Income Taxation

Abstract

After a brief survey of the empirical findings on income taxation in the US and German economies in Sect. 5.2, you learn about the substantial welfare costs that are associated with the taxation of labor income. In Sect. 5.3, these costs are computed in both partial and general equilibrium. As one result, the deadweight loss of labor income taxation in Germany is found to be twice as high as the one in the US. In Sect. 5.4, the seminal result from optimal taxation that capital income should not be taxed in the long run is derived and discussed critically. Section 5.5 estimates the US Laffer curve and shows that the US government, in contrast to many European governments, can still raise its revenues from labor and capital income taxation by approximately 10% of GDP. In Sect. 5.6, the quantitative effects of higher taxes on economic growth are derived in a Dynamic General Equilibrium (DGE) model and are shown to be substantially higher than those typically found in growth regressions. Finally, we demonstrate that stochastic taxes improve the time series properties of the real business cycle (RBC) model with respect to the volatility of aggregate demand components and the dynamics of labor and wages in Sect. 5.7.

Appendices

Appendix 5.1: Derivation of the Chamley-Judd Result

The optimality conditions for the derivation of (5.22) with respect to $\tau ^K_t$ and $\tau ^L_t$ are given by (after the replacement of the factor prices r and w by their marginal products F _K and F _L, respectively):

$$\displaystyle \begin{aligned} \psi_t K_t -\mu_{2t-1} u_{C_{t}} =0, \end{aligned} $$

(5.83)

and

$$\displaystyle \begin{aligned} \psi_t L_t -\mu_{1t} u_{C_t} =0. \end{aligned} $$

(5.84)

The optimality condition with respect to K _t is represented by:

$$\displaystyle \begin{aligned} \begin{aligned} &\psi_t \left[ \tau^K_t \left(F_{K_t}+K_t F_{K_tK_t}\right)+\tau^L_t F_{L_tK_t} L_t \right]+\theta_t \left[ F_{K_t}+1-\delta\right]-\frac{1}{\beta}\theta_{t-1}\\ &+\mu_{1t} u_{C_t} \left[ 1-\tau^L_t \right] F_{L_t K_t} +\mu_{2t-1} u_{C_t} \left[ 1-\tau^K_t\right]F_{K_tK_t}=0. \end{aligned} \end{aligned}$$

Inserting Eqs. (5.83) and (5.84) into the above equation and noticing that for a constant-returns to scale production function, F _KK K + F _LK L = 0, holds,⁶⁴ we derive

$$\displaystyle \begin{aligned} \psi_t \tau^K_t F_{K_t} + \theta_t \left[ F_{K_t}+1-\delta\right]-\frac{1}{\beta}\theta_{t-1}=0. \end{aligned}$$

In steady state, all variables are constant: K _t = K, C _t = C, L _t = L, θ _t = θ, and ψ _t = ψ:

$$\displaystyle \begin{aligned} \psi \tau^K F_{K} + \theta \left[ F_{K}+1-\delta\right]-\frac{1}{\beta}\theta=0.\end{aligned} $$

(5.85)

In addition, the Euler equation of the household

$$\displaystyle \begin{aligned} \beta=\frac{1}{1-\delta +\left(1-\tau^K\right) F_K}\end{aligned}$$

can be substituted into (5.85), yielding:

$$\displaystyle \begin{aligned} \left[\psi+\theta\right] \tau^K F_K=0\end{aligned}$$

This equation is only fulfilled if τ ^K = 0.

Appendix 5.2: Data Sources

In addition to the macroeconomic data presented in Appendices 2.4 and 4.6, we introduce the following variables in our empirical analysis:

• Tax Revenue The data for Fig. 5.1 are retrieved from OECD Revenue Statistics 1965–2015: OECD (2018), Tax revenue (indicator). doi: 10.1787/d98b8cf5-en (Accessed on 26 March 2018).

  https://data.oecd.org/tax/tax-revenue.htm.

  The data for Table 5.1 are retrieved from OECD Revenue Statistics 1965–2016 (2017). and OECD (2018), Tax on corporate profits (indicator). doi: 10.1787/d30cc412-en (Accessed on 26 March 2018).

  The shares of Italian taxes on goods and services in GDP and total revenues are retrieved from the OECD: OECD (2018), Tax on goods and services (indicator). doi: 10.1787/40b85101-en (Accessed on 26 March 2018).

  https://data.oecd.org/tax/tax-on-goods-and-services.htm#indicator-chart.

• Income Tax Rates The US income tax schedule presented in Fig. 5.3 is generated with data from the Tax Foundation (Accessed on 26 March 2018).

  http://taxfoundation.org/article/2016-tax-brackets.

  For Germany, the respective data (as presented in Figs. 5.4 and 5.5) are retrieved from the web page of the Bundesministerium für Finanzen (Ministry of Finance) (Accessed on 26 March 2018).

  https://www.bmf-steuerrechner.de/ekst/.

Problems

5.1

Recompute the model of Sect. 5.3.2. Instead of (5.5), use the following additive utility function:

$$\displaystyle \begin{aligned} u(C_t,L_t) = \frac{C_t^{1-\sigma}}{1-\sigma} - \nu_0 \frac{L_t^{1+\frac{1}{\nu_1}}}{{1+\frac{1}{\nu_1}}}, \end{aligned}$$

where ν ₁ denotes the Frisch elasticity of labor supply. Set the Frisch labor supply elasticity equal to 0.2, ν ₁ = 0.2. All other parameters are set as in Sect. 5.3.2.

• 1. Calibrate ν ₀ such that the steady-state labor supply is equal to L = 0.3.

• 2. Compute the welfare effects of an increase in τ ^L from 23% to 24% in partial equilibrium and in general equilibrium (for both the steady state and the transition). Are the consumption equivalent changes close to one another and insensitive to the Frisch labor supply elasticity?

5.2

This problem follows Prescott (2004). Assume that instantaneous utility is logarithmic. Lifetime utility is given by

$$\displaystyle \begin{aligned} U=\sum_{t=0}^\infty \beta^t \left[\ln C_t + \iota \ln (1-L_t)\right].\end{aligned}$$

The capital stock accumulates according to

$$\displaystyle \begin{aligned} K_{t+1}=(1-\delta) K_t + I_t,\end{aligned}$$

and production is Cobb-Douglas:

$$\displaystyle \begin{aligned} Y_t = A_t K_t^\alpha L_t^{1-\alpha}.\end{aligned}$$

In goods market equilibrium,

$$\displaystyle \begin{aligned} Y_t = C_t + G_t + I_t.\end{aligned}$$

The household’s budget constraint is represented by

$$\displaystyle \begin{aligned} (1+\tau^C) C_t + (1+\tau^I) I_t = (1-\tau^L) w_t L_t + (1-\tau^K) (r_t -\delta) K_t + \delta K_t + Tr_t,\end{aligned}$$

where τ ^C, τ ^I, and τ ^K denote the tax rates on consumption, investment, and capital income, respectively. Capital depreciation is tax-exempt. The government spends tax revenues, τ ^C C _t + τ ^I I _t + τ ^L w _t L _t + τ ^K(r _t − δ)K _t, on government consumption G _t and transfers Tr _t. Assume that α = 0.36, δ = 0.10, and β = 0.96.

• 1. Derive the first-order conditions of the household and the firm. Notice that the wedge on labor income also depends on the consumption tax rate τ ^C.

• 2. Calibrate the parameter ι such that the steady-state labor supply is equal to L = 0.3 for τ ^I = 0, τ ^K = 0.42, τ ^L = 0.23, and τ ^C = 0.26. τ ^C is set such that $\frac {\tau ^C+\tau ^L}{1+\tau ^L}=0.40$ as given in Table 2 of Prescott (2004) for the US economy during the periods 1970–1974 and 1993–1996. Compare the values of the endogenous variables C, L, K, and Y  with those in the the case in which τ ^L and τ ^C increase to the Italian values of the tax rates, τ ^C = 0.485 and τ ^L = 0.429.⁶⁵ How large is the change in the steady-state labor supply? Does this account for the observation that Americans worked 56% more hours than Italians during this period?

5.3

Show that the Chamley-Judd result also holds for the wealth tax. Use the budget constraint (5.25) to derive the first-order conditions of the household optimization problem:

$$\displaystyle \begin{aligned} \lambda_t (1-\tau^L_t) w_t&=-\frac{\partial u}{\partial L_t} =-u_{L_t}, \end{aligned} $$

(5.86a)

$$\displaystyle \begin{aligned} \lambda_t & =\frac{\partial u}{\partial C_t}=u_{C_t}, \end{aligned} $$

(5.86b)

$$\displaystyle \begin{aligned} \lambda_{t} & =\beta \lambda_{t+1}\left(1+r_{t+1}-\delta-\tau^V_{t+1}\right). \end{aligned} $$

(5.86c)

Show that these conditions are equivalent to those presented in (5.19) if

$$\displaystyle \begin{aligned} \tau^K_t r_{t} = \tau^V_{t},\end{aligned}$$

implying that the tax revenues are the same in both cases:

$$\displaystyle \begin{aligned} \tau^K r K = \tau^V K.\end{aligned}$$

5.4

Consider the Laffer curve in Sect. 5.5.

• 1. Show that the equilibrium conditions (5.38) hold.

• 2. Recompute the Laffer curves in Fig. 5.14.

• 3. Compute the sensitivity of the results with respect to a Frisch labor supply elasticity ν ₁ = 0.3.

• 4. Recompute Fig. 5.15 for the case in which depreciation is not tax-deductible.

5.5

Show that a replacement of the income tax τ with a consumption tax τ ^C increases the economic growth rate in the decentralized economy of the endogenous growth model with the public input good presented in Sect. 5.6.1.

5.6

Productive Government Expenditures as a Stock Variable In contrast to (5.39), assume that production uses public capital $K^G_t$ as an input:

$$\displaystyle \begin{aligned} Y_t = F(K_t,L_t,G_t) = A L_t^{1-\alpha} K_t^\alpha \left(K^G_t\right)^{1-\alpha}.\end{aligned} $$

(5.87)

Public capital accumulates according to:

$$\displaystyle \begin{aligned} K^G_{t+1} = (1-\delta) K^G_t + I^G_t.\end{aligned}$$

Public investment $I^G_t$ is financed by a production tax:

$$\displaystyle \begin{aligned} I^G_t = \tau Y_t.\end{aligned}$$

Compute (1) the maximum growth rate in the decentralized economy and (2) the Pareto-efficient growth rate.

5.7

Congestion Effects and Productive Government Expenditures (follows Turnovsky 1996 ) To introduce congestion effects, we distinguish between the capital that is used in the production of the individual firm k and the aggregate capital K. The ratio k∕K measures the size of the individual firm relative to the economy. Accordingly, the public expenditures G imply the following service g to the individual firm:

$$\displaystyle \begin{aligned} g_t = G_t \left( \frac{k_t}{K_t}\right)^{1-\sigma^G},\end{aligned}$$

where σ ^G ∈ [0, 1] denotes the degree of congestion.

Assume that the production of the individual firm is represented by

$$\displaystyle \begin{aligned} y_t = A l_t^{1-\alpha} k_t^\alpha g_t^{1-\alpha}\end{aligned}$$

Derive the equilibrium growth rate in the decentralized economy in which K _t = k _t, l _t = L, and g _t = G _t. In addition, assume that the individual firm does not consider the impact of its investment decision on G _t and K _t.

5.8

Analyze the effects of capital income taxation on the growth rate in the Lucas supply-side model if depreciation can be deducted from capital income taxation. Derive the growth-rate effects of capital income taxation and compare it to the case presented in Sect. 5.6.2.

5.9

Introduce adjustment costs of capital in the RBC model with stochastic taxes presented in Sect. 5.7. Use the specification of the adjustment cost function (4.54) with the parameterization ζ = 3.0. How do adjustment costs affect the results presented in Table 5.5?