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Econ-414 · economics · Economics

Causes of wage inequality and job polarization

LLM status Partially solved unverified

Statement

Original atlas ID: L4 (entry 24). Classification: Editorial model-based historical attribution problem.

Original atlas question: How much comes from technology, trade, institutions, firm sorting and rents?

Definitions and assumptions

Fix two dates and a consistently defined population. For a fixed integer $K\geq1$, let $z=(z_1,\ldots,z_K)$ collect announced exogenous inputs such as technology, trade costs, labor-market institutions, and product demand. Worker-firm sorting, firm rents, and occupational employment are endogenous outcomes of an equilibrium model $M$, rather than independent shocks by definition. Require equilibrium existence and a stated selection rule. Let $F_M(z)$ be the distribution of log hourly wages among employed workers and let $s_M(z)$ be employment shares in occupations ranked using baseline wages. Specify treatment of nonemployment separately. Choose a distributional functional $T$, for example a finite variance or the difference between defined quantiles.

Formal research question

For baseline and final inputs $z^0,z^1$, define $z^S$ by using $z_k^1$ when $k\in S$ and $z_k^0$ otherwise. A transparent allocation of the modeled change is \[ \phi_k(M)=\sum_{S\subseteq\{1,\ldots,K\}\setminus\{k\}} \frac{|S|!(K-|S|-1)!}{K!} \left[T(F_M(z^{S\cup\{k\}}))-T(F_M(z^S))\right]. \] For all models matching an announced set of wage, mobility, production, and policy moments, identify or bound each $\phi_k$ and the corresponding changes in $s_M$. Test whether the same structural restrictions predict out-of-sample responses to independently identified technology, trade, and institutional shocks.

Interpretation and scope

The sum of these allocations equals the model's total change, but the allocation depends on counterfactual inputs and the interaction convention. It is not an assumption-free answer to how much each historical cause contributed. A worker-firm wage decomposition is descriptive unless mobility selection and firm effects are causally identified. Endogenous sorting and rents should either remain equilibrium responses or be assigned explicitly intervenable structural inputs. Employment polarization is growth in high- and low-ranked occupation shares relative to middle-ranked shares; it does not logically imply any particular change in wage variance. Labor-force composition and hours can also change observed wage distributions, so report fixed-population and observed-population targets separately.

Source audit and status

[S1] supplies historical supply-demand analysis, [S2] employment-polarization evidence, and [S3] a task-assignment framework. None states the exact decomposition displayed here as a conjecture. The unified, empirically validated attribution is an editorial research target. Existing historical results remain results; their presence neither settles every counterfactual nor proves a universal 2026 frontier.

References

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  1. Lawrence F. Katz and Kevin M. Murphy (1992). Changes in Relative Wages, 1963--1987: Supply and Demand Factors. The Quarterly Journal of Economics 107(1), 35--78. https://doi.org/10.2307/2118323. Locator: Primary publisher, working-paper, or author record: bibliographic information and abstract; full-text theorem verification is not claimed. Role: Supply-demand account of relative wages; historical evidence rather than a unique causal decomposition.
  2. David H. Autor, Lawrence F. Katz, and Melissa S. Kearney (2006). The Polarization of the U.S. Labor Market. American Economic Review 96(2), 189--194. https://doi.org/10.1257/000282806777212620. Locator: Primary publisher, working-paper, or author record: bibliographic information and abstract; full-text theorem verification is not claimed. Role: Employment-polarization evidence; separates employment shares from wage inequality.
  3. Daron Acemoglu and David H. Autor (2011). Skills, Tasks and Technologies: Implications for Employment and Earnings. Handbook of Labor Economics 4B, chapter 12. https://www.sciencedirect.com/science/chapter/handbook/pii/S0169721811024105. Locator: Primary publisher, working-paper, or author record: bibliographic information and abstract; full-text theorem verification is not claimed. Role: Task assignment and technology framework for specified counterfactuals.

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-414-A1 · Partial result · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-414

Causes of wage inequality and job polarization

Target and what exact attribution requires

The catalogue proposes Shapley allocations of a modeled change in wage inequality across exogenous technology, trade and institutional inputs. This note derives the complete two-input identified set under transparent counterfactual restrictions, a finite linear-program formulation for more inputs, and error bounds for evaluating a fixed structural model. It also shows why employment polarization does not determine wage inequality. Historical attribution remains unresolved without identifying the hybrid counterfactuals.

Write $v(S)=T(F_M(z^S))$ for each subset of the $K$ exogenous input changes. Sorting, rents and occupations remain endogenous responses inside $M$. For a random uniformly ordered permutation $\pi$, let $P_k(\pi)$ be the set of inputs preceding $k$. Then \[ \phi_k={\mathbb{E}}_\pi[v(P_k(\pi)\cup\{k\})-v(P_k(\pi))]. \tag{1} \] There are $|S|!(K-|S|-1)!$ permutations with predecessor set $S$, so (1) equals the formula in the problem. Summing the marginal increments along each permutation telescopes from $v(\varnothing)$ to $v(\{1,\ldots,K\})$. Consequently \[ \sum_k\phi_k=v(\{1,\ldots,K\})-v(\varnothing). \tag{2} \] This accounting identity does not identify a single summand from the two endpoints.

A sharp two-input calculation

Normalize the observed inequality endpoints to $v(\varnothing)=0$ and $v(\{1,2\})=1$. Let the two unobserved hybrid economies have inequality $a=v(\{1\})$ and $b=v(\{2\})$. Then \[ \phi_1=(a+1-b)/2,\qquad \phi_2=(b+1-a)/2. \tag{3} \]
Proposition: If the only restrictions on the hybrids are nonnegativity of variance $a,b\ge0$, the identified set is the entire line $\{(t,1-t):t\in{\mathbb{R}}\}$. If additionally both inputs monotonically increase inequality on the four corners, the sharp set is the segment $\{(t,1-t):0\le t\le1\}$.
Proof: For arbitrary $t$, choose $a=\max(2t-1,0)$ and $b=\max(1-2t,0)$, giving $a-b=2t-1$ and (3). To demonstrate realizable inequality values, use two equally weighted worker types with log wages $\pm\sqrt{v(S)}$ at each corner. The wage variance is exactly $v(S)$ and wages themselves are strictly positive. A competitive benchmark can support these wages by specifying each type's marginal product at each exogenous corner to be its wage, with perfectly inelastic type supplies and linear production. Since no further technological restrictions link the four corners, every construction is admissible in this deliberately broad class. All models have the same observed endpoint wage distributions. Monotonicity restricts $a,b$ to $[0,1]$, yielding $0\le\phi_1\le1$. Every point in that interval is attained, for example with $a=t,b=1-t$. ∎
This is an explicit nonidentification result in a permissive equilibrium class, not a claim that any preferred calibrated task model can fit arbitrary corners. More economic structure can narrow the set, and its identifying contribution should be shown rather than hidden in the decomposition convention.

For illustration, $(a,b)=(0.9,0.1)$ assigns $(0.9,0.1)$ of the change to the two inputs, while $(a,b)=(0.1,0.9)$ reverses those conclusions. Both respect monotonicity and exactly fit the same observed total change. Data on one historical transition cannot choose between them without additional variation.

Joint bounds for a finite counterfactual model

Enumerate the $2^K$ corner values in a vector $v$ and define the known matrix $A$ by the weights in (1), so $\phi=Av$. Suppose credible calibration or experimental restrictions reduce to a nonempty compact polytope $\mathcal V=\{v:Cv\le d,Ev=f\}$. For any linear attribution contrast $q^{\top}\phi$, its sharp bounds within this finite corner class are the linear-program minimum and maximum of $q^{\top}Av$ on $\mathcal V$. Every feasible $v$ must be realizable by the maintained structural class for these bounds to be called sharp for that class; otherwise they are only outer bounds. Compactness proves attainment, and linearity proves convexity of the image $A\mathcal V$.

Reporting separate coordinate intervals loses the joint efficiency constraint (2). For the two-input example, both coordinates range over $[0,1]$, but $(0,0)$ and $(1,1)$ are impossible. A confidence region for all counterfactual moments can be propagated through the same optimization, retaining dependence and common structural parameters. Fitting each corner independently can admit counterfactual combinations no single equilibrium model supports.

Computation errors for a fixed model

If every evaluated corner obeys $|\widehat v(S)-v(S)|\le\varepsilon$, then \[ |\widehat\phi_k-\phi_k|\le2\varepsilon. \tag{4} \] Indeed each marginal difference has error at most $2\varepsilon$ and the nonnegative weights in (1) sum to one. If endpoints are exact, total attribution error sums to zero even though the coordinate bounds remain $2\varepsilon$. Numerical equilibrium multiplicity must be resolved by the same declared selector at every corner; arbitrary solver branch switching is not covered by a small tolerance unless its effect on $v$ is separately bounded.

When exhaustive evaluation is infeasible, sample $m$ independent permutations. Assume every marginal contribution lies in $[-B,B]$ and evaluate each path exactly. The sample mean estimate satisfies, simultaneously for all $k$, \[ |\widehat\phi_k-\phi_k| \le B\sqrt{\frac{2\log(2K/\alpha)}m} \] with probability at least $1-\alpha$, by Hoeffding and a union bound. Common permutations induce cross-coordinate dependence but preserve exact pathwise summing to the total change and do not invalidate the union bound. Add $2\varepsilon$ if each used corner has a uniform evaluation error. This controls simulation error conditional on the model; it says nothing about structural misspecification or causal identification.

Polarization and inequality can move differently

Take three occupations ordered by baseline log wages $(-1,0,1)$ with shares $(1/4,1/2,1/4)$. Baseline variance is $1/2$. Move shares to $(0.4,0.2,0.4)$, so both tails expand at the middle's expense. If log wages stay fixed, variance rises to $0.8$. If the new log wages are instead $(-0.5,0,0.5)$, the same employment polarization produces variance $0.2$, a decline. All occupations retain their baseline ordering. The mean is zero in every example, so this is not a changing-mean artifact.

Nonemployment and hours selection add further targets. A distribution of hourly wages conditional on employment need not track a fixed population's annual earnings inequality. The full exercise must state whose outcomes and which functional are being decomposed before attributing shocks.

Remaining empirical task

A defensible attribution needs exogenous changes that identify hybrid response functions, validated worker-firm matching or production restrictions, and out-of-sample predictions. The exact accounting and error bounds above make those dependencies explicit. They do not decide which historical mechanism explains observed wage inequality.

Completion Estimate

45%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt derives sharp two-input Shapley attribution sets, a finite counterfactual linear program, and rigorous evaluation and permutation-sampling error bounds. Empirically justified hybrid equilibria, linked wage and occupational responses, and out-of-sample validation still prevent a full historical attribution.

References

  • [source] D. H. Autor, L. F. Katz and M. S. Kearney (2006), The Polarization of the U.S. Labor Market, AER 96(2), 189--194. https://www.aeaweb.org/articles?id=10.1257/000282806777212620. Publisher identity checked; this is context for employment polarization, not a source for the newly specified counterfactual examples or proofs above.

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