Inequality and technological change

Macera, Manuel; Tsujiyama, Hitoshi Article Inequality and technological change Quantitative Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Macera, Manuel; Tsujiyama, Hitoshi (2024) : Inequality and technological change, Quantitative Economics, ISSN 1759-7331, The Econometric Society, New Haven, CT, Vol. 15, Iss. 2, pp. 427-451, https://doi.org/10.3982/QE1693 This Version is available at: https://hdl.handle.net/10419/320305 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/4.0/ Quantitative Economics 15 (2024), 427–451 1759-7331/20240427 Inequality and technological change Manuel Macera Department of Economics, Universidad Torcuato Di Tella Hitoshi Tsujiyama School of Economics, University of Surrey We study how technological change affects betweenand within-education-group inequality in the United States. We develop a model with heterogeneous workers and firms in which the demand for skills is characterized by firms’ recruiting behavior. We use the model to quantify the relative contribution of two types of technological change that affect the relative demand for skilled labor: technological change in firm-specific productivity and technological change in labor productivity. We find that technological change in labor productivity, in the form of higher returns to skill in production, is the main driver of the increase in betweenand within-group inequality. Technological change in firm productivity, in the form of higher firm productivity dispersion, plays a less important role in explaining rising inequality, except for the increase in within-group inequality for workers without a college degree. Keywords. Inequality, skill-biased technological change, firm productivity dispersion, skills, education, frictional labor markets, sorting. JEL classification. E24, I24, J23, J24, J31. 1. Introduction A striking feature of the trend in inequality in the United States in recent decades has been its increase both across and within education groups (betweenand within-group inequality).1A major driver of rising inequality is skill-biased technological change (SBTC), broadly understood as a change in production technology that increases the relative demand for skilled labor. While these changes are generally difficult to measure empirically, a recent paper by Decker, Haltiwanger, Jarmin, and Miranda (2020)rigorously estimates one particular form of SBTC, that is, rising firm productivity dispersion.2 Manuel Macera: [email protected] Hitoshi Tsujiyama: [email protected] We thank Naoki Aizawa, Cristina Arellano, Antoine Camous, Nicola Fuchs-Schündeln, Jonathan Heathcote, Marek Ignaszak, Leo Kaas, Loukas Karabarbounis, John Kennan, Georgi Kocharkov, Chiara Lacava, Ilse Lindenlaub, Alexander Ludwig, Zachary Mahone, Giuseppe Moscarini, Andreas Mueller, Roberto Pinheiro, Fabrizio Perri, Ctirad Slavik, Naoki Takayama, Satoshi Tanaka, Gianluca Violante, Minchul Yum, and three anonymous referees for useful comments and discussions. 1See, for example, Goldin and Katz (2007). 2They use the U.S. Census Bureau’s Longitudinal Business Database, which covers the universe of private nonfarm establishments in the U.S., linked to manufacturing data from the Census of Manufacturers ©2024 The Authors. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at http://qeconomics.org.https://doi.org/10.3982/QE1693 428 Macera and Tsujiyama Quantitative Economics 15 (2024) This paper studies the importance of this particular technological change in explaining changes in inequality across different groups of workers with different skill levels. Technological change has both direct and indirect effects on inequality. The direct effects operate through marginal products. When firm productivity and worker productivity are complements in production, technological change in the form of higher returns to skill or higher firm productivity dispersion naturally translates into greater inequality. The indirect effects emerge when these changes alter firms’ recruiting behavior and wage policies, affecting not only the demand for skills but also how workers are matched with firms in the presence of labor market frictions. To study these effects, we develop a tractable equilibrium model with heterogeneous workers and firms in which the demand for skills is characterized by firms’ recruiting behavior. In the model, workers are exogenously given a level of schooling, either college or noncollege, and within each schooling level, they differ in skill. On average, college workers are more skilled. Firms also differ by productivity level. Labor markets are frictional and segmented by educational attainment. Matching occurs randomly, and workers can search on the job. Technology-skill complementarity in production implies that higher-productivity firms benefit more from hiring high-skilled workers than lowproductivity firms. The model features sorting of firms in equilibrium. High-productivity firms recruit more intensively in the college submarket, where workers tend to be more skilled, whereas low-productivity firms focus on the noncollege submarket to avoid competing with more productive rivals. This results in the assortative matching of high-skilled workers with high-productivity firms. In this framework, we consider two types of SBTC: technological change in firm productivity, in the form of higher firm productivity dispersion, and technological change in labor productivity, in the form of higher returns to skill in production. The former increases wage inequality within and between groups because firms that hire college graduates become relatively more productive. The sorting mechanism is also at play; more high-productivity firms recruit and bid more aggressively for college workers—stronger firm sorting. The latter technological change raises the relative productivity of college workers and skilled workers. Using the IPUMS-CPS, we calibrate the model to the United States in 1980 and 2015. The main drivers of increasing earnings inequality are technological changes in firm productivity and labor productivity, both of which constitute SBTC. We rely on an external estimate from Decker et al. (2020) for the change in firm productivity dispersion and residually calibrate the change in the returns to skill. The model matches nicely betweenand within-group inequality measures. The calibrated model also matches untargeted moments such as distributions of firm size and higher-order wages moments. and the Annual Survey of Manufacturers. This allows them to estimate firm productivity dispersion for 1981–2013. There are studies documenting the link between rising inequality and increasing dispersion in firm productivity, for example, Faggio, Salvanes, and Van Reenen (2010), Card, Heining, and Kline (2013), Dunne, Foster, Haltiwanger, and Troske (2004), Barth, Bryson, Davis, and Freeman (2016), and Song, Price, Guvenen, Bloom, and von Wachter (2019). Quantitative Economics 15 (2024) Inequality and technological change 429 To understand the channels through which technological change affects inequality, we perform the following counterfactual exercise. We take the 1980 economy and separately add higher firm productivity dispersion and an increase in the returns to skill, letting firms adjust their skill demand through recruiting behavior, but holding fixed the schooling attainment distribution to that in 1980. This exercise allows us to calculate the relative contribution of each technological change that constitutes SBTC. We then add these changes together to see how SBTC as a whole affects the change in inequality. Our main finding is that technological change in labor productivity, in the form of higher returns to skill in production, is the main driver of the increase in betweenand within-group inequality. In contrast, technological change in firm productivity, in the form of higher firm productivity dispersion, plays a less important role in explaining rising inequality, except for the increase in within-group inequality for workers without a college degree. Specifically, technological change in labor productivity explains 87% of the increase in between-group inequality and 66% and 52% of the increase in withingroup inequality for college and noncollege workers, respectively. The contribution of technological change in firm productivity to the change in these inequality measures is 21%, 23%, and 50%, respectively. Total SBTC, which combines these two technological changes, explains almost all of the observed increase in inequality, while non-SBTC changes, such as changes in the distribution of educational attainment, play a limited role. To elucidate the economic forces behind these results, we decompose the measures of inequality. In the case of between-group inequality, we derive a novel equation that decomposes it into differences in average firms’ wages across education groups (firm pay difference) and differences in average labor productivity in production across education groups (labor productivity difference). Higher firm productivity dispersion raises between-group inequality through an increase in the firm pay difference. There are more high-productivity firms that have a high demand for skills and intensively recruit college workers as a result of firm sorting. This leads to stiffer competition in the college labor market, and thus a larger firm pay difference. On the other hand, an increase in the returns to skill in production rises between-group inequality mainly through an increase in the labor productivity difference between college and noncollege workers. Since college workers are more skilled, this technological change rises their marginal product more. For within-group inequality, following Postel-Vinay and Robin (2002), we decompose it into labor productivity variation (person effect), wage policy variation across firms (firm effect), and wage policy variation within each firm (friction effect). We find that the increase in within-group inequality is largely explained by an increase in the person effect for both education groups. The increase in the person effect is driven by the increase in the returns to skill, which generates greater variation in workers’ productivity, especially for college workers, as they are more skilled than noncollege workers. The increase in the firm effect and the friction effect is mainly driven by higher firm productivity dispersion, as it increases not only the variation of wages across firms but also that of outside offers. 430 Macera and Tsujiyama Quantitative Economics 15 (2024) Overall, we find that technological change in labor productivity explains the lion’s share of rising inequality because our calibration implies a substantial change in the returns to skill in production. On the other hand, technological change in firm productivity plays a less important role in explaining rising inequality because the change in firm productivity dispersion is relatively small and the model implies only limited amplification through labor market frictions. Related literature It is well known that SBTC is a major driving force for the increasing inequality (e.g., Acemoglu (2002)). We consider two forms of SBTC that may affect inequality. One is the standard technological change in labor productivity such as an increase in the returns to skill in production. The other is increasing dispersion in firm productivity, for which there is an independent measure by Decker et al. (2020), who use the U.S. Census Bureau’s Longitudinal Business Database linked to manufacturing data from the Census of Manufacturers and the Annual Survey of Manufacturers and estimate establishmentlevel productivity from 1981 to 2013. Haltiwanger and Spletzer (2020) show that rising between-firm inequality is mainly due to rising dispersion across industries. The presence of assortative matching has been widely documented in recent literature (e.g., Lise, Meghir, and Robin (2016), Hagedorn, Hann Law, and Manovskii (2017), Lopes de Melo (2018), Bagger and Lentz (2018), Song et al. (2019)). On the theoretical side, assortative matching results from factors such as production complementarity between worker and job characteristics (Lise, Meghir, and Robin (2016), Lopes de Melo (2018)), heterogeneity in search technology (Bagger, Fontaine, Postel-Vinay, and Robin (2014), Bagger and Lentz (2018)), or firms’ screening (Helpman, Itskhoki, and Redding (2010)). In our model, assortative matching occurs by a different mechanism: labor market segmentation and sorting of firms. Using employer–employee matched data, Engbom and Moser (2017) show that higher education degrees help sorting toward highwage firms and that this sorting explains a substantial part of the returns to college. Empirical studies typically decompose wage variation through the estimation of Mincerian wage functions that include worker and firm fixed effects (see, e.g., Abowd, Kramarz, and Margolis (1999), Card, Heining, and Kline (2013)). Relative to this literature, wage decomposition based on structural models has the advantage of being able to address the potential biases of a reduced-form wage decomposition. For example, by considering the dynamics of worker mobility, Postel-Vinay and Robin (2002) find much less variation in worker fixed effects than do static error-component models that typically attribute historical wage differences to worker fixed effects. Our paper contributes to the decomposition literature based on structural models by explicitly modeling the selection of firms that create jobs for different education levels and the systematic variation in labor market risk with education, each of which potentially causes systematic bias in reduced-form wage estimation. Our results are broadly consistent with the findings of several empirical studies. Our findings suggest that the change in the labor productivity difference is important for explaining the trends in both between-group inequality (Hendricks and Schoellman Quantitative Economics 15 (2024) Inequality and technological change 431 (2014)) and within-group inequality (Taber (2001), Lemieux (2006)). By explicitly considering employer heterogeneity, from which the previous papers abstract, we also find that changes in firm characteristics are also important in explaining the trends in inequality, especially for noncollege workers. This echoes the results of recent papers that emphasize the role of firms in accounting for earnings inequality (Card, Heining, and Kline (2013), Song et al. (2019)). 2. The model 2.1 Environment Time is continuous and infinite. There is measure one of heterogeneous workers and measure one of heterogeneous firms. Workers face a constant birth/death rate μ, whereas firms live forever. Both types of agents are risk neutral and discount the future at a common discount rate r.Weuseρ=r+μto denote the worker’s effective discount rate. Workers are characterized by exogenous states (z,s),wherez∈Zdenotes the skill level, and sdenotes one of two levels of schooling: noncollege (NC) and college (CL). Denote by the joint distribution of worker type and by sthe marginal distribution of skills conditional on educational level. Firms are heterogeneous in their productivity level p∈P≡[b,p],anddenotes the distribution of firms’ type. Labor market The labor market is segmented by schooling levels. In each submarket s∈{NC, CL}, workers and firms are matched randomly, and production takes place. Firms can post vacancies in both submarkets, but workers can only participate in the submarket corresponding to their educational level. Jobs created in submarket sare destroyed at an exogenous rate δs. Production The marginal product of a firm with productivity p(henceforth, p-firm) of hiring an additional worker with type (z,s)(henceforth, (z,s)-worker) is given by p· A(z,s),whereA(z,s)are efficiency units of labor.3The total output of a p-firm is equal to ptimes the sum of its employees’ efficiency units of labor across both submarkets. Wage determination A worker participating in submarket scontacts a firm at rate λs. The type of the firm is drawn from the sampling distribution Fs. Upon matching, productivities are revealed to both parties, and the worker and the firm bargain over the wage under complete information. We briefly describe our wage bargaining framework, which follows closely Cahuc, Postel-Vinay, and Robin (2006). Without loss of generality, we assume that wages are set in terms of efficiency units of labor. Consider a (z,s)-worker who is contacted by a pfirm. If the worker is unemployed, bargaining results in a wage φs 0(p).Iftheworkeris employed by a firm with productivity p<p, the new firm poaches the worker from 3Note that Ais a function of sbecause we will also consider education-specific technological change, for example, a change in the role of the social network in production, access to which may depend on the level of schooling. However, it is difficult to distinguish such technological changes from a change in returns to college (see, e.g., Krusell, Ohanian, Rìos-Rull, and Violante (2000)). We will return to this point in Section 4.1. 432 Macera and Tsujiyama Quantitative Economics 15 (2024) the incumbent, and bargaining results in a wage φs(p,p), where the first argument denotes the productivity of the last employer.4If the worker is employed by a firm with productivity p≥p, the incumbent can successfully deter poaching. In this case, however, bargaining resumes and results in a wage raise from the current wage wto φs(p,p)if and only if p∈(gs(w,p),p],wheregs(w,p)is the productivity level satisfying φs(gs(w,p),p)=w. Firms Let πs(w,p,z)denote the profit for a p-firm hiring a (z,s)-worker at wage w. Thisprofitmustsatisfy ρπs(w,p,z)=ρ(p−w)·A(z,s)−δs+λs1−Fs(p)πs(w,p,z) +λsp gs(w,p)πsφs(x,p),p,z−πs(w,p,z)dFs(x).(1) The first term on the right-hand side is the (normalized) flow profit. The second term is the expected capital loss that stems from either separation or poaching. The third term is the expected capital loss caused by the wage raise necessary to deter poaching. To hire a worker with schooling level s, a firm must contact her in the corresponding submarket. Once a contact is made, there are three possible scenarios: the firm hires an unemployed worker, it hires a worker previously employed at a firm with lower productivity, or it fails to hire a worker because she is already employed at a firm with higher productivity. The expected profit per worker contacted s(p)thus satisfies rs(p)=usz∈Z πsφs 0(p),p,zds(z) +(1−us)z∈Zp b πsφs(x,p),p,zls(x)dxds(z),(2) where usis the unemployment rate in submarket s,andls(p)is the mass of workers employed by a generic p-firm. The first (second) term on the right side represents the expected profit, conditional on an unemployed (employed) worker being hired. Since the production technology displays constant returns to scale, the recruiting decision is made separately for each submarket. Denote the contact frequency in submarket sas ηsvs,wherevsis a measure of recruiting effort, call it “vacancies” as in Mortensen (2003), and ηsis the aggregate efficiency of recruiting effort.5For each s,a p-firm chooses a recruiting effort policy vs(p)to solve max vsηsvss(p)−χ 2v2 s,(3) where the second term represents recruiting costs. 4In the class of labor search models with on-the-job search and wage renegotiation, wages typically depend on the individual history of past offers. In the present model, however, pis a sufficient statistic for this history. 5Since vsis a measure of recruiting effort made at a point in time, an unfilled “vacancy” is immediately destroyed. Quantitative Economics 15 (2024) Inequality and technological change 433 Sampling distribution The probability of being contacted with a p-firm in submarket sis proportional to the number of vacancies these firms create. Thus, the sampling distribution function is given by Fs(p)=x≤p vs(x)d(x) x∈P vs(x)d(x) .(4) Contact rates and recruiting efficiency Following Mortensen (2003), we assume that the aggregate flow of contacts per period in submarket sis proportional to the product of aggregate recruiting effort and workers’ aggregate contact rate, that is, Ms=ηsλs.Hence, in our model, the following must hold for s∈{NC, CL}:6 λs=x∈P vs(x)d z∈Z d(z,s) ,(5) ηs=1. (6) Equilibrium Astationary equilibrium consists of wage functions {φs 0,φs}, policies and value functions for the firms {vs,πs,s}, contact rates and sampling distributions {λs,Fs}, with a distribution of workers such that: (i) {φs 0,φs}solve the wage bargaining problem; (ii) {vs}solves the recruiting decision problems; (iii) {πs,s}satisfies the respective recursive equations; and (iv) {λs,Fs}are consistent with individual choices. Skill-biased technological change (SBTC) The literature defines SBTC broadly as a general term for any change in production technology that increases the relative demand for skilled labor. We consider two types of technological change that constitute SBTC: technological change in firm productivity corresponding to a change in , and technological change in labor productivity corresponding to a change in the function A(z,s). 2.2 Inequality measures and decomposition Our goal is to study how technological change affects betweenand within-group inequality. To understand the underlying economic forces, we propose two analytical decomposition equations for these inequality moments. 6To see this, notice that since each p-firm contacts a worker at rate ηsvs(p), the aggregate flow of vacancies must also satisfy Ms=msηsx∈Pvs(x)d(x),wheremsis the ratio of the measure of firms to that of workers searching a job in submarket s,thatis,ms=(z∈Zd(z,s))−1. For this to be consistent with the expression in the main text, equations (5) and (6)musthold. 434 Macera and Tsujiyama Quantitative Economics 15 (2024) We use w(z,s,p,p)to denote the actual wage paid to a (z,s)-worker at a p-firm with the best outside offer made by a p-firm. We can write7 wz,s,p,p=A(z,s)φsp,p.(7) Between-group inequality We measure between-group inequality as the difference in average log wages between CL and NC. Applying this definition to equation (7), we obtain our first decomposition equation: E[logw|CL]−E[logw|NC]   BG inequality =Elogφsp,p|CL−Elogφsp,p|NC   Firm pay difference +ElogA(z,s)|CL−ElogA(z,s)|NC   Labor productivity difference .(8) According to this equation, between-group wage inequality equals the sum of two components. The first is the difference in the conditional mean of log wages measured in efficiency units, which we label firm pay difference. The second is the difference in the conditional mean of log efficiency units of labor, which we label labor productivity difference. This decomposition clarifies how the change in between-group inequality is shaped by the two types of SBTC. Technological change in firm productivity (change in )affects the firm pay difference, but not the labor productivity difference if skill formation is exogenous.8In contrast, technological change in labor productivity (change in A)has not only a direct effect on the labor productivity difference between the two education groups, but also an indirect effect on the firm pay difference via a change in firms’ recruiting behavior. Within-group inequality We measure within-group wage inequality as the conditional variance of log wages. Applying this definition to equation (7) and using the law of total variance, we obtain our second decomposition equation: Var[log w|s]   WG inequality =Varlog A(z,s)|s   Person effect +VarElogφsp,p|p|s   Firm effect +EVarlog φsp,p|p|s   Friction effect ,(9) where the labels of each component on the right-hand side follow Postel-Vinay and Robin (2002). The person effect measures heterogeneity in labor productivity. The firm effect measures variation in average log wages across firms. The friction effect measures average within-firm log wage variation, independent of the person effect. The latter effect arises purely because two identical workers in the same firm may have different wage offer histories. 7This equation also holds for the newly employed by setting p=b. 8In the working paper version of this paper, which is available upon request, we endogenize the schooling decisions. The main results are materially unchanged. Quantitative Economics 15 (2024) Inequality and technological change 441 Table 2. Model performance. Targeted Moments Data Model 1980 2015 1980 2015 Labor market conditions Unemployment rate: NC 5.0 5.5 5.0 5.5 Unemployment rate: CL 1.8 2.4 1.8 2.4 Unemployment duration (months) 3.5 6.4 3.4 6.3 Inequality measures Between-group: CL/NC 34.5 52.2 34.5 52.1 Within-group: NC 15.2 27.1 16.0 26.4 Within-group: CL 21.1 33.7 20.6 34.2 Note: We use 5-year averages for empirical moments to mitigate cyclical fluctuations. Between-group inequality is measured as the difference in average log earnings (log point). Within-group inequality is defined as the conditional variance of log earnings multiplied by 100. twice the average skill are 30% more productive in 2015 than they were in 1980. We get ϕCL,2015 =1.09, which implies that the returns to college increase by 9%. Finally, we obtain β=0.31. Cahuc, Postel-Vinay, and Robin (2006) estimate the bargaining parameter for different occupations and industries using French data. Excluding the exceptionally high estimate of 0.98 for executives, managers, and engineers in the construction sector, the remaining 15 estimates range from 0.00 to 0.38. In their preferred specification, Flinn and Mullins (2015) estimate the bargaining parameter of 0.25, which is close to our estimate, although the setups are not fully comparable. 3.3 Model performance Table 2 displays the fit of the model for both 1980 and 2015. The model well replicates the increasing trends in betweenand within-group inequality measures. Our model thus successfully captures key aspects of earnings distributions for different groups of the population. Our calibration is also externally validated by empirical moments that are not targeted. The success in matching these moments lends credence to the model’s predictions on the wage inequality trend. Figure 1 plots the distribution of firm size in 1980 and in 2015. The distribution in the data is constructed using the Business Dynamics Statistics (The Census Bureau (2023a)). The model does a good job in replicating the overall shape of the distribution, especially, in capturing the significant fraction of small firms. The distribution in 2015 is hardly different from that in 1980 in the data, which is also captured by the model. For the wage distribution, in the data, the wage ratio of 50th percentile to 10th percentile (P50/P10) changes from 1.90 to 2.27 and P90/P50 from 1.70 to 2.40 between 1980 and 2015. Our model captures this increasing trend: P50/P10 changes from 1.71 to 2.05 and P90/P50 from 1.68 to 1.95, although the magnitude of the increase is somewhat smaller. 442 Macera and Tsujiyama Quantitative Economics 15 (2024) Figure 1. Firm size distribution. The figure plots the firm size distribution in the model against that in the data from Business Dynamics Statistics for 1980 and 2015. We group large firms with more than 99 employees into one category as their density is tiny. The firm size in the model is obtained by normalizing the average size to that in the data (19.8 in 1980, 23.9 in 2015). 4. Quantitative analysis In our calibration, SBTC drives the change in inequality over time. As discussed in the Introduction and Section 2.1, our notion of SBTC consists of two parts: technological change in firm productivity, in the form of higher firm productivity dispersion (a decrease in γ), and technological change in labor productivity, in the form of an increase in the returns to skill (an increase in θand ϕCL). In this section, we assume that these technological changes are exogenous shocks to the economy and analyze how they affect betweenand within-group inequality. To understand the channels through which technological change affects inequality, we conduct two counterfactual experiments. First, we take the 1980 economy and separately add the change in firm productivity dispersion or in the returns to skill, holding the schooling attainment distribution fixed at that of 1980. This exercise allows us to calculate the relative contribution of each technological change that constitutes SBTC. Second, we add these changes together to see how SBTC as a whole affects the change in inequality and whether there is any interaction between the two types of technological change. Note that simply adding SBTC to the 1980 economy does not result in the 2015 economy, because we still keep the education attainment distribution and the labor market parameters (δs,χ) at their 1980 values. Thus, this exercise also allows us to calculate the contribution of these non-SBTC changes, which turns out to be small. 4.1 Changes in inequality Table 3 reports the changes in betweenand within-group inequality measures between 1980 and 2015. The row labeled “SBTC” presents the results for the second counterfactual experiment, and the rows “Firm productivity” and “Labor productivity” show the results for the first counterfactual experiment. Quantitative Economics 15 (2024) Inequality and technological change 443 Table 3. Counterfactual analysis. BG WG: NC WG: CL Level Pct. Level Pct. Level Pct. SBTC 19.2 109% 10.6 101% 11.9 88% Firm Productivity 3.7 21% 5.2 50% 3.1 23% Labor Productivity 15.4 87% 5.4 52% 9.0 66% Model: 1980–2015 17.6 10.4 13.6 Note:BG denotes the change in between-group inequality of college workers (CL) relative to noncollege workers (NC) between 1980 and 2015. WG corresponds to the change in within-group inequality for each education group. The last row shows the change in each inequality moment predicted by the model between 1980 and 2015. Between-group inequality With SBTC, between-group inequality increases by 19.2 log points, which is 9% larger than the predicted increase of 17.6 log points between 1980 and 2015. This result confirms that SBTC is the main driver of the increase in betweengroup inequality. The non-SBTC changes mitigate the effect of SBTC. Both the increasing dispersion of firm productivity and the increase in the returns to skill contribute to the higher between-group inequality, accounting for 21% and 87% of the total change, respectively. This means that the latter is the main technological change responsible for the higher between-group inequality. There is little interaction between the two technological changes. Within-group inequality With SBTC, within-group inequality increases by 10.6 for NC workers and 11.9 for CL workers, accounting for 101% and 88% of the predicted increase between 1980 and 2015, respectively. Thus, SBTC is also the main driver of the increase in within-group inequality. The increasing dispersion of firm productivity explains 50% of the total change for NC workers and 23% for CL workers. For the increase in the returns to skill, the contribution increases to 52% and 66%, respectively. Thus, this technological change is again most responsible for the higher within-group inequality, although the change in the distribution of firm productivity is equally important for the increase in within-group inequality for NC workers. Robustness As we discussed in Section 2.1, it is difficult to distinguish the change in ϕCL from a change in returns to college (see, e.g., Krusell et al. (2000)). Our result in Table 3 is based on the somewhat strong assumption that the education-specific change (i.e., the 9% increase in ϕCL) is all related to production technology, and thus considered part of SBTC. One can imagine the other extreme, where the education-specific change is entirely due to an improvement in the productivity-enhancing effect of college.20 In this case, the change in the returns to college is attributed to the non-SBTC change, reducing the 20This interpretation corresponds to another calibration that is isomorphic to our baseline calibration. In that calibration, similar to our baseline calibration, we assume that the skill of CL workers follows a Pareto log-normal distribution, z=z1·z2,wherez1∼LN(μz,σ2)and z2∼P(x,α).Here,xis a scale parameter of the Pareto distribution, and in our baseline calibration, we implicitly assumed x=1 throughout the time. In the isomorphic calibration, we instead assume that ϕCL =1butxincreases to 1.09 in 2015. 444 Macera and Tsujiyama Quantitative Economics 15 (2024) Table 4. Decomposition: between-group inequality. BG Firm Pay Diff. Labor Prod. Diff. SBTC 19.2 4.3 15.0 Firm Productivity 3.7 3.7 0.0 Labor Productivity 15.4 0.4 15.0 Model: 1980–2015 17.6 2.6 15.0 Note: The decomposition is based on equation (8). contribution of the technological change in labor productivity. However, even in this extreme case, we find qualitatively the same results. Technological change in labor productivity, now due only to the change in θ, explains 26% of the predicted change in between-group inequality, more than the contribution of technological change in firm productivity. Total SBTC accounts for 51%, so it is again the main driver of change in between-group inequality. For within-group inequality for CL workers, the contribution of technological change in labor productivity falls only slightly from 66% to 59%, and SBTC explains 82% of the observed change, compared to 88% before. The result for within-group inequality for NC workers is unchanged by construction. 4.2 Decomposition To elucidate the economic forces behind these changes in the inequality measures, we use our decomposition equations (8)–(9). Between-group inequality Table 4 shows the decomposition of the change in betweengroup inequality based on equation (8), in which we decompose the change in betweengroup inequality into the firm pay difference and the labor productivity difference. They increase by 2.6 and 15.0 log points, respectively, accounting for 15% and 85% of the overall increase between 1980 and 2015. Thus, the college premium increases mainly because the labor productivity gap between CL workers and NC workers widens. The increase in the labor productivity difference is entirely due to the technological change in labor productivity. That is, the increase in the returns to skill widens the labor productivity gap between CL workers and NC workers. In contrast, increasing firm productivity dispersion does not affect the labor productivity difference. With SBTC, the firm pay difference increases by 4.3 log points. Most of the increase comes from increasing productivity dispersion. In 2015, there are more highproductivity firms that have high demand for skills. Wages increase in both submarkets but disproportionately more in the CL market, because these high-productivity firms intensively recruit CL workers, resulting in stronger firm sorting and a larger firm pay difference. The increase in the returns to skill also strengthens firm sorting as it increases the size of the match surplus more in the CL market. The non-SBTC changes attenuate the effect of SBTC, reducing the firm pay difference by 1.7 log points. Since a larger (smaller) supply of CL (NC) workers makes it easier Quantitative Economics 15 (2024) Inequality and technological change 445 Table 5. Decomposition: within-group inequality. WG: NC Person Firm Friction SBTC 10.6 5.4 2.3 2.9 Firm Productivity 5.2 0.0 2.3 2.9 Labor Productivity 5.4 5.4 −0.0 −0.0 Model: 1980–2015 10.4 5.4 2.4 2.6 WG: CL Person Firm Friction SBTC 11.9 9.1 1.1 1.8 Firm Productivity 3.1 0.0 1.2 1.9 Labor Productivity 9.0 9.1 −0.1 −0.1 Model: 1980–2015 13.6 9.1 2.1 2.4 Note: The decomposition is based on equation (9). (harder) for firms to contact these workers, the logic of competitive labor markets suggests that the firm pay difference should fall in response. This is the relative quantity effect at work. Within-group inequality Table 5 shows the decomposition of the change in withingroup inequality based on equation (9), in which we decompose the increase in withingroup inequality into the person, firm, and friction effects. They explain 52%, 23%, and 25% of the predicted change in within-group inequality for NC workers and 67%, 15%, and 18% for CL workers, respectively. Thus, the increase in within-group inequality is largely driven by the increase in variation in labor productivity within educational groups. The increase in the person effect is entirely driven by the increase in the returns to skill. Larger returns translate into a larger person effect (equation (9)), and the effect of higher θis stronger for CL workers because they are more skilled than NC workers. The increase in the firm and friction effect is mainly driven by the increase in the firm productivity dispersion. Wages per efficiency unit of labor depend on the productivity of the employer (equation (11)), and thus, wage variation across firms hinges on the properties of the productivity distribution. In addition, the appearance of many highproductivity firms increases the variation of outside offers, and thus the likelihood of climbing the ladder within less productive firms, which leads to higher frictional wage variation. Hence, the value of the firm and the frictional effects are higher. 5. Conclusions In this paper, we develop an equilibrium model in which skill demand is characterized by firms’ recruiting behavior. The model features firm sorting and assortative matching of high-skilled workers with high-productivity firms. It provides novel decomposition equations that can be used to study how different forms of SBTC shape betweenand within-group inequality over time in the United States. 446 Macera and Tsujiyama Quantitative Economics 15 (2024) The main result of our analysis is that the increase in betweenand within-group inequality is largely driven by the change in the returns to skill in production, while higher firm productivity dispersion plays an important role in explaining the increase in within-group inequality of noncollege workers. Appendix A: Derivation of Equation (12) We derive the equation by guess and verify. We suppress the schooling level s. Rearranging equation (1) and applying integration by parts, we obtain (ρ+δ+λ)π(w,p)=ρ(p−w)−λp g(w,p) πwφ(x,p),pφ1(x,p)F(x)dx. (16) Differentiating both sides with respect to wand applying Leibniz’s rule yields (ρ+δ+λ)πw(w,p)=−ρ+λφ1g(w,p),pgw(w,p)πw(w,p)Fg(w,p). Since, by the definition of g(w,p),wehaveφ1(g(w,p),p)gw(w,p)=1, we can write πw(w,p)=− ρ ρ+δ+λ1−Fg(w,p). Plugging this expression back into equation (16) and noting ρ(p−w)=ρp g(w,p) φ1(x,p)dx, we have π(w,p)=ρp g(w,p) φ1(x,p) ρ+δ+λ1−F(x)dx. (17) From equation (11), the derivative of the wage equation with respect to the first argument is φ1p,p=(1−β)ρ+δ+λ1−Fp ρ+δ+λβ1−Fp. Plugging this into equation (17) yields equation (12). Appendix B: Data:Current Population Survey (CPS) We use the Current Population Survey (CPS), relying on IPUMS-CPS (Flood et al. (2018)). The CPS is a nationally representative data set that provides important demographic and employment information. Our sample is composed of white males aged 25–55. We drop women from the sample because the educational attainment and labor force participation of women have changed dramatically in recent decades for various reasons, not only those on which we focus in this paper. We also drop nonparticipants in the labor force and samples with missing observations. The sample weights provided are used in computing the empirical moments. Quantitative Economics 15 (2024) Inequality and technological change 447 Figure B.1. Betweenand within-group earnings inequality. Panel A plots the between-group inequality measured as the difference in average log earnings between college workers (CL) and noncollege workers (NC) (log point). Panel B plots the within-group inequality of CL and NC measured as the variance of log earnings. All values are 5-year-centered moving averages. We first define the education categories. To do so, we use a variable for educational attainment provided by IPUMS-CPS.21 We define high school dropouts as those with fewer than 12 years of completed schooling or those without a high school diploma; high school graduates as those having 12 years of completed schooling and not reporting no diploma; some college attendees as those with any schooling beyond 12 years and less than 4 years of college; and college graduates as those with 4 or more years of completed schooling. We define noncollege workers as a sum of high school graduates and some college attendees. We do not use high school dropouts in the analyses. We measure experience as years of schooling subtracted from age minus 5. In the Mincerian regressions, we include cubic controls for experience and controls for industry codes. For earnings inequality measures, we use those working full-time (40+weeks and 35+usual hours per week) for wages and salary in the private labor force. Self-employed workers are excluded. All amounts are adjusted to 2015 U.S. dollars using the CPI (Organization for Economic Co-operation and Development (2023)). We impute average hourly wages for each observation using reported work weeks and usual hours per week. We then drop those with imputed hourly wages falling below one-half of the federal minimum wage (U.S. Department of Labor (2023)). We follow Autor, Katz, and Kearney (2008) for top coding. Prior to 1988, wage and salary incomes were collected in a single variable. After 1988, they were reported as two separate variables, corresponding to primary and secondary earnings. For each of these 21This variable, called EDUC, is constructed from two other variables, HIGRADE and EDUC99. HIGRADE, available prior to 1992, gives only the respondent’s highest grade completed, whereas EDUC99, available since 1992, also provides data on highest degree or diploma attained. In EDUC, the categories of HIGRADE are given the same codes as their equivalents in EDUC99. 448 Macera and Tsujiyama Quantitative Economics 15 (2024) Figure C.1. Numerical comparative statics: The figure shows the numerical comparative statics for the inequality moments in 2015 with respect to θ2015 and ϕCL,2015. BG 15 is the between– group inequality in 2015, WGNC 15 is the within-group inequality for noncollege workers in 2015, and WGCL 15 is the within-group inequality for college workers in 2015. variables, top-coded values are simply reported at the top-code maximum, except for the primary earnings variable in 1996 or later. For those, top-coded values are assigned the mean of all top-coded earners, and we reassign the top-coded value. We then multiply the top-coded earnings value by 1.5. After 1988, we simply sum the two earnings values to calculate total wage and salary earnings. Figure B.1 displays the trends in the betweenand within-group inequality. 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