Research question

• Sweden’s workforce is becoming better educated.
• Large manufacturing firms employ a more educated workforce.
• College premia and sales-to-variable-cost ratios evolve differently across firm sizes.

How does skill supply affect technology adoption, competition, and firm size?

How do markups affect technology adoption and wage inequality?

Firm heterogeneity and skill premia

• Markup heterogeneity and aggregate productivity:
  • De Loecker, Eeckhout, and Unger (2020), Baqaee and Farhi (2020), Edmond, Midrigan, and Xu (2023).
  • Abstracts from skill heterogeneity.
• Skill premia and skill-biased technical change:
  • Katz and Murphy (1992), Krusell et al. (2000), Autor, Goldin, and Katz (2020).
  • Abstracts from market power.

We propose a model that bridges the two.

Reading guide

• Empirical evidence: Swedish education, wage premia, and firm cost ratios.
• Model and calibration: Firm technology choice with product and labor market power.
• Counterfactuals: Higher skill supply and the removal of markup heterogeneity.

Work in progress. Empirical evidence from November 2025 and model results from March 2026.

College share of employment

College premium

College share by firm size: change since 1997

College share by firm size: levels

College premium by firm size: change since 1997

Sales to variable costs, levels

Sales to variable costs: growth relative to small firms

Interpreting sales to variable costs

• Lower returns to scale.
  • Fixed costs spread over greater volume.
• Rising markups.
• Higher capital share of sales.

\[\frac{\text{Sales}}{\text{Variable Costs}} = \left(\frac{\text{RTS}}{Markup} - \frac{\text{Cost of Capital}}{Sales}\right)^{-1}\]

Empirical motivation for the model

• Larger firms have higher college shares, with larger increases over time.
• College premia rise more in larger firms.
• Sales-to-variable-cost ratios rise more in the largest firms relative to small firms.

The model links skill-biased technology choice to firm size and markups.

These patterns motivate the mechanism. The counterfactuals quantify its implications under the model’s assumptions.

Setup

• Heterogeneous firms.
  • Ex-ante variation in total factor productivity (TFP).
  • Monopolistic: Markups.
  • Skill-biased technology (SBT) choice: Skill Premium.
• Households.
  • Two types of labor, skilled and unskilled.
  • Monopsonistic labor markets.
    • Firm type-specific wages and skill-premia.

Product market

• Monopolistic sectors \(j \in J\) with continuous mass of \(n_j\) firms.
  • Free entry to learn firm-TFP, determines firm mass.
  • Fixed operating cost, determines productivity distribution cutoff.
• Demand as in Klenow and Willis (2016).
• Cobb Douglas final good aggregator across sectors: \(Y = \prod_j Y_j^{\theta_j}\).

Klenow and Willis (2016) demand

Demand curve given by:

\[ \Upsilon'(\frac{Y_{ij}}{Y_{j}}) = \frac{\bar{\sigma}-1}{\bar{\sigma}} \exp\left(\frac{1 - \left(\frac{Y_{ij}}{Y_{j}}\right)^{\frac{\varepsilon}{\bar{\sigma}}}}{\varepsilon}\right), \bar{\sigma} > 1\]

• Demand elasticities vary with relative size.

\[ \sigma(\frac{Y_{ij}}{Y_{j}}) = \bar{\sigma} \left(\frac{Y_{ij}}{Y_{j}} \right)^{-\frac{\varepsilon}{\bar{\sigma}}}\]

• Optimal markups increase with size:

\[\color{salmon}{\mu_{i}} = \frac{\bar{\sigma}}{\bar{\sigma} - \left( \frac{Y_{i}^{*}}{Y^*_j}\right)^{\frac{\epsilon}{\bar{\sigma}}}}\]

Production technology

\[Y_{i} = \Omega_{i} K_{i}^{\alpha_k} M_{i}^{\alpha_m} \tilde{H}_{i}^{\alpha_{\tilde{h}}}\]

\[\tilde{H}_{i} = \left( \color{darkcyan}{E_i} H_{i}^{\frac{\rho-1}{\rho}} + L_{i}^{\frac{\rho - 1}{\rho}} \right)^{\frac{\rho}{\rho - 1}}\]

• Ex-ante heterogeneity in TFP: \(\Omega_{i}\).
• Skill-specific technology \(\color{darkcyan}{E_i}\).
• Firms choose heterogeneous \(\color{darkcyan}{E_i}\), inducing skill-firm sorting.
  • Skill premium disproportionately higher in firms with high \(\color{darkcyan}{E_i}\).

Cost of SBT

\[f(\color{darkcyan}{E_{i}}) = \gamma_0 + \frac{\gamma_1}{1 + \gamma_2} \left( \frac{\color{darkcyan}{E_{i}}}{\bar{E}} - 1 \right)^{1 + \gamma_2}\]

• Fixed cost, independent of output.
• \(\gamma_0 > 0\). paid by all firms, determines TFP-cutoff.
• \(\gamma_1 > 0\), cost increases in amount.
• \(\gamma_2 > 0 \rightarrow\) convex in \(\color{darkcyan}{E_{i}}\).

Firm problem, step 1

Choose employment and intermediates, \(H_{i}, L_{i}, K_i, M_i\), conditional on \(\color{darkcyan}{E_{i}}\):

\[\max_{H_{i}, L_{i}, K_i, M_i} P_{i} Y_{i} - W_{h,i}H_{i} - W_{l,i}L_{i} - K_i - M_i\]

subject to labor supply, product demand, and production function.

Firm problem, step 2

Choose \(\color{darkcyan}{E_i}\), taking as given optimal input choices conditional on \(\color{darkcyan}{E_{i}}\):

\[\max_{\color{darkcyan}{E_{i}}} \Pi_{i}^{*}(\color{darkcyan}{E_{i}}) - f(\color{darkcyan}{E_{i}})\]

\(\Pi_{i}^{*}\) denotes firm profits under optimal input choices for a given technology \(E_{i}\).

Optimal skill technology adoption

Characterized by:

\[\left(1 - \frac{1}{\color{salmon}{\mu_i^{*}}} \right) \frac{d \log R_i^{*}}{d \log \color{darkcyan}{E_i}} +\left( \frac{1}{\color{salmon}{\mu_i^{*}}}\right) \frac{d \log \color{salmon}{\mu_i^{*}}}{d \log \color{darkcyan}{E_i}} =\frac{f'(\color{darkcyan}{E_i})\color{darkcyan}{E_i}}{R_i^{*}}\]

\(\rightarrow\) Firms with high markups place greater weight on scale than margins.

Labor market

• Isoelastic monopsonistic labor supply to firm \(i\), sector \(j\).

• Enables firm-level wages.

• \[\frac{H_{i}}{H} = \left(\frac{W_{h,i}}{ W_h } \right)^{\eta}\]

• \[\frac{L_{i}}{L} = \left(\frac{W_{l,i}}{ W_l } \right)^{\eta}\]

Equilibrium labor shares

• \(\frac{W_{l,i} L_{i}}{R_i} = \lambda \color{salmon}{\mu_i}^{-1} \cdot \left( 1 - \frac{\color{darkcyan}{E_i}}{\color{darkcyan}{E_i} + \left(\frac{W_{h,i}}{W_{l,i}}\right)^{\rho - 1}} \right)\)

• \(\frac{W_{h,i} H_i}{R_i} = \lambda \color{salmon}{\mu_i}^{-1} \cdot \frac{\color{darkcyan}{E_i}}{\color{darkcyan}{E_i} + \left(\frac{W_{h,i}}{W_{l,i}}\right)^{\rho - 1}}\)

Markups:

\(\rightarrow\) influence the output elasticities.

\(\rightarrow\) depress the unskilled labor share.

\(\rightarrow\) ambiguously affect the skilled labor share.

Equilibrium

• Goods market clearing: \(Y = M + K + \text{Entry costs} + \text{SBT costs} + \text{Fixed Op. cost}\).
• High skill labor market clearing: \(\bar{H} = \sum_j \int_0^{n_j} H_i di\).
• Low skill labor market clearing: \(\bar{L} = \sum_j \int_0^{n_j} L_i di\).
• Free entry condition: \(\text{Entry Cost}_j = \mathbb{E} \left[ \Pi_{ij} \right]\)
• Zero (operating) profit condition: \(\Pi_{ij} = 0\) for the marginal active firm.
• Firms choose inputs and SBT to maximize profits.
• Households maximize utility.

Skill intensity fit

Two counterfactual experiments

• Increasing the skilled population share.
  • Effects on firm-size and markups distribution.
• Undoing markups heterogeneity.
  • Effects on skill premium and SBT-adoption.

Higher Skill Share

• Keep parameters from baseline.
• Exogenous change in population composition.
  • From: \(H/L = 0.2\).
  • To: \(H/L = 0.3\).
• Compare three economies.
  1. Baseline.
  2. New equilibrium.
  3. New equilibrium, holding fixed SBT from baseline.

SBT-distribution

Skill Premium

Firm size

Markups

High Skill Revenue Share

Low Skill Revenue Share

Experiment

Are markups important for the choice of technology adoption?

• Size-dependent firm-subsidies like Edmond, Midrigan, and Xu (2023).
  • Paid for by household taxes.
• Compare baseline against two economies without markup heterogeneity:
  • Optimal SBT-choice: firms choose their SBT freely.
  • Fixed SBT: Firms’ SBT is fixed to baseline.

Markups

SBT-distribution

Firm size

Skill premium

High Skill Revenue Share

Low Skill Revenue Share