Welfare implications of non-unitary time discounting

Abstract

This study proposes a model of non-unitary time discounting and examines its welfare implications. A key feature of our model lies in the disparity of time discounting between multiple distinct goods, which induces an individual’s preference reversals even though she normally discounts her future utilities for each good. After characterizing the time-consistent decision-making by such an individual, we compare welfare achieved in the market economy and welfare in the planner’s allocation from the perspective of all selves across time. Under certain situations, the selves in early periods strictly prefer the social planner’s allocation, whereas the selves in future periods strictly prefer the market equilibrium. Therefore, the welfare implications of our model are quite different from those in the canonical discounting model and in models of other time-inconsistent preferences.

Appendices

Appendix A: Proof of Proposition 1

We restate the optimization problem of the self in a period:

$$\begin{aligned} V(a)=\max _{c, x, a'}\left\{ u(c)+v(x)+\beta _c V_{c}(a')+\beta _x V_{x} (a') \ \big | \ a'=R a - c-px \right\} , \end{aligned}$$

(A.1)

where $V_c$ and $V_x$ are recursively defined by the following functional equations:

$$\begin{aligned} V_c (a)=u(\phi _c(a))+\beta _c V_c(g(a)), \quad V_x (a) =v(\phi _x(a))+\beta _x V_x(g(a)). \end{aligned}$$

(A.2)

The first-order conditions (FOCs) of the problem (A.1) are

$$\begin{aligned} \dfrac{v'(x)}{u'(c)}&=p, \end{aligned}$$

(A.3)

$$\begin{aligned} u'(c)&=\beta _c \frac{\mathrm{d}V_c(a')}{\mathrm{d}a'}+\beta _x \frac{\mathrm{d}V_x(a')}{\mathrm{d}a'}.&\end{aligned}$$

(A.4)

Differentiating $V_c(a)$ and $V_x(a)$ in (A.2) with respect to a and adding these together, we obtain

$$\begin{aligned} \dfrac{\mathrm{d}}{\mathrm{d}a} \left( V_c(a)+V_x(a)\right) =u'(c) \dfrac{\mathrm{d}\phi _c(a)}{\mathrm{d}a} +v'(x) \dfrac{\mathrm{d}\phi _x(a)}{\mathrm{d}a} + \left( \beta _c \dfrac{\mathrm{d}V_c(a')}{\mathrm{d}a'}+ \beta _x \dfrac{\mathrm{d}V_x(a')}{\mathrm{d}a'}\right) \dfrac{\mathrm{d}g(a)}{\mathrm{d}a} . \end{aligned}$$

Substituting (A.3) and (A.4) into the right-hand side of this equation, we obtain

$$\begin{aligned} \dfrac{\mathrm{d}}{\mathrm{d}a} \left( V_c(a)+V_x(a)\right) =u'(c) \left( \dfrac{\mathrm{d}\phi _c(a)}{\mathrm{d}a} + p \dfrac{\mathrm{d}\phi _x(a)}{\mathrm{d}a}+\dfrac{\mathrm{d}g(a)}{\mathrm{d}a} \right) . \end{aligned}$$

(A.5)

Since g(a) is defined as $g(a)\equiv Ra-\phi _c(a)-p\phi _x(a)$,

$$\begin{aligned} R = \dfrac{\mathrm{d}\phi _c(a)}{\mathrm{d}a} + p \dfrac{\mathrm{d}\phi _x(a)}{\mathrm{d}a}+\dfrac{\mathrm{d}g(a)}{\mathrm{d}a}. \end{aligned}$$

(A.6)

From (A.5) and (A.6), we obtain the following equation, which appears in the main body:

$$\begin{aligned} \dfrac{\mathrm{d}}{\mathrm{d}a} \left( V_c(a)+V_x(a)\right) =u'(\phi _c(a))R. \end{aligned}$$

(A.7)

Substituting (A.7) into (A.4) and evaluating it in period t yields the following Euler equation:

$$\begin{aligned} u'(c_t)= \beta _c u'(c_{t+1}) R- (\beta _c-\beta _x ) \dfrac{\mathrm{d}V_x(a_{t+1})}{\mathrm{d}a_{t+1}}. \end{aligned}$$

Using (A.3) and (A.7), the above equation is also expressed as

$$\begin{aligned} v'(x_t)= \beta _x v'(x_{t+1}) R+ (\beta _c-\beta _x) p \dfrac{\mathrm{d}V_c(a_{t+1})}{\mathrm{d}a_{t+1}}. \end{aligned}$$

Appendix B: Proof of Lemma 1

To show this lemma, we guess $\phi _c(a, X)$ and $V_l(a, X)$ as follows:

$$\begin{aligned} \phi _c(a, X)=\mu _c R(X) a, \quad V_l(a, X)=b_l(X)+d_l \ln a, \end{aligned}$$

(B.1)

where $\mu _c$, $b_l(X)$, and $d_l$ are the parameters that we have to solve for. From (7), we have

$$\begin{aligned} 1-\phi _l(a, X)=\dfrac{\zeta \mu _c R(X) a}{w(X)}. \end{aligned}$$

(B.2)

Using this result and the budget constraint (2) in the main body, we obtain

$$\begin{aligned} a'=g(a, X)=\gamma R(X) a, \end{aligned}$$

(B.3)

where

$$\begin{aligned} \gamma =1-(1+\zeta )\mu _c. \end{aligned}$$

(B.4)

Using the above guesses and (B.3), we can rewrite the Euler equation (8) as follows:

$$\begin{aligned} \dfrac{1}{\mu _c R(X)a}&=\beta _c \dfrac{1}{\mu _c a'}+ (\beta _l -\beta _c)d_l \dfrac{1}{a'} \\&= \left( \beta _c \dfrac{1}{\mu _c}+ (\beta _l -\beta _c)d_l \right) \dfrac{1}{\gamma R(X)a}, \end{aligned}$$

which results in

$$\begin{aligned} \gamma =\beta _c +\mu _c (\beta _l-\beta _c)d_l. \end{aligned}$$

(B.5)

At the same time, $d_l$ (as well as $b_l(X)$) must satisfy the following functional equation:

$$\begin{aligned} V_l(a, X)= \zeta \ln (1-\phi _l(a, X))+ \beta _l V_l(g(a, X), G(X)) \end{aligned}$$

Substituting (B.1)–(B.3) into the above equation yields

$$\begin{aligned} b_l (X)+d_l \ln a = \zeta \ln a + \beta _l d_l \ln a +\text {other terms}, \end{aligned}$$

(B.6)

which results in $d_l= \zeta /(1-\beta _l)$. Substituting this into (B.5),

$$\begin{aligned} \gamma =\beta _c+ \zeta \mu _c \dfrac{\beta _l-\beta _c}{1-\beta _l}. \end{aligned}$$

(B.7)

From (B.4) and (B.7), we finally obtain

$$\begin{aligned} \gamma =\dfrac{\beta _c+\lambda }{1+\lambda }, \end{aligned}$$

where

$$\begin{aligned} \lambda \equiv \dfrac{\zeta }{1+\zeta }\dfrac{\beta _l-\beta _c}{1-\beta _l}. \end{aligned}$$

Appendix C: Derivation of Eq. (9)

When $\delta =1$, human wealth in the main body is expressed as

$$\begin{aligned} \sum _{t'=t}^\infty \dfrac{w_{t'}}{ \prod _{\nu =t}^{t'} (1+r_\nu -\delta )}&= \sum _{t'=t}^\infty \dfrac{w_{t'}}{ \prod _{\nu =t}^{t'} r_\nu } \nonumber \\&=\sum _{t'=t}^\infty \left( \dfrac{1}{ \prod _{\nu =t}^{t'-1} r_\nu } \dfrac{w_{t'}}{r_{t'}} \right) . \end{aligned}$$

(C.1)

From Eq. (3),

$$\begin{aligned} \dfrac{w_{t'}}{r_{t'}}=\dfrac{1-\alpha }{\alpha } X_{t'}. \end{aligned}$$

(C.2)

To simplify the expression of $\prod _{\nu =t}^{t'-1} r_\nu$, we first calculate $\sum _{\nu =t}^{t'-1} \ln r_\nu$.

$$\begin{aligned} \sum _{\nu =t}^{t'-1} \ln r_\nu&= \sum _{\nu =t}^{t'-1} \ln \left( A \alpha X_\nu ^{\alpha -1}\right) \quad (\because \text {Eq. } (3))\nonumber \\&= (\alpha -1) \sum _{\nu =t}^{t'-1} \ln X_\nu + (t'-t) \ln (A \alpha ). \end{aligned}$$

(C.3)

Since we guess that $X_{t+1}=s^{\mathrm{eqm}}A X_t^\alpha$ for all t,

$$\begin{aligned} \ln X_\nu = \ln (s^{\mathrm{eqm}} A)+ \alpha \ln X_{\nu -1} \Leftrightarrow \ln X_\nu =\alpha ^{\nu -t} \ln X_t + \frac{1-\alpha ^{\nu -t}}{1-\alpha } \ln (s^{\mathrm{eqm}}A). \end{aligned}$$

(C.4)

Then,

$$\begin{aligned} \sum _{\nu =t}^{t'-1} \ln X_\nu&= \left( \sum _{\nu =t}^{t'-1} \alpha ^{\nu -t} \right) \ln X_t + \dfrac{1}{1-\alpha } \left[ (t'-t)- \sum _{\nu =t}^{t'-1} \alpha ^{\nu -t} \right] \ln (s^{\mathrm{eqm}}A) \\&= \frac{1-\alpha ^{t'-t}}{1-\alpha } \ln X_t + \dfrac{1}{1-\alpha }\left[ (t'-t)- \frac{1-\alpha ^{t'-t}}{1-\alpha } \right] \ln (s^{\mathrm{eqm}}A) \\&= \dfrac{1}{1-\alpha }\left[ \ln X_t - \ln X_{t'} + (t'-t) \ln (s^{\mathrm{eqm}}A)\right] , \end{aligned}$$

where we use the fact that $\ln X_{t'}$ satisfies Eq. (C.4) with $\nu =t'$. Substituting this into the right-hand side of Eq. (C.3) yields

$$\begin{aligned} \sum _{\nu =t}^{t'-1} \ln r_\nu&= - \ln X_t + \ln X_{t'} -(t'-t) \ln (s^{\mathrm{eqm}}A) + (t'-t) \ln (A \alpha ) \\&= \ln X_{t'}-\ln X_t + (t'-t) \ln (\alpha /s^{\mathrm{eqm}}). \end{aligned}$$

Since $\prod _{\nu =t}^{t'-1} r_\nu = \exp \left[ \sum _{\nu =t}^{t'-1} \ln r_\nu \right]$, the above equation implies

$$\begin{aligned} \prod _{\nu =t}^{t'-1} r_\nu = \dfrac{X_{t'}}{X_t} \left( \dfrac{\alpha }{s^{\mathrm{eqm}}} \right) ^{t'-t}. \end{aligned}$$

(C.5)

Finally, substituting (C.2) and (C.5) into (C.1), we obtain

$$\begin{aligned} \sum _{t'=t}^\infty \dfrac{1}{ \prod _{\nu =t}^{t'-1} r_\nu } \dfrac{w_{t'}}{r_{t'}}&=\sum _{t'=t}^\infty \dfrac{\dfrac{1-\alpha }{\alpha } X_{t'}}{\dfrac{X_{t'}}{X_t} \left( \dfrac{\alpha }{s^{\mathrm{eqm}}} \right) ^{t'-t}} \nonumber \\&= \dfrac{1-\alpha }{\alpha } X_t \sum _{t'=t}^\infty \left( \dfrac{ s^{\mathrm{eqm}} }{\alpha } \right) ^{t'-t}. \end{aligned}$$

(C.6)

If $s^{\mathrm{eqm}}\ge \alpha$, (C.6) implies that the human wealth of the household is infinity in any period, which should be excluded. Since we guess that $s^{\mathrm{eqm}} <\alpha$ (and show that this inequality holds), Eq. (C.6) gives

$$\begin{aligned} \sum _{t'=t}^\infty \dfrac{1}{ \prod _{\nu =t}^{t'-1} r_\nu } \dfrac{w_{t'}}{r_{t'}} = \dfrac{1-\alpha }{\alpha -s^{\mathrm{eqm}}} X_t. \end{aligned}$$

(C.7)

Since $X_t=k_t/l^{\mathrm{eqm}}$ from the market-clearing conditions for labor and assets, we obtain Eq. (9).

Appendix D: Proof of Lemma 2

We state the planner’s problem again here.

$$\begin{aligned} V^\mathrm{{sp}} (k)=\max _{k', l} \left\{ \ln (Ak^{\alpha }l^{1-\alpha }-k^{\prime })+\zeta \ln (1-l)+\beta _cV^\mathrm{{sp}}_c (k^{\prime })+ \beta _l V^\mathrm{{sp}}_l (k^{\prime })\right\} . \end{aligned}$$

(D.1)

Assume that the planner in a period expects that if the value of capital is given by k, the next self’s decisions about savings and labor supply are

$$\begin{aligned} k' = g^\mathrm{{sp}}(k), \quad l=\phi _l^\mathrm{{sp}}(k). \end{aligned}$$

Then, functions $V^\mathrm{{sp}}_c(k)$ and $V^\mathrm{{sp}}_l(k)$ are given by the following functional equations:

$$\begin{aligned} V^\mathrm{{sp}}_c(k)&=\ln \left[ Ak^{\alpha }(\phi _l^\mathrm{{sp}}(k))^{1-\alpha }-g^\mathrm{{sp}}(k)\right] +\beta _c V^\mathrm{{sp}}_c (g^\mathrm{{sp}}(k)), \end{aligned}$$

(D.2)

$$\begin{aligned} V^\mathrm{{sp}}_l(k)&=\zeta \ln (1-\phi _l^\mathrm{{sp}}(k))+\beta _l V^\mathrm{{sp}}_l (g^\mathrm{{sp}}(k)). \end{aligned}$$

(D.3)

The FOCs of the problem in (D.1) with respect to $k'$ and l are given by

$$\begin{aligned}&\dfrac{1}{Ak^{\alpha }l^{1-\alpha }-k^{\prime }}=\beta _c \dfrac{\partial V^\mathrm{{sp}}_c(k')}{\partial k'}+\beta _l \dfrac{\partial V^\mathrm{{sp}}_l(k')}{\partial k'}, \end{aligned}$$

(D.4)

$$\begin{aligned}&\dfrac{(1-\alpha )Ak^{\alpha }l^{-\alpha }}{Ak^{\alpha }l^{1-\alpha }-k^{\prime }}-\zeta \dfrac{1}{1-l}=0. \end{aligned}$$

(D.5)

respectively. We make the following guess for $V^\mathrm{{sp}}_i$ ($i\in \{c, l \}$):

$$\begin{aligned} V^\mathrm{{sp}}_i(k)=a_i+d_i \ln k. \end{aligned}$$

From (D.4), we obtain

$$\begin{aligned} g^\mathrm{{sp}}(k)=\dfrac{\Psi }{1+\Psi }Ak^{\alpha }l^{1-\alpha }, \end{aligned}$$

(D.6)

where $\Psi$ is defined as

$$\begin{aligned} \Psi \equiv \beta _c d_c+\beta _l d_l. \end{aligned}$$

(D.7)

From (D.5) and (D.6), we obtain

$$\begin{aligned} \phi _l^\mathrm{{sp}}(k)=l^\mathrm{{sp}}, \quad \text { where } l^\mathrm{{sp}}=\dfrac{(1-\alpha )(1+\Psi )}{\zeta +(1-\alpha )(1+\Psi )}. \end{aligned}$$

(D.8)

Since the value of $\Psi$ is still unknown, we substitute (D.6), (D.8), and the guess for $V^\mathrm{{sp}}_j$ into (D.2) and (D.3):

$$\begin{aligned} a_c+d_c\ln k&=\ln \left( \frac{1}{1+\Psi }Ak^{\alpha }(l^\mathrm{{sp}})^{1-\alpha }\right) +\beta _c\left[ a_{c}+d_{c}\ln \left( \dfrac{\Psi }{1+\Psi }Ak^{\alpha }(l^\mathrm{{sp}})^{1-\alpha }\right) \right] , \\ a_l+d_l \ln k&=\zeta \ln (1-l^\mathrm{{sp}})+\beta _l \left[ a_l+d_l \ln \left( \dfrac{\Psi }{1+\Psi }Ak^{\alpha } (l^\mathrm{{sp}})^{1-\alpha }\right) \right] . \end{aligned}$$

Since the coefficients of both sides must be equal, we can show that

$$\begin{aligned} d_c=\frac{\alpha }{1-\beta _{c}\alpha }, \quad d_l=0. \end{aligned}$$

Then, $d_l=0$ implies

$$\begin{aligned} \dfrac{dV_l^\mathrm{{sp}}(k)}{dk}=0. \end{aligned}$$

Substituting this result into the definition of $\Psi$, we have

$$\begin{aligned} \Psi =\dfrac{\beta _c \alpha }{1-\beta _c\alpha }. \end{aligned}$$

Finally, substituting the obtained value of $\Psi$ into (D.6) and (D.8), we obtain

$$\begin{aligned} l^\mathrm{{sp}}=\dfrac{1-\alpha }{1-\alpha +\zeta (1-\beta _c \alpha )}, \end{aligned}$$

and

$$\begin{aligned} g^\mathrm{{sp}}(k)= \beta _c \alpha A k^{\alpha } (l^\mathrm{{sp}})^{1-\alpha } \Leftrightarrow s^\mathrm{{sp}}=\beta _c \alpha . \end{aligned}$$

Appendix E: Proof of Proposition 3

If the saving rate and labor supply are determined so that they are constant over time, we calculate the utility of a self in period t as follows.

$$\begin{aligned} U_t&=\sum _{t'=t}^{\infty }\left\{ \beta _c^{t'-t} \ln [(1-s)A (k_{t'})^\alpha l^{1-\alpha }]+ \beta _l^{t'-t} \zeta \ln (1-l)\right\} \nonumber \\&= \dfrac{1}{1-\beta _c}\left[ \ln (1-s)+\ln \left( Al^{1-\alpha } \right) \right] +\dfrac{\zeta }{1-\beta _l} \ln (1-l) +\sum _{t'=t}^\infty \beta _c^{t'-t} \alpha \ln k_{t'}. \end{aligned}$$

(E.1)

From Eq. (12) in the main body, we have

$$\begin{aligned} \ln k_{t'} =\alpha ^{t'-t} \ln k_t+\dfrac{1-\alpha ^{t'-t}}{1-\alpha } \ln (sAl^{1-\alpha }), \end{aligned}$$

where $k_t$ is historically given for the self in period t. Substituting this result into the last term in (E.1) yields

$$\begin{aligned} U_t&= \dfrac{1}{1-\beta _c}\ln (1-s)+\frac{\zeta }{1-\beta _{l}}\ln (1-l) \nonumber +\dfrac{\alpha }{1- \beta _c \alpha }\ln k+ \dfrac{\beta _c \alpha \ln s +\ln A+(1-\alpha ) \ln l}{(1-\beta _c)(1-\beta _c \alpha )} \nonumber \\&=W(s, l, k). \end{aligned}$$

(E.2)

Note that the following identity holds:

$$\begin{aligned} V^j (k)\equiv W(s^j, l^j, k), \quad j\in \{\mathrm{eqm, sp}\}. \end{aligned}$$

To obtain this proposition, we first show the following two lemmas.

Lemma E.1

There exists a unique pair $(s^*, l^*)$ that maximizes W(s, l, k).

Proof

Since this function is strictly concave in (s, l), the necessary and sufficient conditions for $(s^*, l^*)$ are given by the following FOCs:

$$\begin{aligned} \dfrac{\partial W}{\partial s}= & {} 0 : \quad \dfrac{1}{1-\beta _c}\dfrac{1}{1-s}=\dfrac{\beta _c \alpha }{(1-\beta _c)(1-\beta _c \alpha )}\dfrac{1}{s}, \\ \dfrac{\partial W}{\partial l}= & {} 0 : \quad \dfrac{\zeta }{1-\beta _l}\dfrac{1}{1-l}=\dfrac{1-\alpha }{(1-\beta _c)(1-\beta _c \alpha )}\dfrac{1}{l}, \end{aligned}$$

which in turn yield

$$\begin{aligned} s^*= & {} \beta _c \alpha , \\ l^*= & {} \dfrac{1-\alpha }{1-\alpha +\zeta \omega (1- \beta _c \alpha )}, \end{aligned}$$

where $\omega \equiv (1-\beta _c)/(1-\beta _l)$, which deviates from unity if and only if $\beta _c\ne \beta _l$. $\square$

We next show the following lemma.

Lemma E.2

$s^\mathrm{{sp}} ( \equiv s^*) \gtreqless s^\mathrm{{eqm}}$ and $l^*\gtreqless l^\mathrm{{sp}} \gtreqless l^\mathrm{{eqm}}$ if and only if $\beta _c \gtreqless \beta _l.$

Proof

From their definitions, it follows that $(s^\mathrm{{eqm}}, l^\mathrm{{eqm}})=(s^\mathrm{{sp}}, l^\mathrm{{sp}})=(s^*, l^*)$ when $\beta _c=\beta _l$. From the proof of Lemma 3 in the main body, we know that $s^\mathrm{{eqm}}$ and $l^\mathrm{{eqm}}$ are strictly increasing with respect to $\beta _l$, whereas $s^\mathrm{{sp}}$ and $l^\mathrm{{sp}}$ are independent of $\beta _l$. Finally, $l^*$ is strictly decreasing with respect to $\beta _l$. Then, we can show that

$$\begin{aligned}&s^\mathrm{{sp}}\equiv s^*\gtreqless s^\mathrm{{eqm}} \Leftrightarrow \beta _c \gtreqless \beta _l, \\&l^*\gtreqless l^\mathrm{{sp}} \gtreqless l^\mathrm{{eqm}} \Leftrightarrow \beta _c \gtreqless \beta _l. \end{aligned}$$

$\square$

Fig. 1

Contours of W(s, l, k) and the interrelationship of $(s^*, l^*)$, $(s^\mathrm{{eqm}}, l^\mathrm{{eqm}})$, and $(s^\mathrm{{sp}}, l^\mathrm{{sp}})$

Having obtained Lemmas E.1 and E.2, we evaluate the ranking between $V^\mathrm{{eqm}} (k)$ and $V^\mathrm{{sp}} (k)$ by using the contours of W(s, l, k) in the s-l plane. In panels (a) and (b) of Fig. 1, $(s^*, l^*)$ is located at point O, and the closed curves represent the contours of W(s, l, k). The value of the welfare evaluation function W(s, l, k) increases as the curves approach O. At any point on the curve passing through point A (B), $V^\mathrm{{eqm}}(k)$ ($V^\mathrm{{sp}}(k)$) is achieved. From Lemma E.2, the distance from points A to O is necessarily farther than that from points B to O, as long as $\beta _c\ne \beta _l$. Furthermore, from this lemma, we can show that points A and B are located in the same quadrant of the coordinate plane, with its origin given by point O. This means that when $\beta _c\ne \beta _l$, the indifference curve passing through point A is always located outside the curve passing through point B. Therefore, we show that this proposition holds.

Appendix F: Proof of Lemma 5

From the definitions of $V^j(k)$ and W in (E.2), $V^j(k)$ is expressed as

$$\begin{aligned} V^j(k)= \dfrac{\alpha }{1-\beta _c \alpha } \ln k+B^j, \quad j\in \{\mathrm{eqm, sp}\}, \end{aligned}$$

where $B^j$ is a collection of other terms, independent of k. The above equation shows that $V^j (k)$ is strictly increasing.

Appendix G: Proof of Proposition 5

When the saving rate and labor supply are constant over time, the steady state of capital is given by

$$\begin{aligned} k=K_{ss}(s, l) \equiv (sA)^{1/(1-\alpha )}l. \end{aligned}$$

(G.1)

Thus, the steady state of k in the market equilibrium is $K_{ss}(s^\mathrm{{eqm}}, l^\mathrm{{eqm}})$, while that under social planning is $K_{ss}(s^\mathrm{{sp}}, l^\mathrm{{sp}})$. We define the function $W_{ss}(s, l)$ as follows:

$$\begin{aligned} W_{ss} (s, l)&\equiv W(s, l, K_{ss}(s, l)) \\&= \frac{1}{1-\beta _{c}}\ln \left[ (1-s)s^{\frac{\alpha }{1-\alpha }}A^{\frac{1}{1-\alpha }}l\right] +\frac{\zeta }{1-\beta _{l}}\ln (1-l). \end{aligned}$$

We verify that $V^j(K_{ss}(s^j, l^j))\equiv W_{ss}(s^j, l^j)$. By simple calculations, we find that $W_{ss}(s, l)$ is maximized at $(s^*_{ss}, l^*_{ss})$, where

$$\begin{aligned} s^*_{ss}=\alpha , \quad l^*_{ss}=\frac{1-\beta _{l}}{1-\beta _{l}+\zeta (1-\beta _{c})}. \end{aligned}$$

To obtain this proposition, we first show the following lemma.

Lemma E.3

Given $\beta _c$, there exists a unique ${{\overline{\beta }}}_l \in (\beta _c, 1)$, such that $s^\mathrm{{sp}}< s^\mathrm{{eqm}} < s^*_{ss}$ and $l^\mathrm{{sp}}< l^\mathrm{{eqm}}< l^*_{ss}$ if and only if $\beta _c<\beta _l < {{\overline{\beta }}}_l$.

Proof

From the proof of Lemma E.2, we show that

$$\begin{aligned} s^\mathrm{{sp}}< s^\mathrm{{eqm}}, \quad l^\mathrm{{sp}} < l^\mathrm{{eqm}} \ \Leftrightarrow \ \beta _l >\beta _c. \end{aligned}$$

Then, our remaining task is to derive the condition under which $s^\mathrm{{eqm}}<s^{*}_{ss}$ and $l^\mathrm{{eqm}}<l^{*}_{ss}$ hold. From its definition, $s^\mathrm{{eqm}}$ converges to $\alpha (=s^*_{ss})$ as $\beta _l \rightarrow 1$. Since $s^\mathrm{{eqm}}$ is strictly increasing with respect to $\beta _l$, we first find that

$$\begin{aligned} s^\mathrm{{eqm}}< s^{*}_{ss} \ \text { for all } \beta _l\in (0, 1). \end{aligned}$$

We next consider the ranking between $l^\mathrm{{eqm}}$ and $l^*_{ss}$.

From its definition, $l^\mathrm{{eqm}}$ is strictly increasing with respect to $\beta _l$ and converges to $1/(1+\zeta )$ as $\beta _l \rightarrow 1$. On the contrary, we can easily verify that $l^{*}_{ss}$ is decreasing with respect to $\beta _l$, and $l^*_{ss}= 1/(1+\zeta )$ when $\beta _l=\beta _c$, and $l^{*}_{ss} \rightarrow 0$ as $\beta _l \rightarrow 1$. There exists a unique ${{\overline{\beta }}}_l \in (\beta _c, 1)$ such that

$$\begin{aligned} l^\mathrm{{eqm}}<l^{*}_{ss} \ \Leftrightarrow \ \beta _l <{{\overline{\beta }}}_l. \end{aligned}$$

These results show that this lemma holds. $\square$

Lemma E.3 shows that if $\beta _c<\beta _l< {{\overline{\beta }}}_l$, $V^\mathrm{{eqm}} (K_{ss}(s^\mathrm{{eqm}}, l^\mathrm{{eqm}}))>V^\mathrm{{sp}}(K_{ss}(s^\mathrm{{sp}}, l^\mathrm{{sp}}))$. If $\beta _l>{{\overline{\beta }}}_l$, it is ambiguous which case yields higher welfare. Therefore, we focus on the former situation.

Since $V^\mathrm{{eqm}} (k_0)<V^\mathrm{{sp}}(k_0)$, there is at least one period, denoted by T, such that $V^\mathrm{{eqm}} (k^\mathrm{{eqm}}_t)> V^\mathrm{{sp}}(k^\mathrm{{sp}}_t)$ if $t\ge T$. Then, we can complete the proof of this proposition by showing that such a period T is unique. To this end, let $q_t$ denote $q_t \equiv \ln k_t$. Then, we can rewrite (12) in the main body as follows:

$$\begin{aligned} (12)&\Leftrightarrow \ln k^j_t =\alpha ^t \ln k_0 +(1-\alpha ^t) \ln \left[ (s^j A)^{1/(1-\alpha )} l^j \right] \nonumber \\&\Leftrightarrow q^j_t= \alpha ^t q_0 +(1-\alpha ^t)q^j_{ss}, \quad j\in \{\mathrm{eqm, sp}\}, \end{aligned}$$

(G.2)

where $q^j_{ss} \equiv \ln \left[ (s^j A)^{1/(1-\alpha )} l^j \right] =\ln K_{ss}(s^j, l^j)$ from Eq. (G.1). Substituting (G.2) into $V^j(k)$, we obtain $V^j(k^j_t)\equiv {\mathcal {V}}^j(t)$, where ${\mathcal {V}}$ is given by

$$\begin{aligned} {\mathcal {V}}^j(t) \equiv \dfrac{\alpha }{1-\beta _c \alpha } [ \alpha ^t q_0 +(1-\alpha ^t)q^j_{ss}]+B^j. \end{aligned}$$

From Proposition 3, we know that ${\mathcal {V}}^\mathrm{{eqm}}(0)\equiv V^\mathrm{{eqm}} (k_0)< V^\mathrm{{sp}} (k_0)\equiv {\mathcal {V}}^\mathrm{{sp}}(0)$ always holds. Moreover, since we consider the case of $\beta _c<\beta _l<{{\overline{\beta }}}_l$, ${\mathcal {V}}^\mathrm{{eqm}}(T)> {\mathcal {V}}^\mathrm{{sp}}(T)$ as $T\rightarrow \infty$. Finally, subtracting ${\mathcal {V}}^\mathrm{{eqm}}(t)$ from ${\mathcal {V}}^\mathrm{{sp}}(t)$ yields

$$\begin{aligned} {\mathcal {V}}^\mathrm{{eqm}}(t)-{\mathcal {V}}^\mathrm{{sp}}(t)=\dfrac{\alpha (1-\alpha ^t)(q^\mathrm{{eqm}}_{ss}-q^\mathrm{{sp}}_{ss})}{1-\beta _c\alpha }+B^\mathrm{{eqm}}-B^\mathrm{{sp}}. \end{aligned}$$

Since the value of $\alpha ^t$ decreases as t increases, ${\mathcal {V}}^\mathrm{{eqm}}(t')-{\mathcal {V}}^\mathrm{{sp}}(t')>{\mathcal {V}}^\mathrm{{eqm}}(t)-{\mathcal {V}}^\mathrm{{sp}}(t)$, for all $t'>t$, if $q^\mathrm{{eqm}}_{ss}-q^\mathrm{{sp}}_{ss}>0$. Note that this condition is automatically satisfied for the case of $\beta _l>\beta _c$. Thus, there exists a unique $T^*>0$, such that ${\mathcal {V}}^\mathrm{{eqm}}(t)>{\mathcal {V}}^\mathrm{{sp}}(t)$ if and only if $t\ge T^*$. This proves Proposition 5.

Appendix H: Analysis of tax policies

1.1 H. 1 Market equilibrium under the time-invariant tax policy

To obtain the time-consistent tax policy in Proposition 6, we first characterize the market equilibrium when the tax rates are constant over time: $\varvec{\tau }_t =\varvec{\tau }$ for all t, which is necessary to show this proposition. Let us introduce the following new functions:

$$\begin{aligned} {{\hat{r}}}(X) \equiv (1-\tau _r)r(X), \quad {{\hat{w}}}(X) \equiv (1-\tau _w)w(X), \end{aligned}$$

where r(X) and w(X) are given by (3), and the new variable:

$$\begin{aligned} {{\hat{k}}}_{t+1} =(1+\tau _i) k_{t+1}. \end{aligned}$$

The household’s budget constraint is then expressed as

$$\begin{aligned} {{\hat{k}}}_{t+1} = {{\hat{r}}}(X_t) k_t+ {{\hat{w}}} (X_t) l_t - c_t. \end{aligned}$$

(H.1)

Lemma H.1

In the market equilibrium with the constant tax policy, the saving rate, labor supply, and capital income tax rate are given by $s^\mathrm{{eqm}}(\varvec{\tau })$, $l^\mathrm{{eqm}}(\varvec{\tau })$, and $\tau _r(\varvec{\tau })$, respectively, where

$$\begin{aligned} s^\mathrm{{eqm}}(\varvec{\tau })&\equiv \dfrac{\psi (\alpha +(1-\alpha ) \tau _w)}{\psi +(1+\zeta )(1+\tau _i)}, \\ l^\mathrm{{eqm}}(\varvec{\tau })&\equiv \dfrac{(1-\tau _w) (1-\alpha ) (1+\zeta +\psi )}{(1+\zeta )[\zeta (1+s^\mathrm{{eqm}}(\varvec{\tau }) \tau _i)+(1-\tau _w)(1-\alpha )(1+\psi )]},\\ \tau _r(\varvec{\tau })&\equiv -\dfrac{\tau _w(1-\alpha )+\tau _i s^\mathrm{{eqm}}(\varvec{\tau })}{\alpha }, \end{aligned}$$

and

$$\begin{aligned} \psi \equiv \dfrac{\beta _c}{1-\beta _c}+\zeta \dfrac{\beta _l}{1-\beta _l}. \end{aligned}$$

Proof

As in the case of the laissez-faire environment in Sect. 3, the self in period t rationally expects the law of motion for the aggregate state $X_t$ as Eq. (5) in the main body. Then, the optimization problem of the self in a period is given by

$$\begin{aligned} V(k, X)=\max _{c, l, {{\hat{k}}}'} \left\{ \ln c+\zeta \ln (1-l) + \beta _c V_c \left[ {{\hat{k}}}'/(1+\tau _i), X' \right] + \beta _l V_l \left[ {{\hat{k}}}'/(1+\tau _i), X' \right] \right\} , \end{aligned}$$

(H.2)

subject to the budget constraint (H.1). Functions $V_c$ and $V_l$ here are defined as the following functional equations:

$$\begin{aligned} V_c(k, X )=\, & {} \ln \phi _c(k, X)+\beta _c V_c \left[ g(k, X)/(1+\tau _i), G(X) \right] , \end{aligned}$$

(H.3)

$$\begin{aligned} V_l (k, X)=\, & {} \zeta \ln \phi _l(k, X)+\beta _l V_l \left[ g(k, X)/(1+\tau _i), G(X) \right] , \end{aligned}$$

(H.4)

respectively, where $\phi _c(\cdot )$ and $\phi _l(\cdot )$ are the policy functions for c and l in this case, and $g(\cdot )$ is given by $g(\cdot )\equiv {{\hat{r}}}(\cdot ) k+{{\hat{w}}}(\cdot ) \phi _l(\cdot )-\phi _c(\cdot )$.

Hereafter, the arguments of the functions are omitted unless doing so would cause confusion. The FOCs of the problem (H.2) are

$$\begin{aligned} \dfrac{1+\tau _i}{c}= & {} \beta _c \dfrac{\partial V_c}{\partial k'}+ \beta _l \dfrac{\partial V_l}{\partial k'}, \end{aligned}$$

(H.5)

$$\begin{aligned} \dfrac{ {{\hat{w}}}}{c}= & {} \dfrac{\zeta }{1-l}. \end{aligned}$$

(H.6)

We derive the equilibrium in this case by use of “guess and verify.” We guess $V_i$ ($i\in \{c, l\}$) and G are given by

$$\begin{aligned} V_i=\, & {} a_i+b_i \ln X+ d_i \ln (k+\varphi X), \\ G=\, & {} sAX^\alpha . \end{aligned}$$

Then, from (H.1), (H.5), and (H.6), we obtain

$$\begin{aligned} l=\phi _l(k, X)&\equiv 1-\dfrac{\zeta }{1+\zeta + \Psi } \dfrac{{{\hat{r}}}}{{{\hat{w}}}} (k+ \Lambda X), \end{aligned}$$

(H.7)

$$\begin{aligned} c=\phi _c(k, X)&\equiv \dfrac{{{\hat{r}}}}{1+\zeta +\Psi } (k+ \Lambda X), \end{aligned}$$

(H.8)

$$\begin{aligned} {{\hat{k}}}'=g(k, X)&\equiv \dfrac{\Psi {{\hat{r}}} }{1+\zeta +\Psi } (k+ \Lambda X)- (1+\tau _i)\varphi G, \end{aligned}$$

(H.9)

where the definition of $\Psi$ is the same as (D.7):

$$\begin{aligned} \Psi \equiv \beta _c d_c+ \beta _l d_l, \end{aligned}$$

and $\Lambda$ is given by

$$\begin{aligned} \Lambda&\equiv \dfrac{{{\hat{w}}}+\varphi G(X)}{{{\hat{r}}}X} \\&=\dfrac{(1-\tau _w)(1-\alpha )+(1+\tau _i)\varphi s}{(1-\tau _r)\alpha }. \end{aligned}$$

Substituting (H.7)–(H.9) into (H.3) and (H.4), we obtain

$$\begin{aligned} b_c= -\dfrac{1-\alpha }{ (1-\beta _c\alpha ) (1-\beta _c)},\quad d_c=\dfrac{1}{1-\beta _c},\quad b_l=-\dfrac{1}{1-\beta _l},\quad d_l=\dfrac{\zeta }{1-\beta _l}, \end{aligned}$$

implying that

$$\begin{aligned} \Psi = \psi \equiv \dfrac{\beta _c}{1-\beta _c}+\zeta \dfrac{\beta _l}{1-\beta _l}. \end{aligned}$$

Meanwhile, because $\Lambda =\varphi$ holds, we have

$$\begin{aligned} \varphi =\dfrac{(1-\tau _w)(1-\alpha )}{(1-\tau _r)\alpha -(1+\tau _i)s}. \end{aligned}$$

(H.10)

Now, we derive the equilibrium saving rate $s^\mathrm{{eqm}}$, labor supply $l^\mathrm{{eqm}}$, and capital income tax rate $\tau _r$. Since $K_{t+1}=k_{t+1}$ in the equilibrium, the household’s saving behavior must be consistent with the law of motion of aggregate capital in the market equilibrium. Recalling that $X_{t+1}=k_{t+1}/l_{t+1}$, this implies

$$\begin{aligned} G(k_t/l_t)l_{t+1} = (1+\tau _i)^{-1}g(k_t, k_t/l_t) \quad \forall k, l. \end{aligned}$$

(H.11)

Using the household’s condition (H.9), the guess $G(k/l)=sA(k/l)^\alpha$, and the guess that labor supply is constant in the equilibrium, we can rewrite this consistency condition (H.11) as

$$\begin{aligned} s l =\dfrac{\psi \alpha (1-\tau _r)}{(1+\tau _i)(1+\zeta +\psi )}(l+\varphi ) -\varphi s, \end{aligned}$$

which in turn yields

$$\begin{aligned} s= \dfrac{1-\tau _r}{1+\tau _i}\dfrac{\psi \alpha }{1+\zeta +\psi }. \end{aligned}$$

(H.12)

Since $\tau _r$ is still to be determined, we use the government’s budget constraint (equation (14) in the main body) and $X'=s AX^\alpha$ to obtain

$$\begin{aligned} \tau _r=-\dfrac{\tau _w(1-\alpha )+\tau _i s}{\alpha }. \end{aligned}$$

(H.13)

Substituting (H.13) into (H.12), we obtain the equilibrium saving rate:

$$\begin{aligned} s=s^\mathrm{{eqm}}(\varvec{\tau })\equiv \dfrac{\psi (\alpha +(1-\alpha ) \tau _w)}{\psi +(1+\zeta )(1+\tau _i)}. \end{aligned}$$

Then, substituting this result back into (H.10) and (H.13) yields

$$\begin{aligned}&\varphi =\varphi ^\mathrm{{eqm}}(\varvec{\tau }) \equiv \dfrac{(1-\tau _w)(1-\alpha )}{\alpha +\tau _w(1-\alpha )-s^\mathrm{{eqm}}(\varvec{\tau })}, \\&\tau _r=\tau _r(\varvec{\tau })\equiv -\dfrac{\tau _w(1-\alpha )+\tau _i s^\mathrm{{eqm}}(\varvec{\tau })}{\alpha }. \end{aligned}$$

Finally, we obtain the equilibrium labor supply $l^\mathrm{{eqm}}$ using the consistency condition $l^\mathrm{{eqm}}=\phi _l(k, k/l^\mathrm{{eqm}})$. From (H.7), this condition is rewritten as

$$\begin{aligned} l=\dfrac{(1-\tau _w)(1-\alpha )(1+\zeta +\psi )-\zeta (1-\tau _r)\alpha \varphi }{(1-\tau _w)(1-\alpha )(1+\zeta +\psi )+\zeta (1-\tau _r) \alpha }. \end{aligned}$$

Then, substituting $\varphi ^\mathrm{{eqm}}(\varvec{\tau })$ and $\tau _r(\varvec{\tau })$ into this equation and rearranging the terms, we obtain

$$\begin{aligned} l=l^\mathrm{{eqm}}(\varvec{\tau })=\dfrac{(1-\tau _w) (1-\alpha ) (1+\zeta +\psi )}{(1+\zeta )[\zeta (1+s^\mathrm{{eqm}}(\varvec{\tau }) \tau _i)+(1-\tau _w)(1-\alpha )(1+\psi )]}. \end{aligned}$$

$\square$

1.2 H.2 Definition of the time-consistent policy

In this section, we formally define the time-consistent tax policy. Suppose that the government in period t sets $\varvec{\tau }_t = \widetilde{\varvec{\tau }}\equiv ({{\widetilde{\tau }}}_w,{{\widetilde{\tau }}}_i)$. while the government in the other periods set the tax rates as $\overline{\varvec{\tau }}=({{\overline{\tau }}}_w, {{\overline{\tau }}}_i)$. If $\widetilde{\varvec{\tau }}\ne \overline{\varvec{\tau }}$, there is unilateral deviation of the government in period t. Let ${{\widetilde{G}}}(X, \widetilde{\varvec{\tau }})$ denote the law of motion of X, which differs from G(X) obtained in the previous section because of the current government’s one-period deviation. By definition, ${{\widetilde{G}}}(X, \overline{\varvec{\tau }})\equiv G(X)$ (i.e., if $\widetilde{\varvec{\tau }}=\overline{\varvec{\tau }}$, they are the same function).

Then, the optimization problem of the household in period t (H.2) is replaced by

$$\begin{aligned} {{\widetilde{V}}}(k, X, \widetilde{ \varvec{\tau }} )&=\max _{c, l, {{\hat{k}}}'} \left\{ \ln c+\zeta \ln (1-l) +\beta _c V_c \left[ (1+{{\widetilde{\tau }}}_i)^{-1}{{\hat{k}}}', {{\widetilde{G}}}(X, \widetilde{\varvec{\tau }}) \right] \nonumber \right. \\&\left. \quad + \beta _l V_l \left[ (1+{{\widetilde{\tau }}}_i)^{-1} {{\hat{k}}}', {{\widetilde{G}}}(X, \widetilde{\varvec{\tau }}) \right] \right\} . \end{aligned}$$

(H.14)

In Eq. (H.14), functions $V_c$ and $V_l$ are the same as (H.3) and (H.4), respectively, meaning that the household’s decision-making is qualitatively the same as that in the previous section. This is simply because each individual makes her decision taking the factor prices and taxes as given.

Next, we consider the government’s decision-making in period t. In contrast to the household’s behavior, the government recognizes that it can affect the equilibrium labor supply. In what follows, we let $l_t={{\widetilde{l}}}^\mathrm{{eqm}}(\widetilde{\varvec{\tau }}, \overline{\varvec{\tau }})$ denote the equilibrium labor supply in period t. Owing to the current government’s deviation, ${{\widetilde{l}}}^\mathrm{{eqm}}(\widetilde{\varvec{\tau }}, \overline{\varvec{\tau }})$ can be a different function from $l^\mathrm{{eqm}} (\overline{\varvec{\tau }})$.

We can now define the time-consistent tax policy.

Definition 1

The sequence $\{\varvec{\tau }_t\}_{t=0}^\infty$, with $\varvec{\tau }_t=\overline{\varvec{\tau }}$ $\forall t=0, 1, 2 \ldots$, is the time-consistent tax policy if

$$\begin{aligned} \forall k_t>0, \forall t=0, 1, 2 \ldots , \quad \overline{\varvec{\tau }}={\mathrm{arg}}\max _{ \widetilde{\varvec{\tau }}} {{\widetilde{V}}} \left[ k_t, \frac{k_t}{{{\widetilde{l}}}^\mathrm{{eqm}}(\widetilde{\varvec{\tau }}, \overline{\varvec{\tau }})}, \widetilde{\varvec{\tau }}\right] . \end{aligned}$$

In other words, the sequence of tax rates $\{ \varvec{\tau }_t\}_{t=0}^\infty$, with $\varvec{\tau }_t=\overline{\varvec{\tau }}$ $\forall t=0, 1, 2 \ldots$ is the time-consistent tax policy if any selves of the government cannot obtain a strictly positive welfare gain by these selves’ unilateral one-shot deviation from $\overline{\varvec{\tau }}$.

1.3 H.3 Proof of Proposition 6

We now show Proposition 6. Suppose that the government in period t deviates from $\overline{\varvec{\tau }}$ and sets $\varvec{\tau }_t= \widetilde{\varvec{\tau }}$. We guess that the law of motion of the aggregate state in this period is given by

$$\begin{aligned} G(X_t)={{\widetilde{s}}}_t AX_t^\alpha , \end{aligned}$$

where ${{\widetilde{s}}}_t$ is the equilibrium saving rate in period t, which can differ from $s^\mathrm{{eqm}}(\overline{\varvec{\tau }})$ by the government’s deviation in this period. Since the pair of tax rates after this period is always given by $\overline{\varvec{\tau }}$, the equilibrium labor supply $l_{t+j}$ ($j=1, 2, \ldots$) is $l^\mathrm{{eqm}}(\overline{\varvec{\tau }})$. Therefore, Eq. (H.11) now implies

$$\begin{aligned}&G(k/l)l^\mathrm{{eqm}}(\overline{\varvec{\tau }})= (1+\tau _i)^{-1}g(k, k/l) \ \forall k, l\nonumber \\&\quad \Leftrightarrow {{\widetilde{s}}}_t l^\mathrm{{eqm}}(\overline{\varvec{\tau }}) =\dfrac{\psi \alpha (1-\tau _r)}{(1+\tau _i)(1+\zeta +\psi )}(l+ {{\widetilde{\varphi }}}_t ) -{{\widetilde{\varphi }}}_t {{\widetilde{s}}}_t, \end{aligned}$$

(H.15)

where ${{\widetilde{\varphi }}}_t$ is the value of $\varphi$ in period t. In addition, by imposing $\varvec{\tau }=\widetilde{\varvec{\tau }}$ and $l_{t+1}=l^\mathrm{{eqm}}(\overline{\varvec{\tau }})$ on the government budget constraint (14), we obtain

$$\begin{aligned}&{{\widetilde{\tau }}}_r r(X_t)k_t+{{\widetilde{\tau }}}_w w(X_t)l_t +{{\widetilde{\tau }}}_i G(X_t) l_{t+1}=0 \nonumber \\&\quad \Leftrightarrow {{\widetilde{\tau }}}_r \alpha +{{\widetilde{\tau }}}_w (1-\alpha )+{{\widetilde{\tau }}}_i {\widetilde{s}}_t \ l^\mathrm{{eqm}} (\overline{\varvec{\tau }})=0. \end{aligned}$$

(H.16)

From (H.10), (H.15), (H.16), and the consistency condition, $\phi _l(k_t, k_t/l_t)=l_t$, the equilibrium labor supply as well as the values of ${{\widetilde{s}}}_t$, ${{\widetilde{\varphi }}}_t$, and ${{\widetilde{\tau }}}_r$ are determined as the functions of both $\widetilde{\varvec{\tau }}$ and $\overline{\varvec{\tau }}$. Let their values be denoted by ${{\widetilde{s}}}^\mathrm{{eqm}}(\widetilde{\varvec{\tau }}, \overline{\varvec{\tau }})$, ${{\widetilde{\varphi }}}^\mathrm{{eqm}}(\widetilde{\varvec{\tau }}, \overline{\varvec{\tau }})$, and ${{\widetilde{\tau }}}_r(\widetilde{\varvec{\tau }}, \overline{\varvec{\tau }})$, respectively. Then,

$$\begin{aligned} k_{t+1}&={{\widetilde{s}}}^\mathrm{{eqm}}(\widetilde{\varvec{\tau }}, \overline{\varvec{\tau }}) Ak_t^\alpha ({{\widetilde{l}}}^\mathrm{{eqm}}(\widetilde{\varvec{\tau }}, \overline{\varvec{\tau }}))^{1-\alpha }, \\ c_t&=(1-{{\widetilde{s}}}^\mathrm{{eqm}}(\widetilde{\varvec{\tau }}, \overline{\varvec{\tau }}))Ak_t^\alpha ({{\widetilde{l}}}^\mathrm{{eqm}}(\widetilde{\varvec{\tau }}, \overline{\varvec{\tau }}))^{1-\alpha }. \end{aligned}$$

Now, we derive the welfare in period t, ${{\widetilde{V}}}[k_t, k_t/{{\widetilde{l}}}^\mathrm{{eqm}}(\cdot ), \widetilde{\varvec{\tau }}]$. Since the value of $\varphi$ is given by $\varphi ^\mathrm{{eqm}}(\overline{\varvec{\tau }})$ in the subsequent periods, the following equation holds:

$$\begin{aligned} k_{t+1}+\varphi X_{t+1}= \left( 1+ \dfrac{\varphi ^\mathrm{{eqm}} (\overline{\varvec{\tau }}) }{ l^\mathrm{{eqm}}(\overline{\varvec{\tau }}) }\right) k_{t+1}. \end{aligned}$$

Substituting these results into Eq. (H.14) yields

$$\begin{aligned} {{\widetilde{V}}} \left[ k_t, \frac{k_t}{{{\widetilde{l}}}^\mathrm{{eqm}}(\widetilde{\varvec{\tau }}, \overline{\varvec{\tau }})}, \widetilde{\varvec{\tau }}\right]&=\ln \left[(1-{{\widetilde{s}}}^\mathrm{{eqm}}(\widetilde{\varvec{\tau }}, \overline{\varvec{\tau }}))Ak_t^\alpha ({{\widetilde{l}}}^\mathrm{{eqm}}(\widetilde{\varvec{\tau }}, \overline{\varvec{\tau }}))^{1-\alpha }\right] +\zeta \ln \left( 1-{{\widetilde{l}}}^\mathrm{{eqm}}(\widetilde{\varvec{\tau}}, \overline{\varvec{\tau}})\right) \\&\quad + \beta _c \left( b_c + d_c \right) \ln \left[ {{\widetilde{s}}}^\mathrm{{eqm}}(\widetilde{\varvec{\tau }}, \overline{\varvec{\tau}}) A k_t^\alpha ({{\widetilde{l}}}^\mathrm{{eqm}}(\widetilde{\varvec{\tau }}, \overline{\varvec{\tau}}) )^{1-\alpha } \right] + \text {other terms}, \end{aligned}$$

where “other terms” represent the collection of all terms independent of $\widetilde{\varvec{\tau }}$. Note that in the proof of Lemma H.1, the values of $b_c$ and $d_c$ are already given and $b_l+d_l=0$ is already verified.

From the above equation, we find that $\widetilde{\varvec{\tau }}$ affects ${{\widetilde{V}}}$ only through ${{\widetilde{s}}}^\mathrm{{eqm}}(\widetilde{\varvec{\tau }}, \overline{\varvec{\tau }})$ and ${{\widetilde{l}}}^\mathrm{{eqm}}(\widetilde{\varvec{\tau }}, \overline{\varvec{\tau }})$. This implies that obtaining ${{\widetilde{V}}}$ by choosing $\widetilde{\varvec{\tau }}$ can also be achieved directly by choosing the values of ${{\widetilde{s}}}^\mathrm{{eqm}}$ and ${{\widetilde{l}}}^\mathrm{{eqm}}$. The FOCs are given by

$$\begin{aligned} \dfrac{\beta _c (b_c+d_c)}{\widetilde s^{\mathrm{eqm}}}= & {} \dfrac{1}{1-\widetilde s^{\mathrm{eqm}}}, \\ \dfrac{(1-\alpha )[1+\beta _c (b_c+d_c)]}{\widetilde l^{\mathrm{eqm}}}= & {} \dfrac{\zeta }{1-\widetilde l^{\mathrm{eqm}}}. \end{aligned}$$

respectively. Letting $({{\overline{s}}}, {{\overline{l}}})$ denote the solutions, we explicitly obtain

$$\begin{aligned} {{\overline{s}}}=\beta _c \alpha , \quad {{\overline{l}}}=\dfrac{1-\alpha }{1-\alpha +\zeta (1-\beta _c \alpha )}. \end{aligned}$$

(H.17)

The tax policy $\overline{\varvec{\tau }}$ is then determined from

$$\begin{aligned} {{\overline{s}}}={{\widetilde{s}}}^\mathrm{{eqm}}(\overline{\varvec{\tau }}, \overline{\varvec{\tau }}), \quad {{\overline{l}}}={{\widetilde{l}}}^\mathrm{{eqm}} ( \overline{\varvec{\tau }}, \overline{\varvec{\tau }}). \end{aligned}$$

Since ${{\widetilde{s}}}^\mathrm{{eqm}}( \varvec{\tau }, \varvec{\tau })=s^\mathrm{{eqm}}(\varvec{\tau })$, and ${{\widetilde{l}}}^\mathrm{{eqm}}(\varvec{\tau }, \varvec{\tau })=l^\mathrm{{eqm}}(\varvec{\tau })$, for all $\varvec{\tau }$, substituting $s^\mathrm{{eqm}}(\varvec{\tau })$ and $l^\mathrm{{eqm}}(\varvec{\tau })$ in Lemma H.1 into (H.17) yields the following two equations:

$$\begin{aligned}&{{\overline{s}}}=s^\mathrm{{eqm}}(\varvec{\tau })\Leftrightarrow \beta _c \alpha = \dfrac{\psi (\alpha +(1-\alpha ) \tau _w)}{\psi +(1+\zeta )(1+\tau _i)}, \\&{{\overline{l}}}=l^\mathrm{{eqm}}(\varvec{\tau })\Leftrightarrow \dfrac{1-\alpha }{1-\alpha +\zeta (1-\beta _c \alpha )}=\dfrac{(1-\tau _w) (1-\alpha ) (1+\zeta +\psi )}{(1+\zeta )[\zeta (1+\beta _c \alpha \tau _i)+(1-\tau _w)(1-\alpha )(1+\psi )}. \end{aligned}$$

The former and latter equations yield

$$\begin{aligned} 1+\tau _i=\dfrac{\psi }{(1+\zeta )\beta _c} \left[ 1-\beta _c+\tau _w(1-\alpha )/\alpha \right] , \end{aligned}$$

and

$$\begin{aligned} 1+\tau _i=\dfrac{(1-\beta _c)\psi }{(1+\zeta )\beta _c}-\dfrac{\tau _w}{\beta _c}\left[ \dfrac{1-\beta _c\alpha }{\alpha }+\dfrac{(1-\beta _c)\psi }{1+\zeta }\right] , \end{aligned}$$

respectively. Then, from the above two equations, we have ${{\overline{\tau }}}_w=0$ and ${{\overline{\tau }}}_i=\dfrac{1-\beta _c}{(1+\zeta )\beta _c}\psi -1$. From the definition of $\psi$, we then have

$$\begin{aligned} {{\overline{\tau }}}_i=\dfrac{\zeta }{1+\zeta } \left( \dfrac{1-\beta _c}{\beta _c}\dfrac{\beta _l}{1-\beta _l}-1\right) . \end{aligned}$$

Finally, substituting ${{\overline{\tau }}}_w=0$ and $s^\mathrm{{eqm}}(\overline{\varvec{\tau }})=\beta _c\alpha$ into (H.13), we obtain ${{\overline{\tau }}}_r=-\beta _c {{\overline{\tau }}}_i$.