Carlos da Costa gratefully acknowledges financial support from CNPq project
301047/2010-3. Marcelo Santos thanks.... All errors are our sole responsibility. 1
...
Age-dependent Taxation with Endogenous Life-cycle Path for Wages∗ Carlos E. da Costa
Marcelo Santos
FGV/EPGE
[email protected]
INSPER
[email protected]
VERY PRELIMINARY Abstract We use an overlapping generations model of life-cycle choices with human capital accumulation to measure the welfare gains from moving to an age-dependent labor income tax system. We compare our results to those in the literature that takes the wage path as exogenous, and investigate the welfare cost of disregarding the endogeneity of wage path when designing the optimal tax schedule. Keywords: Age-dependent taxes; Human Capital Accumulation J.E.L. codes: E6; H3; J2.
1
Introduction
The use of age-dependent income taxes has received a lot of attention from public finance economists lately. Erosa and Gervais (2002); Garriga (2003) have shown how capital income taxes in an overlapping generations economy depend on whether labor income taxes may be made a function of agents ages. More recently, work on optimal labor income taxation has highlighted the welfare impact of moving to a system of age-dependent taxes: e.g., Weinzierl (2011); Farhi and Werning (2010). A common characteristic of all these papers is the use of an exogenous stochastic process for individuals’ wages. Although it is recognized by the literature that the exogeneity of an individual’s productivity along his or her life-cycle is not an accurate description of how one’s ∗
Carlos da Costa gratefully acknowledges financial support from CNPq project 301047/2010-3. Marcelo Santos thanks.... All errors are our sole responsibility.
1
productivity evolves, it is usually argued that the simplification allowed by this assumption is more than worth its cost. Underlying this simplification is the view that it does not generate important quantitative departures from what is believed to be a better description of the evolution of productivity along the life-cycle. Referring to the use of an exogenous path for wages along the life-cycle, Weinzierl (2011), for example, argues that “The specific results of this paper therefore require that a substantial portion of variation of wages with age is inelastic to taxes. A few considerations suggest that this requirement’s effects on the paper’s results may be limited.” Despite Weinzierl’s (2011) claim, evidence from elsewhere in the literature suggests the opposite. Indeed, recent work by Kapiˇcka (2011) that endogenizes human capital formation yields policy prescriptions that are the exact opposite of what one finds in Weinzierl’s (2011): while Weinzierl (2011) prescribes taxes that increase with age, Kapiˇcka (2011) suggests that they should decrease. While the temporary substitutability versus long term complementarity that drives Kapiˇcka’s (2011) results is absent from Weinzierl’s (2011), the increased dispersion of skills due to idiosyncratic shocks that drives Weinzierl’s (2011) results is not present in Kapiˇcka’s (2011) work. The goal of this paper is to study age dependent taxes in a model where both human capital accumulation and idiosyncratic risks. To do so, we calibrate an overlapping generations economy where agents are heterogeneous at birth with respect to their skills which evolve due to a combination of exogenous shocks and endogenous human capital accumulation. We consider two different models of human capital accumulation: learning-by-doing and learning-or-doing. The latter is particularly important if one is to get a sense of which of the two results discussed in the previous paragraph are more relevant, because it is this model that creates the type of temporary substitutability and long run complementarity that underlies Kapiˇcka’s (2011) result. Because our setting differs in many aspects with respect to those in the literature we address, we evaluate the impact of endogenizing human capital formation, by calculating optimal taxes when make the productivity path exogenous. We also calculate the costs of not taking into account the endogeneity of wage path when one designs the optimal wage dependent tax schedule. Because the main reason for using age-dependent taxes is to exploit the changes in both average and cross-sectional variance along the life-cycle, by taking these aspects as exogenous to the model may potentially lead to significant deviations from optimality. 2
An important aspect of our calibration is how to deal with social security. The point being that social security introduces an element of age dependence in the labor consumption-wedge that must be taken into account when we derive optimal policies.
2 2.1
The Environment Demography
The economy is inhabited by a continuum of measure one of individuals who live for a finite, albeit random, number of periods. Uncertainty regarding the time of death is captured by the fact that everyone faces a probability ψt+1 of surviving to the age t + 1 conditional on being alive at age t, with ψT +1 = 0. In each time period, a new generation is born. The age profile of the population, denoted by {µt }tT=1 , is modeled by assuming that the fraction of agents at the age t in ψ the population is given by the following law of motion µt = (1+tg ) µt−1 and satisfies n
∑tT=1 µt = 1, where gn denotes the population growth rate. What is the associated fertility rate?
2.2
Households
Preferences Individuals derive utility from consumption, ct , and leisure, Lt . Preferences over random paths of (ct , Lt ) over the life cycle are represented by: " E
T
∑β
t =1
t −1
t
∏ ψk
!
# u(ct , Lt ) ,
(1)
k =1
where β is the subjective discount factor, and E is the expectation operator conditional on information at birth. In most of what follows temporary utility will be of the form of a standard CobbDouglas utility function: ρ ( c t 1− ρ L t )1− γ . (2) u(ct , Lt ) = 1−γ Individuals choose labor supply, consumption human capital formation and asset accumulation to maximize their objective (1) subject to budget constraints that we 3
shall explain next. Budget Constraint An individual who works l hours and has human capital h generates a total of lh exp(u + z) efficiency units of labor, which is paid at a rate wt . Earnings are, in this case, y = wlh exp(u + z). While u ∼ N (0, σu2 ) is realized at birth and retained throughout life, z evolves according to an AR(1) process given by zt = ϕz zt−1 + ε t with innovations ε t ∼ N (0, σε2 ). Term u aims at capturing the heterogeneity at birth (one’s most crucial lottery) whereas z is the relevant uncertainty from the perspective of each person’s choices. Parameter ϕz generates exogenous persistence in life-cycle earnings. Labor productivity shocks are independent across agents and, as a consequence, there is no uncertainty regarding the aggregate labor endowment even though there is uncertainty at the individual level. Gross earnings received at age t are taxed at a flat, age-dependent, rate τw,t . This age dependence will be key in our investigation. As it turns, however, τw,t is not the only form of age-dependent taxation: Social Security introduces wedges in the consumption-leisure margin that vary with distance from retirement. Following the Social Security legislation, contribution to social security for an individual who ears yt is τss min {wlt ht exp(u + zt ), ymax } , where ymax defines not only a contribution ceiling but also a benefit ceiling. After-tax labor income for an individual with productivity lh exp(u + z) who supplies labor lt is, therefore, yt = (1 − τw,t )wlt ht exp(u + zt ) − τss min {wlt ht exp(u + zt ), ymax } ,
(3)
when economy wages are w in the economy. Retirement In out model economy, retirement is mandatory. Thus, we assume that at age Tn individuals leave the labor force and start colecting social security benefits. Let b( x ) denote these benefits, where x is the average past earnings, which is calculated by taking into account individual earnings up to age Tn . We specify the
4
following law of motion for x: x t +1 =
xt (t − 1) + min {wlt ht exp(u + zt ), ymax } , t = 1, ..., Tn t
(4)
The function b( x ) is the benefit that agents are entitled to at full retirement age. It is a piecewise linear function, which is specified in accordance with the rules of the U.S. social security system: if x ≤ y1 θ1 x b( x ) = θ1 y1 + θ2 ( x − y1 ) if y1 < x ≤ y2 θ1 y1 + θ2 (y2 − y1 ) + θ3 ( x − y2 ) if y2 < x ≤ y max
(5)
where 0 ≤ θ3 < θ2 < θ1 and (y1 , y2 , y3 ) are the kinks of the benefit schedule. Thus, up to an average earnings level of y1 , retirees are entitled to θ1 x, so that θ1 corresponds to the retirement replacement rate in this case. If the average past earnings are greater than y1 but smaller than y2 , retirees will earn θ1 y1 + θ2 ( x − y1 ), and finally if the past earnings are greater than y2 but below y max, they will receive θ1 y1 + θ2 ( y2 − y1 ) + θ3 ( x − y2 ). Asset Accumulation Individuals can resort to self-insurance to protect themselves against the uncertainty on labor income. Indeed, besides choosing the amount of time to supply to the labor market, they can trade an asset subject to an exogenous lower bound on asset holdings. We assume that this asset, which is denoted by at , takes the form of capital, following Aiyagari (1994). Thus, savings may be precautionary and allow partial insurance against idiosyncratic shocks. Agents are not allowed to incur debt at any age, so that the amount of assets carried over from age t to t + 1 is such that at+1 ≥ 0. Furthermore, given that there is no altruistic bequest motive and death is certain at the age T + 1, agents who survive until age T consume all their available resources, that is, a T +1 = 0. Lifespan uncertainty implies that a fraction of the population leaves accidental bequests, which, for simplicity, are assumed to be distributed to all surviving individuals in a lump-sum fashion. Individuals in this economy also pay capital income tax, τk , and consumption tax τc . Given the considerations above, we can write the budget constraint facing an in-
5
dividual in our model economy as follows. at+1 = [1 + r (1 − τk )] at + (1 − It )yt + e + It b( x ) − (1 + τc )ct
(6)
where It is an indicator function equal to 1 if t ≥ Tn and 0 otherwise and e is the lump-sum transfers due to the accidental bequests. Human Capital Accumulation We adopt two alternative formulations for human capital accumulation. In the first case, leaning-by-doing, individuals accumulate human capital by working. Thus, the law of motion for human capital is given by: h0 = κ (lh)φ + (1 − δh )h
(7)
where δh is the human capital depreciation rate, (κ, φ) are parameters and the time constraint is Lt + lt = 1. In the second case, leaning-or-doing, an agent acquires human capital by spending time training in periods in which he is also working. Thus, the law of motion for human capital can be written as follows: h0 = κ (sh)φ + (1 − δh )h
(8)
with the time constraint given by Lt + lt + st = 1, where s denotes time spent on training. These two approaches are likely to have different implications on the design of the optimal tax policy. Under the leaning-by-doing framework, individuals have an incentive to supply labor less elastically earlier in their life. This is so because work not only generates income in the current period but also increases their future labor productivity. On the other hand, the learning-or-doing approach implies that time spent on training works as a substitute for labor and thus entails a larger elasticity for labor supply of the young. Recursive formulation of individuals’ problems Let Vt (ωt ) denote the value function of an individual aged t, where ωt = ( at , u, zt , ht , xt ) ∈ Ω is the individual state space. In addition, considering that agents die for sure at age T and that there is no altruistic link across generations, we have that VT +1 (ωT +1 ) = 0. Thus, the choice 6
problem of individuals aged t under the leaning-by-doing approach can be recursively represented as follows:1 Vt (ω ) = Max : u(c, 1 − l ) + βψt+1 Ez0 Vt+1 (ω 0 ) l,a0 ≥0
(9)
subject to (6), (7) and (4), where ω 0 = ( a0 , u, z0 , h0 , x 0 ). The same problem under the learning-or-doing approach is given by: Vt (ω ) = Max : u(c, 1 − l − s) + βψt+1 Ez0 Vt+1 (ω 0 ) l,s,a0 ≥0
(10)
subject to (6), (8) and (4), where ω 0 = ( a0 , u, z0 , h0 , x 0 ) It should be stressed that we have imposed non-negativity of assets. We have thus take an extreme (though plausible) position with regards to capital markets. Relaxing a little the assumption by allowing some exogenous limit is likely to have little effect on our conclusions.
2.3
Technology
Technology for producing the consumption good is given by a Cobb-Douglass with constant returns to scale, Y = BK α N 1−α , where K is aggregate capital, N is aggregate efficient units of labor and B is a scale parameter. The standing representative firm solves every period the problem max Kt ,Nt
n
BKtα Nt1−α
o
− wt Nt − rt Kt ,
where rt is the period t rental rate of capital and wt is wage. The first order conditions for the firm’s profit maximization problem are,
(1 − α) BKtα Nt−α = wt , 1 In
(11)
order to simplify the notation, I have suppressed the subscript for age from both the state and control variables.
7
and αBKtα−1 Nt−α = Kt .
2.4
(12)
Government
In our economy, the government runs a social security system, wherein pension benefits are financed through an exogenous tax Tss (y) = τss min {y, ymax }. The amount of benefit received by each retired agent depends on his or her individual average lifetime earnings through the concave, piecewise linear function (5). Additionally, the government levies proportional taxes on consumption, τc , labor income, τw , and capital income, τk , to finance an exogenous stream of expenditures, Gt t . The nature of our exercises will be to let labor income tax, τw,t , depend on age by adopt the following parameterization: τw,t = ξ 1 t + ξ 2
t2 100
(13)
where (ξ 1 , ξ 2 ) are parameters. The problem is that, as we vary τw according to 13 the Government’s budget set may be violated. We then allow τc to adjust to ensure that government budget constraint is satisfied in equilibrium. This introduces an important subtelty to interpret the results. The relevant wedge between consumption and leisure depends on both the consumption tax and the labor income taxes—where we must include the social security contribution as well.2 Hence, in interpreting the results we must take into account the consumption taxes to define the relevant wedge. Finally, we assume that the government collects the accidental bequests and transfers it to all agents in the economy on a lump-sum basis.
2.5
Recursive competitive equilibrium
At each point of time, agents are heterogeneous with respect to age t and to state ω = ( a, u, z, h, x ) ∈ Ω. The agents’ distribution at age t across states ω is described by 2 Calculating
the relevant wedge for the social security contribution is not a trivial task. The issue is that part of the contribution returns as a benefit in the future. Both the market imperfection, a ≥ 0, and uncertainty means that we must ’price’ these benefits using each individual’s dtate price deflator.
8
a measure of probability λt defined on subsets of the state space Ω. Let (Ω, z(Ω), λt ) be a space of probability, where z(Ω) is the Borel σ-algebra on Ω: for each η ⊂ z(Ω), λt (η ) denotes the fraction of agents aged t that are in η. The transition from age t to age t + 1 is governed by the transition function Qt (ω, η ), which depends on the decision rules and on the exogenous stochastic process for z. The function Qt (ω, η ) gives the probability of an agent at age t and state ω to transit to the set η at age t + 1. The definition of a recursive competitive equilibrium for the economy with the human capital accumulation based on learning-by-doing is as follows.3 Definition 1. Given the policy parameters, a recursive competitive equilibrium for this economy is a collection of value functions {Vt (ω )} , policy functions for individual asset holdings d a,t (ω ), for consumption dc,t (ω ), for labor supply dlw ,t (ω ), prices {w, r }, age dependent but time-invariant measures of agents λt (ω ), transfers e and a tax on consumption τc such that: (i) {d a,t (ω ), dlw ,t (ω ), dc,t (ω )} solve the dynamic problems in (9); (ii) The individual and aggregate behaviors are consistent, that is: T
K=
∑ µt
t =1 T
N=
∑ µt
t =1
ˆ d a,t (ω )dλt Ω
ˆ
dlw ,t (ω )ht (ω ) exp(u + zt )dλt Ω
(iii) {w, r } are such that they satisfy the optimum conditions (12) and (11); (iv) The final good market clears: T
∑ µt
t =1
ˆ
{dc,t (ω ) + [d a,t (ω ) − (1 − δ)d a,t−1 (ω )]}dλt = K α N 1−α Ω
(v) Given the decision rules, λt (ω ) satisfies the following law of motion: ˆ Qt (ω, η )dλt ∀η ⊂ z(Ω)
λ t +1 ( η ) = Ω
3 In the case of the learning-or-doing approach,
there is a small change in the definition in which we take into account the policy function for the time spent on human capital accumulation.
9
(vi) The distribution of accidental bequests is given by: T
e=
∑ µt
t =1
ˆ
(1 − ψt+1 )d a,t (ω )dλt Ω
(vii) Taxes are such that the government’s every period: T
τc C = G + SSB − τk rK −
∑ µt
t =1
ˆ τw,t dlw ,t (ω )ht (ω ) exp(u + zt )dλt , Ω
where SSB is the social security balance and C denotes the aggregate consumption. Before we move on to the calibration of our economy, a couple of things are worth mentioning. First, we kept τc C in the left hand side of the last equation in the definition above to draw attention to the fact that this is where we will adjust tax policy to guarantee budget balancedness in our exercises. Second, item (vii) is redundant if conditions (i)–(vi) hold.
3
Calibration
The population age profile {µt }tT=1 depends on the population growth rate gn , the survival probabilities ψt and the maximum age T that an agent can live. In this economy, a period corresponds to one year and an agent can live 75 years, so T = 75. Additionally, we assumed that an individual is born at age 16, so that the real maximum age is 90 years. Data on survival probability by age were extracted from Bell and Miller (2005) and are shown in Figure 1. Given the survival probabilities, the population growth rate is chosen so that the age distribution in the model replicates the dependency ratio observed in the data. By setting gn = 0.0105, the model generates a dependency ratio of 17.27%, which is close to the dependency ratio observed in the data for 2000. Values for preference parameters β, γ, and ρ are summarized in Table 2. The intertemporal discount rate, β, was set to 1. On a yearly basis, this value is consistent with a capital–output ratio of 3.0. The parameter ρ was chosen in such a way that the average fraction of time that individuals spend working is consistent with the empir-
10
ency ratio observed in the data for 2000.
Survival Probability 1
0.95
0.9
0.85
0.8
0.75
0.7 10
20
30
40
50 Age
60
70
80
90
Figure 1: Survival probability by age 2000
ical evidence — approximately 30%.4 For Cobb-Douglas preferences, the coefficient Figure 1: Survival probability by age - 2000 of relative risk aversion is given by 1 − ρ + ργ and the Frisch elasticity for leisure is 1−ρ+ργ given by . Given a ρ = 0.62 , the value reported in Table 1 entail a value of γ of the preference parameters (; ; ) are summarized in Table 2: The intertem2.24 for the coefficient of relative risk aversion and of 0.74 for the Frisch Elasticity unt rate, ; was to 1:On yearly thiswith value consistent withina Auerbach capitalñ and forset leisure. Theseavalues arebasis, consistent theisempirical evidence (1987); Rustchosen and Phelan (1997);aDomeij and Flodén (2006). fraction of o of 3.0. The Kotlikoff parameter was in such way that the average
ndividuals spend working is consistent with the empirical evidence, which suggests
approximately 30%.3 For Cobb-Douglas preferences, the coe¢cient of relative risk Table 1: Parameters
1+ given by 1 + and elasticity :σGiven 2 β the γ Frisch ρ α δ forκleisure φ is δgiven ϕby σε2 z h u , the value reported in Table 1 entail a value of 2:24 for the coe¢cient of relative
3.0 0.62 0.36 0.054 0.15 0.50 0.05values 0.93 are 0.016consistent 0.10 on and of 0:74 for the1.0Frisch Elasticity for leisure. These
mpirical evidence in Auerbach and Kotliko§ (1987), Rust and Phelan (1997) and
The values of technological parameters (α, δ) are also summarized in Table 1. We
d FlodÈn (2006). chose a value for α based on U.S. time series data from the National Income and
I/Y Product Accounts (NIPA). The depreciation rate, in turn, is obtained by δ = K/Y − g. nstance, Juster and Sta§ord (1991).
We set the investment-product ratio I/Y equal to 0.25 and the capital-product ratio K/Y equal to 3.0. The economic growth rate, g, is constant and consistent with the average growth rate of GDP 9 over the second half of the last century. Based on data from Penn-World Table, we set g equal to 2.7%, which yields a depreciation rate of 4 See,
for instance, Juster and Stafford (1991).
11
5.4%. The paramaters (κ, φ, δh )... TO BE DONE. The term exp(u + zt ) captures the stochastic component of the individual’s labor income, where the underline stochastic processes are characterized by the parameters ( ϕz , σε2 , σu2 ). Several authors have estimated similar stochastic processes for labor productivity. The values presented in Table 1 are consistent with the estimates of Kaplan (2011). In the United States the old-age benefit payable to the worker upon retirement at full retirement age is called the primary insurance amount (PIA). The PIA is derived from the worker’s annual taxable earnings, averaged over a period that encompasses most of the worker’s adult years. Until the late 1970s, the average monthly wage (AMW) was the earnings measure generally used. For workers first eligible for benefits after 1978, average indexed monthly earnings (AIME) have replaced the AMW as the usually applicable earnings measure. In our context, both AMW and AIME are given by (4). The complete parameterization of the benefits function requires the specification of values for the parameters {θ1 , θ2 , θ3 , y1 , y2 , y max}. The parameters (y1 , y2 ) correspond to the bend points applied in the formula of calculation of the PIA, whereas (θ1 , θ2 , θ3 ) determine the replacement rate applied in each one of the intervals defined by the bend points. We use the bend points applied to calculate the PIA for workers who were first eligible in 1979 or later according to Social Security Bulletin (2001). In 2000, the PIA equaled 90% of first $531 of AIME, 32% of next $2671 and 15% of AIME over $3202. We multiply these values by 12, adapting to the annual base of the model and then normalize the result dividing it by the average annual wage.5 Figure 2 plots the benefit function obtained for the benchmark economy. The horizontal axis corresponds to the average past earnings, x, and the vertical axis corresponds to the benefit. Note that we have normalized the average past earnings to the average labor income, ym . Thus, for example, if an individual has x equal to ym , his benefit will be 42% of that value. Remember that ymax corresponds to the level of earnings above which earnings in Social Security covered employment is neither taxable nor creditable for benefit computation purposes. In 2000, the maximum taxable annual was $76200. We, then, divided this value by the average annual wage to obtain ymax = 2.34ym . In addition, 5 According
to the Social Security Bulletin (2001), the average annual wage was $36,564 in 2000.
12
0.7
0.6
Benefits
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1 1.5 2 Multiples of average labor income
2.5
3
Figure 2: Benefits by multiples of average labor income
in 2000 American workers covered by the social security system contributed 10.6% of Figure 2: BeneÖts by multiples of average labor income their wages to Old-Age and Survivors Insurance (OASI). Thus, we set τss = 0.106. Finally, we specify the others parameters related to government activity. First, member that ymax corresponds the level ofG,earnings which earnings in under So- the we set governmenttoconsumption, to 18% ofabove the output of the economy curity covered employment is neither taxablea labor nor creditable computation baseline calibration. We assume income tax for ratebeneÖt of 14% and a capital income rate of 27%. The consumption is determined in such a waythis that value the governes. In 2000, thetax maximum taxable annual wastax $76200. We, then, divided ment budget balances in equilibrium, which implies a tax rate of nearly 9% in the average annual wage to obtain ymax = 2:34ym . In addition, in 2000 American workbenchmark economy.
ered by the social security system contributed 10.6% of their wages to Old-Age and
ors Insurance (OASI). Thus, we set ss = 0:106.
4
Results
ally, we specify the others parameters related to government activity. First, we set
Figure 3 displays hours and optimal taxes for the benchmark economy with exogenous wages and for the learning-by-doing economy. Hours worked peak earlier . We assume aand labor income taxvalue rate for of 14% and a capital income rate 27%:for the reach a higher the learning-by-doing model. tax Taxes areoflower reaches 46 years of age, and become higher nsumption tax learning-by-doing is determined inmodel such auntil wayone that the government budget balances in afterwards.a tax rate of nearly 9% in the benchmark economy. rium, which implies The learning-by-doing model does not display the pattern of temporary substiording to the Social Security Bulletin (2001), the average annual wage was $36,564 in 2000. tutability/long run complementarity that generates the results in Kapiˇcka (2011). In fact, learning-by-doing makes it stronger the case for age-increasing marginal labor 11 income taxes.
ment consumption, G, to 18% of the output of the economy under the baseline cali-
13
4 Results B) Optimum Taxation
A) Hour Worked
0.25
0.4 0.35
0.2 Labor income tax
Hours worked
0.3 0.25 0.2 0.15 Learning-by-doing Benchmark
0.1
0.15 Learning-by-doing Benchmark
0.1
0.05 0 15
20
25
30
35
40 Age
45
50
55
60
0.05 15
65
20
25
30
35
40 Age
45
50
55
60
65
Figure 3: Hours worked and optimum taxation - exogenous human capital versus learning-by-doing.
Figure 3: Hours worked and optimum taxation - exogenous human capital versus
5
Conclusion
learning-by-doing.
In this paper we use a simple overlapping generations model to measure the costs of 5 Conclusions disregarding the endogeneity in the wage profile along the business cycle when one calculates the optimal age-dependent income tax. To be done Since the current social security system implicitly defines age dependent labor income taxes, we consider various possibilities for how to change the system. We find that...
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