Taxing capital is a good idea: The role of idiosyncratic risk in an OLG model

Journal of Economic Dynamics & Control 52 (2015) 258–269

Contents lists available at ScienceDirect

Journal of Economic Dynamics & Control journal homepage: www.elsevier.com/locate/jedc

Taxing capital is a good idea: The role of idiosyncratic risk in an OLG model$ Ryoji Hiraguchi a,n, Akihisa Shibata b a b

Faculty of Law and Economics, Chiba University, 1-33, Yayoi-cho, Inage-ku, Chiba, Japan Institute of Economic Research, Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto 606-8501, Japan

a r t i c l e in f o

abstract

Article history: Received 10 January 2014 Received in revised form 6 September 2014 Accepted 12 December 2014 Available online 1 January 2015

We investigate an overlapping generations (OLG) model in which agents who live for two periods receive idiosyncratic productivity shocks when they are old. We show that, around zero tax equilibria, we can always construct a combination of a small capital tax and a lump-sum transfer that are Pareto-improving. As Dávila et al. (Econometrica (2012)) show, a capital reduction in one period raises the welfare level of agents who are old in that period, but lowers that of the young agents, because it reduces their wages. We show that the government can compensate for these wage losses by additionally taxing the old agents, such that their welfare gains remain positive. Our result is unchanged when earnings are uncertain at young age. & 2014 Elsevier B.V. All rights reserved.

JEL classification: E5 Keywords: Idiosyncratic risk Capital tax Incomplete markets Overlapping generations

1. Introduction It is widely known that in most of the infinite horizon, representative agent models with complete markets, the optimal capital tax rate is zero (see Chamley, 1986; Judd, 1985, 1987). On the basis of these findings, Atkeson et al. (1999) and Chari and Kehoe (1999) suggest that taxing capital income should be avoided. A capital tax distorts decisions on current consumption and future consumption more severely than a labor income tax. Fiscal policies relying on capital income taxation lead to higher welfare costs than policies without such a tax. Recently, several authors investigated the validity of the zero capital tax principle in an economy with incomplete markets and agent heterogeneity. By introducing uninsurable labor income shocks and credit constraints into a model with infinitely lived agents, Aiyagari (1995) shows that the steady state optimal capital income tax rate is positive. Dávila et al. (2012) investigate a two-period production economy in which agents are identical ex-ante, but are heterogenous ex-post, that is, their second-period labor endowment is subject to idiosyncratic shocks. They find that the marginal effect of an increase in capital on each individual's expected utility is proportional to the covariance of the marginal utility of second period consumption and the labor endowment shock, the sign of which is negative. This means that the welfare effect of a reduction in capital is strictly positive. Dávila et al. (2012) then prove that a linear capital tax reduces capital and improves

☆ This research is financially supported by Grant-in-Aid for Specially Promoted Research (No. 23000001) and the G-COE program of Osaka University, “Human Behavior and Socioeconomic Dynamics.” n Corresponding author. E-mail addresses: [email protected] (R. Hiraguchi), [email protected] (A. Shibata).

http://dx.doi.org/10.1016/j.jedc.2014.12.003 0165-1889/& 2014 Elsevier B.V. All rights reserved.

R. Hiraguchi, A. Shibata / Journal of Economic Dynamics & Control 52 (2015) 258–269

259

social welfare. They also investigate an infinite horizon version of the model with borrowing constraints and find numerically that a positive capital tax rate is still welfare-improving. A recent paper by Gottardi et al. (2012) extends the framework developed by Dávila et al. (2012) to allow for idiosyncratic shocks to investment and an endogenous labor supply. They show that when shocks to labor productivity are larger than are those to investment, the optimal capital tax rate is positive. Moreover, in line with Dávila et al. (2012) and Gottardi et al. (2012), Carvajal and Polemarchakis (2011), incorporate uninsurable shocks into an overlapping generations (OLG) economy, as developed by Diamond (1965). In their model, agents who live for two periods are born in each period, and are subject to idiosyncratic productivity shocks when they are old. They restrict their attention to the steady state equilibrium and study the marginal effect of a capital increase on steady state welfare. As a result, they find that the welfare effect consists of two terms; the first representing an intergenerational effect, proportional to the difference between the real interest rate and the population growth rate, and the second representing welfare losses owing to the idiosyncratic shocks. As is well known, the intergenerational effect is observed in homogenous agent OLG models such as that in the standard Diamond (1965) model. The second term, which is always negative, is because of idiosyncratic shocks and is almost the same as the effect found by Dávila et al. (2012). Without any shocks, we only observe the first term. In this case, the famous golden rule applies, which states that when the real interest rate is lower than the population growth rate, there is over-accumulation of the capital stock in the steady state. On the basis of their findings, Carvajal and Polemarchakis (2011) show that when an economy is subject to idiosyncratic labor endowment shocks, the golden rule argument should be modified. In other words, even when the real interest rate exceeds the population growth rate, a reduction in capital can raise steady state welfare. However, whether such a capital reduction in an OLG model will be a Pareto improvement in the economy is ambiguous.3 This is because a capital reduction in one period may benefit old agents in that period, but would be harmful to young agents, as it lowers their wage income. In this paper, we investigate an overlapping generation economy in which agents who live for two periods are born in each period.4 Agents are ex-ante identical but are subject to idiosyncratic labor endowment shocks when they are old. This set-up is closely related to that of Carvajal and Polemarchakis (2011). However, we do not limit our attention to steady states and we study whether capital income taxation can raise the utility of one generation without affecting that of other generations. This paper shows that, for any competitive equilibrium allocation in which the government can only use a lump-sum transfer, we can find a combination of lump-sum transfer and linear capital tax that Pareto-improves the market equilibrium. Here, we assume that the lump-sum transfer can be age specific. Just as Dávila et al. (2012) show, a capital reduction in any period, for example period 1, raises the utility of old agents, who are born in period 0. Meanwhile, the capital reduction lowers the wage income of young agents in period 1 and reduces their welfare. However, the benefit from the capital reduction for the old agents is so large that their welfare gain is still positive, even if they are taxed additionally to compensate for the young agents’ wage losses. Since this policy change in period 1 only affects the old and young agents in that period, it Pareto-improves the economy. The desirability of the capital income tax is unchanged when the individuals supply labor endogenously. As an extension, we study a model where young agents are subject to the productivity shock. We first analytically show that small positive capital income tax is still Pareto-improving as long as the utility function is logarithmic. The agent who receive a bad earnings shock (say low-type agent) when he is young saves less than the one who gets a good earning shock (say high-type agent). This implies that the low-type pays less capital income tax than the high-type. Therefore linear capital income tax works as a redistribution from the high-type agent to low-type agent. Although the capital income tax is distortionary, the first order effect is small around the zero capital tax equilibrium. Thus the small capital income tax is Pareto-improving. We next obtained the optimal capital income tax rate in a numerical example where both the young agent and the old agent face the earning risk. As in Dávila et al. (2012), we investigate the unemployment economy where the productivity shock represents unemployment risk. We found that the annualized optimal capital income tax rate is 0.16% and the welfare gain (as measured by the consumption equivalent) is 0.024%. It should be noted that the welfare effect of capital taxation differs significantly from that of government bonds. As Diamond (1965), Ihori (1978) and Tirole (1985) show, government bonds reduce the equilibrium capital level. In this sense, their effect can be said to be similar to that of capital taxation. However, the welfare improvement seen in our economy requires an intertemporal wedge between current and future consumption, which cannot be created by government bonds. Hence, they cannot be a substitute for the capital tax in our model with agent heterogeneity. Our paper is related to Garriga (2003) who investigates the optimal capital tax policy in the overlapping generations model. In Garriga (2003), social welfare is defined as a weighted sum of the utility of all generations. He shows that if agedependent labor income tax is not available, the optimal capital tax is not zero. Here we show that even if age-dependent transfer is available, the optimal capital tax is not zero when the agents are subject to the earning risk. Our paper also relates to that of Conesa et al. (2009), which investigate the optimal capital and income taxes in an overlapping generation economy, in which agents have an elastic labor supply and receive idiosyncratic productivity shocks

3 Carvajal and Polemarchakis (2011) also state that a capital reduction can be Pareto-improving. However, because they focus on stationary equilibria and do not study intergenerational heterogeneity, our analysis differs from theirs significantly. Moreover, we show that a tax and transfers policy implements the improvement as a competitive outcome. They do not discuss this point. 4 Fiorini (2008) investigates the effect of idiosyncratic shocks on the optimal policy in a monetary OLG economy and finds that in some situations inflation can raise steady state welfare.

260

R. Hiraguchi, A. Shibata / Journal of Economic Dynamics & Control 52 (2015) 258–269

in each period. Conesa et al. (2009) find that when agents are heterogeneous in their productivity, the optimal capital income tax rate is positive. Although this result seems to be very similar to our owns, there are two significant differences between our study and that of Conesa et al. (2009). First, Conesa et al. (2009) allow labor income tax to be non-linear, whereas, in our paper, we only consider a lump-sum transfer. Second, they focus on steady state welfare, as do Carvajal and Polemarchakis (2011), whereas we treat all generations differently and consider welfare improvements in the Pareto sense.5 This paper is organized as follows. Section 2 describes the basic structure of the model and characterizes the competitive equilibrium. Section 3 shows that a capital tax can improve social welfare. Section 4 describes a fiscal policy that raises the utility of every generation. Section 5 studies a case with endogenous labor supply. Section 6 investigates the age-dependent tax as a substitute for a capital tax. Section 6 concludes the paper. The Appendix presents proofs of our propositions. 2. The model 2.1. Production Time is discrete and ranges from 0 to þ 1. A continuum of ex-ante identical individuals, with measure N t ¼ N0 ð1 þ nÞt , is born in period t, where n is the population growth rate and N0 is the number of agents born in period 0. They live for two periods. In period 0, there are initial old agents, with measure N  1 ¼ N0 =ð1 þ nÞ who make no savings decisions. We call the group of individuals born in period t (t Z  1) generation t. Agents supply their labor when they are young and old. In period t, generations t  1 and t work. The labor supply of the young, l, is constant, but that of the old, e, is stochastic and takes a value ez 4 0 with probability π z 4 0 (z ¼ 1; 2; …; Z). The P number of possible states is denoted by Zð Z2Þ. Here, we assume ez o ez þ 1 (z ¼ 1; 2; …Z  1) and Zz ¼ 1 π z ¼ 1. If the labor z endowment of an old agent is e , we say that the agent's state is z. By the law of large numbers, the total amount of labor P being provided at time t is Lt ¼ E½eNt  1 þlNt ¼ fð1 þ nÞl þ egNt  1 , where e ¼ E½e ¼ Zz ¼ 1 π z ez . We normalize the units of labor so that the following holds: ð1 þ nÞl þ e ¼ 1:

ð1Þ

Under this normalization, the level of total labor in a period is equal to the size of the previous generation: Lt ¼ N t  1

for all t Z0:

ð2Þ

Later we prove that under (2) the equilibrium level of individual savings coincides with the capital-labor ratio. There are many identical firms. The production function F has constant returns to scale and is given by FðK t ; Lt Þ  f ðkt ÞLt , where Kt is aggregate capital in period t, kt ¼ K t =Lt denotes the capital-labor ratio in period t, and the per-capital production 0 ″ 0 function, f, satisfies f ð0Þ ¼ 0, f 4 0, f o 0, limk-1 f ðkÞ ¼ þ 1, and limk-0 f ðkÞ ¼ þ1. Factor markets are perfectly 0 competitive; the equilibrium wage rate in period t, is wt ¼ wðkt Þ  f ðkt Þ kt f ðkt Þ, and the capital rental rate in period t is 0 Rt ¼ Rðkt Þ  f ðkt Þ þ 1  δ, where δ is the rate of capital depreciation. The initial old agents are endowed with the initial aggregate capital, K0, which is exogenous. Thus, the level of individual capital holdings is K 0 =N  1 ¼ k0 . 2.2. The consumer's problem For all t Z0, generation t chooses their level of savings and maximizes their expected utility U t ¼ uðct Þ þ βE½uðdt þ 1;z Þ; where u is the instantaneous utility function, β is the discount factor, ct is the agent's consumption when he is young, and dt þ 1;z denotes his consumption when he is old and the state is z. The utility function u satisfies u0 4 0, u″ o 0, and limc-0 u0 ðcÞ ¼ þ 1. The budget constraints when an agent is young and old are, respectively, ct ¼ wt l  st þ T yt ; dt þ 1;z ¼ wt þ 1 ez þ ð1  τt þ 1 ÞRt þ 1 st þ T otþ 1 : y

o

Here, st is the level of individual savings in period t, Tt (Tt ) is the lump-sum transfer to the young (old) in period t and τt þ 1 is the linear capital income tax rate in period tþ1. Here, the transfer is age-specific.6 The Euler equation is given by u0 ðct Þ ¼ ð1  τt þ 1 ÞRt þ 1 β E½u0 ðdt þ 1;z Þ:

ð3Þ

5 Dávila (2012) studies the role of capital tax in a deterministic OLG model. He shows that a capital tax can improve stationary welfare if old agents are subject to the lump-sum tax. However, his conclusion does not hold when young agents are also taxed. We show that when agents receive idiosyncratic shocks, a capital tax is welfare improving, even if the government uses lump-sum transfers on young and old. 6 Age-specific tax instruments are also considered by Erosa and Gervais (2002). They show that if the government is not able to condition labor taxes on age then a positive capital income tax rate can be optimal. In contrast, in this paper, we show that if the government conditions lump-sum transfers on age, then a linear capital tax with a small tax rate can always be Pareto-improving.

R. Hiraguchi, A. Shibata / Journal of Economic Dynamics & Control 52 (2015) 258–269

261

The initial old agents do not make any intertemporal choices. Their utility with labor endowment ez U  1 ¼ uðw0 ez þ ð1  τ0 ÞR0 k0 þ T o0 Þ is exogenous. 2.3. Competitive equilibrium Here, we characterize the competitive equilibrium. We first define an allocation. 1

Definition 1. An allocation is a sequence fkt ; ct ; ðdt;z ÞZz ¼ 1 gt ¼ 0 , where kt is the capital-labor ratio in period t, ct is the consumption of generation t when they are young, and dt;z is the consumption of generation t 1 when they are old, and the state is z. By the law of large numbers, the allocation is feasible if and only if Nt ct þ N t  1 E½dt;z  þK t þ 1 ¼ FðK t ; Lt Þ þ ð1  δÞK t ; for all t Z 0. Dividing both sides of the feasibility condition by Lt ¼ Nt  1 , we obtain ð1 þnÞct þ E½dt;z  þ ð1 þnÞkt þ 1 ¼ f ðkt Þ þ ð1  δÞkt :

ð4Þ

1

The government policy is a sequence fτt ; T yt ; T ot gt ¼ 0 . The budget constraint of the government is τt Rt K t ¼ N t T yt þ Nt  1 T ot . Dividing both sides of the equation by Nt  1 , we get

τt Rt kt ¼ ð1 þ nÞT yt þ T ot ;

ð5Þ

In a competitive equilibrium, the aggregate level of savings is equal to aggregate capital holdings, that is, st Nt ¼ K t þ 1 . Using Eq. (2), we obtain st ¼

K t þ 1 Lt þ 1 ¼ kt þ 1 : Lt þ 1 Nt

ð6Þ

Therefore, the value of the level of individual savings in one period is equal to the capital-labor ratio in the next period. We now formally define the competitive equilibrium. 1

Definition 2. A competitive equilibrium consists of an allocation, fkt ; ct ; ðdt;z ÞZz ¼ 1 gt ¼ 0 , and a government policy, 1 fτt ; T yt ; T ot gt ¼ 0 , that satisfy the Euler equation, (3), the government budget constraint, (5), and the following budget constraints of generation t: ct ¼ wt l  kt þ 1 þT yt ;

ð7Þ

dt;z ¼ wt ez þð1  τt ÞRt kt þ T ot :

ð8Þ

Here, Rt and wt are the factor prices and the initial capital K 0 is given. It is easy to verify that Eqs. (5), (7), and (8) imply the resource constraint (4). 3. The role of capital taxation In this section, we show that for any competitive equilibrium allocation without capital taxation there exists another competitive equilibrium allocation with a linear capital tax that Pareto dominates the original allocation. 3.1. Capital reduction 1

Let us pick any competitive equilibrium allocation A ¼ fkt ; ct ; ðdt;z ÞZz ¼ 1 gt ¼ 0 under the zero capital tax policy 1 G ¼ f0; T yt ; T ot gt ¼ 0 which satisfies the government budget constraint 0 ¼ ð1 þnÞT yt þ T ot . This allocation is characterized by the following three equations: ct ¼ wt l  kt þ 1 þT yt ;

ð9Þ

dt;z ¼ wt ez þRt kt þ T ot ;

ð10Þ

u0 ðct Þ ¼ β Rt þ 1 E½u0 ðdt þ 1;z Þ:

ð11Þ

We next consider the following reallocations in periods 0 and 1: Reallocation in period 0: Reduce the savings level of each young individual (generation 0) from k1 to x, with k1 4 x. Reallocation in period 1: Transfer m1 ðxÞ ¼ ð1 þ nÞðw1  wðxÞÞl units of consumption good from each old individual (generation 0) to the young (generation 1). Following the first reallocation, the capital-labor ratio is changed to x. This raises the consumption of all agents in generation 0 by k1 x 4 0 when they are young. Since the population of generation 1 is ð1 þnÞ times larger than that of

262

R. Hiraguchi, A. Shibata / Journal of Economic Dynamics & Control 52 (2015) 258–269

generation 0, the second reallocation raises the income of each individual in generation 1 by m1 ðxÞ=ð1 þnÞ. Note that generations other than 0 and 1 are not affected by this allocation change. The sequence of the capital-labor ratio after the reallocation is given by fk0 ; x; k2 ; k3 ; …g. The income transfer from generation 0 to generation 1 is balanced and thus the allocation following the reallocation is feasible. However, at this point, it is not clear whether we can implement the new allocation as a competitive equilibrium. Later we show that this is possible. The state z consumption bundle of generation 0 depends on x and is written as c0 ðxÞ ¼ c0 þk1 x; d1;z ðxÞ ¼ wðxÞez þRðxÞx m1 ðxÞ þT o1 : It is obvious that c0 ðk1 Þ ¼ c0 and d1;z ðk1 Þ ¼ d1;z . The reallocations do not alter the level of savings of generation 1 and it is equal to k2. Thus his consumption when he is young is wðxÞl  k2 þT y1 þ ð1 þ nÞ  1 m1 ðxÞ ¼ w1 l k2 þ T y1 , which is the same the original level of consumption c1. This means that the loss in his wage income, owing to the capital reduction, is perfectly compensated by the subsidy m1 ðxÞ=ð1 þ nÞ. Since the allocation following period 2 is unchanged, the state z consumption bundle of generation t is ðc0 ðxÞ; d1;z ðxÞÞ when t¼0, and ðct ; dt þ 1;z Þ when t Z1. It should be noted we do not yet know whether generation t can optimally choose the new consumption bundle under certain tax instruments. 3.2. Welfare improvement We now show that the new allocation with lower capital raises the utility of generation 0. The utility depends on x, and, when we write it in the form of v0 ðxÞ, is given by v0 ðxÞ ¼ uðc0 þ k1  xÞ þ βE½uðd1;z ðxÞÞ: From the assumption normalizing the labor unit, (1), the level of the old period consumption d1;z ðxÞ ¼ wðxÞfez þ ð1 þ nÞlg þ RðxÞx w1 lð1 þ nÞ þT o1 can be re-written as d1;z ðxÞ ¼ wðxÞðez  E½ez Þ þf ðxÞ þ ð1  δÞx  w1 lð1 þnÞ þ T o1 : Differentiating the welfare function v0 by x yields v00 ðxÞ ¼  u0 ðc0 þk1  xÞ þ βE½fw0 ðxÞðez  E½ez Þþ RðxÞgu0 ðd1;z ðxÞÞ:

ð12Þ

The first order condition at the original allocation A is u ðc0 Þ ¼ βE½R1 u ðd1;z Þ. Thus when we evaluate Eq. (12) at x ¼ k1 , we obtain 0

v00 ðk1 Þ ¼ βw0 ðk1 Þcovðez ; u0 ðd1;z ÞÞ o 0:

0

ð13Þ

In Eq. (13) the strict inequality holds, because the labor endowment shock and the old period consumption is positively correlated, and the marginal utility, u0 ðÞ, is a decreasing function of consumption. Thus, as long as the new capital level, x, is less than k1, but is sufficiently close to k1, we have vðxÞ 4 vðk1 Þ. This implies that the new allocation improves the welfare of generation 0. As we have already shown above, the welfare of other generations are unaffected by the allocation change. Therefore, the new allocation is welfare improving in the Pareto sense. Our conclusion regarding the desirability of capital reduction is closely related to the result obtained by Dávila et al. (2012), in their investigation of a two-period model with idiosyncratic shocks. The main difference between our OLG model and theirs is that, in Dávila et al. (2012), new agents do not enter the economy in the second period, and thus, they do not consider the negative effect of a capital reduction caused by one generation on the next. In our set-up, when we decrease the capital holdings of one generation, their utility increases, however, the utility of the next generation is reduced. Therefore, a Pareto-improvement cannot be obtained through capital reductions alone. In our model, the birth of young agents in period 1 reduces the negative welfare effect of capital reduction on the old agents. Capital reduction in period 1 or equivalently saving reduction in period 0 increases consumption in period 0, but it lowers the wage income in period 1. In the two-period model, the wage reduction in period 1 affects only one generation who is born in period 0. In the OLG model, both the old agents and the young agents work in period 1. Therefore, the wage loss is shared by the two generations, and the negative effect on the old agents is weakened. Therefore, even if we perfectly compensate for the wage loss experienced by generation 1, we can still raise the welfare of generation 0. Of course, if n¼  1 and the new generation is not born in period 1, our model coincides with Dávila et al. (2012) and the income transfer from generation 0 to generation 1, m1 ðxÞ ¼ ð1 þnÞðw1  wðxÞÞl becomes zero. Our result is also related to that of Carvajal and Polemarchakis (2011), who investigate the welfare of agents in an OLG model. The main difference between the work of Carvajal and Polemarchakis (2011) and our owns is that unlike us, they focus on steady state welfare. When the analysis is restricted to steady states, a capital reduction affects the wage level both when an agent is young and when he is old. In page 20, they argue “… the golden rule may fail, and in an economy where the interest rate is higher than the growth of population, it may be that a Pareto improvement requires for every generation to save less.” We show that, regardless of whether the interest rate is higher than the population growth rate, we can always construct a combination of a small linear capital tax and lump-sum transfer that can raise the welfare of one generation without affecting that of other generations around zero tax equilibria.

R. Hiraguchi, A. Shibata / Journal of Economic Dynamics & Control 52 (2015) 258–269

263 1

Finally, we fix the new capital labor ratio in period 1, x1, such that vðx1 Þ 4 vðk1 Þ and let A^ ¼ fk^ t ; c^ t ; ðd^ t;z ÞZz ¼ 1 gt ¼ 0 denote the new allocation. This is given by 1

fk^ t gt ¼ 0 ¼ fk0 ; x1 ; k2 ; k3 ; …g;

ð14Þ

1

fc^ t gt ¼ 0 ¼ fc0 ðx1 Þ; c1 ; c2 ; c3 ; …g;

ð15Þ

1

fd^ t;z gt ¼ 0 ¼ fd0;z ; d1;z ðx1 Þ; d2;z ; d3;z …g:

ð16Þ

3.3. Implementation with capital taxation ^ can be implemented as a competitive equilibrium using a linear In this section, we show that the new allocation, A, y o 1 capital tax. Let us consider the government policy, G^ ¼ fτ^ t ; T^ t ; T^ t gt ¼ 0 , such that ! 8 > u0 ðc^ 0 Þ m1 > y o ^ <   1 if t ¼ 1; ; T m1 þ τ^ 1 R 1 x1 ; T1 þ y o 1þn 1 ð17Þ τ^ t ; T^ t ; T^ t ¼ R^ 1 βE½u0 ðd^ 1;z Þ > > : ð0; T y ; T o Þ if t a1; t

t

where m1 ¼ m1 ðx1 Þ ¼ ð1 þnÞðw1  wðx1 ÞÞl denotes the income transfer from the old to the young, and R^ 1 ¼ Rðx1 Þ. The policy G^ y o differs from G only when t¼1. It satisfies the government budget constraint because ð1 þ nÞT^ 1 þ T^ 1  τ^ 1 R^ 1 x1 ¼ y o ð1 þ nÞT 1 þT 1 ¼ 0. To show that policy G^ implements A^ as a competitive equilibrium, we must first confirm that, the state z consumption bundle of agents in generation t, ðc^ t ; d^ t þ 1;z Þ, is on their budget constraint and satisfies the Euler equation, if 1 the sequence of capital labor ratio is fk^ t gt ¼ 0 . We must also verify that the level of savings of individuals in generation t ^ coincides with k t þ 1 for all t. As the policy change only affects generations 0 and 1, in the following analysis, we focus on their utility maximization problems. We first investigate the problem of generation 0 when the capital labor ratio in period 1 is x1. If an agent in o generation 0 has a level of savings equal to s, his state z consumption bundle is ðc^ 0 þk1  s; wðx1 Þez þ ð1  τ^ 1 ÞR^ 1 s þ T^ 1 Þ. If s ¼ x1 , ^ ^ one can easily check that the bundle coincides with ðc 0 ; d 1;z Þ and then it is budget-feasible. Moreover, Eq. (17) implies that it satisfies the consumption Euler equation. Therefore, generation 0 optimally chooses the state z consumption bundle ðc^ 0 ; d^ 1;z Þ and his optimal level of savings is x1. Second, we study generation 1. His state z consumption bundle when his level of y savings is s is written as ðwðxÞl  s þ T^ 1 ; w2 ez þR2 s þT o2 Þ. The bundle coincides with ðc^ 1 ; d^ 2;z Þ ¼ ðc1 ; d2;z Þ when s¼k2. Moreover, as there is no capital tax in period 2, the first order condition on consumption coincides with Eq. (11). Therefore, generation 1 optimally chooses ðc^ 1 ; d^ 2;z Þ and the optimal level of savings is equal to k2. From these arguments, we obtain the following proposition. Proposition 1. Any equilibrium allocation with no capital income taxation can be Pareto-improved by a linear capital tax with a small enough tax rate. Eq. (17) shows that in the new competitive equilibrium allocation, the capital tax rate in period 1 is non-zero in general. However, the sign of the tax rate is uncertain. We now assume with regard to the per-capita production function, f. 0

″

Assumption 1. The per-capita production function, f(k), satisfies f ðkÞ þ f ðkÞk 4 0. α

For example, the Cobb–Douglas function, f ðkÞ ¼ Ak , with A 40 and α o 1 always satisfies Assumption 1, as ″ α1 f ðkÞ þ f ðkÞk ¼ α2 Ak 4 0. We have the following proposition regarding the sign of the linear capital tax rate. 0

Proposition 2. Under Assumption 1, an introduction of a small enough positive capital income tax Pareto improves the competitive equilibrium allocation. Proof. See the Appendix.

□

3.4. Government bonds As many authors including Diamond (1965), Ihori (1978), and Tirole (1985), have shown, government bonds can also reduce the level of capital. In this sense, government bonds may have an effect similar to that of the linear capital tax. However, government bonds cannot be a substitute for a linear capital tax. To show this, let us consider a zero capital tax 1 1 policy, f0; T yt ; T ot gt ¼ 0 , together with a debt policy, fbt gt ¼ 0 , where bt denotes the per-capita bond holding of generation t 1. 1 Z Given the two policies, the sequence fkt ; ct ; ðdt;z Þz ¼ 1 gt ¼ 0 is a competitive equilibrium allocation if and only if it satisfies (i) the budget constraint of generation t, ct ¼ wt l  kt þ 1 bt þ 1 þ T yt ; dt þ 1;z ¼ wt þ 1 ez þ Rt þ 1 ðkt þ 1 þ bt þ 1 Þ þ T otþ 1 ;

264

R. Hiraguchi, A. Shibata / Journal of Economic Dynamics & Control 52 (2015) 258–269

(ii) the Euler equation, u0 ðct Þ ¼ Rt þ 1 βE½u0 ðdt þ 1;z Þ;

ð18Þ

and (iii) the government's budget constraint, bt þ 1 ¼ Rt bt þT yt þT ot ; where wt ¼ wðkt Þ and Rt ¼ Rðkt Þ. bo 1 For such an allocation, we can easily see that a zero capital tax policy without government bonds, f0; T by t ; T t gt ¼ 0 ¼ 1 f0; T yt  bt þ 1 ; T ot þRt bt gt ¼ 0 , implements the same allocation as a competitive equilibrium. This implies that the public debt can play a similar role to lump-sum transfers,7 but it cannot be a substitute for a linear capital tax. As we see in Eq. (17), a welfare improvement requires a distortion to the Euler equation; however, public debt cannot generate an intertemporal wedge, as Eq. (18) shows. Therefore, for any competitive equilibrium allocation that involves government bonds but not capital taxation, we can Pareto-improve the allocation by introducing a linear capital tax. 3.5. Productivity shock on young agents So far we have assumed that only the old agents are subject to the productivity shocks. In this section, we consider a case where earnings are uncertain at young age. The labor supply of the old, e, is constant, but that of the young, l, is stochastic q and takes a value l 40 with probability π q 4 0 (q ¼ 1; 2; …; Q ). To simplify our analysis, here we do not consider the P q population growth. The total amount of labor is Lt ¼ ðe þ lÞNt  1 , where l ¼ E½l ¼ Qq ¼ 1 π q l . As in the previous section, we normalize the units of labor so that l þe ¼ 1. Generation t who gets a labor endowment shock lq when he is young maximizes his utility U t;q ¼ uðct;q Þ þ βuðdt þ 1;q Þ, where ct;q (dt þ 1;q ) is his consumption when he is young (old). We let st;q denote his level of savings. The budget constraints are q

ct;q ¼ wt l  st;q þ T yt ; dt þ 1;q ¼ wt þ 1 e þ ð1  τt þ 1 ÞRt þ 1 st;q þT otþ 1 : 1

1

, and a government policy, fτt ; T yt ; T ot gt ¼ 0 , that satisfy A competitive equilibrium consists of an allocation, fkt ; ðct;q ; dt;q ÞQq ¼ 1 g t¼0 0 0 the Euler equation u ðct;q Þ ¼ ð1  τt þ 1 ÞRt þ 1 βu ðdt þ 1;q Þ, the government budget constraint (5), and the budget constraints of the consumers. In a competitive equilibrium, the aggregate level of savings is equal to capital holdings and then kt ¼ E½st;q . We now show that for any equilibrium allocation without capital taxation there exists another equilibrium allocation 1 with a capital tax that Pareto dominates the original allocation. Take an equilibrium allocation fkt ; ðct;q ; dt;q ÞQq ¼ 1 g under t¼0 y o 1 zero capital tax policy f0; T t ; T t gt ¼ 0 . We impose capital income tax τ in period 1 and re-distribute the capital tax revenue to the young individuals as a lump-sum transfer. Suppose that the capital level changes from k1 to x. The wage income loss of q generation 1 in state q is ðw1 wðxÞÞl with w1 ¼ wðk1 Þ and then the average income loss is ðw1 wðxÞÞl. As in the previous section, we additionally tax generation 0 by ðw1  wðxÞÞl units to compensate for the wage loss. His utility is U q ¼ uðw0 l  s1;q þ T y0 Þ þ βuðwðxÞ þð1  τÞRðxÞs1;q þ τRðxÞx þ T o1  w1 lÞ: P The capital labor ratio and the individual savings satisfies x ¼ Qq ¼ 1 π q s1;q . The Euler equation is q

u0 ðw0 l s1;q þ T y0 Þ ¼ βð1  τÞRðxÞu0 ðwðxÞ þð1  τÞRðxÞs1;q þ τRðxÞx þ T o1  w1 lÞ q

ð19Þ

Differentiating the utility Uq by the capital tax τ and evaluating it when τ ¼ 0 yields       1dU q  dx ¼ fw0 ðk1 Þ þ R0 ðk1 Þs1;q g   R1 s1;q þ R1 k1 u0 d1;q :  dτ τ ¼ 0 β dτ τ ¼ 0 P where R1 ¼ Rðk1 Þ. As w0 ðxÞ ¼ R0 ðxÞx, the welfare of generation 1, U ¼ Qq ¼ 1 π q U 1;q satisfies       dU dR cov s1;q ; u0 d1;q ; ¼  β R1   dτ dτ τ ¼ 0 Clearly the covariance term covðs1;q ; u0 ðd1;q ÞÞ is negative. In general, it is not easy to determine the sign of the term R1  dR=dττ ¼ 0 . The next proposition shows that the term is positive, and then the capital tax is welfare-improving when uðcÞ ¼ ln c. Proposition 3. Suppose that the young agents are subject to the productivity shock. An introduction of a small positive capital income tax Pareto improves the economy when the utility function is logarithmic. Proof. See the Appendix.

□

The agent who receives a bad earnings shock (say low-type agent) saves less than the one who got a good earning shock (say high-type agent). This implies that the low-type pays less capital income tax than the high-type. The linear capital income tax works as a redistribution from the high-type agent to low-type agent. Although the capital income tax is 7

Ihori (1978) shows this point in the standard OLG model.

R. Hiraguchi, A. Shibata / Journal of Economic Dynamics & Control 52 (2015) 258–269

265

distortionary, the first order effect is small around the zero capital tax equilibrium. Thus the capital income tax is Paretoimproving as long as it is small. Let k^ 1 ð o k1 Þ denote a new capital-labor ratio which is welfare-improving. One can easily check that the optimization problem of generation 1 is unchanged if he is subject to the linear wage subsidy τw ¼ ðwðk1 Þ=wðk^ 1 ÞÞ  1 when he is young. We do not get analytical results on the optimal capital tax when both the young agents and old agents are subject to the idiosyncratic shocks. In the next section, we obtain the optimal capital tax rate in a numerical example where earnings are uncertain at both ages. 3.6. Numerical analysis We now turn to a numerical analysis. We follow Song et al. (2012) and set the length of a period to 30 years. We assume that capital share α is 0.33, the annualized capital depreciation rate is 0.08, the annualized population growth rate is 1%, and that the period utility function is logarithmic. We set the discount factor β ¼0.802 to match the annualized capital output ratio of 3. As in Dávila et al. (2012), we investigate the unemployment economy where the productivity shock shows unemployment risk. We assume that both young and old agents are subject to the unemployment risk. The shock takes two values e1 and e2 , (0 oe1 o e2 ) with a very low value of unemployment. Dávila et al. (2012) assume that the relative skill level e2 =e1 is 100 and the unemployment probability (i.e., π1) is 5%. Dávila et al. (2012) assume that the relative skill level e2 =e1 is 100 and the unemployment probability (i.e., π1) is 5%. Here we assume that the unemployment probability is 0.73% and relative skill level is 20.0 so that the coefficient of variation of the labor supply is equal to the one in Dávila et al. (2012). Suppose that the economy is originally on the steady state without any transfer and capital tax. We then use the capital n n n n n tax and lump-sum transfer in period 1 to raise the welfare of generation 0. Let An ¼ fk ; cn1 ; cn2 ; d11 ; d12 ; d21 ; d22 g denote the n n steady state allocation, where cz is consumption of the young agent in state z (z¼1,2) and dzz0 is consumption of the old 1 agent who is in state z when he is young and in state z0 when he is old. Also let A^ ¼ fk^ t ; c^ t1 ; c^ t2 ; d^ t11 ; d^ t12 ; d^ t21 ; d^ t22 gt ¼ 0 denote the equilibrium allocation with capital income tax, where c^ tz is consumption of the young agent in state z at time t and d^ tzz0 is consumption of the old agent at time t who is in state z when he is young and is in state z0 when he is old. We measure the welfare gain from having positive capital income tax by consumption equivalent λ, which is defined by n Ez;z0 ½uðð1 þ λÞcnz Þ þ βuðð1 þ λÞdzz0 Þ ¼ Ez;z0 ½uðc^ 0z Þ þ βuðd^ 1zz0 Þ:

Fig. 1 represents the welfare gain λ as a function of the annualized capital income tax rate. It shows that the annualized optimal capital income tax rate is 0.23%, and the welfare gain is 0.05%. The tax rate and the welfare gain are small. 4. Welfare improvement of all generations 1

In Section 3, we showed that for any equilibrium allocation A ¼ fkt ; ct ; ðdt;z ÞZz ¼ 1 gt ¼ 0 under the zero capital tax policy 1 1 , there exists another equilibrium allocation A^ ¼ fk^ t ; c^ t ; ðd^ t;z ÞZ g with k^ 1 ¼ x1 in which the welfare of f0; T y ; T o g t

z¼1 t¼0

t t¼0

generation 0 improves and the ones of other generations are unchanged. In this section, we show that by using the capital n

n

1

income tax and lump-sum transfer for every period, we can construct an equilibrium allocation An ¼ fkt ; cnt ; ðdt;z ÞZz ¼ 1 gt ¼ 0 , in which all individuals are welfare-improved. Here we assume that only the old agents are subject to the productivity shocks. ^ we consider the following step which raises the welfare of the initial old: First, given A,

Step 0: In period 0, transfer income from generation 0 to the initial old by ð1 þ nÞϵ units per each young individual by reducing his consumption, where ϵ 4 0. If ϵ is sufficiently small, welfare of generation 0 is still higher than the one under the original allocation A. Fix such ϵ and 1 1 1 let A1 ¼ fkt ; c1t ; ðdt;z ÞZz ¼ 1 gt ¼ 0 denote a new allocation after Step 0. For both the initial olds and the generation 0, A1 is better

Welfare gain (%)

0.06

0.04

0.02

0

0

0.1

0.2

0.3

Capital income tax rate (%)

Fig. 1. Welfare improving capital tax.

0.4

266

R. Hiraguchi, A. Shibata / Journal of Economic Dynamics & Control 52 (2015) 258–269

than the original allocation A, but these two allocations are indifferent for the other agents. The allocation A1 is written as 8 > < ðk0 ; c0 ð1 þ nÞϵ þk1 x1 ; d0;z þ ϵÞ if t ¼ 0; 1 1 1 if t ¼ 1 ð20Þ ðkt ; ct ; dt;z Þ ¼ ðx1 ; c1 ; d1;z ðx1 ÞÞ > : ðk ; c ; d Þ if t 4i t

t

t;z

Note that the level of generation 0's savings, x1 is already defined in Section 3. The allocation A1 can no longer be ^ decentralized by the policy G. We repeat similar reallocations to raise the welfare of every generation. First of all, given the levels of x1 ; x2 ; …, and xi for 1 i i some integer i Z1, let the allocation Ai ¼ fkt ; cit ; ðdt;z ÞZz ¼ 1 gt ¼ 0 be such that 8 ðk0 ; c0  ð1 þ nÞϵ þ k1  x1 ; d0;z þ ϵÞ if t ¼ 0; > > > > < ðxt ; ct þ kt þ 1 xt þ 1 ; dt;z ðxt ÞÞ if 1 rt r i 1; i i ðkt ; cit ; dt;z Þ ¼ ; c ; d ðx ÞÞ if t ¼ i ðx > i i i;z i > > > : ðkt ; ct ; dt;z Þ if t 4 i where dt;z ðxÞ ¼ wðxÞez þ RðxÞx  T ot  ðwt  wðxÞÞð1 þ nÞl and wt is the wage rate under the original allocation A. Since the value of x1 is fixed, clearly A1 has already been determined. Suppose that the allocation Ai is determined for some integer i. Then consider the following reallocation on Ai that improves welfare of generation i without changing the welfare of other individuals: Step i: Find xi þ 1 so that the following reallocations raise generation i's welfare: (i) In period i, set the level of savings of generation i (young) to xi þ 1 4 0. (ii) In period iþ1, transfer income from generation i (old) to generation iþ1 (young) by mi þ 1 ¼ ð1 þ nÞðwi þ 1  wðxi þ 1 ÞÞl 40. We have already shown in the previous section that such xi þ 1 always exists and that Step i does not affect the utility of 1 iþ1 iþ1 generation who is born after period iþ1. After Step i, the allocation Ai þ 1 ¼ fkt ; cit þ 1 ; ðdt;z ÞZz ¼ 1 gt ¼ 0 is determined. 1 n n Z n n Continuing Step i for every integer i, we get the allocation A ¼ fkt ; ct ; ðdt;z Þz ¼ 1 gt ¼ 0 such that ( ðk0 ; c0 ð1 þ nÞϵ þ k1 x1 ; d0;z þ ϵÞ if t ¼ 0; n n ðkt ; cnt ; dt;z Þ ¼ ð21Þ if t Z1: ðxt ; ct þ kt þ 1 xt þ 1 ; dt;z ðxt ÞÞ Since Step i improves the welfare of generation i while the one of the other generations is unchanged, the welfare of every generation is higher at An than at A. The next proposition shows that new allocation can be decentralized. 1

Proposition 4. The government policy fτnt ; T tny ; T tno gt ¼ 0 such that 8 y o > > ð0; T 0  ð1 þnÞϵ; ϵ þT 0 Þ !  n ny no  < u0 ðcnt Þ τt ; T t ; T t ¼ y n n n o mt > ; þ T ; τ R k þT m 1  t t t t t t > n : Rnt þ 1 β E½u0 ðdt þ 1;z Þ 1 þ n n

n

if t ¼ 0; ð22Þ

if t Z1; n

n

1

where mt ¼ ð1 þnÞðwt  wnt Þl, wnt ¼ wðkt Þ and Rnt ¼ Rðkt Þ, implements the allocation An ¼ fkt ; cnt ; ðdt;z ÞZz ¼ 1 gt ¼ 0 as a competitive equilibrium allocation. Thus it raises the equilibrium welfare of every generation. Proof. See the Appendix.

□

5. Endogenous labor supply In this section, we endogenize the labor supply decision to show that capital income tax still Pareto-improves the economy. To simplify the model, here we assume that there is no population growth and Nt ¼ 1 for all t. Generation t chooses their level of savings and labor supply and maximizes their expected utility h  n i  t þ 1;z : U ¼ uðct Þ  g ðlt Þ þ βE u dt þ 1;z  g z e Here g is the disutility from labor, lt is his labor supply when he is young and nt þ 1;z is his effective labor supply when he is old and his skill shock is ez. The function g satisfies gð0Þ ¼ g 0 ð0Þ ¼ 0, and g 0 ðnÞ 4 0 and g ″ ðnÞ 40 for n 4 0. The budget constraints are ct ¼ wt lt st þ T yt ; dt þ 1;z ¼ wt þ 1 nt þ 1;z þð1  τt þ 1 ÞRt þ 1 st þ T otþ 1 : The initial old chooses his labor supply and maximizes his utility uðd0;z Þ  gðn0;z =ez Þ subject to the budget constraint 1 d0;z ¼ w0 n0;z þ ð1  τ0 ÞR0 K 0 þ T o0 . Here the competitive equilibrium allocation is denoted as A ¼ fkt ; ct ; lt ; ðdt;z ; nt;z ÞZz ¼ 1 gt ¼ 0 . As in the previous section, kt ¼ K t =Lt denotes the capital-labor ratio, but the total labor is now Lt ¼ lt þ E½nt;z .

R. Hiraguchi, A. Shibata / Journal of Economic Dynamics & Control 52 (2015) 258–269 n

n

n

267

1

Let us pick any competitive equilibrium allocation A ¼ fkt ; cnt ; lt ; ðdt;z ; nnt;z ÞZz ¼ 1 gt ¼ 0 under the zero capital tax policy 1 f0; T yt ; T ot gt ¼ 0 . The first order conditions in period t is uðcnt Þ ¼ βRðkt þ 1 ÞE½u0 ðdt þ 1;z Þ

ð23Þ

 n n wðkt Þu0 ðcnt Þ ¼ g 0 lt

ð24Þ

n

n

n

n

ez wðkt þ 1 Þu0 ðdt þ 1;z Þ ¼ g 0 ðnnt þ 1;z =ez Þ

ð25Þ

We now lower the capital-labor ratio in period 1 from k1 to x, compensate for the income loss of generation 1, δðxÞ ¼ n n fwðk1 Þ wðxÞgl1 by taxing generation 0, and choose n1;z so that the first order condition on labor supply of the old agent, (25) n continues to hold. We do not change the labor supply of the young individuals lt . The consumption and labor supply of generation 0 when they are old, ðd1;z ; n1;z Þ depends on x, and when we write the allocation as ðd1;z ðxÞ; n1;z ðxÞÞ, we have n

d1;z ðxÞ ¼ wðxÞn1;z ðxÞ þ RðxÞK 1 ðxÞ þT o1  δðxÞ;   for all z; ez wðxÞu0 ðd1;z ðxÞÞ ¼ g 0 n1;z ðxÞ=ez n

n

n

where K 1 ðxÞ ¼ xfl1 þE½n1;z ðxÞg is the aggregate capital in period 1. Their consumption when they are young, c0 ðxÞ ¼ wðk0 Þl0  n K 1 ðxÞ þ T o0 also depends on x. Thus the welfare of generation 0, vðxÞ ¼ uðc0 ðxÞÞ gðl0 Þ þ βE½uðd1;z ðxÞÞ gðn1;z ðxÞ=ez Þ is also a n function of x. We differentiate v(x) and evaluate it when x ¼ k1 to obtain v0 ðk1 Þ ¼ β w0 ðk1 Þcov½nn1;z ; u0 ðd1;z Þ o0 n

n

n

ð26Þ

Therefore when we lower the capital labor ratio, we can improve the welfare of generation 0 without affecting generation 1. n n Now fix a capital-labor ratio k^ 1 o k1 such that vðk^ 1 Þ 4 vðk1 Þ, and let ðc^ 0 ; d^ 1;z ; n^ 1;z Þ ¼ ðc0 ðk^ 1 Þ; d1;z ðk^ 1 Þ; n1;z ðk^ 1 ÞÞ denote the new allocation. Then consider the capital income tax on generation 0, τ1 and the labor income tax on generation 0, τw 0 , and the w one on generation 1, τ1 that satisfy the following: u0 ðc^ 0 Þ ¼ ð1  τ1 ÞβRðk^ 1 ÞE½u0 ðd^ 1;z Þ;   n 0 ^ 0 n ð1  τw 0 Þwðk0 Þu ðc 0 Þ ¼ g l0 ;   0 n 0 n ^ ð1  τw 1 Þwðk 1 Þu ðc1 Þ ¼ g l1 : We obtain the following proposition. Proposition 5. When the labor supply decision is endogenous, a combination of the capital income tax, the distortionary labor income tax on the young individuals and a lump-sum transfer can Pareto-improve the economy as long as the capital tax is small. □

Proof. See the Appendix.

Therefore the introduction of the labor-leisure choice does not change the desirability of the capital income tax, but we also need a linear labor income tax on the young. 6. Age-dependent tax Erosa and Gervais (2002) and Garriga (2003) investigate the OLG models with elastic labor supply and show that a tax on capital can mimic an age-dependent tax on labor. In Section 5, we show that a combination of the age-dependent income tax and the capital income tax is Pareto-improving when the labor supply is elastic. However, at this point we do not know whether the age-dependent income tax can solely Pareto-improves the economy. When the labor supply is inelastic as we assume in Section 2, labor income tax cannot generate a distortionary effect. Thus the age-dependent income tax cannot be a substitute for the capital 1 income tax. However, age-dependent consumption tax is useful. Here we show that the allocation A^ ¼ fk^ t ; c^ t ; ðd^ t;z ÞZz ¼ 1 gt ¼ 0 which 1 Z is defined in Eqs. (14)–(16), and Pareto-dominates the original allocation A ¼ fkt ; ct ; ðdt;z Þz ¼ 1 gt ¼ 0 can be implemented as a competitive equilibrium if we use a linear consumption tax on young. The budget constraint becomes ð1 þ τct Þct ¼ wt l st þ T yt ; dt þ 1;z ¼ wt þ 1 ez þRt þ 1 st þT otþ 1 ; where τt is the consumption tax on the young agents in period t. The Euler equation in period 0 is u0 ðc0 Þ ¼ ð1 þ τc0 ÞR1 βE½u0 ðd1;z Þ. y o 1 It is easy to see that the allocation A^ can be decentralized by the following policy G~ ¼ fτ~ ct ; T~ t ; T~ t gt ¼ 0 : 8 ! > u0 ðc^ 0 Þ > y c o > 1 ^ ~ c þ τ ; T ; T if t ¼ 0; > 0 0 1 > 0 > > R^ 1 β E½u0 ðd^ 1;z Þ <   y o

ð27Þ τ~ ct ; T~ t ; T~ t ¼ m1 y o > ; T þ m if t ¼ 1; 0; T > 1 1 1 > 1 þn > > > > : ð0; T y ; T o Þ if t 4 1; c

t

t

268

R. Hiraguchi, A. Shibata / Journal of Economic Dynamics & Control 52 (2015) 258–269

The agents live for two periods and then their savings are proportional to the consumption when they are old. Thus capital income tax is very close to the tax on the old consumption. 7. Conclusion In this paper, we investigate an overlapping generations model in which agents receive idiosyncratic productivity shocks when they are old. We show that, around zero tax equilibria, we can always find a combination of a small capital tax and a lump-sum transfer that is Pareto-improving. As shown by previous studies, capital reduction in a period raises the welfare of agents who are old in that period. However, it lowers the welfare of young agents because it reduces their wage income. We show that the government could compensate for the wage loss of the young agents by additionally taxing the old generation. As a future study, we would like to investigate whether the age-dependent income tax can mimic the capital income tax when the labor supply is elastic.

Acknowledgement We thank two anonymous referees for their careful readings and comments. We also thank SAET Conference participants for their comments. Appendix A. Proof of Proposition 2 Under Assumption 1, ðRðxÞxÞ0 40. Thus d1;z ðxÞ ¼ wðxÞez þ RðxÞx þðwðxÞ w1 Þlð1 þnÞ þ T o1 is an increasing function of x and we have d1;z 4 d^ 1;z ¼ d1;z ðx1 Þ. Moreover, we have c0 o c^ 0 and R1 o R^ 1 ¼ Rðx1 Þ. Therefore, the capital tax satisfies 1¼

u0 ðc0 Þ u0 ðc^ 0 Þ 4 ¼ 1  τ^ 1 ; 0 R1 βE½u ðd1;z Þ R^ 1 βE½u0 ðd^ 1;z Þ

and this implies τ^ 1 4 0.

□

Appendix B. Proof of Proposition 3 If uðxÞ ¼ ln x, the first order condition (19) is re-expressed as wðxÞ þð1  τÞRðxÞs1;q þ τRðxÞx þ T o1  w1 l ¼ βð1  τÞRðxÞðw0 l  s1;q þT y0 Þ: q

When we take the expectation of this equation, we obtain f ðxÞ þ T o1  w1 l ¼ βð1  τÞRðxÞðw0 l  x þ T y0 Þ:

ð28Þ

This equation is written as f ðxÞ þ T o1  w1 l RðxÞðw0 l þ T y0  xÞ

¼ β ð1  τ Þ

As the left-hand side of the equation is an increasing function of x, dx=dτ o 0. Differentiating Eq. (28) by τ and evaluating it when τ ¼0, we get    1 dx 1 þ β R1 dR ¼ R þ : 1 dτ w0 l  x þT y dτ τ ¼ 0 0

This is strictly negative since dx=dτ. Thus dU=dτ o0.

□

Appendix C. Proof of Proposition 4 1

First of all, the new tax policy fτnt ; T tny ; T tno gt ¼ 0 clearly satisfies the government budget constraint. Next, when the capitaln labor ratio in period t is kt and the individual level of savings is s, his state z consumption bundle ðct ðsÞ; dt þ 1;z ðsÞÞ is given by ct ðsÞ ¼ wnt l  s þ mt þ T yt ¼ wt þ 1 l s þ T yt ; dt þ 1;z ðsÞ ¼ wnt þ 1 ez þ ð1  τnt þ 1 ÞRnt þ 1 s þ τnt þ 1 Rnt þ 1 kt þ 1 þT otþ 1  ð1 þ nÞmt þ 1 : n

n

n

n

n

n

n

If s ¼ kt þ 1 , cðkt þ 1 Þ ¼ ct þ kt þ 1  kt þ 1 ¼ cnt and dðkt þ 1 Þ ¼ wnt þ 1 ez þ Rnt þ 1 kt þ 1 þ T otþ 1  ð1 þnÞmt þ 1 ¼ dt þ 1;z . Thus the state z n n consumption bundle ðcnt ; dt þ 1;z Þ is feasible. Finally, it satisfies the Euler equation u0 ðcnt Þ ¼ ð1  τnt þ 1 ÞRnt þ 1 βE½u0 ðdt þ 1;z Þ. n Therefore the generation t optimally chooses the level of savings kt þ 1 . □

R. Hiraguchi, A. Shibata / Journal of Economic Dynamics & Control 52 (2015) 258–269

269

Appendix D. Proof of Proposition 5 o n n Assume that the lump-sum transfer on generation 1 when he is young is T^ 0 ¼ T o0 þ τw 0 wðk0 Þl0 , ythe one when he is old is o n ^ ^ ^ ^ ¼ T 1  δðk 1 Þ þ τ1 Rðk 1 ÞK 1 ðk 1 Þ and the lump-sum transfer on generation 1 when he is young is T^ 1 ¼ T y1 þ δðk^ 1 Þ þ τw 1 wðk 1 Þn1 . The new budget constraints under the distortionary labor and capital income taxes are given by o T^ 1

n ^o c0 ¼ ð1  τw 0 Þwðk0 Þl0 s0 þ T 0 ; o d1;z ¼ wðk^ 1 Þn1;z þð1  τ1 ÞRðk^ 1 Þs0 þ T^ 1 ; y c1 ¼ ð1  τw Þwðk^ 1 Þl1  s1 þ T^ ;

1

1

n We can easily check that generation 0 optimally chooses c0 ¼ c^ 0 ; l0 ¼ l0 ; s0 ¼ K 1 ðk^ 1 Þ; n1;z ¼ n^ 1;z and d1;z ¼ d^ 1;z and that n generation 1 optimally chooses c1 ¼ cn1 , s1 ¼ K n2 and l1 ¼ l1 . Thus the new allocation can be sustained as a competitive equilibrium. □

References Aiyagari, S.R., 1995. Optimal capital income taxation with incomplete markets, borrowing constraints, and constant discounting. J. Polit. Econ. 103, 1158–1175. Atkeson, A., Chari, V.V., Kehoe, P., 1999. Taxing capital income: a bad idea. Fed. Reserv. Bank Minneap. Q. Rev. 23, 3–17. Carvajal, A., Polemarchakis, H., 2011. Idiosyncratic risk and financial policy. J. Econ. Theory 146, 1569–1597. Chamley, C., 1986. Optimal taxation of capital income in general equilibrium with infinite lives. Econometrica 54, 607–622. Chari, V.V., Kehoe, P., 1999. Optimal fiscal and monetary policy. In: Taylor, J., Woodford, M. (Eds.), Handbook of Macroeconomics, vol. 1C, Elsevier, Amsterdam. Conesa, J., Kitao, S., Krueger, D., 2009. Taxing capital? Not a bad idea after all!. Am. Econ. Rev. 99, 25–48. Dávila, J., 2012. The taxation of capital returns in overlapping generations models. J. Macroecon. 34, 441–453. Dávila, J., Hong, J.H., Krusell, P., Rios-Rull, J.V., 2012. Constrained efficiency in the neoclassical growth model with uninsurable idiosyncratic shocks. Econometrica 80, 2431–2467. Diamond, P., 1965. National debt in a neoclassical model. Am. Econ. Rev. 55, 1126–1150. Erosa, A., Gervais, M., 2002. Optimal taxation in life-cycle economies. J. Econ. Theory 105, 338–369. Fiorini, L., 2008. Overlapping generations and idiosyncratic risk: can prices reveal the best policy? J. Math. Econ. 44, 1312–1320. Garriga, C., 2003. Optimal fiscal policy in overlapping generations models, mimeo. Gottardi, P., Kajii, A., Nakajima, T., 2012. Constrained inefficiency and optimal taxation with uninsurable risks, KIER Discussion Paper 694, Kyoto University. Ihori, T., 1978. The golden rule and the role of government in a life cycle growth model. Am. Econ. Rev. 68, 389–396. Judd, K., 1985. Redistributive taxation in a simple perfect foresight model. J. Publ. Econ. 28, 59–83. Judd, K., 1987. The welfare cost of factor taxation in a perfect-foresight model. J. Polit. Econ. 95, 675–709. Song, Z., Storesletten, K., Zilibotti, F., 2012. Rotten parents and disciplined children: a politico-economic theory of public expenditure and debt. Econometrica 80, 2785–2803. Tirole, J., 1985. Asset bubbles and overlapping generations. Econometrica 53, 1499–1528.