Minimum wages in an automating economy

Eckardt, Marcel Steffen Article — Published Version Minimum wages in an automating economy Journal of Public Economic Theory Provided in Cooperation with: John Wiley & Sons Suggested Citation: Eckardt, Marcel Steffen (2021) : Minimum wages in an automating economy, Journal of Public Economic Theory, ISSN 1467-9779, Wiley, Hoboken, NJ, Vol. 24, Iss. 1, pp. 58-91, https://doi.org/10.1111/jpet.12528 This Version is available at: https://hdl.handle.net/10419/284799 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/ J Public Econ Theory. 2022;24:58–91.58 | wileyonlinelibrary.com/journal/jpet Received: 10 November 2020 | Revised: 9 April 2021 | Accepted: 16 May 2021 DOI: 10.1111/jpet.12528 ORIGINAL ARTICLE Minimum wages in an automating economy Marcel Steffen Eckardt Department of Law and Economics, Technical University of Darmstadt, Darmstadt, Germany Correspondence Marcel Steffen Eckardt, Department of Law and Economics, Technical University of Darmstadt, Hochschulstraße 1, D‐64289 Darmstadt, Germany. Email: [email protected] Abstract We explore the suitability of the minimum wage as a policy instrument for reducing emerging income inequality created by new technologies. For this, we implement a binding minimum wage in a task‐based framework, in which tasks are conducted by machines, low‐skill, and high‐skill workers. In this framework, an increasing minimum wage reduces the inequality between the low‐skill wage and the other factor prices, whereas the share of income of low‐skill workers in the national income is nonincreasing. Then, we analyze the impact of an automating economy along the extensive and intensive margins. In a setting with a minimum wage, it can be shown that automation at the extensive margin and the creation of new, labor‐ intensive tasks do not increase the aggregate output in general, as the displacement of low‐skill workers counteracts the positive effects of cost‐savings. Finally, we highlight a potential trade‐off between less inequality of the factor prices and greater inequality of the income distribution when a minimum wage is introduced into an automating economy. KEYWORDS automation, displacement effects, employment, inequality, labor demand, minimum wage, tasks, wages JEL CLASSIFICATION E25, J20, J31, J38 This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. © 2021 The Authors. Journal of Public Economic Theory published by Wiley Periodicals LLC 1|INTRODUCTION In the last decades, new technologies, such as artificial intelligence and robots and other forms of automation, have been rapidly developing. These new technologies will likely have a significant impact on the economy. In particular, the labor market will change fundamentally in the future (e.g., Brynjolfsson & McAfee, 2014; Ford, 2015). Frey and Osborne (2017) explore the susceptibility of jobs in relation to automation and estimate that approximately 47% of current jobs in the United States (US) could be automated within one or two decades. Empirical studies show that automation has a substantial impact on routine tasks, leads to a polarized labor force, and increases inequality in the economy (e.g., Acemoglu & Restrepo, 2020a; Autor, 2015; Autor & Dorn, 2013; Autor et al., 2003,2015; Goos & Manning, 2007; Graetz & Michaels, 2018). Moreover, Goos et al. (2019) emphasize that the adjustment costs from automation on unemployed job seekers are unequally distributed between low‐skill and high‐skill workers. To reduce the emerging inequality various policy instruments, such as taxing robots, a basic universal income, or a minimum wage, are discussed (e.g., Acemoglu et al., 2020; Costinot & Werning, 2018; Freeman, 2015; Furman, 2019; Guerreiro et al., 2017; McAfee & Brynjolfsson, 2016; Thuemmel, 2018). Little is known, however, about the effects of the minimum wage in conjunction with automation. In one of the few existing studies, Lordan and Neumark (2018) empirically show that higher minimum wages reduce employment in automatable jobs. Moreover, they emphasize that there are groups of workers, such as older and less‐skilled workers, that are frequently ignored in the empirical literature on the effects of minimum wages. However, it appears that scarcely any theoretical work exists on the effects of a minimum wage in a task‐ based framework, in which tasks are increasingly conducted by machines replacing low‐skill workers. One exception is the work by Aaronson and Phelan (2019), who develop a theoretical framework based on tasks to test for the labor market consequences of minimum wages. The aim of this paper is to explore the effects of a binding minimum wage on aggregate output, employment, factor prices, and various measures of the income distribution in an automating economy. To analyze the labor market effects of a minimum wage in conjunction with automation, we build on the work of Acemoglu and Restrepo (2018a,2018b,2018d) and that of Acemoglu and Autor (2011), both of which are interconnected, and are based on Zeira (1998) and Acemoglu and Zilibotti (2001). A task‐based framework takes up the notion of labor markets that can be empirically characterized by the task content of jobs (e.g., Goos et al., 2019). Theoretically, a task‐based framework allows us to model automation along intensive and extensive margins (Acemoglu & Restrepo, 2018c), also with respect to the effects that may arise in the interplay with the introduction of a minimum wage. In our task‐based framework, tasks in a unit interval are conducted by machines, low‐skill, and high‐skill workers. The range of tasks that machines and low‐skill workers can produce is bounded by exogenous thresholds. The assumption of comparative advantage for each production factor on a subset of the tasks leads to a simple allocation of the factors. Hence, our task interval is divided into three intervals with increasing complexity, where machines produce the tasks in the first interval, low‐skill workers produce the tasks in the middle interval, and high‐skill workers produce the tasks in the last interval. By assuming a fixed and inelastic supply of machines, low‐skill, and high‐skill workers, we implement a minimum wage that is higher than the equilibrium low‐skill wage and determine the new equilibrium. ECKARDT | 59 Under an automating economy, we understand the impact of technological progress on changing exogenously the thresholds of tasks, the productivity of production factors, and the task interval. More precisely, we consider the consequences of increasing automation, whereby we distinguish between automation at the extensive and intensive margins. Automation at the extensive margin increases the measure of tasks that machines are able to produce whereas automation at the intensive margin (which we call the “deepening of automation”) means that the productivity of machines on the tasks increases (e.g., by replacing older machines with newer ones). Analogously to the increasing automation, technological progress can enable low‐ skill and high‐skill workers to raise their productivity in their tasks and their range of tasks. Furthermore, technological progress can lead to the creation of new, more complex tasks that are labor‐intensive, as assumed above. First, we analyze the effects of an increase in the minimum wage and show that the low‐ skill employment, and thus the aggregate output, decrease. Moreover, a higher minimum wage reduces the inequality between the low‐skill and the high‐skill wage, as well as the rental rate of machines. In other words, machines and high‐skill workers become relatively cheaper than low‐skill workers: this can lead to a displacement of low‐skill workers wherein the share of their income in the national income decreases. Furthermore, the expected low‐skill wage decreases. Here, we consider the expected low‐skill wage as a group‐specific welfare measure of low‐skill workers, as the income of the low‐skill workers who retain their job increases, whereas the income of those who become unemployed falls to zero. Subsequently, we explore the effects of an automating economy in the presence of a minimum wage. We start by increasing automation at the extensive margin. Machines displace low‐skill workers if they are relatively cheaper in performing the new tasks. On the one hand, this increases the aggregate output that has a positive impact on the labor demand. On the other hand, the displacement of low‐skill workers from the tasks, which are automated, reduces the demand for low‐skill workers and, consequently, the low‐skill employment that has a negative impact on the aggregate output. We observe two counteracting effects of the displacement of low‐skill workers on the aggregate output and on low‐skill employment. The magnitude of these effects decides whether the impacts on the aggregate output and low‐skill employment are positive or negative. In particular, it is possible that the impact on the aggregate output is positive and the impact on low‐skill employment is negative. Therefore, in contrast to Acemoglu and Restrepo (2018a,2018b,2018d), the impact on aggregate output and low‐skill employment is ambiguous whenever there is a displacement of low‐skill workers in the economy with a binding minimum wage. We show that automation at the intensive margin raises the aggregate output, the rental rate, and the high‐skill wage. Consequently, as the low‐skill wage is fixed by the minimum wage, the inequality between the low‐skill wage and the other factor prices increases. Furthermore, low‐skill workers can displace high‐skill workers in some cases and this can be greater than the displacement of machines. This is contrary to the model without a minimum wage (e.g., Acemoglu & Restrepo, 2018a). Moreover, the sign of the effect of the deepening of automation on the employment is ambiguous. Analogously, we can consider an increase in the productivity of low‐skill or high‐skill workers. If the productivity of high‐skill workers increases, the mechanisms are similar and lead to the same results. If the productivity of low‐skill workers increases, their employment increases and the share of income of low‐skill workers in the national income is nondecreasing, as low‐skill workers can displace machines and high‐ skill workers in some cases. 60 | ECKARDT Moreover, we consider the effects of an expansion of the range of tasks that low‐skill workers are able to perform. We show that this expansion raises the aggregate output, the employment, and the share of income of low‐skill workers in the national income if low‐skill workers displace high‐skill workers by expanding their skills. The sign of the effect on the ratio between the low‐skill and the high‐skill wage is ambiguous. By analyzing the impact of the creation of new tasks, we show that the impact on the aggregate output and low‐skill employment is ambiguous, similar to the behavior in the case of automation at the extensive margin. For the same reason as before, this result also differs from the model without a minimum wage (Acemoglu & Restrepo, 2018a). Our contribution relates to a large literature that theoretically explores the labor market effects of minimum wages in the standard neoclassical model (e.g., Borjas, 2012), emphasizing the possible employment‐enhancing effects of minimum wages in monopsony (e.g., Manning, 1995; Robinson, 1969), or focusing on the interplay of minimum wages with search frictions (e.g., Burdett & Mortensen, 1998; Pissarides, 2000). For the most part, minimum wage effects are analyzed with respect to employment, but there are also contributions that look into more encompassing welfare measures (e.g., Gerritsen & Jacobs, 2020; Lavecchia, 2020). We contribute to the existing literature by focusing on the interplay of minimum wages and automation in a task‐based framework. In addition to showing the effects of a minimum wage on employment, we also analyze the effects of a minimum wage on various measures of wage and income distribution. There is only a limited number of contributions to the literature on the effects of minimum wages using a task‐based framework. Building on the task‐based framework of Autor et al. (2003), Aaronson and Phelan (2019) explore the effect of an increase in the minimum wage on the employment of routine and nonroutine workers. They show that an increase in the minimum wage decreases the employment of routine workers, who are either displaced by nonroutine workers or capital. Chu et al. (2020) endogenize R&D and automation to analyze the impact of a minimum wage on unemployment, high‐skill workers, economic growth, R&D, and automation in a Schumpeterian growth model. Our contribution differs from the aforementioned approaches to analyze the effects of a minimum wage in a task‐based framework. Our tasks do not distinguish between routine and nonroutine work, but may be conducted by machines, low‐skill, or high‐skill workers. Moreover, R&D is not endogenous in our model, but automation may occur in different forms, which yields distinct labor market effects when a minimum wage is introduced. In particular, this paper explores, in contrast to the above‐mentioned papers, the effects of five different types of automation on aggregate output, employment, factor prices, and various measures of the income distribution in the presence of a minimum wage. The paper is organized as follows. In Section 2we introduce our model with a binding minimum wage. The comparative statics are explored in Section 3, while Section 4discusses generalizations. In Section 5we conclude. Details of the proofs can be found in Appendix Aand the Online Appendix. 2|THE MODEL This section introduces the theoretical framework for exploring the economic consequences of introducing a minimum wage in an automating economy. We use a task‐based framework similar to Acemoglu and Autor (2011) and Acemoglu and Restrepo (2018a,2018b). ECKARDT | 61 The unique final good Y is aggregated by combining the outputs of tasks in a unit measure task interval NN [ −1, ] , ≥ N1 , according to a Cobb‐Douglas function that is given by ∫ () Yyxx= exp ln ( ) d , N N −1 (1) where yx() denotes the output of a task x . We postulate higher‐indexed tasks as more complex. This enables us to model the creation of new, more complex tasks by shifting our task interval through an increase in N (Acemoglu & Restrepo, 2018a,2018d). The final good is produced by machines (capital), low‐skill, and high‐skill workers in a competitive market. We assume that the supply of machines, denoted by K , as well as the supply of low‐skill and high‐skill workers, denoted by L and H, respectively, are positive, fixed, and inelastic. Moreover, there are thresholds I and S (with N IS N −1< < < ), where I denotes the technological frontier of automation and S denotes the different abilities between low‐skill and high‐skill workers. High‐skill workers can produce each task. The following output function summarizes our previous assumptions: ⎧ ⎨ ⎪ ⎩ ⎪ ∈ ∈ ∈ yx γxhx γxlx γxkx x N I γxhx γxlx x IS γxhx x SN ()= () ()+ ()()+ () () if [ −1,], () ()+ ()() if (, ], () () if (, ], HLK HL H where γx( ) i,∈ i KLH{,, } , denotes the productivity function of the corresponding factor and kx( ) , l x() ,hx() denote the demand for machines, low‐skill, and high‐skill workers, respectively, for some task ∈xN N[−1, ] . The next assumption introduces comparative advantages into our model. Assumption 2.1 (Comparative advantage). Assume that the productivity functions γi , ∈ i KLH{,, } , and the ratios ≔γγ γ 1 L K and ≔γ γ γ 2 H L are continuously differentiable and strictly increasing. Moreover, assume the following domains and ranges: ⎡ ⎣ ⎡ ⎣ ∞→ ∞ ∞→ ∞ ) ) γγγγ γ ,, ,:0, (0, ), :0, (1,). KLH1 2 1 The increasing productivity functions correspond to the postulation of the increased complexity of higher‐indexed tasks. The properties of the ratios imply that high‐skill workers have a comparative advantage relative to machines and low‐skill workers in higher‐indexed tasks, as well as that low‐skill workers have a comparative advantage relative to machines in higher‐ indexed tasks. This Assumption 2.1 also simplifies the allocation of tasks with respect to all production factors within the task interval. The feature that γ 2 is greater than 1 will guarantee that the high‐skill wage is always greater than the low‐skill wage. We denote the equilibrium rental rate (or the cost of machines) by R , the equilibrium low‐ skill and high‐skill wage by W Land WH , respectively. Moreover, we introduce a binding minimum wage ∈ W WW(, ) mL0. 2 1 For our considerations, it is not necessary that the productivity functions γγγ,, KLH are increasing. Moreover, this assumption is not necessary, but it makes the comparative statics easier. 2 The low‐skill wage is bounded above by W0 (see Appendix A). 62 | ECKARDT Appendix Ashows (under the assumption that we choose the final good as numeraire) that for any ≥ K LH N IS,, >0, 1,, with N IS N −1< < < there is a unique equilibrium that is characterized by thresholds IS ( *,* ) with N IS N −1< *<*<, the demand for low‐skill workers ⎛ ⎝ ⎜⎞ ⎠ ⎟ ⋅⋅ () LB WSI=1*−* , m m SI 1 1− *+* (2) and the aggregate output ⎛ ⎝ ⎜⎞ ⎠ ⎟ ⋅YB W =1 , m SI SI *−* 1− *+* (3) where ⎜⎟ ⎜⎟ ⎜⎟ ⎛ ⎝ ⎜⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ ⎞ ⎠ ⎟ ∫∫∫ ⋅⋅B γxx γxx γxx K IN H NS = exp ln ( ) d + ln ( ) d + ln ( ) d *−+1 − * . N I KI S LS N H IN NS −1 * * * * *−+1 −*SI 1 1− *+* Moreover, the rental rate, the minimum wage, and high‐skill wage satisfy the equations: ⋅⋅⋅RY IN KWY SI LWY NS H =*−+1 ,= *−*,= −* . m m H(4) The market‐clearing conditions are given by ∫∫∫ K kxx LL lxx H hxx= ( )d , > = ( )d , = ( )d . N I mI S S N −1 * * * *(5) Finally, Appendix Aindicates that the machines produce the tasks within NI [ −1, *], the low‐skill workers within IS ( *,*], and the high‐skill workers within SN ( *,] . The measure of these sets are the respective shares in the national income si,∈ i KLH{,, } , namely: ⋅ ⋅ ⋅ sKR YIN sLW YSI sHW YNS == *−+1, == *−*, ==− *, K Lmm HH (6) where the terms on the right‐hand side are obtained from (4). Figure 1illustrates the equilibrium outcomes of our model. Computing the equilibrium in our model without minimum wage, we can show that the equilibrium low‐skill wage is continuous and strictly decreasing with respect to the supply of low‐skill workers (i.e., equal to the demand). This is visualized on the right‐hand side of Figure 1. Moreover, Figure 1 indicates that the low‐skill wage is bounded above by W0 ,whichisdefinedasthelow‐skill wage where the relative prices of low‐skill and high‐skill workers are equal at some particular task J .Here, J divides the sets of tasks in an economy where only machines and high‐skill workers produce the final good. The allocation of tasks to the production factors characterized by the functions I * and S* with respect to low‐skill employment are plotted on the left‐ hand side in Figure 1. We can see there that the measure of low‐skill tasks (which is equal to s L ) is nondecreasing if the low‐skill employment increases and zero if there is no low‐skill employment. Introducing a binding minimum wage ∈ W WW(, ) mL0into our model, we obtain ECKARDT | 63 a unique demand for low‐skill workers L m and corresponding thresholds IS ( *,* ) in the new equilibrium (see Figure 1). Now, the demand for low‐skill workers L m is smaller than the supply of low‐skill workers L and, consequently, the low‐skill labor market is not clearing (see Equation 5). Contrary to this, the market for high‐skill workers is clearing as the minimum wage is not binding for high‐skill workers ( W W> H m ). This is a consequence of Assumption 2.1 that high‐skill workers have a higher productivity than low‐skill workers for each task ( γγ> H L ). 3|THE EFFECTS OF A MINIMUM WAGE This section analyzes the model introduced in the last section. We are interested in the effects of a binding minimum wage on the aggregate output, employment, factor prices, and various measures of the income distribution. The analysis requires us to distinguish the following four cases concerning the relationship between the factor prices and the ratio of the productivity functions as the corresponding thresholds IS ( *,* ) are not differentiable in general at the transitions between the cases. These cases are illustrated in Figure 2. •In the first case, machines and low‐skill workers are limited by their exogenous thresholds. This means that machines are relatively cheaper than (or equally expensive as) low‐skill and high‐skill workers for each task in the set I [ 0, ] and low‐skill workers are relatively cheaper than (or equally expensive as) high‐skill workers for each task in the set IS ( ,] . Then, the inequalities ≥γI( ) W R1 mand ≥γS( ) W W2 H m hold and IS IS ( *,*)=(, ) follows. •In the second case, only machines are limited by their exogenous threshold I and there is a threshold S S ˆ<such that the relative prices of low‐skill and high‐skill workers are equal at the task S ˆ. Then, it holds that ≥γI( ) W R1 mand γS=( ˆ ) W W2 H m. The allocation of tasks is characterized by IS IS ( *,*)=(,ˆ ) . •The third case is similar to the second. Here, the low‐skill workers are limited by their exogenous threshold S and there is a threshold I I ˆ<such that the relative prices of FIGURE 1 The low‐skill wage and the corresponding allocation of factors (represented by the thresholds I * and S* ) with respect to the supply of (or demand for) low‐skill workers 64 | ECKARDT machines and low‐skill workers are equal at the task I ˆ . Therefore, it holds that γI=( ˆ) W R1 m and ≥γS( ) W W2 H m and IS IS ( *,*)=( ˆ, ) . •In the fourth case, there are thresholds I I ˜<and S S ˜<such that the relative prices of machines and low‐skill workers are equal at the task I ˜ and the relative prices of low‐skill and high‐skill workers are equal at the task S ˜. The equations γI=( ˜) W R1 mand γS=( ˜ ) W W2 H mare satisfied. This allocation of tasks is characterized by IS IS ( *,*)=( ˜,˜ ) . The values I ˆ , S ˆ,I ˜ , and S ˜are unique (for fixed I and S ) and the unique endogenous thresholds are given by I S IS IS IS IS ( *,*)=min{(, ),(,ˆ), (ˆ,),( ˜,˜)} . 3 Note that from now on we consider strict inequalities in the first, second, and third case because, as already mentioned, the transitions between the cases are not differentiable in general. In the sequel, we analyze the effects of a change in the minimum wage and an automating economy. As described in the introduction, we think of an automating economy along several dimensions; namely, changing exogenous thresholds, productivity functions, and the task interval (creation of new, labor‐intensive tasks). Each analysis starts with the consideration of the FIGURE 2 The tasks are allocated to the factor that has the lowest relative price 3 The minimum mi n is defined component‐by‐component and corresponds to one of the four couples. ECKARDT | 65 note that the rental rate increases more strongly than the high‐skill wage (if it increases at all). Second, the share of income of machines in the national income increases, too. Moreover, we get distributional effects between the low‐skill and high‐skill workers. If the output effect is negative, the high‐skill wage decreases. This reduces the inequality between the wages. Then, the employment effect is negative and, consequently, the expected low‐skill wage decreases. If S S *=ˆ, high‐skill workers displace the low‐skill workers, which further reduces the low‐skill share in the national income and raises the high‐skill share in the national income. When the output effect is positive, the wage inequality increases and in the additional case where S S *=ˆ, the low‐skill workers displace the high‐skill workers. Consequently, this FIGURE 3 The movement on the curve describes the cost‐saving effect. If the cost‐saving effect increases the demand for low‐skill workers more strongly than it is reduced by the displacement effect, the curve of the aggregate output is shifted up and pushes the output effect up. If this is not the case, the curve of the aggregate output is shifted down and will be either still above or below its initial value 72 | ECKARDT reduces the inequality in the income distribution. Because the ripple effect is not bounded, it is possible that it is larger than one and, therefore, the share of income of low‐skill workers in the national income increases. However, as seen in the previous subsection, there is a trade‐off between the inequality of the factor prices and the inequality of the income distribution. Further, from the above discussion it follows that the signs of the effects on the ratio between the high‐skill wage or the rental rate and the expected low‐skill wage are ambiguous, which is summarized in the following corollary. Corollary 3.10. Let Assumption 2.1 hold. Further, let N = 1 , K LH,, >0 , IS 0 << < 1 , and ∈ W WW(, ) mL0.If ∈I S IS IS ( *,*) {(, ),(,ˆ)}, then it holds that ⎜⎟ ⎜⎟ ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ ⋅⋅⋅ ⋅ ⋅⋅⋅ ⋅ ⋅ ⋅ () () I L Hsss Iss I I L Ksss Iss I d d=1d d−d d, d d=1d d−d d. W uW L LHHL R uW L LKKL (1 − ) 2 (1 − ) 2 H m m 3.2.2 |Deepening of automation The second type of automation is automation at the intensive margin, meaning that the productivity (function) of machines performing tasks increases. As in Acemoglu and Restrepo (2018a), we call it the “deepening of automation”. It occurs when machines are developed further or replaced with newer, more productive machines. Here, we do not limit ourselves to I I *= , in contrast to the previous case. We assume that the productivity function is proportional to the factor‐augmenting technologies, that is, ⋅∈γx Aψx i KLH()= () for { , , } , iii(8) where A>0 i ,∈ i KLH{,, } , denotes the factor‐augmenting technology, respectively, and ψ i , ∈ i KLH{,, } , has the same properties as γi in Assumption 2.1. Moreover, we denote ψ =ψ ψ 1 L K and ψ =ψ ψ 2 H L . Consequently, when analyzing the deepening of automation, we have an increase in A K . We now consider the ripple effects in the following lemma. Lemma 3.11. Let Assumption 2.1 hold. Further, let N = 1 , K LH,, >0 , IS 0 << < 1 , and ∈ W WW(, ) mL0. •If IS IS ( *,*)=(,ˆ ) , then it holds that ⋅⋅ ⋅⋅⋅ S A S YγSHW dˆ d=(1 − ˆ) +′( ˆ)>0 . K I SI Y A I SI m 1− ˆ+ 1− ˆ+2 K ECKARDT | 73 •If IS IS ( *,*)=( ˆ, ) , then it holds that ⋅⋅ ⋅⋅ ⋅ () I A Iγ I AψII ψI dˆ d= ˆ(ˆ) ′(ˆ)ˆ+( ˆ) >0. K S SI LS SI 1− 1− +ˆ1 1 1− 1− +ˆ •If IS IS ( *,*)=( ˜,˜ ) , then it holds that I A S A d˜ d>0and d˜ d>0 . KK Proof. The proof follows by totally differentiating the equation γS=( ˆ ) W W2 H mwith respect to A K and S ˆ, the equation γI=( ˆ) W R1 m with respect to A K and I ˆ , and the equations γI=( ˜) W R1 mand γS=( ˜ ) W W2 H mwith respect to A K ,I ˜ , and S ˜. The details can be found in Part IV of the Online Appendix. □ Lemma 3.11 shows that the thresholds of the tasks are increasing if they are not at the extensive margins. This is intuitive as the machines become better at their tasks and we expect them to displace low‐skill workers as in the model without a minimum wage. To understand the mechanism behind this lemma, we need the results of the following proposition and corollary. Proposition 3.12. Let Assumption 2.1 hold. Further, let N = 1 , K LH,, >0 , IS 0 << < 1 , and ∈ W WW(, ) mL0. Then, it holds that ⎛ ⎝ ⎜⎞ ⎠ ⎟ ⎛ ⎝ ⎜⎞ ⎠ ⎟ ⋅⋅ ⋅ ⋅ L ASI Y W IS I A I A S A Y A Y SI I A I A S A d d=1 1− *+* *(*−*)−d* d+d* d, d d=1− *+* *−d* d+d* d>0. m KmKKK KKKK In particular, if ∈I S IS IS ( *,*) {(, ),(,ˆ)}, then it follows that > 0 L A d d m K. Proof. The proof follows by differentiating (3), the fact that ⋅LY= mSI W *−* m, and from Lemma 3.11. The details can be found in Part IV of the Online Appendix. □ In contrast to our previous findings, introducing a minimum wage does not alter the economic consequences of an increase in the aggregate output. Deepening of automation creates a cost‐saving effect. Consequently, the aggregate output increases. The impact on the employment of low‐skill workers is ambiguous due to the possible displacement of low‐skill workers by machines that decreases the demand for low‐skill workers and counteracts the output effect. In the following corollary, we consider the effects on the factor prices. Corollary 3.13. Let Assumption 2.1 hold. Further, let N = 1 , K LH,, >0 , IS 0 << < 1 , and ∈ W WW(, ) mL0. Then, it holds that 74 | ECKARDT ⋅⋅ ⋅ ⋅ ⋅⋅ ⋅ ⋅ ⋅⋅⋅ ⋅ ≤ () () () () () () W ASI Y HSI A I AIS A R ASI Y KII A S ASI A A K HI IS ASI A d d=1 1− *+*(1 − *)*−d* d−*d* d>0, d d=1 1− *+***+d* d+(1− *)d* d>0, d d=− 1 (*)*d* d+(1− *)d* d0. H KKKK KKKK W R KKK 2 H First, we note that the high‐skill wage increases due to the positive output effect. This explains the displacement of the high‐skill workers at the endogenous threshold S ˆor S ˜. Besides, the wage inequality increases. Second, the rental rate does not increase as strongly as the factor‐augmenting technology. 7 This explains the other ripple effect. The sign of the effect on the ratio between the rental rate and high‐skill wage is ambiguous. Figure 4presents the allocation of tasks (and the ripple effects). There, the curves of the relative factor prices are plotted. An increase in A K leads first to a downshift in the curve of the relative price of the machines. The positive output effect raises the rental rate and shifts this curve upwards but it still remains below the initial curve. The same holds for the curve of the relative price of high‐skill workers. Consequently, we can conclude that machines become relatively cheaper than the other production factors as their relative price curve is below the initial curve. Moreover, low‐skill workers become relatively cheaper than high‐skill workers as their relative price curve does not change. A further implication of Lemma 3.11 is that the share of income of machines (high‐skill workers) in the national income is nondecreasing (nonincreasing) as long as the minimum wage is binding. However, the sign of the effect on the share of income of low‐skill workers in the national income is ambiguous. Thus, it can also be positive, which cannot happen in the model without a minimum wage and leads to an increasing low‐skill employment. 8 The following corollary summarizes these results. Corollary 3.14. Let Assumption 2.1 hold. Further, let N = 1 , K LH,, >0 , IS 0 << < 1 , and ∈ W WW(, ) mL0. Then, it holds that ≤≥ s A S A I A s A S A s A I A d d=d* d−d* d,d d=− d* d0, and d d=d* d0. L KKK H KK K KK As the sign of the effect on the share of income of low‐skill workers in the national income is ambiguous, the signs of the effect on the ratios between it and the other shares of income are ambiguous, too, see Corollary 3.15. Corollary 3.15. Let Assumption 2.1 hold. Further, let N = 1 , K LH,, >0 , IS 0 << < 1 , and ∈ W WW(, ) mL0. Then, it holds that 7 It holds that ∕ () Add<0 R AKK. 8 In the model without a minimum wage, it holds that ≥≥0 I AK S AK d* d d* d. ECKARDT | 75 ⎛ ⎝ ⎜⎞ ⎠ ⎟ ⎛ ⎝ ⎜⎞ ⎠ ⎟ ⋅⋅⋅ ⋅ ⋅⋅⋅ ⋅ ⋅ ⋅ () () A L Hsss Ass A A L Ksss Ass A d d=1d d−d d, d d=1d d−d d. W uW KL LH K HL K R uW KL LK K KL K (1 − ) 2 (1 − ) 2 H m m 3.2.3 |Deepening of skills Analogously to the deepening of automation, we can also consider an upshift in the productivity function of low‐skill or high‐skill workers. By assuming the same representation of the productivity functions as in (8), we model these shifts with an increase in the labor‐ augmenting technologies A L and A H . The mechanisms are analogous to the case of deepening of automation. An increase in a labor‐augmenting technology (A L or A H ) decreases the relative price of the corresponding factor. This creates a cost‐saving effect that raises the aggregate output. On the one hand, increasing A H reverses the effects of the deepening of automation on the thresholds of the tasks and thus on the shares in the national income. The other effects remain the same. On the other hand, if A L increases, low‐skill workers displace machines if ≠I I *, and they displace high‐skill workers if ≠ S S * . This leads to a nondecreasing share of income of low‐skill workers in the national income. Moreover, the sign of the effects on the high‐skill wage and the rental rate are ambiguous, whereas the employment effect is positive. The expected low‐skill wage defined in Equation (7) increases at least as much as the rental rate or the high‐skill wage. 3.2.4 |Expanding skills In our model, we assume that low‐skill workers cannot perform tasks above S . Now we explore what happens when an increase in S occurs, which corresponds to an expansion of the skill FIGURE 4 The development of the relative factor prices for increasing AKand the consequences on the allocation of the factors to tasks 76 | ECKARDT range of low‐skill workers. Therefore, we limit our considerations to the essential case S S *=, where the low‐skill workers are relatively cheaper than the high‐skill workers at the task S , namely, < W γS W γS() () m L H H . 9 We start to determine the ripple effect in the case II *=ˆ. Lemma 3.16 (Ripple effect). Let Assumption 2.1 hold. Further, let N = 1 , K LH,, >0 , IS 0 << < 1 , and ∈ W WW(, ) mL0. If IS IS ( *,*)=( ˆ, ) , then it holds that ⎛ ⎝ ⎜⎛ ⎝ ⎜⎞ ⎠ ⎟⎞ ⎠ ⎟ ⋅⋅ ⋅⋅ ⋅⋅ I SγII γI dˆ d=− ln + 1 ′(ˆ)ˆ+( ˆ)<0 . γI I SI Y WγS S SI (ˆ)ˆ 1− +ˆ 1 () 11 1− 1− +ˆ S H m 1 1− 2 Proof. The proof follows by totally differentiating the equation γI=( ˆ) W R1 m with respect to S and I ˆ . The details can be found in Part IV of the Online Appendix. □ To explain this negative ripple effect, which differs from the framework without a (binding) minimum wage, we need the following proposition. 10 Proposition 3.17. Let Assumption 2.1 hold. Further, let N = 1 , K LH,, >0 , IS 0 << < 1 , and ∈ W WW(, ) mL0. •If IS IS ( *,*)=(, ) , then it holds that ⎛ ⎝ ⎜ ⎜ ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎞ ⎠ ⎟ ⎟ ⋅ ⋅⋅⋅ L S SI W Y S Y W Y S Y SI Y WγS d d=−d d+>0, d d=1− + ln 1 () +1 >0. m mm S H m 1− 2 •If IS IS ( *,*)=( ˆ, ) , then it holds that ⎛ ⎝ ⎜⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎜ ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎞ ⎠ ⎟ ⎟ ⎛ ⎝ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⋅⋅ ⋅⋅⋅⋅ ⋅⋅ ⋅ L S SI W Y S Y W I S Y S Y SI Y WγS γII γI d d=−ˆd d+1− dˆ d>0, d d=1− + ln 1 () +1 1+ ′(ˆ)ˆ+( ˆ)>0. m mm S H m γI I SI S SI 1− 2 (ˆ)ˆ 1− +ˆ 11 1− 1− +ˆ 1 Proof. The proof follows by differentiating (3), the fact that ⋅LY= mSI W *−* m, and from Lemma 3.16. The details can be found in Part IV of the Online Appendix. □ This proposition implies that we get a positive output and employment effect by increasing S . The reason for this is twofold: a cost‐saving effect, and a reinstatement effect of the low‐skill workers. Both raise the output, the demand for low‐skill workers, and thus the employment. 9 The case ≠ S S * is self‐explanatory. Then, all effects of an increase in S are zero. 10 In the model without a minimum wage, it holds that ∈(0, 1) I S dˆ d . ECKARDT | 77 The mechanism is as follows. Due to the cost‐saving effect, the production of the final good becomes cheaper, leading to an increase in the aggregate output. This raises the demand for low‐skill workers (i.e., the employment), which again raises the aggregate output. At the same time, the reinstatement effect, which is the opposite of the displacement effect, increases the demand for low‐skill workers, too. Due to the increase in the low‐skill employment, there is a further positive output effect. Figure 5shows these output effects. The ripple effect strengthens both effects of Proposition 3.17 as the displacement of machines by low‐skill workers creates an employment effect and thus an output effect. It follows from an increase in the rental rate, which we see together with the other effects of the factor prices in the following corollary. Corollary 3.18. Let Assumption 2.1 hold. Further, let N = 1 , K LH,, >0 , IS 0 << < 1 , and ∈ W WW(, ) mL0.If ∈IS IS IS ( *,*){(,),( ˆ,)} , then it holds that ⎜⎟ ⎛ ⎝ ⎞ ⎠ ⋅ ⋅⋅ ⋅⋅ ⋅ () W S S H Y S Y H R S I K Y S Y K I S S K HI SI S d d=1− d d−, d d=*d d+d* d>0, d d=− 1 (*)1+(1− ) d* d. H W R 2 H In particular, it holds that () <0 S d d WH Rif ≠III *= ˆ . As emphasized before, the effect on the rental rate is positive, which explains the ripple effect in Lemma 3.16. Consequently, the inequality between the low‐skill wage and the rental rate increases. The signs of the effects on the high‐skill wage and the ratio between the rental rate and the high‐skill wage are ambiguous. These depend on the size of the output effect. Due to the displacement of the high‐skill workers and perhaps also the machines, the share of income of the low‐skill workers in the national income increases. The other shares in the national income are nonincreasing, whereas the share of income of machines in the national income is nondecreasing in the model without a minimum wage. We summarize this in the following corollary. FIGURE 5 First, the movement on the curve describes the cost‐saving effect. The upshift in the curve is a consequence of the employment effect as a result of the first output effect and the reinstatement effect 78 | ECKARDT Corollary 3.19. Let Assumption 2.1 hold. Further, let N = 1 , K LH,, >0 , IS 0 << < 1 , and ∈ W WW(, ) mL0.If ∈IS IS IS ( *,*){(,),( ˆ,)} , then it holds that ≥≤ s S I S s S s S I S d d=1− d* d1, d d=−1,and d d=d* d0 . LHK As a consequence of Corollary 3.19, the expected low‐skill wage increases more strongly than the rental rate and the high‐skill wage. This result is summarized in the following corollary, which also shows that an increase in S reduces the inequality in the income distribution. Corollary 3.20. Let Assumption 2.1 hold. Further, let N = 1 , K LH,, >0 , IS 0 << < 1 , and ∈ W WW(, ) mL0.If ∈IS IS IS ( *,*){(,),( ˆ,)} , then it holds that ⎜⎟ ⎜⎟ ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ ⋅⋅⋅ ⋅ ⋅⋅⋅ ⋅ ⋅ ⋅ () () S L Hsss Sss S S L Ksss Sss S d d=1d d−d d<0, d d=1d d−d d<0. W uW L LHHL R uW L LKKL (1 − ) 2 (1 − ) 2 H m m 3.2.5 |Creation of new tasks Acemoglu and Restrepo (2018a,2018d) implement the creation of new, labor‐intensive tasks in their task‐based framework by an increase in N and show a powerful counteracting force of these new tasks regarding the rapid automation. We implement the creation of new tasks in our framework with a binding minimum wage. For the analysis, we need a further assumption. Assumption 3.21. Let Assumption 2.1 hold. Further, let ≥ N1 , K LH,, >0 , IS 0 << < 1 , and ∈ W WW(, ) mL0hold such that W γN R γN() <(−1) H HK holds. This assumption ensures that an increase in N creates a cost‐saving effect and thus the possibility of a positive output effect. After an infinitesimal right‐shift of the task interval, the new tasks can be produced relatively cheaper than the old destroyed tasks. The power of this cost‐saving effect is very important for the following ripple effects. Lemma 3.22 (Ripple effects). Let Assumption 2.1 and 3.21 hold. Further, let ≥ N1 , K LH,, >0 , IS 0 << < 1 , and ∈ W WW(, ) mL0. •If IS IS ( *,*)=(,ˆ ) , then it holds that ⎜⎟ ⎛ ⎝ ⎞ ⎠ ⋅⋅⋅⋅ ⋅⋅ () S NγSH dˆ d= 1+ ln ′(ˆ)+ >0 . Y W NS SI H NS IN K γN γN Y W IN SI −ˆ 1− ˆ+− ˆ −+1 () (−1) 2 1+ − 1− ˆ+ m H K m ECKARDT | 79 •If IS IS ( *,*)=( ˆ, ) , then it holds that ⋅ ⋅⋅⋅ ⋅⋅ () I NγI γI γ I I N dˆ d=− ( ˆ) −1 + ln (ˆ)+′( ˆ)( ˆ−+1) . IN SI H NS IN K γN γN NS SI 1 ˆ−+1 1− +ˆ− ˆ−+1 () (−1) 1 − 1− +ˆ1 H K •If IS IS ( *,*)=( ˜,˜ ) , then it holds that the sign of I N d˜ d is ambiguous and the sign of S N d˜ dis positive. Proof. The proof follows by totally differentiating the equation γS=( ˆ ) W W2 H mwith respect to N and S ˆ, the equation γI=( ˆ) W R1 m with respect to N and I ˆ , and the equations γI=( ˜) W R1 mand γS=( ˜ ) W W2 H mwith respect to N ,I ˜ , and S ˜. The details can be found in Part IV of the Online Appendix. □ If ≠ S S * , low‐skill workers displace high‐skill workers. Furthermore, the sign of the effect on I * is ambiguous due to the relation between the cost‐saving effect and the displacement effect of machines from the old destroyed tasks. This relation corresponds to the change in the rental rate and can be seen with the following proposition and corollary. Proposition 3.23. Let Assumption 2.1 and 3.21 hold. Further, let ≥ N1 , K LH,, >0 , IS 0 << < 1 , and ∈ W WW(, ) mL0. Then, it holds that ⎛ ⎝ ⎜ ⎜ ⎛ ⎝ ⎜⎞ ⎠ ⎟ ⎞ ⎠ ⎟ ⎟ ⎛ ⎝ ⎜ ⎜ ⎛ ⎝ ⎜⎞ ⎠ ⎟ ⎞ ⎠ ⎟ ⎟ ⋅⋅ ⋅ ⋅ ⋅ ⋅⋅⋅ L NSI Y WSI H NS IN K γN γN I N S N Y N Y SI H NS IN K γN γN I N S N d d=1 1− *+*(*−*)ln −* *−+1 () (−1)−d* d+d* d, d d=1− *+*ln −* *−+1 () (−1)−d* d+d* d. m m H K H K In particular, if ∈I S IS IS ( *,*) {(, ),(,ˆ)}, then it follows that Y N L N d d>0and d d>0. m Proof. The proof follows by differentiating (3), the fact that ⋅LY= mSI W *−* m, and from Lemma 3.22. The details can be found in Part IV of the Online Appendix. □ The consequence of this proposition is that the output effect, as well as the employment effect, are not positive in general. If I I *= , then they are positive, whereas their signs are ambiguous in the other cases. These results are contrary to the framework without a (binding) minimum wage, where the output effect is equal to the cost‐saving effect and, consequently, positive by assumption. Moreover, in the model without minimum wage, both ripple effects are positive and the flexible low‐skillwageabsorbsthechangeinthedemand for low‐skill workers. Due to the right‐shift of the task interval, the set of tasks conducted by machines decreases. This reduces the rental rate if the cost‐saving effect cannot balance the right‐shift. If this is the case (the rental rate decreases) and ≠I I *, machines displace low‐skill workers. Without the cost‐saving effect, this displacement effect reduces the employment and, thus, the aggregate output. Therefore, the signs of the output and the employment effect depend on the power of the cost‐saving and the displacement effect and on the question of which of the two dominates 80 | ECKARDT (as they are counteracting). In particular, it is possible that the aggregate output increases and the low‐skill employment decreases at the same time. This is illustrated in Figure 6. Analogously to the “so‐so”automation, we talk about the “so‐so”new tasks if the cost‐saving effect is too small to create an output effect. In the other cases, the output and the employment effect are positive as there is no displacement of low‐skill workers. Moreover, high‐skill workers benefit from the creation of new tasks as in the model without a minimum wage because their wage increases. Consequently, there is a reinstatement effect of the low‐skill workers if S S *=ˆor S S *=˜. Therefore, low‐skill employment increases and thus also the aggregate output. The next corollary summarizes this discussion. Corollary 3.24. Let Assumptions 2.1 and 3.21 hold. Further, let ≥ N1 , K LH,, >0 , IS 0 << < 1 , and ∈ W WW(, ) mL0. Then, it holds that ⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟ ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎜ ⎜ ⎜ ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ ⎞ ⎠ ⎟ ⎟ ⎟ ⋅⋅ ⋅⋅ ⋅⋅ ⋅ () () W N NS H Y N Y H S N R N IN K Y N Y K I N N K HI N S N NS IN I N d d=−*d d+1− d* d>0, d d=*−+1d d+d* d−1, d d=1 *−+1 1− d* d−−* *−+1 d* d−1 . H W R 2 H In particular, if ∈IS IS IS IS ( *,*){(,),( ˆ,),( ˜,˜)}, it holds that FIGURE 6 Both panels plot the allocation of tasks. If ≠I I * and the rental rate decreases ( N increases to N ′ and Y increases to Y ′ ), the curve of the relative price of machines is shifted down and I * increases to I* ′ . This creates a displacement effect of low‐skill workers. In the left panel, the cost‐saving effect dominates the displacement effect by focusing on the demand for low‐skill workers. Then, the low‐skill employment increases Lmto L′ m . Therefore, the aggregate output increases further to Y″. Consequently, the rental rate also increases and shifts the curve of the relative price of machines up. Then, the low‐skill workers displace machines and generate a reinstatement effect that raises the low‐skill employment and the aggregate output, and so on. Finally, there is a positive output and employment effect. In the right panel, the displacement effect dominates the cost‐saving effect by focusing on the demand for low‐skill workers. Therefore, the low‐skill employment Lm decreases to L′ m and the aggregate output also decreases to Y″. Consequently, the rental rate decreases further and shifts the curve of relative price of machines downwards. Then, the mechanism starts again, and so on. At the end, there is a negative‐employment effect and the output effect can be either positive or negative ECKARDT | 81 Proof. We set, without loss of generality, N = 1 . As the existence and uniqueness of the function ω is clear from Proposition A.1, we have to show that ω is continuous and strictly decreasing. For this, we consider the pairs IS IS IS IS ( ,),(, ˆ), (ˆ,),( ˜,˜ ) from Proposition A.1. If these pairs are continuous, then IS ( *,* ) and ω are continuous because ω LW()= L is a composition of continuous functions. The continuity of each pair follows from the implicit function theorem. In addition, the second property follows by calculating the slope of W L. 13 □ Now, to determine the inverse function of ω , we show that the low‐skill wage is bounded and that the equilibrium of the model with low‐skill workers converges to the equilibrium of the model without low‐skill workers. Proposition A.3. Let ≥ N1 ,L= 0 , and Assumption 2.1 hold. Then, there exists for any K H,>0 ,∈IN N(−1, ) a unique equilibrium that is characterized by a threshold ∈ J NI(−1,] and the aggregate output ∫∫ ⋅⋅ () ()() Yγxxγxx= exp ln ( ) d + ln ( ) d . N J KJ N H K JN JN H NJ NJ 0−1 −+1 −+1 − − Moreover, it holds that ⋅⋅RY JN KWY NJ H =−+1 and = − . H00 ,00 In particular, it holds that ↘↘ ↘ IJ S YY lim *==lim *, lim = LL L 00 00 and ↘WW γJ Wlim = ()=: , LL H 0 ,0 2 0(A3) where Y ,I * , and S* are taken from the model in Proposition A.1 with L> 0 . Proof. The first part is analogous to the proof of Proposition A.1. To show the convergence, we have to study the behavior of the threshold IS ( *,* ) if L increases. Similar to Lemma 3.1 (consider Lemma A.2) it follows that I * is nonincreasing and S* is nondecreasing. Thus, S I *− * is nondecreasing, too. It holds that ⋅≤ ≤ ⋅ I KγI SI L S HγS *(*)*−*1− *1 (*) . 1 2 As SI L *−*is bounded by ⋅Hγ 1 (0) 2 , the convergences ∈∞ ↘↘ () SI SI L lim *−*=0 and lim *−*(0, ) LL00 13 The function WL is not differentiable in general at the transitions between the pairs IS IS IS IS(, ),( ˆ,),(, ˆ), (˜,˜). 88 | ECKARDT follow. From the uniqueness of the equilibrium, we conclude ↘↘ IJ Slim *==lim * LL00 and thus ↘YYlim = . L00 For small L , it even holds that ⋅ SI L S HγS *−*=1− *1 (*) . 2 Consequently, it holds that ⋅⋅ ↘↘ SI L S Hγ S J Hγ J lim *−*=lim 1− * (*)=1− () LL00 22 and thus (A3) follows. □ Lemma A.2 and Proposition A.3 imply the following corollary that gives the (unique) necessary supply of low‐skill workers in a competitive market for a fixed arbitrary low‐skill wage between 0 and W0 . Corollary A.4. Let Assumption 2.1 hold. Further, let ≥ N1 , K H,>0 , N IS N −1< < < . Then, there is a unique function →∞ ↦ ωW ωW L : (0, ) (0, ), :. L −1 0 −1 In particular, ω − 1 is continuous and strictly decreasing. A.1.1 |Proof of Equations (2)–(4) Let ≥ N1 , K LH,, >0 ,∈∈IN NSIN(−1,), (, ) and ∈ W WW(, ) mL0. As the supply and demand for low‐skill workers are identical in an equilibrium of a competitive market, Corollary A.4 implies that the demand for low‐skill workers ∈LωW L=()(0,) mm −1 if the minimum wage W m is binding. By construction, we get a new equilibrium IS ( *,* ) and Y for K LH I ,,, m and S , where I S IS IS IS IS ( *,*)=min{(, ),(,ˆ), (ˆ,),( ˜,˜)} and ISIS ˆ,ˆ,˜,˜satisfy ⋅⋅ ⋅⋅ SI IN K LγI NS SI L HγS SI IN K LγI NS SI L HγS −ˆ ˆ−+1 =( ˆ), −ˆ ˆ−=( ˆ), ˜−˜ ˆ−+1 =( ˜), and −˜ ˜−˜=( ˜), m m m m 12 12 ∫∫∫ ⋅⋅⋅ () ()()() Y γxx γxx γxx= exp ln ( ) d + ln ( ) d + ln ( ) d , N I KI S LS N H K IN IN L SI SI H NS NS −1 * * * **−+1 *−+1 *−* *−* −* −* m(A4) and ⋅⋅⋅RY IN KWY SI LWY NS H =*−+1 ,= *−*,= −* m m H analogously to before. Consequently, (4) holds. Then, it holds that ECKARDT | 89 ⎜⎟ ⎜⎟ ⎜⎟⎜⎟ ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ ∫∫∫ ⋅ ⋅⋅⋅ W γxx γxx γxx K IN L SI H NS SI L = exp ln ( ) d + ln ( ) d + ln ( ) d *−+1 *−*−* *−*. mN I KI S LS N H IN mSI NS m −1 * * * * *−+ 1 *−*−* Thus, we get (2) if we solve it for L m . Moreover, we get (3) from (A4) and (2). A.2 | Comparison of the model with and without minimum wage Table A1 summarizes the differences between the model with and without a minimum wage. TABLE A1 Differences of partial effects of automation between the model with and without a (binding) minimum wage With a minimum wage Without a minimum wage Automation at the extensive margin () sign S I dˆ dis ambiguous ∈(0, 1 ) S I dˆ d () sign Y I d dis ambiguous >0 Y I d d ⎛ ⎝ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎛ ⎝ ⎜⎞ ⎠ ⎟ sign I d d WH Wmis ambiguous ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ > 0 I d d WH WL ( ) sign s I d d Lis ambiguous <0 s I d d L Deepening of automation >0 S A dˆ dK = 0 S A dˆ dK ⎛ ⎝ ⎜⎞ ⎠ ⎟⎛ ⎝ ⎜⎞ ⎠ ⎟ ,>0 AA d d d d WH Wm K R Wm K ⎛ ⎝ ⎜⎞ ⎠ ⎟⎛ ⎝ ⎜⎞ ⎠ ⎟ ≥, 0 AA d d d d WH WL K R WL K ( ) sign s A d d L Kis ambiguous ≤0 s A d d L K Deepening of skills ⎛ ⎝ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎛ ⎝ ⎜⎞ ⎠ ⎟ sign A d d WH Wm Lis ambiguous ⎛ ⎝ ⎜⎞ ⎠ ⎟ ≤0 A d d WH WL L ⎛ ⎝ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎛ ⎝ ⎜⎞ ⎠ ⎟ sign A d d R Wm Lis ambiguous ⎛ ⎝ ⎜⎞ ⎠ ⎟ ≤0 A d d R WL L <0 I A dˆ dH = 0 I A dˆ dH ⎛ ⎝ ⎜⎞ ⎠ ⎟⎛ ⎝ ⎜⎞ ⎠ ⎟ ,>0 AA d d d d WH Wm H R Wm H ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ≥,0 AA d d d d WH WL H R WL H () sign s A d d L His ambiguous ≤0 s A d d L H 90 | ECKARDT Expanding skills ≤0 * I S d d∈[0, 1 ) * I S d d ⎛ ⎝ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎛ ⎝ ⎜⎞ ⎠ ⎟ sign S d d WH Wmis ambiguous ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ <0 S d d WH WL ≤0 s S d d K≥0 s S d d K Creation of new tasks ( ) sign * I N d dis ambiguous ∈[0, 1 ) * I S d d () sign Y N d dis ambiguous >0 Y S d d ⎛ ⎝ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎛ ⎝ ⎜⎞ ⎠ ⎟ sign N d d R Wmis ambiguous ⎛ ⎝ ⎜⎞ ⎠ ⎟ <0 N d d R WL () sign s N d d Kis ambiguous <0 s N d d K ( ) sign s N d d H is ambiguous >0 s N d d H Notes: This table shows differences of partial effects of automation on selected outcome variables for the cases where partial effects differ between a model with and without a minimum wage. Whenever partial effects do not differ, results are not compared. ECKARDT | 91