Optimal fiscal policy with a balanced-budget restriction: revisiting Chamley and Barro

Abstract

Barro (J Polit Econ 98:S103–S126, 1990) established an endogenous growth model with taxed-financed public services that affect production or utility. Since there is only an income tax, the tax rate must be positive under a balanced-budget restriction. Then there arises a question as to whether the famous zero-capital-tax result of Chamley (Econometrica 54:607–622, 1986) applies in the Barro model if the government is allowed to tax capital and labor incomes, respectively. To examine this question, we develop a generalized Barro model, which combines characteristics of the Chamley model by incorporating elastic labor supply and capital and labor income taxation. By using the dual approach, we derive simple formulas for optimal income taxation and the second-best rules for public consumption and production services. In particular, we demonstrate that both capital and labor incomes should be taxed when the economy grows along a balanced-growth path. The intuition behind our positive capital tax result is that, under the balanced-budget constraint, the government has to minimize each time’s tax distortion by equating the marginal excess burdens of capital and labor taxation at any time.

Appendices

Appendix 1: The proof of Propositions 2 & 3

To prove Proposition 2, it is more convenient to take the agent’s after-tax income as a function of $(c,l,k)$. First, from Eq. (16), we obtain

$$\overline{r}k + \overline{w}l = \frac{{\rho k + \overline{w}l - \Theta c}}{1 - \Theta },$$

where $\Theta \ne 1$ and the superscript “*” is omitted for convenience. Then, by substituting Eq. (17) into the above equation, we derive the agent’s after-tax income as

$$\overline{r}k + \overline{w}l = \frac{{\rho k - {{U_{l} l} \mathord{\left/ {\vphantom {{U_{l} l} {U_{c} }}} \right. \kern-0pt} {U_{c} }} - \Theta c}}{1 - \Theta } \equiv {\rm Z}(c,l,k).$$

(35)

By using Eq. (35), we rewrite the balanced-budget constraint (20) as

$$y - {\rm Z}(c,l,k) - g - h = 0.$$

(36)

Therefore, the government’s Hamiltonian (23) can be rewritten as

$$H^{G} = V(\overline{r},\overline{w},k,h) + \lambda (y - c - g - h) + \mu [y - {\rm Z}(c,l,k) - g - h].$$

The first-order conditions for $\overline{r}$ and $\overline{w}$ are,

$$[U_{c} + U_{l} l_{c} + \lambda (f_{l} l_{c} - 1) + \mu (f_{l} l_{c} - {\rm Z}_{c} - {\rm Z}_{l} l_{c} )]c_{{\overline{r}}} = 0,$$

(37)

$$[U_{c} + U_{l} l_{c} + \lambda (f_{l} l_{c} - 1) + \mu (f_{l} l_{c} - {\rm Z}_{c} - Z_{l} l_{c} )]c_{{\overline{w}}} + [U_{l} + \lambda f_{l} + \mu (f_{l} - {\rm Z}_{l} )]l_{{\overline{w}}} = 0.$$

(38)

Since $c_{{\overline{r}}} \ne 0$, from Eq. (37), we have

$$U_{c} + U_{l} l_{c} + \lambda (f_{l} l_{c} - 1) + \mu (f_{l} l_{c} - {\rm Z}_{c} - {\rm Z}_{l} l_{c} ) = 0.$$

(39)

Substituting the above result to Eq. (38) yields

$$U_{l} + \lambda f_{l} + \mu (f_{l} - {\rm Z}_{l} ) = 0.$$

(40)

Then, by substituting Eqs. (40) and (9) to Eq. (39), we obtain

$$q - \lambda = \mu {\rm Z}_{c}.$$

(41)

To derive the optimal tax wedge on capital income, we first derive the costate equation for $\lambda$ as

$$\dot{\lambda } = \lambda \rho - [U_{c} + U_{l} l_{c} + \lambda (f_{l} l_{c} - 1) + \mu (f_{l} l_{c} - {\rm Z}_{c} - {\rm Z}_{l} l_{c} )]c_{k} - \lambda f_{k} - \mu (f_{k} - {\rm Z}_{k} ).$$

We then substitute Eq. (37) and ${\rm Z}_{k} = {\rho \mathord{\left/ {\vphantom {\rho {(1 - \Theta )}}} \right. \kern-0pt} {(1 - \Theta )}}$ (according to Eq. (35)) into the above equation to obtain

$$\frac{{\dot{\lambda }}}{\lambda } = \rho - f_{k} - \frac{\mu }{\lambda }(f_{k} - \frac{\rho }{1 - \Theta }).$$

By equating ${{\dot{\lambda }} \mathord{\left/ {\vphantom {{\dot{\lambda }} \lambda }} \right. \kern-0pt} \lambda }{{ = \dot{q}} \mathord{\left/ {\vphantom {{ = \dot{q}} q}} \right. \kern-0pt} q}$, we derive the formula for optimal capital income tax in this case as

$$r - \overline{r} = f_{k} - \overline{r} = \frac{\mu }{\lambda }(\frac{\rho }{1 - \Theta } - f_{k} ).$$

(42)

In Appendix 2, we derive the absolute value of the marginal excess burden of distortionary taxation as

$$\frac{\mu }{1 - \Theta } > 0.$$

Thus, we must have

$$\mu \left\{ \begin{gathered} > 0\begin{array}{*{20}c} {\begin{array}{*{20}c} {} & {{\text{if}}} \\ \end{array} } & {0 < \Theta < 1} \\ \end{array} \hfill \\ < 0\begin{array}{*{20}c} {} & {{\text{if}}} & {\Theta > 1} \\ \end{array} \hfill \\ \end{gathered} \right..$$

If $\Theta > 1$, it is easy to see the optimal tax wedge on capital income $r - \overline{r} > 0$ according to the formula (42). If $0 < \Theta < 1$, we can also show that the optimal capital tax wedge must be positive. To see this, we first using Eq. (14) to derive that $c(t) = c(0) \cdot e^{{[(\overline{r} - \rho )/\Theta ]t}}$. Since the economy goes on a balanced-growth path, we must have $l(t) = l(0)$ and $h(t) = h(0) \cdot e^{{[(\overline{r} - \rho )/\Theta ]t}}$. We then substitute the above results into the instantaneous utility (12) to derive

$$U(c,l,h) = U(0) \cdot \exp \{ [(\overline{r} - \rho )/\Theta ] \cdot (1 - \Theta ) \cdot t\} ,$$

(43)

where $U(0) = \{ [c(0) \cdot h(0)^{\chi } \cdot v(l)]^{1 - \sigma } + \beta h(0)^{1 - \sigma } \} /(1 - \sigma )$, $\Theta = \sigma - \chi (1 - \sigma )$, and $\chi > 0$ if $\beta = 0$ or $\beta > 0$ if $\chi = 0$. Then, we substitute Eq. (43) into Eq. (1) to derive the agent’s overall utility as

$$W = \int_{0}^{\infty } {U(0) \cdot \exp \{ [(\overline{r} - \rho )/\Theta ] \cdot (1 - \Theta ) \cdot t - \rho t\} } {\text{d}}t.$$

The condition that utility should be bounded ensures that $\rho > [(\overline{r} - \rho )/\Theta ] \cdot (1 - \Theta )$. Hence, we obtain ${\rho \mathord{\left/ {\vphantom {\rho {(1 - \Theta )}}} \right. \kern-0pt} {(1 - \Theta )}} > \overline{r}$ when $0 < \Theta < 1$. However, if the capital tax wedge is non-positive, then, according to the formula for optimal capital taxation (42), we must have $\overline{r} \ge f_{k} \ge \rho /(1 - \Theta )$, which violates the condition for bounded utility. Therefore, the optimal tax wedge on capital income $r - \overline{r}$ must be positive to ensure the utility is bounded when $0 < \Theta < 1$. In sum, no matter whether $0 < \Theta < 1$ or $\Theta > 1$, the optimal tax on capital income is positive.

Meanwhile, using the result of Eqs. (17), we get the optimal tax wedge on labor income as

$$w - \overline{w} = f_{l} - \overline{w} = f_{l} + \frac{{U_{l} }}{{U_{c} }}.$$

Thus, we prove Proposition 2.

Meanwhile, the first-order conditions with respect to public consumption and production services h and g are, respectively,

$$\frac{\partial H}{{\partial h}} = U_{h} - \lambda - \mu = 0,$$

$$\frac{\partial H}{{\partial g}} = \lambda (f_{g} - 1) + \mu (f_{g} - 1) = 0.$$

Then, we easily derive the second-best rules for tax-financed public consumption and production services, which are, respectively,

$$U_{h} = \lambda + \mu ,$$

$$f_{g} = 1.$$

Thus, we prove Proposition 3.

Appendix 2: The derivation of the marginal social value of capital and the marginal excess burden of distortionary taxation

Below the government’s Hamiltonian (23) in the text, we state that $\lambda > 0$ represents the marginal social value of capital and ${\mu \mathord{\left/ {\vphantom {\mu {(1 - \Theta )}}} \right. \kern-0pt} {(1 - \Theta )}}$ represents the absolute value of the marginal excess burden of distortionary taxation. In this appendix, we provide the derivation of the marginal social value of capital and the marginal excess burden of taxation. The methods we use here are proposed by Chiang (1992, pp. 206–207) and Atkinson and Stern (1974, pp. 126–127).

Since the marginal excess burden is also known as the marginal value of replacing lump-sum taxation with distortionary taxation (Chamley, 1986, p. 611; Atkinson & Stern, 1974, p. 126), it would be helpful to demonstrate the excess burden by examine the effect of allowing the government to use lump-sum taxation at level $T(t)$ for each point in time. We assume that $T(t) \ge 0$ is exogenously given and grows at the same rate with k.

Then, the government’s budget constraint (20) is now formulated as

$$y - \overline{r}k - \overline{w}l - g - h + T = 0.$$

(44)

Correspondingly, the agent’s budget constraint (2) turns to

$$\dot{k} = \overline{r}k + \overline{w}l - c - T.$$

In the case of $\Theta \ne 1$, after some calculation, similar to that in the main text, it is easy to obtain the agent’s utility-maximization solutions to $(c,l)$ as

$$c = c(\overline{r},\overline{w},T,k),$$

$$l = l(\overline{w},c),$$

where we omit the superscript “*” for convenience. Hence, the indirect utility function (21) now turns to

$$V(\overline{r},\overline{w},k,h,T) = U[c(\overline{r},\overline{w},k,T),l[c(\overline{r},\overline{w},k,T),\overline{w}],h].$$

(45)

Meanwhile, we obtain that

$$\overline{r}k + \overline{w}l = \frac{{\rho k - {{U_{l} l} \mathord{\left/ {\vphantom {{U_{l} l} {U_{c} }}} \right. \kern-0pt} {U_{c} }} - \Theta (c + T)}}{1 - \Theta } \equiv {\rm Z}(c,l,k,T).$$

(46)

By using Eq. (46), we rewrite the balanced-budget constraint (44) as

$$y - {\rm Z}(c,l,k,T) - g - h + T = 0.$$

(47)

Then, the government’s optimal fiscal policy problem is to maximize the (indirect) overall welfare

$$\int_{0}^{\infty } {V(\overline{r},\overline{w},k,h,T)e^{ - \rho t} } {\text{d}}t,$$

(48)

by controlling the policy variables $g$, $h$, $\overline{r}$, and $\overline{w}$ subject to the aggregate feasibility constraint (7) and the government’s balanced-budget constraint (44).

The Lagrangean of the above problem is

$$L = \int_{0}^{\infty } {e^{ - \rho t} V(\overline{r},\overline{w},k,h,T)} {\text{d}}t + \int_{0}^{\infty } {\lambda_{p} [y - c - g - h - \mathop k\limits^{.} } ]{\text{d}}t + \int_{0}^{\infty } {\mu_{p} [y - {\rm Z}(c,l,k,T) - g - h} + T]{\text{d}}t,$$

where $\lambda_{p}$ and $\mu_{p}$ are the multipliers associated with the aggregate feasibility constraint and the government’s balanced-budget constraint (in the present value), respectively. Further, after some calculation, we can rewrite the Lagrangean as

$$\begin{aligned} L & = \int_{0}^{\infty } {\{ e^{ - \rho t} V(\overline{r},\overline{w},k,h,T) + \mu_{p} [y - {\rm Z}(c,l,k,T) - g - h + T]} + \lambda_{p} [y - c - g - h] + k\mathop {\lambda_{p} }\limits^{.} \} {\text{d}}t \\ \,\,\,\,\,\, - \lambda_{p} (\infty )k(\infty ) + \lambda_{p} (0)k(0) \\ = \int_{0}^{\infty } {\{ e^{ - \rho t} H^{G} (t)} + k(t)\dot{\lambda }_{p} (t)\} dt - \lambda_{p} (\infty )k(\infty ) + \lambda_{p} (0)k(0), \\ \end{aligned}$$

(49)

where $H^{G} (t)$ is the current value Hamiltonian with the form

$$\begin{aligned} H^{G} (t) & = V[\overline{r}(t),\overline{w}(t),k(t),h(t),T(t)] + \lambda (t)[y(t) - c(t) - g(t) - h(t)] \\ \,\,\,\,\, + \mu (t)[y(t) - {\rm Z}[c(t),l(t),k(t),T(t) - g(t) - h(t) + T(t)], \\ \end{aligned}$$

where $\lambda (t) = \lambda_{p} (t)e^{\rho t}$ and $\mu = \mu_{p} e^{\rho t}$ are the multipliers in the current value.

We assume the solution to the government’s optimal fiscal policy problem exists. Thus we can derive the agent’s maximum utility (in the present value) as a value function of the exogenous variables (i.e., the initial capital $k(0) = k_{0}$ and the time path of the lump-sum tax $T(t)$), denoted by $M[k_{0} ,T(t)]$.

According to the Envelope Theorem, we obtain the marginal social value of the initial capital as

$$\frac{\partial M}{{\partial k_{0} }} = \frac{\partial L}{{\partial k_{0} }} = \lambda_{p} (0).$$

Further, if the initial time is not zero but some particular point of time $\overline{t}$, then the value function becomes $M(k_{{\overline{t}}} ,T)$, where $k_{{\overline{t}}}$ is the initial capital at time $\overline{t}$. In this case, we have

$$\frac{\partial M}{{\partial k_{{\overline{t}}} }} = \frac{\partial L}{{\partial k_{{\overline{t}}} }} = \lambda_{p} (\overline{t}).$$

Note that $\lambda (t) = \lambda_{p} (t)e^{\rho t}$ is just the current value of $\lambda_{p} (t)$. Thus, we regard $\lambda (t)$ for any t as the marginal social value of capital at that particular point of time (in the current value).

In the same time, we can also derive the marginal social value of replacing a unit of distorting taxes with a unit of lump-sum tax at some particular time $\overline{t}$ as

$$\frac{\partial M}{{\partial T(\overline{t})}} = \frac{\partial L}{{\partial T(\overline{t})}} = e^{{ - \rho \overline{t}}} \frac{{\partial H^{G} (\overline{t})}}{{\partial T(\overline{t})}},$$

where ${{\partial H^{G} } \mathord{\left/ {\vphantom {{\partial H^{G} } {\partial T}}} \right. \kern-0pt} {\partial T}} = [U_{c} + U_{l} l_{c} + \lambda (f_{l} l_{c} - 1) + \mu (f_{l} l_{c} - {\rm Z}_{c} - {\rm Z}_{l} l_{c} )]c_{T} + \mu (1 - {\rm Z}_{T} )$. Note that, in the existence of lump sum taxation, the first-order condition (39) still holds and $Z_{T} = {{ - \Theta } \mathord{\left/ {\vphantom {{ - \Theta } {(1 - \Theta )}}} \right. \kern-0pt} {(1 - \Theta )}}$, and hence ${{\partial H^{G} } \mathord{\left/ {\vphantom {{\partial H^{G} } {\partial T}}} \right. \kern-0pt} {\partial T}} = {\mu \mathord{\left/ {\vphantom {\mu {(1 - \Theta )}}} \right. \kern-0pt} {(1 - \Theta )}}$. Therefore, we obtain

$$\frac{\partial M}{{\partial T(\overline{t})}} = e^{{ - \rho \overline{t}}} \frac{{\partial H^{G} (\overline{t})}}{{\partial T(\overline{t})}} = e^{{ - \rho \overline{t}}} \frac{{\mu (\overline{t})}}{1 - \Theta } = \frac{{\mu_{p} (\overline{t})}}{1 - \Theta }.$$

Since $\mu (t) = \mu_{p} (t)e^{\rho t}$ is the current value of $\mu_{p} (t)$, we get ${{\mu (t)} \mathord{\left/ {\vphantom {{\mu (t)} {(1 - \Theta )}}} \right. \kern-0pt} {(1 - \Theta )}}$ for any t as the current value of the marginal value of lump sum taxation at that particular time. It is also equal to the absolute value of the marginal excess burden of distortionary taxation and must be positive as long as distorting taxes exist.

Appendix 3: The proof of Propositions 4 & 5

In the case of $\Theta = 1$, the government’s Hamiltonian can be written as

$$\tilde{H}^{G} = V(\overline{w},k,h) + \lambda (y - c - g - h) + \mu (y - \overline{r}k - \overline{w}l - g - h).$$

The first-order condition for $\overline{r}$ is,

$$- \mu k = 0,$$

which implies that

$$\mu = 0.$$

In intuition, since $\Theta = 1$ in this case, $\overline{r}$ does not affect c and l, and then it affects welfare only through the balanced-budget constraint (20). Thus, its marginal effect on welfare is $- \mu k$, which should be zero at the optimum. Therefore, we get $\mu = 0.$

Using this result, we derive the first-order condition for $\overline{w}$ as

$$[U_{c} \tilde{c}_{l} + U_{l} + \lambda (f_{l} - \tilde{c}_{l} )]\tilde{l}_{{\overline{w}}} = 0,$$

which requires that

$$U_{c} \tilde{c}_{l} + U_{l} + \lambda (f_{l} - \tilde{c}_{l} ) = 0.$$

Then, according to the above equation and $\mu = 0$, we derive the costate equation for $\lambda$ as

$$\dot{\lambda } = \lambda \rho - U_{c} \tilde{c}_{k} - \lambda (f_{k} - \tilde{c}_{k} ).$$

Equating ${{\dot{\lambda }} \mathord{\left/ {\vphantom {{\dot{\lambda }} \lambda }} \right. \kern-0pt} \lambda }{{ = \dot{q}} \mathord{\left/ {\vphantom {{ = \dot{q}} q}} \right. \kern-0pt} q}$ yields the formula for optimal capital income tax in this case as

$$r - \overline{r} = f_{k} - \overline{r} = (1 - \frac{q}{\lambda })\tilde{c}_{k} .$$

(50)

Note that $\lambda$ represents the marginal social value of capital, which must be larger than the marginal private value of capital, q, as long as there exist distorting taxes. Meanwhile, according to the agent’s consumption function $c = \tilde{c}(l,k)$, we obtain $\tilde{c}_{k} > 0$ because c and k grow at the same rate along the balanced-growth path. Therefore, we must have $r - \overline{r} = (1 - q/\lambda ) \cdot \tilde{c}_{k} > 0$, which indicates that the optimal capital income tax should also be positive in this case.

Meanwhile, just as in the case of $\Theta \ne 1$, the optimal wedge on labor income is also given by

$$w - \overline{w} = f_{l} - \overline{w} = f_{l} + \frac{{U_{l} }}{{U_{c} }}.$$

Further, by calculating the first-order conditions for $h$ and $g$ and using the result of $\mu = 0$, we easily derive the conditions for optimal provision of public consumption and production services in this case as follows,

$$U_{h} = \lambda ,$$

$$f_{g} = 1.$$

Thus, we end the proof of Propositions 4 and 5.

Appendix 4: The derivation of the limit of the tax formula (24) as $\Theta \to 1$

In this appendix, we first derive the marginal excess burden of distortionary taxation in the case of $\Theta = 1$ and then, by using this result, derive the limit of the capital tax formula (24) as $\Theta \to 1$.

First, to derive the marginal excess burden of distortionary taxation in the case of $\Theta = 1$, we introduce lump taxation at level $T(t) \ge 0$ into this case, just as we do in Appendix 2.

In the existence of lump taxation, Eq. (25) in the text now turns to

$$c = \rho k + \overline{w}l - T,$$

(51)

where $\overline{w} = w(c,l)$. After some calculation, we derive the agent’s private consumption and labor supply as

$$c = \tilde{c}(l,k,T),$$

(52)

$$l = \tilde{l}(\overline{w},k,T),$$

and hence the agent’s indirect utility as

$$\tilde{V}(\overline{w},k,h,T) = U\{ \tilde{c}[\tilde{l}(\overline{w},k,T),k,T],\tilde{l}(\overline{w},k,T),h\} .$$

The government’s Hamiltonian now turns to

$$\tilde{H}^{G} = V(\overline{w},k,h,T) + \lambda (y - c - g - h) + \mu (y - \overline{r}k - \overline{w}l - g - h + T).$$

We assume the optimal solution to the government’s problem exists. Thus, we can derive the agent’s maximum utility, denoted by the value function $\tilde{M}[k_{0} ,T(t)].$

Then, by using the results of $\mu = 0$, $U_{c} \tilde{c}_{l} + U_{l} + \lambda (f_{l} - \tilde{c}_{l} ) = 0$, and $U_{c} = q$ (see Appendix 3 and Eq. (9) in the text), we obtain

$$\frac{{\partial \tilde{M}}}{{\partial T(\overline{t})}} = e^{{ - \rho \overline{t}}} \frac{{\partial \tilde{H}^{G} (\overline{t})}}{{\partial T(\overline{t})}} = e^{{ - \rho \overline{t}}} (U_{c} - \lambda )\tilde{c}_{T} = e^{{ - \rho \overline{t}}} (q - \lambda )\tilde{c}_{T} .$$

Therefore, we derive the absolute value of the marginal excess burden of distortionary taxation as $(q - \lambda )\tilde{c}_{T}$ (in the current value).

Note that, in the case of $\Theta \ne 1$, the marginal excess burden in the current value is ${\mu \mathord{\left/ {\vphantom {\mu {(1 - \Theta )}}} \right. \kern-0pt} {(1 - \Theta )}}$ (see Appendix 2); while in the present case of $\Theta = 1$, it is $(q - \lambda )\tilde{c}_{T}$. Therefore, we must have

$$\mathop {\lim }\limits_{\Theta \to 1} \frac{\mu }{1 - \Theta } = (q - \lambda )\tilde{c}_{T} .$$

We then use the above result to derive of the limit of the capital tax formula (24) as $\Theta \to 1$, which is

$$\mathop {\lim }\limits_{\Theta \to 1} (r - \overline{r}) = \mathop {\lim }\limits_{\Theta \to 1} \frac{\mu }{\lambda }(\frac{\rho }{1 - \Theta } - f_{k} ) = \mathop {\lim }\limits_{\Theta \to 1} \frac{\mu }{1 - \Theta } \cdot \frac{\rho }{\lambda } = (\frac{q}{\lambda } - 1)\tilde{c}_{T} \rho .$$

(53)

Meanwhile, according to Eqs. (51) and (52), it is easy to show that

$$\tilde{c}_{k} (l,k,T) = - \rho \tilde{c}_{T} (l,k,T).$$

We substitute the above result into Eq. (53) to obtain

$$\mathop {\lim }\limits_{\Theta \to 1} (r - \overline{r}) = \mathop {\lim }\limits_{\Theta \to 1} \frac{\mu }{\lambda }(\frac{\rho }{1 - \Theta } - f_{k} ) = (1 - \frac{q}{\lambda })\tilde{c}_{k} ,$$

which is just the capital tax formula (27) in the case of $\Theta = 1.$

Appendix 5: The derivation of Eqs. (30) and (31) and Eqs. (32)–(34)

In the example, given the paths of policy variables, the representative agent’s Hamiltonian (8) can be written as

$$H^{C} = \frac{{[c(1 - l)^{\gamma } ]^{1 - \sigma } }}{1 - \sigma } + \frac{{\beta h^{1 - \sigma } }}{1 - \sigma } + q(\overline{r}k + \overline{w}l - c).$$

The first-order conditions for $c$ and $l$ are

$$U_{c} = c^{ - \sigma } (1 - l)^{\gamma (1 - \sigma )} = q,$$

$$U_{l} = - \gamma c^{1 - \sigma } (1 - l)^{\gamma (1 - \sigma ) - 1} = - q\overline{w}.$$

Combining the above two conditions, we obtain

$$\overline{w} = \frac{\gamma c}{{1 - l}}.$$

(54)

The costate equation for q can be written as

$$\frac{{\dot{q}}}{q} = \frac{{\dot{u}_{c} }}{{u_{c} }} = - \sigma \cdot \frac{{\dot{c}}}{c} = \rho - \overline{r}.$$

As is discussed in the general model, the economy from the beginning grows along a balanced path. Hence, by using ${{\dot{c}} \mathord{\left/ {\vphantom {{\dot{c}} c}} \right. \kern-0pt} c} = {{\dot{k}} \mathord{\left/ {\vphantom {{\dot{k}} k}} \right. \kern-0pt} k}$, we obtain

$$\frac{{\overline{r} - \rho }}{\sigma } = \overline{r} + \frac{{\overline{w}l}}{k} - \frac{c}{k}.$$

The above equation can be rewritten as

$$c = (1 - \frac{1}{\sigma })\overline{r}k + \overline{w}l + \frac{\rho k}{\sigma }.$$

(55)

We begin with the case of $\sigma = 1$. In this case, the utility function (28) can be rewritten as  $U(c,l,h) = \ln c + \gamma \ln (1 - l) + \beta \ln h$.

Substituting Eq. (54) into Eq. (55) yields the agent’s private consumption function

$$\tilde{c}(l,k) = \frac{\rho k}{{1 - {{\gamma l} \mathord{\left/ {\vphantom {{\gamma l} {(1 - l)}}} \right. \kern-0pt} {(1 - l)}}}}.$$

(56)

We then combine Eqs. (54) and (56) to derive the agent’s labor supply function

$$\tilde{l}(\overline{w},k) = 1 - \frac{{\gamma (1 + {{\rho k} \mathord{\left/ {\vphantom {{\rho k} {\overline{w}}}} \right. \kern-0pt} {\overline{w}}})}}{1 + \gamma }.$$

Then we obtain the indirect utility function in this case as

$$\tilde{V}(\overline{w},k,h) = \ln \tilde{c}[k,\tilde{l}(\overline{w},k)] + \gamma \ln [1 - \tilde{l}(\overline{w},k)] + \beta \ln h.$$

(57)

The government’s problem is to maximize the agent’s lifetime utility (26), where $\tilde{V}(\overline{w},k,h)$ is given by Eq. (57), subject to the resource constraint (7) and the balanced-budget constraint (20). The Hamiltonian of this problem is

$$\tilde{H}^{G} = \ln \tilde{c}[k,\tilde{l}(\overline{w},k)] + \gamma \ln [1 - \tilde{l}(\overline{w},k)] + \beta \ln h + \lambda (y - c - g - h) + \mu (y - \overline{r}k - \overline{w}l - g - h).$$

By calculating the first-order conditions for the policy variables, we directly obtain Eqs. (30) and (31) in the main text.

We then go back to Eq. (55) to deal with the case of $\sigma \ne 1$. First, according to Eq. (54), we get

$$l = 1 - \frac{\gamma c}{{\overline{w}}}.$$

(58)

We then substitute (58) into (55) to derive the agent’s consumption function as

$$c = c(\overline{r},\overline{w},k) = \frac{{(1 - \frac{1}{\sigma })\overline{r}k + \overline{w} + \frac{\rho k}{\sigma }}}{1 + \gamma }.$$

Meanwhile, according to Eq. (58), the agent’s labor supply function is

$$l\left( {\overline{w},c(\overline{r},\overline{w},k)} \right) = 1 - \frac{{\gamma c(\overline{r},\overline{w},k)}}{{\overline{w}}}.$$

Substituting the above two response functions into the agent’s instantaneous utility function (28) yields the indirect instantaneous utility function with the form

$$V(\overline{r},\overline{w},k,h) = \frac{{\left\{ {c(\overline{r},\overline{w},k)\left[ {1 - l\left( {\overline{w},c(\overline{r},\overline{w},k)} \right)} \right]^{\gamma } } \right\}^{1 - \sigma } }}{1 - \sigma } + \frac{{\beta h^{1 - \sigma } }}{1 - \sigma }.$$

(59)

The government’s problem is to maximize the agent’s indirect lifetime utility (22), where $V(\overline{r},\overline{w},k,h)$ is given by Eq. (59), subject to the resource constraint (7) and the balanced-budget constraint (36), where

$${\rm Z}(c,l,k) = \overline{r}k + \overline{w}l = \frac{{\rho k + \left[ {{{\gamma l} \mathord{\left/ {\vphantom {{\gamma l} {(1 - l) - \sigma }}} \right. \kern-0pt} {(1 - l) - \sigma }}} \right]c}}{1 - \sigma }.$$

Thus, in this case, the government’s Hamiltonian can be written as

$$H^{G} = \frac{{\left\{ {c(\overline{r},\overline{w},k)\left[ {1 - l\left( {\overline{w},c(\overline{r},\overline{w},k)} \right)} \right]^{\gamma } } \right\}^{1 - \sigma } }}{1 - \sigma } + \frac{{\beta h^{1 - \sigma } }}{1 - \sigma } + \lambda (y - c - g - h) + \mu [y - {\rm Z}(c,l,k) - g - h].$$

By solving the government’s first-order conditions, we just obtain Eqs. (32)–(34) in the main text.