P_M(O)=P_O\}$ and the identified set $\{\int_B\log(w_i^C(M)/w_{i,j(i)}(M))\,d\mu(i\mid B):M\in\mathcal M(P_O)\}$. A bound is sharp only if every permitted value is attained or approached by such models. Fix a declared family $\mathcal M_0$ before observing the target: its members have measurable preferences and offer sets, differentiable technologies, feasible budgets, a wage-setting game, and a selected equilibrium for both the baseline and intervention $C$. The family includes explicit restrictions on amenities, skills, incentive contracts, productivity error, shock exogeneity, and exclusion restrictions; these restrictions are inputs to the problem. Only $\mathcal M_0\cap\{M:P_M(O)=P_O\}$ is admissible. Choose a fixed baseline-worker subset $B$ of positive $\mu$-mass on which both wages are positive and log wage changes integrable under every compared model, and use $\mu(\cdot\mid B)$ in the log target. Define $w_i^C$ at selected job $J_i^C$, not at a vanished match. Nonemployment is a separate population outcome. The requested output is the identified set and its infimum/supremum, conditional on these inputs, with a proof of attainable or limiting sharpness.

Variants and assumption changes

  1. Firm-level version (editorial specialization): hold outside options fixed and impose differentiable labor supply $L_j(w)>0$, $L'_j(w)>0$, and profits $R_j(L_j(w))-wL_j(w)$. At an interior optimum, $p_j/w_j=1+1/\varepsilon_j$, where $\varepsilon_j=w_jL'_j/L_j$. Identify $\varepsilon_j$ using wage and employment responses rather than wage passthrough alone; this identity is already known, not a new conjecture.
  2. Equilibrium version (editorial extension): allow entry, amenities, and reallocations to respond to $C$. Replace the same-match comparison by the distribution of $(J_i^C,w_i^C)$, and jointly bound wage gains and the probability of nonemployment. This changes both the estimand and the restrictions needed to identify it.

References and source audit

Support and access. Metadata and full chapter checked. Passthrough alone is insufficient under the stated model; the counterfactual wage-component estimand is editorial. Locator: Sections 1 (pp. 658--660) and 3.1.2 (p. 684).
  1. Patrick Kline. 2025. Labor Market Monopsony: Fundamentals and Frontiers. Introduction, printed pp. 658–660; section 3.1.2, printed p. 684. https://eml.berkeley.edu/~pkline/papers/HoLE_vol6_monopsony.pdf. DOI: https://doi.org/10.1016/bs.heslab.2025.07.007.

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-189-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-189

Measuring employer wage-setting power Sharp competition-wage bounds in a declared labor-supply model

Problem ID: J42-185. Status: Unresolved; rigorous restricted results.

Short recap and the counterfactual being identified

The target is the log change in the same baseline worker's selected wage when employer wage-setting power is removed, with production, preferences, and resources held fixed. An observed productivity--wage wedge is not in general that counterfactual change. We solve the identified-set calculation in an explicit one-employer family with isoelastic supply and decreasing marginal revenue product, and show exactly which additional responses identify its primitives. We do not solve the general sorting, amenities, contract, entry, or equilibrium-selection problem.

Kline [kline], Section 3.1.2, shows that productivity passthrough alone need not identify wage-setting power; Section 3.1.3 discusses using employment responses as well. Those passages were inspected in the full chapter. Our calculation uses a more restrictive model to turn this measurement issue into sharp bounds for a specified competition intervention. The familiar monopsony first-order condition is not presented as a new result.

Preferences, technology, and the intervention

Fix observed baseline wage $w_0>0$ and employment $L_0>0$. Parameters lie in a known rectangle \[ e\in[\underline e,\overline e]\subset(0,\infty),\qquad h\in[\underline h,\overline h]\subset[0,1). \] Here $e$ is a labor-supply elasticity and $h$ is the elasticity of the marginal product with respect to employment, taken with a minus sign. Workers have unit labor endowments, identical productive skill, no job amenities, no incentive pay, and quasilinear utility equal to wage minus a reservation cost $v$ when employed. There is one employer and an outside option normalized to utility zero. On the relevant wage range the measure of workers with $v\leq w$ is \[ L(w)=L_0(w/w_0)^e,\qquad w(L)=w_0(L/L_0)^{1/e}. \tag{1} \] One can implement this with a finite population measure on reservation costs: choose a common sufficiently large $\overline L$ and use quantile function $v(u)=w_0(u/L_0)^{1/e}$ for $u\in[0,\overline L]$. Choose $\overline L$ larger than all baseline and competitive employment levels computed below. Compactness of the parameter rectangle makes this possible. Extend supply by saturation above its largest reservation cost.

The firm's differentiable revenue on $L>0$ has marginal product \[ p(L)=p_0(L/L_0)^{-h},\qquad R(L)=K+\frac{p_0L_0}{1-h}(L/L_0)^{1-h},\qquad p_0=w_0(1+1/e). \tag{2} \] The constant $K$ is output from a fixed input; it has no effect on employment choices. The limit at $L=0$ exists since $h<1$. Revenue is output in a numeraire good with exogenous unit price; this is a technology, not an endogenous product-market price schedule. Fixed inputs and $K$ are held fixed under the competition intervention. The baseline employer posts a uniform wage and hires every worker willing to accept it. Profit is $R(L(w))-wL(w)$, with no entry or wage discrimination.

Define intervention $C$ to replace wage posting by price-taking labor demand, with the same firm technology and worker supply. A competitive wage $w^C$ clears $L(w^C)$ against a profit-maximizing choice of $L$ at that wage. If $h=0$, demand at wage $p_0$ is a correspondence; the declared selector chooses the market-clearing supply quantity. The wage remains unique. Firm profits accrue to an owner; wage payments and profits are feasible transfers out of output, and profits are nonnegative in the construction with $K\geq0$ below.

Let $B$ be the fixed set of baseline employed workers, those with reservation cost at most $w_0$. The target is their average log wage change. All workers in $B$ remain employed under $C$, as proved below; their selected job stays at the same employer. Thus there is no undefined log wage or composition change on $B$. New entrants are reported separately.

Theorem (Equilibrium and the counterfactual wage): For every parameter pair in the rectangle the unique baseline employment optimum is $L_0$ and its wage is $w_0$. Competitive employment and wage satisfy \[ \frac{L^C}{L_0}=(1+1/e)^{e/(1+eh)},\qquad \frac{w^C}{w_0}=(1+1/e)^{1/(1+eh)}. \tag{3} \] Every baseline worker is retained and the target is \[ m(e,h)=\frac{\log(1+1/e)}{1+eh}. \tag{4} \] The increase in employment is $L_0[(1+1/e)^{e/(1+eh)}-1]$.
Proof: Below the supply cap, choosing a wage is equivalent to choosing employment. The marginal wage bill is \[ \frac{d}{dL}[w(L)L]=(1+1/e)w_0(L/L_0)^{1/e}. \] The ratio of marginal revenue (2) to this marginal expenditure is $(L/L_0)^{-(h+1/e)}$. Hence profit is strictly increasing for $L<L_0$ and strictly decreasing for $L>L_0$, up to the cap. Wages above the cap's clearing wage only increase the wage bill with no extra output, so they cannot improve profit. This proves the unique baseline choice.

A competitive equilibrium obeys $p(L^C)=w(L^C)$, giving $(L^C/L_0)^{h+1/e}=1+1/e$. Both sides of the intersection are monotone in opposite directions, and supply is strictly increasing, so the intersection and wage are unique. When $h>0$, concavity of $R$ makes this firm's unique demand optimum at $w^C$. When $h=0$, $w^C=p_0$ makes the firm indifferent across feasible employment and the specified quantity is profit maximizing. The cap was chosen beyond this intersection. Equation (1) then gives the second formula in (3). Since $1+1/e>1$, both employment and wage increase. Workers with reservation cost at most $w_0$ still prefer employment, proving retention of the fixed baseline group. Everyone receives the common new wage, so its log ratio is (4). The employment formula follows by subtraction. ∎

The baseline technological wedge is $\log(p_0/w_0)=\log(1+1/e)$, whereas the wage gain divides that wedge by $1+eh$. Endogenous expansion reduces marginal productivity when $h>0$. Holding the baseline marginal product fixed when computing the competitive wage would therefore answer a different intervention.

Sharp identified sets from baseline records

Specify the observable law to contain the common baseline wage, employment, and total revenue $R_0$, but neither reservation costs nor a direct observation of marginal product. Worker labels among the employed can be assigned identically across models. Suppose \[ R_0\geq\max_{e,h}\frac{w_0(1+1/e)L_0}{1-h}. \] For each pair choose $K=R_0-p_0L_0/(1-h)\geq0$. Thus every pair has the same observed baseline records. The unobserved technology and reservation-cost distribution may differ across compatible models; within each model they remain fixed under $C$.
Theorem (Attainable sharp wage-gain bounds): Under exactly this observation scheme and primitive rectangle, the sharp identified set is the interval \[ \left[ \frac{\log(1+1/\overline e)}{1+\overline e\,\overline h}, \frac{\log(1+1/\underline e)}{1+\underline e\,\underline h} \right]. \tag{5} \] If marginal product $p_0>w_0$ is also exactly observed, then $e=(p_0/w_0-1)^{-1}$ is identified when it lies in the rectangle, and the sharp interval is \[ \left[\frac{\log(p_0/w_0)}{1+e\overline h}, \frac{\log(p_0/w_0)}{1+e\underline h}\right]. \tag{6} \] If that implied elasticity is outside the rectangle, no model is compatible.
Proof: Equation (4) is continuous. Its numerator is positive and strictly decreasing in $e$, while its positive denominator is nondecreasing in $e$ and in $h$. It is strictly decreasing in $h$ because $e>0$. These facts place its minimum and maximum at the opposite corners shown in (5). Every parameter pair is observationally compatible by the explicit $K$ construction, and the equilibrium proof establishes a complete admissible model for each pair. The rectangle is connected; along a line segment joining the two corners, continuity and the intermediate value theorem attain every value between the endpoints. Thus (5) is a sharp set, not merely coordinate bounds on parameters. With observed $p_0$, equation (2) uniquely determines $e$. Varying $h$ continuously over its interval proves (6) by the same construction. ∎

The production-information qualification matters. Observing the entire function $R(L)$ over the relevant range would identify its derivative and curvature in this family; such data cannot be treated as equivalent to observing the single revenue level $R_0$. Likewise, if the fixed-input component $K$ were separately known, not every rectangle point would remain compatible with the revenue observation.

Which exogenous responses remove the ambiguity?

Let a scalar productivity shifter $z$ multiply the variable part of revenue in (2) by $e^z$, with supply, preferences, and $K$ fixed. Consider a neighborhood of $z=0$ where no capacity constraint binds. Assume the structural response to $z$ is observed, for example because $z$ is exogenously assigned and has no direct effect on supply or amenities. Define local elasticities \[ \rho=\left.\frac{d\log w(z)}{dz}\right|_{z=0},\qquad \nu=\left.\frac{d\log L(z)}{dz}\right|_{z=0}. \]
Proposition (Wage and employment responses): In this family \[ \rho=\frac1{1+eh},\qquad \nu=\frac e{1+eh}. \] Thus joint observation of $(\rho,\nu)$, with $\rho,\nu>0$, identifies \[ e=\nu/\rho,\qquad h=(1-\rho)/\nu,\qquad m=\rho\log(1+\rho/\nu). \tag{7} \] Observation of $\rho$ alone leaves the feasible elasticity set \[ E_\rho=\{e\in[\underline e,\overline e]: (\rho^{-1}-1)/e\in[\underline h,\overline h]\}. \] If this set is a nonempty interval $[e_-,e_+]$, the sharp gain interval is $[\rho\log(1+1/e_+),\rho\log(1+1/e_-)]$.
Proof: The baseline first-order equation under the shifter is $e^z(L(z)/L_0)^{-h}=(L(z)/L_0)^{1/e}$. Hence $\log(L(z)/L_0)=ez/(1+eh)$ and, using supply, $\log(w(z)/w_0)=z/(1+eh)$. Differentiation gives $\rho,\nu$, and algebra gives (7). If only $\rho$ is observed, its equation restricts the product $eh$ and gives precisely $E_\rho$. For $\rho=1$, this requires $h=0$; otherwise $\rho\in(0,1)$ and interval intersection with the inequalities on $(\rho^{-1}-1)/e$ gives an interval or the empty set. Every compatible pair has the identical wage response $w(z)=w_0e^{\rho z}$ on a common sufficiently small neighborhood. Baseline $K$ is adjusted as before. Equation (4) equals $\rho\log(1+1/e)$ along this feasible set and is strictly decreasing and continuous in $e$, proving the sharp interval assertion. ∎

The wage-only statement deliberately does not suppose employment responses are also hidden in the observable law: once $\nu$ is observed the joint formula applies. A reduced-form wage response is useful only under the exclusion that the shock does not also change supply or amenities. There is no theorem here identifying $e$ from an arbitrary wage shock.

Economic restrictions and the unresolved target

The bounds are informative because skill, amenity, incentive, and productivity-measurement confounding have been ruled out by explicit primitive restrictions; they have not been removed by a statistical identity. No baseline worker becomes nonemployed in this family, so that mass is exactly zero. The intervention adds workers from the same fixed population and keeps the same production technology and numeraire resources. It is a partial-equilibrium labor market with a declared price-taking benchmark, not an economy-wide entry or reallocation model.

A full solution to the catalogue question must allow model-consistent amenity and incentive responses, multiple employers and selected jobs, measurement error in marginal products, and equilibrium changes in outside options. Bounds (5)--(7) cannot be transported to those settings without rebuilding the observable equivalence and sharpness arguments. The present result locates a concrete remaining ambiguity even in the smaller family: a perfectly measured baseline wedge need not determine the competitive wage when the employment response changes marginal productivity.

References

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