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Econ-427 · economics · Economics

Separating discrimination mechanisms

LLM status Open unverified

Statement

Original subject group: Labor markets and work

Audit classification: Background; proposed extension. Authors, title, year and NBER identifier checked. The study separates customer and manager channels in baseball; statistical-discrimination and intersectional variants are editorial.

Definitions and precise research target

Complete editorial problem. Fix a hiring setting with applicants, employers, and customers. Let $g_i$ be applicant $i$'s recorded identity vector and $s_i$ independently measured productive characteristics. Let $H_i(d,c,q)$ indicate hiring under experimentally specified employer information $d$, customer exposure $c$, and productivity signal precision $q$. Employer prejudice concerns identity-dependent preferences holding expected productivity and customer revenue fixed; customer bias concerns the revenue consequences of customer exposure; statistical discrimination concerns decisions based on imperfect productivity beliefs. For identity groups $g$ and $g'$, define conditional hiring gaps at common $s$, and use orthogonal variation in $d,c,q$ to estimate which channels generate their changes. The identification task is to find assumptions and feasible variations that distinguish these channels, allowing interactions rather than forcing an additive decomposition. A policy arm $p$ has causal effect $E[H_i(p)-H_i(0)\mid g_i=g]$, and should also be evaluated for earnings and firm output. Identity labels themselves need not be manipulable. Correspondence tests identify responses to perceived identities under their design; they do not automatically establish motives. Quantify what remains observationally equivalent, especially across intersecting identities and beyond the studied occupation.

Definitions and admissible scope. An identity cue $\ell$ is a manipulable label that employers perceive; the underlying recorded identity $g$ is not itself assumed manipulable. Specify employer posterior $E[s\mid\ell,q]$, customer revenue $r(s,\ell,c)$, and any identity-dependent employer utility $v(\ell)$. Distinct triples can generate the same hiring decision, which defines observational equivalence. Let $H_i(\ell,c,q)\in\{0,1\}$ be the outcome under a complete randomized design. The cue effect $E[H_i(\ell,c,q)-H_i(\ell',c,q)]$ is identified by randomized cues with common applicant profiles. Separating motives requires restrictions on how $c$ affects customer demand and how $q$ affects beliefs; a factorial interaction alone is not a unique channel decomposition. Productivity and revenue measurements must be held comparable across groups. The signal input $d$ contains the perceived identity cue $\ell$ and any other randomized applicant information, whereas $q$ specifies the known precision of a productivity signal and $c$ the customer-exposure regime. A model supplies the signal experiment, employer prior and update rule, customer-revenue function, employer objective, capacity constraint, and choice rule. At fixed productive profile $s$, define the recorded-group gap as $E[H\mid g,s]-E[H\mid g',s]$; this is descriptive unless an admissible intervention and selection assumptions connect it to a causal contrast. The mathematical target is the set of such models generating all observed cell probabilities and policy responses, together with conditions under which the intended channels are separately identified.

Variants and assumption changes

  1. Customer versus manager channels (source-based scope): restrict to the two channels studied in the cited baseball setting and compare choices across expected customer exposure. Their empirical separation is an existing result, while transport to another industry is an editorial extension.
  2. Statistical and intersecting-identity extension (editorial): randomize verified productivity-signal precision and multiple cues jointly. Characterize the set of taste, customer, and belief models fitting all hiring probabilities instead of forcing an additive decomposition.

References and source audit

Support and access. Authors, title, year and NBER identifier checked. The study separates customer and manager channels in baseball; statistical-discrimination and intersectional variants are editorial. Locator: BFI abstract, January 29, 2025.
  1. Majid Ahmadi; Gwen-Jirō Clochard; Jeff Lachman; John A. List. 2025. Toward an Understanding of Discrimination When Multiple Channels Exist. Author abstract, opening and final paragraphs, dated January 29, 2025. https://bfi.uchicago.edu/working-papers/toward-an-understanding-of-discrimination-when-multiple-channels-exist/. DOI: https://doi.org/10.3386/w33391.

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-427-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-427

What a factorial hiring experiment identifies about discrimination channels

Problem ID: J71-423. Status: Unresolved. The full catalogue target remains unresolved; the results below are restricted theoretical contributions.

Cue responses and motives are different targets

The catalogue distinguishes a manipulable perceived-identity cue from recorded identity and asks when employer prejudice, customer demand and productivity beliefs can be separated. The source on multiple channels studies an empirical customer-versus-manager setting [source]. We give a finite signal-and-choice model that is identified under explicit exclusions, then construct observationally equivalent alternatives when those exclusions are removed. No empirical attribution or hiring-policy recommendation follows without the assumptions and data.

Consider a fixed productive profile $s$ and randomized cue $\ell\in\{0,1\}$. Randomize customer exposure $c\in\{0,1\}$ and a verified signal experiment indexed by precision $q$. A known precision weight $\rho_q\in[0,1]$ enters the employer's posterior mean \[ \mu_\ell(q)=\rho_qs+(1-\rho_q)m_\ell. \tag{1} \] Here $m_\ell$ is the employer's cue-dependent prior mean; (1) is an imposed linear updating rule. It can arise from a normal signal with common known prior variance, but that particular derivation is not needed. Assume it is stable across exposure arms.

Employer taste is $v_\ell$, customer-revenue effect is $c\kappa_\ell$, and common wage or reservation cost is $w$. Hiring follows \[ H={\mathbf 1}\{\mu_\ell(q)+c\kappa_\ell+v_\ell-w+\varepsilon\ge0\}. \tag{2} \] Let the known strictly increasing distribution function $F$ satisfy ${\mathbb{P}}(\varepsilon\ge-u)=F(u)$. The shock distribution and scale are common across arms and independent of random assignment. Capacity constraints and competition between applicants are absent: each vacancy is a separate binary decision.

Identification under explicit exclusions

Randomization and consistency identify cell probabilities $\pi_{\ell cq}={\mathbb{E}}[H\mid\ell,c,q]\in(0,1)$. Transform them to indices \[ z_{\ell cq}=F^{-1}(\pi_{\ell cq})-\rho_qs+w =(1-\rho_q)m_\ell+c\kappa_\ell+v_\ell. \tag{3} \] The transformation requires a known $F$ and interior probabilities. It is not valid to interpret raw probability interactions as the same additive index effects.
Theorem (Separating the three coefficients): Suppose a perfect-information arm has $\rho_{q_H}=1$, another arm has $\rho_{q_L}<1$, and both customer exposures are supported for both cues. Then \[ v_\ell=z_{\ell,0,q_H},\qquad \kappa_\ell=z_{\ell,1,q_H}-z_{\ell,0,q_H},\qquad m_\ell=\frac{z_{\ell,0,q_L}-z_{\ell,0,q_H}} {1-\rho_{q_L}}. \tag{4} \] These are the unique coefficients generating the observed cells in model (1)--(2).
Proof: At perfect information, the prior term in (3) vanishes. Setting $c=0$ identifies $v_\ell$, and subtracting exposures identifies $\kappa_\ell$. At imperfect information and $c=0$, subtract the already identified taste intercept and divide by its nonzero prior weight. This proves existence of at most one coefficient triple, while substitution into every cell checks compatibility. Remaining cells are overidentifying restrictions, not additional free parameters. ∎

The relevant cue-gap interpretation is \[ \Delta z_{cq}=(1-\rho_q)\Delta m+c\Delta\kappa+\Delta v. \tag{5} \] If $w$ is unknown but common across cues, the absolute taste levels in (4) have an unknown common intercept; cue differences remain identified because $w$ cancels. Likewise the shock scale must be normalized before cardinal coefficients are meaningful. Two distinct imperfect precision levels suffice algebraically to fit the line in $1-\rho_q$, but interpreting its extrapolated intercept as prejudice requires the same structural exclusions all the way to full information.

A worked probability-scale check

Use the logistic $F(u)=1/(1+e^{-u})$, $s=w=0$ and $\rho_L=1/2$, $\rho_H=1$. Let cue 0 have $(m_0,\kappa_0,v_0)=(0,0,0)$ and cue 1 have $(-2,-1,-1/2)$. The transformed cue-1 indices are \[ z_{1,0,H}=-1/2,\quad z_{1,1,H}=-3/2,\quad z_{1,0,L}=-3/2,\quad z_{1,1,L}=-5/2. \] The corresponding hiring probabilities are $F$ at those four values; cue 0 is hired with probability $1/2$ in every arm. Formula (4) exactly recovers the chosen coefficients. The customer effect on the index is always $-1$, yet its hiring-probability effect differs by precision because the logistic function is nonlinear. Thus a factorial probability interaction may arise even when the latent channels enter additively. Conversely a zero probability interaction need not certify an absence of structural interaction in a different choice model.

These values are only a worked synthetic table. They do not measure discrimination in a real occupation or show that recorded identities can be randomized. The design randomizes perceived cues on otherwise common applicant profiles.

Exact observational equivalence without invariance

Now allow employer utility to depend on precision and exposure, written $v_\ell(c,q)$, and retain the same observed cell probabilities. For arbitrary constants $d_\ell,e_\ell$, define \[ m'_\ell=m_\ell+d_\ell,\quad \kappa'_\ell=\kappa_\ell+e_\ell,\quad v'_\ell(c,q)=v_\ell(c,q)-(1-\rho_q)d_\ell-ce_\ell. \tag{6} \] Substitution into (3) leaves every index and therefore every hiring probability unchanged. Yet the assigned belief, customer and taste contributions change. This establishes a continuum of observationally equivalent motive decompositions even with complete factorial randomization.

Perfect information eliminates the $d_\ell$ term at $q_H$, but does not by itself validate exposure invariance or exclude identity-dependent productivity scales, applicant-side costs or alternative objectives. The necessary scientific question is why customer exposure should change revenue while leaving employer utility and signal interpretation invariant. Direct customer-demand measurements and incentive-compatible posterior elicitations can help assess that restriction, but they are additional observed variables with their own measurement laws.

Overidentification, intersections and performance outcomes

Within the restricted model, (3) implies two concrete checks: the exposure difference is $\kappa_\ell$ at every precision level, and the zero-exposure index is affine in $1-\rho_q$. Three or more precision levels overidentify that affine form. A failure rejects the restricted model; success does not prove its psychological interpretation, since alternatives like (6) still exist in a larger class.

For two independently randomized identity cues $(\ell_1,\ell_2)$, define a separate coefficient triple for each of the four cue combinations. The same formulas identify those triples if the exclusions hold. A contrast such as $v_{11}-v_{10}-v_{01}+v_{00}$ measures an interaction in the declared employer-utility index. It is not meaningful to assign it uniquely to one recorded identity without an attribution convention. Randomizing one cue alone while allowing the other to vary endogenously does not produce this factorial contrast.

Finally, hiring is not the complete policy outcome. Earnings and output measured only among hires condition on a post-treatment selection event. A policy changing the threshold in (2) changes who is observed, so mean output among hires may move even with unchanged applicant productivity. Evaluating workforce output requires the baseline applicant population, hiring capacity, wages, rejected-applicant destinations and firm production accounting. These are outside the binary independent-vacancy model.

The remaining full task is to defend feasible experiments and exclusions in an occupation and characterize the larger identified set when they fail. This attempt gives a solved restricted identification model, direct overidentifying restrictions and an explicit nonidentification transformation. It leaves the broad empirical discrimination-channel problem unresolved.

Completion Estimate

55%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt identifies taste, customer-revenue, and belief coefficients under a specified signal and choice model, with overidentifying restrictions and an exact invariance-breaking equivalence. Defending those exclusions, handling competitive hiring capacity, and evaluating earnings and firm output remain necessary for the broader discrimination-mechanism target.

References

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