The timing of optimal capital income tax reforms: The role of intangible capital investment

Conesa, Juan Carlos; Domínguez, Begoña Article The timing of optimal capital income tax reforms: The role of intangible capital investment SERIEs - Journal of the Spanish Economic Association Provided in Cooperation with: Spanish Economic Association Suggested Citation: Conesa, Juan Carlos; Domínguez, Begoña (2019) : The timing of optimal capital income tax reforms: The role of intangible capital investment, SERIEs - Journal of the Spanish Economic Association, ISSN 1869-4195, Springer, Heidelberg, Vol. 10, Iss. 3/4, pp. 419-438, https://doi.org/10.1007/s13209-019-0199-3 This Version is available at: https://hdl.handle.net/10419/286506 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/4.0/ SERIEs (2019) 10:419–438 https://doi.org/10.1007/s13209-019-0199-3 ORIGINAL ARTICLE The timing of optimal capital income tax reforms: the role of intangible capital investment Juan Carlos Conesa1·Begoña Domínguez2 Received: 11 February 2019 / Accepted: 3 July 2019 / Published online: 16 July 2019 © The Author(s) 2019 Abstract This paper studies the role of intangible capital investment in the timing of optimal capital income tax reforms. Within an infinitely lived worker–capitalist model as in Judd (J Public Econ 28:59–83, 1985), we consider two different economies: one in which capitalists devote physical investment, management time and intangible capital investment to build capital; and a second one in which capitalists do not need to devote intangible capital investment. We perform a Pareto-improving Ramsey tax reform and compare the optimal paths of corporate and dividend taxes during the transition with and without intangible capital. Without intangible investment, optimal corporate income taxes are set at 100% for 10years and then fall to 0%, while optimal dividend taxes are set to 78% initially and follow a steep decline to 37% over 10 years. With intangible investment, optimal corporate income taxes are set to −20% initially and slowly converge toward zero, while dividend income taxes are set to 61% initially and follow a slow decline to 27% over 50years. Keywords Optimal policy ·Capital taxes ·Intangible investment JEL Classification E62 ·H23 ·H25 1 Introduction In this paper, we evaluate the impact that the existence of intangible investment has for the outcome of optimal capital income tax reforms in an environment where redistribuBJuan Carlos Conesa [email protected] Begoña Domínguez b[email protected] 1Department of Economics, Stony Brook University, Stony Brook, NY 11794, USA 2School of Economics, The University of Queensland, Colin Clark Building (39), St Lucia, Brisbane, QLD 4072, Australia 123 420 SERIEs (2019) 10:419–438 tive concerns matter. We consider an infinitely-lived two-agent model, where workers provide labor to firms, while firms are run by capitalists that invest tangible and intangible resources, and provide management time in order to build productive capital. We consider two forms of capital taxation: corporate taxes and dividend taxes. The key difference between these forms of taxation lies in their differential tax treatment. Tangible investment is tax deductible from dividend income taxation, while this is not the case for corporate income taxes. In contrast, intangible investment is immediately expensed and then tax deductible from both corporate and dividend income taxation. Intangible investment by firms includes explicit expenses like R&D, IT, or marketing. Besides those explicit investments, there are additional expenses or allocation of resources by firms that contribute toward building a company’s reputation and market value, such as building and maintaining a customer base, training and maintaining the labor force, etc. We distinguish between two types of such allocation of resources. When we think of the allocation of explicit resources, we refer to it as intangible investment, and when we think of the allocation of hours of work, we refer to it as managerial time. The expenses in intangible investment are large and have been found to be important for productivity and hours, see McGrattan and Prescott (2005). Managerial time is also large, given the fraction of total hours worked in non-production activities. Conesa and Domínguez (2013) study optimal capital and labor taxes in a representative agent economy, as in Chamley (1986), and find that intangible investment matters for the optimal tax prescriptions and for the time inconsistency of optimal tax reforms. While Conesa and Domínguez (2018) already add redistributive concerns to the analysis, in this paper we focus on the role of intangible investment for the timing of optimal capital taxes. In this environment, we perform a Pareto-improving Ramsey tax reform. We characterize the resulting optimal paths of corporate and dividend taxes and compare them with those for an economy with no intangible investment. Our main results are as follows. Without intangible investment, the Ramsey corporate income tax is set at its maximum (we assume it is 100%) for the first 10years and then set to zero permanently, while the Ramsey dividend tax rate is set to 78% in period 0 and steeply declines to 37% over the first 10 years. With intangible investment, the optimal capital tax prescriptions change substantially. As expected, the presence of intangible investment limits the ability of the tax authority to confiscate initial wealth through high corporate income taxes. The Ramsey corporate income tax rate falls on impact to −20% in period 0 and converges very slowly toward zero. The Ramsey dividend income tax rate is set to 61% and declines also very slowly toward 27%. The overall transition takes around 50years. Intangible investment does not affect the timing of optimal labor tax rates but affects the level. Without intangible investment, optimal labor tax rates are roughly zero, while with intangible investment, optimal labor tax rates are about 13%. Overall, the presence of intangible capital decreases substantially the magnitude of income redistribution, not only in the short run but also in the long run. Following up with the public finance tradition (see Auerbach 2002), there are a number of papers that consider different forms of capital taxation and tax deductions. Abel (2007) studies optimal capital taxes with tax deductible purchases. Anagnostopoulos et al. (2012) study the effect of dividend and capital gains tax cuts in an 123 SERIEs (2019) 10:419–438 421 incomplete market model. Strulik and Trimborn (2012) also consider these forms of taxation when performing a dynamic scoring exercise. Other papers also examine optimal policy in similar heterogeneous agent economies. Armenter (2007) studies Markov-perfect time-consistent optimal policies in a similar environment. Bassetto (2014) studies optimal policy in an economy with rentiers and tax payers. The above papers, however, do not consider intangible investments. The role of intangibles has not been widely studied. Recently, Lev (2018) points at the continued growth of intangible investments as the key characteristic of developed economies. A recent paper by Peters and Taylor (2017) finds that intangible capital adjusts more slowly to changes in investment opportunities. This may explain our finding of Ramsey reforms with a longer transition in the presence of intangible investment. The rest of the paper is organized as follows. Section 2presents the model economy. Section 3describes the optimal policy problem and develops the main analytical and numerical results. Section 4concludes. The appendix includes further derivations. 2Themodel In this section we present our general model economy. Our benchmark model extends the framework of Conesa and Domínguez (2018) by incorporating intangible investment. Time is discrete and denoted by t,with t=0,1,2,.... As in Judd (1985), the economy is populated by a benevolent government and two types of infinitely-lived agents: capitalists and workers. All agents are identical within type. Workers (agents of type 1) supply labor to firms and cannot save. Capitalists (agents of type 2) own firms and can save.1In addition, capitalists dedicate management time and intangible investment to the firm in order to build new capital. Population is normalized to one and is composed of a proportion κ1of workers and a proportion κ2=1−κ1of capitalists. We now describe the representative worker. Given the discount factor β∈(0,1), the worker’s preferences are given by U1= ∞  t=0 βtu1(c1,t,n1,t), (1) where c1,tdenotes the worker’s consumption, and n1,tthe quantity of labor supplied. The utility function u1satisfies standard conditions; i.e., u1is assumed to be strictly increasing (decreasing) in consumption (labor), strictly concave (convex) in consumption (labor), and twice continuously differentiable. For simplicity of exposition, we assume u1to be separable between consumption and labor. In any given period t,the budget constraint of the representative worker is given by 1As in Conesa and Domínguez (2018), capitalists may choose to supply raw labor instead of or in addition to managerial effort. For Pareto-improving reforms in our benchmark parameterization, this option is irrelevant. However, this option matters for Pareto weights on the capitalists that are very low, see our sensitivity analysis. 123 422 SERIEs (2019) 10:419–438 c1,t=(1−τn t)wtn1,t,(2) where wages are denoted by wtand labor income tax rates by τn t. The optimization problem of the representative worker is as follows. Given policies, prices and other agents’ choices, each worker chooses c1,t,n1,t∞ t=0to maximize her welfare (1) subject to her budget constraint (2). This problem yields the following optimal consumption-labor decision: −u1n,t=u1c,t1−τn twt.(3) In the above equation and in what follows, we denote partial derivatives with a subscript. We now turn to describe the representative capitalist. The capitalist’s preferences are given by U2= ∞  t=0 βtu2(c2,t,e2,t), (4) where c2,tdenotes the capitalist’s consumption, and e2,tis the quantity of management time supplied. We assume that the utility function u2satisfies the same general properties as u1.Each capitalist owns an equal amount of capital kt>0 and employs an equal amount of labor nt=κ1 κ2n1,t,to run a firm that produces a general consumption/investment good using the technology f(kt,nt),where fis increasing, concave and satisfies the Inada conditions. In any period t,the budget constraint of the representative capitalist is given by: c2,t+bt+1=t+Rb tbt,(5) where bt+1denotes government bond purchases, Rb tthe return on government bonds and tthe after-tax dividend income, which takes the form: t=1−τd t1−τk tf(kt,nt)−wtnt−xu,t+τk tδkt−xm,t.(6) Dividend income tis taxed at the rate τd t.Dividend income includes corporate income, f(kt,nt)−wtnt−xu,t, that is taxed at the rate τk t. Notice that there is a capital depreciation allowance equal to the depreciation of the capital stock and that the only source of tangible investment is retained earnings. Capitalists can invest in both tangible xm,tand intangible investments xu,t.Tangible investment is fully and immediately tax deductible from dividend income but not from corporate income, while intangible investment is fully and immediately tax deductible for all forms of income. These two forms of investments, together with management time, are required in order to produce new capital2: kt+1=Ixm,t,xu,t,e2,t+(1−δ)kt.(7) 2Here we think of capital as a composite that reflects overall productive capacity and would equal the value of the firm in equilibrium. This simplifies the analysis substantially. In Conesa and Domínguez (2013), instead, we kept track of two different types of capital. 123 SERIEs (2019) 10:419–438 423 The function Iis assumed to be increasing, concave, homogeneous of degree 1, continuously differentiable, and satisfies the Inada conditions. In equation (7), we assume that intangible investments xu,tand managerial time are necessary to transform tangible resources into new productive capital kt. Later on, we will be comparing this situation with an economy where intangible investments are not required, that is, where Ixm,t,xu,t,e2,t=Ixm,t,e2,t. Given policies, prices and other agents’ choices, a capitalist chooses c2,t,e2,t,xm,t, xu,t,nt,kt+1,bt+1∞ t=0to maximize her welfare (4) subject to her budget constraint (5), the technology for new capital (7), and the no-Ponzi game conditions on capital and bonds. The optimality conditions for this problem are given by [c2,t]u2c,t=ξt, [e2,t]−u2e,t=Ie,tϕt, [xm,t]ϕtIm,t=1−τd tξt, [xu,t]ϕtIu,t=1−τd t1−τk tξt, [nt]fn,t=wt, [kt+1]ϕt=βξt+11+1−τk t+1(rt+1−δ)+βϕt+1, [bt+1]ξt=βξt+1Rb t+1, and the transversality conditions, where ξtand ϕtdenote the multipliers on (5) and (7), respectively. The above optimality conditions can be summarized in a non-arbitrage condition on bonds, the labor hiring decision fn,t=wt,and −u2e,t Ie,t =1−τd tu2c,t Im,t ,(8) Iu,t=1−τk tIm,t,(9) 1−τd tu2c,t Im,t =β1−τd t+1Im,t+11−τk t+1fk,t+1−δ+δ +(1−δ)]u2c,t+1 Im,t+1 .(10) From the above equations, we see that dividend taxes distort management time and the timing of investment, while corporate taxes distort the intra-period allocation of investment between tangibles and intangibles, and the intertemporal decision to invest. Notice that in the absence of managerial effort a constant dividend tax would not be distortionary, and as such it could be used to confiscate as much of initial wealth as desired. Substituting dividend and corporate taxes from the first two optimality conditions, the Euler condition (10) can be written as −u2e,t Ie,t =−βIu,t+1fk,t+1−δ+Im,t+1δ+(1−δ)u2e,t+1 Ie,t+1 .(11) 123 424 SERIEs (2019) 10:419–438 The government is benevolent with preferences given by γ1κ1U1+γ2κ2U2,(12) where γ1is the Pareto weight on workers and γ2=1−γ1on capitalists. The government must finance an exogenous and wasteful government consumption, gt>0, per period and the debt repayments through taxes and new bond issuance. We assume the present value of all government expenditures and liabilities is large enough to require distortionary taxation and low enough to guarantee that the Ramsey allocation converges to an interior steady state.3In each period t,the government’s budget constraint is given by κ1τn twtn1,t+κ2τk t(1−τd t)fk,tkt−xu,t−δkt +τd tfk,tkt−xu,t−xm,t+bt+1 =gt+κ2Rb tbt, where corporate tax rates are bounded above by 100%, τk t≤1,in all periods. Finally, the resource constraint of the economy is κ1c1,t+κ2c2,t+κ2xm,t+xu,t+gt=κ2f(kt,nt). (13) For a given policy and initial conditions, the competitive equilibrium of this economy is characterized by the budget constraint and the optimality condition for the workers, (2)–(3), the budget constraint and optimal conditions for the capitalists, (5)–(10), fn,t=wt,and the non-arbitrage condition on bonds, together with feasibility (13), market clearing and the transversality conditions. 3 The Ramsey tax reform This section performs a Ramsey tax reform in our economy. We first present the Ramsey problem by the government at date 0, next characterize the Ramsey tax plan analytically, and then we resort to numerical methods. As usual in this literature, we assume that there is a commitment technology so that future governments follow the Ramsey policy plan prescribed by the government in period 0. In addition, we assume that the initial after-tax interest rate on bonds Rb 0is given. Here we follow the primal approach and use the workers’ and capitalists’ optimality conditions to substitute prices and taxes in their respective budget constraints. As a 3Recently, Straub and Werning (2018) re-examine Judd (1985) and Chamley (1986)’s results. They find that the optimal capital tax rate maybe set at the upper bound forever whenever the intertemporal elasticity of substitution is less than one and the initial government debt is large enough. As mentioned by the authors, they do not allow for consumption taxes [as considered by Chari et al. (2016)] and dividend taxes with tax deductible investment [as considered in Conesa and Domínguez (2018) and in this paper], since these instruments provide alternative ways to tax initial wealth. 123 SERIEs (2019) 10:419–438 425 result and as shown in the “Appendix”, we obtain the following implementability conditions: u1c,tc1,t+u1n,tn1,t=0,(14) for the worker in each period, and ∞  t=0 βtu2c,tc2t+u2e,te2,t=W0,(15) for the capitalist in its life-time, where W0=− u2e,0 Ie,0Im,0Iu,0 Im,0fk,0−δ+δ+(1−δ)k0+u2c,0Rb 0b0. In addition, the upper bound on capital tax rates can be written in period 0 as Iu,0 Im,0≥0 and in each period t≥1as −u2e,t−1 Ie,t−1 ≥−βIm,tδ+(1−δ)u2e,t Ie,t .(16) For a given allocation, labor tax rates follow from (3). However, for the capitalists, we have three optimality conditions (8)–(10) as functions of two taxes: the dividend and corporate tax rates. These tax rates follow, respectively, from (8) and (9). Then, in our economy with intangible investment, an allocation needs to additionally satisfy the Euler condition (10), which takes the form of the decentralization condition after substituting taxes (11). The Ramsey problem is defined as follows: Given the exogenous stream of government spending {gt}∞ t=0and initial conditions for capital, bonds and the aftertax return on bonds, the government at date 0 chooses the sequences c1,t,c2,t, n1,t,e2,t,xm,t,xu,t,kt+1∞ t=0to maximize the social welfare function (12) subject to the resource constraint (13), the production of new capital (7), the implementability conditions of the workers (14) in each period, the life-time implementability condition of the capitalists (15), the decentralization condition (11) in each period, and the upper bound on capital tax rates (16). In our economy, the upper bound on capital tax rates (16) does not bind provided a sufficient distortion on intangible investment which is needed to build capital. The Lagrangian and the resulting first-order conditions are relegated to the “Appendix”. Combining the optimality conditions from the Ramsey problem with those from the household problem, we obtain the following optimal Ramsey taxes in periods t≥1: τn t=1−Z1c,t Z1n,t +1 Z1n,t ψtJb n,t fn,tu1c,t ,(17) 123 426 SERIEs (2019) 10:419–438 τd t=1−Z2c,t Z2e,t +ψt+1 u2c,tZ2e,tJa m,t−Ja e,t Im,t Ie,t −ψt u2c,tZ2e,tJb m,t−Jb e,t Im,t Ie,t,(18) τk t=Ie,t Im,tZ2e,tu2e,tψt+1Ja m,t−Ja u,t−Ja e,t Im,t Ie,t +Ja e,t Iu,t Ie,t +ψtJb m,t−Jb u,t−Jb e,t Im,t Ie,t +Jb e,t Iu,t Ie,t,(19) where Zjq,t=γj+λj,t1+ujqq,tqj,t ujq,t,Ja x,t=∂u2e,t Ie,t ∂xt,and Jb x,t= ∂[Iu,t(fk,t−δ)+Im,tδ+(1−δ)]u2e,t Ie,t ∂xt,with φt,ψ t,μ t,λ 1,t,and λ2, respectively, denoting the multipliers on (7), (11), (13), (14), and (15). Furthermore, the Ramsey allocation also satisfies u1c,tZ1c,t=u2c,tZ2c,t. To further characterize the optimal taxes, we consider utility functions of the form: ui(ci,t,ni,t+ei,t)=ci,t1−σi 1−σi −θini,t+ei,t1+χi 1+χi ,(20) where σiis the inverse of the intertemporal elasticity of consumption, χiis the inverse of the Frisch labor supply elasticity, and θiis the disutility of work for an agent of type i, with e1,t=0 and n1,t>0,and e2,t>0 and n2,t=0. From the above Ramsey taxes (17)–(19), it is clear that the optimal tax rates during the transition are affected by the decentralization constraint (11), which involves the marginal gain and marginal cost of producing a new unit of capital. If the decentralization constraint (11) stops binding, i.e., ψt=0,4the optimal tax rates become τn t=1−Z1c,t Z1n,t ≡1−γ1+λ1,t(1−σ1) γ1+λ1,t(1+χ1), τd t=1−Z2c,t Z2e,t ≡1−γ2+λ2(1−σ2) γ2+λ2(1+χ2), τk t=0. In addition, we find u1c,tγ1+λ1,t(1−σ1)=u2c,tγ2+λ1,t(1−σ2).The above findings are consistent with the results of Albanesi and Armenter (2012) as there are no permanent intertemporal distortions. Once the decentralization constraint (11) does not bind, the long-run optimal level of the corporate income tax is zero while the optimal level of the labor (dividend) income tax depends on the Pareto weight on workers (capitalist), the preference parameters of the workers (capitalists), and the distortionary cost of taxation levied on the workers (capitalists). To further characterize the optimal taxes during the transition and in the long run, we resort to numerical methods. 4In our numerical exercise, we find that the decentralization constraint (11) does not bind in the long run. 123 SERIEs (2019) 10:419–438 433 ad be cf Fig. 7 Ramsey taxes and allocation for ρu=−0.25 Fig. 8a–c shows that now there is more redistribution toward workers. More specifically, without intangibles, optimal corporate tax rates remain at 100% for more periods, dividend tax rates are larger and converge to 72% in the long run and labor tax rates provide subsidies along the transition and in the long run (coveraging to −23%). With intangibles, Fig. 8a shows that corporate taxes provide large subsidies (of around 40%) 123 434 SERIEs (2019) 10:419–438 ad be cf Fig. 8 Ramsey taxes and allocation for γ1=0.75 and γ2=0.25 that slowly coverage toward zero in the long run. In Fig. 8b, we see that, with intangibles, most of the redistribution comes through high taxes rates on dividend income (65% in the long run) and very low and negative taxes on labor income (−6% in the long run). As with a lower Pareto weight on workers, the presence of intangibles limits the extent of redistribution through taxation. 123 SERIEs (2019) 10:419–438 435 4 Conclusions In this paper we have examined the optimal timing of corporate and dividend tax rates in a Pareto-improving reform and how this timing is affected by the presence of intangible investment. Overall, we find that the presence of intangible investment affects radically the timing of optimal capital taxes. With intangible investment, optimal capital taxes are much lower, smoother and follow a long transition toward their steady-state levels. Our exercise has considered a Pareto-improving reform in which workers and capitalists all gain in terms of welfare from the tax changes. Our paper suggests that the optimal prescriptions of that reform are very different if the policymaker ignores the effects of intangible investment. Once the policymaker takes into account the effect of intangibles, the degree of confiscatory capital taxation is substantially lower, and therefore the ability to redistribute across agents is severely affected. There are several limitations to our analysis, among them the assumption of a representative firm. In a world with young growing firms that finance their activities by issuing equity, the fiscal treatment of firm’s earnings has additional distortionary impact on the financing of growing firms and thus in their entry decisions, see Gourio and Miao (2010) and Erosa and González (2019). Compliance with ethical standards Conflict of interest The authors declare that they have no conflict of interest. Ethical standard The authors declare that this research article is in compliance with the ethical standards of SERIEs and our home institutions. The authors do not need an ethical approval as the paper has a purely theoretical content. Informed consent This article does not contain any information that requires informed consent. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. 5 Appendix: Ramsey problem In this Appendix, we first derive the implementability conditions of the worker and of the capitalist, then we set up the Ramsey problem and derive its first-order conditions. The per-period implementability condition of the worker is obtained by multiplying the worker’s budget constraint (2)byu1c,tin each period and substituting in the labor income tax rate from the consumption-leisure decision (3), which yields u1c,tc1,t+u1n,tn1,t=0. 123 436 SERIEs (2019) 10:419–438 To derive the capitalists’ implementability condition, we first multiply the capitalist’s budget constraint (5)byβtu2c,t, impose wt=fn,t,1−τk t=Iu,t Im,t,and add over time, to get ∞  [t=0] βtu2c,tc2,t+ ∞  [t=0] βt1−τd tu2c,tIu,t Im,t xu,t+xm,t = ∞  [t=0] βtu2c,t1−τd tIu,t Im,t fk,t+τk tδkt +u2c,0Rb 0b0+ ∞  [t=0] βtβu2c,t+1Rb t+1−u2c,tbt+1.(21) We then multiply the investment production (7)byβt1−τd tu2c,t Im,tand add over time to find ∞  [t=0] βt1−τd tu2c,t Im,tkt+1−Ixm,t,xu,t,e2,t−(1−δ)kt=0.(22) We now add (21) and (22) together to obtain ∞  [t=0] βtu2c,tc2,t+1−τd tu2c,tIu,t Im,t xu,t+xm,t−It Im,t =1−τd 0u2c,0 Im,0Iu,0 Im,0 fk,0+τk 0δ+1−δk0+u2c,0Rb 0b0 + ∞  [t=0] βtβ1−τd t+1u2c,t+1 Im,t+1Iu,t+1 Im,t+1 fk,t+1+τk t+1δ+1−δ −1−τd tu2c,t Im,tkt+1+ ∞  [t=0] βtβu2c,t+1Rb t+1−u2c,tbt+1. The terms in brackets in the last line become zero after, respectively, imposing the first-order condition for capital (10) and the one for bonds. Next, substituting dividend taxes from the optimality condition (8) in the above equation, we find ∞  [t=0] βtu2c,tc2,t+u2e,tIt Ie,t −Iu,t Ie,t xu,t−Im,t Ie,t xm,t =u2e,0 Ie,0Iu,0 Im,0 fk,0+τk 0δ+1−δk0+u2c,0Rb 0b0, which, given e2,t=It Ie,t−Im,t Ie,txm,t−Iu,t Ie,txu,t,as Iis homogeneous of degree 1, becomes the implementability condition of the capitalist (15). 123 SERIEs (2019) 10:419–438 437 The Lagrangian for the Ramsey problem for the government at date 0 is L= ∞  t=0 βtγ1κ1u1(c1,t,n1,t)+γ2κ2u2(c2,t,e2,t) + ∞  t=0 βtμtκ2f(kt,nt)−κ1c1,t−κ2c2,t−κ2xm,t+xu,t−gt + ∞  t=0 βtφtκ2Ixm,t,xu,t,e2,t+(1−δ)kt−kt+1 + ∞  t=0 βtλ1,tκ1u1c,tc1,t+u1n,tn1,t+λ2κ2∞  t=0 βtu2c,tc2,t+u2e,te2,t−W2,0 + ∞  t=1 βt−1ψtκ2u2e,t−1 Ie,t−1−βIu,tfk,t−δ+Im,tδ+(1−δ)u2e,t Ie,t +ς0κ2Iu,0 Im,0+ ∞  t=1 βtςtκ2−βIm,tδ+(1−δ)u2e,t Ie,t+u2e,t−1 Ie,t−1, where μt,φ t,λ 1,t,λ 2,ψ t,and ςt, respectively, denote the multipliers on (13), (7), (14), (15), (11) and on Iu,0 Im,0≥0 in period 0 and (16) in periods t≥1. The presence of intangible investment makes corporate taxes distortionary in every period and the upper bound on capital taxes not to bind. Then the first-order conditions for the Ramsey problem for all periods t≥1are [c1,t]u1c,tZ1c,t=μt, [c2,t]u2c,tZ2c,t=μt, [n1,t]u1n,tZ2n,t=−fn,tμt+ψtJb n,t, [e2,t]u2e,tZ2e,t=−Ie,tφt−ψt+1Ja e,t+ψtJb e,t, [xm,t]Im,tφt=μt−ψt+1Ja m,t+ψtJb m,t, [xu,t]Iu,tφt=μt−ψt+1Ja u,t+ψtJb u,t, [kt+1]φt=βμt+1fk,t+1+β(1−δ)φt+1−βψt+1Jb k,t+1, where Zjq,t=γj+λj,t1+ujqq,tqj,t ujq,t,Ja x,t=∂u2e,t Ie,t ∂xt,and Jb x,t= ∂[Iu,t(fk,t−δ)+Im,tδ+(1−δ)]u2e,t Ie,t ∂xt. References Abel AB (2007) Optimal capital income taxation. Unpublished Manuscript Aguiar M, Bils M (2015) Has consumption inequality mirrored income inequality? Am Econ Rev 105:2725– 2756 Anagnostopoulos A, Cárceles E, Lin D (2012) Dividend and capital gains taxation under incomplete markets. J Monet Econ 59:599–611 Albanesi S, Armenter R (2012) Intertemporal distortions in the second best. Rev Econ Stud 79:1271–1307 Armenter R (2007) Time consistent fiscal policy and heterogeneous agents. Rev Econ Dyn 10:31–54 Asker J, Farre-Mensa J, Ljungqvist A (2015) Corporate investment and stock market listing: a puzzle? Rev Financ Stud 28:342–390 123 438 SERIEs (2019) 10:419–438 Auerbach AJ (2002) Taxation and corporate financial policy. In: Auerbach AJ, Feldstein M (eds) Handbook of public economics, vol 3. Elsevier, Amsterdam, pp 1251–1292 Bassetto M (2014) Optimal fiscal policy with heterogeneous agents. Quant Econ 5:675–704 Chamley C (1986) Optimal taxation of capital income in general equilibrium with infinite lives. Econometrica 54:607–622 Chari VV, Nicolini JP, Teles P (2016) More on the optimal taxation of capital. Unpublished Manuscript Chetty R, Guren A, Manoli D, Weber A (2012) Does indivisible labor explain the difference between micro and macro elasticities? A meta-analysis of extensive margin elasticities. NBER Macroecon Annu 27:1–56 Conesa JC, Domínguez B (2013) Intangible investment and Ramsey capital taxation. J Monet Econ 60:983– 995 Conesa J. C, Domínguez B (2018) Capital taxes and redistribution: the role of management time and tax deductible investment. Stony Brook working paper series, 18-10 Erosa A, González B (2019) Taxation and the life cycle of firms. J Monet Econ (forthcoming) Gourio F, Miao J (2010) Firm heterogeneity and the long-run effects of dividend tax reform. Am Econ J Macroecon 2:131–168 Judd KL (1985) Redistributive taxation in a simple perfect foresight model. J Public Econ 28:59–83 Lev B (2018) Intangibles. SSRN working papers McGrattan ER, Prescott EC (2005) Taxes, regulations and the value of U.S. and U.K. corporations. Rev Econ Stud 72:767–796 Peters R, Taylor L (2017) Intangible capital and the investment-q relation. J Financ Econ 123:251–272 Straub L, Werning I (2018) Positive long run capital taxation: Chamley–Judd revisited. Unpublished Manuscript Strulik H, Trimborn T (2012) Laffer strikes again: dynamic scoring of capital taxes. Eur Econ Rev 56:1180– 1199 Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. 123