Optimal Time-Consistent Taxation with Overlapping Generations Steve Ambler∗ October 2002

∗

CIRPEE, UQAM, C.P. 8888, Succ. Centre-ville, Montr´eal, Qu´ebec, Canada H3C 3P8, phone (514) 987-3000 ext. 8372, fax (514) 987-8494, email [email protected]. I would like to thank the CRSH and the Fonds FCAR for generous financial support, the European University Institute for its generous financial support and hospitality during my sabbatical year (1999-2000), and Pascal St. Amour, Christian Zimmermann, seminar participants at the University of Montreal, York University and Laval University for helpful comments. The usual caveat applies.

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Optimal Time-Consistent Taxation with Overlapping Generations Abstract The paper analyzes optimal time-consistent taxation in an overlapping generations model with two-period lived households. The government chooses tax rates and borrowing to finance an exogenous stream of expenditures. It cannot commit to future policies, so announced policies that are not time consistent are not credible. Dynamic programming is used to derive Markov-perfect equilibria. In contrast to optimal fiscal policy in representative-agent models with commitment, optimal capital income tax rates are positive in the long run, and bounded below one in the short run for a wide range of parameter values. Key words: Optimal Fiscal Policy, Time Consistency, Overlapping Generations JEL classification codes: E61, E62, C63

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1

Introduction

There is an extensive literature on optimal second-best taxation in dynamic general equilibrium (DGE) models.1 Most researchers have focused on the representativeagent neoclassical growth model and solutions to the so-called Ramsey problem (1927), in which the government can precommit to its future policies. Representative-agent models of optimal taxation (RAMOT) lead to striking predictions. In deterministic models, the optimal tax rate on capital income converges to zero in the long run. In models of endogenous growth with human capital, the optimal tax rates on all forms of capital income converge to zero. Although lump-sum taxation is not permitted, the government can achieve the firstbest social optimum if it can undertake a capital levy, taxing capital income at a high enough rate to accumulate assets to finance its future expenditures: in order to make the optimal taxation problem non-trivial, arbitrary limits must be set on capital income tax rates at the beginning of the government’s optimal program. The government’s optimal plan is not time consistent: if allowed to re-optimize, it can achieve a first-best outcome (from that point onward) by undertaking a capital levy.2 Finally, in stochastic versions of the problem, the average tax rate on capital income converges to zero, but the tax rates on income from capital and financial 1

The possibility of lump-sum taxation is excluded so that the first-best social optimum cannot be attained. See Ambler (1999), Atkeson, Chari and Kehoe (1999), Chari, Christiano and Kehoe (1995), Chari and Kehoe (1998), and Judd (1999) for surveys of the literature. 2 For this reason, I would argue that explicit mechanisms such as trigger strategies to support the precommitment equilibrium are implausible, since they must lead to outcomes that are worse than the first-best outcome. Papers that examine the use of trigger strategies, such as Chari and Kehoe (1990) and Stokey (1991), do so in the context of models that preclude the possibility of attaining the first-best outcome.

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assets are extremely volatile: in response to unexpected shocks which affect the expected present value of the government’s tax revenues, it is optimal to adjust rates of taxation on assets which are inelastically supplied. The predictions of the RAMOT approach clearly diverge from observed behavior. Capital income tax rates are generally quite high in the data. This divergence has led some researchers to examine optimal taxation in environments richer than representative-agent models with government precommitment. For example, Klein and Rios-Rull (1999) study a representative-agent in which the government cannot precommit to future tax rates, and cannot borrow. They find that in such a setting, optimal capital income tax rates are considerably higher than in the data in the long run. This paper explores the implications of a model which combines two key departrues from the assumptions used by the RAMOT approach. Specifically, the paper analyzes optimal taxation without government precommitment in an overlapping generations model. I impose no restrictions on the government’s ability to borrow. Capital ownership is not uniform across cohorts: with life expectancies of two periods and no bequests, capital is held only by old agents. In this setting, taxing capital at very high rates is not optimal because of its effects on the intergenerational distribution of income. It is therefore not necessary to impose arbitrary restrictions on rates of capital income taxation in the very short run, even if capital is supplied inelastically. Since the government cannot precommit to future rates of taxation, capital income tax rates do not converge to zero in the long run: with precommitment, the government could achieve its distributional goals 4

by taxing (exclusively or primarily) each cohort’s labor income. The model is fairly simple, but its solution is not. I derive the optimal linear feedback rule for an approximation to the government’s problem, with a quadratic approximation to its value function and linearized laws of motion for the economy’s state variables. I use iterative techniques to derive the government’s feedback rule. At each iteration, the government’s feedback rule is updated as are private agents’ optimal linear feedback rules given the government’s current feedback rule.3 The paper is structured as follows. In the following section, I briefly review some of the relevant literature on optimal taxation in macroeconomic models. The model is presented in the third section. The fourth section describes in detail the government’s optimal policy problem. The model’s calibration and the numerical solution methodology are discussed in the fifth section. The sixth section outlines the results. The seventh section concludes.

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Relevant Literature

The classic result on the convergence of capital income tax rates to zero was first derived by Chamley (1985, 1986) and Judd (1985, 1987), using non-stochastic versions of the neoclassical growth model and extending the work of Arrow and Kurz (1970) and Ramsey (1928). Optimal taxation in stochastic models was stud3

I conjecture that one of the reasons that the use of Ramsey problems has been so popular in the literature is that they allow researchers to use the so-called primal approach (see Ambler, 1999, or Chari and Kehoe, 1998, for details), which simplifies the calculation of optimal policies. Using this approach, it is possible to solve directly for the optimal allocations in the economy. The tax rates and prices that support these allocations can be derived once these allocations are known.

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ied by Zhu (1992) and in a series of papers by Chari, Christiano and Kehoe (1991, 1994, 1995). Jones, Manuelli and Rossi (1993, 1997) analyzed optimal capital income taxation in endogenous growth models with both physical and human capital. There are at least three important threads in the literature which diverge from the basic assumptions Kydland and Prescott (1980), in their seminal paper on the time inconsistency of optimal policy, include an appendix in which they describe how to calculate a time consistent strategy using dynamic programming methods. Distinguish clearly between among threads in the literature. 1) Models with overlapping generations: Atkinson and Sandmo (1980), Park (1991), Erosa and Gervais (1998), Auerbach and Kotlikoff (1987), Scarth (1999). 2) Models of optimal policy without commitment: Kydland and Prescott (1979, 1980), Benhabib and Rustichini (1997), Benhabib, Rustichini and Velasco (1996), Krusell, Kurus¸c¸u and Smith (2000), Ambler and Paquet (1996), Lucas and Stokey (1983), Klein and Rios Rull (2000), Hansen, Epple and Roberds (1985), Cohen and Michel (1988), Fischer (1980), Sargent (1987, section 15.5), Oudiz and Sachs (1985), Miller and Salmon (1985), Calvo and Obstfeld (1988). 3) Models which try to show how to support the time-inconsistent equilibrium with trigger strategies. Note the importance of Calvo and Obstfeld (1988), which deals with overlapping generations. It allows for lump-sum taxation, and derives conditions under which social welfare functions are compatible with time consistency due to preferences. 6

Justify the lack of commitment assumption. Refer to Chari and Kehoe (1990), Stokey (1991). Models in which a long run tax rate of zero on capital income is not optimal in the long run have started to appear in the literature quite recently. Benhabib and Rustichini (1997) and Benhabib, Rustichini and Velasco (1996) study optimal taxation in representative agent models without commitment. They consider optimal government strategies which are robust at time t to any possible deviations from time t on, in the sense that any deviation worsens social welfare.4 They derive the result that, in order to create the necessary incentives for the government not to deviate from its optimal plan, capital income taxation rates must be strongly negative. Klein and Rios-Rull (1999) study a representative agent model in which the government must balance its budget each period without borrowing. Optimal capital income taxation rates are nonzero in the long run, and no additional restriction is necessary on initial capital income tax rates, since the government is prohibited at all times from building up arbitrarily large claims on the private sector. The stochastic properties of tax rates in their model are closer to the data than in similar models with unlimited government borrowing. In an important recent paper, Erosa and Gervais (1998) consider a model with overlapping generations of households with arbitrary, finite planning horizons. They show analytically that optimal fiscal policy with commitment involves a tax rate on capital income that does not generally converge to zero in the long 4

As in representative agent models of optimal taxation with commitment, they impose a limit on the rate of taxation of capital income at the start of the optimal program

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run, in contrast to most representative agent models. However, when preferences are additively separable in consumption and leisure, optimal capital income tax rates converge to zero after an initial period when they are positive. Atkeson, Chari and Kehoe (1999) demonstrate a similar result. In addition, in calibrated versions of their model, Erosa and Gervais show that capital income tax rates are quantitatively quite small in the long run (less than three percent). Overlapping generations models have also been used to examine the effects of exogenous (not demonstrably optimal) changes in fiscal policy on economic equilibrium and welfare. These models have included overlapping generations models in which agents have finite horizons (Auerbach and Kotlikoff, 1987; Ventura, 1996), and models which use the Yaari (1965) assumption of a constant probability of death or the Weil (1989) assumption of overlapping generations of agents with infinite horizons (James, 1994; Scarth, 1999).

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The Model

The economy consists of overlapping generations of two-period lived individuals. The total population of the economy is constant. The size of each cohort is normalized to equal one. There are no bequests, and individuals are born without assets. Agents work, consume, and save in the first period of their lives. They are retired in the last period of their lives, and use their accumulated savings to finance consumption. Perfectly competitive firms rent labor from young agents and capital from old agents and produce goods which are sold to the young and old and to the government. The government finances an exogenous stream of expen8

ditures by taxing labor and capital income and by borrowing. Agents have perfect foresight. However, the model is used to analyze once-and-for-all changes to the time path of the model’s main exogenous variable (the level of technology).

3.1

Households

An individual born at date t lives for two periods. He works, consumes and saves in the first period and consumes in the second period. He maximizes the following utility function: Ut =

1 γ 1 cy,t (1−σ) − nt (1+φ) + β co,t+1 (1−σ) , 1−σ 1+φ 1−σ

(1)

subject to the following flow budget constraint in the first period: cy,t = nt Wt (1 − τn,t ) − kt+1 − bt+1 ,

(2)

and the second-period flow budget constraint given by: co,t+1 = (1 + rt+1 )bt+1 + Rt+1 kt+1 (1 − τk,t+1 ).

(3)

The variable definitions are as follows: cy,t is consumption when young, nt denotes hours worked by young agents, co,t+1 is consumption when old, kt+1 is capital acquired when young, bt+1 denotes bonds acquired when young, Wt is the real wage rate, rt is the net real interest rate on bonds, Rt is the real rental rate on capital, τk,t is the tax rate on capital income, and τn,t is the tax rate on labor income. Old agents cannot borrow in equilibrium, since this would violate their lifetime budget constraints. Young agents are all born without assets and have homo9

geneous preferences, so in equilibrium the only bonds that they hold are government bonds. This is imposed on the model as an equilibrium condition. Capital owned by old agents depreciates completely, so that old agents die without assets.

3.2

Firms

A representative competitive firm rents labor from young agents and capital from old agents and produces output according to a standard aggregate production function given by yt = Zt nt α kt (1−α) ,

(4)

where Zt is the level of technical progress. Profit maximization leads to the following standard optimality conditions which determine the real wage rate and the rental rate of capital:

3.3

Wt = αZt nt (α−1) kt (1−α) ,

(5)

Rt = (1 − α)Zt nt α kt −α .

(6)

The Government

The government finances an exogenous (from its point of view) stream of spending by taxing capital and labor income and buy borrowing. Its expenditures are denoted by Gt . The government’s flow budget constraint is given by: Gt + (1 + rt )Bt = Wt nt τn,t + Rt kt τk,t + Bt+1 ,

(7)

where Bt is the stock of government debt (which consists of one-period bonds). The government does not have access to lump sum taxation. We abstract from 10

other forms of taxation such as consumption taxes. In a slightly different context, Judd (1999) shows that adding consumption taxes does not give the government an independent policy instrument. It would imply that several different combinations of tax rates could be used to achieve the same equilibrium. We ignore this possibility in the interests of simplicity. The government’s social welfare function is given by SWt =

∞ X i=0

(

δ i−1

cy,t−1+i (1−σ) γnt−i+1 (1+φ) βco,t+i (1−σ) − + 1−σ 1+φ 1−σ

)

(8)

The social welfare function is just the sum of the lifetime utilities of individual cohorts, discounted relative to their dates of birth using the social discount rate δ. Calvo and Obstfeld (1988) showed that the utility of individual cohorts must be discounted relative to the beginnings of their lives in order to avoid problems of time inconsistency arising solely from the specification of the government’s preferences as opposed to time inconsistency arising from constraints imposed on the financing of the government’s spending.5 In the calibration of the model I set the social discount rate δ equal to private agents’ discount rate, but in principle the two could be different. The terms in cy,t−1 and nt−1 cannot be affected by the government’s policies at time t. They act as a constant terms which shift the level of the social welfare function: omitting them would not change the government’s optimal policies, but they bring out more clearly the logic of discounting cohorts’ lifetime utilities relative to their dates of birth. 5

See Ambler and Desruelle (1991) for a discussion of how time inconsistency arises in dynamic games when agents share the same objectives but face different constraints.

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3.4

Maximization by the Young

In the absence of uncertainty, bonds and capital are perfect substitutes and must have the same after-tax rate of return. We can write the young agent’s maximizing problem as follows: (

max

at+1 ,nt

(nt Wt (1 − τn,t ) − at+1 )(1−σ) 1−σ

γnt (1+φ) β (Rt+1 (1 − τk,t+1 )at+1 )(1−σ) − + 1−σ 1+φ ,

)

(9)

where at+1 = bt+1 + kt+1 . The first order conditions are: (nt Wt (1 − τn,t ) − at+1 )−σ Wt (1 − τn,t ) = γnt φ ,

(10)

(nt Wt (1 − τn,t ) − at+1 )−σ = β (Rt+1 (1 − τk,t+1 )at+1 )−σ Rt+1 (1 − τk,t+1 ). (11)

3.5

Equilibrium

Competitive equilibrium in the model is defined in the standard way. All private agents maximize their utilities for given government policy rules and given prices. Markets clear given these prices. In addition, we impose the requirement that the government’s policy rules maximize social welfare subject to the laws of motion of the economy which are derived by imposing aggregate consistency constraints on individual agents’ optimality conditions. Aggregate consistency constraints stipulate that individual agents’ choice values such as future holdings of financial assets and capital, hours worked, consumption, etc., are equal in equilibrium to their per capita aggregate counterparts.

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The model’s dynamics can be reduced to a system of the form: St+1 = S(Zt , St , ut ),

(12)

where St is a vector of endogenous state variables at time t, Zt is a vector of exogenous state variables, and ut is a vector of government policy variables. The exogenous state vector is given by Zt = {1, zt } . The vector includes a constant term in order to facilitate the numerical solution of the model. The vector of endogenous state variables is St = {kt , bt } , The vector of government policy variables is ut = {τnt , Bt+1 } . One of the laws of motion for the state variables is just bt+1 = Bt+1 , which has the effect of imposing the equilibrium condition that all bonds held by young agents are government bonds. By using dynamic programming to derive agents’ feedback rules, the model’s dynamics can be written in a form which does not depend on forward-looking relative prices. This makes the government’s optimal policy problem recursive, so that standard dynamic programming techniques can be used to solve it. 13

3.6

Uniqueness

We do not have an analytical proof of the uniqueness of competitive equilibrium in our model. However, uniqueness is assured in our numerical simulations for the following reasons. 1. When solving for the model’s steady state, it can easily be reduced to one nonlinear equation in the capital stock. For given steady-state values of τn , B and z, and for the parameter values used in both the base-case simulations and sensitivity analyses, it can easily be shown numerically that this equation has a unique solution. 2. The first order conditions for private agents can also be reduced to a nonlinear equation in the future level of the capital stock, once the aggregate consistency conditions are imposed. For given paths for τnt+i , Bt+i+1 and zt+i , for all the parameter values used in the numerical simulations, and for given levels of kt , bt and rt , it can easily be shown that there is a unique value of kt+1 that satisfies this equation. 3. When the laws of motion in (12) are linearized around their steady-state values, the linearized dynamical system has the desired stability properties for all parameter values considered in the paper. The system’s three state variables are predetermined, and the system has three stable eigenvalues. For given initial conditions on the values of the state variables, the linearized system converges along a unique path towards its steady state.

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4. The government’s optimal policy problem is solved using a quadratic approximation to its social welfare function and linearized laws of motion for the economy’s state variables. By construction, such linear-quadratic problems have unique solutions.

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Characterizing Optimal Policies

Following Ambler and Cardia (1997) and Ambler and Paquet (1996, 1997), the government solves its optimal policy problem using dynamic programming methods and assumes that all future governments will do the same. Klein and Rios-Rull (1999) use a similar formulation of the government’s optimal policy problem in the context of a representative agent model with a period-by-period budget constraint. As they note, this approach can trace its origins to the appendix to Kydland and Prescott (1980). Because the government’s and private agent’s policy rules depend only on the current state of the economy, equilibrium in the model is sometimes known as Markov-perfect equilibrium: see Bernheim and Ray (1989) and Maskin and Tirole (1993). The optimal policy problem does not have an analytical solution. We find an approximate numerical solution to the problem using a quadratic approximation to the government’s social welfare function given by equation (8) and a linear approximation to the economy’s laws of motion as described in equation (12). The algorithm, based on Ambler and Paquet (1997), can be summarized as follows: • Initialize the government’s reaction function.

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• Do until convergence of the government’s reaction function. – Calculate the steady state of the model consistent with the government’s reaction function and the laws of motion given in (12). – Calculate a linear approximation of the economy’s equations of motion (12) around the steady state equilibrium. – Calculate a quadratic approximation of the economy’s social welfare function (8) around the steady state equilibrium. – Using standard techniques,6 calculate the government’s optimal linear reaction function. – Check for convergence, based on the size of the change in the government’s reaction function coefficients between the current iteration and the previous iteration. • End do.

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Calibration

Table 1 summarizes the base-case parameter values used to generate the numerical simulation results. I calibrate the model for periods which are twenty-five years long, which means that agents’ life expectancies once they start working are fifty years. The private subjective discount rate (β) and the depreciation rate of capital (which is assumed to be equal to 100%) both reflect the choice of period 6

See Hansen and Prescott (1995) and Ambler and Paquet (1997) for details.

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length. The value of β is the main factor determining the steady-state real interest rate, which is equal to 1.3% annually, which is reasonable given that the model abstracts from a positive trend rate of growth in technology. I set the discount rate in the social welfare function (δ) equal to private agents’ discount rates. The weight on labor in the aggregate function (α) is given a standard value based on labor’s share of national revenue in the data for most industrialized economies. The value of θ pins down the elasticity of labor supply (equal to 1/(θ − 1)), which is equal to one half. The low value once again reflects the choice of period length: I consider the sensitivity of the results to the value of θ below. I choose the value of γ so that labor supply in the long run is equal to 0.33. If total available time is normalized to equal one, this means that young agents spend one third of their available time working in the steady state. The value of the AR(1) parameter of the law of motion for technology is chosen to reflect stochastic processes that are highly persistent at a quarterly frequency. The standard deviation of technology shocks matches previous studies in the literature. The choice of g means that government spending on goods and services is equal to 20% of real output. The constant term in the law of motion for technology normalizes the level of technology. Changing it rescales the equilibrium level of output, but does not affect the model’s properties in any significant way.

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Results

The model’s steady state properties are summarized in Table 2. The model’s predictions concerning the response of tax rates to fluctuations in the level of tech17

nology are given in Table 3, along with predictions from the model of Klein and Rios-Rull (1999) and the empirical behavior of U.S. tax rates.

6.1

Steady State Properties

The optimal steady state tax rates on labor and capital income are respectively equal to 0.165 and 0.203. The capital tax rate is quantitatively important, and exceeds the ratio of government spending to output, which is calibrated to equal 0.20. Total tax revenues in the steady state are insufficient to finance government spending. In order to finance the difference between spending on goods and services and tax revenue, the government must receive a positive income from its holdings of assets. The optimal level of government debt in the long run is equal to -0.008, which is approximately equal to 6% of steady state real output. The steady state labor and capital income tax rates are both below the average marginal tax rates in the U.S. data (see Table 3), although government expenditure is calibrated to approximately match its share in output. This can be explained by the negative level of government debt in the steady state in the model. Table 2 also gives the equilibrium level of output in the steady state, and the decomposition of aggregate demand into consumption, investment, and government spending on goods and services. The ratios of consumption, output, and investment to output are not far from those in the data for industrialized economies. The capital to output ratio in annualized terms (25k/y) is equal to 5.22. The annualized real rate of interest on bonds ((1 + r)(1/25) − 1) is equal to 0.013. The capital rental rate gives the same after-tax rate of return from financial bonds and

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physical capital. We have β(1 + r) = 1.033 in the steady state. Equation (11) implies that there is a slightly upward-sloping profile for the marginal utility of consumption for each cohort of agents.

6.2

The Response of Tax Rates to Technology Shocks

The optimal feedback rule for the government is given by: "

τnt Bt+1 "

+

#

"

=

0.017 0.391 0.010 −0.047

5.139 0.001 0.000 −0.635 0.000 0.000

#

#"

1 zt

#

kt bt rt

(13)

The optimal labor income tax rate responds positively to both technology shocks and the level of the capital stock, while the level of government debt responds negatively to both of these variables. Figure 1 gives the response of the labor income and capital income tax rates to a positive one-standard-deviation technology shock. As can be inferred from the signs of the coefficient matrices of the feedback rule (13), the labor income tax rate increases in response to a positive technology shock and then decreases gradually to its steady state level. The optimal capital income tax rate jumps up when the shock hits, and then decreases sharply in the following period, approaching its steady state level from below. The responses of optimal tax rates to technology shocks are qualitatively similar to those obtained by Klein and Rios-Rull (1999) in a representative-agent model of time consistent taxation without borrowing,7 7

They exclude the possibility of borrowing. As we note in the introduction, allowing the government to set the future level of government debt (including a negative level of debt) would allow it to attain the first best social optimum.

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except that they do not obtain an initial positive response of the capital income tax rate. The difference is explained by the fact that Klein and Rios-Rull assume that the government can commit to its (state-contingent) capital income tax rate at least for the following period.8 The first column in Table 3 shows the variance, standard deviations and the coefficients of variation of capital and labor income tax rates for the model’s basecase parameter values.9 We compare our results to those in Klein and Rios-Rull (1999). The second column in the table shows results on the volatility of tax rates in their representative-agent model with commitment by the government, and the last column shows results for their model without commitment, both for the case where the period length is one year. The mean capital and labor income tax rates in our model are close to those in the data, and closer to their values in the data than those predicted by the Klein and Rios-Rull model, except in the case of the labor income tax rate with no commitment. The standard deviations of tax rates in our model are quite close to those in the U.S. data, and close to those in Klein and Rios-Rull for the version of their model without commitment. Their results show that optimal capital income tax rates in models with precommitment are highly variable compared to labor income tax rates. The latter are extremely smooth when the government can precommit to its optimal policy. This is a standard feature of models of optimal taxation with commitment. As summarized by Klein 8

There is also a quantitative difference. Klein and Rios-Rull show results for stochastic simulations in which the stochastic process governing technology is a two-state Markov process. 9 The coefficients of variation are just the absolute values of the ratios of the standard deviations to the means.

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and Rios-Rull (p.20), “under commitment the burden of taxation is borne almost completely by labor while capital taxation accommodates all surprises.” In representative agent models of optimal taxation with government borrowing, such as Chari, Christiano and Kehoe (1994), capital income tax rates are generally even more volatile.

6.3

Sensitivity Analysis

Table 4 shows how the steady-state labor income tax rate, capital income tax rate and level of government debt vary with some of the key parameters in the model. We vary each of the parameters while holding all other parameters constant at their values in the base case scenario which yields the steady state properties summarized in Table 2. We calibrate the model so that labor supply in the base case scenario is equal to 0.33, but this level changes as we vary the values of the model’s parameters while holding the value of γ constant. In each of the three panels of the table, the column in bold gives the model’s predictions for the base-case parameter values. The table shows that the basic result that the capital income tax rate is quantitatively important is quite robust. In only one case is the capital income tax rate below 10% in the steady state. This is for a labor supply elasticity equal to two (θ = 1.5), which is implausibly high given the length of periods in our model. In all other cases, the capital income tax rate is within four percentage points of the ratio of public spending to output, calibrated to equal 0.2 in the model. The central panel shows that both labor and capital income tax rates are negatively

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related to the elasticity of labor supply. The top panel of the table shows that tax rates are not very sensitive to agents’ subjective discount rates. The relationship is non-monotonic for both the labor income tax rate and the capital income tax rate. The lower panel shows that tax rates are also not very sensitive to the degree of risk aversion in the utility function as measured by σ. One result that is surprisingly robust is that for all parameter values considered in the table, the optimal level of government debt in the long run is negative. However, the optimal size of net government assets is quite small: for the parameter values considered in the table, the ratio of government assets to GDP never exceeds 10%.

7 Conclusions We have used an overlapping generations model with two-period lived agents to analyze optimal fiscal policy. Optimal capital income tax rates are much higher than in previous studies on optimal taxation in dynamic general equilibrium models, and closer to the average marginal tax rates on capital income in industrialized countries. We do not need to impose arbitrary limits on capital income tax rates in the short run in order to avoid the capital levy problem. The model predicts an optimal capital income tax rate which is much more volatile than that of the optimal labor income tax rate. The relative volatility of capital income taxes is lower than in representative agent models of optimal taxation with precommitment, but it is still higher than in industrialized countries. The model also predicts, for a wide range of parameter values, that the optimal level of government debt in the steady state is negative. This is perhaps the major divergence between the predic22

tions of the model and the data, but. Aiyagari and McGrattan (1994) show that the optimal level of government debt in a model of heterogeneous agents subject to liquidity constraints is positive. The availability of government bonds allows liquidity-constrained agents greater consumption-smoothing possibilities. Their model one of the few in the literature capable of rationalizing a positive level of debt as the outcome of maximizing behavior by the government. Clearly, the use of a model where one period represents approximately twentyfive years inhibits our ability to judge the model by its ability to match the data. The next step will be to extend the analysis to overlapping generations models calibrated to annual data. Rios-Rull (1995) describes how to solve such models with exogenous government policies. The solution of optimal policy problems in such a context is more difficult by an order of magnitude than the model considered in this paper because of the large number of state variables. This is reserved for future work.

References Aiyagari, Rao and Ellen McGrattan (1995), “The Optimum Quantity of Debt”, Staff Report 203, Federal Reserve Bank of Minneapolis Ambler, Steve (1999) “Les mod`eles a` agent repr´esentatif et la politique de taxation optimale”, Actualit´e e´ conomique: Revue d’analyse e´ conomique 75, 539-557. Available as cahier de recherche 91, Center for Research on Economic Fluctuations and Employment, Universit´e du Qu´ebec a` Montr´eal http://ideas.uqam.ca/ideas/data/Papers/crecrefwp91.html Ambler, Steve and Emanuela Cardia (1997), “Optimal Government Spending in a Business Cycle Model”, Business Cycles and Macroeconomic Stability: Should We Rebuild Built-In Stabilizers? Jean-Olivier Hairault, Pierre-Yves H´enin and Franck Portier, editors, Kluwer Academic Press, 31-53 Ambler, Steve and Dominique Desruelle (1991), “Time Inconsistency in TimeDependent Team Games”, Economics Letters 37, 1-6 Ambler, Steve and Alain Paquet (1996), “Fiscal Spending Shocks, Endogenous 23

Government Spending and Real Business Cycles”, Journal of Economic Dynamics and Control 20, 237-256 Ambler, Steve and Alain Paquet (1997), “Recursive Methods for Computing Equilibria of General Equilibrium Dynamic Stackelberg Games”, Economic Modelling 14, 155-173 Arjona, Roman (2000), “A Dynamic General Equilibrium Analysis of the Spanish Social Security System”, PhD dissertation, European University Institute Arrow, Kenneth J. and Mordecai Kurz (1970), Public Investment, the Rate of Return, and Optimal Fiscal Policy. Baltimore, Johns Hopkins Atkeson, Andrew, V.V. Chari and Patrick Kehoe (1999), “Taxing Capital Income: A Bad Idea”, Federal Reserve Bank of Minneapolis Quarterly Review 23, 3-17 Atkinson, Anthony B. and A. Sandmo (1980), “Welfare Implications of the Taxations of Savings”, Economic Journal 90, 529-549 Auerbach, Alan and Laurence Kotlikoff (1987), Dynamic Fiscal Policy. Cambridge, Cambridge University Press Benhabib, Jess and Aldo Rustichini (1997), “Optimal Taxes without Commitment”, Journal of Economic Theory 77, 231-259 Benhabib, Jess, Aldo Rustichini and Andr´es Velasco (1996), “Public Capital and Optimal Taxes without Commitment”, C.V. Starr Center working paper 96-19, New York University Bernheim, B. Douglas and Debraj Ray (1989), “Markov Perfect Equilibria in Altruistic Growth Economies with Production Uncertainty”, Journal of Economic Theory 47, 195-202 Calvo, Guillermo and Maurice Obstfeld (1988), “Optimal Time-Consistent Fiscal Policy with Finite Lifetimes: Analysis and Extensions”, in Elhanan Helpman, Assaf Razin and Efraim Sadka, eds., Economic Effects of the Government Budget. Cambridge, MA, MIT Press Chamley, C.P. (1985), “Efficient Taxation in a Stylized Model of Intertemporal General Equilibrium”, International Economic Review 26, 451-468 Chamley, C.P. (1986), “Optimal Taxation of Income in General Equilibrium with Infinite Lives”, Econometrica 54, 607-622 Chari, V.V., Lawrence J. Christiano and Patrick J. Kehoe (1991), “Optimal Fiscal and Monetary Policy: Some Recent Results”, Journal of Money, Credit and Banking 23, 519-539 Chari, V.V., Lawrence J. Christiano and Patrick J. Kehoe (1994), “Optimal Fiscal Policy in a Business Cycle Model”, Journal of Political Economy 102, 617652 Chari, V.V., Lawrence J. Christiano and Patrick J. Kehoe (1995), “Policy Analysis in Business Cycle Models” in Thomas F. Cooley, ed., Frontiers of Business Cycle Research. Princeton, Princeton University Press Chari, V.V. and Patrick J. Kehoe (1990), “Sustainable Plans”, Journal of Political Economy 98, 783-802 Chari, V.V. and Patrick J. Kehoe (1998), “Optimal Fiscal and Monetary Policy”, Staff Report 251, Federal Reserve Bank of Minneapolis, July XXXX update reference as chapter of Macro handbook. 24

Cohen, Daniel and Philippe Michel (1988), “How Should Control Theory Be Used to Calculate a Time-Consistent Government Policy?”, Review of Economic Studies 55, 263-274 Erosa, Andr´es and Martin Gervais (1998), “Optimal Taxation in Life-Cycle Economies”, mimeo, University of Western Ontario Erosa, Andr´es and Martin Gervais (2001), “On the Optimality of Age-Dependent Taxes and the U.S. Tax System”, mimeo, Federal Reserve Bank of Richmond Fischer, Stanley (1980), “Dynamic Inconsistency, Cooperation, and the Benevolent Dissembling Government”, Journal of Economic Dynamics and Control 2, 93-107 Greenwood, Jeremy, Zvi Hercowitz and Gregory Huffman (1988), “Investment, Capacity Utilization and the Real Business Cycle”, American Economic Review 78, 402-417 Hansen, Gary D. and Edward C. Prescott (1995), “Recursive Methods for Computing Equilibria of Business Cycle Models” in Thomas F. Cooley (ed.), Frontiers of Business Cycle Research. Princeton, N.J., Princeton University Press James, Steven (1994), “Debt Reduction with Distorting Taxes and Incomplete Ricardianism: A Computable Dynamic General Equilibrium Analysis”, in William Robson and William Scarth (eds.), Deficit Reduction: What Pain, What Gain?. Toronto, C.D. Howe Institute Jones, L.E., R. Manuelli and P.E. Rossi (1993), “Optimal Taxation in Models of Endogenous Growth”, Journal of Political Economy 101, 485-517 Jones, L.E., R. Manuelli and P.E. Rossi (1997), “On the Optimal Taxation of Capital Income”, Journal of Economic Theory 73, 93-117 Judd, Kenneth L. (1985), “Redistributive Taxation in a Simple Perfect Foresight Model”, Journal of Public Economics 28, 59-83 Judd, Kenneth L. (1987), “The Welfare Cost of Factor Taxation in a Perfect Foresight Model”, Journal of Political Economy 95, 675-709 Judd, Kenneth L. (1999), “Optimal Taxation and Spending in General Competitive Growth Models”, Journal of Public Economics 71, 1-26 Klein, Paul and Jos´e-Victor Rios-Rull (2000), “Time-Consistent Optimal Fiscal Policy”, mimeo, University of Western Ontario Krusell, Per and Antony A. Smith (2000), “Consumption-Savings Decisions with Quasi-Geometric Discounting”, mimeo, University of Rochester Krusell, Per, Burhanettin Kurus¸c¸u and Antony A. Smith (2000), “Equilibrium Welfare and Government Policy with Quasi-Geometric Discounting”, mimeo, University of Rochester Kydland, Finn E. and Edward C. Prescott (1979), “Rules Rather than Discretion: The Inconsistency of Optimal Plans”, Journal of Political Economy 85, 473491 Kydland, Finn E. and Edward C. Prescott (1980), “Dynamic Optimal Taxation, Rational Expectations, and Optimal Control”, Journal of Economic Dynamics and Control 2, 79-91 Lucas, Robert E. and Nancy L. Stokey (1983), “Optimal Fiscal and Monetary Policy in an Economy without Capital”, Journal of Monetary Economics 12, 25

55-93 Maskin, E. and J. Tirole (1993), “Markov Perfect Equilibrium”, mimeo, Harvard University Mendoza, Enrique G., Assaf Razin and Linda L. Tesar (1994), “Effective Tax Rates in Macroeconomics: Cross-Country Estimates of Tax Rates on Factor Incomes and Consumption”, Journal of Monetary Economics 34, 297-323 Miller, Marcus and Mark Salmon (1985), “Policy Coordination and Dynamic Games”, in Willem Buiter and Richard Marston, eds., International Economic Policy Coordination. Cambridge, Cambridge University Press Oudiz, Gilles and Jeffrey Sachs (1985), “Policy Coordination and Dynamic Games”, in Willem Buiter and Richard Marston, eds., International Economic Policy Coordination. Cambridge, Cambridge University Press Park, No-Ho (1991), “Steady-State Solutions of Optimal Tax Mixes in an Overlapping-Generations Model”, Journal of Public Economics 46, 227-246 Ramsey, Frank (1927), “A Contribution to the Optimal Theory of Taxation”, Economic Journal 37, 47-61 Ramsey, Frank (1928), “A Mathematical Theory of Saving”, Economic Journal 38, 543-559 Rios-Rull, Jos´e Victor (1995), “Models with Heterogeneous Agents”, in T.F. Cooley, ed., Frontiers of Business Cycle Research. Princeton, Princeton University Press Sargent, Thomas J. (1987), Macroeconomic Theory Scarth, William (1999), “Alternatives for Raising Living Standards”, mimeo, McMaster University Stokey, Nancy (1991), “Credible Public Policy”, Journal of Economic Dynamics and Control 15, 627-657 Ventura, Gustavo (1996), “Flat Tax Reform: A Quantitative Assessment”, mimeo, University of Illinois Weil, Philippe (!989), “Overlapping Generations of Infinitely-Lived Agents”, Journal of Public Economics 38, 183-198 Yaari, Menahem E. (1965), “Uncertain Lifetime, Life Insurance, and the Theory of the Consumer”, Review of Economic Studies 32, 137-150 Zhu, X. (1992), “Optimal Fiscal Policy in a Stochastic Growth Model”, Journal of Economic Theory 58, 250-289

26

Table 1: Parameter Values Parameter Value β 0.75 δ 0.75 α 0.64 θ 3.00 σ 1.75 γ 0.66 ρz 0.40 g 0.20 z 1.00 σz 0.01

Table 2: Steady State Properties Variable τn τk B y g k cy co c/y i/y g/y 25 · k/y b/y n w R r

27

Value 0.165 0.203 -0.008 0.138 0.028 0.029 0.052 0.029 0.591 0.208 0.200 5.220 -0.058 0.333 0.265 1.727 0.377

Table 3: Variability of Tax Rates Model Klein/Rios-Rull (Commitment)

Klein/Rios-Rull (No Commitment)

Data

Capital Income Tax mean s.d. c.v.

0.203 0.110 0.543

-0.002 0.180 85.71

0.650 0.110 0.170

0.283 0.088 0.309

Labor Income Tax mean s.d. c.v.

0.165 0.030 0.181

0.310 0.009 0.028

0.120 0.031 0.250

0.248 0.024 0.097

s.d.: standard deviation c.v.: coefficient of variation The second and third columns are from Klein and Rios-Rull (1999, Tables 3 and 4). The last column is from Chari, Christiano and Kehoe (1995, Table 12.4).

28

Table 4: Sensitivity Analysis Sensitivity to β β τn τk B

0.450 0.159 0.239 -0.001

0.550 0.650 0.157 0.159 0.221 0.209 -0.002 -0.005

0.750 0.165 0.203 -0.008

0.850 0.179 0.200 -0.012

0.950 0.187 0.202 -0.016

θ τn τk B

1.500 0.147 0.063 -0.003

2.000 0.155 0.161 -0.007

3.000 0.165 0.203 -0.008

4.000 0.171 0.213 -0.007

5.000 0.175 0.218 -0.006

6.000 0.177 0.220 -0.005

σ τn τk B

1.250 0.169 0.192 -0.009

1.500 0.167 0.198 -0.008

1.750 0.165 0.203 -0.008

2.000 0.163 0.207 -0.008

2.250 0.162 0.210 -0.007

2.500 0.161 0.213 -0.007

Sensitivity to θ

Sensitivity to σ

29

∗

CIRPEE, UQAM, C.P. 8888, Succ. Centre-ville, Montr´eal, Qu´ebec, Canada H3C 3P8, phone (514) 987-3000 ext. 8372, fax (514) 987-8494, email [email protected]. I would like to thank the CRSH and the Fonds FCAR for generous financial support, the European University Institute for its generous financial support and hospitality during my sabbatical year (1999-2000), and Pascal St. Amour, Christian Zimmermann, seminar participants at the University of Montreal, York University and Laval University for helpful comments. The usual caveat applies.

1

Optimal Time-Consistent Taxation with Overlapping Generations Abstract The paper analyzes optimal time-consistent taxation in an overlapping generations model with two-period lived households. The government chooses tax rates and borrowing to finance an exogenous stream of expenditures. It cannot commit to future policies, so announced policies that are not time consistent are not credible. Dynamic programming is used to derive Markov-perfect equilibria. In contrast to optimal fiscal policy in representative-agent models with commitment, optimal capital income tax rates are positive in the long run, and bounded below one in the short run for a wide range of parameter values. Key words: Optimal Fiscal Policy, Time Consistency, Overlapping Generations JEL classification codes: E61, E62, C63

2

1

Introduction

There is an extensive literature on optimal second-best taxation in dynamic general equilibrium (DGE) models.1 Most researchers have focused on the representativeagent neoclassical growth model and solutions to the so-called Ramsey problem (1927), in which the government can precommit to its future policies. Representative-agent models of optimal taxation (RAMOT) lead to striking predictions. In deterministic models, the optimal tax rate on capital income converges to zero in the long run. In models of endogenous growth with human capital, the optimal tax rates on all forms of capital income converge to zero. Although lump-sum taxation is not permitted, the government can achieve the firstbest social optimum if it can undertake a capital levy, taxing capital income at a high enough rate to accumulate assets to finance its future expenditures: in order to make the optimal taxation problem non-trivial, arbitrary limits must be set on capital income tax rates at the beginning of the government’s optimal program. The government’s optimal plan is not time consistent: if allowed to re-optimize, it can achieve a first-best outcome (from that point onward) by undertaking a capital levy.2 Finally, in stochastic versions of the problem, the average tax rate on capital income converges to zero, but the tax rates on income from capital and financial 1

The possibility of lump-sum taxation is excluded so that the first-best social optimum cannot be attained. See Ambler (1999), Atkeson, Chari and Kehoe (1999), Chari, Christiano and Kehoe (1995), Chari and Kehoe (1998), and Judd (1999) for surveys of the literature. 2 For this reason, I would argue that explicit mechanisms such as trigger strategies to support the precommitment equilibrium are implausible, since they must lead to outcomes that are worse than the first-best outcome. Papers that examine the use of trigger strategies, such as Chari and Kehoe (1990) and Stokey (1991), do so in the context of models that preclude the possibility of attaining the first-best outcome.

3

assets are extremely volatile: in response to unexpected shocks which affect the expected present value of the government’s tax revenues, it is optimal to adjust rates of taxation on assets which are inelastically supplied. The predictions of the RAMOT approach clearly diverge from observed behavior. Capital income tax rates are generally quite high in the data. This divergence has led some researchers to examine optimal taxation in environments richer than representative-agent models with government precommitment. For example, Klein and Rios-Rull (1999) study a representative-agent in which the government cannot precommit to future tax rates, and cannot borrow. They find that in such a setting, optimal capital income tax rates are considerably higher than in the data in the long run. This paper explores the implications of a model which combines two key departrues from the assumptions used by the RAMOT approach. Specifically, the paper analyzes optimal taxation without government precommitment in an overlapping generations model. I impose no restrictions on the government’s ability to borrow. Capital ownership is not uniform across cohorts: with life expectancies of two periods and no bequests, capital is held only by old agents. In this setting, taxing capital at very high rates is not optimal because of its effects on the intergenerational distribution of income. It is therefore not necessary to impose arbitrary restrictions on rates of capital income taxation in the very short run, even if capital is supplied inelastically. Since the government cannot precommit to future rates of taxation, capital income tax rates do not converge to zero in the long run: with precommitment, the government could achieve its distributional goals 4

by taxing (exclusively or primarily) each cohort’s labor income. The model is fairly simple, but its solution is not. I derive the optimal linear feedback rule for an approximation to the government’s problem, with a quadratic approximation to its value function and linearized laws of motion for the economy’s state variables. I use iterative techniques to derive the government’s feedback rule. At each iteration, the government’s feedback rule is updated as are private agents’ optimal linear feedback rules given the government’s current feedback rule.3 The paper is structured as follows. In the following section, I briefly review some of the relevant literature on optimal taxation in macroeconomic models. The model is presented in the third section. The fourth section describes in detail the government’s optimal policy problem. The model’s calibration and the numerical solution methodology are discussed in the fifth section. The sixth section outlines the results. The seventh section concludes.

2

Relevant Literature

The classic result on the convergence of capital income tax rates to zero was first derived by Chamley (1985, 1986) and Judd (1985, 1987), using non-stochastic versions of the neoclassical growth model and extending the work of Arrow and Kurz (1970) and Ramsey (1928). Optimal taxation in stochastic models was stud3

I conjecture that one of the reasons that the use of Ramsey problems has been so popular in the literature is that they allow researchers to use the so-called primal approach (see Ambler, 1999, or Chari and Kehoe, 1998, for details), which simplifies the calculation of optimal policies. Using this approach, it is possible to solve directly for the optimal allocations in the economy. The tax rates and prices that support these allocations can be derived once these allocations are known.

5

ied by Zhu (1992) and in a series of papers by Chari, Christiano and Kehoe (1991, 1994, 1995). Jones, Manuelli and Rossi (1993, 1997) analyzed optimal capital income taxation in endogenous growth models with both physical and human capital. There are at least three important threads in the literature which diverge from the basic assumptions Kydland and Prescott (1980), in their seminal paper on the time inconsistency of optimal policy, include an appendix in which they describe how to calculate a time consistent strategy using dynamic programming methods. Distinguish clearly between among threads in the literature. 1) Models with overlapping generations: Atkinson and Sandmo (1980), Park (1991), Erosa and Gervais (1998), Auerbach and Kotlikoff (1987), Scarth (1999). 2) Models of optimal policy without commitment: Kydland and Prescott (1979, 1980), Benhabib and Rustichini (1997), Benhabib, Rustichini and Velasco (1996), Krusell, Kurus¸c¸u and Smith (2000), Ambler and Paquet (1996), Lucas and Stokey (1983), Klein and Rios Rull (2000), Hansen, Epple and Roberds (1985), Cohen and Michel (1988), Fischer (1980), Sargent (1987, section 15.5), Oudiz and Sachs (1985), Miller and Salmon (1985), Calvo and Obstfeld (1988). 3) Models which try to show how to support the time-inconsistent equilibrium with trigger strategies. Note the importance of Calvo and Obstfeld (1988), which deals with overlapping generations. It allows for lump-sum taxation, and derives conditions under which social welfare functions are compatible with time consistency due to preferences. 6

Justify the lack of commitment assumption. Refer to Chari and Kehoe (1990), Stokey (1991). Models in which a long run tax rate of zero on capital income is not optimal in the long run have started to appear in the literature quite recently. Benhabib and Rustichini (1997) and Benhabib, Rustichini and Velasco (1996) study optimal taxation in representative agent models without commitment. They consider optimal government strategies which are robust at time t to any possible deviations from time t on, in the sense that any deviation worsens social welfare.4 They derive the result that, in order to create the necessary incentives for the government not to deviate from its optimal plan, capital income taxation rates must be strongly negative. Klein and Rios-Rull (1999) study a representative agent model in which the government must balance its budget each period without borrowing. Optimal capital income taxation rates are nonzero in the long run, and no additional restriction is necessary on initial capital income tax rates, since the government is prohibited at all times from building up arbitrarily large claims on the private sector. The stochastic properties of tax rates in their model are closer to the data than in similar models with unlimited government borrowing. In an important recent paper, Erosa and Gervais (1998) consider a model with overlapping generations of households with arbitrary, finite planning horizons. They show analytically that optimal fiscal policy with commitment involves a tax rate on capital income that does not generally converge to zero in the long 4

As in representative agent models of optimal taxation with commitment, they impose a limit on the rate of taxation of capital income at the start of the optimal program

7

run, in contrast to most representative agent models. However, when preferences are additively separable in consumption and leisure, optimal capital income tax rates converge to zero after an initial period when they are positive. Atkeson, Chari and Kehoe (1999) demonstrate a similar result. In addition, in calibrated versions of their model, Erosa and Gervais show that capital income tax rates are quantitatively quite small in the long run (less than three percent). Overlapping generations models have also been used to examine the effects of exogenous (not demonstrably optimal) changes in fiscal policy on economic equilibrium and welfare. These models have included overlapping generations models in which agents have finite horizons (Auerbach and Kotlikoff, 1987; Ventura, 1996), and models which use the Yaari (1965) assumption of a constant probability of death or the Weil (1989) assumption of overlapping generations of agents with infinite horizons (James, 1994; Scarth, 1999).

3

The Model

The economy consists of overlapping generations of two-period lived individuals. The total population of the economy is constant. The size of each cohort is normalized to equal one. There are no bequests, and individuals are born without assets. Agents work, consume, and save in the first period of their lives. They are retired in the last period of their lives, and use their accumulated savings to finance consumption. Perfectly competitive firms rent labor from young agents and capital from old agents and produce goods which are sold to the young and old and to the government. The government finances an exogenous stream of expen8

ditures by taxing labor and capital income and by borrowing. Agents have perfect foresight. However, the model is used to analyze once-and-for-all changes to the time path of the model’s main exogenous variable (the level of technology).

3.1

Households

An individual born at date t lives for two periods. He works, consumes and saves in the first period and consumes in the second period. He maximizes the following utility function: Ut =

1 γ 1 cy,t (1−σ) − nt (1+φ) + β co,t+1 (1−σ) , 1−σ 1+φ 1−σ

(1)

subject to the following flow budget constraint in the first period: cy,t = nt Wt (1 − τn,t ) − kt+1 − bt+1 ,

(2)

and the second-period flow budget constraint given by: co,t+1 = (1 + rt+1 )bt+1 + Rt+1 kt+1 (1 − τk,t+1 ).

(3)

The variable definitions are as follows: cy,t is consumption when young, nt denotes hours worked by young agents, co,t+1 is consumption when old, kt+1 is capital acquired when young, bt+1 denotes bonds acquired when young, Wt is the real wage rate, rt is the net real interest rate on bonds, Rt is the real rental rate on capital, τk,t is the tax rate on capital income, and τn,t is the tax rate on labor income. Old agents cannot borrow in equilibrium, since this would violate their lifetime budget constraints. Young agents are all born without assets and have homo9

geneous preferences, so in equilibrium the only bonds that they hold are government bonds. This is imposed on the model as an equilibrium condition. Capital owned by old agents depreciates completely, so that old agents die without assets.

3.2

Firms

A representative competitive firm rents labor from young agents and capital from old agents and produces output according to a standard aggregate production function given by yt = Zt nt α kt (1−α) ,

(4)

where Zt is the level of technical progress. Profit maximization leads to the following standard optimality conditions which determine the real wage rate and the rental rate of capital:

3.3

Wt = αZt nt (α−1) kt (1−α) ,

(5)

Rt = (1 − α)Zt nt α kt −α .

(6)

The Government

The government finances an exogenous (from its point of view) stream of spending by taxing capital and labor income and buy borrowing. Its expenditures are denoted by Gt . The government’s flow budget constraint is given by: Gt + (1 + rt )Bt = Wt nt τn,t + Rt kt τk,t + Bt+1 ,

(7)

where Bt is the stock of government debt (which consists of one-period bonds). The government does not have access to lump sum taxation. We abstract from 10

other forms of taxation such as consumption taxes. In a slightly different context, Judd (1999) shows that adding consumption taxes does not give the government an independent policy instrument. It would imply that several different combinations of tax rates could be used to achieve the same equilibrium. We ignore this possibility in the interests of simplicity. The government’s social welfare function is given by SWt =

∞ X i=0

(

δ i−1

cy,t−1+i (1−σ) γnt−i+1 (1+φ) βco,t+i (1−σ) − + 1−σ 1+φ 1−σ

)

(8)

The social welfare function is just the sum of the lifetime utilities of individual cohorts, discounted relative to their dates of birth using the social discount rate δ. Calvo and Obstfeld (1988) showed that the utility of individual cohorts must be discounted relative to the beginnings of their lives in order to avoid problems of time inconsistency arising solely from the specification of the government’s preferences as opposed to time inconsistency arising from constraints imposed on the financing of the government’s spending.5 In the calibration of the model I set the social discount rate δ equal to private agents’ discount rate, but in principle the two could be different. The terms in cy,t−1 and nt−1 cannot be affected by the government’s policies at time t. They act as a constant terms which shift the level of the social welfare function: omitting them would not change the government’s optimal policies, but they bring out more clearly the logic of discounting cohorts’ lifetime utilities relative to their dates of birth. 5

See Ambler and Desruelle (1991) for a discussion of how time inconsistency arises in dynamic games when agents share the same objectives but face different constraints.

11

3.4

Maximization by the Young

In the absence of uncertainty, bonds and capital are perfect substitutes and must have the same after-tax rate of return. We can write the young agent’s maximizing problem as follows: (

max

at+1 ,nt

(nt Wt (1 − τn,t ) − at+1 )(1−σ) 1−σ

γnt (1+φ) β (Rt+1 (1 − τk,t+1 )at+1 )(1−σ) − + 1−σ 1+φ ,

)

(9)

where at+1 = bt+1 + kt+1 . The first order conditions are: (nt Wt (1 − τn,t ) − at+1 )−σ Wt (1 − τn,t ) = γnt φ ,

(10)

(nt Wt (1 − τn,t ) − at+1 )−σ = β (Rt+1 (1 − τk,t+1 )at+1 )−σ Rt+1 (1 − τk,t+1 ). (11)

3.5

Equilibrium

Competitive equilibrium in the model is defined in the standard way. All private agents maximize their utilities for given government policy rules and given prices. Markets clear given these prices. In addition, we impose the requirement that the government’s policy rules maximize social welfare subject to the laws of motion of the economy which are derived by imposing aggregate consistency constraints on individual agents’ optimality conditions. Aggregate consistency constraints stipulate that individual agents’ choice values such as future holdings of financial assets and capital, hours worked, consumption, etc., are equal in equilibrium to their per capita aggregate counterparts.

12

The model’s dynamics can be reduced to a system of the form: St+1 = S(Zt , St , ut ),

(12)

where St is a vector of endogenous state variables at time t, Zt is a vector of exogenous state variables, and ut is a vector of government policy variables. The exogenous state vector is given by Zt = {1, zt } . The vector includes a constant term in order to facilitate the numerical solution of the model. The vector of endogenous state variables is St = {kt , bt } , The vector of government policy variables is ut = {τnt , Bt+1 } . One of the laws of motion for the state variables is just bt+1 = Bt+1 , which has the effect of imposing the equilibrium condition that all bonds held by young agents are government bonds. By using dynamic programming to derive agents’ feedback rules, the model’s dynamics can be written in a form which does not depend on forward-looking relative prices. This makes the government’s optimal policy problem recursive, so that standard dynamic programming techniques can be used to solve it. 13

3.6

Uniqueness

We do not have an analytical proof of the uniqueness of competitive equilibrium in our model. However, uniqueness is assured in our numerical simulations for the following reasons. 1. When solving for the model’s steady state, it can easily be reduced to one nonlinear equation in the capital stock. For given steady-state values of τn , B and z, and for the parameter values used in both the base-case simulations and sensitivity analyses, it can easily be shown numerically that this equation has a unique solution. 2. The first order conditions for private agents can also be reduced to a nonlinear equation in the future level of the capital stock, once the aggregate consistency conditions are imposed. For given paths for τnt+i , Bt+i+1 and zt+i , for all the parameter values used in the numerical simulations, and for given levels of kt , bt and rt , it can easily be shown that there is a unique value of kt+1 that satisfies this equation. 3. When the laws of motion in (12) are linearized around their steady-state values, the linearized dynamical system has the desired stability properties for all parameter values considered in the paper. The system’s three state variables are predetermined, and the system has three stable eigenvalues. For given initial conditions on the values of the state variables, the linearized system converges along a unique path towards its steady state.

14

4. The government’s optimal policy problem is solved using a quadratic approximation to its social welfare function and linearized laws of motion for the economy’s state variables. By construction, such linear-quadratic problems have unique solutions.

4

Characterizing Optimal Policies

Following Ambler and Cardia (1997) and Ambler and Paquet (1996, 1997), the government solves its optimal policy problem using dynamic programming methods and assumes that all future governments will do the same. Klein and Rios-Rull (1999) use a similar formulation of the government’s optimal policy problem in the context of a representative agent model with a period-by-period budget constraint. As they note, this approach can trace its origins to the appendix to Kydland and Prescott (1980). Because the government’s and private agent’s policy rules depend only on the current state of the economy, equilibrium in the model is sometimes known as Markov-perfect equilibrium: see Bernheim and Ray (1989) and Maskin and Tirole (1993). The optimal policy problem does not have an analytical solution. We find an approximate numerical solution to the problem using a quadratic approximation to the government’s social welfare function given by equation (8) and a linear approximation to the economy’s laws of motion as described in equation (12). The algorithm, based on Ambler and Paquet (1997), can be summarized as follows: • Initialize the government’s reaction function.

15

• Do until convergence of the government’s reaction function. – Calculate the steady state of the model consistent with the government’s reaction function and the laws of motion given in (12). – Calculate a linear approximation of the economy’s equations of motion (12) around the steady state equilibrium. – Calculate a quadratic approximation of the economy’s social welfare function (8) around the steady state equilibrium. – Using standard techniques,6 calculate the government’s optimal linear reaction function. – Check for convergence, based on the size of the change in the government’s reaction function coefficients between the current iteration and the previous iteration. • End do.

5

Calibration

Table 1 summarizes the base-case parameter values used to generate the numerical simulation results. I calibrate the model for periods which are twenty-five years long, which means that agents’ life expectancies once they start working are fifty years. The private subjective discount rate (β) and the depreciation rate of capital (which is assumed to be equal to 100%) both reflect the choice of period 6

See Hansen and Prescott (1995) and Ambler and Paquet (1997) for details.

16

length. The value of β is the main factor determining the steady-state real interest rate, which is equal to 1.3% annually, which is reasonable given that the model abstracts from a positive trend rate of growth in technology. I set the discount rate in the social welfare function (δ) equal to private agents’ discount rates. The weight on labor in the aggregate function (α) is given a standard value based on labor’s share of national revenue in the data for most industrialized economies. The value of θ pins down the elasticity of labor supply (equal to 1/(θ − 1)), which is equal to one half. The low value once again reflects the choice of period length: I consider the sensitivity of the results to the value of θ below. I choose the value of γ so that labor supply in the long run is equal to 0.33. If total available time is normalized to equal one, this means that young agents spend one third of their available time working in the steady state. The value of the AR(1) parameter of the law of motion for technology is chosen to reflect stochastic processes that are highly persistent at a quarterly frequency. The standard deviation of technology shocks matches previous studies in the literature. The choice of g means that government spending on goods and services is equal to 20% of real output. The constant term in the law of motion for technology normalizes the level of technology. Changing it rescales the equilibrium level of output, but does not affect the model’s properties in any significant way.

6

Results

The model’s steady state properties are summarized in Table 2. The model’s predictions concerning the response of tax rates to fluctuations in the level of tech17

nology are given in Table 3, along with predictions from the model of Klein and Rios-Rull (1999) and the empirical behavior of U.S. tax rates.

6.1

Steady State Properties

The optimal steady state tax rates on labor and capital income are respectively equal to 0.165 and 0.203. The capital tax rate is quantitatively important, and exceeds the ratio of government spending to output, which is calibrated to equal 0.20. Total tax revenues in the steady state are insufficient to finance government spending. In order to finance the difference between spending on goods and services and tax revenue, the government must receive a positive income from its holdings of assets. The optimal level of government debt in the long run is equal to -0.008, which is approximately equal to 6% of steady state real output. The steady state labor and capital income tax rates are both below the average marginal tax rates in the U.S. data (see Table 3), although government expenditure is calibrated to approximately match its share in output. This can be explained by the negative level of government debt in the steady state in the model. Table 2 also gives the equilibrium level of output in the steady state, and the decomposition of aggregate demand into consumption, investment, and government spending on goods and services. The ratios of consumption, output, and investment to output are not far from those in the data for industrialized economies. The capital to output ratio in annualized terms (25k/y) is equal to 5.22. The annualized real rate of interest on bonds ((1 + r)(1/25) − 1) is equal to 0.013. The capital rental rate gives the same after-tax rate of return from financial bonds and

18

physical capital. We have β(1 + r) = 1.033 in the steady state. Equation (11) implies that there is a slightly upward-sloping profile for the marginal utility of consumption for each cohort of agents.

6.2

The Response of Tax Rates to Technology Shocks

The optimal feedback rule for the government is given by: "

τnt Bt+1 "

+

#

"

=

0.017 0.391 0.010 −0.047

5.139 0.001 0.000 −0.635 0.000 0.000

#

#"

1 zt

#

kt bt rt

(13)

The optimal labor income tax rate responds positively to both technology shocks and the level of the capital stock, while the level of government debt responds negatively to both of these variables. Figure 1 gives the response of the labor income and capital income tax rates to a positive one-standard-deviation technology shock. As can be inferred from the signs of the coefficient matrices of the feedback rule (13), the labor income tax rate increases in response to a positive technology shock and then decreases gradually to its steady state level. The optimal capital income tax rate jumps up when the shock hits, and then decreases sharply in the following period, approaching its steady state level from below. The responses of optimal tax rates to technology shocks are qualitatively similar to those obtained by Klein and Rios-Rull (1999) in a representative-agent model of time consistent taxation without borrowing,7 7

They exclude the possibility of borrowing. As we note in the introduction, allowing the government to set the future level of government debt (including a negative level of debt) would allow it to attain the first best social optimum.

19

except that they do not obtain an initial positive response of the capital income tax rate. The difference is explained by the fact that Klein and Rios-Rull assume that the government can commit to its (state-contingent) capital income tax rate at least for the following period.8 The first column in Table 3 shows the variance, standard deviations and the coefficients of variation of capital and labor income tax rates for the model’s basecase parameter values.9 We compare our results to those in Klein and Rios-Rull (1999). The second column in the table shows results on the volatility of tax rates in their representative-agent model with commitment by the government, and the last column shows results for their model without commitment, both for the case where the period length is one year. The mean capital and labor income tax rates in our model are close to those in the data, and closer to their values in the data than those predicted by the Klein and Rios-Rull model, except in the case of the labor income tax rate with no commitment. The standard deviations of tax rates in our model are quite close to those in the U.S. data, and close to those in Klein and Rios-Rull for the version of their model without commitment. Their results show that optimal capital income tax rates in models with precommitment are highly variable compared to labor income tax rates. The latter are extremely smooth when the government can precommit to its optimal policy. This is a standard feature of models of optimal taxation with commitment. As summarized by Klein 8

There is also a quantitative difference. Klein and Rios-Rull show results for stochastic simulations in which the stochastic process governing technology is a two-state Markov process. 9 The coefficients of variation are just the absolute values of the ratios of the standard deviations to the means.

20

and Rios-Rull (p.20), “under commitment the burden of taxation is borne almost completely by labor while capital taxation accommodates all surprises.” In representative agent models of optimal taxation with government borrowing, such as Chari, Christiano and Kehoe (1994), capital income tax rates are generally even more volatile.

6.3

Sensitivity Analysis

Table 4 shows how the steady-state labor income tax rate, capital income tax rate and level of government debt vary with some of the key parameters in the model. We vary each of the parameters while holding all other parameters constant at their values in the base case scenario which yields the steady state properties summarized in Table 2. We calibrate the model so that labor supply in the base case scenario is equal to 0.33, but this level changes as we vary the values of the model’s parameters while holding the value of γ constant. In each of the three panels of the table, the column in bold gives the model’s predictions for the base-case parameter values. The table shows that the basic result that the capital income tax rate is quantitatively important is quite robust. In only one case is the capital income tax rate below 10% in the steady state. This is for a labor supply elasticity equal to two (θ = 1.5), which is implausibly high given the length of periods in our model. In all other cases, the capital income tax rate is within four percentage points of the ratio of public spending to output, calibrated to equal 0.2 in the model. The central panel shows that both labor and capital income tax rates are negatively

21

related to the elasticity of labor supply. The top panel of the table shows that tax rates are not very sensitive to agents’ subjective discount rates. The relationship is non-monotonic for both the labor income tax rate and the capital income tax rate. The lower panel shows that tax rates are also not very sensitive to the degree of risk aversion in the utility function as measured by σ. One result that is surprisingly robust is that for all parameter values considered in the table, the optimal level of government debt in the long run is negative. However, the optimal size of net government assets is quite small: for the parameter values considered in the table, the ratio of government assets to GDP never exceeds 10%.

7 Conclusions We have used an overlapping generations model with two-period lived agents to analyze optimal fiscal policy. Optimal capital income tax rates are much higher than in previous studies on optimal taxation in dynamic general equilibrium models, and closer to the average marginal tax rates on capital income in industrialized countries. We do not need to impose arbitrary limits on capital income tax rates in the short run in order to avoid the capital levy problem. The model predicts an optimal capital income tax rate which is much more volatile than that of the optimal labor income tax rate. The relative volatility of capital income taxes is lower than in representative agent models of optimal taxation with precommitment, but it is still higher than in industrialized countries. The model also predicts, for a wide range of parameter values, that the optimal level of government debt in the steady state is negative. This is perhaps the major divergence between the predic22

tions of the model and the data, but. Aiyagari and McGrattan (1994) show that the optimal level of government debt in a model of heterogeneous agents subject to liquidity constraints is positive. The availability of government bonds allows liquidity-constrained agents greater consumption-smoothing possibilities. Their model one of the few in the literature capable of rationalizing a positive level of debt as the outcome of maximizing behavior by the government. Clearly, the use of a model where one period represents approximately twentyfive years inhibits our ability to judge the model by its ability to match the data. The next step will be to extend the analysis to overlapping generations models calibrated to annual data. Rios-Rull (1995) describes how to solve such models with exogenous government policies. The solution of optimal policy problems in such a context is more difficult by an order of magnitude than the model considered in this paper because of the large number of state variables. This is reserved for future work.

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Cohen, Daniel and Philippe Michel (1988), “How Should Control Theory Be Used to Calculate a Time-Consistent Government Policy?”, Review of Economic Studies 55, 263-274 Erosa, Andr´es and Martin Gervais (1998), “Optimal Taxation in Life-Cycle Economies”, mimeo, University of Western Ontario Erosa, Andr´es and Martin Gervais (2001), “On the Optimality of Age-Dependent Taxes and the U.S. Tax System”, mimeo, Federal Reserve Bank of Richmond Fischer, Stanley (1980), “Dynamic Inconsistency, Cooperation, and the Benevolent Dissembling Government”, Journal of Economic Dynamics and Control 2, 93-107 Greenwood, Jeremy, Zvi Hercowitz and Gregory Huffman (1988), “Investment, Capacity Utilization and the Real Business Cycle”, American Economic Review 78, 402-417 Hansen, Gary D. and Edward C. Prescott (1995), “Recursive Methods for Computing Equilibria of Business Cycle Models” in Thomas F. Cooley (ed.), Frontiers of Business Cycle Research. Princeton, N.J., Princeton University Press James, Steven (1994), “Debt Reduction with Distorting Taxes and Incomplete Ricardianism: A Computable Dynamic General Equilibrium Analysis”, in William Robson and William Scarth (eds.), Deficit Reduction: What Pain, What Gain?. Toronto, C.D. Howe Institute Jones, L.E., R. Manuelli and P.E. Rossi (1993), “Optimal Taxation in Models of Endogenous Growth”, Journal of Political Economy 101, 485-517 Jones, L.E., R. Manuelli and P.E. Rossi (1997), “On the Optimal Taxation of Capital Income”, Journal of Economic Theory 73, 93-117 Judd, Kenneth L. (1985), “Redistributive Taxation in a Simple Perfect Foresight Model”, Journal of Public Economics 28, 59-83 Judd, Kenneth L. (1987), “The Welfare Cost of Factor Taxation in a Perfect Foresight Model”, Journal of Political Economy 95, 675-709 Judd, Kenneth L. (1999), “Optimal Taxation and Spending in General Competitive Growth Models”, Journal of Public Economics 71, 1-26 Klein, Paul and Jos´e-Victor Rios-Rull (2000), “Time-Consistent Optimal Fiscal Policy”, mimeo, University of Western Ontario Krusell, Per and Antony A. Smith (2000), “Consumption-Savings Decisions with Quasi-Geometric Discounting”, mimeo, University of Rochester Krusell, Per, Burhanettin Kurus¸c¸u and Antony A. Smith (2000), “Equilibrium Welfare and Government Policy with Quasi-Geometric Discounting”, mimeo, University of Rochester Kydland, Finn E. and Edward C. Prescott (1979), “Rules Rather than Discretion: The Inconsistency of Optimal Plans”, Journal of Political Economy 85, 473491 Kydland, Finn E. and Edward C. Prescott (1980), “Dynamic Optimal Taxation, Rational Expectations, and Optimal Control”, Journal of Economic Dynamics and Control 2, 79-91 Lucas, Robert E. and Nancy L. Stokey (1983), “Optimal Fiscal and Monetary Policy in an Economy without Capital”, Journal of Monetary Economics 12, 25

55-93 Maskin, E. and J. Tirole (1993), “Markov Perfect Equilibrium”, mimeo, Harvard University Mendoza, Enrique G., Assaf Razin and Linda L. Tesar (1994), “Effective Tax Rates in Macroeconomics: Cross-Country Estimates of Tax Rates on Factor Incomes and Consumption”, Journal of Monetary Economics 34, 297-323 Miller, Marcus and Mark Salmon (1985), “Policy Coordination and Dynamic Games”, in Willem Buiter and Richard Marston, eds., International Economic Policy Coordination. Cambridge, Cambridge University Press Oudiz, Gilles and Jeffrey Sachs (1985), “Policy Coordination and Dynamic Games”, in Willem Buiter and Richard Marston, eds., International Economic Policy Coordination. Cambridge, Cambridge University Press Park, No-Ho (1991), “Steady-State Solutions of Optimal Tax Mixes in an Overlapping-Generations Model”, Journal of Public Economics 46, 227-246 Ramsey, Frank (1927), “A Contribution to the Optimal Theory of Taxation”, Economic Journal 37, 47-61 Ramsey, Frank (1928), “A Mathematical Theory of Saving”, Economic Journal 38, 543-559 Rios-Rull, Jos´e Victor (1995), “Models with Heterogeneous Agents”, in T.F. Cooley, ed., Frontiers of Business Cycle Research. Princeton, Princeton University Press Sargent, Thomas J. (1987), Macroeconomic Theory Scarth, William (1999), “Alternatives for Raising Living Standards”, mimeo, McMaster University Stokey, Nancy (1991), “Credible Public Policy”, Journal of Economic Dynamics and Control 15, 627-657 Ventura, Gustavo (1996), “Flat Tax Reform: A Quantitative Assessment”, mimeo, University of Illinois Weil, Philippe (!989), “Overlapping Generations of Infinitely-Lived Agents”, Journal of Public Economics 38, 183-198 Yaari, Menahem E. (1965), “Uncertain Lifetime, Life Insurance, and the Theory of the Consumer”, Review of Economic Studies 32, 137-150 Zhu, X. (1992), “Optimal Fiscal Policy in a Stochastic Growth Model”, Journal of Economic Theory 58, 250-289

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Table 1: Parameter Values Parameter Value β 0.75 δ 0.75 α 0.64 θ 3.00 σ 1.75 γ 0.66 ρz 0.40 g 0.20 z 1.00 σz 0.01

Table 2: Steady State Properties Variable τn τk B y g k cy co c/y i/y g/y 25 · k/y b/y n w R r

27

Value 0.165 0.203 -0.008 0.138 0.028 0.029 0.052 0.029 0.591 0.208 0.200 5.220 -0.058 0.333 0.265 1.727 0.377

Table 3: Variability of Tax Rates Model Klein/Rios-Rull (Commitment)

Klein/Rios-Rull (No Commitment)

Data

Capital Income Tax mean s.d. c.v.

0.203 0.110 0.543

-0.002 0.180 85.71

0.650 0.110 0.170

0.283 0.088 0.309

Labor Income Tax mean s.d. c.v.

0.165 0.030 0.181

0.310 0.009 0.028

0.120 0.031 0.250

0.248 0.024 0.097

s.d.: standard deviation c.v.: coefficient of variation The second and third columns are from Klein and Rios-Rull (1999, Tables 3 and 4). The last column is from Chari, Christiano and Kehoe (1995, Table 12.4).

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Table 4: Sensitivity Analysis Sensitivity to β β τn τk B

0.450 0.159 0.239 -0.001

0.550 0.650 0.157 0.159 0.221 0.209 -0.002 -0.005

0.750 0.165 0.203 -0.008

0.850 0.179 0.200 -0.012

0.950 0.187 0.202 -0.016

θ τn τk B

1.500 0.147 0.063 -0.003

2.000 0.155 0.161 -0.007

3.000 0.165 0.203 -0.008

4.000 0.171 0.213 -0.007

5.000 0.175 0.218 -0.006

6.000 0.177 0.220 -0.005

σ τn τk B

1.250 0.169 0.192 -0.009

1.500 0.167 0.198 -0.008

1.750 0.165 0.203 -0.008

2.000 0.163 0.207 -0.008

2.250 0.162 0.210 -0.007

2.500 0.161 0.213 -0.007

Sensitivity to θ

Sensitivity to σ

29