Generalized compensation principle

Schulz, Karl; Tsyvinski, Aleh; Werquin, Nicolas Article Generalized compensation principle Theoretical Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Schulz, Karl; Tsyvinski, Aleh; Werquin, Nicolas (2023) : Generalized compensation principle, Theoretical Economics, ISSN 1555-7561, The Econometric Society, New Haven, CT, Vol. 18, Iss. 4, pp. 1665-1710, https://doi.org/10.3982/TE3971 This Version is available at: https://hdl.handle.net/10419/296450 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/4.0/ Theoretical Economics 18 (2023), 1665–1710 1555-7561/20231665 Generalized compensation principle Karl Schulz Department of Economics, University of St. Gallen Aleh Tsyvinski Department of Economics, Yale University Nicolas Werquin Economic Research Department, Federal Reserve Bank of Chicago Economic disruptions generally create winners and losers. The compensation problem consists of designing a reform of the existing income tax system that offsets the welfare losses of the latter by redistributing the gains of the former. We derive a formula for the compensating tax reform and its impact on the government budget when only distortionary tax instruments are available and wages are determined endogenously in general equilibrium. We apply this result to the compensation of robotization in the United States. Keywords. Compensation principle, distortionary taxation, general equilibrium, wage disruption. JEL classification. D61, D63, H21, H31. Introduction Economic disruptions, for instance, technological change, opening to international trade, inflows of immigration, or exogenous price shocks, generally create winners and losers, i.e., real wage and welfare gains for some individuals and welfare losses for others. The welfare compensation problem consists of designing a reform of the tax-and- transfer system that offsets the losses by redistributing the winners’ gains. We solve this Karl Schulz: [email protected] Aleh Tsyvinski: [email protected] Nicolas Werquin: [email protected] Opinions expressed in this article are those of the authors and do not necessarily reflect the views of the Federal Reserve Bank of Chicago or the Federal Reserve System. We are particularly grateful to Dominik Sachs. We thank Andy Atkeson, Gadi Barlevy, Marco Bassetto, Joydeep Bhattacharya, Don Brown, Ariel Burstein, Jeff Campbell, Raj Chetty, Wolfgang Dauth, Georgy Egorov, Eduardo Faingold, Antoine Ferey, Axelle Ferriere, Sebastian Findeisen, François Gourio, Nathan Hendren, Jim Hines, Marek Kapicka, Louis Kaplow, Rohan Kekre, Eungsik Kim, Helen Koshy, Nicolas Lambert, Tim Lee, Etienne Lehmann, Jesse Perla, Pascual Restrepo, Emmanuel Saez, Stefanie Stantcheva, Kjetil Storesletten, Christopher Tonetti, and Gianluca Violante for useful comments, and Craig Epstein for excellent research assistance. This research received financial support from the ADEMU network grant, part of the EU H2020 program (Grant 649396). Karl Schulz gratefully acknowledges financial support from the basic research fund at the University of St. Gallen. ©2023 The Authors. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at https://econtheory.org.https://doi.org/10.3982/TE3971 1666 Schulz, Tsyvinski, and Werquin Theoretical Economics 18 (2023) problem in an environment where only distortionary taxes are available, and wages are determined endogenously in general equilibrium. The traditional public finance literature (Kaldor (1939), Hicks (1939,1940)) shows that in an economy where individualized lump-sum taxes are available, the tax reform that redistributes the welfare gains and losses caused by a disruption is straightforward: It simply consists of raising (resp., lowering) the lump-sum tax liability of agents whose welfare increases (resp., decreases) from the shock by an amount equal to their compensating variation. This standard Kaldor–Hicks approach is flawed, however. First, because of asymmetric information, as in Mirrlees (1971), the only tax instrument at the government’s disposal, the labor income tax, is distortionary. Second, many economic shocks require explicitly modeling the endogeneity of wages. Consider, for example, an inflow of low-skilled immigration, i.e., an exogenous (relative) increase in the total supply of low-skilled labor. In partial equilibrium, i.e., if wages were exogenous, this would not affect the individual utility of resident workers. However, in general equilibrium, this disruption lowers the wage of low-skilled workers whose marginal product of labor is decreasing and raises the wage of high-skilled workers whose labor is complementary to the tasks performed by the incoming workers; see, e.g., Card (2009). Therefore, immigration flows have nontrivial welfare consequences only because the endogeneity of wages is explicitly taken into account. Similarly, the impact of automation on inequality can be understood as a race between education—the supply of high-skilled workers—and technology; see, e.g., Katz and Murphy (1992). In both of these examples, movements in the relative labor supplies of different skills fundamentally drive trends in relative wages. As a result, standard public finance models in which labor supply is endogenous but wages are exogenous cannot properly account for the welfare implications of these disruptions. Now suppose that in response to the disruption, the government implements a tax reform that aims to compensate the welfare losses of agents whose wages are adversely impacted. Since the only available policy tools are distortionary taxes, such a reform affects workers’ labor supply choices. These labor supply adjustments impact individuals’ wages and utility by the same general equilibrium forces we just described. The resulting welfare effects themselves need to be accounted for and compensated. But this can only be done through the distortionary tax code, which creates further welfare gains and losses, and so on. Hence the combination of distortionary taxes and endogenous wages leads to an a priori complex fixed point problem for the compensating tax reform. We start by analyzing the welfare compensation problem in a partial-equilibrium environment where wages are exogenous. We show that the design of the compensating tax reform that brings every agent’s utility back to its pre-disruption level is simple, even when distortionary income taxes are the only available instrument. The key insight here is that individual utility is only affected by the average tax rates of the reform; that is, the changes in marginal tax rates do not impact welfare. This follows from an envelope theorem argument: The marginal tax rate that individuals face affects their indirect utility only through their optimal labor supply decision so that the corresponding welfare effect is second order. As a consequence, it is straightforward to show that a suitably Theoretical Economics 18 (2023) Generalized compensation principle 1667 designed adjustment in the average tax rate is sufficient to achieve exact welfare compensation. Namely, one that exactly cancels out the after-tax income gain or loss caused by the exogenous disruption, regardless of the marginal tax rate changes it induces. The analysis becomes significantly more complex when distortionary taxes are coupled with general-equilibrium forces. In this case, despite the envelope theorem, endogenous changes in labor supply do matter for welfare through their impact on wages, resulting from the decreasing marginal productivities and the production complementarities. Therefore, in general equilibrium, because of the labor supply responses that they generate, the tax reform’s marginal rates directly affect the agent’s utility, even conditional on the average tax rate change. As a result, to determine the compensating tax reform, we must solve for its average and marginal rates simultaneously. This is the key difference from the partial-equilibrium environment and the main technical challenge of our paper. We show that the solution to the welfare compensation problem can be formalized as the solution to an integro-differential equation. Our first main result is to derive a formula for the compensating tax reform in general equilibrium in terms of elasticity variables that can be measured empirically. This formula is valid for arbitrary preferences, initial tax code, production function, and wage disruptions as long as they are marginal; that is, our tax reform compensates for the first-order welfare effects caused by general disruptions. Our second main result is to derive a formula for the fiscal surplus (or deficit), i.e., the impact of the disruption and its compensation on the government budget. Thus, our analysis generalizes the traditional Kaldor–Hicks criterion and provides a simple test to determine whether economic shocks or policies are compensable, that is, whether offsetting the individual welfare changes using only distortionary tax instruments is budget-feasible. More generally, the value of the fiscal surplus (not only its sign) provides a relevant monetary measure of the aggregate welfare gains or losses from the disruption. The main economic insight of our general-equilibrium compensation formula is that whenever the exogenous disruption features a sharp nonlinearity around some income level (say, a large wage drop), the compensating policy smoothes out the distortions by spreading the tax rebates over the entire range of incomes below that level. More specifically, exact compensation is achieved via a progressive tax reform over that range of incomes, with monotonic reductions in marginal and average tax rates. The rate of progressivity of the compensating tax reform—i.e., how fast the average tax rate grows with income—is given by the ratio between the labor demand and labor supply elasticities, net of the rate of progressivity of the initial tax code. These results stand in contrast to the partial-equilibrium compensation, which tracks the nonlinearities of the wage disruption one-for-one. To understand this result and derive further analytical properties of the compensation, we apply our formula to several simple disruptions. We assume that the production function is constant elasticity of substitution (CES) and consider first a disruption that affects all wages uniformly. In this case, the compensating tax reform in general equilibrium coincides with the partial-equilibrium compensation. This follows from the fact 1668 Schulz, Tsyvinski, and Werquin Theoretical Economics 18 (2023) that the endogenous wage responses caused by the decreasing marginal product of labor and the skill complementarities in production exactly offset each other—a consequence of Euler’s homogeneous function theorem—thus removing the need to adjust the partial-equilibrium policy. Now consider the polar opposite case, where a single skill is adversely affected by the disruption, thus creating a sharp nonlinearity in wage losses. In partial equilibrium, the compensation would grant a large tax rebate to the corresponding income level. However, doing so would involve large movements in the marginal tax rates around that income level, which would cause sizeable unintended welfare consequences in general equilibrium. Instead, to offset these welfare effects from wage responses, the appropriate policy smoothes the tax changes by progressively reducing the tax liabilities of all incomes below that of the disrupted agent. When the marginal product of labor is decreasing, the tax reform must ensure that the (negative) welfare effects caused by a reduction in any worker’s marginal tax rate are offset by the (positive) welfare effects of reducing her average tax rate. If, as empirically relevant, the ratio of the elasticities of labor demand and labor supply is larger than the rate of progressivity of the preexisting tax code, the reduction in the marginal tax rate at each income level must be compensated by an even larger reduction in the average tax rate. Thus, the tax rebates on earnings below that of the disrupted worker are exponentially growing, i.e., progressive. Next, skill complementarities in production generate additional indirect wage adjustments that also need to be compensated. The marginal tax rates of this second round of compensation cause, in turn, further wage and welfare changes, which themselves require compensation, and so on. We generally solve this fixed point problem by defining inductively a sequence of functions that each capture a round of general-equilibrium wage changes and their compensation. In other words, when the shock hits, we adjust the tax schedule to compensate for it, ignoring production complementarities. We then compute the first round of general equilibrium effects on wages, compensate for them again, and so forth until convergence. If the production function is CES, this series boils down to a uniform shift of the marginal tax rates, adding to the progressive component described in the previous paragraph. We then apply our compensating formula, under a CES technology, to disruptions that affect all incomes uniformly above (or below) a threshold or over an interior range of incomes. We show that one can analytically decompose the compensation of such disruptions into the sum of three elements: first, the partial-equilibrium reform that tracks income gains and losses one for one; second, a correction for the decreasing marginal product of labor that features progressively growing tax changes—at a rate given by the simple combination of elasticities described above—on all incomes below each sharp nonlinearity in wage gains and losses; third, a correction for the cross-wage complementarities that amounts to a uniform shift in tax rates. We then quantitatively explore the robustness of the compensating tax reform to the size of the labor supply and demand elasticities, the initial tax schedule, and the (nonmarginal) size of the disruption. We show, in particular, that our tax reform compensates for at least 95% (resp., 78%, 53%) of the welfare losses of a disruption that leads to 1% (resp., 5%, 10%) wage losses. Theoretical Economics 18 (2023) Generalized compensation principle 1669 We finally apply our theory in the context of the robotization of the U.S. economy between 1990 and 2007. Acemoglu and Restrepo (2020) estimate the impact of an additional robot per one thousand workers on the wages of different skills—roughly the amount observed in the United States between these dates. The closed-form solution we derive allows us to easily evaluate the compensating reform quantitatively. For instance, we find that an additional robot per thousand workers requires compensating agents at the 10th (resp., 85th) percentile of the wage distribution by 97% of their income loss (resp., 132% of the income gain) from the disruption. This represents a 0.7 percentage point (resp., 0.08 percentage point (pp)) decrease in their average tax rate and generates a $145 budget deficit for the government. Related literature Our theoretical analysis builds on Kaplow (2004,2012)andHendren (2020), who extend the Kaldor–Hicks principle to the case of distortionary taxes in partial equilibrium using inverse-optimum weights (see, e.g., Jacobs, Jongen, and Zoutman (2017)). Our main contribution is the analysis of the general equilibrium environment in which wages are endogenous. Guesnerie (1998), Itskhoki (2008), and Antras, de Gortari, and Itskhoki (2016) study compensating tax reforms and the welfare implications of trade liberalization in a general-equilibrium framework similar to ours. They restrict the analysis to specific classes of distortionary taxes and tax reforms, however: linear for Guesnerie (1998) and with a constant rate of progressivity (as in Bénabou (2002), Heathcote, Storesletten, and Violante (2017)) for Antras, de Gortari, and Itskhoki (2016). While we do not consider a sophisticated trade model, we solve the compensation problem by allowing for arbitrarily nonlinear tax schedules and nonlinear tax reforms. The generality of the tax reforms, in particular, is necessary to ensure that every agent’s welfare is compensated for. Andersen and Bhattacharya (2017,2020)andAndersen, Bhattacharya, and Liu (2020) extend the Kaldor–Hicks approach to dynamic overlapping generations (OLG) settings; they focus on achieving generation-by-generation Pareto neutrality via taxation and debt, and do not consider intra-generational heterogeneity. More broadly, our model is within the class of Mirrleesian economies in general equilibrium. Stiglitz (1982), Rothschild and Scheuer (2013), and Sachs, Tsyvinski, and Werquin (2020) study optimal taxes in this environment for given production and social welfare functions. Ales, Kurnaz, and Sleet (2015), Guerreiro, Rebelo, and Teles (2017), Uwe (2018), Costinot and Werning (2018), Hosseini and Shourideh (2018), Beraja and Zorzi (2021) characterize optimal income taxes, robot taxation, or trade policies following disruptions. Costinot and Werning (2018), in particular, derive optimal robot taxes by studying, like us, tax changes that keep utility unchanged. In contrast to these papers, our goal is to study the specific tax reform that achieves such compensation in general equilibrium. Finally, our paper is related to the literature that analyzes the set of Pareto efficient taxes—an important alternative to the standard optimal tax problem that does not require positing a social welfare function; see, e.g., Werning (2007), Scheuer and Werning (2017), Bierbrauer and Boyer (2014), Lorenz and Sachs (2016), Bierbrauer, Boyer, and Hansen (2020). We discuss in more detail the relationship to the optimal and Pareto efficient taxation literature in Section 4.1. Finally, from a technical viewpoint, our derivations are based on the general-equilibrium tax incidence analysis of Sachs, Tsyvinski, and Werquin (2020). However, this paper does not address the 1670 Schulz, Tsyvinski, and Werquin Theoretical Economics 18 (2023) compensation problem, which requires solving not only for labor supply changes in response to a given tax reform, but also for the tax reform itself. Outline In Section 1, we set up the model and define the welfare compensation problem. In Section 2, we solve for the compensating tax reform and the fiscal surplus in partial and general equilibrium. In Section 3, we analyze the compensating tax reform considering various examples of disruptions and an empirical application to the robot disruption. Section 4concludes with a discussion of the differences between the compensation approach and the standard optimal taxation approach. The proofs are gathered in the Appendix. 1. Welfare compensation problem 1.1 Initial equilibrium There is a continuum of measure 1 of individuals indexed by their skill i∈[0, 1].Inthe initial (undisrupted) economy, agents iearn a pre-tax wage rate wi∈R+that they take as given. Without loss of generality, we order skills so that wages wiare increasing in i. Thus, the skill index i∈[0, 1]can be interpreted as the agent’s percentile in the wage distribution of the initial economy. Agents with skill ihave preferences over consumption cand labor supply lthat are represented by the utility function ui(c,l). They choose effort liand earn pre-tax income yi=wili. Under standard assumptions on preferences, income yi=wiliis strictly increasing in i, so that there are one-to-one maps between skills i, wages wi, and incomes yiin the initial equilibrium.1We assume that incomes yibelong to an interval [y,¯ y]⊂R+and have a continuous density f(·). The government levies a nonlinear income tax. The tax schedule T:R+→Ris twice continuously differentiable. Agents iconsume their after-tax income ci=yi−T(yi). Their indirect utility Uiis thus given by Ui=uiwili−T(wili),li,(1) where the labor supply lisatisfies the first-order condition2 − ∂ui ∂l wili−T(wili),li ∂ui ∂c wili−T(wili),li =1−T(wili)wi.(2) There is a continuum of mass 1 of identical firms whose inputs in production are the aggregate labor supplies Ljof all types j∈[0, 1]. The production function has constant 1This is the case, for instance, if agents have a common utility function uthat satisfies the Spence– Mirrlees condition. Importantly, because we focus on marginal perturbations, this ordering of wages need not be preserved by the disruption and the tax reform. 2We assume that this equation has a unique solution. Theoretical Economics 18 (2023) Generalized compensation principle 1671 returns to scale and is denoted by F(L),whereL≡{Lj}j∈[0,1]. In equilibrium, firms earn no profits and the wage wiis equal to the marginal product of labor of skill i, wi=∂F ∂Li (L).(3) We finally denote government revenue by R=1 0 T(wili)di.(4) For future reference, we define the local rate of progressivity of the tax schedule at income yias (minus) the elasticity of the retention rate ri=1−T(yi)with respect to gross income yi,thatis,p(yi)≡−∂ln(1−T(yi))/∂ lnyi. 1.2 Wage disruption and tax reform Consider an exogenous perturbation ˆ wE={ˆ wE i}i∈[0,1]of the wage distribution w= {wi}i∈[0,1],where ˆ wE i∈Rfor all i.Thatis,thewageofagentichanges, on impact, from wito wi(1+μˆ wE i),whereμ>0 is a constant. Such a disruption can be caused by various exogenous shocks, e.g., technological change, which affects the production function F, or immigration flows, which modify the relative shares of different skills in the economy.3Without loss of generality, we normalize supi∈[0,1]|ˆ wE i|=1.4Thus, the map {ˆ wE i}i∈[0,1]defines the (infinite-dimensional) direction of the disruption, while the scalar μparametrizes its size. Following the disruption, the government can implement an arbitrarily nonlinear tax reform ˆ T(·), whereby the statutory tax payment at income yichanges from T(yi)to T(yi)+μˆ T(yi).5 In response to the wage disruption ˆ wEand the tax reform ˆ T, individuals optimally adjust their labor supply. In general equilibrium, these decisions impact their wages, which in turn further modify their labor supply choices, and so on. We denote by μˆ wi and μˆ lithe total endogenous percentage changes in the wage and labor supply of individual ibetween the initial and the perturbed equilibria. Thus, the wages and labor supplies in the disrupted economy are, respectively, given by wi(1+μˆ wE i+μˆ wi)and li(1+μˆ li). We define agent i’s compensating variation μˆ Uias the change in utility between the initial and the perturbed equilibria, normalized by the (initial) marginal utility of consumption ∂ui/∂c so as to obtain a monetary measure of the welfare gains and losses. Finally, we denote by μˆ Rthe change in government revenue caused by the disruption and the tax reform, or fiscal surplus. 3For instance, the wage disruption implied by a change in the production function from Fto ˜ Fis given by μˆ wE i≡1 wi[∂˜ F/∂Li−∂F/∂Li]for all i. 4Throughout the paper, we focus on continuously differentiable functions i→ ˆ wE ion [0, 1]. 5In Section 2.4, we assume that the tax reforms ˆ Tthat the government can implement belong to the Banach space of functions that are continuously differentiable and bounded, with bounded first derivative. 1672 Schulz, Tsyvinski, and Werquin Theoretical Economics 18 (2023) 1.3 Compensation problem Compensating tax reform The welfare compensation problem consists of designing a reform ˆ Tof the existing tax code that offsets the welfare gains and losses of the wage disruption μˆ wE. Hence, the tax reform ˆ Tmust be designed such that each agent’s compensating variation is equal to zero: ˆ Ui=0∀i∈[0, 1].(5) We say that the disruption {ˆ wE i}i∈[0,1]is compensable if the fiscal surplus is nonnegative, i.e., ˆ R≥0. Marginal wage disruptions In this paper, we characterize analytically the solution to the welfare compensation problem for marginal wage disruptions, i.e., as μ→0. Thus, our exercise consists of designing and evaluating the fiscal impact of a tax reform ˆ Tthat compensates the first-order welfare effects of a small wage disruption in the direction ˆ wE.InSection3.4, we explore quantitatively how our compensating tax reform fares against large shocks. Aggregate gains of disruptions If a disruption is compensable, then it is possible to find a reform of the initial tax code Tthat achieves a strict Pareto improvement.6Conversely, it is possible that a disruption generates strictly positive aggregate gains, both in terms of gross incomes and government revenue, but that these gains are not compensable (i.e., the fiscal surplus ˆ Ris negative) if the labor supply distortions that the compensation would generate outweigh these gains. More generally, the value of the fiscal surplus, not only its sign, carries important information: It provides a metric that allows us to compare, in monetary units, the aggregate welfare gains (or losses) of different economic shocks. For example, suppose that a given disruption (say, automation) generates more revenue, after implementing the compensating tax reform, than another (say, an inflow of immigration). It follows that the government can achieve a strictly better Pareto improvement from the former shock. Remark: A more general problem It is natural to wonder what a compensating tax reform would be if the government’s objective were to compensate all agents to make their welfare at least as large (rather than exactly as large) as in the initial economy, i.e., ˆ Ui≥0 for all i. To address this problem, we can directly specify the nonzero welfare improvements (or losses) ˆ Ui=hi∈Rthat one wants to achieve for each skill level. We then solve the compensation problem by replacing 0 with hiin the right-hand side of (5). The differential equation derived in Lemma 2below now features the exogenous function h. The corresponding tax reform and fiscal surplus can then be straightforwardly derived following identical steps as in the proofs of Propositions 1and 2. 6For instance, the government can redistribute lump sum the budget surplus uniformly to all workers. Theoretical Economics 18 (2023) Generalized compensation principle 1679 and the fiscal surplus reads as14 ˆ R=Eyˆ E(y)−ET(y)yy y εd y εd z (y,z)εr zˆ (z)dz, (18) with ˆ (z)≡d[φ−1 zˆ E(z)+(z)]/d lnz. In these expressions, we let (y,z)=εd z εr zzexp−z y εd x εr xxdx and (z)=∞ n=1(n)(z)is defined inductively, for all n≥1,as (n)(z)=¯ y y z,yεd yφy(n−1)(y)−¯ y z (y,x)(n−1)(x)dxdy, with (0)(z)≡φ−1 zˆ E(z). If the production function is CES, (z)is a constant E[y(φy(0)(y)−¯ y y(y,x)(0)(x)dx)]/E[y¯ y y(y,x)dx]. Formulas (17)and(18) depend only on the exogenous wage disruption ˆ E(or ˆ wE) and on variables that are observed in the pre-disruption economy: statutory marginal tax rates, elasticities of labor supply εr y,εn y,εw y, elasticities of labor demand εd y, and elasticities of substitution between skills y,z(or γy,z). It is thus straightforward to implement such a tax reform in practice. In Section 3, we analyze the shape of the compensation (17) in detail, and study various examples and an empirical application. Before proceeding, we describe the structure of the compensation formula (17). It expresses the tax reform as a series of partial compensations. Suppose first that the marginal product of labor is decreasing but that skills are perfect substitutes in production, so the cross-wage elasticities z,yare equal to 0. In this case, the compensation of the exogenous wage disruption (0)(z)≡φ−1 zˆ E(z)reduces to (1− T(y))yy y(y,z)(0)(z)dz (we analyze this expression in the next section). For a general production function, this compensation and the cross-wage effects z,ygenerate further wage changes for agent zgiven by (1)(z). These must be compensated by (1−T(y))yy y(y,z)(1)(z)dz (second round of “compensating the compensation”), thus leading to further changes in wages and so on. Repeating this procedure for all n≥2 yields the full compensation (17), where each term (n)(z)in the series captures the (cross-)wage changes caused by the (n−1)th round of partial compensation. In other words, when the exogenous shock hits, we adjust the tax schedule to compensate for it, ignoring the endogeneity of wages due to production complementarities, i.e., treating each labor market with its own labor demand curve and decreasing marginal product of labor, independently of the others. We then compute the first round of general-equilibrium effects on wages, naively compensate for them again, and so forth, until we have settled. When the production function is CES, this iterative procedure 14This expressions assumes for simplicity that ¯ y→∞; the general expression is derived in the proof in the Appendix. 1680 Schulz, Tsyvinski, and Werquin Theoretical Economics 18 (2023) becomes particularly simple: In this case, each intermediate round of compensation leads to uniform wage changes across workers (i.e., constant (n)(·)), so the series (·) collapses to a constant. Remark: Extensive margin of labor supply Our results extend to a setting where, in addition to adjusting their labor effort on the intensive margin, workers can respond to wage disruptions and tax changes by deciding to enter or exit the labor force. Suppose that agents differ along two dimensions: their skill i∈[0, 1], as in the previous sections, and their fixed cost of participating in the labor force κ∈R+. These two characteristics can be arbitrarily correlated in the population. An agent with types (i,κ)has idiosyncratic preferences over consumption cand labor supply ldescribed by ui(c,l)−κI{l>0}, where I{l>0}is an indicator function equal to 1 if the agent is employed. Agents iparticipate if their fixed cost of work κis smaller than a threshold κiequal to the difference between the utility conditional on employment, ui[wili−T(wili),li], and the utility conditional on unemployment, ui[−T(0),0 ]. We can easily show that the tax reform derived in Proposition 2, along with a fixed unemployment transfer −T(0), continues to solve the compensation problem in this setting. Indeed, this reform leaves unchanged the worker’s utility both conditional on employment and on unemployment, so that no agent switches participation status after the disruption and its compensation.15 3. Analysis of the compensating tax reform In this section, we analyze the economic implications of Proposition 2by applying the compensation (17) to various disruptions. In Sections 3.1,3.2,and3.3, we study three benchmark classes of disruptions: first, those that affect all skills uniformly; second, those that change the wage of a single skill; third, those that involve an interval of skills (e.g., the middle class or the top decile). These special cases help establish the main principles of welfare compensation in general equilibrium. In Section 3.4,weevaluate the robustness of our results to the size of the behavioral elasticities, the shape of the initial tax schedule, and the size of (nonmarginal) disruptions. Finally, Section 3.5 turns to a concrete empirical application: the compensation of robots in the United States. Unless stated otherwise, we impose the following assumption throughout this section. Assumption 1. The initial (pre-disruption) production function is CES. Preferences take the form u(c,l)=c1−η/(1−η)−l1+1/e/(1+1/e)with e>0and η≥0. The initial tax schedule has a constant rate of progressivity (CRP), i.e., T(y)=y−((1−τ)/(1−p))y1−p with τ∈Rand p<1. Assumption 1ensures that the rate of progressivity p(y)=p, the Hicksian elasticity er i=e/(1+(1−p)eη), the income effect parameter en i=(1−p)ηer i, the labor supply elasticities εr i,εn i,εw i, the labor demand elasticity εd i, and the elasticity of substitution between skills, are all constant. 15This argument implies in particular that the values of the elasticities of participation with respect to the tax rates (which otherwise would matter to determine the endogenous wage adjustments ˆ wi) are irrelevant for the construction of the compensating tax reform. Theoretical Economics 18 (2023) Generalized compensation principle 1681 3.1 Uniform disruptions We first study a perturbation that reduces the wages of all workers by the same amount in percentage terms. Corollary 1. Suppose that Assumption 1holds. Consider a uniform wage disruption, so that ˆ wE i≡ˆ wEfor all i∈[0, 1]. Then the general-equilibrium compensation (17)coincides with the partial-equilibrium compensation (11). To show this result, notice first that the partial- and general-equilibrium compensations coincide if and only if the endogenous wage adjustments ˆ wiin (9) vanish; that is, if −(1/εd)ˆ li+1 0γij ˆ ljdj =0. Now Euler’s homogeneous function theorem imposes that −1/εd+1 0γij dj =0. Thus, it suffices to prove that, under Assumption 1, the labor supply response to the disruption and the compensation is uniform, i.e., ˆ lE i=ˆ lE jfor all i,j. But this is straightforward to show by plugging (11) into (8). Therefore, for a uniform wage disruption and the assumed preferences and tax schedule, the own- and cross-wage effects just offset each other, thus yielding zero general-equilibrium wage adjustments. As a result, the partial-equilibrium (PE) tax reform achieves exact welfare compensation even in the general-equilibrium (GE) environment. The uniform disruption and its compensation are represented in Figure 1.Wecalibrate the elasticity of labor supply to e=0.33 (Chetty (2012)) and the elasticity of substitution between skills to εd=∞(partial equilibrium) or εd=1.5 (Katz and Murphy (1992), Card and Lemieux (2001), Card (2009)). We suppose moreover that there are no income effects on labor supply: η=0. We take a rate of progressivity of the initial tax schedule equal to p=0.15 and a level parameter of τ=−3(Heathcote, Storesletten, and Violante (2017)). The marginal tax rate is thus increasing with income: It is equal to 9% at $20,000, to 23% at $60,000, and to 29% at $100,000. We match the U.S. annual earnings distribution by positing a (truncated) log-normal distribution below $150,000 with mean 10 and variance 0.95 and appending a Pareto distribution with a tail parameter that decreases from a=2.5 at $150,000 to a=1.5 for all incomes above $350,000 (Diamond and Saez (2011)). As in Saez (2001), we infer the wage distribution from the observed earnings distribution and the individuals’ first-order conditions (see Sachs, Tsyvinski, and Werquin (2020) for details on the extension of this method to the generalequilibrium setting). The left panel shows that the disruption reduces the wage of all agents by 1%. These wage losses translate into pre-tax income losses represented by the black curve in the right panel: e.g., workers with income equal to $60,000 (respectively, $100,000, $500,000) before the disruption suffer pre-tax earnings losses of $600 (resp., $1000, $5000). The blue and red curves in the right panel show the compensation in partial and general equilibrium, respectively. Recall that the decrease in the agent’s average tax rate implied by the tax reform mirrors the after-tax income losses due to the wage disruption. Since the initial tax schedule is progressive, this implies that the compensation is flatter than the gross income losses: Losing a dollar of pre-tax income does not hurt higher-paid workers as much, since they retain a smaller share 1−T(y)of that dollar. Quantitatively, 1682 Schulz, Tsyvinski, and Werquin Theoretical Economics 18 (2023) Figure 1. Uniform disruption and compensation. a pre-tax income loss of $1000 at $100,000 (respectively, $5000 at $500,000) translates into an after-tax income loss—and thus requires a reduction in tax payment—of $712 (resp., $2796). 3.2 Dirac disruptions Our second polar case consists of a disruption that affects only the wage of agents with a given skill i∗and corresponding income y∗. Formally, we let ˆ wE(y)=−δ(y−y∗),where δ(·)is the Dirac delta function.16 Corollary 2. Suppose that Assumption 1holds and let ¯ y→∞. Consider a Dirac wage disruption at income y∗,sothat ˆ wE i≡−δ(yi−y∗)for all i∈[0, 1]. Then the generalequilibrium compensation is given by ˆ T(y) y=−εd εr(1−τ)yεd/εr−p yεd/εr+1 ∗ I{y≤y∗}+C(1−τ)y−p, (19) where C=y∗f(y∗)/Ez−(εd/εr)E[(z/Ez)(z/y∗)εd/εr+1I{z≤y∗}]is a constant. To understand this result, first ignore the cross-wage complementarities. In this case, (19)withC=0 follows from the first-order linear ordinary differential equation (ODE) (14), which reduces to ˆ T(y) y=εd φεrˆ T(y)∀y<y ∗. (20) This equation requires that the change in average tax rates is proportional to the change in marginal tax rates at every income level below y∗. Intuitively, suppose that the government naively compensates for the partial-equilibrium disruption by reducing the tax 16Note that this perturbation is not differentiable. We approximate it with a sequence of smooth wage disruptions centered around income y∗. Theoretical Economics 18 (2023) Generalized compensation principle 1683 Figure 2. Dirac disruption and compensation. liability of agent y∗. It must then reduce the marginal tax rates of those with lower incomes y<y ∗, i.e., ˆ T(y)<0. However, in general equilibrium, this reduction in MTR raises their labor supply and, hence, lowers their wage, thereby causing welfare losses that are proportional to the ratio of elasticities of labor supply and demand. These welfare losses must be offset by welfare gains of equal magnitude through reductions in their average tax rates ˆ T(y)/y < 0. If εd/εr>por, equivalently εd/φεr>1,17 the average tax rate must fall more than one-for-one in response to a marginal tax rate cut. However, this mechanically lowers the tax bill of agents with slightly higher income, thus requiring an even larger cut in their marginal tax rate, and so on. This “race” between the MTR and the ATR leads to exponentially decreasing tax rates on [y,y∗),capturedby ˆ T(y)/y ∝−yεd/εr−pin the solution to the ODE (19). That is, the compensating tax reform is progressive at a rate given by the ratio of elasticities of labor demand and labor supply εd/εr, net of the rate of progressivity pof the preexisting tax code. Finally, accounting for the cross-wage effects adds the correction C(1−T(y))yto the compensating tax reform in (19). It is easy to show that this amounts to raising the parameter τof the baseline CRP tax schedule by an amount ˆτ/(1−τ)=(1−p)C.That is, skill complementarities require a uniform percentage shift in tax rates over the entire income distribution. Figure 2illustrates these results. We construct a 1% wage disruption μˆ wE(y∗)at income level y∗=$60,000.18 This leads to a pre-tax income loss of y∗μˆ wE(y∗)=$600 (black curve in the right panel) and an after-tax income loss of (1−T(y∗))y∗μˆ wE(y∗)= $461 (blue curve in the right panel). The compensating tax reform in partial equilibrium tracks the after-tax income losses: It leaves the tax liabilities of all agents y= y∗ unchanged while reducing the tax bill of income y∗by a large amount (blue curve). In general equilibrium, the compensation (red curve) accounts for the additional wage adjustments induced by the disruption and tax changes. In particular, the wage loss at 17Empirically, the inequality εd/εr>pis clearly satisfied, since we have p≈0.15, εr≈0.3, and εd≥0.5. 18The calibration is the same as in Figure 1. 1684 Schulz, Tsyvinski, and Werquin Theoretical Economics 18 (2023) income y∗lowers these agents’ labor supply, marginally increasing their wages and reducing the compensation necessary to keep utility unchanged. At the same time, all other income levels need to be compensated because the labor supply reductions at y∗ adversely affect their wages via production complementarities. The key insight from Figure 2is that the partial-equilibrium compensation creates large movements in marginal tax rates around income y∗, which yield sizeable unintended welfare consequences in general equilibrium. For instance, such a tax reform would make agents with income just below y∗strictly worse off because of the very sharp decrease in their marginal tax rate, which raises labor supply and lowers their wages and welfare. Instead, the accurate compensation reduces the tax payment of the disrupted agent at y∗by a much smaller amount while at the same time granting substantial tax rebates to incomes below y∗even though the disruption did not initially hurt them.19 Finally, agents with an income higher than y∗also face tax cuts; these are barely noticeable in Figure 2, however, since the disruption affects a small number of workers and, thus, generates minor cross-wage effects. 3.3 Interval disruptions We finally consider disruptions intermediate between the two polar (uniform and Dirac) cases studied above, and affect a nontrivial range of workers; e.g., all incomes above a threshold or all incomes within a given interval. Corollary 3. Suppose that Assumption 1holds and let ¯ y→∞. Consider a wage disruption that affects uniformly all skills above i∗, with corresponding income y∗;thus, ˆ wE(y)≡−I{y≥y∗}. Then the general-equilibrium compensation is given by ˆ T(y) y=1−T(y)ˆ wE(y)−y y∗εd/εr I{y≤y∗}+C, (21) where C=E[(z/Ez)(z/y∗)εd/εrI{z≤y∗}]is a positive constant. More generally, consider a disruption that affects all skills in an interval [iL,iH]uniformly, with corresponding incomes [yL,yH];thus, ˆ wE(y)≡−I{yL≤y≤yH}. Then the general-equilibrium compensation is given by ˆ T(y) y=1−T(y)ˆ wE(y)−y yLεd/εr I{y≤yL}+y yHεd/εr I{y≤yH}+C, (22) where C=E[(z/Ez)(z/yL)εd/εrI{z≤yL}]−E[(z/Ez)(z/yH)εd/εrI{z≤yH}]is a constant. 19Note also that the compensation peaks at an income y∗∗ that is strictly below the income y∗that incurs the largest wage loss. Indeed, by definition, the agent y∗∗ with the highest tax reduction has a zero marginal tax rate change. Thus, an agent with a slightly higher income gets almost the same total tax rebate (the difference between the two is second order since ˆ T(y∗∗ )=0) and a strictly higher marginal tax rate change (the difference is first order if ˆ T(y∗∗ )>0), and, hence, a strictly higher compensation. This explains why we must have y∗∗ <y ∗. Theoretical Economics 18 (2023) Generalized compensation principle 1685 Formula (21) characterizes the compensation for a disruption that hurts all workers above an income threshold y∗. The compensation can be decomposed as the sum of three terms. The first is the partial-equilibrium compensation derived in Proposition 1. Appropriately normalized by the net-of-tax rate (1−T(y))yto account for the redistribution already achieved by the existing tax code, this term tracks the exogenous wage losses ˆ wE(y)one-for-one. The second term in (21) corrects for the own-wage effects caused by the decreasing marginal product of labor. It reduces the tax liabilities below the disrupted incomes, i.e., on [0, y∗], and has the same shape as in the case of a Dirac disruption in Corollary 2.In particular, its rate of progressivity—that is, the rate at which the compensation’s average and marginal rates fall with income—is equal to the ratio of elasticities of labor demand and labor supply, εd/εr, net of the rate of progressivity pof the initial tax code. Finally, the third term in (21) compensates for the cross-wage effects caused by skill complementarities in production. It amounts to a uniform increase in average tax rates at all income levels, above and beyond the partial-equilibrium compensation and the progressive correction we just described. Again, this last element of the compensation is similar to the corresponding term in Corollary 2. The compensation for a disruption that affects an interior interval of skills, given by (22), follows again from a similar logic, except that there are now two progressive corrections due to own-wage effects, captured by the second and third terms on the right-hand side: The former spreads tax cuts across workers y≤yLto compensate for the sharp wage loss at income yL, while the latter has the opposite sign and compensates for the sharp reversal at income yH. Figure 3shows this decomposition graphically for a uniform interval disruption that affects workers in the income range yL=$20,000 and yH=$100,000, with corresponding income losses shown in black. The left panel shows the partial-equilibrium (dashed blue) and general-equilibrium (solid red) compensations. The right panel decomposes the latter into its four components: the partial-equilibrium compensation (dashed blue), the progressive corrections for the decreasing marginal product of labor (solid and dotted curves), and the uniform correction for the cross-wage complementarities (dashed red). Figure 4constructs the compensation of smoothed-out versions of the same perturbation. In the top panels, a smooth disruption hurts middle-class workers and peaks at y∗=$60,000. In the bottom panels, the disruption uniformly affects all workers with earnings higher than $120,000. The top and bottom left panels depict the direct and total wage losses ˆ wE(y)and ˆ E(y), respectively. The black curves in the top and bottom right panels show the corresponding gross earnings losses. The key insights described in the previous sections carry over to these cases. First, the compensation in partial equilibrium tracks the shape of the after-tax earnings losses due to the exogenous disruption. These are represented by the dashed blue curves in the top and bottom right panels. As before, a given percentage change in the wage leads to more considerable earnings losses at higher income levels, although these losses are dampened by the progressivity of the initial tax code. In general equilibrium, the compensation—represented by the solid red curves in the top and bottom 1686 Schulz, Tsyvinski, and Werquin Theoretical Economics 18 (2023) Figure 3. Compensation of an interval disruption: decomposition. Figure 4. Interval disruptions and compensation. Theoretical Economics 18 (2023) Generalized compensation principle 1687 right panels—accounts for the additional wage adjustments induced by the disruption and tax changes. The robust finding is that whenever the partial-equilibrium compensation implies sharp changes in marginal tax rates and substantial unintended welfare effects, the general-equilibrium forces smooth out such nonlinearities. They spread the tax changes over all the lower income levels, as captured by the progressive terms in (21)and(22). The top right panel of Figure 4also shows that the general-equilibrium compensation reduces the tax rates even for the indirectly affected, high-income workers. This reflects the compensation for the general-equilibrium wage changes due to cross-skill complementarities. 3.4 Robustness of the results In this section, we evaluate the robustness of our results to the values of the labor supply and demand elasticities, the shape of the baseline tax schedule, and the size of the exogenous disruption. Throughout this section, we focus on the middle-class disruption studied in the top panel of Figure 4. Behavioral elasticities The top left panel of Figure 5displays the compensation for different values of the Frisch elasticity e∈{0.25, 0.33, 0.5}, otherwise keeping the same calibration as in the previous sections. The top right panel of Figure 5plots the compensation for various values of the income effect on labor supply, η∈{0, 0.25, 0.5}. While the partial-equilibrium compensation is unaffected by these different behavioral elasticities, the general-equilibrium compensation is sensitive to the values of eand η.Recall from Lemma 2that income effects increase the welfare cost of raising an individual’s total tax liability: They make the agents work more, which reduces their wage. As a result, higher income effects move the general-equilibrium compensation closer to the partial-equilibrium one. Moreover, the endogenous wage adjustments are driven by the magnitude of the labor supply responses to tax and wage changes, which are in turn determined by the Frisch elasticity. Accordingly, the compensation in general equilibrium is closer to the partial-equilibrium compensation for smaller values of e. The bottom panel of Figure 5displays the compensations for different labor demand elasticities εd∈{0.5, 1.5, 2.5, ∞}. This exercise shows that the magnitude of generalequilibrium effects plays a critical role for the compensation of the middle-class disruption. A smaller value of εdimplies stronger own- and cross-wage effects, lowering the middle class’s compensation and raising the tax cuts for higher incomes. Conversely, as εdgrows larger, the compensation converges to the partial-equilibrium case (εd=∞). Baseline tax schedule In our previous simulations, we assumed that the initial tax schedule had a constant rate of tax progressivity (CRP). This may be unrealistic for at least two reasons: The phasing out of low-income transfers may lead to high marginal tax rates at the bottom (rather than negative tax rates in the case of a CRP tax code), and the tax rates converge to a value lower than 100% at the top. We now evaluate (17)for alternative tax codes. To illustrate the impact of these two features, we use the optimal Mirrlees tax schedule (in partial or general equilibrium) as the baseline tax code. As is well known, the optimal tax schedule has high marginal tax rates at the bottom, and the 1688 Schulz, Tsyvinski, and Werquin Theoretical Economics 18 (2023) Figure 5. Robustness to the behavioral elasticities. tax rate at the top is bounded away from 1, with an overall U-shape for the marginal tax rates. Figure 6plots the compensation of the middle-class disruption studied above. The left panel (resp., right panel) shows the compensation in partial equilibrium (resp., general equilibrium) for these alternative baseline tax schemes. The shape of the compensating tax reform is qualitatively robust to the initial tax scheme: it always follows the shape of the disruption in dollar values. However, the size of the compensation is sensitive to the preexisting tax rates. The CRP tax scheme has the lowest marginal tax rates in the depicted income range and, hence, the highest retention rates. Accordingly, the tax cuts are largest in this case. In contrast, the tax changes under the optimal Mirrlees tax schedules are substantially smaller. The optimal tax schedule in general equilibrium features lower marginal tax rates and, hence, higher retention rates than the Mirrlees optimum in partial equilibrium, so that the compensation under the former tax code is slightly closer to that obtained under a CRP tax scheme. Properly accounting for the schedule of marginal tax rates in the preexisting economy is therefore important for the design of the welfare compensation. Theoretical Economics 18 (2023) Generalized compensation principle 1695 approach of Saez and Stantcheva (2016) to dynamic environments, provide useful steps in these directions. Conclusion The classic policy question of compensating winners and losers from an economic disruption becomes quite involved when the environment features distortionary taxes and general-equilibrium responses. At the same time, both of these considerations are important in many applied and policy settings (e.g., to compensate for the adverse effects of technical change). We derive and analyze a general closed-form formula for the design of the welfare-compensating tax reform and its impact on the government budget. This equation is straightforward to implement in practical applications. Appendix Definition of the perturbed equilibrium. After a disruption and a tax reform, the perturbed indirect utility of agent iis given by ˜ Ui=ui˜ wi˜ li−T(˜ wi˜ li)−μˆ T(˜ wi˜ li),˜ li, (23) where the equilibrium labor supplies ˜ li=li(1+μˆ li)and wages ˜ wi=wi(1+μˆ wE i+μˆ wi) are defined by the perturbed first-order condition −u i,l˜ wi˜ li−T(˜ wi˜ li)−μˆ T(˜ wi˜ li),˜ li u i,c˜ wi˜ li−T(˜ wi˜ li)−μˆ T(˜ wi˜ li),˜ li=1−T(˜ wi˜ li)−μˆ T(˜ wi˜ li)˜ wi(24) and the perturbed wage equation ˜ wi=˜ F iLj(1+μˆ lj)j∈[0,1]. (25) The perturbed government revenue is given by ˜ R=1 0T(˜ wi˜ li)+μˆ T(˜ wi˜ li)di. (26) Proof of (6). Thechangeinutilityofagentiin response to the disruption and tax reform is given by μˆ Ui≡˜ Ui−Ui=ui˜ wi˜ li−T(˜ wi˜ li)−μˆ T(˜ wi˜ li),˜ li−uiwili−T(wili),li, where ˜ wi=wi(1+μˆ wE i+μˆ wi)and ˜ li=li(1+μˆ li). A first-order Taylor expansion of this equation around the initial equilibrium (as μ→0) yields ˜ Ui−Ui=μ1−T(yi)yiˆ li+yiˆ wE i+yiˆ wi−ˆ T(yi)u i,c+μliˆ liu i,l+o(μ). (27) However, the first-order condition (2), or the envelope theorem, implies (1− T(yi))yiˆ liu i,c+liˆ liu i,l=0. We thus obtain (6). 1696 Schulz, Tsyvinski, and Werquin Theoretical Economics 18 (2023) Proof of (7). The perturbed first-order condition of agent iin response to the disruption and tax reform is given by 0=1−T(˜ wi˜ li)−μˆ T(˜ wi˜ li)˜ wiu i,c˜ wi˜ li−T(˜ wi˜ li)−μˆ T(˜ wi˜ li),˜ li +u i,l˜ wi˜ li−T(˜ wi˜ li)−μˆ T(˜ wi˜ li),˜ li. A first-order Taylor expansion of this equation around the initial equilibrium (as μ→0) gives 0=1−T(yi)2w2 iu i,cc +21−T(yi)wiu i,cl +u i,ll −w2 iT(yi)u i,cliˆ li +1−T(yi)2wiliu i,cc +1−T(yi)liu i,cl +1−T(yi)−wiliT(yi)u i,c ×wiˆ wE i+ˆ wi −wiu i,cˆ T(yi)−1−T(yi)wiu i,cc +u i,clˆ T(yi). The Hicksian (compensated) labor supply elasticity er iand the income effect parameter en iare, respectively, equal to (see, e.g., Saez (2001, p. 227)) er i= u i,l li u i,l u i,c2 u i,cc −2u i,l u i,cu i,cl +u i,ll , en i=u i,l u i,c2 u i,cc −u i,l u i,cu i,cl u i,l u i,c2 u i,cc −2u i,l u i,cu i,cl +u i,ll . (28) Solving the previous equation for ˆ lithen implies ˆ li=1−p(yi)er i−en i 1+p(yi)er iˆ wE i+ˆ wi −er i 1+p(yi)er i ˆ T(yi) 1−T(yi)+en i 1+p(yi)er i ˆ T(yi) 1−T(yi)yi . Using the definitions of the elasticities along the nonlinear budget constraint εr i,εn i,εw i leads to (7). Proof of (9). Consider an exogenous disruption μˆ FEof the production function and ataxreformμˆ T,withμ>0. The corresponding wage disruption is defined by ˆ wE i=∂ˆ FE ∂Li{Lj}j∈[0,1]. Theoretical Economics 18 (2023) Generalized compensation principle 1697 Denote by μˆ wiand μˆ lithe first-order endogenous percentage changes as μ→0inthe wage and labor supply of type i,andlet ˜ wi=wi(1+μˆ wE i+μˆ wi)and ˜ li=li(1+μˆ li).In the perturbed equilibrium, the wage is equal to the marginal product of the labor of the corresponding type: ˜ wi=∂F+μˆ FE ∂LiLj(1+μˆ lj)j∈[0,1]. The Gateaux derivative of the wage functional is given by ˆ wi≡lim μ→0 1 μwi˜ wi−wi−μˆ wE i =lim μ→0 1 μwi∂F+μˆ FE ∂LiLj(1+μˆ lj)j∈[0,1] −∂F ∂Li{Lj}j∈[0,1]−μ∂ˆ FE ∂Li{Lj}j∈[0,1]. This expression is equal to ˆ wi=1 wi1 0 ˆ ljLj ∂2F(L) ∂Li∂Lj dj. The own-wage (or inverse labor demand) and cross-wage elasticities are defined by Lj wi ∂wi ∂Lj ≡γij −1 εd j δ(j−i) for all i,j. In particular, when the production function is CES, the cross-wage elasticities are given by, for i= j, Lj wi ∂2F(L) ∂Li∂Lj =Lj wi ∂ ∂LjθiL−1/εd i1 0 θjL1−1/εd jdj1 εd−1 =1 εd θjL1−1/εd j 1 0 θkL1−1/εd kdk =1 εd wjLj F(L)≡γj, and the own-wage elasticities by Li wi ∂2F(L) ∂L2 i =Li wi ∂ ∂LiθiL−1/εd i1 0 θjL1−1/εd jdj1 εd−1 =γi−1 εd 1 wi θiL−1/εd i1 0 θjL1−1/εd jdj1 εd−1δ(0)=γi−1 εdδ(0). 1698 Schulz, Tsyvinski, and Werquin Theoretical Economics 18 (2023) Substituting into the formula for ˆ wileads to ˆ wi=1 0 ˆ ljγij −1 εd j δ(j−i)dj, which leads to (9). Proof of (10). The effect of the wage disruption and the corresponding compensating tax reform on government budget is given by ˆ R=lim μ→0 1 μ1 0T(˜ wi˜ li)+μˆ T(˜ wi˜ li)di−1 0 T(wili)di. A first-order Taylor expansion around the initial equilibrium easily leads to (10). Proof of Lemma 1. This lemma follows from Sachs, Tsyvinski, and Werquin (2020); for completeness, we give its proof here. Substituting for ˆ wiinto (7) using (9) leads to ˆ li=φiˆ lE i+φiεw i1 0 γij ˆ ljdj, (29) where we let φi=1 1+εw i/εd i and ˆ lE i=εw iˆ wE i−εr i ˆ T(yi) 1−T(yi)+εn i ˆ T(yi) 1−T(yi)yi . (30) This is a Fredholm integral equation in {ˆ li}i∈[0,1]. To solve for the labor supply changes for a general production function, substitute for ˆ ljin the integral to obtain ˆ li=φiˆ lE i+φiεw i1 0 γij φjˆ lE j+φjεw j1 0 γjkˆ lkdkdj =φiˆ lE i+φiεw i1 0 γij φjˆ lE jdj+φiεw i1 01 0 γikφkεw kγkj dkˆ ljdj ≡φiˆ lE i+φiεw i1 0 γij φjˆ lE jdj+φiεw i1 0 (1) ij ˆ ljdj, where (0) ij =γij and (1) ij =1 0(0) ik φkεw kγkj dk. By induction, it is easy to show that, for all N≥0, ˆ li=φiˆ lE i+φiεw i1 0N  n=0 (n) ij φjˆ lE jdj+φiεw i1 0 (N+1) ij ˆ ljdj, where, for all n≥0, (n+1) ij =1 0(n) ik φkεw kγkj dk. The condition 1 01 0|φiεw iγij |2didj < 1 ensures that the series N n=0(n) ij converges as N→∞. This implies (13). Note that we Theoretical Economics 18 (2023) Generalized compensation principle 1699 can write the endogenous wage changes as ˆ wi=φi εd i−εw iˆ wE i+εr i ˆ T(yi) 1−T(yi)−εn i ˆ T(yi) 1−T(yi)yi +φi1 0 ij φjεw jˆ wE j−εr j ˆ T(yj) 1−T(yj)+εn j ˆ T(yj) 1−T(yj)yjdj, (31) which follows from (7)and(13). Finally, if the initial production function is CES, the cross-wage elasticities γij depend only on j. Multiplying both sides of (29)byγiand integrating from 0 to 1 then leads to 1 0 γiˆ lidi =1 0 γiφiˆ lE idi +1 0 γiφiεw idi1 0 γjˆ ljdj =1 0 γiφiˆ lE idi 1−1 0 γiφiεw idi . Substituting this expression into (29)yields ˆ li=φiˆ lE i+φiεw i1 0 jφjˆ lE jdj, where j≡γj/(1−1 0γkφkεw kdk). Using the expression of the cross-wage elasticities γk=yk/(εdEy),wecanwrite1−1 0γkφkεw kdk =1 0(yk/Ey)(1−φkεw k/εd)dk. Using φk=1/(1+εw k/εd)finally gives j=γj/(1 0φkyk/Eydk ). Proof of Lemma 2. Substitute for ˆ wE i+ˆ wiin (6) using (7)toget ˆ T(yi)=1 εw i1−T(yi)yiˆ li+εr i εw i yiˆ T(yi)−εn i εw i ˆ T(yi). Using the expression we derived above for ˆ lileads to ˆ T(yi)=1 εw i1−T(yi)yiφiˆ lE i+εr i εw i yiˆ T(yi)−εn i εw i ˆ T(yi) +1−T(yi)yiφi1 0 ij φjˆ lE jdj. Replacing the partial-equilibrium labor supply changes ˆ lE iwith their expression (30)allows us to rewrite this equation as ˆ T(yi)=1−T(yi)yiφiˆ wE i+1 0 ij φjεw jˆ wE jdj +εr i/εd i 1+εw i/εd i yiˆ T(yi)−εn i/εd i 1+εw i/εd i ˆ T(yi) 1700 Schulz, Tsyvinski, and Werquin Theoretical Economics 18 (2023) −1−T(yi)yiφi1 0 ij φjεr j ˆ T(yj) 1−T(yj)−εn j ˆ T(yj) 1−T(yj)yjdj. This leads to (14). Rearranging and summing over all agents leads to 1 0 ˆ T(yi) 1−T(yi)=1 0 yiˆ E idi +1 0 φi εd i yiεr i ˆ T(yi) 1−T(yi)−εn i ˆ T(yi) 1−T(yi)yidi −1 0 φiyiidi. The last integral in this expression can be rewritten as 1 0 φiyiidi =1 01 0 φiyiij diφjεr j ˆ T(yj) 1−T(yj)−εn j ˆ T(yj) 1−T(yj)yjdj. An application of Euler’s homogeneous function theorem (see Lemma 2, (24) in Sachs, Tsyvinski, and Werquin (2020)) implies that 1 0φiyiij di =(1/εd j)yj.Wethusobtain E[ˆ T(yi)/(1−T(yi))]=E[yiˆ E i]. Proof of Proposition 1.Equation(11) is a special case of (14) obtained by setting ij =0 and letting εd→∞. Using this formula for the compensating tax reform in partial equilibrium along with ˆ wi=0, the fiscal surplus (10) can be expressed as ˆ R=1 0ˆ wE i+T(yi)ˆ liyidi. Differentiate ˆ T(y)with respect to yin (11) to obtain the marginal tax rates of the compensating tax reform. Letting y i≡dyi/di,weobtain ˆ T(yi)=1 y i−y iT(yi)yiˆ wE i+1−T(yi)y iˆ wE i+1−T(yi)yi dˆ wE i di . Using p(yi)=yiT(yi)/(1−T(yi)),wecanthuswrite ˆ li=εw iˆ wE i−εr i1−p(yi)ˆ wE i+yi y i dˆ wE i di +εn iˆ wE i=−εr i yi y i dˆ wE i di , where we used the fact that εw i=(1−p(yi))εr i−εn i. Substituting into the above expression for ˆ Rand changing variables from skills to incomes leads to (12). Proof of Proposition 2. Since there is a one-to-one map between skills iand incomes yi, we can change variables to express the ODE (14) in terms of incomes. We obtain ˆ T(y)−1−p(y)+εd y εr yˆ T(y) y=− 1−T(y)εd y εr y A(y), Theoretical Economics 18 (2023) Generalized compensation principle 1701 whereweused1/(φiεr i/εd i)=1−p(yi)+(εd i−εn i)/εr iand where A(y)≡φ−1 yˆ E(y)+¯ y y y,zφz−εr z ˆ T(z) 1−T(z)+εn z ˆ T(z) 1−T(z)zdz =φ−1 yˆ E(y)+(y), with ˆ E(y)=φyˆ wE(y)+φy¯ y y y,zφzεw zˆ wE(y)dz. We can solve this equation as a first-order ODE. The general solution to the homogeneous equation is given by ˆ TH(y)=Ce−¯ y y(1−p(z)+εd z εr z)dz z=C1−T(y)y 1−T(¯ y)¯ ye−¯ y y εd z εr z dz z, where Cis a constant and where the second equality uses the fact that p(z)/z = T(z)/(1−T(z)),sothaty x(1−p(z))dz/z =log[((1−T(y))y)/((1−T(x))x)]. Using the method of variation of the parameter, we find a particular solution of the form ˆ TP(y)=C(y)1−T(y)y 1−T(¯ y)¯ ye−¯ y y εd z εr z dz z, where the function C(y)satisfies C(y) 1−T(¯ y)¯ y=¯ y y εd x εr x e¯ y x εd z εr z dz zA(x)dx x. The general solution to (14)isthusequalto ˆ T(y)=1−T(y)y¯ y y (y,x)A(x)dx +C1−T(y)y 1−T(¯ y)¯ ye−¯ y y εd z εr z dz z, where (y,x)=(εd x/εr xx)e−x y(εd z/εr z)dz/z. If the initial tax schedule is Pareto efficient, the tax reform should be ˆ T(·)=0 in the absence of a disruption ( ˆ E(·)=0). (Note that as ¯ y→∞, the last term in the previous expression converges to zero for any value of C.) If the production function is CES, y,zdoes not depend on yand, hence, (y)is equal to a constant ∈R. To find , recall that E[ˆ T(y)/(1−T(y))]=E[yˆ E(y)].Substituting the solution to the ODE into this condition (setting C=0) yields = Eyˆ E(y)−Ey¯ y y (y,x)φ−1 xˆ E(x)dx Ey¯ y y (y,x)dx. 1702 Schulz, Tsyvinski, and Werquin Theoretical Economics 18 (2023) For a general production function, we can use the ODE and insert its solution into the definition of the auxiliary function A(·)to rewrite it as A(y)=φ−1 yˆ E(y)−¯ y y y,zφzεw z+εd zˆ T(z) 1−T(z)z−εd zA(z)dz =φ−1 yˆ E(y)+¯ y y y,zφzεd zA(z)dz −¯ y y y,zεd z¯ y z (z,x)A(x)dxdz, where the second equality uses φz(εw z+εd z)=εd z. Inverting the order of the two integrals in the last line implies that this expression can be rewritten as A(y)=φ−1 yˆ E(y)+¯ y yy,zφzεd z−z y (x,z)y,xεd xdxA(z)dz. However, this is a standard linear Fredholm integral equation, with kernel equal to K(0) y,z, where K(0) y,z≡y,zεd zφz−z y (x,z)y,xεd xdx. Assume that [y,¯ y]2K(0) y,z 2dy dz < 1, which ensures the convergence of the series ∞ n=0K(n) y,zdefined below. Following steps analogous to the proof of Lemma 1,weget A(y)=φ−1 yˆ E(y)+¯ y y∞  n=0 K(n) y,zφ−1 zˆ E(z)dz, with K(n) y,z=¯ y yK(n−1) y,xK(0) x,zdx for all n. Inverting the integrals one more time leads to ¯ y y K(0) y,zφ−1 zˆ E(z)dz =¯ y y y,zεd zˆ E(z)dz −¯ y y y,zεd z¯ y z (z,x)φ−1 xˆ E(x)dxdz ≡¯ y y λ(0)(y,z)dz, where we denote λ(0)(y,z)=y,zεd zφzφ−1 zˆ E(z)−¯ y z (z,x)φ−1 xˆ E(x)dx. Theoretical Economics 18 (2023) Generalized compensation principle 1703 Now, for any n≥1, we can write ¯ y y K(n) y,zφ−1 zˆ E(z)dz =¯ y y K(n−1) y,x¯ y y K(0) x,zφ−1 zˆ E(z)dzdx =¯ y y K(n−1) y,z¯ y y λ(0)(z,x)dxdz, so that ∞  n=0¯ y y K(n) y,zφ−1 zˆ E(z)dz =¯ y y λ(0)(y,z)dz + ∞  n=1¯ y y K(n−1) y,z¯ y y λ(0)(z,x)dxdz =(1)(y)+ ∞  n=0¯ y y K(n) y,z(1)(z)dz, wherewedenote (1)(y)≡¯ y y λ(0)(y,z)dz =¯ y y y,zεd zφzφ−1 zˆ E(z)−¯ y z (z,x)φ−1 xˆ E(x)dxdz. By induction, repeating the above steps for n≥2 leads to ∞  n=0¯ y y K(n) y,zφ−1 zˆ E(z)dz= N  n=1 (n)(y)+ ∞  n=0¯ y y K(n) y,z(N)(z)dz for all N,where,foralln≥2, (n)(y)=¯ y y K(0) y,z(n−1)(z)dz =¯ y y y,zεd zφz(n−1)(z)−¯ y z (z,x)(n−1)(x)dxdz. Assuming that the series converges as N→∞, we finally obtain A(y)=φ−1 yˆ E(y)+ ∞  n=1 (n)(y). 1704 Schulz, Tsyvinski, and Werquin Theoretical Economics 18 (2023) For completeness, let us compute (z)from the series representation when the productionisCES.Inthiscase,recallthaty,z=1/(εdE[yφy])zf (z),sothat (1)(y)≡1 E[yφy]¯ y y zφzφ−1 zˆ E(z)−¯ y z (z,x)φ−1 xˆ E(x)dxf(z)dz =1 E[yφy]Ezˆ E(z)− Ez¯ y z (z,x)φ−1 xˆ E(x)dx E[yφy]. Note that (1)(y)≡(1)is a constant that does not depend on y. By induction, assuming that (n−1)(z)is a constant, we get, for any n≥2, (n)(y)=1 E[yφy]¯ y y zφz(n−1)−¯ y z (z,x)(n−1)dxf(z)dz =(n−1) E[yφy]−Ez¯ y z (z,x)dx E[yφy], which is a constant. We thus obtain ∞  n=1 (n)= Ezˆ E(z)−Ez¯ y z (z,x)φ−1 xˆ E(x)dx E[yφy] + ∞  n=21− Ez¯ y z (z,x)dx E[yφy](n−1). Solving for ≡∞ n=1(n)leads to ∞  n=1 (n)= Ezˆ E(z)−Ez¯ y z (z,x)φ−1 xˆ E(x)dx Ez¯ y z (z,x)dx, which is indeed the expression we found above. We finally compute the fiscal surplus (10). Substituting for ˆ liusing (7) and for ˆ wE i+ˆ wi using (6)inthisexpression,wecanwrite ˆ R=1 0 ˆ T(yi)di −1 0 T(yi)yiεr i ˆ T(yi) 1−T(yi)−1+εw i+εn iˆ T(yi) 1−T(yi)yidi. The ODE (14) can be rewritten as ˆ T(yi) 1−T(yi)=1−p(yi)+εd i εr iˆ T(yi) 1−T(yi)yi −εd i εr i φ−1 iˆ E i−εd i εr i i.