Optimal Taxation of Robots

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1 Optimal Taxation of Robots Ue Thuemmel University of Zurich June 208 Abstract I study optimal taxation of robots and labor income in a model in hich robots substitute for routine ork and complement non-routine ork. The model features intensive-margin labor supply, endogenous ages and occupational choice. I sho that it is optimal to distort firms use of robots, thereby violating production efficiency. Hoever, it is not obvious hether robots should be taxed or subsidized. The optimal policy exploits general equilibrium effects to compress the age distribution hich relaxes incentive constraints and raises elfare. If robots polarize the age distribution, a tax on robots compresses ages at the top but raises inequality at the bottom. The sign of the robot tax depends on hich of the to effects dominate. In addition, occupational choice partly offsets age compression, thereby limiting the effectiveness of a robot tax. I use the model for quantitative analysis based on US data. In the short-run, in hich occupations are fixed, the optimal robot tax is positive and sizable, but its elfare impact is negligible. In the medium-run, ith occupational choice, the optimal robot tax and its elfare impact are diminished further and approach zero as the price of robots continues to fall. Key ords: Optimal taxation, Input taxation, Production efficiency, Technological change, Robots, Inequality, General equilibrium, Multidimensional heterogeneity JEL-Codes: D3, D33, D50, H2, H23, H24, H25, J24, J3, O33 Created on :25:24 Git commit: 44693c6330d3b32a2080f87ed ae0 I am especially grateful to Florian Scheuer, Bas Jacobs, and Björn Brügemann for very helpful conversations and comments. I also thank Eric Bartelsman, Michael Best, Nicholas Bloom, Raj Chetty, Iulian Ciobica, David Dorn, Aart Gerritsen, Renato Gomes, Jonathan Heathcote, Gee Hee Hong, Hugo Hopenhayn, Caroline Hoxby, Albert Jan Hummel, Guido Imbens, Pete Kleno, Wojciech Kopczuk, Tom Krebs, Dirk Krüger, Per Krusell, Simas Kucinskas, Sang Yoon Lee, Moritz Lenel, Agnieszka Markieicz, Magne Mogstad, Serdar Ozkan, Pascual Restrepo, Michael Saunders, Isaac Sorkin, Kevin Spiritus, Frank Wolak, Floris Zoutman and seminar participants at Erasmus University Rotterdam, VU Amsterdam and CPB Netherlands for valuable comments and suggestions. This paper has benefited from a research visit at Stanford University supported by the C. Willems Stichting. All errors are my on. Address: University of Zurich, Department of Economics, Schoenberggasse, 800 Zurich, Sitzerland. ue.thuemmel@uzh.ch. Web:

2 Introduction Public concern about the distributional consequences of automation is groing see e.g. Ford, 205; Brynjolfsson and McAfee, 204; Frey et al., 207. It is feared that the rise of the robots is going to disrupt the labor market and ill lead to massive income inequality. These concerns have raised the question ho policy should respond to automation. Some policy makers and opinion leaders have suggested a tax on robots.,2 Is this a good idea? This paper tries to anser this question. I find that in general, the optimal robot tax is not zero, and that its sign can be ambiguous. Quantitatively, the optimal robot tax is positive and sizable in the short-run ith fixed occupations, but is diminished in the medium-run ith occupational choice approaching zero as robots get cheaper. To reach these conclusions, I embed a model of labor market polarization similar to Autor and Dorn 203 in an optimal taxation frameork based on Rothschild and Scheuer 203, 204. Robots substitute for routine ork and complement non-routine ork. 3 Individuals choose ho much and in hich occupation to ork, firms hire orkers and buy robots, and the government maximizes elfare by setting a non-linear income tax schedule and by taxing or subsidizing robots. Under the realistic assumption that income taxes may not be conditioned on occupation, production efficiency is no longer optimal. Instead, it is desirable to tax or subsidize robots in order to compress ages across occupations. Wage compression relaxes orkers incentives to imitate one another hich makes redistribution less distortionary and raises elfare. The optimal policy exploits that robots affect ages in routine and non-routine occupations differently: non-routine orkers gain more from robots than routine orkers. Suppose that non-routine orkers earn higher ages than routine orkers, then a tax on robots compresses the age distribution and is therefore optimal. Hoever, empirically, orkers in non-routine occupations are not only concentrated at the top of the age distribution but also at the bottom see e.g. Acemoglu and Autor, 20. In such a setting, robots iden inequality at the top, but decrease inequality at the bottom. Whether robots should ultimately be taxed or subsidized then depends on several factors: ho exactly robots affect ages across occupations, ho strongly incentive constraints are impacted by changing ages, and ho the government values redistribution at different parts of the age distribution. Additional effects need to be taken into account if there is age heterogeneity ithin occupations and if occupational choice is endogenous. A tax on robots leads to a reallocation of labor supply ithin occupations; and if age distributions across occupations overlap, this affects ho much incentive constraints are relaxed. Finally, since a tax on robots drives up ages in routine occupations relative to See e.g. this quote from a Draft report by the Committee on Legal Affairs of the European Parliament. Bearing in mind the effects that the development and deployment of robotics and AI might have on employment and, consequently, on the viability of the social security systems of the Member States, consideration should be given to the possible need to introduce corporate reporting requirements on the extent and proportion of the contribution of robotics and AI to the economic results of a company for the purpose of taxation and social security contributions; takes the vie that in the light of the possible effects on the labour market of robotics and AI a general basic income should be seriously considered, and invites all Member States to do so; 2 Bill Gates has advocated for a tax on robots. See bill-gates-the-robot-that-takes-your-job-should-pay-taxes/ 3 Much of the optimal tax literature has focused on perfectly substitutable orkers in the spirit of Mirrlees 97. Exceptions are Stiglitz 982 and more recently Rothschild and Scheuer 203, 204; Ales et al

3 non-routine occupations, some non-routine orkers find it beneficial to sitch to routine ork. Such occupational shifting partly offsets the age compression induced by the robot tax. As a result, the robot tax becomes a less effective instrument. To assess the optimal policy quantitatively, I bring the model to data for the US economy. The model is calibrated such as to generate a realistic distribution of ages. Moreover, I relate the model to the empirical evidence on the effect of robots on ages from Acemoglu and Restrepo 207. Optimal marginal income taxes follo the knon U-shaped pattern. In the short-run ith fixed occupations, marginal tax rates are loer at the top and higher at the bottom if robots can be taxed. In the medium-run ith occupational choice, the optimal income tax is is hardly affected by the presence of a tax on robots. The short-run optimal robot tax is around 4% and its elfare impact is very small. In the medium-run ith occupational choice, the tax is in the order of 0.4% and its elfare impact is negligible. Moreover, it approaches zero as robots get cheaper and are used more. It is thus doubtful that taxing robots ould be a good idea. The remainder of the paper is structured as follos: Section 2 discusses the related literature. Section 3 sets up a simplified model ith discrete orker types and ithout occupational choice to illustrate the main mechanism. Section 4 introduces continuous types and occupational choice and characterizes the optimal robot tax. Section 5 studies the quantitative implications of the model. Section 6 discusses the results and concludes. Proofs and additional material are contained in an Appendix. 2 Related literature Optimal taxation and technological change. Fe papers have investigated the question ho taxes should respond to technological change. Most closely related is Guerreiro et al. 207, ho in parallel and independent ork also ask hether robots should be taxed. Their model is inspired by Stiglitz 982 and features to discrete types of orkers routine and non-routine ho are assigned tasks. In addition, some tasks are performed by robots. If labor income is taxed non-linearly, they find that it is optimal to tax robots at a rate of up to 0% provided that some tasks are still performed by routine labor. The rationale for taxing robots is the same as in this paper: compressing the age distribution to relax incentive constraints. Guerreiro et al. then replace non-linear labor income taxation by a parametric tax schedule as in Heathcote et al. 207, augmented by a lump-sum rebate. In this setting, the optimal robot tax can get as high as 30%. This paper differs from theirs in important ays. First, by considering three groups of occupations, I allo for age polarization. The empirical literature on the labor market effects of technological change has highlighted that routine orkers are found in the middle of the income distribution see e.g. Acemoglu and Autor, 20. Under these conditions, the sign of the robot tax is ambiguous. Second, my model features heterogeneity ithin occupations and thus generates a realistic income distribution. Third, the model features occupational choice, connecting to the literature on employment polarization see e.g. Acemoglu and Autor, 20; Autor and Dorn, 203; Goos et al., 204a; Cortes, 206. Occupational choice is an important adjustment margin in the medium-run. In contrast, in the model by Guerreiro et al. 207, individuals ho are substituted for by robots simply drop out of the labor market and as the share of routine orkers approaches zero, so does the robot tax. In my model, the optimal robot tax also approaches zero as the share of routine orkers becomes small. Hoever, I find that the mere possibility of occupational choice already substantially diminishes the robot tax, even if the share of routine orkers is still large. Finally, my quantitative analysis goes beyond that of Guerreiro et al. by targeting the 3

4 key moments hich matter for the optimal robot tax. Gasteiger and Prettner 207 also study taxation of robots, but focus on the mediumrun implications for groth in an OLG model. Automation depresses ages, and since individuals only save out of labor income not out of asset income, it also depresses investment and thus medium-run groth. A tax on robots has the opposite effect. Hemous and Olsen 208 study the impact of automation on groth, innovation and inequality. They also investigate the impact of to types of taxes: a tax on automation innovation, and a tax on the use of automation technology. Both taxes initially make routine orkers better off. Hoever, in the long run, a tax on automation innovation harms routine orkers. This is because routine orkers benefit from being employed in the production of nely introduced products, hich happens less if automation innovation is taxed. In contrast, a tax on the use of automation has benefits for routine orkers even in the long run. Slavík and Yazici 204 give a similar argument for taxing equipment capital as this paper does for taxing robots. Due to capital-skill complementarity see Krusell et al., 2000, a tax on equipment capital depresses the skill-premium, thereby relaxing incentive constraints. In contrast, structures capital, hich is equally complementary to lo and high-skilled labor, should not be taxed. In their quantitative analysis for the US economy, they find an optimal tax on equipment capital of almost 40%. Moreover, they find large elfare gains of moving from non-differentiated to differentiated capital taxation. While in their dynamic model the returns to capital are taxed, in my static model the stock of robots is taxed hich may explain the smaller magnitude of taxes. One reason for the different elfare implications is that I study the effect of introducing a robot tax into a system hich taxes labor income optimally, hereas Slavík and Yazici start out from the current US tax system in hich this is not the case. The implications of technological change for tax policy are also analyzed by Ales et al. 205 ho study a model in hich individuals are assigned to tasks based on comparative advantage. They ask ho marginal tax rates should optimally have been set in the 2000s compared to the 970s, based on changes in the US distribution of ages over a set of occupations. Over this period, labor market polarization has led to relative losses for middle income orkers. As a consequence, the optimal tax reform eases the burden for these orkers. In my model, technological change in this case robotization also polarizes the income distribution, and a tax on robots has relative benefits for middle income orkers. Production efficiency. A tax on robots violates production efficiency. This paper is thus related to the Production Efficiency Theorem Diamond and Mirrlees, 97 hich states that production decisions should not be distorted, provided that the government can tax all production factors inputs and outputs linearly and at different rates. In addition, the Atkinson-Stiglitz Theorem Atkinson and Stiglitz, 976 states that if utility is eakly separable beteen consumption and leisure and the government can use a nonlinear income tax, commodity taxes should not be used for redistribution. Combining the to theorems implies that neither consumption nor production should be distorted for redistributive reasons, provided the government can tax labor income non-linearly and has access to sufficient instruments to tax inputs and outputs. This implication has subsequently been put in perspective by Naito 999; Saez 2004; Naito 2004; Jacobs 205 ho all study settings hich feature less tax instruments than required for achieving production efficiency. Also in this paper, the set of tax instruments is too restricted for production efficiency to be optimal. In particular, income taxes may not be conditioned on 4

5 occupation. 4 In a related setting, Scheuer 204 studies optimal taxation of labor income and entrepreneurial profits. He shos that hen labor income and profits are subject to the same non-linear tax schedule, it is optimal to distort production efficiency in order to compress ages differentially. Production efficiency is restored if labor income and profits can be subject to different tax schedules. 5 Robots and the labor market. A recent empirical literature studies the impact of robots on the labor market. Using data on industrial robots from the International Federation of Robotics IFR, 204, Acemoglu and Restrepo 207 exploit variation in exposure to robots across US commuting zones to identify the causal effect of industrial robots on employment and ages beteen 990 and I use their results to inform the quantitative analysis. Other articles hich study the impact of robots on labor markets are Graetz and Michaels 208 for a panel of 7 countries and Dauth et al. 207 for Germany. Also related is the literature hich studies labor market polarization see e.g. Goos et al., 204b and the disappearance of routine jobs see e.g. Cortes et al., Model ith discrete types and no occupational choice To develop intuition, I first discuss a simple model hich features discrete types and abstracts from occupational choice. The model extends Stiglitz 982 to three sectors or occupations and features endogenous ages. The model illustrates the key arguments for taxing robots. Hoever, it is too stylized for a quantitative analysis, and by abstracting from occupational choice leaves out an important adjustment margin. I discuss a richer model ith continuous types and occupational choice later. 3. Setup 3.. Workers, occupations and preferences There are three types of orkers i I {M, R, C} ith corresponding mass f i. A orker s type corresponds to his occupation, here M refers to an occupation hich requires manual non-routine labor, R refers to an occupation requiring routine labor, and C denotes a cognitive non-routine occupation. The distinction beteen routine and non-routine occupations is motived by the empirical literature hich has established that in recent decades technology has substituted for routine ork, and has complemented non-routine ork see e.g. Autor et al., Moreover, the literature on labor market polarization suggests to distinguish beteen lo-skilled and high-skilled non-routine occupations see e.g. Cortes, 206; Cortes et al., 207. Workers derive utility from consumption c and disutility from 4 Saez 2004 refers to this as a violation of the labor types observability assumption. 5 Lozachmeur et al. 206 set up a model in hich orkers ith continuously distributed ability choose both, intensive margin labor supply and occupation, as they do in this paper. They then study optimal occupation-specific non-linear income taxation and sho that occupational choice is optimally distorted. They refer to this as distortion of production efficiency. Hoever this interpretation is debatable, as all marginal rates of transformation are equalized in their model. Also in Scheuer 204 occupational choice is distorted in the presence of occupation-specific taxes hoever, production is efficient. The distortion of occupational choice in my model complicates the derivation of optimal occupation-specific taxes, hich I therefore leave for future research. 5

6 labor supply l, according to the quasi-linear utility function here ε is the labor supply elasticity Technology U c, l = c l+ ε +, ε Denote by L L M, L R, L C the vector of aggregate labor supplies ith L j = f j l j for all i I. Let B denote robots. The final good is produced by a representative firm according to a constant returns to scale production function Y L, B. The firm maximizes profits by choosing the amount of total labor of each type i I and the number of robots, taking ages i and the price of robots p as given. Normalizing the price of the final good to one, the firm s profit maximization problem is Denote the marginal products of total effective labor as max Y L, B i L i pb. 2 L,B i J Y i L, K Y L, K L i i I, and the marginal product ith respect to robots as Y B L, B Y L, B. B In equilibrium e then have i L, K = Y i L, B i I and p = Y B L, B. Unless stated otherise, I assume throughout that robots are better substitutes for routine ork than for non-routine ork. More specifically, I make the folloing assumption Assumption. A marginal increase in the amount of robots raises the factor price of non-routine labor relative to routine labor. > 0, B B YM L,B Y R L,B YC L,B Y R L,B > 0. Due to constant returns to scale, equilibrium profits are zero. Robots are produced linearly ith the final good, according to B x = x, 3 q here I denote by x the amount of the final good allocated to the production of robots, and here /q is a productivity shifter. 6 In the absence of taxes, e then have p = q in equilibrium. Later, hen taxes drive a edge beteen p and q, I refer to q as the producer price of robots and to p as the user price of robots. 6 Due to constant returns to scale, in equilibrium neither the final goods producer nor the robot producer make any profits. I thus do not need to specify firm onership. 6

7 3..3 Government and tax instruments There is a benevolent government hich cares about social elfare W f M ψ M V M + f R ψ R V R + f C ψ C V C, 4 here ψ i is the Pareto eight attached to orkers of type i, here the eights satisfy fi ψ i =, and V i U c i, l i are indirect utilities. While the government is aare of the structure of the economy, it cannot observe an individual s occupation. This assumption is satisfied by real orld tax systems hich also do not condition taxes on occupation, for example, because enforcement may be difficult. 7 Hoever, the government can observe individual income and consumption, as ell as the value of robots purchased by the final goods producer. Accordingly, I assume that the government has access to to tax instruments: a non-linear income tax, and a tax on the value of robots. Denote by y i i l i gross labor income earned by an individual of type i. The government levies a non-linear income tax T y on gross labor income y. Taking the age and income tax schedule as given, a orker of type i then maximizes utility by choosing consumption and labor supply subject to a budget constraint: max U c i, l i s.t. c i i l i T i l i. 5 c j,l j The value of robots purchased by the final goods producer is given by qb, on hich the government may levy a proportional tax τ, to hich I refer as robot tax. 8 The user price of robots is then p = + τ q. While throughout this paper I refer to τ as a tax on robots, I highlight that τ may be negative, and may thus be a subsidy. The government faces a budget constraint f M T y M + f R T y R + f C T y C + τqb = 0, 6 stating that by raising tax revenue ith the income tax and taxes on robots and other capital, it must break even Optimal policy The government chooses tax instruments T and τ such as to maximize social elfare 4 subject to budget constraint 6. To characterize optimal taxes, I follo the conventional approach of first solving for the optimal allocation from a mechanism design problem. Afterards, prices and optimal taxes are determined hich decentralize the allocation. In a direct mechanism, orkers announce their type i, and then get assigned consumption c i and labor supply l i. Here, I consider the equivalent problem in hich instead of consumption, the planner allocates indirect utilities V i and define c V i, l i as the inverse of U c i, l i ith respect to its first argument. The allocation must induce orkers to truthfully report their type and thus needs to be incentive compatible. Since there is no heterogeneity of types ithin occupations, the 7 In this simple model, types and occupations coincide. Occupation-specific income taxes ould thus correspond to individualized lump-sum taxes, hich ould allo the government to achieve the first-best. In this case, the optimal robot tax ould be zero. Later, I relax the assumption that types and occupations coincide. 8 I focus on a linear tax on robots, since ith a non-linear tax and constant returns to scale there ould be incentives for firms to break up into parts until each part faces the same minimum tax burden. With linear taxes, such incentives are absent. 9 Introducing an exogenous revenue requirement does not change the analysis. 7

8 only ay in hich orkers can imitate one another is by mimicking incomes of orkers in other occupations. I assume that parameters of the model are such that C > R > M is satisfied. Moreover, I limit attention to those cases in hich only the donard adjacent incentive constraints may be binding, hile all other incentive constraints are slack. 0 To induce a cognitive orker to truthfully report his type, the folloing must hold R L, B V C U c V R, l R, l R, 7 C L, B here l R R C is the amount of labor hich a cognitive orker needs to supply to mimic the income of a routine orker. Similarly, a routine orker has to be prevented from mimicking the income of manual orkers, and thus M L, B V R U c V M, l M, l M. 8 R L, B 3.2. Separation into inner and outer problem I follo Rothschild and Scheuer 203, 204 and separate the mechanism design problem into an inner problem and an outer problem. In the inner problem, the planner takes the tuple of inputs L, B as given and maximizes elfare W L, B over {V i, l i } i I subject to constraints specified belo. In the outer problem, the planner chooses the vectors L = L M, L R, L C and robots B such that W L, B is maximized. The mechanism design problem can thus be ritten as max W L, B max f M ψ M V M + f R ψ R V R + f C ψ C V C 9 L,B {V i,l i } i I subject to the incentive constraints 7 and 8, the consistency conditions and the resource constraint f i l i L i = 0 i I 0 Y L, B f M c V M, l M f R c V R, l R f C c V C, l C qb = 0. The consistency conditions 0 restate the definition of aggregate labor supplies. Since the inner and outer problem separate optimization over individual labor supplies and aggregate labor supplies, including the consistency conditions ensures that the definition of aggregate labor supplies remains satisfied. The final term in the resource constraint captures that x = qb units of the final good have to be used to produce B robots Optimal robot tax I first characterize the optimal robot tax by using that in the outer problem at the optimum W L, B / B = 0, hence a change in robots may not lead to a change in elfare. Proposition. With quasi-linear utility as specified in, for any given Pareto eights ψ i 0 hich satisfy i I f iψ i =, and ith /q the productivity shifter in robot production, the optimal tax on robots is characterized by 0 This case is the relevant one for gaining intuition hich carries over to the continuous-type model. While I could characterize optimal policy in this simple frameork ithout this separation, the approach ill turn out to be useful in the full model. Already applying it here leads to expressions hich can be easily compared to those in the full model, as the structure of the problem remains the same. 8

9 τqb =ε C / R,B I CR ε M / R,B I RM 2 ith elasticities of relative ages ith respect to the number of robots defined as and incentive effects Proof. See Appendix A.. ε C / R,B C/ R B ε M / R,B M/ R B B > 0, C / R B > 0, M / R + R ε I CR f C ψ C l R, C M I RM f M ψ M l M R + ε. The left-hand side of 2, τqb, is the tax revenue raised ith the robot tax. Ceteris paribus, the robot tax is thus larger in magnitude, the smaller the cost of producing robots, q, and the loer the number of robots, B. At the optimum, robot tax revenue is equal to the difference in incentive effects I CR and I MR, eighted by the respective elasticity terms, ε C / R,B and ε M / R,B. The elasticity terms capture the percentage increase in ages of non-routine orkers relative to the age of routine orkers due to a one-percent increase in the number of robots. By Assumption : Since robots substitute for routine ork, an increase in robots raises the age of non-routine orkers relative to routine orkers. As a consequence, ε C / R,B > 0 and ε M / R,B > 0. The incentive effects I CR and I RM capture ho incentive constraints 7 and 8 are affected by a marginal increase in robots, and ho this, in turn, affects elfare. I first focus on I CR. Raising the number of robots increases C / R, and since C > R, age inequality at the top of the age distribution rises. Suppose that elfare eights are regular, and thus decrease ith income, leading to ψ C <. The government thus attaches a loer-than-average eight to cognitive non-routine orkers. In this case, it is desirable to redistribute income from cognitive non-routine orkers to orkers ho earn less. The increase in age inequality at the top then tightens the incentive constraint 7: cognitive non-routine orkers no need to put in less labor than before to imitate the income of a routine orker. This tightening of 7 makes income redistribution ith the income tax more distortive, hich limits redistribution and loers elfare. The robot tax has the opposite effect. By increasing the user price of robots p, the equilibrium number of robots falls. As a consequence, C / R drops, hich corresponds to a reduction in age inequality at the top of the age distribution; and to a relaxation of incentive constraint 7. Relaxing 7 is elfare improving. It is no less distortive to use the income tax to redistribute income from cognitive non-routine orkers to other orkers. Consequently, I find a positive incentive effect I CR > 0 if and only if ψ C <. Moreover, ceteris paribus, a larger incentive effect I CR calls for a higher tax on robots. The magnitude of I CR positively depends on l R R / C, hich is the amount of labor hich a cognitive orker has to supply to imitate the income of a routine orker. Since C > R, the incentive effect is larger, the smaller the age gap beteen cognitive and routine orkers. Ceteris paribus, larger age inequality at the top of the distribution thus 9

10 calls for a loer tax on robots. This counterintuitive result is driven by the utility function being convex in labor supply. Suppose a cognitive orker ere to imitate the income of a routine orker and that doing so required little effort l C = l R R / C. Marginally increasing the necessary effort by loering the age gap beteen C and R ould no reduce utility by du. In contrast, suppose that imitating the income of a routine orker required higher effort l C = l R R / C > l C. Due to U l < 0, U ll > 0, the same marginal reduction in the age gap and hence the same marginal increase in the level of effort required to imitate no reduces utility by du > du. Increasing the effort required to imitate is thus more discouraging and hence more effective in relaxing the incentive constraint if imitating is more costly to begin ith, that is, if C is close to R. The magnitude of the incentive effect also depends on the labor supply elasticity ε. Finally, the eighting of incentive effect I CR by elasticity ε C / R,K is again intuitive. The elasticity captures ho effective taxing robots, and thus decreasing their number, is in reducing the age gap C / R. I no turn to the second term on the right-hand side of 2. The incentive effect I RM captures ho reducing the age gap beteen R and M affects elfare via the incentive constraint 8. With regular elfare eights e have ψ M >, hence the government values redistributing income to manual non-routine orkers. In this case, loering the gap beteen R and M relaxes 8, and makes it less distortive to redistribute income ith the income tax, captured by I RM > 0. Again, the magnitude of I RM is larger if it takes more effort for a routine orker to imitate the income of a non-routine orker, that is, if the age gap beteen R and M is small. The eighting ith elasticity ε M / R,B > 0 captures ho effective the robot tax is in changing the age gap beteen R and M. The minus sign is crucial: a tax on robots increases the age of routine orkers relative to manual non-routine orkers. As inequality at the bottom of the age distribution increases, the incentive constraint 8 tightens. Redistributing to manual nonroutine orkers ith the income tax becomes more distortive, thereby loering elfare. This effect, ceteris paribus, calls for a loer tax on robots. To summarize: With regular elfare eights, a tax on robots decreases age inequality beteen cognitive and routine orkers, thereby relaxing incentive constraint 7 and increasing elfare. At the same time, a tax on robots raises the age gap beteen routine and manual orkers, hich tightens incentive constraint 8 and loers elfare. Due to these opposing forces, the sign of the robot tax is ambiguous. If the first effect dominates, robots should be taxed, hereas if the second effect is more important, robots should be subsidized. In general, the robot tax is non-zero. It is thus optimal to distort production efficiency in order to relax incentive constraints. While taxing robots distorts the firm s production choices, thereby loering output, the elfare-increasing effect of relaxing incentive constraints by making income redistribution less distortive dominates. Ceteris paribus, several factors make it more likely for the optimal robot tax to be positive: if robots increase age inequality at the top of the distribution more than they reduce inequality at the bottom; if the share of cognitive non-routine orkers is large, hereas the share of manual non-routine orkers is small; if the age gap beteen cognitive and routine orkers is small, hereas the age gap beteen routine and manual orkers is large; finally, if the government attaches relatively little eight ψ C and ψ M to cognitive orkers and manual orkers, respectively. The final point can be restated as the government attaching relatively much eight ψ R to routine orkers. This is intuitive: as routine orkers gain relative to non-routine orkers hen robots are taxed, putting relatively much eight on them calls for a larger tax on robots. 0

11 Only in special cases is the tax on robots zero. First, it is zero if the effect of relaxing incentive constraints at the top of the age distribution exactly cancels against the effect of tightening incentive constraints at the bottom. Second, it is zero if the incentive effects are zero. This ill be the case if income taxation is non-distortive, for example because the government can levy personalized lump-sum taxes, or because labor supply is fixed, corresponding to ε 0. Finally, if in contrast to hat I have assumed so far, robots are equally complementary to labor in all occupations, the optimal robot tax is zero since in this case ε C / R = ε R / M = 0. 2 The ambiguous sign of the robot tax is in contrast to Guerreiro et al. 207 ho argue that the robot tax should be positive. This is due to Guerreiro et al. considering only to groups of orkers: routine and non-routine. In their model, taxing robots unambiguously relaxes the single binding incentive constraint. My result highlights that aggregating orkers into just to groups can be misleading as it masks heterogeneous effects of robots on ages along the income distribution. The empirical literature see e.g. Autor and Dorn, 203 finds that routine orkers are not found at the very bottom of the income distribution. Instead, those ho earn least often perform non-routine manual ork hich is hard to automate. In this case, taxing robots ill iden inequality at the bottom of the age distribution, thereby locally tightening incentive constraints. If these effects are taken into account, the sign of the robot tax becomes ambiguous Optimal income taxes I no use the inner problem, taking L and B as given, to characterize optimal marginal income taxes. Proposition 2. Let µ denote the multiplier on the resource constraint. Let µη CR be the multiplier on incentive constraint 7 and µη RM the multiplier on 8. Define µξ i as the multiplier on the consistency condition for occupation i in 0. With quasi-linear utility as specified in, for any given Pareto eights ψ i 0 hich satisfy i I f iψ i =, the optimal marginal income tax rates satisfy T M + ξ M YM T M = ψ M M R + ε 3 T R + ξ R YR T R = f C f R ψ C R C + ε 4 ith T C = ξ C Y C, 5 ξ i = ε R / C,L i I CR + ε M / R,L i I RM, here the semi-elasticities of relative ages ith respect to L j are defined as ε R / C,L i L i R C and ε M / R,L i M R. L i R M 2 Along the same lines, Slavík and Yazici 204 sho that structures capital hich is equally complementary to labor in all occupations should not be taxed. C R

12 Proof. See Appendix A.2 First, note that each expressions for optimal marginal income tax rates features a correction for general equilibrium effects, ξ i /Y i. Suppose for the moment that general equilibrium effects are absent. We then have ξ i = 0 i I. Moreover, assume that elfare eights satisfy ψ M > and ψ C <, as ill be the case ith a elfarist government hich attaches higher eights to individuals earning loer incomes. As a consequence, marginal tax rates T M and T R are positive. This is in line ith the function of marginal income tax rates: the role of a marginal tax rate at income y is to redistribute income from individuals earning more than y to individuals earnings equal to or less than y. Consider 3: the social marginal value of distributing income from f R routine orkers and f C cognitive orkers to a mass of f M manual orkers is ψ M = f M [f R ψ R + f C ψ C ]. Similarly, in 4, the term f R f C ψ C captures the social marginal value of redistributing income from f C cognitive orkers to a mass of f R routine orkers. Hoever, marginal tax rates distort labor supply, hich is captured by the terms M / R +/ε and R / M +/ε. With ε > 0 both terms are smaller than and thus scale don marginal tax rates, hereas in the absence of labor supply responses ε 0 and both terms tend to. Finally, ithout general equilibrium effects T C = 0. This is the famous result of no distortion at the top. Since there are no individuals ho earn more than cognitive orkers, setting a positive marginal tax rate has no distributional benefits but ould distort labor supply. It is thus optimal to set a marginal tax rate of zero. No consider the case ith general equilibrium effects. Under special conditions, it is possible to sign the multiplier terms ξ M and ξ C. Corollary. With general equilibrium effects, the multiplier terms ξ M and ξ C can be signed as follos: if if L M R C < 0, ξ M < 0, L C M R > 0, ξ C > 0. Proof. See Appendix A.2.4 Suppose that ε R / C,L M < 0 and thus ξ M < 0. In this case T M is larger than in the absence of general equilibrium effects. The intuition has again to do ith the relaxation of incentive constraints. A higher marginal tax rate T M discourages labor supply of manual orkers, hich increases their age relative to routine orkers, thereby relaxing incentive constraint 7. Moreover, since by assumption ε R / C,L M < 0, a reduction in the supply of manual labor also increases the age of routine orkers relative to cognitive orkers, hich relaxes incentive constraint 7. A similar reasoning applies to the case in hich ε M / R,L C > 0 and thus ξ C > 0. No T C becomes negative, as in Stiglitz 982, hich encourages labor supply of cognitive orkers. As a result, their age drops relative to routine orkers, hereas, by assumption, the age of manual orkers increases relative to routine orkers. Again, this overall age compression relaxes incentive constraints, and is therefore elfare improving. Whether the marginal income tax for routine orkers is scaled up or don in the presence of general equilibrium effects is ambiguous. Consider an increase in T R. As a result, labor supply of routine orkers falls, raising their age relative to manual and cognitive orkers. This change has opposing effects on incentive constraints: it relaxes 7 but tightens 8. Whether an increase in T R is desirable thus depends on hich of the to effects is more relevant for elfare. 2

13 3.2.4 Effect of robot tax on marginal income taxes Ho ould marginal income taxes be affected if the government could not tax robots? To investigate this, I impose the additional constraint Y B L, B = q, hich corresponds to τ = 0. The expressions in 3, 4 and 5 are not affected by the absence of the robot tax. Hoever, the multipliers on the consistency conditions are no different. Corollary 2. Let preferences be as in. In the absence of the robot tax, let µκ denote the multiplier on the additional constraint Y B L, B q = 0. Let µξ i be the multiplier on the consistency condition for occupation i, µη CR the multiplier on incentive constraint 7 and µη RM the multiplier on incentive constraint 8. The folloing holds at any Pareto optimum. ξ i = ε R / C,L i I CR + ε M / R,L i I RM + κ Y B L, B L i i, ith κ Y B L, B B =ε B C / R,K I CR ε M / R,K I MR. 6 Without the robot tax, ξ i is adjusted by κ Y B L, B / L j. Note that the righthand-side of 6 is the same as in the expression for the optimal robot tax 2. Since Y B L, B / B < 0, κ thus has the opposite sign of the optimal robot tax hich ould result if I ere not to rule out robot taxation. Suppose that the optimal robot tax ould be positive, and thus κ < 0. I first focus on the unambiguous cases i {M, C}. We then have Y B L, B / L i > 0 and hence ξ i is loer in the absence of the robot tax. It thus follos that ith the robot tax, both T M and T C are loer. Intuitively, a tax on robots loers the ages of manual and cognitive orkers, hich induces them to decrease their labor supply and leads to a loss in output. Loer marginal income tax rates encourage labor supply and thus partly offset the loss in output. Moreover, in the case of cognitive orkers, the drop in ages hich results from increased labor supply further compresses the age distribution. No consider routine orkers. The sign of Y B L, B / L R is ambiguous, and as a result it is not clear in hich direction the marginal income tax for routine orkers is adjusted if robots are taxed. Suppose that Y B L, B / L R > 0, hich ill be the case if the difference in elasticities of substitution beteen robots and routine orkers on the one hand and robots and cognitive or manual orkers on the other hand is not too large. ξ R is no loer ithout the robot tax, and thus, T R ill be loer if robots can be taxed. 4 Continuous types and occupational choice Having developed intuition, I no extend the model to allo for continuous types as ell as for occupational choice. 4. Setup 4.. Skill Heterogeneity There is a unit mass of individuals. Each individual is characterized by three-dimensional skill-vector θ Θ Θ M Θ R Θ C, ith Θ i [θ i, θ i ] and i I {M, R, C}. As before, each dimension of skill determines an individual s productivity in one of the three occupations: manual non-routine M, routine R, and cognitive non-routine C. Skills 3

14 are distributed according to a continuous cumulative distribution function F : Θ [0, ] ith corresponding density f Technology The final good is produced using aggregate effective labor in the three occupations, L L M, L R, L C, and robots B according to a constant returns to scale production function Y L, B. As before, robots are better substitutes for routine labor than for non-routine labor, and Assumption holds. Aggregate effective labor is no defined as L M θ M l θ df θ, L R θ R l θ df θ, L C θ C l θ df θ, M R here M, R and C are the sets of individuals θ orking in occupations M, R and C, respectively. As before, robots are produced linearly ith the final good, as described in Section Preferences and occupational choice Individuals derive utility from consumption c and disutility from supplying labor l according to the strictly concave utility function U c, l ith U c > 0, U l < 0. Let y denote an individual s gross income and the age. I assume that U satisfies the standard single-crossing property, that is, the marginal rate of substitution beteen income and consumption, U lc, y, decreases in. Individuals choose their occupation according to U cc, y a Roy 95 model, such that their age is maximized and, in equilibrium, given by θ = max {Y M L, B θ M, Y R L, B θ R, Y C L, B θ C }, here the subscript indicates that the age for individual θ is pinned don by factor inputs L and B. 4.2 Optimal taxation 4.2. Reducing dimensionality I follo Rothschild and Scheuer 203 to reduce the dimensionality of the problem: Given factor inputs L and B, age rates and sectoral choice are determined, and the threedimensional heterogeneity in skill can be reduced to one-dimensional heterogeneity in ages. The distribution of skills F θ corresponds to the age distribution F L,B = F ith occupational age densities f M L,K = f R L,K = f C L,K = Y M L, B Y R L, B Y C L, B Y M L, B, /YC L,B /YR L,B θ C θ R /YC L,B /YM L,B θ C θ M /YR L,B /YM L,B θ R θ M 4 Y R L, B, f f f C Y C L, B Y M L, B, θ R, θ C dθ R dθ C, θ M, Y R L, B, θ C dθ M dθ C, θ M, θ R, Y C L, B dθ M dθ R.

15 The age density for occupation i at age is thus obtained by evaluating the skill density at that skill θ i hich corresponds to earning in occupation i, θ i = /Y i L, B, and by integrating over the mass of individuals hose skill is not sufficient to earn more in any occupations other than i. I denote the support of the age distribution for any given L, B by [, ], here L,B = θ M, θ R, θ C and = θm, θ R, θ C are the ages earned by the least and most skilled individuals, respectively. The overall age density is given by f = f M L,B + f R L,B + f C L,B. Like in the discrete model, the social planner attaches general cumulative Pareto eights to types, hich I no denote by Ψ θ ith the corresponding density ψ θ. Since for given L, B there is a unique mapping from types to ages, Pareto eights can be ritten as function of ages, Ψ L,B, ith occupation-specific densities ψ M L,B, ψr L,B, and ψ C L,B such that the overall density is ψ L,B = ψ M L,B + ψr L,B + ψc L,B. Moreover, once L, B is fixed, all endogenous variables hich depend on an individual s type can be ritten in terms of ages. This feature allos for a handy separation of the planner problem Separation into inner and outer problem As in Rothschild and Scheuer 203, I separate the planner problem into an inner problem hich maximizes elfare over labor supply and indirect utilities for given L and B, and an outer problem hich maximizes elfare over L and B. Inner problem Denote by V θ indirect utility of type θ. Social elfare is defined as an integral over eighted indirect utilities Θ V θ dψ θ, hich given L, B can be ritten as V dψ L,B. As is common, I use the primal approach in hich the planner L,B maximizes social elfare by directly choosing an allocation of indirect utilities and labor supplies, subject to incentive and resource constraints. In addition, the allocation needs to be consistent ith L and B, hich is ensured by consistency conditions. Apart from these consistency conditions, the problem is a standard Mirrlees 97 problem. I define the inner problem as W L, B max V dψ L,B V,l subject to V + U l c V, l, l l = 0 [, ] 7 L,B l fl,b M d L M =0 8 Y M L, B Y R L, B Y C L, B l f R L,B d L R =0 9 l f C L,B d L C =0 20 l c V, l f L,B d + Y B L, B qb =

16 Here, 7 is the set of incentive constraints, 8, 9, 20 are the consistency conditions for occupations M, R and C, respectively. Equation 2 is the resource constraint and the continuous-type equivalent of. Outer problem In the outer problem, the planner chooses inputs L and B such that elfare is maximized, that is, he solves max L,B W L, B. It is useful that W L, B corresponds to the value of the Lagrangian of the inner problem, evaluated at optimal indirect utilities and labor supplies Optimal robot tax I obtain a condition for the optimal robot tax from the outer problem by differentiating the maximized Lagrangian ith respect to robots, B. As in the simple model, the optimal allocation can be implemented using a linear tax on the value of robots. To see this, note that since firms maximize profits, they equate the marginal return to robots ith the price of robots. Under laissez-faire, e thus have Y B L, B = q. 22 Hoever, given L, the planner might ant to distort the choice of robots such that 22 is no longer satisfied. By setting a linear tax τ on the value of robots, profit maximization of the firm leads to Y B L, B = + τ q. 23 Since Y B L, B is strictly monotone in B, for each B there exists a unique robot tax τ such that 23 holds. For given optimal L, the optimal robot tax τ thus uniquely implements the optimal B. Using 23, I characterize the optimal robot tax as follos. Proposition 3. Let µ denote the multiplier on the resource constraint and µξ j the multiplier on the consistency condition for occupation j J. Denote by q j E the income share in occupation j. The optimal tax on robots, τ, is characterized by τqb = ε YC /Y R,B L, B I C L, B R C L, B + ξ i C Ci L, B + S Ci L, B i I +ε YM /Y R,B L, B I M L, B R M L, B + ξ i C Mi L, B + S Mi L, B, i I 24 here for i {M, R, C} I i L, B R i L, B and for i, j {M, R, C} C ij L, B V µ η u c V d d f i L,B d, f L,B ψ i L,B ψ L,B f L,B i ψ L,B d, f L,B L,B 2 l Cov q Y j L, B L,B, i q j L,B f L,B d, 6

17 and S ij as defined in the Appendix. The elasticities of equilibrium factor prices ith respect to the amount of robots are defined as and ε YM /Y R,B L, B Y M L, B /Y R L, B B ε YC /Y R,B L, B Y C L, B /Y R L, B B Proof. See Appendix B B Y M L, B /Y R L, B > 0, B Y C L, B /Y R L, B > 0. The expression in 24 characterizes the optimal tax revenue raised ith the robot tax. First, note the similarity beteen 24 and the corresponding expression in the simplified model, 2. In both cases, the effect of robots on relative skill prices plays a crucial role. 3 According to my definition of robots, skill prices in non-routine occupations increase relative to the skill price in routine occupations as the number of robots increases. As a consequence, both elasticities are positive. As in the simple model, the elasticities multiply the incentive effects I i. In contrast to the simple model, the elasticities no also multiply additional terms hich are due to heterogeneous types and occupational choice. Folloing Rothschild and Scheuer 203 I refer to these as redistributive effects R i, effort reallocation effects C ij and occupational shift effects S ij here i, j {M, R, C}. I discuss them in turn. Incentive effects. Since types are continuous, the discrete-type incentive constraints in the simple model have been replaced by a local incentive constraint 7. The incentive effect captures ho this local incentive constraint is affected by changes in relative skill prices and ho these changes, in turn, affect elfare. Suppose that the incentive constraint is donard-binding, as in the simplified model, and thus η 0. Since indirect utilities V increase in, the sign of I i L, B is determined by d d. f i L,B f L,B This term captures ho the share of individuals earning age in occupation i changes ith a marginal increase in. In other ords, the sign of I j is determined by the ay in hich orkers in occupation i are concentrated at certain ages. 4 For example, consider I C hich measures ho a loering of relative skill prices Y C /Y R affects elfare via the incentive constraint. Suppose that orkers in cognitive occupations are concentrated at high ages, as seen in the data. We then have that the term is going d d f C L,B f L,B to be positive at most, and increasingly so for high, capturing the concentration of cognitive orkers at high ages. As a consequence, e ill find I C > 0. A loering of 3 In the simplified model elasticities of relative ages coincide ith elasticities of relative skill prices. Due to heterogeneity of ages ithin occupations, this is no longer the case here. 4 Note that if I i 0 for some occupation i, it has to hold that there is at least one other occupation j i, for hich I j 0. To see this, rite I i L, B = = = j i η u c V d d η u c V d d η u c V d d f i L,B d f L,B fl,b j i f j L,B d f L,B f j L,B d. f L,B 7

18 Y C /Y R thus increases elfare. To understand hy, note that a decrease in the skill price Y C relative to Y R loers ages of cognitive orkers relative to routine orkers. Since cognitive orkers are concentrated at high ages, this corresponds to a compression of ages at the top of the age distribution. With locally donard-binding incentive constraints, such a compression of ages makes it more costly for cognitive orkers to imitate types ho earn marginally loer incomes in routine occupations hich locally relaxes incentive constraints, thereby increasing elfare. Ceteris paribus, the larger I C, the higher the optimal tax on robots. Next, consider I M and suppose that manual non-routine orkers are concentrated at lo ages, as observed empirically. The term d d f M L,B f L,B is going to be negative at most, and increasingly so for high, leading to I C < 0. The negative sign captures that a tax on robots loers elfare by locally tightening incentive constraints at the bottom of the age distribution. To see this, note that taxing robots increases the gap beteen Y R and Y M. As a result, age inequality at the bottom of the age distribution increases, hich makes it more attractive for routine orkers to imitate types ho earn marginally loer incomes as manual non-routine orkers. As in the simple model, a tax on robots thus has opposing effects on incentive constraints at the top and at the bottom of the age distribution. As a consequence, the sign of the robot tax is again ambiguous. Redistributive effects. The redistributive effects R i L, B capture the government s preferences for the different occupations. If all the planner cares about is orkers utilities, irrespective of the distribution of orkers across occupations, R i L, B = 0 for all i I. This ill for example be the case ith relative elfare eights Ψ F. If, in contrast, the government cares more about individuals in occupation j than in occupation i j, e have that R j L, B > R i L, B. 5 For example, suppose that for some reason the government cares more about the utilities of orkers in routine occupations than in nonroutine occupations, such that e have R R > 0 and R C < 0, R M < 0. Ceteris paribus this ill call for a positive tax on robots. The intuition is straightforard: by taxing robots the government increases the ages of routine orkers relative to non-routine orkers, hich according to the government s preferences raises elfare. Effects on the consistency conditions. The final to sets of effects arise from the fact that a change in the number of robots, via changing relative skill prices, affects the consistency conditions. This, in turn, affects incentive constraints, and thus elfare. The social elfare impact of relaxing the consistency condition for occupation i is given by µξ i, and since µ > 0, the sign of the effect is determined by ξ i. Effort reallocation effects. The effort reallocation effects C ij capture ho the changes in relative skill prices induced by the robot tax lead to a reallocation of labor supply ithin occupations, and thereby to a reallocation of labor supply across occupations at a given. The term ql,b i θ captures the share of income hich individual θ earns in occupation i. The covariance of income shares across occupations i and j is then computed conditional on age. Suppose the share of income earned in occupation i increases in, hile it decreases in for occupation j. In other ords, suppose that high incomes are concentrated in occupation i, hile lo incomes are concentrated in occupation j. In that case, Cov ql,b i, qj L,B < 0. If labor supply is increasing in, it follos that 5 Using the same argument as before, if R j L, B 0 for some j, then there exists at least one i j ith R i L, B 0. 8

19 C ji L, B < 0. Suppose that ξ j > 0, hence there are elfare benefits of encouraging labor supply in occupation j via general equilibrium effects on the age distribution. In this case, C ji L, B < 0 calls for a ceteris paribus loer robot tax. To understand hy, note that compressing ages in occupation i relative to occupation j leads individuals to reallocate efforts in such a ay that the ratio of aggregate effort in occupation i relative to occupation j decreases. This reallocation of aggregate effort raises the age rate in occupation j relative to occupation i, hich orks against age compression, thereby limiting the amount of compression hich can be achieved by raising the robot tax. Occupational shift effects. Finally, there is a set of occupational shift effects S ij L, B hich capture the change in income in occupation j as individuals move from occupation i to j in response to a marginal increase in the number of robots, hile holding effort schedules as ell as ages fixed. To elucidate: Suppose a marginal increase in robots leads to a rise in Y j L, B /Y i L, B. This change in relative factor prices induces some individuals to sitch from occupation i to j, thereby reducing income earned in i and increasing income earned in j. The occupational shift effect isolates that change in income hich is due to individuals sitching occupation, hile keeping fixed the schedules of effort as ell as the age densities.if, instead, one anted to compute the total change in income due to individuals sitching, one ould also have to account for changes in effort schedules and age densities. Hoever, these effects have already been taken into account in the effort reallocation effects. The sectoral shift effect S ij is negative if individuals move from occupation i to occupation j, thereby reducing income and more importantly effective labor in i relative to j. As a result of the reduction in relative aggregate labor supplies, age compression beteen i and j is partly offset, as the age rate in occupation i rises relative to occupation j. Again, if ξ j > 0, this effect ill ceteris paribus call for a loer robot tax. While I cannot in sign the multipliers ξ i for the general case, I sho that in the absence of incentive and redistributive effects, e have ξ i = 0 i I. Corollary 3. In the absence of incentive effects and redistributive effects, that is I i = R i = 0 i I, it follo that multipliers ξ i = 0 i I, hich implies that the optimal robot tax is zero τ = 0. Proof. See Appendix B.. The corollary implies that in the absence of incentive and redistributive effects, the optimal robot tax is zero. The effort-reallocation and occupational-shift effects are thus only relevant if either incentive or redistributive effects are present. The optimal robot tax is also zero if robots are equally complementary to labor in all occupations. In the latter case e have ε YM /Y R,B = ε YC /Y R,B = Marginal income taxes Turning to optimal marginal income taxes, I note that the only difference ith the inner problem in Rothschild and Scheuer 204 is the presence of the term qb in the resource constraint. In the inner problem this term is kept fixed and does therefore not influence the optimal allocation of indirect utilities and labor. As a consequence, the characterization of optimal marginal income taxes is the same as in Rothschild and Scheuer 204. Proposition 4. Let µ be the multiplier on the resource constraint and denote by µξ i the multiplier on the consistency condition pertaining to occupation i. Let ε u be the uncompensated labor supply elasticity and ε c the compensated labor supply elasticity. Given L 9

20 and B, optimal marginal tax rates are characterized by T = ξ j fl,b i Y i L, B f L,B i I ith η = ψ L,B z f L,B z + η f L,B U c z z exp εu s µ ε c s + ε u ε c dy s f L,B z dz, y s here y = l, and µˆη = µη /U c is the multiplier on the incentive constraint. The terms and ε c and ε u are the compensated and uncompensated labor supply elasticities, respectively. Proof. See Rothschild and Scheuer 204 As highlighted by Rothschild and Scheuer 204, the formula closely resembles the optimal income tax expression in the standard Mirrlees model. The only difference is the ξ i f i L,B correction term i I Y i L,B f L,B, hich adjusts retention rates T to account for general equilibrium effects. The fact that the expression for optimal marginal income tax rates in my model is the same as in Rothschild and Scheuer 204 does not mean that taxes on robots and other capital do not interact ith optimal income taxes. As in the simple model, the presence of these taxes changes the multipliers ξ i. In contrast to the simple model, it is no not anymore possible to solve for ξ i and I thus only study the effect of the robot tax on marginal income taxes as part of the quantitative analysis belo. 5 Quantitative analysis In this section, I study optimal taxes on robots and labor income quantitatively. To do so, I first calibrate the model to the US economy for a given tax system. The calibrated model is then used for optimal tax analysis. I choose 993 as the base year for the calibration, since this is the first year for hich data on industrial robots is available from the International Federation of Robotics. 6 The calibration proceeds in several steps. First, I obtain the skill distribution F θ as ell as skill prices Y i from data on ages and occupations. Then, based on F θ and Y i, and based on assumptions on individual labor supply, aggregate labor supplies L i are computed for a given parametric tax system. After this step, five parameters of the production function remain to be calibrated. Moreover, the initial value of robots needs to be set. Using L i as inputs, the parameters are calibrated against three internal targets: Y M, Y R, and Y C ; and to external targets: the elasticity of the average routine age ith respect to robots, and the price-elasticity of firm s use of robots. An additional restriction pins don the initial amount of robots. By matching skill prices Y i, the model generates a realistic age distribution. Moreover, by matching the external targets, the model captures the age response to an increase in robots, as ell as the response of firms to changes in robot prices and thus to a tax on robots. 6 An industrial robot is defined as an automatically controlled, reprogrammable, multipurpose manipulator programmable in three or more axes, hich can be either fixed in place or mobile for use in industrial automation applications. See 2.pdf 20

21 5. Skill-distribution I follo Heckman and Honoré 990 and Heckman and Sedlacek 990 by assuming that skills θ follo a joint log-normal distribution. Moreover, I normalize the mean to be the zero vector. Log skills are then distributed according to ln θ M ln θ R ln θ C N 0 0 0, σ 2 M ρ MR σ M σ R ρ MC σ M σ C ρ MR σ M σ R σ 2 R σ RC σ R σ C ρ MC σ M σ C ρ RC σ R σ C σ 2 C, 25 here I denote by σ i the standard deviation of latent skill for occupation i and by ρ ij the correlation beteen skills in occupations i and j. To estimate the parameters in 25, I use that according to the model, log ages in occupation i are given by ln i = ln Y i + ln θ i, 26 here Y i is the equilibrium skill price in occupation i. Assuming that log skills have zero mean, I can estimate both, skill prices and the parameters in 25, using data on ages and occupations. 7 According to the Roy model, an individual chooses that occupation in hich he earns the highest age. Selection thus needs to be taken into account hen deriving the distribution of observed ages ithin occupations. For example, due to selection, the mean of observed log ages does not correspond to the mean log skill price. Bi and Mukherjea 200 provide the distribution of the minimum of a tri-variate Normal distribution. Here, I need the distribution of the maximum of a tri-variate Normal hich follos easily. The log-likelihood function for estimating the parameters of 25 is provided in Appendix C. 5.. Data To estimate the skill-distribution, I use data on ages and occupational choice from the Current Population Survey CPS Merged Outgoing Rotation Groups MORG as prepared by the National Bureau of Economic Research NBER. 8 The data cover the years from 979 to 206. I focus on the year 993. I restrict the sample to individuals of age 6 to 64. Hourly ages are based on eekly earnings divided by usual hours orked per eek, here I exclude those individuals ho ork less than 35 hours. Like Autor and Dorn 203, I correct for top-coded eekly earnings by multiplying them by.5. All ages are converted into 206 dollar values using the personal consumption expenditures chain-type price index. 9 Occupations are categorized into three groups: manual non-routine, routine, and cognitive non-routine folloing Acemoglu and Autor I provide an overvie of the occupations contained in the three categories in Table. More details of the occupation classification and summary statistics are provided in Data Appendix E. 7 Without the assumption of a zero mean of log skills, skill prices cannot be identified from data on ages and occupational choice alone, as log skill prices cannot be distinguished from means of the log skill-distribution. I use skill prices as internal calibration targets. They do not have an interpretation in themselves. 8 See 9 I obtain the price index from 20 Acemoglu and Autor 20 use the labels abstract for cognitive non-routine and services for manual non-routine. See Cortes, 206 for the same classification and labels as used in this paper. 2

22 Table : Occupation classification based on Acemoglu and Autor 20 Three-group classification Cognitive non-routine Routine Manual non-routine Contained occupations Managers Professionals Technicians Sales Office and admin Production, craft and repair Operators, fabricators and laborers Protective service Food prep, buildings and grounds, cleaning Personal care and personal services Table 2: Calibration - Skill-distribution parameters and skill-prices σ M σ R σ C ρ MR ρ MC ρ RC Y M Y R Y C Note: Parameters obtained from ML estimation based on 55 using CPS MORG data for 993. Section E for a discussion of the data. See 5..2 Estimation I estimate the parameters of 25 by Maximum Likelihood. The results are given in Table 2. To assess the goodness of fit, I compute the estimated density of log ages for each occupation and plot it against a histogram of the data in Figure. Both line up ell, and the model thus generates a realistic age distribution. 2 The plots also reveal considerable overlap of age distributions across occupations. The effort reallocation effects discussed above ill thus play a role. 5.2 Production function I model the production function similar to Autor and Dorn 203 ho study the role of ICT capital for age polarization. Like this paper, they distinguish beteen manual non-routine, routine and cognitive non-routine labor. 22 Their production function has the feature that ICT capital substitutes for routine labor and complements non-routine labor, thereby affecting ages in the three occupations differentially. Hoever, hereas in Autor and Dorn 203 production of services and goods takes place in separate sectors, in my 2 A log-normal distribution is a good description of most of the age distribution. Hoever, ages at the top are better described by a Pareto distribution. In optimal tax analysis it is therefore common practice to append a Pareto tail to the top of the distribution. While this is crucial for the analysis of income taxes at the top, it is unlikely to change results for the optimal robot tax, hich is the focus of this paper. Since appending a Pareto tail to the ability distribution ould further complicate the estimation, this paper focuses on the log-normal distribution. 22 Autor and Dorn 203 refer to manual non-routine as Services and to cognitive non-routine as Abstract. 22

23 density log age log age log age a manual non-rout. b rout. c non-rout. cogn. Figure : Goodness of fit: Estimated density vs. data model a single consumption good is produced by combining all factors according to 23 Y L, B = A L α ML β C µl ρ ρ R ρ ρ ρ α β + µb ρ. 27 The inputs are manual non-routine labor L M, routine labor L R, cognitive non-routine labor L C, and robots B. Parameter A is a productivity shifter, α determines the income share going to manual labor, β, governs the share of income that goes to cognitive labor and 0 < µ < is the eight given to routine labor as compared to robots. Finally, ρ is the elasticity of substitution beteen robots and routine labor. 24 Ceteris paribus: if ρ >, an increase in the amount of robots leads the factor price of routine labor to fall relative to non-routine labor. Moreover, if ρ α + β >, the factor price for routine orkers drops in absolute terms as the number of robots increases. 25 The partial-equilibrium effect of robots on relative skill-prices is summarized by the elasticities ε YM /Y R,B = ε YC /Y R,B = µ ρ L /ρ R B > 0, ρ µ L /ρ R B + µl RB /ρ hich enter the expression for the optimal robot tax. The production function 27 thus satisfies Assumption. Moreover, it implies that in partial equilibrium, factor prices for manual non-routine labor and cognitive non-routine labor are symmetrically affected by a marginal change in robots. The same effect of robots on factor prices ould result if one instead considered a production structure and preferences as in Autor and Dorn 203, and by setting the elasticity of substitution beteen goods and services in consumption to unity but no the effect on the factor price for manual non-routine labor ould run via demand. 26 Having specified the production technology, I no proceed ith calibrating aggregate labor inputs. To do so, I need to specify preferences and individual labor supply as ell as the tax system. 23 In Autor and Dorn 203 output of goods and services is combined on the consumption side. In contrast, one can interpret 27 as combining output of different sectors on the production side ithout having to consider to commodities hile generating the same pattern of factor-price responses 24 Cortes et al. 207 also use a similar production function hen studying the role of automation for disappearing routine jobs. 25 Note that these conditions hold in a partial equilibrium setting. In the model, the choice of robots is endogenous, as is labor supply and occupational choice. Still, the conditions give an indication for the ranges of parameters required to generate certain age responses. 26 Herrendorf et al. 203 find that a consumption elasticity of substitution beteen manufacturing and services of close to one is in line ith the structural transformation observed in the US. 23

24 5.2. Preferences and labor supply Preferences over consumption and labor supply are quasi-linear and given by U c, l = c l+ ε +, ε here c is consumption, l is individual labor supply and ε is the labor supply elasticity. Unless indicated otherise, I set ε = The aggregate supply of labor in occupation i is specified as L i = l f i d i I. Y i Let T y denote the marginal income tax rate at income y. Using that preferences are quasi-linear, optimal labor supply l is then implicitly given by the solution to l ε = T y. 28 I use the parametric tax function proposed by Heathcote et al. 207 as an approximation to the US income tax schedule, ith T y = τ λy τ, 29 here τ is referred to as the progressivity parameter, hile λ can be used to calibrate total tax revenue. Based on 29, marginal tax rates are given by T y = τ λy τ. 30 Using that y = l and substituting 30 in 28, one can explicitly solve for l = [ τ λ τ ] ε +τε. Heathcote et al. 207 estimate τ = 0.8 based on data from the PSID and using the NBER TAXSIM. The value of λ is not reported in their paper, but can be inferred from their Figure hich features a marginal tax rate of 50% at an income of USD, implying λ Finally, the level of income generated by my model does not coincide ith annual income. To have the appropriate marginal tax rates apply to incomes in my model, I introduce a scaling parameter κ such that the median income in my model multiplied by κ corresponds to the median income in the data, hich implies κ With the scaling factor κ, I no have l = [ τ λκ τ τ ] ε +τε. 3 All parameters for computing labor supply are summarized in Table Remaining parameters Having calibrated aggregate labor supplies L i, it remains to calibrate the parameters of the production function and to set the amount of robots. I target skill prices Y M, Y R and Y C for the model to generate a realistic age distribution. I no turn to the external targets hich primarily govern the model s response to 27 See e.g. Blundell and Macurdy, 999; Meghir and Phillips, 2008 for a discussion on labor supply elasticities. 28 The median income is roughly 50000$. 24

25 Table 3: Calibration - Parameters for labor supply equation 3 Param. Value Explanation Source ε 0.3 Labor Supply Elasticity Blundell and Macurdy 999; Meghir and Phillips 2008 τ 0.8 Tax Progressivity Heathcote et al. 207 λ 6.56 Revenue Parameter Inferred from Heathcote et al. 207 κ 494 Income Scaling Parameter Computed based on median income a change in the amount of robots and a tax on robots. The empirical evidence regarding the impact of robots on ages is still scarce. Using data from the International Federation of Robotics IFR, 204, Acemoglu and Restrepo 207 estimate the effect of a change in exogenous exposure to industrial robots beteen 993 and 2007 on ages and employment at the level of US commuting zones. They argue that at the level of the aggregate economy, an additional robot per thousand orkers reduces ages by 0.5%. I convert this semi-elasticity into an elasticity by using that there ere 0.36 robots per thousand orkers in 993, hich implies an elasticity of Before targeting this elasticity, several comments are in order. First, in my model, the adoption of robots is endogenous. What is exogenous, hoever, is the productivity in robot production hich is one-to-one related to the producer price of robots. I therefore generate an exogenous change in robots by changing the producer price of robots. Second, I highlight that my model cannot generate a drop in average ages as the result of robot adoption: hile it is possible for ages of routine orkers to fall, orkers in non-routine occupations alays gain from robots. Does this mean that my model is inconsistent ith the results by Acemoglu and Restrepo 207? Not necessarily. They estimate local effects of robot exposure on ages, and then aggregate these effects up to the national level. While in their aggregation, they account for trade across commuting zones, they do not account for the possibility that some of the impact of robots on ages might not be local. For example, it is likely that not all parts of a product s value chain are located in the same commuting zone. The gains from adopting robots might thus be found in commuting zones hich are not directly affected by robots. Moreover, these gains might accrue to orkers in non-routine occupations. In fact, it might be that those orkers hose ages fall due to robot exposure in a commuting zone are concentrated in routine occupations. Acemoglu and Restrepo find a negative employment impact on occupations hich I classify as routine in this paper. In contrast, the effect of employment in non-routine occupations is small or insignificant. I therefore interpret the negative impact of robots on the average age as an effect on the average age of routine orkers, hich I target in the calibration. Since Acemoglu and Restrepo do not find evidence for occupational shifting, I do not allo for occupational choice in the calibration but ill allo for it later hen studying optimal taxation. I also target the impact of a change in the price of robots on robot adoption. This is an important moment since a tax on robots affects robot adoption via the same channel: by changing the consumer price of robots. According to data from the IFR 2006, the quality adjusted price of robots dropped by about 60% beteen 993 and 2005 the latest date for hich I have quality adjusted price data, hile the stock of robots roughly tripled. For lack of better evidence, I treat the price change in the data as exogenous and as the only driver of robot adoption, implying a price elasticity of robot adoption 25

26 Table 4: Calibration - Production function parameters Parameter A α β µ ρ Value Note: Parameters obtained from calibration to match the folloing targets: factor prices Y M, Y R, Y C, the elasticity of the average routine age rt. robots, the elasticity of robot adoption rt. the price of robots, and a drop in robot prices consistent ith the elasticities. of 5. Finally, for given parameters of the production function, the unit in hich robots are measured determines the return to robots. Hoever, there is no clear target for the return to robots. Although the IFR computes a price index for robots, it is only useful for assessing relative price changes. Instead of taking the initial amount of robots from the data, I therefore calibrate it such that a one-percent drop in the price of robots is in line ith the targeted elasticities. All targets, internal and external, are matched perfectly. I report the calibrated parameters in Table 4. The value of the productivity shifter A has no clear interpretation. It is required to match levels of skill prices. The share in total income going to manual orkers is 6%, hereas 45% goes to cognitive orkers. The eight given to routine orkers as compared to robots is close to. Finally, the elasticity of substitution beteen robots and routine ork is 4.9 and thus quite a bit higher than. The high elasticity is required in order to generate a negative age effect for routine orkers. Table 5 summarizes the elasticities for relative factor prices and average ages hich are implied by the calibration both for the model ithout occupational choice, as ell as for a model ith occupational choice. 29 As in the calibration, elasticities ith respect to the amount of robots are computed by exogenously loering the price of robots by %, and by then using the implied equilibrium responses. As discussed above, in partial equilibrium a change in robots affects factor prices for manual and cognitive labor symmetrically, and thus ε YM /Y R,B = ε YC/YR. With fixed occupational choice, general equilibrium elasticities,b and partial equilibrium elasticities coincide. Moreover, changes in average ages are driven by changes in factor prices only, and e therefore observe the same effect of robots on average ages for manual and cognitive orkers, M and C. The average age of routine orkers, R, falls, as required to match the targeted elasticity. 30 The effect of robots on the overall average is slightly positive. If occupational choice is possible, ages respond generally less to a change in robots. Routine orkers ho see their ages drop due to more robots being used, sitch to other occupations in this case mostly manual nonroutine occupations hich dampens both the decline of routine ages and the increase of non-routine ages The model ith occupational choice is based on the same calibrated parameters as the model ithout occupational choice. The only difference is that individuals may sitch occupation. 30 L,B The average age in occupation i is given by i fl,b i L,B d fl,b i d. 3 This pattern is in line ith Cortes et al., 207, ho find that those demographic groups hich are associated ith a decline in routine employment also account for an increase in employment in non-routine manual occupations as ell as for an increase in non-employment. 26

27 Occ. Choice Table 5: Calibration Results: Implied Elasticities ε GE Y M /Y R,B ε GE Y C /Y R,B ε GE,B ε GE M,B ε GE R,B ε GE C,B No Yes Note: ε GE Y M /Y R,B and ε GE Y C /Y R,B are the general equilibrium GE equivalents of ε YM /Y R,B and ε YC /Y R,B. ε,b GE is the GE elasticity of the average age rt. robots. ε GE i,b is the GE elasticity of the average age in occupation i {M, R, C} rt. robots. Elasticities are computed by loering the price of robots by % and by then computing the general equilibrium response of robots and average ages given the calibrated parameters, and given a parametric tax system. Without occupational choice, general and partial equilibrium elasticities coincide. 5.3 Social Welfare Weights I follo Rothschild and Scheuer 203 and assume relative social elfare eights according to Ψ = F r, here r parametrizes the government s desire to redistribute. Here, r = corresponds to utilitarian preferences, hich combined ith quasi-linear utility ould imply no redistribution. As r, the elfare eights approach that of a Ralsian social planner. I set r =.3 hich leads to marginal tax rates in the range of those observed empirically. 5.4 Optimal tax results I compute the optimal policy for the short-run scenario ithout occupational choice and the medium-run scenario ith occupational choice. The to scenarios only differ by one factor: hether or not individuals respond to the introduction of a robot tax by sitching occupation. To achieve this, I obtain results for the short-run as follos: first, equilibrium is computed for a model ith occupational choice, but in hich it is not possible to tax robots; next, I fix occupational choice, but allo for a tax on robots. In the short-run, the optimal tax on robots is 4.08%. I obtain the elfare impact of introducing a robot tax by computing the consumption equivalent. The optimal robot tax is orth 0.003% of GDP, hich based on US per capita GDP in 206 translates into.73$ per person per year. In contrast, in the medium-run ith occupational choice, the optimal robot tax is 0.42%. Intuitively, individuals adjust their occupation in response to the age changes brought about by the robot tax, thereby partly offsetting those changes. With occupational choice, the robot tax is thus less effective in compressing ages, and is therefore optimally smaller. Occupational choice lets the elfare gains evaporate: they are no reduced to a share of % of GDP or 0.0$ per person per year. Figure 2 plots optimal marginal income tax rates both for the case in hich robots can and cannot be taxed. First, note that marginal tax rates follo the common U-shape as for example in Saez 200. In the short-run, ithout occupational choice, the presence of a robot tax affects marginal taxes: tax rates are no higher at lo to medium ages, and loer at high ages. For the simple model in Section 3, I derive that marginal income taxes in the presence of a robot tax are ceteris paribus loer for manual and cognitive orkers. The effect on marginal income taxes for routine orkers is ambiguous. As cognitive orkers are concentrated at high ages, the result of loer taxes seems to carry over to the full model. Moreover, in terms of the simple model, loer marginal tax rates for manual orkers seem to be overturned by higher marginal taxes for routine orkers. Of course, the equivalence ith the simple model is imperfect as age distributions no 27

28 ith robot tax ithout robot tax ith robot tax ithout robot tax Marginal Tax Rate Marginal Tax Rate Wage a06c94eb762039e64a707d0d0eaac97 a short-run ithout occupational choice Wage a06c94eb762039e64a707d0d0eaac97 b medium-run ith occupational choice Figure 2: Effect of robot tax on optimal marginal income taxes Note: Results for moderate redistributive preferences ith r=.3. overlap occupations. If individuals can sitch occupation, the ability to tax robots does not visibly change optimal marginal income taxes, hich is not surprising, given that the optimal robot tax is very small. Next, I simulate a drop in the price of robots and compute the optimal policy as ell as the corresponding equilibrium. Figure 3 shos results for a scenario in hich employment shares are fixed at their initial level. Ruling out composition effects, average ages are driven by changing factor prices alone. The average age of routine orkers decreases, hereas average ages for non-routine occupations increase.the optimal robot tax increases from around 4% to 5% as robots get cheaper, but starts to fall once average ages of manual non-routine orkers overtake those of routine orkers. If individuals cannot sitch occupation at all, the tax thus remains sizable as long as routine orkers earn more than non-routine orkers. I no turn to the medium-run. Figure 4 shos the response of employment shares and average ages by occupation, assuming that occupational sitching is possible. As robots get cheaper, they substitute for more and more routine orkers, ho then sitch to either manual or cognitive occupations. As in the data, employment in cognitive occupations rises fastest. As the price of robots falls, the average age of routine orkers and cognitive orkers stays virtually constant, hereas manual orkers gain. Constant ages of routine and cognitive orkers can be explained by composition effects: as individuals move from routine to cognitive ork, the skill composition in cognitive occupations orsens, hile it improves in routine occupations thereby offsetting effects on ages driven by skill prices. Eventually, productivity gains due to cheaper robots dominate composition effects and average ages increase for all occupations. Moreover, as average earnings of manual and routine orkers become more equal, inequality falls. With occupational choice, the optimal robot tax is already lo initially, and drops ith the robot price, as its distributional benefits decrease. Eventually, the robot tax converges to zero, together ith the employment share in routine occupations. To summarize: ith occupational choice, not only is the optimal robot tax very small, it eventually approaches zero as robots become cheaper. 28

29 cognitive manual routine Robot Tax in pct Pct. drop in robot price a Average age Pct. drop in robot price b Robot Tax e309e7a eb8978c0c334a0ddbffa2 Figure 3: Effect of robot price drop on employment and ages: ithout occupational choice cognitive manual routine Robot Tax in pct Pct. drop in robot price Pct. drop in robot price Pct. drop in robot price e309e7a eb8978c0c334a0ddbffa2 a Employment shares b Average age c Robot Tax Figure 4: Effect of robot price drop on employment, ages and the robot tax: ith occupational choice 29

30 5.5 Robustness In this section, I discuss ho strongly the results depend on the response of routine ages to an increase in robots. While the baseline calibration makes use of the best empirical evidence available, there is arguably still a lot of uncertainty regarding the effect of robots on ages. Since I consider this calibration target as both important and the least certain, I investigate ho results change if the targeted elasticity is different. In particular, I study ho a response of routine ages to robots hich is 0, 20 and 30 times as large as in the baseline affects the results. All other calibration targets are kept the same, for the model to generate the same income distribution as in the baseline version, and for taxes to have the same effect on firms choice of robots. I report the results in Table 6 in the Appendix. As before, I compute the optimal robot tax both for the short-run ithout occupational choice, and for the medium-run ith occupational choice. I find that robot taxes are loer, the more responsive ages are to a change in robots. This is intuitive: the more ages respond to a change in robots, the loer a tax on robots is needed to achieve the desired age compression. I also compute elfare for both, the short- and medium-run. In the short-run, the elfare benefits per capita no increase to around 20$ in the most elastic case. Hoever, the medium-run elfare impact remains in the order of $ per person per year. The finding that in the medium-run, a tax on robots has negligible elfare effect is thus robust, even if robots have a much larger impact on ages than the available empirical evidence suggests. If at all, a tax on robots might only be orth considering in the short-run. 6 Conclusion I study the optimal taxation of robots and labor income in a model in hich robots substitute for routine ork and complement non-routine ork. The model features intensivemargin labor supply, endogenous ages and occupational choice. I find that in general, the optimal robot tax is not zero and that its sign can be ambiguous. The robot tax is used to exploit general equilibrium effects for age compression. Such compression relaxes incentive constraints, thereby making income redistribution less distortionary. The ambiguous sign of the optimal robot tax results from routine orkers being located in the middle of the income distribution, as documented by the empirical literature. Taxing robots reduces inequality at high incomes, thereby locally relaxing incentive constraints; but it increases inequality at lo incomes, and thus locally tightens incentive constraints. Whether robots should be taxed or subsidized then depends on several factors: ho robots affect ages across occupations, ho strongly incentive constraints are impacted by changing ages, and ho the government values redistribution at different parts of the age distribution. Additional effects need to be considered if there is heterogeneity ithin occupations and if occupational choice is endogenous. I bring the model to data for the US economy to assess the optimal policy quantitatively. The calibration matches the distribution of ages in the data. Moreover, it is informed by the empirical effect of robots on ages. I distinguish beteen a short-run scenario, in hich occupations are fixed, and a medium-run scenario in hich individuals can sitch occupation in response to the introduction of a robot tax. In the short-run, the optimal tax on the stock of robots is in the order of 4% and the consumption-equivalent elfare gain from introducing the tax is about.20$ per person per year. In the mediumrun, as individuals sitch occupation in response to a tax on robots, its effectiveness in compressing ages is reduced. The optimal robot tax and the elfare impact are then 30

31 much smaller: 0.4% and 0.0$. If the price of robots falls, the optimal robot tax goes to zero together ith the share of routine orkers. The negligible elfare impact in the medium-run is robust, even if ages respond much more strongly to an increase in robots than suggested by the data. The benefits of a robot tax in the short-run increase more substantially ith the age response, but remain very modest overall. Additional caveats to taxing robots exist, as this paper has abstracted from a number of important factors. With a tax on robots come considerable administrative costs as it requires to classify machinery into robots and non-robots. Moreover, I have abstracted from implications hich a tax on robots ould have in an open economy. As any tax on capital, a tax on robots could impact a firm s location choice ith additional implications for inequality and elfare. In light of the negligible elfare gains from a robot tax, it is doubtful that taxing robots ould be a good idea. 3

32 Appendix A Derivations for simple model A. Optimal robot tax Assuming that only the adjacent donard-binding incentive constraints are relevant, maximized social elfare is given by the Lagrangian L =f M ψ M V M + f R ψ R V R + f C ψ C V C Y R L, B +µη CR V C U c R V R, l R, l R Y C L, B Y M L, B +µη RM V R U c M V M, l M, l M Y R L, B +µξ M f M l M L M +µξ R f R l R L R +µξ C f C l C L C +µ Y L, B i I f i c i V i, l i qb, 32 ith I {M, R, C}, here µ is the multiplier on the resource constraint, µη CR is the multiplier on the incentive constraint for cognitive orkers, and µη RM is the multiplier on the incentive constraint for routine orkers. Moreover, µξ i is the multiplier on the consistency condition for occupation i. To find an expression for the optimal robot tax, differentiate the Lagrangian ith respect to B to obtain L B = µη CRU l c R V R, l R, l R Y R L, B Y C L, B µη RM U l c M V M, l M, l M Y M L, B Y R L, B + µ Y B L, B q. YR L, B B Y C L, B l M l R B YM L, B Y R L, B Define elasticities of relative skill prices ith respect to robots as ε YC /Y R,B = ε YR /Y C,B YR L, B YC L, B B Y C L, B Y R L, B B, ε YM /Y R,B YM L, B YR L, B B Y R L, B Y M L, B B. Use that Y B L, B = p = + τ q Y B L, B q = τq, set 33 equal to zero, rearrange and divide by µ to get Y R L, B τqb = η CR U l c R V R, l R, l R Y C L, B + η RM U l c M V M, l M, l M Y M L, B Y R L, B l R Y R L, B Y C L, B ε Y C /Y R,B Y M L, B l M Y R L, B ε Y M /Y R,B

33 No define the incentive effects in a similar ay as Rothschild and Scheuer 203 using that i = Y i and suppressing some arguments R R I CR η CR U l c R, l R l R 34 C C and to rite A.. M M I RM η RM U l c M, l M l M. 35 R R τqb =ε YC /Y R,BI CR ε YM /Y R,BI RM. 36 Expressions for incentive effects To obtain expressions for the incentive effects 34 and 35, one needs to derive expressions for the multipliers η CR and η RM. To do so, differentiate the Lagrangian in 32 ith respect to indirect utilities to obtain L V M = f M ψ M µη RM U c M c M c M, l M f M. R V M Equating to zero, using that c M / V M = /U c c M, l M and rearranging yields η RM = µ f M ψ M U c c M, l M U c c M, l M. M R Analogously, I obtain L V R = f R ψ R µη CR U c R c R c R, l R + µη RM f R, C V R hich after equating to zero, substituting for c R / V R = /U c c R, l R and rearranging yields η RM = µ f R U c c R, l R ψ R R + ˆη CR U c c R, l R. C Finally, I get L V C = f C ψ C + µη CR f C c C V C, hich after equating to zero, using c C / V C = /U c c C, l C and rearranging becomes η CR = µ f C U c c C, l C ψ C. With quasi-linear utility U c, l = c l+ ε +, ε e have µ = and U c =. As a result, one obtains η CR = f C ψ C, and η RM = f M ψ M = f R ψ R + η CR = f R ψ R + f C ψ C. 33

34 Moreover, quasi-linear utility leads to such that the incentive effects are U l = l ε + R ε I CR = f C ψ C l R, 37 C and + M ε I RM = f M ψ M l M. 38 R A.2 Optimal income tax To derive expressions for the optimal marginal income tax rates, I differentiate the Lagrangian 32 ith respect to individual labor supplies l i. A.2. First consider L l M = µη RM Use that Expression for T M [U c c V M, l M, l M M + µξ M f M + µf M M c l M R. c VM, l M + U l l M ] M M c V M, l M, l M R R 39 c = U l c V i, l i, l i l i U c c V i, l i, l i = i T i, 40 here the last step is based on the definition of marginal tax rates. Substituting in 39, I obtain ] L M M M = µη RM [U c c V M, l M, l M M T M + Ul c V M, l M, l M l M R R R + µξ M f M + µf M M M T M Setting equal to zero, dividing by µ and collecting terms yields T M + ξ M YM T M = M η RM [U c c V M, l M, l M f M R U c c V M, l M, l M U l c V M, l M, l M U l With quasi-linear utility the expression becomes T M + ξ M YM T M = ψ M c V M, l M, l M M R M R + ε. M R ]. 34

35 A.2.2 Expression for T R The first-order condition ith respect to l R is R [U c c V R, l R, l R I thus obtain L l R = µη CR + µξ R f R + µf R R c l R T R + ξ R YR T R C. c l R + U l = R η CR [U c c V R, l R, l R f R U c c V R, l R, l R U l c V R, l R, l R U l hich ith quasi-linear utility translates to A.2.3 T R + ξ R YR T R Expression for T C = f C f R ψ C C c V R, l R, l R R C c V R, l R, l R R C Finally, the first order condition ith respect to l C is A.2.4 L l c =µξ C f C + µf C R C C c l C Setting equal to zero, using 40 and rearranging leads to Expressions for multipliers ξ i T C = ξ C Y C. + ε.. R C ], R To derive expressions for the multipliers ξ i I differentiate the Lagrangian 32 ith respect to aggregate effective labor supplies L i and obtain L R = µη CR U l c V R, l R, l R L i C µη RM U l c V M, l M, l M M R µξ i + µ j {M,R,C} l R R L i L i l M Y j f j l j + Y B B. L i L i C M By Euler s Theorem, the last ro is zero. Using that at the optimum a change in L i has no effect on elfare and rearranging, I obtain R R ξ i = η CR U l c V R, l R, l R l R C L i C η RM U l c V M, l M, l M M R l M L i M R R. C ] 35

36 Using the definitions of the incentive effects 37 and 38, I arrive at ξ i =I CR L i +I RM L i R C C M R R R M. No define the semi-elasticities of relative ages ith respect to L i as ε R / C,L i L i R C C R and to rite ε M / R,L i L i M R R M ξ i = ε R / C,L i I CR + ε M / R,L i I RM. 4 Signing the multipliers and The sign of the multiplier is determined by the terms R L i C L i M R. The sign of ξ i is unambiguous if both of these terms have the same sign. Consider ξ M. We have M L M R < 0. A sufficient condition for ξ M < 0 is thus R L M C < 0. No consider ξ C. We have R L C C > 0, and hence M L C R > 0 is a sufficient condition for ξ C > 0. Finally, since R L R C < 0, and L R M R > 0 the sign of ξ R is ambiguous and depends on the magnitudes of the different terms in 4. B Optimal tax on robots ith continuous types and occupational choice In order to derive an expression for the optimal tax on robots, I use that at the optimum, a marginal change in robots B, has no first-order elfare effect. The elfare effect of a marginal change in B corresponds to differentiating the Lagrangian of the inner problem, evaluated at the optimal allocation, ith respect to B. The Lagrangian of the inner 36

37 problem is given by L = V c, l dψ + µ η U l c V, l, l l d L,B + µξ M L M l fl,b M d Y M L, B L,B + µξ R L R l fl,b R d Y R L, B + µξ C L C Y C L, B + µ Y K L, B K + l f C L,B d l f d c V, l f d qb The impact of a marginal change in B can be decomposed into four effects. First, there is a direct effect on the resource constraint. The other three effects result from the impact hich a change in B has on ages: The second effect is the direct result from a change in ages, leading to changes in l and V. The third effect is the indirect result of changing ages. As ages change, individuals move along the schedules l and V. Finally, as relative skill prices change, some individuals sitch occupation, hich has an effect on the consistency conditions. Instead of computing the effects by holding the schedules l and V fixed, using the envelope theorem, I follo Rothschild and Scheuer 204 and construct variations in the schedules l and V hich simplify the derivations. The idea is as follos: Instead of having to take into account that changes in B alter the age densities, the adjustment of l and V is chosen such that it offsets at each changes hich otherise ould require adjusting the densities. Denote the adjusted schedules by l and Ṽ. Schedule variations In hat follos, I first derive l and Ṽ hich requires some preparation: Indicate occupations by i I {M, R, C} and denote by β i B L, B Y i L, B B Y i L, B, the semi-elasticity of the skill-price Y i L, B ith respect to B. ql,b i θ such that { ql,b i, if θ orks in i θ = 0, otherise. Using that ages are given by θ = max {Y M L, B θ M, Y R L, B θ R, Y C L, B θ C }, the age of individual θ can thus be ritten as θ = i I q i L,B θ θy i L, B.. 42 I define the indicator 37

38 The semi-elasticity of ages ith respect to B for individual θ is thus θ B θ = i I = i I q i L,B θ Y i L, B B q i L,B θ β i B L, B. Y i L, B 43 Like Rothschild and Scheuer 204, I no construct l and Ṽ such that at each, changes in average labor supply and indirect utility hich come from individuals shifting along l and V are offset. I focus on deriving l. The derivations for Ṽ are analogue. First, consider an individual θ. A change db, hich leads to a change in θ, causes θ to adjust labor supply by l θ db = l ql,b i θ βb i L, B db, B i I here I use 43. No, note that the same age can be earned by different individuals if age distributions overlap across occupations. In order to compute the average adjustment in labor supply, e need to compute the expected adjustment over all types θ earning, that is l E [ ql,b i θ ] βb i L, B db = l fl,b i f L,B βi B L, B db, i I i I ] here I use that E [ql,b i θ corresponds to the share of individuals ho earn in occupation i, that is f i /f. I no obtain the adjusted schedule l by subtracting the change in l induced by a change db from l, that is l l l δ B L,B db, here I define δ B L,B i I f i L,B f L,B βi B L, B. The adjusted schedule Ṽ is Ṽ = V V δl,b B db. Since l and V are chosen optimally, by the envelope theorem, a marginal adjustment to l and Ṽ brought about by a marginal change db has no first-order effect on elfare. I no consider the effect of a change db on the different parts of the Lagrangian. Effect on objective The objective is given by V ψ L,B d. First, there is a direct effect, hich can be derived as above and hich is given by V ψl,b i i I ψ L,B βi B L, B db. Second, e need to adjust the V schedule to Ṽ by subtracting V δl,b i db to offset the indirect effect, leading to ψ i βb i L, B V L,B ψ L,B f L,B i db. f L,B i I 38

39 Integrating over all leads to i I β i B hich I rerite as V µ i I ith the redistributive effect defined as R i L, B V µ ψ i L,B ψ L,B f L,B i ψ L,B d db, f L,B β i B L, B R i L, B db, ψ i L,B ψ L,B f L,B i ψ L,B d. f L,B Incentive constraint effect The incentive constraint is as in Rothschild and Scheuer 204, ho sho that the schedule modification to l and Ṽ changes V U l c, l l / by V dδl,b i /d db. Integrating over all ages then leads to the folloing effect on the incentive constraint i I βb i L, B µ ith the incentive effect I i L, B η U c V d d f i L,B d db = µ βb i L, B I i L, B db, f L,B i I η U c V d d f i L,B d. f L,B Resource constraint effect The expression in 42 pertaining to the resource constraint is µ Y B L, B B + l f d c V, l f d B. 44 First, a change in B has a direct effect on Y B L, B B B, Y B L, B B + Y B L, B. 45 B Second, there is a direct effect on in the first integrand, leading to δ B L,B l f d. 46 Next, there are direct effects on l and V, and thus on c V, l. Hoever, these effects are exactly canceled out by varying the schedules to l and Ṽ to offset 39

40 the indirect effect. Use δl,b B = i βi fl,b i B to rerite 46 as fl,b i = i I = i I = i I δ i B l f d β i B L, B Y i L, B B Y i L, B L i. B Y i L, B l f i L,B d l f i L,B d Y i L,B No use that due to linear homogeneity of the production function, i I B L i + Y B L,B B B = 0, hence adding 45 and 46 and multiplying by µ yields the resource constraint effect Consistency constraint effects µ Y B L, B. Effort-reallocation effect Consider the consistency condition for M. The derivations for the other consistency conditions are analogue. First, rather than riting the condition in terms of ages, rite it in terms of types θ as θ M l M θ df θ L M. Θ No use that in the Roy model individuals fully specialize and rite l M θ = l q M L,B θ, here = Y M L, B θ M. The integrand can then be ritten as θ M q M L,B θ l. 47 A change in B affects the expression via three channels: First, there is a direct effect on l as a change in B affects ages. Second, there is an indirect effect, as due to a change in ages, individuals move along the schedule l. Third, a change in B affects relative skill prices across sectors, and thus occupational choice, captured by ql,b M θ. Here, I focus on the first to effects. The third effect ill be discussed as occupational shift effect belo. For a single individual θ, the direct effect changes 47 to θ M ql,b M θ l ql,b i θ βb i L, B db i I = βb i L, B l 2 ql,b M θ ql,b i θ db. Y M L, B i I 40

41 To compute the effect at = Y M L, B θ M, one needs to take the expectation over all individuals θ ho earn, leading to Y M L, B βb i L, B l 2 E [ ql,b M θ ql,b i θ ] db. 48 i I To offset the indirect effect on 47, I change the schedule l to l by subtracting l i I E [ q i L,B θ ] β i B L, B db, leading to θ M q M L,B θ l i I E [ q i L,B θ ] β i B L, B db. Again, this expression is for a single individual. To compute the effect at, I take the expectation over all θ earning, hich by the la of iterated expectations yields θ M l i I E [ q M L,B θ ] E [ q i L,B θ ] β i B L, B db = Y M L, B βb i L, B l 2 E [ ql,b M θ ] E [ ql,b i θ ] db. i I Combine 48 and 49 to arrive at βb i L, B l 2 Y M L, B i I [ E q M L,B θ ql,b i θ ] E [ ql,b M θ ] E [ ql,b i θ ] db = βb i L, B l 2 Cov [ ql,b i θ, ql,b M θ ] db. Y M L, B i I Integrate over ages to obtain βb i L,B L, B l 2 Cov [ ql,b i θ, ql,b M θ ] f d db, Y M L, B i I and define C im and generally for j I C ij L,B l 2 Cov [ ql,b i θ, ql,b M θ ] f d, Y M L, B L,B ] l 2 Cov [ql,b i θ, q j L,B Y j L, B θ f d. The effort-reallocation effect for the consistency condition hich corresponds to occupation j I is then µξ j βb i L, B C ij db. i I 4

42 No effort-reallocation effect if age distributions do not overlap No suppose that age distributions do not overlap sectors, that is, ql,b i θ = 0 for i M. The expression in 50 then becomes Y M L, B βm B L, B l 2 E [ q M L,B θ q M L,B θ ] E [ q M L,B θ ] E [ q M L,B θ ] db. No use that ql,b M θ qm L,B θ = qm L,B θ. Moreover, ith no overlap of distributions all individuals ho earn = Y M L, B θ M are in occupation M, and thus ql,b M θ =. As a result, the term in parenthesis becomes E [ ql,b M θ ql,b M θ ] E [ ql,b M θ ] E [ ql,b M θ ] =E [ ql,b M θ ] E [ ql,b M θ ] E [ ql,b M θ ] =, and hence there is no effort-reallocation effect if age distributions do not overlap across occupations. Occupational-shift effect To derive the impact of occupational change on the consistency conditions, I focus on the effect on the condition for occupation M. The derivations for the other consistency conditions are analogue. Focus on Y M L, B y, ith income y l. I first derive ho income y earned in occupation M changes due to occupational shifts in response to an increase in B. A change in B alters skill prices Y i L, B, hich in turn affect occupational choice. Write the impact of a marginal increase in B on skill price Y i L, B as Y i L, B βb i L, B. No, first consider ho a marginal increase in Y R L, B affects occupational choice, and thus income y earned in occupation M. As Y R L, B increases, individuals are going to shift from occupation M to R. Since I consider a marginal change in Y R, I focus on those individuals ho are indifferent beteen M and R, hich implies that they earn the same age in both occupations, and thus θ M Y M L, B = θ R Y R L, B θ R = θ M Y M L, B Y R L, B. 5 Moreover, individuals need to be better off orking in occupation M than orking in occupation C, thus θ M Y M L, B θ C Y C L, B θ C θ M Y M L, B Y C L, B. 52 Having characterized the affected individuals, I next, consider ho a change in relative prices due to a change in Y R, Y R, affects conditions 5 and 52. We get θr Y M L, B = θ M Y R L, B + θ YM L, B M Y R, Y R Y R L, B hile 52 is not affected. 42

43 a Before increase in YR b After increase in YR Figure 5: Illustration of occupational shifting due to increase in YR Note: The axes correspond to the three dimensions of skill, θm, θr, θc. The green volume corresponds to the mass of manual orkers, the yello volume to the mass of cognitive orkers, and the blue volume to the mass of routine orkers. As YR increases, manual and cognitive orkers move into routine occupations. At a given point θm, θc, θr, geometrically, the height of the polyhedron of individuals changing from occupation M to occupation R due to an increase in YR is given by YM L, B θr θr = θm YR. YR YR L, B The density at this point is YM L, B f θm, θr, θc = f θm, θm, θc. YR L, B In order to compute the mass of individuals ho sitch from occupation M to R at a L,B given θm ith θr = θm YYM, e first need to integrate over all values θc for hich R L,B L,B individuals do not ork in occupation C. This range of values is given by [θc, θm YYM. C L,B Integrating over the density yields Z θm YM L,B YC L,B f θc YM L, B θm, θm, θc YR L, B dθc, hich at a given θm corresponds to a slice of the surface of indifference beteen sectors M and R. To arrive at the first expression for the mass of sitchers, e need to multiply θ, leading to this slice of the surface by the height of the polyhedron of sitchers, θm M θm YR YM L, B YR L, B Z θm YR YM L,B YC L,B θc f YM L, B θm, θm, θc YR L, B dθc. In order to compute the income moving from occupation M to occupation R, due to YR, rite income as θm YM L, B ` θm YM L, B, 43