Losing Welfare by Getting Transfers∗ Marcel Fischer†

Bjarne Astrup Jensen†

This version: March 10, 2014

∗ We

are grateful for helpful comments and suggestions from David Lando, Kristian Miltersen, Lasse Heje Pedersen, Peter Norman Sørensen, and seminar participants at Copenhagen Business School. † Copenhagen Business School, Department of Finance, Solbjerg Plads 3, DK-2000 Frederiksberg, Denmark. Marcel Fischer, phone: +45-3815-3628, e-mail: [email protected]; Bjarne Astrup Jensen, phone: +45-3815-3614, e-mail: [email protected].

Abstract Losing Welfare by Getting Transfers This article studies the general equilibrium impact of a redistributive tax system on consumption and investment decisions in a production economy. We show in closed form that taxing households’ income to finance transfers from richer to poorer households can result in Pareto inefficient aggregate production, which in turn can result in welfare losses for net recipients of transfer income. JEL Classification Codes: G11, E21, H24 Key Words: redistributive tax system, portfolio choice, real investment

1

Introduction

Redistribution from richer to poorer households through a redistributive tax system is a wide-spread phenomenon. For instance, the U.S. government has spent more than 1.3 trillion dollars on wealth transfers via social security and income security in 2012. However, surprisingly little is known about the effects of redistributive tax systems on consumption, investments, and welfare in general equilibrium. Two notable exemptions are Sialm (2006) and Fischer and Jensen (2014) who focus on an endowment economy, where aggregate production is exogenously given. In our work, we instead focus on a production economy, where aggregate production is endogenously determined such as in Cox, Ingersoll, and Ross (1985). We extend this line of research by integrating a redistributive tax system in such models. Our results show that taxing households’ income and redistributing tax revenues can heavily alter the intertemporal paths of aggregate consumption and production. More specifically, confirming the classical empirical evidence in Boskin (1978), our results show that aggregate production decreases as tax rates increase. Even when all tax revenues are immediately redistributed to the households in the economy, aggregate production can become inefficient and thus have a negative impact on welfare. On the individual household level, welfare is affected through two channels. First, the redistribution of income from richer to poorer households implies progressive effective tax rates and has the desired effect of increasing poorer households welfare at the cost of reducing richer households’. Second, the Pareto inefficient level of aggregate production leads to welfare losses for all households. Overall, redistribution has the desired effect of increasing poorer households’ welfare levels at the expense of decreasing richer households’. However, the Pareto inefficient aggregate production also implies that even households that are net recipients, in the sense that they receive transfers in excess of their tax payments, can be better off in the absence of a redistributive tax system. This effect is particularly pronounced for middle-class households. Throughout the last century, individual income taxation has been introduced in most industrialized countries around the world. Simultaneously, social insurance programs and income support programs for households with low incomes have been implemented. Research on optimal taxation goes back to Mirrlees (1971) who studies optimal taxation of labor income when households differ by their abilities to and their efforts made to earn

1

labor income.1 However, this literature is often not concerned with distributional issues, which are the focus of our work. Other strands of literature focus on how taxes affect household behavior and asset prices. Recent contributions include Dai, Maydew, Shackelford, and Zhang (2008), Sialm (2009), Barro and Redlick (2011), Pástor and Veronesi (2012, 2013), and Mertens and Ravn (2013). Despite all the efforts that have been made, still relatively little is known about how a redistributive tax system that taxes income and redistributes tax revenues affects welfare in general equilibrium – especially on the individual household level. Our work contributes to the literature by deriving these welfare effects in closed form. Matching the empirical evidence in Barro and Redlick (2011), it shows a direct impact of redistribution on intertemporal behavior. Specifically, it demonstrates that households increase immediate consumption at the cost of reducing the future GDP. This is so, because income taxation alters the pricing kernel by changing the relative price between present and future consumption in such a way that future consumption becomes relatively more expensive. As a consequence, redistributive tax systems can result in a Pareto inefficient intertemporal allocation of resources, thus leading to welfare losses on the aggregate level despite all tax revenues being immediately returned to the households. These welfare effects can be so pronounced that even net recipients of transfer income can be better off in a setting without a redistributive tax system. This effect is particularly relevant for middle-class households that may lose welfare by getting transfers. The remainder of this paper is organized as follows. Section 2 introduces our general equilibrium model and its closed form solution. Section 3 illustrates the quantitative effects. Section 4 concludes. Formal derivations and proofs are provided in the appendix.

2 2.1

The Model The Economy

We consider a stylized model of a production economy with n households and a financial market in which two assets can be traded. First, households can trade a locally risk-free 1

If the government taxes income attempting to tax households with high abilities, households are discouraged from exerting effort. Saez (2002) therefore argues that an optimal tax system imposes a negative tax rate on poorer households’ income and imposes a large range of income over which it is phased out. Similarly, Golosov, Troshkin, Tsyvinski, and Weinzierl (2013) suggest taxing goods preferred by individuals with high abilities and Weinzierl (2011) proposes age-dependent taxes.

2

asset paying a pre-tax return of rt from time t to t+1. This asset comes in zero net supply; i.e., if some household wants to hold a positive fraction of its wealth in the risk-free asset, the market equilibrium has to bring about an interest rate that makes the remaining households willing to issue such a risk-free asset. Second, households can invest in a real investment opportunity that represents ownership to production means. The initial shares of production means of the households are denoted by α0−,j > 0, j = 1,2, . . . ,n, i.e., Pn j=1 α0−,j = 1. Production means are perishable in the sense that an investment only generates an output for the coming period. This output can either be consumed or be reinvested in preparation for consumption in the following periods. Hence, the only way of storing for future consumption is by reinvesting some of the present consumption opportunity in the production process. In our production economy, aggregate consumption is thus the result of households’ intertemporal decisions about how to allocate the available production result every period between immediate consumption and investments to produce future consumption. Risk is modeled by assuming that the production process is stochastically homogeneous of degree one, cf. the classical production economy model in Cox, Ingersoll, and Ross (1985). Aggregate production is distributed to households in proportion to their ownership. The output produced and available at time at time t depends on the evolution of the economy. a a It is given by the product Gt It−1 , where It−1 denotes the aggregate investment made at time t−1, and Gt > 1 is the gross growth factor per unit of investment made at time t−1.2 For simplicity, we assume that the growth rates are i.i.d. copies of a binomial variable G.3

2.2

The Redistributive Tax System

Following Fischer and Jensen (2014), we consider a government that wants to reduce the disparity in lifetime consumption opportunities across households by imposing transfers from households with relative high endowments to households with relatively low endowments. However, the government recognizes that such transfers are associated with friction costs, and there may also be political preferences as to the desired extent of redistribution. The higher the degree of redistribution, the higher the implicit costs associated with such transfers. 2 3

More generally, we use the superscript a to denote aggregate quantities. Neither the i.i.d. assumption nor the binomial nature of the growth factors are necessary restrictions. The model can be solved for in more generality. These assumptions are solely made in order to avoid an excessive amount of notation.

3

In real life, ad hoc redistributions of wealth are typically not a feasible policy instrument available to governments. Therefore, redistributions from richer to poorer households are usually spread out over longer time horizons and are implemented by a repeated taxation of households and a redistribution of tax revenues in an attempt to harmonize lifetime consumption opportunities. Taking the present value of future taxes and transfers into account, such a redistribution mechanism increases the initial endowments of the relatively poor (the net recipients of transfers) and decreases those of the relatively rich (the net payers of transfers). That is, it leads to similar effects as a one-time redistribution of initial endowments. To introduce a tax mechanism that reduces disparities in lifetime consumption opportunities, we identify the consumption opportunity of a given household j at time t by its net income, i.e., its share of the production outcome. We let N It−,j denote the value of this net income in period t before any redistribution has taken place. The aggregate a net income in the economy is (G − 1)It−1 , and the government acts to harmonize the consumption opportunities given by this net income, but faces friction costs or political constraints as to the extent of this harmonization. With a quadratic objective function, the government’s problem at time t becomes: 2 n  X 1 a min n N It+,j − (G − 1)It−1 + k (N It+,j − N It−,j )2 {N It+,j }j=1 n j=1 s.t. n X

a N It+,j = (G − 1)It−1

(1)

(2)

j=1

N It+,1 ≤ N It+,2 ≤ . . . ≤ N It+,n ,

(3)

where N It+,j is the net income of household j after redistribution, k ≥ 0 indicates the severeness of the friction costs, and the inequalities in (3) make sure that the ascending ordering of households with respect to their net income before and after taking taxes and redistributions into account remains unchanged.4 The solution to the government’s 4

Other work, such as Krusell, Quadrini, and Ríos-Rull (1996) determines the optimal tax rate using political-equilibrium theory.

4

optimization problem in Equations (1) to (3) is a linear feedback rule: 1 1 k a ⇔ N It−,j + (G − 1) It−1 1+k  1+kn  1 1 a = . N It−,j − (G − 1) It−1 1+k n

N It+,j = N It−,j − N It+,j

(4) (5)

This redistribution mechanism can be implemented by taxing household income at the 1 combined with an equal allocation of tax revenues to the households constant rate τ = 1+k in the economy. The net result of such a redistributive tax system is that households with an income below (above) the average are net recipients of (contributors to) the taxation and redistribution mechanism. When k = 0, reflecting the absence of friction costs, the optimal tax rate is 100%, and there is a complete harmonization of net income.5 When k → ∞, the optimal tax rate goes to zero. In that limiting case, attempting a harmonization is infinitely costly or undesirable. When k = 1, the optimal tax rate is 50%. The idea of using a linear redistribution scheme is not only optimal for a government with a quadratic objective function, it is also analytically convenient and widely used in public economics. It goes back to earlier work by Romer (1975) and Meltzer and Richard (1981) and was more recently used in, e.g., Alesina and Angeletos (2005). The optimal redistribution mechanism summarized in Equation (5) implies that the government neither builds up wealth nor debt. Within the time horizon of our model, any government debt must be settled through tax payments by the same households.6 Consequently, government debt would never be considered net wealth by the households, cf. also the reasoning in Barro’s seminal work (Barro, 1974). Overall, the redistribution mechanism in Equation (5) makes sure that the goal of harmonizing lifetime consumption opportunities is attained. However, by taxing investment profits, the redistribution mechanism affects the households’ marginal investment decision.

2.3

The Household’s Optimization Problem

Each household maximizes its present discounted utility from consumption subject to its intertemporal budget constraint. Households have time-additive CRRA utility functions 5

Note that due to differences in initial endowments, this does not imply a perfect harmonization of lifetime consumption opportunities. 6 We explicitly disregard the possibility that the government can embark on a Ponzi scheme and ignore its long-run budget constraint.

5

Table 1 Definition of variables Variable ρ γ It,j α0−,j αt,j βt,j Ct,j τ Rt Gt bt R Wt,j n N

Description The households’ common utility discount factor The households’ common relative risk aversion coefficient Household j’s investment in the production process at time t Household j’s initial endowment Household j’s share of aggregate investments Number of units of risk-free asset held by household j from time t to t+1 Household j’s consumption at time t Tax rate Gross risk-free rate from time t to t+1 Gross growth factor of investment from time t−1 to t Gross risk-free rate after tax from time t to t+1 Household j’s wealth level at time t before consumption Number of households in the economy Length of investment horizon

with risk aversion parameter γ ≥ 0, i.e., the utility from a consumption of C is given by

U (C) =

  C 1−γ 1−γ

if γ 6= 1

(6)

ln (C) if γ = 1. We summarize the notation and variables used in Table 1. The optimization problem for household j is then given by max

t=N −1 {{Ct,j }t=N } t=0 ,{It,j ,βt,j }t=0

U (C0,j ) +

N X

ρt E0 [U (Ct,j )]

(7)

t=1

s.t. Wt,j = Ct,j + It,j + βt,j

(8)

τ a bt−1 Wt,j = It−1,j (Gt − (Gt − 1) τ ) + (Gt − 1) It−1 + βt−1,j R n IN,j = βN,j = 0

(9) (10)

P a where W0,j is household j’s (exogenously given) initial endowment and It−1 = ni=1 Ii,t−1 is the aggregate investment made at time t − 1. Equation (8) is the household’s budget constraint. Equation (9) describes the household’s evolution of wealth. It shows that the household’s entering wealth level at time t before consumption consists of three terms. The

6

first term is the household’s investment in the previous period including its profits after taxes. The second term is the household’s transfer income. The third term describes the household’s holding in the risk-free asset including its interest after taxes over the previous period. Equation (9) can be rewritten as Wt,j = It−1,j Gt − (Gt −

a τ 1) It−1

 αt−1,j

1 − n

 bt−1 , + βt−1,j R

(11)

a where αt−1,j = It−1,j /It−1 is household j’s share of aggregate investment. The second term in Equation (11) reveals that the statutory tax rate, τ , is multiplied by (αt−1,j − 1/n), thus implying a dependency of the effective tax rate on the household’s share of aggregate investment. Specifically, the effective tax rate increases in αt−1,j . That is, richer households, that hold larger shares of the risky asset, face higher effective tax rates. In other words, even though statutory tax rates are constant in our model, effective tax rates are progressive.

2.4

The Closed-Form Solution

In this section, we derive households’ general equilibrium consumption-investment strategies in closed form. The results are summarized in Theorem 1: Theorem 1. Assume that the growth factors Gt are i.i.d. copies of a binomial variable G and G > 1. We denote the two possible realizations of G as G+ and G− . Then the general equilibrium solution to the optimization problems for n households differing only in their initial endowments, as stated in Equations (7) to (10), is as follows: 1. Aggregate consumption and investment follow binomial processes as shown in (12), a where the fraction Ft of total output Wta = It−1 Gt at time t that is reinvested is state-independent: Cta = (1 − Ft )Wta , Ita = Ft Wta .

(12)

2. The fraction Ft of wealth reinvested from time t to t + 1 decreases over time with the natural boundary condition FN = 0. Ft follows the backwards difference equation Ft =

1 , 1 + (1 − Ft+1 )H

7

(13)

where H=

h h  ρ −1/γ h
−γ ii−1/γ
−γ
1−γ i
1−γ + G− . (14) + τ G+ + G− (1 − τ ) G+ 2

The solution to (13) is

Ft =

 

1−H N −t 1−H N −t+1

for H 6= 1



N −t N −t+1

for H = 1.

(15)

3. The growth rate of consumption is the same for all households and follows the i.i.d. binomial process with distribution Ct+1,j d 1 = G. Ct,j H

(16)

It decreases as the tax rate increases. The pricing kernel and the equivalent martingale measure Q can be expressed in terms of the growth factor:  ρ

a Ct+1 Cta

−γ

= ρH γ G−γ

(17) −γ

−γ

q=

(G− ) (G+ ) , 1 − q = . (G+ )−γ + (G− )−γ (G+ )−γ + (G− )−γ

(18)

b is constant over time, but it is a decreasing 4. The gross risk-free rate after tax, R, function of the tax rate τ : 1−γ

1−γ

+ + (G− ) b = (1 − τ ) (G ) R + τ = (1 − τ )EQ [G] + τ. (G+ )−γ + (G− )−γ

(19)

The gross risk-free rate before tax, R, is constant over time and independent of the tax rate τ : 1−γ 1−γ (G+ ) + (G− ) R= = EQ [G] . (20) (G+ )−γ + (G− )−γ 5. The allocation of market risk is in accordance with a linear sharing rule relative to the wealth distribution after tax. The risk-free asset plays a role in order to establish

8

the linear sharing rule. The position in the risk-free asset is determined by βt,j

τ = Ita b R



1 − αt,j n

 (21)

where αt,j ≡ It,j /Ita is household j’s share of aggregate investment. Household j’s position in the risk-free asset only depends on its share of aggregate investments in the risky asset, αt,j , and aggregate investments, Ita , but otherwise not on the distribution of wealth among other households. 6. The individual household’s consumption policy is given by a constant share of aggregate production, which we denote by ωj : Ct,j τ ωj ≡ = αN −1,j (1 − τ ) + = a (1 − Ft )It−1 Gt n

 αN −1,j

1 − n

 (1 − τ ) +

1 . n

(22)

The consumption share ωj can be expressed in closed form as W0,j Z−1 − 1 + ωj = W0a Z−1



W0 1 − a n W0

 =

1 W0,j Z−1 − 1 1 + , Z−1 W0a Z−1 n

(23)

where Z−1 is given by the decreasing sequence of variables Zt following the backwards b difference equation in (24) with Y ≡ R/R: Zt = 1 − Ft+1 + Ft+1 Y Zt+1 , ZN = 0.

(24)

The explicit solution to this difference equation is7 Zt =

N X

" Y j−(t+1)

j−1 Y

# Fp (1 − Fj ) .

(25)

p=t+1

j=t+1

Since Z−1 > 1, the consumption share is a weighted average of the share of wealth in the initial wealth distribution and the uniform distribution of wealth. In particular, it is otherwise independent of the distribution of wealth among other households. 7

We adopt the standard convention that the sum (product) over an empty index set is zero (one).

9

7. The equity share, αt,j , is given by the following dynamic relations:     1 1 (1 − τ ) = Zt ωj − αt,j − n n αt,j

⇔

1 Zt 1 = + n Z−1 1 − τ



W0,j 1 − a W0 n

 .

(26)

It increases (decreases) over time for households that are net recipients of (net contributors to) transfer income. Household j’s share of wealth invested only depends on its initial share of aggregate wealth, but otherwise not on the distribution of wealth among other households. The share of wealth invested, αt,j + βt,j , is given by αt,j + βt,j

1 = + Y Zt n



W0,j 1 − a W0 n

 .

(27)

If the initial endowment is below (above) the average, the share of wealth invested remains below (above) the initial endowment over the entire investment horizon. 8. The level of the received net transfer payment for each household at time t is  τ

 1 a − αt−1,j It−1,j (G − 1) . n

(28)

There is a fixed relation, independent of time and state, between the net transfer payments received in the boom and the bust states, respectively. This ratio is given by (G+ −1)/(G− −1). 9. The sequence of aggregate consumption is given by " W0a (1 − F0 ),F0 G1 (1 − F1 ),F0 F1 G1 G2 (1 − F2 ), . . . ,

t−1 Y

!

#

(Fj Gj+1 ) (1 − Ft ) =

j=0

t=N

 " W0a

t Y j=1

! Gj

H N −t (1 − H) 1 − H N +1

#t=N =

W0a

t  Y  Gj  j=1

t=0

!

H N −t    N P Hi i=0

10

t=1

.

(29)

10. The utility from aggregate consumption is given by U

HN W0a PN i i=0 H

!

j N  X Y j=0

H


  Y N +1 N 1 − H (1 − H) H
. = U W0a Y 1 − H N +1 1− H

(30)

For the log-investor (γ = 1), the variable H simplifies to H = Yρ , and the utility from aggregate consumption can be expressed as   N 1 − ρN +1 a H (1 − H) log W0 + 1−ρ 1 − H N +1  −    + G ρ − ρN +1 (1 + N (1 − ρ)) G + log 0.5 log (ρ 6= 1) (31) H H (1 − ρ)2    +  −   N G G N (N + 1) a Y (1 − Y ) + 0.5 log + log (ρ = 1). (N + 1) log W0 N +1 1−Y H H 2 (32) 11. Increasing the tax rate lowers the intertemporal growth path of consumption and also lowers reinvestment rates as the values of Ft decrease for all t as a result of an increasing tax rate. As a consequence, the utility from aggregate consumption, as given in Equations (30) to (32), decreases with an increasing tax rate. 12. Household j’s utility from consumption is given by the expressions in Equations (30) to (32) with W0a substituted by ωj W0a . Proof [Theorem 1] The details of the derivations are found in Appendix A. In a world without taxation and redistribution, the households’ initial endowments, determine the lifetime consumption opportunities. Given the initial endowments, the intertemporal profile of consumption in a Pareto optimum is determined by each household’s optimal consumption-investment policy in general equilibrium. As soon as the redistributive tax system is introduced, households deviate from the Pareto optimal intertemporal allocation of consumption opportunities (Theorem 1, item 3). Matching the empirical evidence in Barro and Redlick (2011), our model predicts that households increase immediate consumption at the cost of reducing the future GDP. Our results in Theorem 1, item 10 show that this has a negative impact on welfare on the aggregate level. Utility from aggregate consumption decreases as the tax rate increases.

11

That is, despite all tax revenues being immediately returned to the households in the economy, aggregate welfare costs are higher, the more redistribution takes place. On the individual household level welfare effects can be decomposed into two components. First, there is a direct effect depending on whether a household is a net recipient or payer of transfer income. Second, the Pareto inefficient intertemporal allocation of consumption has a negative welfare effect that is independent from the individual household’s endowment. As we demonstrate in more detail in our numerical results in section 3, this second effect can be so strong that even net recipients of transfer income can be better off in a world without taxation and transfer income. In the absence of a redistributive tax system, the asset allocation policy is static. All households own a share of aggregate production equal to their initial endowment, and there is no need for a bond market. In the presence of a redistributive tax system, however, the asset allocation policy becomes dynamic. It calls for a time-varying equity share, where net recipients of transfer income reduce their equity position below their initial endowment share.8 It reflects, that tax revenues are subject to macroeconomic risk; hence, net recipients of transfer income are already exposed to that risk via this channel. Furthermore, the bond market is necessary in order to implement the optimal policy. Net recipients of transfer income have long bond market positions, whereas net contributors have short bond market positions.

3

Quantitative Effects

In this section, we turn to quantifying the impact of the redistributive tax system on the intertemporal allocation of aggregate consumption as well as welfare effects on both aggregate and individual level.

3.1

Calibration

For the numerical examples presented in this section, we consider households with logpreferences, i.e., γ = 1, and a utility discount factor of ρ = 1. The length of the investment horizon is set to N = 60 periods. For the investment, we assume an expected return of 5% and a volatility of 2%, thus implying real gross returns of G+ = 1.07 and G− = 1.03.9 8 9

See also Fischer and Jensen (2014) that document a similar effect in an endowment economy. Our results are qualitatively robust to varying our base-case parameter values. We therefore restrict the presentation of our numerical examples to these base-case parameter values.

12

Table 2 Base-case parameter values Description

Parameter

Value

γ ρ

1 1 1.07 1.03 60

Degree of risk aversion Utility discount factor Expected real gross return investment boom Volatility real gross return investment bust Length of investment horizon

G+ G− N

Aggregate consumption relative to no−tax case

Figure 1 Aggregate consumption relative to no-tax case

4 3 2 1 0 0

0 20 50

40 100 60

Tax rate (in %)

Time

This figure shows the intertemporal evolution of aggregate consumption relative to the no-tax case.

We refer to this set of parameters as our base-case parameters throughout. They are summarized in Table 2.

3.2

Aggregate Consumption

Having introduced our base-case parameter choice, we next turn to quantitatively illustrating the impact of the redistributive tax system on aggregate consumption over the investment horizon. In Figure 1, we depict aggregate consumption relative to the case without taxation and redistribution as a function of time and the tax rate. Confirming our analytical results from Theorem 1, Figure 1 shows that the redistributive tax system with an income tax alters the intertemporal allocation of aggregate consumption. More specifically, the redistributive tax system decreases the after-tax return from

13

Change aggregate wealth equivalent (in %)

Figure 2 Change in aggregate wealth equivalent 0 −5 −10 −15 −20 −25 −30 0

20

40 60 Tax rate (in %)

80

100

This figure shows the impact of a redistributive tax system on the aggregate wealth equivalent as a function of the tax rate.

investing, thus favoring early over late consumption. Quantitatively, the intertemporal reallocations of aggregate consumption are huge. For instance, at a tax rate of τ = 20%, aggregate consumption increases by more than 30% at time t = 0, whereas it decreases by more than 25% at time t = 60. These effects are further amplified for higher levels of the tax rate. The intertemporal reallocation of consumption also has important welfare implications, because it causes deviations from the Pareto optimal intertemporal allocation of resources attained for a tax rate of τ = 0%. We investigate these welfare effects on both the aggregate and the individual level in more detail throughout the next two sections.

3.3

Welfare Effects on Aggregate Level

Our results in Theorem 1, item 10 show that once an income tax is imposed, the redistributive tax system results in a Pareto inefficient intertemporal allocation of aggregate consumption and thus in welfare losses. In this section, we quantify welfare losses from aggregate consumption in monetary terms through the wealth equivalent. We ask which change in initial aggregate wealth, W0a , is required in a setting with no taxes to attain the same level of utility from aggregate consumption as in a setting with taxation and redistribution. Figure 2 depicts the change in the aggregate wealth equivalent as a function of the tax rate. 14

Aggregate welfare costs from the redistributive income tax system increase progressively in the tax rate once an income tax is imposed. For instance, while a tax rate of 20% implies a decrease in the aggregate wealth equivalent by 1.4%, a tax rate of 30% results in a decrease by 3.1%. That is, in the absence of taxation and redistribution, the same utility from aggregate consumption can be attained with a 1.4% or 3.1% lower level of initial wealth, W0a , than with a 20% or a 30% tax rate. Overall, our results in Figure 2 show the magnitude of the welfare losses from aggregate consumption arising from the Pareto inefficient intertemporal allocation of consumption opportunities. Welfare losses are highest for rather extreme values of the tax rate and are more modest for levels of the tax rate than can be found in current tax laws around the world. We next turn to demonstrating that welfare effects on the individual household level can be of a much higher order of magnitude. We also demonstrate that recipients of transfer income can be better off in a setting without taxation and redistribution.

3.4

Welfare Effects on Household Level

In this section, we illustrate the welfare consequences of redistributive tax systems under taxation of income and consumption, respectively, on the individual household level. Theorem 1 shows that these welfare consequences solely depend on a household’s initial endowment relative to the average household’s initial endowment, but otherwise not on the distribution of wealth among households or the number of households. It is thus sufficient to differentiate cases based on households’ initial endowments relative to the average endowment. Overall, welfare is affected through two channels. First, a Pareto inefficient intertemporal allocation of consumption opportunities has a negative impact on all households. Second, the redistribution of consumption opportunities has a positive impact on welfare for the net recipients of transfers. Our results in Figure 3 depict the changes in individual households’ wealth equivalents for households with different levels of initial endowments as a function of the tax rate. That is, we report what change in the initial endowment at time t = 0 is necessary in the absence of a redistributive tax system to make the household as well off as with the redistributive tax system. The upper two graphs in Figure 3 show welfare effects for relatively poor households with initial endowments of 10% and 50% of the average initial endowment, respectively. Given that these households receive significant amounts of transfers, it is not surprising that they 15

Figure 3 Changes in individual households’ wealth equivalents 50% of average initial endowment Change individual wealth equivalent (in %)

Change individual wealth equivalent (in %)

10% of average initial endowment 450 400 350 300 250 200 150 100 50 0 0

20

40 60 Tax rate (in %)

80

40 35 30 25 20 15 10 5 0 0

100

0 −5 −10 −15 −20 −25 −30 0

20

40 60 Tax rate (in %)

80

40 60 Tax rate (in %)

80

100

150% of average initial endowment

5

Change individual wealth equivalent (in %)

Change individual wealth equivalent (in %)

95% of average initial endowment

20

100

0

−10

−20

−30

−40

−50 0

20

40 60 Tax rate (in %)

80

100

This figure shows the impact of a redistributive tax system on individual households’ wealth equivalents as a function of the tax rate.

benefit from the redistributive tax system. However, the increases in the individual wealth equivalents are not linear in the tax rate, reflecting the negative welfare consequences from the Pareto inefficient intertemporal allocation of consumption increase as the tax rate does. This effect is so strong, that welfare effects under income taxation are not monotonically increasing, but peaking at tax rates of 78% and 54%, respectively. The two lower graphs in Figure 3 depict welfare effects for a middle-class household with an initial endowment of 95% of the average initial endowment and a richer household with an initial endowment of 150% of the average initial endowment. That is, the former household is still a net recipient of transfer income, while the latter is a net contributor to the redistributive tax system.

16

For the middle-class household that is a net recipient of transfer income, welfare effects are negative for a wide range of tax rates. That is, the negative consequences of the Pareto inefficient intertemporal allocation of consumption outweigh the positive welfare effects of the received transfers. For this household, welfare effects peak at a tax rate of 9% and become negative at a tax rate of 18%. That is, despite being net recipient of transfer income, these households prefer living in a world without a redistributive tax system compared to the redistributive tax system with income taxation when the tax rate exceeds 18%. These households essentially lose welfare by getting transfers due to the negative side-effects of the Pareto inefficient intertemporal allocation of consumption. For the household with an endowment of 150% above the average one, welfare effects are always negative, reflecting that this household is a net contributor to the system. Overall, the redistributive tax system improves poorer households’ welfare at the expense of decreasing richer households’. However, our results also stress that the redistributive tax system has important side-effects. Specifically, it causes a Pareto inefficient intertemporal allocation of consumption that can even lead to welfare losses for net recipients of transfer income, such as middle-class households.

4

Conclusion

In this paper, we investigate the general equilibrium implications of a redistributive tax system in a production economy in closed form. In our model, households have CRRA preferences and only differ by their initial financial endowments. The redistributive tax system aims at reducing the disparity across households in lifetime consumption opportunities by making richer households net contributors to poorer households. Implementing such a redistributive tax system affects households’ consumption and investment behavior. In contrast to the classical “no trade in financial assets” result in the asset pricing and asset allocation literature with identical CRRA preferences, our results show that such a redistributive tax system causes agents to engage in active trading. This is so, because the transfer of consumption opportunities simultaneously involves a transfer of financial risk that households hedge against by active trading. In particular, poorer households invest less in risky assets than in the absence of the redistributive tax system. Simultaneously, the redistributive tax system results in Pareto inefficient aggregate production despite all tax revenues being immediately redistributed to the households in the economy. The inefficiency reflects that the redistributive tax system alters the pricing kernel and thereby favors present over future consumption. As a consequence of this Pareto 17

inefficiency, even households that are net recipients may be better off in the absence of a redistributive tax system.

References Alesina, A., and G.-M. Angeletos, 2005, “Fairness and Redistribution,” American Economic Review, 95(4), 960–980. Barro, R., 1974, “Are Government Bonds Net Wealth?” Journal of Political Economy, 82(6), 1095–1117. Barro, R. J., and C. J. Redlick, 2011, “Macroeconomic Effects from Government Purchases and Taxes,” Quarterly Journal of Economics, 126(1), 51–102. Boskin, M. J., 1978, “Taxation, Saving, and the Rate of Interest,” Journal of Political Economy, 86(2), 3–27. Bronstein, I., K. Semendyayev, G. Musiol, and H. Mühling, 2007, Handbook of Mathematics (5th ed.), Springer. Cox, J. C., J. E. Ingersoll, and S. A. Ross, 1985, “An Intertemporal General Equilibrium Model of Asset Prices,” Econometrica, 53(2), 363–384. Dai, Z., E. Maydew, D. A. Shackelford, and H. H. Zhang, 2008, “Capital Gains Taxes and Asset Prices: Capitalization or Lock-in?” Journal of Finance, 63(2), 709–742. Fischer, M., and B. A. Jensen, 2014, “Taxation, Transfer Income and Stock Market Participation,” Review of Finance, forthcoming. Golosov, M., M. Troshkin, A. Tsyvinski, and M. Weinzierl, 2013, “Preference Heterogeneity and Optimal Capital Income Taxation,” Journal of Public Economics, 97(1), 160–175. Krusell, P., V. Quadrini, and J.-V. Ríos-Rull, 1996, “Are Consumption Taxes Really Better than Income Taxes?” Journal of Monetary Economics, 37(3), 475–503. Meltzer, A. H., and S. F. Richard, 1981, “A Rational Theory of the Size of Government,” Journal of Politial Economy, 89(5), 914–927.

18

Mertens, K., and M. O. Ravn, 2013, “The Dynamic Effects of Personal and Corporate Income Tax Changes in the United States,” American Economic Review, 103(4), 1212– 1247. Mirrlees, J. A., 1971, “An Exploration in the Theory of Optimum Income Taxation,” Review of Economic Studies, 38(2), 175–208. Pástor, L., and P. Veronesi, 2012, “Uncertainty about Government Policy and Stock Prices,” Journal of Finance, 67(4), 1219–1264. Pástor, L., and P. Veronesi, 2013, “Political Uncertainty and Risk Premia,” Journal of Financial Economics, 110(3), 520–545. Romer, T., 1975, “Individual Welfare, Majority Voting, and the Properties of a Linear Income Tax,” Journal of Public Economics, 4(2), 163–185. Saez, E., 2002, “Optimal Income Transfer Programs: Intensive versus Extensive Labor Supply Responses,” Quarterly Journal of Economics, 117(3), 1039–1073. Sialm, C., 2006, “Stochastic Taxation and Asset Pricing in Dynamic General Equilibrium,” Journal of Economic Dynamics and Control, 30(3), 511–540. Sialm, C., 2009, “Tax Changes and Asset Pricing,” American Economic Review, 99(4), 1356–1383. Weinzierl, M., 2011, “The Surprising Power of Age-Dependent Taxes,” Review of Economic Studies, 78(4), 1490–1518.

19

Appendix A

Proof of Theorem 1

The optimization problem stated in Equations (7) to (10) is equivalent to max

t=N −1 } {{Ct,j }t=N t=0 ,{It,j ,βt,j }t=0

U (C0,j ) +

N X

ρt E0 [U (Ct,j )]

(A.1)

t=1

s.t. 

Ct,j = It−1,j (1 − τ ) +

a τ It−1

n



Gt − It,j − βt,j

  1 a bt−1 (A.2) + It−1,j − It−1 τ + βt−1,j R n

C0,j = W0,j − I0,j − β0,j

(A.3)

IN,j = βN,j = 0.

(A.4)

For a general number of periods N , we have the following Lagrangian for each of the household optimization problems given in (7) to (10): L =U (C0,j ) +

N X

ρt E0 [U (Ct,j )] − λ0 (C0,j − W0−,j − I0 β0,j ) −

t=1

N X

bt−1 > < λt,j ,βt,j − βt−1,j R

t=1

N −1 X



 τ a 1 a − < λt,j ,Ct,j − It−1,j (1 − τ ) + It−1 Gt − It−1,j − It−1,j τ + It,j > n n t=1     τ 1 (A.5) − < λN,j ,CN,j − IN −1,j (1 − τ ) + INa −1 GN − IN −1,j − INa −1 τ > , n n 



where < , > is the inner product over the states. The first-order conditions are: λt,j =

 ρ t 2

−γ Ct,j

t = 0,1, . . . ,N

1 λt,j = Et [λt+1,j ((1 − τ ) Gt+1 + τ )] t = 0,1, . . . ,N − 1 2 1 bt Et [λt+1,j ] λt,j = R t = 0,1, . . . ,N − 1 2 plus the original constraints.

20

[Ct,j ]

(A.6)

[It,j ]

(A.7)

[βt,j ]

(A.8)

Proof of items 1, 2, and 3 To continue, observe that the first-order conditions are homogeneous in the following t=N −1 −1 sense: If a given solution, ({Ct,j }t=N ; {βt,j }t=N ; {λt,j }t=N t=0 t=0 ; {It,j }t=0 t=0 ), satisfies the conditions for a given value of W0,j , then t=N −1 −1 (x{Ct,j }t=N ; x{βt,j }t=N ; x−γ {λt,j }t=N t=0 ; x{It,j }t=0 t=0 t=0 )

also satisfies the conditions for x · W0,j .10 This proportionality property implies that the fraction of total output consumed, 1 − Ft , and complimentary, reinvested, Ft , is identical across states. The marginal utilities of consumption, together with the time preference parameter ρ, form the – uniquely determined – pricing kernel ρ (Ct+1 /Ct )−γ , which is identical across individual households. Hence, the growth proces for consumption is the same for all household and equal to the aggregate growth rate. This also implies a linear sharing rule. From (A.6) it then follows that ρ λt+1,j = λt,j 2



Ct+1,j Ct,j

−γ

ρ = 2



a Ct+1 Cta

−γ .

(A.9)

Combining this with the evolution of wealth we have the following relations, where we added a time index to the growth factor to avoid confusion: a a Cta = (1 − Ft )It−1 Gt , Ct+1 = (1 − Ft+1 )Ita Gt+1

(A.10)

Ita = Ft Wta

(A.11)

a Ct+1 1 − Ft+1 Ita Gt+1 1 − Ft+1 = = Ft Gt+1 . a a Ct 1 − Ft It−1 Gt 1 − Ft

(A.12)

and therefore

and

Since the terms in front of Gt+1 are state independent, this determines the martingale measure as −γ −γ (G+ ) (G− ) q= , 1−q = , (A.13) (G+ )−γ + (G− )−γ (G+ )−γ + (G− )−γ 10

Observe also that the Lagrangian is homogeneous of degree 1 − γ.

21

which verifies Equation (18). To verify (13) and (14) we proceed as follows, using the first order condition (A.7) in aggregate form, cf. (A.9): "  #  −γ a Ct+1 λt+1,j [(1 − τ )Gt+1 + τ ] = Et ρ 1 = Et 2 [(1 − τ )Gt+1 + τ ] λt,j Cta  −γ   ρ (1 − Ft+1 )Ita G+ t+1 + (1 − τ )G + τ + = t+1 −γ 2 (Cta )  −γ   ρ (1 − Ft+1 )Ita G− t+1 − (1 − τ )G + τ . t+1 −γ 2 (Cta ) 

(A.14)

(A.15)

We can isolate Cta from here and get Cta = Ita (1 − Ft+1 )H  ρ −1/γ h h h

i

ii−1/γ + 1−γ − 1−γ + −γ − −γ H= (1 − τ ) G +τ G + G + G 2 Wta = Cta + Ita = Ita [1 + (1 − Ft+1 )H]

(A.16) (A.17)

Ita =

1 Wa 1 + (1 − Ft+1 )H t

(A.18)

Cta =

(1 − Ft+1 )H Wta 1 + (1 − Ft+1 )H

(A.19)

Ft =

1 . 1 + (1 − Ft+1 )H

(A.20)

This proves (13) and (14). It remains to be shown that the solution for Ft in (15) produces a decreasing sequence over time. For H = 1 this is obvious. Assume H < 1. Then 1 − H N −t 1 − H N −t−1 > ⇔ 1 − H N −t+1 1 − H N −t (1 − H N −t )2 > (1 − H N −t−1 )(1 − H N −t+1 ) ⇔ Ft > Ft+1 ⇔

1 + H 2N −2t − 2H N −t > 1 − H N −t−1 − H N −t+1 + H 2N −2t ⇔ (H − 1)2 > 0,

(A.21)

which is obviously true. The proof for the case H > 1 is analogous. To prove (16) and (17) we first observe that the growth rate of consumption is identical across households, cf., e.g., Equation (A.9). By elementary algebraic manipulations on

22

Equation (13), which defines the backward difference equation for Ft , it follows that 1 1 − Ft+1 Ft = , 1 − Ft H

(A.22)

which proves Equation (16). Equation (17) is a direct consequence of this.

Proof of item 4 The interest rate after tax is determined using the pricing kernel (17):
i ρ γ h +
−γ − −γ −1 b + G G R = H 2  + −γ −γ  (G ) + (G− ) ρ2      = 2 ρ (1 − τ ) (G+ )1−γ + (G− )1−γ + τ (G+ )−γ + (G− )−γ 1 1 = = . 1−γ 1−γ + − ) +(G ) (1 − τ )EQ [G] + τ + τ (1 − τ ) (G −γ −γ (G+ ) +(G− )

(A.23) (A.24) (A.25)

This proves (19). The discount factor before tax in (20) follows directly from its definition.

Proof of item 5 The linear sharing rule – and, consequently, the bond positions in (21) – is a result of the fact that the growth rate of consumption is the same for all individuals. In order for the sharing rule to be linear, it is necessary to eliminate predictable terms from the budget equation. These terms are the last two of the five entering terms in the budget equation (A.2): 

Ct,j = It−1,j (1 − τ ) +

a τ It−1

n



Gt − It,j − βt,j

  1 a b + It−1,j − It−1 τ + βt−1,j R. n

(A.26)

Hence, it is necessary to choose the bond market position as shown in (21): βt,j

τ = b R



1 a I − It,j n t



τ = Ita b R



 1 − αt,j , t = 0,1, . . . ,N − 1. n

23

(A.27)

Proof of items 6 and 7 We make use of the optimal bond position as given in (21) to eliminate the bond-position variables βt,j and βt−1,j in (A.2). For t = 1,2, . . . ,N − 1 we then have the relations:   τ 1 a τ a  I − It ⇒ Ct = It−1 (1 − τ ) + It−1 G − It − b n t n R    τ τ Ita 1 Ct Ita = αt−1 (1 − τ ) + − − αt a − αt a a b It−1 G G n It−1 n It−1 G R 

(A.28) ⇒ (A.29)

  τ 1 τ ωj (1 − Ft ) = αt−1 (1 − τ ) + − αt F t − F t − αt b n n R       1 1 R(1 − τ ) 1 (1 − Ft ) = αt−1 − (1 − τ ) − αt − Ft ωj − . b n n n R 

⇒

(A.30) (A.31)

By backwards induction we obtain the solution claimed in (26). We make use of the usual conventions that the sum (product) over the empty set is zero (one). The constant ratio b is denoted by Y for ease of notation: R/R     1 1 (1 − τ ) = Zt ωj − αt − n n

(A.32)

Zt = 1 − Ft+1 + Ft+1 Y Zt+1 ZN −1 = 1 Zt =

(A.33) (A.34)

j−1 Y

"

N X

Y

j=t+1

j−t−1

# Fp (1 − Fj ) .

(A.35)

p=t+1

Since the interest rate is positive and Y > 1, this produces a decreasing sequence of Zt ’s. This follows by backwards induction: Since ZN −1 = 1 we have ZN −2 = 1 − FN −1 + FN −1 Y > 1 which gives the first step in the induction argument. To proceed we first show that the sequence Y Zt − 1 is positive. This is so, because Y Zt − 1 = Y Ft+1 (Y Zt+1 − 1) + (Y − 1).

(A.36)

Plugging in t = N − 1 results in Y −1. By backwards induction, Y Zt − 1 is the sum of two

24

positive terms. Assume now that Zt+1 > Zt+2 > . . . ZN −2 > 1. Then since the sequence of Ft ’s is decreasing we have Zt = 1 − Ft+1 + Ft+1 Y Zt+1 = 1 + Ft+1 (Y Zt+1 − 1) > 1 + Ft+2 (Y Zt+2 − 1) = Zt+1 , which verifies Equation (24). Once we know α0,j , we can determine the consumption share ωj from (A.32). Knowing the initial distribution of wealth and the fact that the fraction of aggregate initial wealth invested is F0 , we get the desired value of α0,j and ultimately (23): C0,j = W0,j − I0,j − β0,j = W0,j = W0,j

 1 − α0,j − n   τ a − I0 α0,j Y (1 − τ ) + b nR α0,j I0a

C0a = (1 − F0 )W0a

τ − I0a b R



(A.37)

⇒

W0,j F0 − ωj = a (1 − F0 )W0 (1 − F0 )

(A.38)   τ α0,j Y (1 − τ ) + . b nR

(A.39)

By suitable manipulations we next arrive at the relations in (A.40) and (A.41):    1 1 W0,j − F0 + Y Z0 ωj − (1 − F0 )ωj = W0a n n     1 W0,j 1 Z−1 ωj − = − ⇒ a n W0 n 1 W0,j Z−1 − 1 1 + , ωj = Z−1 W0a Z−1 n which verifies Equation (23).

25

⇒

(A.40) (A.41) (A.42)

Proof of items 8, 9, and 10 The relation (28) follows directly from the budget equation, since there is no net transfer payment related to the bond holdings. The sequence of aggregate consumption is: " W0a (1 − F0 ),W0a F0 G1 (1 − F1 ),W0a F0 F1 G1 G2 (1 − F2 ), . . . ,W0a

t Y j=1

Gj

# " t−1 Y

# Fj (1 − Ft ).

j=0

We use the short-hand notation Gt for the product of t independent copies of the i.i.d. growth factors. This can be calculated explicitly from (15). Consider the case H 6= 1: W0a Gt

N −t H N −t (1 − H) a t H . = W G P 0 N i 1 − H N +1 H i=0

(A.43)

For H=1 the limiting value of this expression is valid. For γ 6= 1, the utility from each individual term is "  #t  1−γ 1−γ N 1 H (1 − H) G a t W0 = (by independence) ρ E0 1−γ 1 − H N +1 H  1−γ   " + 1−γ  − 1−γ #t N 1 ρ t G G H (1 − H) W0a + = N +1 1−γ 1−H 2 H H  1−γ  t N 1 Y a H (1 − H) W0 . N +1 1−γ 1−H H

(A.44)

(A.45) (A.46)

After summing from t = 0 to t = N , the utility from aggregate consumption is given by the result stated in (30). For the log-investor, the utility from each individual term is " ρt E0

" t  #!# !    N N Y Gj H H G t a t a + tρ E0 log log W0 PN = ρ log W0 PN . i i H H i=0 H i=0 H j=1 (A.47)

For ρ = 1 the result in (32) is straightforward. For ρ 6= 1, the claim holds, because (see, e.g., Bronstein, Semendyayev, Musiol, and Mühling, 2007) N X t=1

tρt =

ρ − ρN +1 (1 + N (1 − ρ)) . (1 − ρ)2

26

(A.48)

Proof of items 11 and 12 Increasing the tax rate lowers the intertemporal growth path of consumption and also lowers reinvestment rates as the values of Ft decrease for all t as a result of an increasing tax rate. We want to show that this leads to a loss in utility from aggregate consumption. The tax rate affects Ft as well as the utility function through H, which also defines the ratio Y /H. we first show that H is an increasing function of τ : 1 EQ [G] − 1 1r ∂H 1 = = > 0, b ∂τ H γ (1 − τ )EQ [G] + τ γR

(A.49)

where r = R − 1 is the net risk-free rate before taxes. In order to prove the claim that the sequence of reinvestment rates Ft is uniformly negatively affected by an increase in the tax rate τ , we use the backwards difference equation (13) together with backwards induction, starting from N − 1: 1 1+H

⇒

∂ −1 ∂H [FN −1 ] = <0 ∂τ (1 + H)2 ∂τ

(A.50)

1 1 + (1 − Ft+1 )H

⇒

∂ H ∂Ft+1 [Ft ] = 2 < 0. ∂τ Ft ∂τ

(A.51)

FN −1 = Ft =

In order to prove that the utility of aggregate consumption is negatively affected by an increase in the tax rate τ , we make use of the following relation: If A and B are both functions of a common variable τ , the product A · B is decreasing as a function of τ , if the sum of the elasticities is negative: (AB)0τ

 = AB

A0τ B0 + τ A B

 .

(A.52)

We first derive the necessary relations for the power term A. Without loss of generality we assume W0a = 1.

∂ ∂τ

1 HN = PN ⇒ PN i −j i=0 H j=0 H !γ−1 !γ−2 N N N X X X 1r H −j = (γ − 1) H −j (−j)H −j b γR j=0 j=0 j=0 = (γ − 1)

N X

!γ−1 H

−j

j=0

27

N X H −j 1r (−j) PN . −j b γR j=0 H j=0

(A.53)

(A.54)

(A.55)

The elasticity for this term is (γ − 1)

N X

1r H −j . j PN −j b γR j=0 H j=0

(A.56)

The relation for the “annuity term” is:
i γ−1 Rρ h +
−γ Y − −γ H + G = G H 2  i  i ∂ Y γ−1 Y r = i. b ∂τ H γ H R

Both of these elasticities contain the factor γ −1, corresponding to the term (1−γ)−1 in front of the entire utility function. Hence, to prove the claimed decrease in the utility from aggregate consumption, we have to show that N X

i

i=0

YN ≡


Y i H

>

YN

N X H −j j HN j=0

i N  X Y H

i=0

HN ≡

N X

(A.57)


Y N +1 1− H = Y 1− H

H −j = H

j=0

1 − H −(N +1) . H −1

(A.58)

(A.59)

This is equivalent to N X

i

i=0

YN ≡


H −i Y YN

N X H −i > i HN i=0

−i N  X H i=0

H1− = Y

Y

(A.60)
H −(N +1) Y . H −1 Y

(A.61)

It is sufficient to prove that the function f : f (H) ≡

N X H −i i HN i=0

28

(A.62)

is a decreasing function of H. This is a consequence of Jensen’s inequality: PN

0

2 −i−1 HN i=0 (−i )H HN2

f (H) =

=

1 H

PN

2

i=0 (−i

−

−i

)H HN +

PN

i=0

P N

iH −i HN0

i=0

iH

HN2 

N 1  X H −i = i H HN i=0

!2

 N −i X H  − i2 HN i=0

−i

(A.63)

2 (A.64)

(A.65)

The sign of the expression in (A.65) does not depend on the term 1/H in front, but only by the expression in brackets. This is of the form E [X 2 ] − (E [X])2 , where X is the random variable taking on values i = 0,1,2, . . . ,N with probabilities H −i /HN . Hence, by Jensen’s inequality f 0 (H) < 0, and the inequality needed to show the decreasing nature of the value of utility from aggregate consumption and thereby item 11 is verified. Since the utility function in the aggregate is identical to the utility function for any individual household, item 12 follows directly by inserting the wealth level of household j, ωj W0a , into this utility function. This completes the proof of Theorem 1.

29