Taxation and parental time allocation under different assumptions on altruism

Abstract

This paper examines the effects of labor income taxation on parental time allocation in an OLG model in which child care arrangements, that is the combination of parental and non-parental time, matter for human capital accumulation. We show that the sign of the impact of labor income taxation on parental time with children and on growth critically depends on the assumption on the altruistic motives behind the choice of devoting time to children.

Appendices

Appendix 1: Proof of Propositions 1 and 2

We stationarize Eqs. (6), (8), (9) and (12)–(14) and focus on a balanced growth path. We denote a stationarized variable with a $\widetilde{}$ . After few manipulations it is possible to show that the equilibrium can be expressed as a function of l and n only, and it is characterized by the following system of equations:

$$\begin{aligned} \varphi _{1}(l,n,\tau )\equiv & {} \frac{1+\theta }{l-\frac{a-n}{\chi }}(1-\tau )-\beta (z)^{\kappa -1}=0 \end{aligned}$$

(23)

$$\begin{aligned} \varphi _{2}(l,n,\tau )\equiv & {} \frac{1+\theta }{l-\frac{a-n}{\chi }}\left( 1-\tau -\frac{1}{\chi }\right) \nonumber \\&-\frac{\text {d}\widetilde{x}}{\text {d}n}\frac{1}{\widetilde{x}}\left( \gamma ^{p}+\gamma ^{fa}\frac{1+\theta }{l-\frac{a-n}{\chi }}l(1-\tau )\right) =0 \end{aligned}$$

(24)

where $\frac{\text {d}\widetilde{x}_{}}{\text {d}n_{}}\frac{1}{\widetilde{x}_{}}=\frac{-\sigma (a-n_{})^{\nu -1}+(1-\sigma )n_{}^{\nu -1}}{\sigma (a-n)^{\nu }+(1-\sigma )n^{\nu }}.$

We apply the implicit function theorem to the system of Eqs. (23 )–(24), and we obtain:

$$\begin{aligned} \frac{\partial l}{\partial \tau }=-\frac{\left| \begin{array}{cc} \frac{\partial \varphi _{1}}{\partial \tau } &{} \frac{\partial \varphi _{1}}{\partial n} \\ \frac{\partial \varphi _{2}}{\partial \tau } &{} \frac{\partial \varphi _{2}}{\partial n} \end{array} \right| }{\left| \begin{array}{cc} \frac{\partial \varphi _{1}}{\partial l} &{} \frac{\partial \varphi _{1}}{\partial n} \\ \frac{\partial \varphi _{2}}{\partial l} &{} \frac{\partial \varphi _{2}}{\partial n} \end{array} \right| } =-\frac{\frac{\partial \varphi _{1}}{\partial \tau }\frac{\partial \varphi _{2}}{\partial n}-\frac{\partial \varphi _{2}}{\partial \tau }\frac{\partial \varphi _{1}}{\partial n}}{\frac{\partial \varphi _{1}}{ \partial l}\frac{\partial \varphi _{2}}{\partial n}-\frac{\partial \varphi _{2}}{\partial l}\frac{\partial \varphi _{1}}{\partial n}} \end{aligned}$$

(25)

and

$$\begin{aligned} \frac{\partial n}{\partial \tau }=-\frac{\left| \begin{array}{cc} \frac{\partial \varphi _{1}}{\partial l} &{} \frac{\partial \varphi _{1}}{ \partial \tau } \\ \frac{\partial \varphi _{2}}{\partial l} &{} \frac{\partial \varphi _{2}}{ \partial \tau } \end{array} \right| }{\left| \begin{array}{cc} \frac{\partial \varphi _{1}}{\partial l} &{} \frac{\partial \varphi _{1}}{ \partial n} \\ \frac{\partial \varphi _{2}}{\partial l} &{} \frac{\partial \varphi _{2}}{ \partial n} \end{array} \right| }-\frac{\frac{\partial \varphi _{1}}{\partial l}\frac{\partial \varphi _{2}}{\partial \tau }-\frac{\partial \varphi _{2}}{\partial l}\frac{\partial \varphi _{1}}{\partial \tau }}{\frac{\partial \varphi _{1}}{\partial l}\frac{\partial \varphi _{2}}{\partial n}-\frac{\partial \varphi _{2}}{\partial l}\frac{\partial \varphi _{1}}{\partial n}}. \end{aligned}$$

(26)

In the case of paternalism (Proposition 1), we have:

$$\begin{aligned} \frac{\partial \varphi _{1}}{\partial \tau }= & {} -\frac{1+\theta }{l-\frac{a-n}{ \chi }} \end{aligned}$$

(27)

$$\begin{aligned} \frac{\partial \varphi _{1}}{\partial n}= & {} -\frac{1+\theta }{\chi \left( l-\frac{ a-n}{\chi }\right) ^2}(1-\tau )-\beta (1-\kappa )z^{\kappa -2} \end{aligned}$$

(28)

$$\begin{aligned} \frac{\partial \varphi _{1}}{\partial l}= & {} -\frac{1+\theta }{\left( l-\frac{a-n}{ \chi }\right) ^2}(1-\tau )-\beta (1-\kappa )z^{\kappa -2} \end{aligned}$$

(29)

$$\begin{aligned} \frac{\partial \varphi _{2}}{\partial \tau }= & {} -\frac{1+\theta }{l-\frac{a-n}{\chi }} \end{aligned}$$

(30)

$$\begin{aligned} \frac{\partial \varphi _{2}}{\partial n}= & {} -\frac{1+\theta }{\chi \left( l-\frac{a-n}{\chi }\right) ^2} \left( 1-\tau -\frac{1}{\chi }\right) -\gamma ^{p}\frac{\text {d}}{\text {d}n}\left( \frac{\text {d}\widetilde{x}}{\text {d}n}\frac{1}{\widetilde{x}}\right) \end{aligned}$$

(31)

$$\begin{aligned} \frac{\partial \varphi _{2}}{\partial l}= & {} -\frac{1+\theta }{\left( l-\frac{a-n}{ \chi }\right) ^2}\left( 1-\tau -\frac{1}{\chi }\right) \end{aligned}$$

(32)

where:

$$\begin{aligned}&\frac{\text {d}}{\text {d}n}\left( \frac{\text {d}\widetilde{x}}{\text {d}n}\frac{1}{\widetilde{x}}\right) \nonumber \\&= \frac{\left[ \sigma (\nu -1)(a-n)^{\nu -2}+(1-\sigma )(\nu -1)n^{\nu -2} \right] \widetilde{x}-\nu \left[ -\sigma (a-n)^{\nu -1}+(1-\sigma )n^{\nu -1} \right] ^{2}}{\widetilde{x}^{2\nu }}\qquad \end{aligned}$$

(33)

with $\widetilde{x}=\left[ \sigma (a-n)^{\nu }+(1-\sigma )(n_{t})^{\nu } \right] ^{\frac{1}{\nu }}$. Note that $\frac{\text {d}}{\text {d}n}\left( \frac{\text {d}\widetilde{x} }{\text {d}n}\frac{1}{\widetilde{x}}\right) <0$ (recall that $\nu <1$).

After a few manipulations, we can write $\frac{\partial l}{\partial \tau }$ in (25) as follows:

$$\begin{aligned}&\frac{\partial l}{\partial \tau }=\nonumber \\&-\frac{-\frac{(1+\theta )^2}{\chi ^2\left( l- \frac{a-n}{\chi }\right) ^3}+\gamma ^p(1+\theta )\frac{\frac{\text {d}}{\text {d}n}\left( \frac{\text {d} \widetilde{x}}{\text {d}n}\frac{1}{\widetilde{x}}\right) }{\left( l-\frac{a-n}{\chi } \right) }-\frac{1+\theta }{\left( l-\frac{a-n}{\chi }\right) }\beta (1-\kappa )z^{ \kappa -2}}{\gamma ^p\frac{\text {d}}{\text {d}n}\left( \frac{\text {d}\widetilde{x}}{\text {d}n}\frac{1}{ \widetilde{x}}\right) \left[ \frac{1+\theta }{\left( l-\frac{a-n}{\chi }\right) ^2 }(1-\tau )\gamma ^p+\beta (1-\kappa )z^{\kappa -2}\right] -\frac{\chi -1}{\chi } \beta (1-\kappa )z^{\kappa -2}\frac{1+\theta }{\left( l-\frac{a-n}{\chi }\right) ^2} \left( 1-\tau -\frac{1}{\chi }\right) }\qquad \nonumber \\ \end{aligned}$$

(34)

Since both the numerator and the denominator of Eq. (34) are negative, we get $\frac{\partial l}{\partial \tau }<0$.

As to $\frac{\partial n}{\partial \tau }$, after a few calculations we can write it as

$$\begin{aligned}&\frac{\partial n}{\partial \tau }=\nonumber \\&-\frac{\frac{\left( 1+\theta \right) ^2}{ \chi \left( l-\frac{a-n}{\chi }\right) ^3}+\beta (1-\kappa )z^{\kappa -2}\frac{ \left( 1+\theta \right) }{\left( l-\frac{a-n}{\chi }\right) }}{\gamma ^p\frac{\text {d}}{\text {d}n }\left( \frac{\text {d}\widetilde{x}}{\text {d}n}\frac{1}{\widetilde{x}}\right) \left[ \frac{ 1+\theta }{\left( l-\frac{a-n}{\chi }\right) ^2}(1-\tau )\gamma ^p+\beta (1- \kappa )z^{\kappa -2}\right] -\frac{\chi -1}{\chi }\beta (1-\kappa )z^{\kappa -2} \frac{1+\theta }{\left( l-\frac{a-n}{\chi }\right) ^2}\left( 1-\tau -\frac{1}{\chi } \right) }.\nonumber \\ \end{aligned}$$

(35)

The denominator of (35) corresponds to the one in Eq. ( 34) and it is negative, whereas the numerator is positive; thus, $\frac{\partial n}{\partial \tau }>0$.

We now focus on the case of full altruism (Proposition 2). Equations (27), (28) and (29) still hold, whereas Eqs. (30), (31) and (32) read as

$$\begin{aligned} \frac{\partial \varphi _{2}}{\partial \tau }= & {} -\frac{1}{\chi l(1-\tau )^{2}} \end{aligned}$$

(36)

$$\begin{aligned} \frac{\partial \varphi _{2}}{\partial n}= & {} -\gamma ^{fa}\frac{\text {d}}{\text {d}n}\left( \frac{\text {d}\widetilde{x}}{\text {d}n}\frac{1}{\widetilde{x}}\right) \end{aligned}$$

(37)

$$\begin{aligned} \frac{\partial \varphi _{2}}{\partial l}= & {} -\frac{1-\tau -\frac{1}{\chi }}{ l^{2}(1-\tau )}. \end{aligned}$$

(38)

Starting from $\frac{\partial l}{\partial \tau }$, it is straightforward to see that both the numerator and the denominator of (25) are negative and thus $\frac{\partial l}{\partial \tau }<0$. The sign of $\frac{\partial n }{\partial \tau }$ is instead ambiguous. To identify a condition such that $\frac{\partial n}{\partial \tau }$ is positive or negative, we first note that $\frac{\partial n}{\partial \tau }$ can be written as follows¹⁴:

$$\begin{aligned} \frac{\partial n}{\partial \tau }=-\frac{\frac{\partial \varphi _{2}}{ \partial \tau }+\frac{\partial \varphi _{2}}{\partial l}\frac{\partial l}{ \partial \tau }}{\frac{\partial \varphi _{2}}{\partial n}}. \end{aligned}$$

(39)

The denominator of (39) is positive. As to the numerator, we substitute from (36) and from (38); rearranging terms we obtain $\frac{\partial n}{\partial \tau }>0$ if and only if:

$$\begin{aligned} \left| \epsilon _{l\tau }\right| <\frac{\tau }{1-\tau }\frac{1}{ \chi \left( 1-\tau -1/\chi \right) }\equiv \bar{\epsilon }_{l\tau }, \end{aligned}$$

(40)

which is condition (16) in the main text.

Appendix 2: Calibration

The calibration follows Casarico and Sommacal (2012).

We interpret each period as having a length of 25 years. We set the time span over which child care is provided to 6 years; thus, $a=6/25=0.24$. This means that $24\%$ of the first period of life is spent receiving child care. Because population is constant, we normalize its size N at 1.

To choose the productivity parameter of the day care sector $\chi$ we look at the child to staff ratio. According to the OECD family database,¹⁵ this ratio changes depending on the country considered and on the age range of the children, being lower for children between 0 and 3 and higher for children between 3 and 5. We set $\chi$ equal to 5, which is at the lower bound of the child to staff ratio in the data mentioned above.¹⁶

We choose the parameters $\beta$ and $\delta$ in the utility function (20) and the parameter $\sigma$ in the production function of the quality of child care (18) to generate a realistic allocation of time between labor l, productive parental time with the child $n^1$ and unproductive parental time with the child $n^2$. Leisure z is residually determined using the time constraint (7).¹⁷ In line with previous research (e.g., Juster 1985; Cardia and Ng 2003; Ragan 2013), we assume that non-personal time available for discretionary use amounts to 100 hours per week. We use the Harmonized European Time Use Survey (HETUS).¹⁸ As a proxy of productive time with the child we use child care performed by parents as a primary activity and as a proxy of unproductive time we use child care performed as a secondary activity, plus the time spent transporting children.¹⁹ The data show that on average $32\%$ of the time endowment is devoted to work, $4\%$ to productive time with the child and $3\%$ to unproductive time with the child.

As to the parameter $\nu$ which governs the elasticity of substitution between $d_{t}h_{t}$ and $n^1_{t}h_{t}$, $\zeta _{\nu }=1/(1-\nu )$, see Eq. (18), the existing estimates refer to the elasticity of substitution between inputs in the production of the general category of home-produced goods. We use a value of $\nu =0.5$, giving an elasticity $\zeta _{\nu }=2$ which belongs to the range of available estimates (see Aguiar et al. 2012).²⁰

With regard to the choice of $\tau$, we use Eurostat data to compute the average of the implicit tax rates²¹ on labor income for the set of countries we mention in footnote 18 and set the policy parameter $\tau$ to $39\%$.

The day care subsidy is set in order to match a ratio of public expenditure on child care and preschool to GDP equal to $0.9\%$, which is the average value for the countries mentioned in footnote 18, according to OECD data.

Table 4 Values of the calibrated parameters (for different values of the wage elasticity of labor supply)

The intertemporal discount factor $\theta$ is chosen in order to have a ratio of gross savings to GDP equal to 0.23. We choose $\alpha$, that is the share of capital income in national product, equal to 0.33. The depreciation rate is set equal to 1, which is a quite common assumption in two-period OLG models.²² We choose the parameter q of the human capital production function to obtain an annual growth rate of GDP per capita ($(1+g)^{\frac{1}{25}}-1$) equal to $2\%$ which is the average value for the set of countries we consider.

The parameter characterizing the degree of paternalism $\gamma ^{p}$ is set equal to the intertemporal discount factor $\theta$ when the case of paternalism is considered (i.e., we assume that the parent gives the same weight to her own consumption and to the human capital of the child), whereas it is set to 0 when we consider the case in which parental child care is motivated by fully altruistic preferences. By the same token we set the parameter characterizing the degree of full altruism $\gamma ^{fa}$ equal to $\theta$ when the case of full altruism is analyzed, whereas it is 0 when we consider the case of paternalistic parental preferences.

We finally choose the parameter $\kappa$ in the utility function (20) to obtain different values for the wage elasticity of labor supply, namely 0.1, 0.2, 0.3, 0.4, 0.5.

The values of all the parameters calibrated according to the procedure specified above are reported in Table 4.