2.1. Observations

There is well-documented evidence that wage inequality both between- and within-observable groups substantially increased over the last few decades. In this section, I summarize several salient features of those two types of wage inequality observed in the recent 40 years, focusing on how they differ when workers are sorted based on their education and occupations. Whereas some were discussed extensively in previous works, one particular fact that the trend of the growth of wage inequality became significantly steeper from 2000 up until now has rarely been explored before.

Figure 1. Real hourly wage for silled and unskilled workers.

Note: Calculation based on CPS ASEC data, 1983–2019. Both panels show various percentiles of the real hourly wage for workers with (skilled) or without (unskilled) a bachelor’s degree. Real hourly wages are calculated using annual wage and salary divided by usual hours worked per year. I adjust for inflation using Consumer Price Index adjustment factors which convert dollar amounts to constant 1999 dollars (CPI99 in IPUMS), so all values are expressed as 1999 dollars. The sample comprises nonfarm wage and salary workers who are 25–64 years old. The dotted lines in the top panel give the slope of wage growth during periods before and after 1999 (vertical line).

First and foremost, the trend of wage growth for skilled workers varies a lot across percentile groups before and after 2000. To be specific, as shown in the top panel of Fig. 1, the growth rate of real wages for above-median college degree holders significantly accelerated around 2000 relative to their lower-tail peers. Table 1 reemphasizes the point by confirming that the gaps in real wage growth between percentiles were not large during 1983–1999 but were substantially widened afterward. For example, one can find in particular that the real wage growth of the top 10th percentile of skilled workers significantly accelerated starting from 2000 compared to their peers, as reflected by the larger difference between the average annual growth before and after 2000. However, the divergence is not present for the group of unskilled workers, within which the growth rates of wages are relatively invariant across percentile groups. This observation is consistent with the finding in Autor et al. (Reference Aghion and Kearney2008) that the bulk of within-group inequality occurs above the median of the wage distribution in recent decades, given that presumably college degree holders represent the upper tail of the wage distribution. One subtle difference of my observation from what they found is essentially the notable acceleration of divergence within the educated group rather than the whole population after 2000.

Second, despite the sharp increase in the college wage premium, the bottom half of skilled workers barely benefited from it. Table 1 makes it clear that the growth of real wage for the first decile of college degree holders was almost zero in the past four decades, and it is probably surprising to note that even the median college degree holders experienced negligible real wage increase compared to their upper-tail peers. This observation, combined with the last one, suggests that in contrast to the conventional view that the increase in the college wage premium benefits most of the college degree holders, it is just the top half of them that benefits. Thus, wage dispersion has been increasingly enlarged since the beginning of the 21st century.

Table 1. Annual % change in real wage

Note: Annual % change is calculated by the cumulative % change over the corresponding period divided by the number of years.

Lastly, unskilled workers saw a mild decline in real wages in the recent 40 years. Although there is a general agreement on this fact in many works based on various sources of dataset,Footnote ⁸ the fact is still relevant and noteworthy in the context of wage inequality because it serves as a natural contrast to the first fact related to within-group inequality. The divergence of wage growth across percentile groups that is salient for the skilled group is not present for the unskilled group at all, which indicates a tied connection between the widening of within-group wage inequality and the specific group of skilled workers.

2.2. Empirical implementation

In this section, I implement an empirical strategy to quantify the magnitude of the change in the trend of wage inequality as observed. Besides, using the task-based framework by ALM, I run the same regression for the subsamples of workers in five occupations to explore which occupations drive the changing trend. One remarkable finding is that the changing trend of wage inequality was entirely driven by one category of task, namely the nonroutine analytic task.

The occupations based on consistent three-digit COC in CPS data are classified as five categories of task groups following the criterion in ALM and David and Dorn (Reference David and Dorn2013). They are nonroutine analytic, nonroutine interactive, routine cognitive, routine manual, and nonroutine manual tasks. They are constructed using the rank of certain measures of task intensity by the Dictionary of Occupational Titles (DOT). Occupations with the highest rank among five measures of task intensity are attributed to corresponding categories.Footnote ⁹

I then run the following regression for skilled workers as well as the subsamples of workers in each of the five occupations defined above:

(1) \begin{equation} WI^{i} = \beta _{0}^{i} + \beta _{1}^{i} \, Year + \beta _{2}^{i} \, \unicode{x1D7D9} (Post2000) + \beta _{3}^{i} \, \unicode{x1D7D9} (Post2000) \times Year + \epsilon ^{i} \end{equation}

where $i$ denotes different (sub) groups of workers and $WI^{i}$ is a set of measures of wage differentials for the whole group and five subgroups obtained by taking log differences between wage percentiles in each group; $Year$ captures the time trend of wage differential; and $\unicode{x1D7D9} (Post2000)$ is an indicator of whether the observation is on or after 2000 (1 if it is and 0 otherwise). Therefore, the coefficients of interest are $\beta _{2}^{i}$ and $\beta _{3}^{i}$ , which reflect the magnitude of the shift in wage inequality and the change of the slopes (trend) after 2000 separately.

The results shown in the first column of Table 2 confirm the observation that the top 10% of skilled workers gained greatly over their peers after 2000. For example, the growth of the 90/10 percentile differential widens about 0.4% per year on average after 2000 relative to before. This is a sizable change of wage structure because on average, the differential grows about 0.2% each year ( $\beta _{1}$ ), which means the magnitude of the widening amounts to two times the previous trend of the growth of wage differential. In some cases, this change of trend is augmented with a one-time upward shift after 2000. Other measures of upper-tail wage inequality also significantly increased after 2000, while measures of lower-tail wage inequality (50–10) did not.

Table 2. Changes in wage inequality after 2000

Note: This table displays the results of the regression (equation (1)) for the whole sample of skilled workers (column 1), as well as the subsamples of workers in the five occupations (columns (2)–(6)). The values on the left and right in each column are estimates of $\beta _{3}$ and $\beta _{2}$ separately. ^***, ^**, and ^* indicate the estimator is significant at 1%, 5%, and 10% level.

I then investigate which category of occupation (task) drives the changing trend of wage inequality for the overall skilled workers by implementing the same regression for the five subgroups of workers defined above, from which we might have a sense of what kinds of skilled workers gained more after 2000 and why. Interestingly, the classification of occupation under the task-based framework shows an informative picture of differential behaviors of wage inequality across occupation categories, although the interpretation is slightly different from ALM which focuses only on the reallocation of task input demand rather than wage inequality.

The results in columns (2)–(6) demonstrate it is essentially just one occupation (task) that contributes to the increases in the growth of overall within-group inequality, namely the nonroutine analytic task. It is appealing to find that the nonroutine analytic task not only underwent a widening of upper-tail wage inequality in trend after 2000, but also the magnitude was more than twice as much compared to the whole skilled group. The change of trend (slope) was also augmented with a sizable one-time upward shift. In contrast, none of the rest of the occupations showed a significant shift in the trend of the wage structure after 2000. If any, there was a notable contraction of wage inequality for the nonroutine manual jobs. Put together, the evidence suggests that the change of the trend of upper-tail wage differential within the skilled group is exclusively driven by that within nonroutine analytic jobs, which also more than offset the opposite force brought about by the contraction of wage inequality within nonroutine manual jobs.

2.3. Educational distribution

Because my empirical analysis mainly focuses on wage inequality within skilled workers who have at least a bachelor’s degree, I am interested in to what extent does this relate to the overall inequality. It turns out the answer that greatly depends on which category of occupation we are looking at. I also investigate which occupations employ the most workers with college degrees. Understanding those helps shed a light on what might drive the changing structure of wage inequality within the skilled group.

One appealing finding is that the bulk of workers consists of skilled workers in the analytic task, which is exactly the task that underwent a changing trend of wage inequality. This suggests a close link between the skill intensity of a task and its structure of wage inequality. Fig. 2 shows a clear hierarchical pattern of the share of skilled workers for the five occupation categories. More than half of the workers doing nonroutine analytic jobs are college degree holder, and the task skill intensity is kept increasing in the recent four decades. For nonroutine interactive jobs, this number is a bit lower. For the rest of the three categories, the skill intensity is greatly lower, although routine manual jobs are well above the other two.

Figure 2. Percentage of college degree holder.

Note: The values are shares of workers with at least a bachelor’s degree in each category of occupation.

Figure 3. Destinations of college degree holders.

Note: The values show what percentage of skilled workers (with at least a bachelor’s degree) goes to each category of occupation, so the five numbers in each year add up to one.

The data on the destinations of skilled workers convey a similar big picture as Fig. 2 with slightly more information. Fig. 3 explores the educational distribution from the other direction, which is what percentage of skilled workers goes to the five categories of occupations separately. It shows most (more than 80%) of the college degree holders work in nonroutine cognitive occupations. It is notably interesting that the percentage of skilled workers working in nonroutine analytic jobs surpassed interactive jobs just around 2000, coinciding with the timing of the change of the wage structure within nonroutine cognitive jobs.Footnote ¹⁰

3.1. Environment: firms and production

There is a representative firm operating a production technology for the final output. The production function is a standard nested CES production function using two task inputs—abstract and routine task:

(2) \begin{equation} Y_{t} = [\mu _{a} N_{at}^{\frac{\gamma _{a} - 1}{\gamma _{a}}} + (1 - \mu _{a}) [(1 - \mu _{r}) K_{t}^{\frac{\gamma _{r} - 1}{\gamma _{r}}} + \mu _{r} N_{rt}^{\frac{\gamma _{r} - 1}{\gamma _{r}}}]^{\frac{\gamma _{r} (\gamma _{a} - 1)}{\gamma _{a}(\gamma _{r} - 1)}}]^{\frac{\gamma _{a}}{\gamma _{a} - 1}} \end{equation}

where the subscript $a$ ( $r$ ) refers to the abstract (routine) task, $N_{at}$ ( $N_{rt}$ ) is efficiency units of labor in the abstract (routine) task, and $K_{t}$ is machines embodied by capital. The parameters include $\gamma _{a}$ ( $\gamma _{r}$ ), which is the corresponding elasticity of substitution between routine and abstract tasks (routine labor and machines), and $\mu _{a}$ ( $\mu _{r}$ ), which is the relative weight of the abstract (routine) labor in production. The terms in the first square bracket indicate that the output of the abstract task is produced simply using the total efficiency units of labor in that task, while for the routine task, the output could be produced by either machines $K$ or labor. That investment-specific technological change can potentially substitute for routine labor and is the key mechanism of shifts in employment across tasks to capture the fundamental task replacement arguments put forth by ALM.

The problem for the firm is

(3) \begin{equation} \max _{K_{t},N_{rt},N_{at}} Y_{t} - r_{t} K_{t} - w_{rt} N_{rt} - w_{at} N_{at} \end{equation}

3.2. Environment: workers

To simplify the household problem, I assume a representative family setup, in which the household owns the capital and is comprised of a continuum of workers of measure one. Each worker supplies one unit of labor inelastically. For Preferences, I assume a standard logarithmic specification for households’ instantaneous utility with discount factor $\beta$ :

(4) \begin{equation} u(c) = \log\!(c_{t}) \end{equation}

The household accumulates capital by the following law of motion:

(5) \begin{equation} K_{t+1} = (1-\delta )K_{t} +q_{t}I_{t} \end{equation}

where $q_{t}$ is the relative price of consumption to investment goods and included in the equation of capital evolution following Greenwood et al. (Reference Greenwood, Hercowitz and Krusell1997).Footnote ¹³ An increase in $q_{t}$ represents increased investment-specific technological progress and is the main driving force of the model. Its movement is assumed to be exogenously determined for now.

In addition, workers are heterogeneous in efficiency units of labor in the abstract task, which is represented by a single index $z$ . This assumption is key in generating a dispersed wage distribution in the abstract task that can be mapped back to the data and evaluated quantitatively. I assume that the distribution of $z$ is simply standard normal. A worker with $z$ provides $\phi _{a}(z)$ efficiency units of labor in the production of abstract task, where $\phi _{a}(z) = e^{\alpha _{a}z}$ , so that the distribution of efficiency units is log-normal. Therefore, wages for a worker with skill level $z$ in the abstract occupation are given by the product of the common skill price, $w_{at}$ , and the heterogeneous efficiency units in the abstract occupation, $\phi _{a}(z)$ . For simplicity, I assume that the abstract skill level does not affect workers’ productivity in the routine task ( $\phi _{r}(z) = 1$ ), so wages in that task are homogeneous. Abstracting from the heterogeneity of workers in the routine task simplifies the structure but will not affect the main focus of the model .

Thus, the representative household problem is given by:

(6) \begin{equation} \max _{C_{t}(z),O_{t}(z),K_{t+1}} \sum _{t=0}^{\infty } \beta ^{t}\int _{z} \log\!(C_{t}(z)) d\Phi (z) \end{equation}

\begin{equation*} s.t. \quad C_{t} + I_{t} = r_{t} K_{t} + \int _{z} W_{t}(z) d\Phi (z) \quad {\rm and} \quad K_{t+1} = (1-\delta )K_{t} +q_{t}I_{t} \end{equation*}

where $K_{t}$ is the household’s total capital holdings, $O_{t}(z)$ is the occupational choice (abstract or routine task) for each individual, and $W_{t}(z)$ , the wage for an individual with $z$ , is obtained by $\sum _{j} \unicode{x1D7D9}(O_{t}(z)=j) w_{jt}\phi _{j}(z)$ .

The representative family setup implies that consumption will be equalized across all workers, which greatly simplifies the model. Since consumption inequality and its labor market implications are not the main focus here, this parsimonious framework would not affect the central point of interest, which is wage inequality.

3.3. Definition of equilibrium

The competitive equilibrium for the economy consists of a set of prices ( $r_{t}, w_{at}, w_{mt}$ ) and allocations ( $C_{t}(z),O_{t}(z)$ , $K_{t},N_{rt},N_{at}$ ), such that:

1. (1) Taking prices ( $r_{t}, w_{at}, w_{mt}$ ) as given, workers choose ( $C_{t}(z),O_{t}(z),K_{t+1}$ ) to solve (6) starting from any initial conditions $K_{0}$ for all $t \geq 1$

2. (2) Taking ( $r_{t}, w_{at}, w_{mt}$ ) as given, the firm chooses ( $K_{t},N_{rt},N_{at}$ ) to solve (2);

3. (3) Labor and output markets clear:

   (7) \begin{equation} N_{jt} = \int _{z} \unicode{x1D7D9}(O_{t}(z)=j) \phi _{j}(z) d\Phi (z) \qquad j = a,r \end{equation}
   and resource constraint:
   (8) \begin{equation} Y_{t} = C_{t} + I_{t} = C_{t} + \frac{1}{q_{t}}(K_{t+1} - (1-\delta )K_{t}) \end{equation}

3.4. Characterization of equilibrium

Since the full path of $q_{t}$ is deterministic and its movement is the primary driving force for the incentives of investment, it is useful to relate the rental rate of capital $r_{t}$ to $q_{t}$ . Therefore, it is convenient to rewrite the Euler equation for the model in the following way:

(9) \begin{equation} r_{t+1} = \frac{1}{q_{t}}\!\left(\frac{C_{t+1}}{\beta C_{t}} - \frac{1-\delta }{\frac{q_{t+1}}{q_{t}}}\right) \end{equation}

A worker’s problem can then be characterized by the Euler equation (9). Note that the rental rate of capital next period $r_{t+1}$ is not only a decreasing function of $q_{t}$ but also an increasing function of the growth rate of $q$ . The first result holds simply because an increase in $q_{t}$ leads to higher investment and thus induces a higher supply of capital in this period, which has a downward pressure on the rental rate next period. The growth rate of $q_{t}$ also affects the rental rate as a high growth rate will lower the relative capital gains on undepreciated capital and thus offsets declines in the rental rate.

Then, I characterize the assignment of workers into tasks based on a comparative advantage structure. The main feature of the equilibrium is that the assignment of labor takes a form of the threshold value. Formally, there exists a unique $z^{*}$ such that workers with $z \gt z^{*}$ choose to work in the abstract task, while workers with $z \lt z^{*}$ choose to work in the routine task.Footnote ¹⁴

The key driving force of the model is that as a technical change takes place, the prices for investment goods drop ( $q$ increases). Cheaper investment goods induce the firm to replace the labor in the routine task by machines. Workers who become obsolete in the routine task have to relocate to the abstract task. From the threshold argument above, we know that the marginal entrants to the abstract task are lower types of $z$ compared to the incumbents. This drives the threshold value $z^{*}$ down and changes the composition of the workers in terms of $z$ in the abstract task. This mechanism of labor assignment and mobility based on comparative advantage is closely related to the classic work by Roy (Reference Roy1951) and many modern treatments of the Roy model.Footnote ¹⁵

Finally, the prices ( $r_{t}$ , $w_{at}$ , $w_{mt}$ ) can be obtained in terms of aggregate variables using the first-order conditions by the firm’s profit maximization problem (3):

(10) \begin{equation} w_{at} = \mu _{a}\!\left(\frac{Y_{t}}{N_{at}}\right)^{\frac{1}{\gamma _{a}}} \end{equation}

(11) \begin{equation} w_{rt}= (1-\mu _{a})\mu _{r}Y_{t}^{\frac{1}{\gamma _{a}}}R_{t}^{\frac{\gamma _{a}-\gamma _{r}}{\gamma _{a}(\gamma _{r}-1)}}\!\left(\frac{1}{N_{rt}}\right)^{\frac{1}{\gamma _{r}}} \end{equation}

(12) \begin{equation} r_{t}= (1-\mu _{a})(1-\mu _{r})Y_{t}^{\frac{1}{\gamma _{a}}}R_{t}^{\frac{\gamma _{a}-\gamma _{r}}{\gamma _{a}(\gamma _{r}-1)}}\!\left(\frac{1}{K_{t}}\right)^{\frac{1}{\gamma _{r}}} \end{equation}

4.1. Exogenous forcing process and model solution

To solve and evaluate the model quantitatively, I construct the entire path of $q_{t}$ following vom Lehn (Reference vom Lehn2020), which used the deflator for investment goods as constructed following Gordon (Reference Gordon1990). As in Greenwood et al. (Reference Greenwood, Hercowitz and Krusell1997), $q_{t}$ is measured as the price of consumption divided by the price of equipment and software investment. I assume that the time series starts in a steady state with $q_{1}$ normalized to 1 in the year 1983. The data are available until 2015, after which I assume the growth rate of $q_{t}$ linearly decays to zero in a few years.Footnote ¹⁶ The entire path of $q_{t}$ and its growth is plotted in Fig. 4.

Figure 4. Time path and growth rate of $q_{t}$ .

Note: The time series starts in a steady state with $q_{1}$ normalized to 1. The data of $q_{t}$ calculated based on the price index of investment goods constructed in Gordon (Reference Gordon1990) is available until 2015, after which it is assumed that the growth rate of $q_{t}$ linearly decays to zero in 12 years. The growth of $q$ in period $t$ is given by $\log\!(q_{t+1}) - \log\!(q_{t})$ .

With the values of $q_{t}$ , I can first obtain any steady state variables using the Euler equation (9). I then calibrate the parameters of the model so that the model solution matches certain empirical counterparts. Lastly, the transition path is solved using a standard shooting algorithm given that we know the initial and terminal steady states. The details of solving the model are in Section 5 and Appendix A.

4.2. Calibration

The parameters to calibrate are the elasticities of substitution ( $\gamma _{a}$ , $\gamma _{r}$ ), share parameters in the production function ( $\mu _{a}$ , $\mu _{r}$ ), the scaling parameter of productivity function in the abstract task ( $\alpha _{a}$ ), capital depreciation rate ( $\delta$ ), and discount factor ( $\beta$ ). Among those parameters, the discount factor and depreciation rates can be set externally following a standard value as estimated in Cummins and Violante (Reference Cummins and Violante2002). This gives $\beta = 0.96$ and $\delta = 0.1$ . Besides, since the structure of production is almost identical to the one in vom Lehn (Reference vom Lehn2020), except that manual jobs are aggregated in the routine task, I use his calibration results for the elasticities of substitution. So we have $\gamma _{a} = 0.43$ and $\gamma _{r} = 1.38$ . $\gamma _{r} \gt 1 \gt \gamma _{a}$ implies that capital is a substitute for the labor input in the routine task, while the routine task and abstract task are a gross complement. The values of elasticities capture the key qualitative attributes of tasks that are necessary for the desired task reallocation and are therefore reasonable results for calibration.

I calibrate the share parameters in the production function ( $\mu _{a}$ , $\mu _{r}$ ) in the way that the labor share of output and the fraction of labor income in routine and abstract jobs in the initial steady state match their empirical counterparts in 1983. $\alpha _{a}$ is calibrated such that the log difference between the 90th percentile and 10th percentile of the wage is matched to the data in 1983. Table 3 summarizes parameter values for the calibration.

Table 3. Calibrated values for model parameters

Note: This table reports the values for calibrated model parameters. They are elasticities of substitution ( $\gamma _{a}$ , $\gamma _{r}$ ), share parameters in the production function ( $\mu _{a}$ , $\mu _{r}$ ), the scaling parameter of productivity function in the abstract task ( $\alpha _{a}$ ), capital depreciation rate ( $\delta$ ), and discount factor ( $\beta$ ).