Skill-Biased Technological Change and Inequality in the U.S.

Since the 1980’s, income inequality has increased markedly and is at the highest level ever since it has been recorded in the U.S. This paper uses an overlapping-generations model with incomplete markets that allows for household heterogeneity and that is calibrated to match the U.S. economy with the purpose to study how skill-biased technological change (SBTC) and changes in taxation quantitatively account for the increase in inequality from 1980 to 2010. We ﬁnd that SBTC and taxation decrease account for 48% of the total increase in the income Gini coefﬁcient. In particular, we conclude that SBTC alone accounted for 42% of the overall increase in income inequality, while changes in the progressivity of the income tax schedule alone accounted for 5,7%.


Introduction
Some argue that we are in the period of a "Forth Industrial Revolution", which moved production function shares. There is an increasing concern in the possible dominance of technology over the human labor: "Automation and AI will lift productivity and economic growth, but millions of people worldwide may need to switch occupations or upgrade skills" ( Manyika et al. (2017)).
Most of the literature focus on the substitution of low-skilled labor for capital (Autor et al. (2003); Acemoglu and Restrepo (2018)). Although, it is essential to have in mind that high-skilled automation can, and will probably be an issue due to artificial intelligence and machine learning. Acemoglu and Restrepo (2016) describe: "If the long-run rental rate of capital relative to the wage is sufficiently low, the long-run equilibrium involves automation of all tasks". Hence, as shown by Acemoglu and Restrepo (2018), low skill-automation will increase wage inequality because people are being substituted by machines or losing their job. A social measure to reduce inequality is using taxation. Saez (2001), claimed that labor tax rates should be U-shaped, separating households with low and higher income distributions, instead of the previous proposed lump-sum taxation (Mirrlees (1971)). Furthermore, Aiyagari (1995) ensures that with incomplete markets and uncertainty, optimal capital taxation is positive.
In this manner, the present article pretends to answer quantitatively how SBTC and taxation changes account for the paths of income inequality in the U.S from 1980 to 2010. Our contribution is similar to Krusell et al. (2000). The authors show that capital-skill complementary changes account for most of the variations on the skill premium. Other related studies also measure wage inequality through skill premium (Heckman et al. (1998)). We focus instead in the entire income distribution, and see how much of the increase in income inequality can be accounted by investment-specific technological change (ISTC).
The model developed in this framework is an overlapping generations economy, with incomplete markets where agents face a persistent uninsurable income shock and a borrowing limit.
Agents are also born with different skills, where one type is complementary with capital and the other substitutable. Investment-specific technological change will thus generate inequality by in-creasing the skill premium. Parallel to the increase in the skill premium, we could also observe a drastic decrease in labor income progressivity (see Figure 5) during the Reagan Administration, precisely at the start of our period of analysis. We also replicate the level and progressivity of the income tax schedule in order to study how much can these changes account for the increase in post-tax income inequality that we observe.
Our model reproduces simultaneously some phenomenon of the U.S. economy from 1980 to 2010, namely: the skill premium rise; a growth in income and wealth inequality; a rise in skilled labor share, and a reduction on the unskilled labor share. We were able to account for 48% of the total change in income inequality. In particular, we show that SBTC alone account for 42%, while taxation alone accounted for 5,7%.
The rest of the work is organized as follows. In Section 2 we discuss some related literature and empirical facts. In Section 3, we present the model and the calibration method and in Section 4 the results. Section 5, concludes the work.

Related literature and facts
It is quite a consensus that labor share has been declining since 1980 (Eden and Gaggl (2018); Karabarbounis and Neiman (2013)). Some recent models attribute the labor share contraction to the substitutability between capital and unskilled labor in the technological production function.
This substitution in the course of investment-specific technological change has been referred as automation.
Particularly, Eden and Gaggl (2018) calibrate an aggregate production function that highlights the interaction between information and communication technology (ICT) and different types of labor for the U.S. economy and find that the decline in the aggregate labor share is explained by the decrease in routine occupations, since the income share of non-routine labor has been rising 1 .
1 It is very important to distinguish between occupation and worker skill type. Some professions are non-routine, although they do not infer a skilled household, i.e., an educated household, for example, an electrician does not have a college degree, although performs a non-routine job. Contrary, diagnosis doctors are skilled, but they perform a routine occupation. Most of the routine occupations are conducted by agents that have a college degree or higher.
For instance, automation can create distinct effects on the economy. On the one hand, it can increase the aggregate welfare, because it pushes up productivity and, as a result, the factor prices change (Acemoglu and Restrepo (2018); Eden and Gaggl (2018)) 2 . But on the other hand, as capital becomes cheaper, or in other words, as investment prices decline, unemployment rates will increase due to a shift in companies' factor demand, which will raise the demand for skilled people and reduce the demand for unskilled people (Acemoglu and Restrepo (2016)). As demand-supply rule takes place, unskilled households will see their wages decrease, although skilled agents will face an increase in their salaries.
In fact, U.S. wage structure shows that since 1970 there is an increase in dispersion in household earnings, especially in different levels of education, age, and experience. Furthermore, Katz et al. (1999) mentioned that the observed wage structure for U.S. seems to translate an increase in inequality. The author summarises several reasons that are attributed to wage inequality: (i) higher demand for more educated people driven by SBTC; (ii) loss in the wage premium paid to less educated people, due to a rising globalisation pressure; (iii) higher dispersion in skills, due to increase of unskilled immigration; (iv) and changes in wage setting norms.
As a consequence, households will pursue different behaviors when they face income risk.
Agents can create an ex-ante response, i.e., in anticipation of the shock they tend to increase their precautionary savings and engage in contracts in which wages are kept constant (Krueger et al. (2016)). Or, after they face a shock, i.e., as an ex-post response to risk, households will make a consumption revision, which will be lower if the income shock is negative, or higher otherwise (Heathcote et al. (2014)). To smooth the shocks, households can change their skills, this is, they can increase their human capital, becoming skilled (Heathcote et al. (2017)).
The U.S. wage pattern is related to technological development because periods of significant technological developments are correlated with high skill premiums. Indeed, SBTC increased the demand for skilled workers since 1980, and this increase explains part of the rise in education wage premium. Furthermore, the more demanded occupations in 2026 will be those that are less likely to be automated and will be more related to social skills, creative thinking, and problem-solving capacities. These non-routine occupations are related primarily with high-skilled jobs which need higher levels of education and have more significant earnings. Figure 1 presents a projection for the growing job positions for 2026, which shows that to have access to most of them it is necessary to incorporate in distinct levels of education. In reality, most of those occupations will require college degrees.
These recent projections support Heckman et al. (1998) who introduce human capital accumulation in an OLG model in order to explain the rise in the wage inequality, measured by the skill premium, without giving an unique role to capital, and conclude that the higher demand for high skilled labor induces a supply response, thus more and more people will go to college as a response to the required features. shows the most demand jobs, where 0 indicates that there is no need for formal education credential; 1 indicates that it is necessary a high school diploma; 2 indicates that it is necessary a Bachelor's degree; 3 for Master's degree; and 4 for Doctoral and advanced degree. Source: U.S. Bureau of Labor Statistics.
The skill premium can be seen as an explanatory variable for the decreasing labor share in the course of SBTC. Murphy and Welch (1992) calculated that the skill premium grows 3,3% each year, ceteris paribus. Furthermore, Krusell et al. (2000) show that there was a decline in 1970 in the skill premium, but in 1980 there was a severe increase. Figure 2 is the representation for the skill premium since 1980 for the U.S. economy, which shows that there is, indeed, an increase that was maintained until 2010, although since then it has been quite steady. The more considerable difference is coming from the college skill premium, calculated as the ratio for bachelor degrees and high school degrees. This problematic of income polarization may continue to increase due to the higher demand for high-wage occupations that can grow more than middle-wage jobs.
As this trend continues, the problem can appear because not everyone has the same opportunity to access to higher education. Thus, inequality surges, because automation leads to unemployment in low-skilled people (Acemoglu and Restrepo (2018)) and because wealthier agents tend to be more educated and older (Krueger et al. (2016)). Figure 4 compares the Gini Index for pre-tax and post-tax income and shows that inequality is rising since 1980. Thus, although taxes are taking influence in reducing inequality, it seems that this has not been entirely effective. Indeed, the income share of the bottom 90% is dropping in the same period. Of course, distinct levels of income correspond to different levels of experience, skills, and productivities, as it will be clarified in section 3.
Taxation can be a force to increase output and consumption because it affects government spending (Ferriere and Navarro (2014)). The Mirrlessian approach concludes that individuals with highest skills have optimal taxation of zero (Diamond (1998)). However, recent studies show that there are welfare gains when we move to a non-linear tax function especially when the government does not observe the skills of the citizens (Gorry and Oberfield (2012)). Thus, the government should set different taxation on workers with different abilities and, in consequence, with different elasticities towards capital. Heathcote et al. (2010) find that the optimal income taxation structure to maximize social welfare is only a two-parameter function, that embraces the level of taxation and the progressivity of tax, as it will be clarified in section 3. Using this, and also, other income taxation approaches, Guner et al. (2014) find that it is possible to reduce the Gini Index from 0,56 to 0,55 only by using labor taxation.
Although this is a useful measure to reduce income inequality, taxation can create an adverse effect. As the government increases progressive taxation, agents have less incentive to work, and they prefer to invest less in skills, which can create even more heterogeneity (Golosov et al. (2016); Heathcote et al. (2017); Stiglitz (1982)).
Progressive taxation is essential to redistribute after-tax income across ex-ante heterogeneous households. Thus an optimal policy can create beneficial effects on society. Krueger et al. (2013) found optimal labor taxation of 34,1% taking into account skilled and unskilled households and, concluded that this taxation leads to a lower Gini index, higher GDP and consumption, and more people deciding to go to college.

Model
The model used is an incomplete markets economy with uninsurable idiosyncratic risk and heterogeneous agents. There are two types of households that derive utility from consumption and leisure. Skilled agents born with particular abilities, such as intelligence, creativity, and physical force, that allow them to execute tasks that are a complement to capital. On the other hand, unskilled households have attributes that enable them to substitute capital. Both types face an idiosyncratic uninsurable stream in the form of wages and make joint decisions about their consumption, savings and hours worked. The model is based on Brinca et al. (2016) and has a bequest motive such as Brinca et al. (2018) and Brinca et al. (2019).

Demographics
The economy is populated by J-1 overlapping generations. Households make their labor decision at 20 years old and finish their work life at 65 when they retire. Such as in Brinca et al. (2016) households have an age-dependent probability of dying, π( j) and, consequently an age-dependent probability of surviving ω( j) = 1 − π( j), where π(100) = 1. Furthermore, the economy will be composed by q=J−1 ∏ q=80 ω(q) retired households, at any given period.

Technology
The economy behaves in perfect competition with constant elasticity of substitution (CES) production function that uses three inputs: capital (K), skilled (S) and unskilled labor (NS). Z t is an intermediate good and Y t is the aggregated production function, as it is specified: where A t is the technology level, A k,t is augmented technological level, φ 1 describes the share of the composite factor, φ 2 is the share of capital, ρ is the elasticity of substitution between capital and skilled labor and σ is the elasticity of substitution between composite factors and unskilled labor.
This doubled-nested production function inserts the single-nested specification where the skilled labor is directly complemented with capital and, unskilled labor is substitutable with composite factors, this is simular with Karabarbounis and Neiman (2013) and Krusell et al. (2000) 3 . Thus, the law of capital motion can be given as: where δ comes for capital depreciation and X t for the gross investment.

Labor income
Households are the labor suppliers, and they are different among them. Each household faces different discount factors, different permanent abilities, different persistent idiosyncratic shocks, and different asset holdings. Moreover, retired households receive a benefit that is given by the government.
Households' productivity will depend on their age, j, permanent ability, a ∼ N(0, σ 2 ε ), which determines if an agent is able to provide skilled or non-skilled labor, and an idiosyncratic productivity shock, u, which follows an AR (1) process, such as: Furthermore, wages will be given as: w( j, a, u) = w s e γ 1 j+γ 2 j 2 +γ 3 j 3 +a+u , s ∈ {S, NS} where γ determines the age profile of wages and w S determines the wages per efficiency unit of the individual 4 . It is important to mention that college education will affect the household labor earnings through wages and a distinct age profile of wages, although it will not affect through an increase in the efficiency unit endowment.

Preferences
Households utility depend on consumption and work time and it is given by: where χ is the notation for disutility if work and η denotes Frisch labor elasticity. As in many macroeconomic models this is a CRRA utility function in order to be able to produce a balanced growth. Retired households face an additional benefit on their income:

Government
The government has a balanced budget and collects revenues to supply the economy with public goods and benefits. Revenues from social security are flat and come from the part of the employer and the employee to finance the retirement benefits, Ψ t .
The government also taxes consumption and capital at constant rates. The labor tax function follows a non-linear form as suggested by Benabou (2002), that incorporates heterogenous households, and diverse studies confirm that it is a good fit for U.S. data (Guner et where y a is the post government income, y is the pre-tax labor income or money income. Parameter θ 1 determines the level of taxation and θ 2 determines the progressivity of the tax. In this sense, it is possible to see that when the progressivity is null, θ 2 = 0, the tax rate will be flat, such that y a = 1 − θ 1 , thus the marginal and average tax rates are equal and this will capture a rise in the overall level of taxation. Hence, if θ 2 = 1 the labor tax function infers complete redistribution; however, when θ 2 > 0(< 0) it implies that the taxation system is progressive (regressive), since marginal rates surpass (are under) average rates.
All tax revenues are used by the government to finance the supply of public goods, G t , public debt interest expenses, r t B t , and other lump-sum transfers, g t . Thus, the government budget constraint will be as follows, where R SS t is the social security revenues and R t are other tax revenues.
Equilibrium Firms maximize their profits in a competitive equilibrium, thus factor prices will be equal to their marginal products: Furthermore, the household's problem can be formulated recursively for a given period. Each household is defined by his age, j, asset position, k, time discount factors β ∈ {β 1 , β 2 }, permanent ability, a, persistent idiosyncratic shock, u, and a time-constant ability to supply a specific labor type s ∈ {S, NS}. Giving his position the household chooses consumption, c, working hours, n, and the assets holding, k ′ . The problem can be summarized in the following form, where Y L is the household labor income after taxes: In this manner, the stationary recursive competitive equilibrium can be defined as follows, where Φ( j, k, β , a, u) is the stationary distribution of agents: 1. Taking factor prices and initial conditions as given, the value function V ( j, k, β , a, u) and the policy functions, c( j, k, β , a, u), k ′ ( j, k, β , a, u) and n( j, k, β , a, u) solve the household's optimization problem.

The government budget balances:
g dΦ + G + rB = τ k r(k + Γ) + τ c c + nτ l nw(a, u, j) 1 +τ SS dΦ 5. The social security system balances: 6. The assets of the dead are uniformly distributed among the living:

Calibration
The model is calibrated to match the U.S. economy in 1980, in light with the method used by Brinca et al. (2016) and Brinca et al. (2018). Preferences and age profile of wages, ρ u and σ ε are setting according to Brinca et al. (2016). The first discount factor is set to match the capital-output ratio in 1980 and the second discount factor is set to match the income share of the bottom 90%.
The distribution parameters, φ 1 and φ 2 are fixed to 0,55 and 0,8, respectively, so that the skill premium and the quantities of labor supplied are close to levels observed in 1980 (Eden and Gaggl (2018)). Furthermore, the elasticity of substitution between capital and skilled labor is 0,43, and the elasticity of substitution between capital and unskilled labor is 2,33.
The disutility of work, χ, and the variance of ability, σ a , are set using the Simulated Method of Moments (SMM).
Furthermore, risk aversion was set to 1,2. We, also assume that capital depreciates at 0,06 and the share of non-routine skills is set to 40%.

Wages
The wage profile through life-cycle represented by equation (5) is calibrated directly from the data. We run equation (15) using data from Luxembourg Income and Wealth Study (LWS).
where j is the age of individual i. To calculate ρ u and σ ε we use PSID data and regressed equation (15), then we use the residuals in order to estimate both parameters. These parameters are kept constant across steady-states 5 .

Preferences
There has been an extensive debate in the literature relative to the value of Frisch elasticity of labor supply, η. The estimates for η are comprehended between 0,5 to 2 6 . We set the Frisch elasticity to 1 as Trabandt and Uhlig (2011).

Taxation
As it is specified in section 3, we use the labour income tax function represented in equation (8), to capture the progressively of both the tax schedule and government transfers. In order to estimate θ 1 and θ 2 we follow the method of Ferriere and Navarro (2014). Thus we fix θ 1 = 0, 85 and θ 2 = 0, 16 for 1980. For 2010 the values of θ 1 and θ 2 are set to 0, 87 and 0, 095, respectively.
The rates for social security are set assuming no progressivity, the taxes on behalf for employer and employee are set to 7,65% for both steady states. Furthermore, capital taxation and consumption taxation are set according to the values obtained by Mendoza et al. (1994), which are τ c = 5, 4% and τ k = 46, 9%. For 2010 these values are 5,5% and 36% for consumption and capital, respectively, following Brinca et al. (2016).

Endogenous calibrated parameters
Since some parameters do not have an empirical counterpart, they are calibrated using SMM.
These parameters are calibrated to match the target values in Brinca et al. (2016), as in Table 1. We choose β 1 , β 2 , χ, σ a and ϕ to minimize the loss function: M m is the moment in the data and M d refers the moments in the model. We have five instruments, and five moments in the data to have an identified system. Table 2 displays the values of the parameters calibrated by SMM.

Results and Discussion
The supply of skills is shaped by many variables, such as demographic trends, preferences and education shifts. Due to technological changes, workers may want to upgrade their skills, as the skill demand increases. Initially, technical change was viewed as factor-neutral, this is, improvements in the TFP leave marginal rates unchanged. However, empirically, we observe a rise in the skill premium, as well as the increase in skilled labor supply, as we show in section 2. Even with a higher supply of skilled people since 1970, wages for skilled people kept rising, which can be observed as pieces of evidence of skill-biased technological change. In fact, Acemoglu and Autor (2011) argue that technical changes are by its nature skill-biased.
Thus, some argue that the changes in production are not just an effect of the decrease in the price of investment, but also an increase in the skill supply. As society keeps getting more educated, employers will prefer to use people's ability to make them even more productive and, as they gain experience they can be more profitable more rapidly than unskilled households. Furthermore, skilled households have an advantage compared with unskilled households, since they give less uncertainty to the employer.
For simplicity, most of the studies assume that production function elasticity of substitution between capital and labor is equal to 1. However, recently, a departure from this assumption has taken place. If the elasticity of substitution between capital and unskilled labor is higher than 1, firms will substitute labor for capital. In this manner, we guarantee that the growth of skilled labor is greater than the growth of unskilled labor. In this sense, if σ > 1, then the two inputs are substitutes.
Thus, the economy will be endogenously augmented through capital, because an increase in A k,t will increase the marginal productivity of capital. This effect occurs jointly with an increase in the skill premium and marginal productivity of skilled labor. However the unskilled labor has lower productivity. Contrary, if the elasticity of substitution between capital and unskilled labor is lower than 1, the two factors will be complements and the demand shift will decrease the skill premium.
This goes according with the results reported by Acemoglu and Autor (2011), Autor et al. (2003), Karabarbounis and Neiman (2013) and Krusell et al. (2000). Krusell et al. (2000) show that the values for the elasticity of substitution between skilled labor and capital are between 0 and 1,2 and the values for the elasticity of substitution between unskilled labor and capital are between 0,5 and 3. Therefore, skilled labor and capital tend to be complements and unskilled labor and capital tend to be substitutes. This interpretation has consequences for taxation because taxes depend on the heterogeneous characteristics of the households. Hence, agents with higher skill level should face higher taxes and unskilled households should face lower taxes, i.e., the lower the substitution between factors the higher should be the tax rate imposed, and vice-versa.
To capture the SBTC, we use capital-augmenting technology, A k,t , as a substitute. We use an elasticity of substitution for skilled labor and capital lower than 1. Thus these factors are complements. Contrary, we set an elasticity of substitution for unskilled labor and capital higher than 1, stating that these factors are substitutes.
Our experiments are as follows. First, we calibrated the model for the U.S. to match the capitaloutput ratio, average hours, and moments of income and wealth distributions for 1980. Then, we changed the tax system according to 2010 values, as referred in section 3. After this change, we compute the changes in the total factor productivity (TFP) and skill-biased technological change to replicate the growth in PIB per capita between 1980 and 2010. We follow Greenwood et al. (1997) and keep the contribution the TFP and SBTC constant and equal to one-half 7 .
With this model, we capture several aspects of the U.S. economy since 1980 to 2010, such as: (i) rising skill premium; (ii) increase in income and wealth Gini coefficient; (iii) decrease in the wealth share owned by the bottom 90% of families (iv) an increase in skilled labor share; (v) a reduction in unskilled labor share; (vi) and, an increase in wage dispersion. Furthermore our model recognises, as expected, that people spend more hours working and the supply of skilled households increased in 2010, due to a decrease in progressive labor taxation.
Our model accounts for 48% of the total increase in the income Gini Index for the period.
Then, we access the contribution of changes in the tax system and changes in the investmentspecific technological change separately, by changing each factor at a time. We find that changes in the tax system account for 5,7% of the total increase in income inequality, while changes in investment-specific technological change account for 42%.

Conclusion
Most of the economists believe that the U.S. wage structure is influenced predominantly for technological shifts, especially since 1980. We use an overlapping generations model with incomplete markets, where agents face persistent uninsurable income shock and a borrowing limit, featuring skill-biased technological change. To generate SBTC we assume that agents born with different abilities, whereby some are endowed with abilities that are complemented by capital and others are endowed with capabilities that are substituted by capital, i.e., we use the substitution of unskilled labor for capital as a reasonable mechanism to explain the skill-biased technological change (Karabarbounis and Neiman (2013); Krusell et al. (2000)).
We calibrated our model to match the U.S. economy in 1980 to answer quantitatively how skilled-biased technological change and taxation explain income inequality in the U.S. from 1980 to 2010. The model captures the severe decrease in labour income progressive taxation; the rise in the skill premium; the increase in income inequality; as well as the increase in the share of the skilled population, opposing to the decrease in the share of unskilled labor. This shows that highskilled workers have, indeed, an advantage in the labor market because they give less uncertainty to the employers. More importantly, we find that changes in taxation and capital-skill complementary jointly account for 48% of the increase in income Gini. Furthermore, we find that SBTC account for 42%, while taxation alone accounted for 5,7%.
An essential introduction to the model can pass for add an endogenous education choice in light withÁbrahám (2008). Before entering in the economy, a household can observe its ability and decide whether to begin to work as an unskilled worker or to attend college. This decision will depend not only on the distribution of agents ability, but also on the initial wealth distribution, taking into account a costly educational choice. Moreover, it is also possible to study an optimal taxation across the transition path between steady-states.
In this sense, the tax wedge for any two incomes (y 1 , y 2 ) is given by 1 − 1 − τ(y 2 ) 1 − τ(y 1 ) = 1 − y 2 y 1 −θ 2 (17) and therefore independent of the scaling parameter θ 1 . In this manner, one can raise average taxes by lowering θ 1 and not the progressivity of the tax code, since the progressivity is uniquely determined by the parameter θ 2 .

Labor tax function calculation
In order to estimate θ 1 and θ 2 we follow Ferriere and Navarro (2014). The authors calculated the progressive tax rate as: We use data from Mertens and Montiel Olea (2018) for AMTR (Annual Marginal Tax Rate). ATR (Annual Tax Rate) is equal to: AT R = TotalTaxLiability

TotalIncome
The data for Total Tax Liability is retrieved from Statistic of Income and Total Income data is retrieved from Piketty and Saez (2003).
Noticing that AMTR is equal to the sum of AMIITR (Average Marginal Individual Income Tax Rate ) and AMPTR (Average Marginal Payroll Tax Rate), the formula was changed using only AMIITR, which incorporates solely tax rate series for the federal individual income tax, because the presented model already incorporates the taxation for social security.
The level of tax rate can be seen as a quantitatively close measure of the average tax rate Ferriere and Navarro (2014). Thus, if we use y = 1 we are assuming that the household income equals to the mean income and we obtained the same values for both measures.   Level of labor tax 0, 849 0, 869 φ 2 Progressivity of labor tax 0, 159 0, 095 1 For capital and consumption taxation in 1980 we use the values from Mendoza et al. (1994) and for 2010 we use the values from Brinca et al. (2016); 2 For labor taxes we use Ferriere and Navarro (2014) method;