Labour market effects of wage inequality and skill-biased technical change

ABSTRACT This paper analyses the effects of wage inequality on labour market development. Relevant theories are ambiguous, just as public debates. We measure the effects of inequality, skill-biased and skill-neutral technical change on hours, productivity and wages in a novel structural vector error correction framework identified by economically motivated long-run restrictions. The results show that structural inequality shocks have a negative impact on hours, productivity and wages. These effects are particularly pronounced at high inequality levels and for inequality below the median wage. Skill-biased technology shocks reduce – unlike skill-neutral ones – hours but increase inequality, productivity and wages.


I. Introduction
Recent years have witnessed intensifying debates on increasing economic inequality worldwide. This brings to the forefront the question of the effects that inequality has on economic growth or the labour market. However, the range of the literature on this issue is vast, as are the differing theoretical hypotheses and channels linking inequality to the macroeconomic variables of interest. On the one hand, there are theoretical considerations stating that wage dispersion is a necessary precondition, i.e. the price for higher incentives, investment, growth rates and employment chances. On the other hand, a range of theories expect a higher level of inequality to impede the opportunities of an important share of the labour force to participate in educational advancement, signifying an obstacle to growth, productivity and employment development. Empirical evidence ranges from finding results in favour of the first (e.g. Forbes (2000), Bowles and Park (2005)) over ambiguous results (e.g. Persson and Tabellini (1994)) to those in favour of the latter (e.g. Panizza (2002)).
However, the lion's share of existing research investigates the links between inequality and economic growth, investment, or political stability (see, for instance, Cingano (2014) or Alesina and Perotti (1996)). Studies that directly investigate the relationship between inequality and labour market outcomes are scarce. Fitzenberger and Garloff (2008), for example, examine the impacts of wage disparity on the level of unemployment. Furthermore, country-specific measures of inequality are available only on a yearly frequency, which makes in-depth structural analyses of shortrun and dynamic effects difficult. Instead, the existing literature often focuses on the cross-sectional dimension via multiple country analyses (see, for instance, Forbes (2000)). Naturally, relationships for single countries cannot be inferred from these studies. Another strand of the literature (e.g. Kölling (2014)) focuses on firm-level data and links wage dispersion within companies to bargaining power, productivity, the profit rate and competitiveness. However, the conclusions to be made concern employment effects at the firm level only and are difficult to transfer to the aggregate level.
We contribute to the literature by proposing a structural macroeconometric framework for analysing economic and labour market effects of wage inequality. Within this framework, causal effects can be identified for single countries, i.e. without recourse to cross-sectional methods. The model provides high flexibility in that it minimizes the set of identifying assumptions and allows for estimating fully dynamic (i.e. short-, medium-and long-run) impacts. In addition, it enables us to simultaneously model the effects of further structural shocks such as skill-biased and skill-neutral technological change. This is particularly important since skill-biased technical change (SBTC) directly favours the skilled over the unskilled and hence has an inherently efficiency-increasing nature. The structural inequality shock in turn captures sources of changes in inequality such as minimum wages, bargaining power of unions, employment regulations or hiring preferences. Importantly, incorporating SBTC into our framework allows to model structural inequality shocks that are closer to how they are treated in the theoretical literature.
For measuring wage inequality and SBTC we can rely on a large sample size using the integrated employment biographies (IEB), a unique dataset based on administrative data of the Federal Employment Agency in Germany. The data range from 1975 to 2014 and allow for us to collect the relevant labour market information of every single employee during his or her career. To measure gross wages, this dataset provides data with high quality and precision compared to survey data. In addition, we can spot changes in overall inequality not only once per year but at any point in time. Logically, inequality is no longer the limiting variable in terms of frequency. As a consequence, the full range of quarterly information stemming from variables such as productivity, hours or wages is at our disposal for a thorough structural macroeconometric investigation. This provides the opportunity for an in-depth analysis of inequality effects in one of the world's most sizable labour markets that also features an exemplary rise in inequality through time, including heated debates on the topic.
We calculate the Gini coefficient as a comprehensive measure of wage inequality. The results show an upward trend in inequality that prevailed for decades but has come to an end and even reversed since 2010, a result also found by Weber (2015). We find that this reversion in wage dispersion is mainly driven by a reduction in inequality in the lower half of the wage distribution.
For identification purposes, we construct a structural vector error correction model (SVECM), i.e. a dynamic cointegrating model with economically motivated short-and long-run restrictions. The analysis is embedded in a framework including major driving forces of the labour market and inequality, productivity shocks and SBTC. Structural inequality shocks are defined as impulses changing wage inequality in the long run (in contrast to other shocks) but not affecting SBTC, which in turn also represents a driving force of inequality. For this purpose, we measure SBTC from time series of relative wages and factor supplies. Importantly, the ambiguity of empirical results on the effects of inequality could stem from the fact that wage dispersion itself, in addition to other factors, can be driven by inherently efficiency-enhancing forces, SBTC representing the prime case (see Katz and Murphy (1992) or Juhn, Murphy, and Pierce (1993), for instance). Hence, we allow for the effects of structural inequality shocks on the labour market variables of interest being discriminated from the effects of SBTC.
The results based on impulse response analysis show that structural inequality shocks have a negative impact on hours worked and additionally reduce productivity. Historical decompositions show that the rise of inequality from 1997 to 2009 reduced productivity by 3.3% and hours by 1.2%. In addition, we find that the adverse effects become stronger with higher inequality levels. Furthermore, wage dispersion has negative labour market effects irrespective of whether it occurs below or above the median wage. Below, however, these negative effects are substantially stronger. Skill-biased technology shocks increase productivity and wages but reduce hours worked and drive up inequality. By contrast, skill-neutral technology shocks have a positive long-run effect on hours worked. The opposing effects of skill-biased and skillneutral technology shocks can contribute to a more comprehensive understanding of the relationship of technology and the labour market (cf. Balleer and van Rens (2013)).
The paper is structured as follows. Different theories linking inequality to growth or labour market variables are discussed in Section II. Section III discusses the variable selection and introduces the data used in this paper. Section IV presents our macroeconomic model and the identification strategy. The results based on the impulse responses and historical decompositions are laid out in Section V. The last section concludes.

II. Theoretical background
The following paragraphs present a short overview of the mechanisms postulated in the literature analysing the relationship between inequality and economic growth or -though only scarcely existingbetween inequality and the labour market.
Theoretical considerations consistent with the incentive hypothesis postulate that wage dispersion is the price for higher investment, growth rates and employment chances. Mirrlees (1971) or Lazear and Rosen (1981), for instance, state that higher dispersion leads to higher incentives for harder work, more investment and higher willingness to take risks to benefit high rates of return. This provides a direct link to aggregate productivity: high skill premia could motivate more people to improve their educational status. Given that high-education workers are more productive, aggregate productivity is also influenced. Kaldor (1955) postulates a positive relationship between inequality and growth through a different mechanism: Based on the finding that the rich typically save more than the poor, higher dispersion raises aggregate savings so that more capital is accumulated.
By contrast, theories in line with the opportunities hypothesis expect a higher level of inequality to impede the opportunities of an important share of the labour force to participate in educational advancement, signifying an obstacle to employment development. Galor and Zeira (1993) formalized the so-called human capital accumulation theory. It depends on imperfect financial markets in which the level of income (or wealth) determines whether an individual can afford profitable investments. On the educational market, the poor do not receive the optimal level of education even though the rate of return on education is high. This type of under-investment is not only negative at the individual level but also for the society, and it harmfully affects future productivity and growth. Similarly, more inequality can exclude the poor from access to health care (Galor and Zeira 1993;Galor and Moav 2004;Galor 2009;van der Weide and Milanovic 2018), which again can harm productivity and growth.
Another strand of the literature, the endogenous fiscal policy theory (see Persson and Tabellini (1994), for example), emphasizes the role of political institutions for the inequality effects: high wage dispersion leads voters to insist on higher tax rates, more regulation and anti-business policies, all of which could harm growth through reduced incentives to invest. However, Kenworthy and McCall (2008) find little support for the median-voter hypothesis in the sense that higher inequality in the market earnings leads to greater generosity in redistributive policy. Related to this theory is the political instability argument, especially for countries with substantial poverty. Alesina and Perotti (1996), for instance, argue that extreme inequality may lead to social unrest and hence be a drag on growth. Nel (2003)'s findings do not support this hypothesis in a clear way. He finds no statistically significant effects of inequality on political stability. However, he argues that high levels of inequality change potential investors' risk perceptions, which negatively affects future growth prospects. In general, democratic legitimacy can be questioned due to rising inequality. Especially for Europe, Kuhn et al. (2016) find that income inequality contributes to europscepticism especially among the low educated.
The theories presented so far link inequality to the labour market only indirectly and more in the longer term. Furthermore, one may argue that a channel via personal educational investment might be less relevant in view of a comprehensive public infrastructure in industrialized (compared to developing) economies. Nonetheless, arguments in the spirit of the opportunities hypothesis open the possibility for short-and medium-run effects as well. The opportunities hypothesis, beyond general education, could also operate through participation in further training, which is by far the weakest for low-wage earners. By the same token, the wage also reflects the value of a job, including longer-term perspectives and development opportunities. In other words, arguments in line with the opportunities hypothesis extend to the quality of jobs, which has immediate consequences for labour market development. Concretely, inequality not only can impede opportunities to participate in educational advancement but can also create a sector of persistent low-quality jobs with limited productivity dynamics and poor career opportunities. This can further be linked to negative employment effects since in such a sector typically layoff risks are higher, productivity might fall short of a more or less rigid wage level, structural problems and unemployment hysteresis are more prevalent and labour force participation is limited.
Studies that directly investigate the relationship between inequality and labour market variables are scarce. Bowles and Park (2005), for instance, investigate how incentives to emulate the rich can influence an individual's decision between labour and leisure, so that greater inequality can lead to longer work hours. The short-and medium-run effects of inequality-reducing interventions are often seen to depend on the economy's level of market failure. While inequality-reducing measures such as minimum wages (e.g. Flinn and Mullins (2015), Acemoglu and Pischke (1999) or Agell and Lommerud (1997)), unemployment benefits (e.g. Atkinson (1999) or Acemoglu and Shimer (1999)) or employment protection (e.g. Pissarides (2001)) can have detrimental effects on efficiency and the labour market outcome in perfectly competitive markets, they might lead to an increase in efficiency and employment in the presence of a certain level of market failure, e.g. monopsonistic or even monopolistic markets, or frictions in the labour market. Fitzenberger and Garloff (2008), for instance, examine the impacts of wage disparity on the level of unemployment, distinguishing between two hypotheses. The frictional hypothesis postulates that both income inequality and unemployment increase if the bargaining power of companies increases, while the heterogeneity hypothesis links the wage of an employee to his/ her marginal productivity. A compression of wages, e.g. by minimum wages, leads to high employment barriers, signifying high entry rates to unemployment and low exit rates out of unemployment. The results do not clearly support either hypothesis. However, the authors state that the frictional hypothesis seems to perform better since they find no negative correlation between unemployment within age/education cells and within-cell wage dispersion.
A further set of theories links wage inequality to a firm's productivity, profit rate and competitiveness (see, e.g. Kölling (2014)). In theory, there should be a positive relationship among these variables if efficiency and tournament wages increase the firm's productivity, while there should be a negative relationship if wage inequality violates fairness beliefs (compare Akerlof and Yellen (1988)) and reduces workers' motivation as well as firms' attractiveness and innovation potential. Under the assumption that the successful companies grow stronger while the unsuccessful ones shrink or close down, one could translate these firm-level-based theories into considerations at the aggregate level. Table 1 summarizes the hypotheses. It shows the direct effects of inequality (i.e. for instance no labour market effects that stem only from growth effects). The wide range of research trying to theoretically capture the mechanisms through which inequality impacts growth allows for both negative or positive effects. The empirical work that aimed at discriminating between these channels has often been ambiguous or inconclusive. Since the existing literature is often occupied with multiple country analyses, relationships for single countries cannot be inferred. In contrast, we will construct a framework for investigating inequality effects within single countries, focusing on Germany. We argue that the ambiguity of hitherto empirical results on the effects of inequality could also stem from the fact that wage dispersion itself, in addition to other factors, can be driven by inherently efficiency-enhancing forces. In this context, SBTC represents the prime case in the literature (see Katz and Murphy (1992), Juhn, Murphy, and Pierce (1993), for instance). Appendix B provides a more detailed discussion on this issue.

III. Variable selection and data
In our model of the economy and the labour market, we use five variables: productivity, wages, hours worked, SBTC, and inequality. We measure these variables as explained in the subsequent paragraphs. All data are either available at a quarterly frequency or are converted from monthly frequency. They range from 1975Q1 to 2014Q4 so that the total number of observations amounts to 160. For adjusting the structural-level shift in 1992Q1 due to the German reunification, we could rely on an overlap of the German and West German macroeconomic time series in 1991, providing a factor that we applied to the time series after the shift. This section explains the respective data sources and methods used for data preparation.

Productivity
Productivity is both an important factor involved by the hypotheses discussed in Section II and a key factor in modern theories of the labour market (e.g. Mortensen and Pissarides (1994)). Therefore, we include this variable in our structural model. We use seasonally adjusted productivity from the Federal Statistical Office (destatis) in Germany. Productivity is measured in terms of real GDP per hours worked by the whole working population (hours being described below). The solid line in Figure 1 shows the development of productivity after taking logs and multiplying by 100. In addition to normal business cycle fluctuations, the slump during the Great Recession of 2008/2009 is clearly visible. By the same token, we observe a certain flattening of productivity growth from about 2002 onwards (compare Hutter and Weber (2021)).

Wages
The second variable in our model is real wage cost as published by the Federal Statistical Office. It comprises the dependent workers' gross hourly wages and salaries plus the employers' social security contributions. The variable represents the wage level as distinct from wage inequality discussed below. The time series is seasonally adjusted and converted to real terms through the GDP deflator. The dashed line in Figure 1 shows the log � 100 of real wage cost in Germany since 1975Q1. The graph might suggest the existence of a long-run relationship between productivity and wages. A gap opened during the wage moderation phase mainly through the 2000s, but in recent years (also beyond the sample), real wage growth again exceeded productivity growth. Based on cointegration tests, we will allow for such a relation in our model. Cointegration between productivity and wages is economically equivalent to the presence of a covariance-stationary labour share. More detailed information on our structural model is presented in Section IV.

Hours
Our preferred variable for measuring the labour market quantity effects of inequality and SBTC is total hours as calculated by the Institute for Employment Research (IAB). It is a holistic measure of labour market activity that, in contrast to the number of dependent workers, considers the employees' working time and is hence able to capture structural effects such as the changing importance of part-time work or minijobs. This choice is in parallel to large strands of the literature measuring influences of technical change on the macroeconomy, e.g. Gali (1999). Figure 2 shows the log � 100 of seasonally adjusted hours worked by all dependent workers. It clearly mirrors the downturn of the German labour market over the 1990s and the recovery since 2005 that is interrupted only temporarily by the Great Recession.

Inequality
We choose the Gini coefficient G given by Equation (1) as our measure of wage inequality.
where N denotes the total number of cross-section individuals. G is equivalent to half of the average absolute gross wage difference of all pairs of employees at a certain point in time, divided by the average wage in order to normalize for scale. Thus, the Gini coefficient is a comprehensive inequality measure that takes the whole wage structure into account. It can take on values between 0 (in case every worker earns the same) and nearly 1 (in case all wages go to a single worker). From a different perspective, G equals 2 times the area between the 45° line signifying a perfectly equal wage distribution and the actual wage distribution given by the Lorenz curve. In order to measure inequality, we use the IEB micro dataset that allows for us to collect wage information of 100% of the workers. Details on these data and the way we calculate inequality are provided in Appendix A.
The solid line in Figure 3 shows the seasonally adjusted Gini coefficient. It reveals that the welldocumented upward trend in wage inequality that prevailed for decades has come to an end and even reversed since 2010, a result also found by Weber (2015). The flattening of SBTC (dotted line) is likely to have facilitated this trend reversal. However, this change is clearly not large enough to account for the marked reduction of inequality. This underlines that inequality is driven by other sources as well.

Skill-biased technical change
The main reason why we model SBTC is to capture an important and inherently efficiency-enhancing source of inequality in our model. 1 We measure SBTC as follows: where SBTC is calculated as relative productivity A of high skilled (H) over low skilled (L) workers. σ denotes the elasticity of substitution between both groups, and w H w L is the skill premium. We obtain observations of relative wages and factor supplies from the Sample of Integrated Labour Market  Biographies (SIAB) of the Institute for Employment Research (IAB). Wages are controlled for demographic factors in Mincer (1974)-type regressions. In Appendix B we provide a theoretical derivation and explain in detail how we calculate SBTC from the micro data. The dotted line in Figure 3 shows the development of seasonally adjusted SBTC with σ ¼ 1:7. SBTC is steepest through the 1990s, which is probably connected to general computerization. However, it markedly flattens in the subsequent decade. This could be explained by the phasing out of the first wave of computerization and the fact that the new digitalization wave did not yet start (compare also Beaudry, Doms, and Lewis (2010) for technology waves).

Model setting
Several features of the interdependence of inequality and the labour market require specific traits of the econometric model. First, we are interested in the response of, for instance, hours or productivity to inequality shocks over time; thus, the model needs to be dynamic, covering the short, medium and long run. Second, we want to isolate structural inequality shocks from skill-biased technology (SBT) shocks, which in return must be disentangled from skill-neutral technology (SNT) shocks. This requires a structural model to be identified by statistical and economic reasoning. The presence of technology shocks leads us to form a dynamic structural model with long-run restrictions that do not pre-empt the results with respect to the hypothesized effects.
Regarding the dynamic model, we start with a vector autoregression (VAR) model to let the data speak as flexibly as possible. This has the advantage of capturing very general interaction of the variables without imposing strong structural assumptions a priori. The VAR explicitly models the dynamics in the data with a lag structure up to q þ 1 quarters as follows: where the n ¼ 5 endogenous variables log of productivity (p), log of wages (w), log of hours worked (h), SBTC and inequality (I) are collected in the vector y t . A � i are n � n coefficient matrices capturing their interlinkages and u t is an n-dimensional vector of white noise errors encompassing the variation of the endogenous variables that is not explicitly modelled in the systematic part. As deterministic terms, we allow for a n � 1 vector of constants c 0 and a linear trend.
Augmented Dicky-Fuller (ADF) tests confirm that our variables follow a (stochastic) trend and hence should be treated as non-stationary. This is why we model the variables in first differences. However, as visualized in Figure 1, there could exist a long term relation between p and p (cointegration). In this case, the labour share w is covariance-stationary. Indeed, this is supported by an ADF test for the labour share, which rejects nonstationarity with a p-value of 3%. Therefore, we allow for a level relationship between p and p, which generalizes a first-difference specification. In the cointegration relation, we also allowed for a linear trend just as in Equation (3), which might already be suggested by the time series developments in Figure 1. For instance, also Carbonero, Offermanns, and Weber (2022) find a falling labour share over our sample period in Germany. In case of cointegration, Granger's representation theorem allows us to rewrite the VAR in Equation (3): where β contains the cointegrating vector and α includes the adjustment coefficients.
Þ to allow for the covariancestationary labour share discussed above.

Identification
The VECM in Equation (4) is still in reduced form. In particular, the correlated error terms in u t do not represent economically interpretable innovations. Since we seek to estimate causal dynamic effects, we need to proceed to a structural model. Thereby, the error terms in Equation (4) can be thought of as linear combinations of the structural shocks. Formally, this can be written as where B is an n � n parameter matrix, and e t represents the vector of structural disturbances that are of interest in our analysis, e.g. inequality and SBT shocks. B contains the initial impacts of the shocks on the respective variables, with diagonal elements normalized to be non-negative. In order to estimate economic effects, the structural shocks have to be determined from Equation (5) i.e. an identification problem has to be solved. Evidently, the matrix B in Equation (5) introduces n 2 ¼ 25 unknown coefficients into the model. The variances of e t are normalized to one, and the cross-correlations between the different structural shocks are assumed zero (as is standard in structural VAR models). This reduces the number of unknowns by nðn þ 1Þ=2 ¼ 15, still leaving nðn À 1Þ=2 ¼ 10 restrictions to impose for identification of the structural form. Such an identification scheme guarantees a structural interpretation of the model while the approach leaves a maximum of flexibility in order to let the data speak.
We address this issue by a set of long-run restrictions. 2 From the VECM moving average representation (Johansen 1995), one gets the matrix of the long-run effects of the reduced-form error terms u t : with ? denoting the orthogonal complement (thus, α 0 α ? ¼ 0, where both α and α ? have full column rank). In detail, the ith row of Ξ � contains the longrun impacts of each of the n residuals in u t . Accordingly, the long-run matrix associated with the fundamental shocks e t results in Ξ :¼ Ξ � B. The elements of this matrix equal the structural impulse responses that are reached when the adjustment processes following a shock are finished.
Once the structural coefficients are identified, they provide the basis for the impulse response analysis, which will be presented in Section V. In what follows, we discuss the long-run restrictions imposed in order to disentangle the structural shocks of interest. As noted above, 10 linearly independent restrictions either in B or in Ξ are needed to exactly identify the model. We define � i;j as the long-term effect of the ith variable on the jth structural shock. For the moment, we implement only long-run restrictions to identify the structural shocks, while in Appendix D, we also show the results for an identification scheme with both short-and long-run restrictions. Throughout the paper, we confine ourselves to a minimal set of restrictions, in line with the flexible characteristic of our model.
We are interested in the effects of technology shocks, SBT shocks and inequality shocks. The remaining two innovations are identified as shocks to labour demand (such as fiscal policy) and supply (such as migration), both of which can of course also be affected by the following three shocks, disentangling them by the standard neoclassical assumption that demand shocks have no long term effect on hours. However, in a robustness check (Appendix D), we abstain from this assumption and simply leave the correlation of the w-and h-residuals unidentified. While we are not interested in these shocks, importantly, they cover independent sources of variation in the wage level, thus facilitating the identification of pure wage inequality shocks that concern the wage structure. In any case, the two shocks are assumed to have no longrun impact on productivity (p) and relative skill productivity (SBTC). This is in line with the standards in the growth literature stating that the only relevant long-term drivers of productivity are technology shocks (compare Gali (1999)).
We assume that inequality is affected in the long run only by its own shocks and -since SBTC is a potential source of I -by SBT shocks. Logically, the usual (i.e. skill-neutral) technology shocks do not drive inequality in the long run (just as the other two aggregate level shocks). Structural inequality shocks can occur, for instance, through changes in the employers' hiring preferences that lead to substandard employment, through a growing weight of finance in the economy, also known as financialisation (compare Meyer (2017)), through the introduction or changes of minimum wages, through performance pay in combination with decreasing union density (compare Barth et al. (2012)), through de-regulation of temporary employment, or through labour market reforms in general (e.g. the Hartz reforms, compare Klinger and Weber (2016)). Furthermore, globalization (not necessarily linked to technological change) can have an impact on inequality if it favours particularly the workers in exporting sectors, or if outsourcing and competition from less developed countries put pressure on specific segments of the labour market such as low-skill production jobs.
In sum, on our model's level of aggregation, inequality-driving forces are divided into SBTC and structural inequality shocks, where the latter comprise inequality-relevant factors. Of course, not all of these factors will have exactly identical economic effects, but we aim to identify an overall effect of inequality. Even if one of its factors should have effects strongly different from the overall shock, then at least we can say that this factor cannot be quantitatively important for the development of inequality.
By contrast, SBTC is defined as being driven only by SBT shocks in the long run. Examples could be the widespread usage of computers or robotics at workplaces or other skill-complementing or lowskill replacing technologies. This is in line with explicitly modelling SBTC as source of inequality, the reasoning followed in our modelling framework. Notwithstanding, the constraint � SBTC;I ¼ 0 could be questioned if a higher endowment with high-education workers -also connected to higher inequality ceteris paribus -leads to a bigger market for skill-biased technologies in the long run (i.e. directed technical change, see Appendix B, or if inequality-changing measures (such as the minimum wage) target certain skill groups more than others and hence affect relative productivity over time. Therefore, as a robustness check (Appendix D), we drop the involved long-run restriction and replace it by the respective short-run restriction. This way we explicitly allow both SBTC and inequality to be mutually affected by their respective shocks in the long run.
The third shock of interest is the normal, i.e. skill-neutral, technology shock. This shock is defined as having no long-run impact on SBTC and on inequality, which yields the two remaining constraints required for identification.
The restrictions discussed above can be summarized in the matrix of long-run effects Ξ. Table 2 gives an overview. Appendix Cdescribes the details of our estimation strategy.

Impulse responses
From the structural model, we estimate impulse responses and confidence intervals using the bootstrap of Hall (1992) with 2.000 replications. Figure 4 shows the impulse responses for a horizon of 16 quarters together with 2/3 confidence intervals. We consider 1 unit shocks. As all variables were multiplied by 100, this implies a technology shock connected to an immediate 1% productivity impact, an SBT shock connected to an immediate 1% impact on SBTC (i.e. the relation of the factor-augmenting technology terms of the high-and low-skilled) and an inequality shock connected to an immediate impact of 1 point on the Gini coefficient (scaled between 0 and 100).
As expected, skill-biased technology shocks increase productivity (Figure 4, upper middle panel). Along with productivity, wages also rise (central panel). However, hours worked are clearly reduced by the SBT shock (lower middle panel). This is consistent with high-skilled workers being more productive than low-skilled workers. Then, if the relative demand for high-skilled is increased, The table shows the identifying assumptions regarding the long-run effects in our structural macroeconometric model. � i;j is defined as the long-term reaction of the i th variable to the j th structural shock. SNT: skill neutral technology, SBT: skill-biased technology. (0): � w;D ¼ 0 and � w;S ¼ 0 are no additional binding restrictions, but zeroes that follow from the cointegration properties.
fewer hours are required to produce a given output. Put differently, the income effect of SBTC seems not to offset the displacement or substitution effect (compare Moore and Ranjan (2005)). In other words, if less productive workers are substituted with more productive ones following an SBT shock, total hours can decrease, while their production impact rises. By the same token, one can think of SBTC invoking skill mismatch and thus longlasting structural unemployment (Restrepo 2015). There is only a weak rebound visible in the impulse response, which could reflect reallocation of labour following an initially distortionary shock. In this context, one might hypothesize that the adjustment to technical change in the German labour market has been limited due to sclerotic structures. However, the labour market reforms in 2003-2005 could have changed this by improving flexibility and reallocation capacity. Indeed, when estimating the model only until 2002, we measure an even more negative reaction of hours to SBT shocks without any rebound. Logically, the period after the reforms is inclined to more advantageous SBT effects. Furthermore, we find that SBT shocks positively affect inequality. This confirms the role of SBTC as source of inequality and can be taken as a plausibility check for our identification scheme. A 1% shock to SBTC increases the Gini coefficient (scaled between 0 and 100) by about 0.011. While this value seems rather limited, the large range of the SBTC variable over the sample (Figure 3) compared to the other variables must be taken into account. In fact, SBTC increased from 1978 (the inequality minimum) until 2009 (the inequality maximum) by 215.4 points. We can calculate a counterfactual in the model by hypothetically neutralizing this rise by negative SBT shocks summing to the same size (which are the only ones affecting SBTC in the long run according to our identification scheme). These additional SBT shocks affect inequality via the long-run impulse response estimated above, resulting in a total effect of À 2:40 points. In other words, without SBTC, the Gini coefficient would have increased only by 3.16 instead of 5.57 points. Still, the better part of the increase of wage inequality in Germany cannot be traced back to technology but to structural inequality shocks. This is in line with a recent finding of Kristal and Cohen (2017) for the US.
Skill-neutral technology shocks naturally increase productivity and wages (Figure 4, upper left and middle left panel), the latter partly with delay. Notably, we also find an increase for hours worked (lower left panel). This positive effect following a 1% technology shock is weak in the short run but increases until the fourth quarter to about 0.4%, before it again reverses. It goes hand in hand with the abovementioned delayed wage reaction and is in line with results from Christiano, Eichenbaum, and Vigfusson (2004), amongst others. However, it stands in contrast to the persistent negative effects reported in Gali (1999) and the subsequent literature. In this context, note that these latter results are based on a single technology shock that implicitly captures both skill-neutral and skill-biased technology shocks (compare also Balleer and van Rens (2013)). The hours effect of the latter has already been shown to be negative above. Logically, responses to overall (intermingled) technology shocks will tend more to the negative area. Indeed, if we eliminate SBTC and inequality from the system and thus estimate a small standard model, the response of hours to the technology shock is negative on impact and insignificant in the following. However, the positive hours effect reached in the complete model is more in line with the expectation from standard search and matching theory that plain productivity shocks foster vacancy creation and therefore employment. This emphasizes the advantage of an approach that disentangles skillneutral and skill-biased technology shocks in order to allow for different effects on the labour market.
As SBTC, structural inequality shocks have a negative impact on hours worked (Figure 4, lower right panel). In addition, they reduce productivity (upper right panel) and wages (middle right panel). These variables drop by 0.57%, and hours by just under half as much, following a shock of one point in the Gini coefficient scaled between 0 and 100. This implies that relevantly sized employment and productivity impacts appeared in the past: From our structural system, we can calculate a historical decomposition which mirrors the contributions of the structural shocks to the stochastic part (i.e. beyond deterministics) of our variables of interest. Figure 5 shows the contributions of inequality shocks to productivity and hours, respectively. Focussing on the period with the strongest inequality increase, 1997-2009, these shocks decreased productivity by 3.3% and hours by 1.2%. By contrast, falling inequality since 2010 strengthened productivity by 1.9% and hours by 0.6% and hence contributed to the strong labour market upswing in Germany.
The results indicate that inequality has adverse impacts on the labour market as implied by the opportunities hypothesis. However, the fact that the responses materialize rather fast supports the reasoning that besides a mechanism via educational investment, channels via low quality of jobs, lacking development perspectives, weak participation in further training or fairness beliefs prove relevant, just as theories considering frictions and market failures. Moreover, there appear to be no counterbalancing effects in terms of efficiency (i.e. productivity) gains, quite the opposite. The finding that SBT shocks have positive and inequality shocks have negative effects on productivity confirms our argument that it is important to model SBTC as an inherently efficiencyenhancing source of inequality separately. In sum, the investigation implies that higher inequality harms employment and productivity in Germany. Naturally, as in all empirical models, these results must not be extrapolated too far beyond the range of observed data. For instance, one cannot infer that complete equality would bring about the most beneficial effects. A series of robustness checks is provided in Appendices D and E.

Inequality below and above the median wage
A comparison of the respective origins of the incentive hypothesis and the opportunities hypothesis leads to the conclusion that the latter has been designed mainly for developing countries since it explicitly addresses the opportunities of the poor. Barro (2000) and Castelló-Climent (2010), for instance, investigate the effects of inequality on growth separately for rich and poor countries and find the relationship to be positive in the former and negative in the latter (compare also Bandyopadhyay and Basu (2005) for a theoretical model on this question). Transferred into the context of industrialized countries, this could imply that the two contradicting hypotheses are not equally important in different parts of the wage distribution (see, e.g. Voitchovsky (2005)).
To shed more light on this issue, we calculate inequality below ('lower inequality', I l ) and above ('upper inequality', I l ) the median wage. This is done by applying Equation (1) separately to all individuals earning less or more than the median wage. Figure 6 shows that the increase in total inequality seems to be driven mainly by an increase in wage dispersion above the median (I l ), at least until the mid-nineties, while the marked decrease in inequality since 2010 comes from reduced wage dispersion below the median (I l ). The two different dynamics by which total inequality is driven raises the question whether the effects of inequality are more pronounced if it occurs in a certain half of the wage distribution.
To investigate potential differing effects, we add the two inequality variables, I l and I u , in our model (in place of overall inequality). Thereby, we model no causal effects between the two inequality measures. While their residuals can be correlated, it appears plausible to trace this correlation back to common factors rather than to bilateral spillover effects. Technically, this is implemented by allowing for correlation between the structural residuals P I l and P I u , which does not influence the impulse responses. Figure 7 shows the resulting responses to the respective structural shocks. The negative labour market effects of inequality shocks are prevailing for both upper and lower inequality. However, the latter has stronger (negative) effects on productivity, wages and hours than overall inequality. An explanation could be connected to the relevance of the opportunities and incentive hypotheses for higher and lower wage groups. The former points to the situation of low-income earners: inequality can impede opportunities to participate on educational advancement and create a sector of persistent low-quality jobs with limited productivity dynamics. Therefore, the opportunities hypothesis is likely to strengthen the adverse effects of lower inequality. In contrast, the incentive hypothesis might be more relevant for higher income jobs, where career paths and development chances are more prevalent. Logically, this would dampen the negative effects of I u .
These findings also relate to the recent debate on the consequences of increasing top income inequality (see, e.g. Auclert and Rognlie (2017)). Although we find a pronounced increase in inequality at the top of the wage distribution, our impulse response analysis clearly shows that much more damaging effects on the labour market would have resulted if this increase in inequality had taken place at levels below the median wage.

Effects at high and low levels of inequality
Inequality is often thought of as influencing the economy differently depending on its level (e.g. Voitchovsky (2005), Galor and Moav (2004) and Banerjee and Duflo (2003)). In detail, the balance of different effects might change with the overall level of inequality. If rather low levels prevail, higher inequality might have positive effects via higher incentives or employment chances. In contrast, once inequality lies too high, further increases might be harmful when educational opportunities are narrowed, problems in lower segments in two-tier labour markets become urging and political opposition grows. Then, the negative effects would dominate.
When considering the development of wage inequality in Figure 3, a strong increase can be observed from 1997 onwards. Therefore, we conduct two separate estimations for the samples 1975-1997 and 1997-2014. Comparing the impulse responses of productivity, wages and hours in each of the two subsamples, it becomes evident that inequality shocks are more Figure 7. Responses of p, w and h to I l and I u shocks. The solid lines show the responses of productivity (upper panels), wages (middle panels), and hours (bottom panels) to 1 unit shocks in wage inequality below (left panels) and above (right panels) the median wage up to 16 quarters. The dotted lines denote Hall (1992)'s 2/3 bootstrapped confidence intervals.
harmful in the second period where the inequality level was higher. A one unit shock to the Gini coefficient lowers productivity and wages by 0.59%, slightly more than in the overall sample. The hours reduction due to structural inequality shocks is −0.78% and hence three times stronger compared to the effect estimated for the whole sample.
In contrast, in the first subsample inequality shocks have more muted effects: they reduce productivity and wages by just above 0.3%. The effect on hours worked is even positive but statistically insignificant. Of course, for evaluating the effects of potential future inequality changes, the estimates from the later period are relevant. This implies that further increases in inequality might be quite harmful. In contrast, employment could benefit from a further reduction of inequality that undoes the increases during the second sample half.

VI. Conclusion
The underlying study analyses the effects of inequality, skill-neutral and skill-biased technical change (SBTC) on the economy and the labour market. We explicitly model SBTC as a source of inequality to isolate structural inequality shocks. We put forward a dynamic cointegrating framework with theory-based (short-and) long-run restrictions for identifying the impacts of inequality on productivity, wages and hours worked.
A structural impulse response analysis revealed that skill-biased technology shocks increase productivity and wages but reduce hours worked and raise inequality. Structural inequality shocks also have a negative impact on hours worked but additionally reduce productivity. These adverse effects are stronger in the second half of the sample that is characterized by higher inequality levels. We find that inequality has negative labour market effects irrespective of whether it occurs below or above the median wage. Below, however, these negative effects are clearly stronger. Hence, it is worth noticing that, while recent debates focus more on top income inequality, inequality at the bottom can have much more damaging labour market effects according to our results. Furthermore, we can show that skill-neutral technology shocks have a positive long-run effect on hours worked. In general, the results indicate that inequality has negative impacts on the labour market as implied by theories in line with the opportunities hypothesis (cf. Galor and Zeira (1993)) or theories allowing for market failures and frictions (cf. Acemoglu and Pischke (1999)). Moreover, there appear to be no counterbalancing effects in terms of efficiency (i.e. productivity) gains, quite the contrary.
The results imply that the rising wage inequality in Germany since the 1990s should not be seen as a precondition for the German labour market upswing of the recent ten years. Instead, higher inequality appears to harm employment and productivity. The employment upswing is more likely connected to those components of the reforms that aimed at enhancing the efficiency of the labour market functioning (compare Hutter et al. (2022), Launov and Wälde (2016), and Klinger and Weber (2016)) and to other factors such as the upward trend of the service sector and high immigration in recent years. Wage moderation as such could also have played a role in strengthening labour demand, but according to our analysis, wage inequality was an obstacle to labour market development. From an international point of view, the results have two important implications: First, they question in how far German policies fostering a low-wage sector e.g. by deregulation and increasing pressure on unemployed should internationally serve as a role model -a view that has become common during the European crisis. And second, they underline that limiting inequality in Germany would strengthen domestic demand and therefore contribute to a reduction of current disequilibria in international trade. Indeed, the declining inequality since the end of the Great Recession is likely to have contributed to expanding employment during a period where slower employment growth due to the phasing-out of the Hartz-reform effects was already expected -and growth of the German current account balance was flatter than before and even negative after 2015.
In disentangling the effects of skill-biased and skillneutral technology shocks (see also Balleer and van Rens (2013)), our analysis contributes to a more comprehensive understanding of the relationship of technology and the labour market (see, e.g. Christiano, Eichenbaum, and Vigfusson (2004), Gali (1999)). Regarding future technological developments, the new wave of intelligent and interconnected digitalization dominates the debate. Clearly, these changes may involve skill bias; compare e.g. Wolter et al. (2016) predicting a rise of qualification needs for Germany. On the one hand, this can be taken as a warning signal for the risk of negative employment effects. This would underline the key role of qualification. On the other hand, our results show that a new wave of SBTC would contribute to overcoming the prevalent productivity slack.
The general construction of the model framework paves the way for further economic analyses of inequality. For instance, the results concerning inequality below or above the median wage could be generalized using the Zenga index and curve which allow for measuring inequality in any part of the wage distribution (e.g. Zenga (1990)). In addition, measuring economic effects of inequality based on data from other countries could shed light on the degree of generality or conditionality of the results. Since rising inequality as in Germany has been a common feature among many countries facing similar trends of deregulation and technological change, these further insights can be expected to contribute to a better understanding of the linkage of economic, political and technological developments. Moreover, the functional form could be extended in order to capture potential nonlinearities in the relationship of inequality and labour market outcomes. Finally, additional differentiation in modelling inequality shocks could elaborate further on the concrete mechanisms at work.
The goods Y L and Y L are produced with unskilled labour, Y H , and skilled labour, Y H , respectively. σ denotes the elasticity of substitution between Y L and Y L .
At the sector level, Y L and Y H are produced by competitive firms as follows: where w L and w H are the wages for the unskilled and skilled workers, respectively. Monopoly power ensures that the prices for the intermediate goods have a markup over the marginal cost, which allows for innovation to be driven by monopoly profits. Innovation comes in form of the introduction of new varieties of intermediates, i.e. A H or A L grows.
With the standard assumption of labour market clearing (which leads to lðiÞ ¼ L=A L and hðiÞ ¼ h=A H ), the skill premium w H =w L reads as follows: Note that σ denotes the elasticity of substitution both between the final goods (Y L =Y H ¼ ðP H =P L Þ σ ) and between the respective factor supplies (L=H ¼ A H =A L ðP H =P L Þ σ ). The skill premium is decreasing in the relative supply of skilled labour and increasing in relative skill productivity A H =A L if σ > 1. It can be shown that on the balanced growth path, the skill bias of technological progress can be written as follows: Equation (11) reveals that market size is the fundamental determinant of technological progress, which is biased towards the abundant factor if σ > 1. Inserting Equation (11) into Equation (10) yields the skill premium at the balanced growth path: The relationship between relative wages and relative labour supply can either be positive or negative. On the one hand, a large supply of one factor depresses its price. On the other hand, a large supply of one factor induces a technology bias in its favour, thereby raising its productivity. The latter effect dominates when σ > 2. Taking the natural logarithm on both sides of Equation (10) and solving for relative skill productivity yields: Since the state of the economy is certainly not on the balanced growth path at every point in time, we do not use Equation (11) to measure SBTC. Instead, we make use of Equation (13), which is also valid beyond the balanced growth path. In particular, it is important to understand the SBTC measure as a typical shock-driven variable that is allowed to be hit by SBTC-relevant changes exogenous to our model. This way, just as it is the case for inequality, it is possible to interpret the structural disturbances obtained by our SVEC model as economically meaningful innovations and to conduct a historical decomposition shedding light onto their relevance over time. Notwithstanding, Appendix E.explicitly incorporates as robustness check an additional potential factor, union power, that especially for Germany is often supposed to have effects on wage development.
Appendix B2. Measuring SBTC According to Equation (13), measuring SBTC requires relative labour supply, the skill premium, and an estimate of σ, the elasticity of substitution between high-and loweducation workers. To estimate the first two ingredients, we use the Sample of Integrated Labour Market Biographies (SIAB). This dataset provides detailed information about an individual's employment history on the German labour market. Basically, SIAB is a 2 percent random sample of the population collected in the Integrated Employment Biographies (IEB) that comprises all (un)employed persons in Germany between 1975 and 2014.
Concerning the labour supply, we use full-time and parttime workers. Appendix Dshows robustness no matter how one deals with the issue of not having hourly wages for parttimers available: One can exclude part-timers only for the calculation of w H and w L , but include them in H and H (our baseline version). Or one can calculate SBTC with full-time workers only. We consider all employees and unemployed (including participants of active labour market policy measures) with completed vocational training or higher education as being high-skilled and all workers without completed vocational training or high school degree as being lowskilled (H and H in Equation (13)). At first, this classification seems to differ from the college vs. no college perspective. However, for the German case, we find it appropriate due to the special role of the dual system of vocational training in Germany (compare Müller and Wolbers 2003). Indeed, it comprises the main part of jobs that require a college degree in other countries. Furthermore, the permeability and cooperation between the vocational training system and tertiary education has increased substantially (compare Wolter and Kerst 2015). In addition, the unemployment rates of workers with vocational training are far closer to the academics than to those without any degree (compare Röttger, Weber, and Weber 2018). In a sense, our approach is similar to the unskilled/skilled classification used in Dustmann and Meghir (2005). Shifts in the labour supply variables in 1992 (reunification) and 2005 (statistical effects of the Hartz reforms) were adjusted in ARIMA models with dummies.
Concerning the skill premium, we rely on the wage information from full-time workers for the reasons described above. 4 In case of multiple employment, only reports of the main job are included. To calculate the skill premium, one must estimate, for each period of time, the average wages for high-education and low-education workers, after controlling for demographic factors. For this purpose, we first run monthly Mincer (1974)-type regressions of the individual's log wage on age, squared age, seniority, squared seniority and dummies for gender, nationality and East Germany, which also controls for composition effects. 5 Note that non-SBTCrelated causes for wage differentials are controlled for, whereas variables such as education, sectors or firm size are intentionally left out in the Mincer-regressions. This fits the needs of our analysis since alongside these dimensions, SBTC unfolds its distortive characteristic. 6 The resulting residual wages for high-education and low-education workers are used to calculate w H and w L of Equation (13).
There is broad consensus in the literature that the elasticity of substitution between high-and low-education workers, σ, ranges between 1.4 and 2. Katz and Murphy (1992), for instance, find a value of σ � 1:4 for US data, whereas Angrist (1995) results on Palestinian skill premia imply σ � 2. Möller (2000) finding of σ � 1:7 for German data naturally will be our preferred estimate for this study. This value is also in accordance with other studies (see, for instance , Hamermesh 1993;or Bound and Johnson 1992). The robustness checks in Appendix Dreveal that the resulting impulse responses are robust with respect to different elasticities of substitution.
Consistent with all the other variables in our model, also SBTC was seasonally adjusted, and statistical structural breaks (such as the reunification break) were eliminated.
• We choose a rich parameterization by doubling the lag length to q ¼ 4. Figure A1 shows the resulting impulse responses of p, w and h to inequality shocks, compared to the baseline model described in Section IV. The reactions of p and w are slightly stronger when using σ ¼ 1:4 for measuring SBTC or when allowing for directed technical change. The reactions of h are marginally weaker for σ ¼ 2:0 and for the setting with directed technical change and -again -slightly stronger for σ ¼ 1:4. The reactions are virtually unaffected by dropping the neoclassical assumption � h;D ¼ 0. Calculating SBTC with full-time workers only does not change the long-run reactions of p and w to inequality shocks. However, the effects on h are a bit stronger. With the higher lag length of q ¼ 4, the reactions of p, w and h to inequality shocks are strengthened (by about 1/3) as compared to the baseline version. In total, Figure  A1 shows that the results are robust to alternative settings and identification schemes, or at any rate not weakened.

Appendix E. Union power
The theoretical approach in Appendix B resembles Solow (1957) way of residually quantifying factor-neutral technical change from measures of aggregate output, capital and labour and an estimate of the elasticity of output to capital. The approach can be broadened to cover an additional potential factor, union power, that especially for Germany is often supposed to have effects on wage development. To investigate this issue, we use data from the IAB establishment panel. They allow for calculating qualification-specific figures on the share of workers covered by collective labour agreements. Although these data are available on an annual frequency and only since 1997, they cover periods when SBTC was still increasing strongly as well as periods when it was flattening. Indeed, Figure A2 confirms that the share of workers subject to collective labour agreements has decreased substantially from 82 percent in 1997 to 59 percent in 2014 (solid line). However, the crucial point that matters for the measurement of SBTC is that high skilled (triangles) and low skilled (rectangles) are affected rather equally by this development. As a consequence, relative union power (coverage of low skilled over coverage of high skilled) has stayed rather constant over time, at values between 0.94 and 0.99 (dotted line).
Enhancing the endogenous growth model of Appendix B with union power in both skill sectors, μ H and μ L , Equation (10) becomes: Solving for A H =A L and converting annual values of μ L μ H to quarterly data by linear interpolation, the resulting alternative SBTC time series has a very similar development compared to that in Figure 3. Furthermore, the impulse responses for the sample from 1997 to 2014 are also similar to those described in Subsection 5.3: Long-run effects of SBT shocks on productivity and wages are positive, on hours worked negative. A one unit shock to the Gini coefficient lowers productivity and wages by 0.63 percent, the hours reduction due to structural inequality shocks is -0.58 percent. Hence, we conclude that union power introduces no important bias in our results for SBTC (but, of course, could still drive the inequality shock, compare Stennek 2019).