The timing of optimal capital income tax reforms: the role of intangible capital investment
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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 10 years 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 50 years.
Keywords
Optimal policy Capital taxes Intangible investmentJEL Classification
E62 H23 H251 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 redistributive 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 10 years 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 50 years. 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 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 2 presents the model economy. Section 3 describes the optimal policy problem and develops the main analytical and numerical results. Section 4 concludes. The appendix includes further derivations.
2 The model
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 \(t=0,1,2,\ldots \). 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.^{1} In 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 \(\kappa _{1}\) of workers and a proportion \(\kappa _{2}=1-\kappa _{1}\) of capitalists.
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 \(R_{0}^{b}\) is given.
The Ramsey problem is defined as follows: Given the exogenous stream of government spending \(\left\{ g_{t}\right\} _{t=0}^{\infty }\) and initial conditions for capital, bonds and the after-tax return on bonds, the government at date 0 chooses the sequences \(\big \{ c_{1,t},c_{2,t}\), \(n_{1,t},e_{2,t},x_{m,t},x_{u,t},k_{t+1}\big \} _{t=0}^{\infty }\) to 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.
3.1 Numerical characterization
In this section we provide a numerical assessment of the timing of the optimal corporate, dividend and labor tax rates presented in the previous section and compare them with the timing of the optimal tax rates in an economy with no intangible investment.
In our calibration, we consider a period to be the equivalent to 1 year. In terms of parameter values, we consider the following. In accordance with McGrattan and Prescott (2005), we assume the population is composed of 90% workers, i.e., \(\kappa _{1}=0.90\), and 10% capitalists, i.e., \(\kappa _{2}=0.10\). We assume preference parameters are the same between workers and capitalists. For both, we assume log utility (\(\sigma _{1}=\sigma _{2}=1\)), and a Frisch labor supply elasticity of 0.75 consistent with Chetty et al. (2012), i.e., \(\chi _{1}=\chi _{2}=1.33.\) In the calibration, we target \(\theta _{1}=\theta _{2}\) to yield \(n_{1}=0.33\) at the initial steady state. Both types of agents also share the same discount factor \(\beta ,\) equal to 0.98, and consistent with an annual interest rate of 2%.
In the final good production function, we set total factor productivity as \(A=4\) for computational convenience. We choose an income share of capital equal to \(\alpha =0.40\) to target a consumption ratio of \(\frac{c_{2,ss}}{c_{1,ss}}=3.0\) in accordance with Aguiar and Bils (2015). The depreciation rate is set to \(\delta =0.08\) to take account of both tangible and intangible investments.^{5} For the investment function, we set \(\rho _{m}=-0.50\) to consider some complementarity between tangible resources \(x_{m,t}\) and the composite of intangible resources and management time. We set \(\rho _{u}=0.25\) to allow for some substitutability between intangible investment, \(x_{u,t}\), and management time, \(e_{2,t}\), and later perform sensitivity with respect to this parameter. Then we calibrate B and \(\mu _{m}\) to match an investment to capital ratio of \(\frac{x_{m,ss}+x_{u,ss}}{k_{ss}}=0.037\), consistent with Asker et al. (2015), and a capital to output ratio of \(\frac{k_{ss}}{f(k_{ss},n_{ss})}=2.73,\) consistent with McGrattan and Prescott (2005) estimations. The parameter \(\mu _{u}\) is set to 0.15, and C is calibrated to target the hours devoted to management time by capitalists, \(e_{2,ss}=0.33.\)
We assume that the economy starts off at an initial steady state. That initial steady state considers a government policy characterized by a government spending to output ratio of 19%, a labor income tax rate of 31.6%, a dividend income tax rate of 29.1%, and a corporate income tax rate of 35%.
Parameter values and initial steady-state allocation and welfare
Benchmark parameters | |||||||||
Preference | \(\beta \) | \(\sigma _{1}=\sigma _{2}\) | \(\theta _{1}=\theta _{2}\) | \(\chi _{1}=\chi _{2}\) | |||||
0.98 | 1.00 | 10.04 | 1.33 | ||||||
Technology | A | \(\alpha \) | \(\delta \) | B | C | \(\rho _{m}\) | \(\rho _{u}\) | \(\mu _{m}\) | \(\mu _{u}\) |
4.0 | 0.40 | 0.08 | 17.76 | 0.84 | \(-0.50\) | 0.25 | 0.38 | 0.15 | |
Policy | \(\tau ^{n}\) | \(\tau ^{k}\) | \(\tau ^{d}\) | \(\frac{g}{y}\) | \(\gamma _{1}\) | \(\gamma _{2}\) | |||
0.316 | 0.35 | 0.291 | 0.19 | 0.25 | 0.75 | ||||
Initial steady-state allocation | |||||||||
Allocation | \(\frac{c_{2}}{c_{1}}\) | \(\frac{e_{2}}{n_{1}}\) | \(n_{1}\) | \(\frac{c}{y}\) | \(\frac{x_{m+}x_{u}}{k}\) | \(\frac{x_{m+}x_{u}}{y}\) | \(\frac{k}{y}\) | ||
3.0 | 1.00 | 0.33 | 0.71 | 0.037 | 0.10 | 2.73 | |||
Initial steady-state welfare | |||||||||
Welfare | \(U_{1}\) | \(U_{2}\) | \(\frac{U_{2}}{U_{1}}\) | ||||||
46.66 | 103.40 | 2.22 |
For both economies, all functions and parameters are now chosen, except for the Pareto weights on workers and capitalists. For \(\sigma _{1}=\sigma _{2}=1,\) the following optimality condition from the Ramsey problem \(u_{1c,t}\left[ \gamma _{1}+\lambda _{1,t}\left( 1-\sigma _{1}\right) \right] =u_{2c,t}\left[ \gamma _{2}+\lambda _{1,t}\left( 1-\sigma _{2}\right) \right] ,\) can be written as \(\frac{c_{2,t}}{c_{1,t}}=\frac{\gamma _{2}}{\gamma _{1}}.\) Then to consider a Pareto-improving reform we choose the Pareto weights so as to keep the consumption ratio at the same level as in the initial steady state, i.e., \(\frac{c_{2,t}}{c_{1,t}}=3.0,\) which implies \(\gamma _{1}=0.25\) and \(\gamma _{2}=0.75.\) As welfare is also affected by labor and time management choices, we later check that these Pareto weights are consistent with a Pareto-improving reform.
The timing of the optimal corporate tax rate is presented in Fig. 1. Without intangible investment, the Ramsey corporate tax is set to the upper bound of 100% for 10 years, and then after a 1 year with 83% taxation, corporate taxes fall to 0% permanently. In contrast, with intangible investment, the Ramsey corporate tax is initially set to − 20%, and from then on the optimal corporate tax follows a slow convergence toward zero that takes 50 years, approximately.
Figure 2 presents the timing of the optimal dividend tax rate. Without intangible investment, the Ramsey dividend tax is set to 78% initially and falls sharply toward 37% over 10 years. With intangible investment, however, the Ramsey dividend tax is initially set to 61% and from then on the optimal dividend tax follows a slow and smooth decline toward 27% that again takes around 50 years.
In Fig. 3 we depict the timing of the optimal labor tax rate. Without intangible investment, the Ramsey labor tax falls to zero initially and converges to − 2% in the long run. The presence of intangible investment does not affect the timing of the optimal labor tax rate, but affects the level. With intangible investment, the Ramsey labor tax rate is set to 17% initially and converges to 11% in the long run.
All in all, we see that when we abstract from intangible capital, the optimal fiscal policy is a confiscatory tax on corporate income (a little bit less so for dividends) for as long as necessary, together with the immediate elimination of labor income taxes. This happens even in a world where the planner puts three times more weight on the welfare of each capitalist than on the welfare of each worker.
In contrast, when we consider intangible capital, confiscatory taxation of corporate income becomes very distortionary. In such a world, the optimal fiscal policy is radically different, with a reduction of corporate income taxes on impact (turning it into a 20% subsidy), and then a long time period until it converges to zero, together with an increase in dividend taxes (not as large as in the case without intangibles) and then it is progressively reduced. Labor income taxes are reduced but not eliminated, both in the short run and in the long run.
Figure 4 shows that intangible investment has very little effect on the capital output ratio. Relative to the initial steady state ratio, the capital output ratio is lower without intangibles than with intangibles.
Figure 5 depicts the investment to output ratios relative to their respective initial steady-state values. Both tangible and intangible investment increase on impact for both economies, with and without intangibles. However, the timing of tangible investment is affected by the presence of intangibles. Without intangibles, tangible investment to output increases for around 10 years and then declines toward its steady-state value, while, with intangibles, the tangible investment to output ratio follows a slow decline toward its steady state.
Figure 6 presents the Ramsey allocation of labor and effort. We find that the presence of intangibles affects the level of labor, but not the timing of the optimal labor supply. However, intangibles do affect the allocation of time management during the transition. Without intangibles, time management increases steeply and quickly toward its steady state, while with intangibles, the optimal time management follows a rather flat and slow increase toward its long-run value.
As shown in Fig. 7a, we find that, with intangibles, the optimal corporate tax falls on impact to 11% and follows a slow decline toward zero. Therefore, when intangible investment and management time are complements (substitutes), corporate income is taxed (subsidized) during the transition. For this level of complementarity, Fig. 7b, c shows that dividend (labor) income is taxed more (less) than when investment and management time are substitutes. In Fig. 7d, we see that, relative to the initial steady state, now the capital to output ratio is higher without intangible investment. Figure 7e shows that, when \(x_{u,t}\), and \(e_{2,t}\) are complementary rather than substitutes, the ratio of intangible capital to output remains below the initial steady-state level rather than above. As shown in Fig. 7f, the optimal levels of labor and effort are not affected very much by the degree of complementarity.
For the new Pareto weights, the Ramsey allocation delivers a consumption ratio of \(\frac{c_{2,t}}{c_{1,t}}=1.05\) independently of whether there are intangibles or not. As expected, 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%) 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.
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).
Footnotes
- 1.
As 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.
- 2.
Here 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.
- 3.
Recently, 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.
- 4.
In our numerical exercise, we find that the decentralization constraint (11) does not bind in the long run.
- 5.
Here we assume that tangible capital depreciates at a rate equal to 6.7 while intangible depreciates at a 10% rate.
- 6.
Conesa and Domínguez (2018) reports results for a wider range of Pareto weights in an economy without intangibles.
Notes
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