On economic growth and automatic stabilizers under linearly progressive income taxation

Journal of Macroeconomics 60 (2019) 378–395

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Journal of Macroeconomics journal homepage: www.elsevier.com/locate/jmacro

On economic growth and automatic stabilizers under linearly progressive income taxation Shu-Hua Chen

T

⁎

Department of Economics, National Taipei University, 151 University Road, San Shia District, New Taipei City, 23741 Taiwan

ARTICLE INFO

ABSTRACT

Keywords: Equilibrium (in)determinacy Endogenous growth Linearly progressive income taxation Productive government spending

I analytically show that the adoption of a linearly progressive income tax scheme destabilizes the endogenously growing economy of Barro (1990) by giving rise to dual balanced growth path equilibria, wherein the low-growth equilibrium exhibits local indeterminacy and belief-driven growth fluctuations, and the high-growth equilibrium displays saddle-path stability. I propose that both a sufficiently high lump-sum taxes-to-capital ratio and a sufficiently high consumption tax rate operate as automatic stabilizers that eliminate the indeterminate low-growth trap, thereby ensuring the existence of a unique determinate balanced growth path equilibrium that displays high output growth. In these cases, both the welfare- and the growth rate-maximizing marginal income tax rate on the high-growth balanced growth path are lower than the elasticity of output with respect to government spending. I further show that, when the marginal income tax rate is optimally set, investment subsidies are unable to eliminate the low-growth trap. Finally, government spending on goods and services cannot serve as an automatic stabilizer whether or not the income tax rate is optimally set.

JEL classification: E62 O41

1. Introduction In a seminal paper in the endogenous growth literature that explores the macroeconomic effects of tax policy, Barro (1990) shows that the constant income tax rate should be set equal to the elasticity of output with respect to government spending so as to attain maximal levels of the output growth rate and social welfare. It is noted that the model economy studied in Barro (1990) displays saddle-path stability and hence is not subject to endogenous growth fluctuations driven by agents’ animal spirits or sunspots. While the assumption of a constant tax rate on households’ taxable income is commonly adopted in the majority of related works, it is not consistent with the progressive income tax policies observed in many developed and developing countries. Several subsequent works on the macroeconomic implications of progressive income taxation are thus motivated.1 Among the theoretical efforts, Chen and Guo (2013) study the (in)stability effects of Guo and Lansing's (1998) non-linear income tax structure in Barro's (1990) model. It is found that, in sharp contrast to traditional Keynesian-type stabilization policies, progressive taxation operates like an automatic

Corresponding author. E-mail address: [email protected]. 1 For example, Yamarik (2001) analyzes the growth implication of a non-linear tax structure in an AK model of endogenous growth with government expenditures that are purely “wasteful.” Li and Sarte (2004) examine the growth and redistributive effects of the Guo-Lansing progressive tax scheme in a two-sector endogenously growing economy with heterogeneous agents and useless government spending. Greiner (2006) explores the growth and stability effects of the Guo-Lansing tax policy in an endogenous-growth setting with public capital and public debt. ⁎

https://doi.org/10.1016/j.jmacro.2019.04.005 Received 29 September 2018; Received in revised form 8 April 2019; Accepted 18 April 2019 Available online 24 April 2019 0164-0704/ © 2019 Elsevier Inc. All rights reserved.

Journal of Macroeconomics 60 (2019) 378–395

S.-H. Chen

destabilizer that magnifies growth fluctuations and thus destabilizes the economy.2 However, the growth and welfare implications of income tax policy are not the focus of the paper. The theoretical framework of this paper differs from that of Chen and Guo (2013) in only one aspect; i.e., I adopt an alternative formulation of the income tax scheme á la Dromel and Pintus (2007). In particular, Dromel and Pintus point out that the feature of continuously increasing average and marginal tax rates á la (Guo and Lansing, 1998) is not consistent with the practice in countries that implement schemes of progressive income taxation. Dromel and Pintus thus propose an alternative fiscal formulation featured by a constant marginal tax rate imposed on the household’s taxable income when it exceeds a fixed exemption threshold – namely, a linearly progressive taxation. By incorporating the fiscal formulation of Dromel and Pintus (2007) into Barro's (1990) determinate one-sector representativeagent model of endogenous growth with inelastic labor supply and productive public expenditures, this paper carries out a comprehensive analytical investigation of the interrelations between a linearly progressive income tax, equilibrium (in)determinacy, and economic growth along the economy’s balanced growth path(s). I show that, in contrast to Chen and Guo (2013) where there is a chance that the economy can maintain saddle-path stability – i.e., when the long-run average income tax rate is lower than the output elasticity of government spending and the degree of tax progressivity is below a critical level – equilibrium determinacy is impossible in this paper, which employs Dromel and Pintus' (2007) linear fiscal formulation. In particular, I find that linearly progressive income taxation leads the model of Barro (1990) to display two interior balanced growth path (BGP) equilibria, wherein the low-growth one is an indeterminate sink that exhibits belief-driven growth fluctuations, and the high-growth one displays saddle-path stability and equilibrium uniqueness. I start from a particular BGP and suppose that agents expect an expansion in future economic activities. Acting upon this belief, the representative household will reduce consumption and increase investment today, which in turn lead to a higher stock of capital in the next period. The consequential effect on the economy’s future rate of return on capital is two-fold. First, because of diminishing returns, the return on capital investment falls (the diminishing-returns effect). Second, as a result of higher real activities, the government collects more tax revenues and hence provides more productive government spending. This increases the return on capital (the tax-revenue effect). Under high-growth BGP, the tax-revenue effect outweighs the diminishing-returns effect, leading to an increase in the return on capital. Agents’ initial optimistic expectations are thus validated as a self-fulfilling equilibrium. By contrast, for the low-growth BGP, the tax-revenue effect exerts a relatively weaker impact on the return on capital investment, hence preventing agents’ rosy anticipation from becoming self-fulfilling. In terms of the growth-rate effect of the marginal income tax rate, I find that it can be negative, positive, or zero, and is closely linked with the local stability properties of a BGP. In particular, when the economy is on the indeterminate low-growth BGP, it displays a negative long-run growth-rate effect of the marginal income tax rate. On the contrary, when the economy is on the determinate high-growth BGP, the nexus between long-run output growth and the marginal income tax rate is positive (negative) under lower (higher) levels of the marginal income tax rate. Hence, in the high-growth BGP equilibrium, there exists an interior positive marginal income tax rate that attains the maximum rate of output growth. I find that, under linearly progressive income taxation, the growth rate-maximizing marginal income tax rate is strictly higher than the elasticity of output with respect to public expenditures. Moreover, the more progressive the income tax schedule is, the higher the growth rate-maximizing marginal income tax rate will be. This result runs in sharp contrast to that of Barro (1990), who shows that maximizing output growth is equivalent to maximizing social welfare. Intuitively, there are two channels through which the marginal income tax rate influences the output growth rate. First, under a given level of the government spending-to-capital ratio which determines the marginal product of capital, a higher marginal income tax rate shrinks the after-tax return on capital (the direct effect). Second, a higher marginal income tax rate raises government spending. This in turn, by raising the equilibrium rental price of capital, reduces the firm’s demand for capital and hence causes reductions in production, tax revenues, and the provision of productive infrastructure. Hence, a higher marginal income tax rate may lead to a rise or a decrease in public expenditures and hence the return on capital investment (the indirect effect). I find that the indirect effect that the marginal income tax rate imposes on the return on capital is negative on the low-growth BGP. This, together with the negative direct effect, gives rise to an inverse relationship between the rates of output growth and marginal income tax. By contrast, a positive indirect effect is present on the high-growth BGP. It turns out that a positive nexus between the rates of output growth and the marginal income tax is present under lower levels of the marginal income tax rate, because of a dominating positive indirect effect. When the marginal income tax is at higher values, the negative direct effect dominates, and hence output growth and the marginal income tax rate become inversely related. In view that the adoption of a linearly progressive income tax schedule leads to the emergence of multiple BGPs with a lowgrowth trap, I propose policy instruments that help to eliminate the indeterminate low-growth BGP, thereby ensuring the existence of a unique determinate BGP that displays a high balanced growth rate. I show that this can be achieved by setting either the lump-sum taxes-to-capital ratio or the consumption tax rate above a critical level. I further find that, when the low-growth BGP is eliminated by either lump-sum taxes or a consumption tax and when the marginal income tax rate is set optimally, both the growth rate- and the

2 The finding about the potential destabilization effect of progressive income taxation is qualitatively identical to that in several recent studies under different analytical settings. For example, Chen and Guo (2016) explore an AK model with fixed hours worked, Chen and Guo (2018) examine an AK model with utility-generating government purchases of goods and services, and Chen and Guo (forthcoming) investigate an endogenously growing economy with variable labor supply and useless public expenditures. By contrast, Christiano and Harrison (1999) obtain that progressive taxation on agents’ labor effort is an automatic stabilizer within the endogenous growth setting of Benhabib and Farmer (1994).

379

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welfare-maximizing marginal income tax rate are strictly lower than the output elasticity of public expenditures. Moreover, the higher the lump-sum taxes-to-output ratio and/or the consumption tax rate are, the lower the growth rate- and the welfare-maximizing marginal income tax rate will be. The welfare-maximizing marginal income tax rate is shown to be higher than the rate that attains the maximum rate of output growth. I further demonstrate that, under a given/fixed level of the marginal income tax rate, a sufficiently high investment subsidy rate helps to eliminate belief-driven growth fluctuations and the low-growth trap. Nonetheless, when the marginal income tax rate is set to achieve the highest level of either output growth or social welfare, the model always displays two BGPs and equilibrium indeterminacy, and hence investment subsidies do not function as an automatic stabilizer. Finally, I find that government consumption is incapable of serving as an automatic stabilizer whether or not the income tax rate is optimally set. This runs in sharp contrast to Guo and Harrison’s (2004) finding that government consumption and lump-sum transfers/taxes display the same stabilizing effect that guarantees saddle-path stability of the model under constant income tax rates. The remainder of this paper is organized as follows. Section 2 describes the model under the income tax formulation á la Dromel and Pintus (2007) and analyzes the equilibrium conditions. Section 3 derives the economy’s balanced growth equilibria and examines the associated local stability properties and growth-rate effects. Section 4 explores fiscal policy instruments that potentially serve as automatic stabilizers. Section 5 quantitatively investigates the determination of the optimal marginal income tax rate and the effectiveness of fiscal policy instruments in regaining equilibrium uniqueness. Section 6 analyzes a combined linear and non-linear income tax scheme, and Section 5 concludes. 2. The economy I incorporate the time-varying version of Dromel and Pintus’ (2007) linearly progressive tax formulation into Barro (1990) onesector representative-agent model of endogenous growth with productive government spending. The economy is populated by a unit measure of identical infinitely-lived households. Each household provides fixed labor supply and maximizes its discounted lifetime utility:

U=

0

t dt ,

log ct e

(1)

where ct is consumption, and ρ > 0 denotes the subjective rate of time preference. I assume that there are no fundamental uncertainties present in the economy. The budget constraint faced by the representative household is given by:

ct + it = yt

(yt

Et ) ,

E0

0 given,

(2)

Tax Paid

where yt is output, it is investment, and Et represents the exemption threshold that is postulated to grow continuously at the rate of

per-capita output on the economy’s BGP, i.e.

y* Et = yt* Et t

=

y,

and hence Et = E0 e

yt

for all t. As in Dromel and Pintus (2007), I analyze the

environment with yt > Et > 0 for all t, and a constant marginal tax rate τ ∈ (0, 1) that is higher than the corresponding average tax

rate given by

(1 ). It follows that the tax schedule under consideration is progressive. Et yt

Investment adds to the stock of physical capital according to the law of motion:

k t = it

(3)

kt , k 0 > 0 given,

where kt is the household’s capital stock, and δ ∈ (0, 1) is the capital depreciation rate. As in Barro (1990), the flow of government spending gt enters the firm’s Cobb-Douglas production technology as an input that is complementary to private capital:

yt = Akt gt1

,

A > 0,

0<

(4)

< 1.

(0, 1), captures the degree of positive external effect Note that the elasticity of output with respect to government spending, 1 that public expenditures exert on the production process, and that the technology (4) exhibits constant returns-to-scale with respect to kt and gt such that sustained economic growth is present in equilibrium. The first-order conditions for the representative agent with respect to the indicated variables and the associated transversality conditions (TVC) are: ct :

kt :

TVC :

1

ct

=

(5)

t,

(1

t

lim e

t

) t

yt kt t kt

=

t

t,

(6)

= 0,

(7)

where (5) equates the marginal utility of consumption to its marginal cost λt, which is the Lagrange multiplier on the household’s budget constraint (2) that also captures the shadow price of capital. Eq. (6) is the Keynes-Ramsey condition that characterizes how 380

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S.-H. Chen

the stock of physical capital evolves over time, and (7) is the transversality condition. The government balances its budget at each point in time. Hence, the instantaneous government budget constraint is given by:

gt = (yt

(8)

Et ).

By combining (8) with the household’s budget constraint (2), together with the law of motion of physical capital (3), I derive the following aggregate resource constraint for the economy: (9)

ct + kt + kt + gt = yt . 3. Balanced growth path

I focus on the economy’s balanced growth path along which output, consumption, physical capital, and government spending exhibit a common, positive constant growth rate denoted as θy. To facilitate the subsequent dynamic analyses, I adopt the following gt ct variable transformations: z t and x t . Per these variable transformations, the model’s equilibrium conditions can be collapsed kt kt into the following autonomous dynamical system:

Axt1

z t = [xt

xt =

1

(1

+ zt

(10)

] zt ,

Et / kt ) Axt

[

Axt1

y

+ xt + z t + ],

(11)

xt . 1 (1 ) > 0 and Et / kt = where A balanced growth equilibrium is characterized by a pair of positive real numbers (z*, x*) that satisfy z t = xt = 0 . It follows from (8), (10), and (11) that: Axt1

z * = A (x *)1

x* + ,

(12)

x * = A (x *)1

E0 ; k0

(13)

and

in addition, the common (positive) rate of economic growth on the balanced growth path is given by: y

) A (x *)1

= (1

(14)

.

To examine the existence and number of the economy’s balanced growth paths in a transparent manner, I let f(x*) ≡ E0 A (x *)1 from (13), and obtain: k 0

) A (x *)

f = (1

> 0 and f =

f < 0; x*

(15)

in addition,

E0 < 0 and f ( ) k0

f (0) =

.

(16)

The equilibrium x* will be located from the (possibly more than one) intersection(s) of f(x*) and the 45-degree line.

(

Based on (15) and (16), Fig. 1 depicts that f(x*) is an upward-sloping concave curve with a negative vertical intercept =

E0 k0

).

Hence, the number of intersections between f(x*) and the 45-degree line in the positive quadrant can be zero, one, or two. I proceed by first deriving the critical level of the marginal income tax rate, denoted as ^, at which f(x*) is tangent to the 45-degree line such that there exists a unique BGP characterized by x^ , and the corresponding equilibrium growth rate is y (x^) . Using (15) with f (x^) = 1 and (13) evaluated at x^ , it is straightforward to show that: 1

x^ = [(1

(17)

) A] ;

in addition, the threshold level ^ is derived as:

^=

1 (1

) A1

1

E0 k0

1

.

(18)

I next find that an increase in τ shifts the locus of f(x*) upwards, because:

f (x *)

=

x*

> 0,

(19)

which implies that the model possesses no (two) balanced growth path(s), provided τ < ( > ) ^ . Hence, any small deviation from the BGP associated with x^ will lead to the disappearance of a BGP equilibrium, or the emergence of dual BGP equilibria. This result indicates that the economy undergoes a saddle-node bifurcation that may cause the hard loss of equilibrium stability, i.e. a radical 381

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S.-H. Chen

Fig. 1. Marginal Income Tax Rate and Possible Multiple BGPs.

qualitative change in the behavior of the dynamical system (10) and (11) takes place, as the marginal income tax rate passes through the threshold value ^ . Fig. 1 shows that when the marginal tax rate is higher than ^ , there exist two interior balanced-growth equilibrium paths in the model characterized by x1* and x 2*, where x1* < x^ < x 2*. Since a higher government spending-to-capital ratio, x*, produces a higher marginal product of capital, physical investment and hence the balanced growth rate will be enhanced (see the second equation of (36)). Therefore, a higher level of x* corresponds to a higher balanced growth rate; hence, y (x1*) < y (x^) < y (x 2*) . In terms of the local stability properties of a BGP, I compute the Jacobian matrix J of the dynamical system (10) and (11) evaluated at (z*, x*). The determinant and trace of the resulting Jacobian matrix are:

Det =

Tr =

(1

) (1 (1

+ (1

)(1

) A (x *) z *E0 f ) k0

) A (x *)1

+

0 when

(1

)(1 (1

f

1,

(20)

) A (x *) E0 f ) k0

.

(21)

The local stability property of the balanced-growth equilibrium path is determined by comparing the eigenvalues of J that have negative real parts with the number of initial conditions in the dynamical system (10) and (11), which is zero, because zt and xt are both non-predetermined jump variables.3 As a result, the BGP displays local determinacy and equilibrium uniqueness when both eigenvalues have positive real parts. If one or two eigenvalues have negative real parts, then the BGP is locally indeterminate (a sink) and can be exploited to generate endogenous growth fluctuations driven by agents’ self-fulfilling expectations or sunspots. Since f (x 2*) < 1, Eqs. (20) and (21) show that, around the high-growth BGP associated with x 2*, the model’s Jacobian matrix J possesses a positive determinant and a positive trace. Thus, this high-growth equilibrium path displays saddle-path stability. On the other hand, in the neighborhood of the low-growth BGP equilibrium associated with x1*, the property that f (x1*) > 1 implies that the determinant of the Jacobian is negative, indicating that the two eigenvalues of J have opposite signs. It follows that the low-growth equilibrium exhibits indeterminacy and sunspots. The following proposition is established based on the above discussion. Proposition 1. The adoption of a linearly progressive income tax scheme leads the endogenously growing economy of Barro (1990) to display dual balanced growth equilibria, wherein the low-growth one is a locally indeterminate sink and the high-growth one exhibits saddle-path stability. 3 As for the initial condition of consumption c0, the period-0 level of government spending g0 (a flow variable) will be endogenously determined g c through the model’s equilibrium conditions. It follows that both z 0 = k0 and x 0 = k0 are not predetermined.

0

382

0

Journal of Macroeconomics 60 (2019) 378–395

S.-H. Chen

The intuition for the above (in)determinacy result can be understood as follows. When agents expect a higher future return on capital, they will reduce consumption and increase investment today. The resulting higher stock of capital in the next period produces two opposing effects on the future rate of return on capital. First, because of diminishing returns, the return on capital investment falls (the diminishing-returns effect). Second, national income increases. The resulting higher tax revenues leads to a higher productive government spending, which in turn promotes the return on capital investment (the tax-revenue effect). On the high-growth BGP, the tax-revenues effect outweighs the diminishing-returns effect, leading to an increase in the return on capital. Hence, agents’ initial optimistic expectations are validated as a self-fulfilling equilibrium. On the contrary, a dominated tax-revenue effect on the low-growth BGP leads to a reduction in the return on capital, thus preventing agents’ expectations from becoming self-fulfilling. Fig. 1 also shows that, if the economy starts at the high-growth (low-growth) BGP equilibrium associated with x 2* (x1*), then an increase in the marginal income tax rate τ raises (lowers) the government spending-to-capital ratio. Intuitively, under a given level of taxable income, a higher τ raises tax revenues and hence increases government spending. This in turn, by raising the equilibrium rental price of capital, induces the firm to cuts its demand for capital; the production of output (which is taxable income) and hence tax revenue and government spending subsequently fall. Because of diminishing returns of government spending in the production of output, the equilibrium rental price of capital increases less on the high-growth BGP associated with a higher x 2*. Hence, on the highgrowth BGP, government spending rises upon an increase in τ. Given that the capital stock is predetermined, this leads to a rise in x 2*. On the contrary, on the low-growth BGP associated with a lower x1*, the equilibrium rental price of capital increases more in response to an increase in τ. A higher τ thus causes a reduction in the government spending-to-capital ratio. In terms of the growth-rate effect of the marginal income tax rate τ, the second equation of (36) implies that there are two channels through which τ influences the BGP’s growth rate. First, under a given level of x*, a higher τ shrinks the after-tax return on capital and hence capital investment (the direct effect). The second effect is whereby a higher τ influences x*, which in turn (positively) affects the rate of return on capital (the indirect effect). It follows that if the economy starts at the high-growth (low-growth) BGP equilibrium associated with x 2* ( x1*), then the indirect effect is a rise (fall) in the government spending-to-capital ratio that stimulates (discourages) capital investment and boosts (suppresses) the balanced growth rate. By putting together the direct and the indirect effects, I derive that output growth is negatively related with the marginal income tax rate τ on the low-growth BGP, because the direct and the indirect effects are both negative: y (x 1*)

(1

=

)(1 (1

)

1

f )

A (x1*)

< 0.

(22)

On the high-growth BGP, however, the direct effect is negative and the indirect effect is positive, giving rise to an ambiguous effect of τ on the output growth rate. By taking total differentiation on (13)–(36), I derive that: y (x 2*)

A (x 2*)

=

(1

f )

) x 2* +

· (1

(1

) 2E0 k0

0.

(23)

The above equation implies that there may exist an interior τ, denoted as ˜, that attains the maximum rate output growth. Before proceeding to the derivation of ˜ on the high-growth BGP under linearly progressive income taxation, I first discuss the case where the tax schedule is flat with E0 = 0 and hence Et = 0 for all t. In this case, I recover Barro's (1990) model with a flat income tax schedule whereby the average and the marginal tax rates are both τ. Substituting E0 = 0 into (13) shows that the ratio of 1 government spending to physical capital remains unchanged over time: x t = ( A) for all t. This in turn implies that the dynamical system (10) and (11) degenerates. Resolving the model with Et = 0 leads to the following single differential equation in zt that describes the equilibrium dynamics:

z t = ( A)

1

A ( A)

1

+ zt

zt ,

(24)

which has a unique interior solution z* that satisfies z t = 0 along the BGP. I then linearize (24) around the BGP and find that its local stability properties are governed by the positive eigenvalue z* > 0. Consequently, the economy exhibits local determinacy since there 1 is no initial condition associated with (24). In addition, by substituting x t = ( A) into (36), the economy’s balanced growth rate can 1 ) A ( A) be expressed as = (1 . It is straightforward to show that, as derived in Barro (1990), the growth-rate . maximizing marginal income tax rate is equal to the elasticity of output with respect to gt, i.e., 1 With the understanding of the properties of Barro (1990) model, I then go back to (23). I first notice that <1 and E0 > 0. I then insert = 1 into (23), and find that: y (x 2*)

= =1

A (x 2*) (1

(1 f ) k0

) 2E0

> 0 if

E0 > 0.

y (x2*)

> 0 for all

(25)

The above equation implies that, under a linearly progressive income tax schedule, the economy’s output growth is still rising at =1 . Therefore, the growth-rate maximizing ˜ will be higher than 1 . y (x2*) = 0 ; and (ii) To derive the level of ˜ (> 1 ), one may note that it satisfies the requirements: (i) the locus of y (x 2*) peaks: the corresponding x 2* constitutes a balanced growth equilibrium: x 2* = f (x 2*) . These two conditions collapse into a single equation as follows: 383

Journal of Macroeconomics 60 (2019) 378–395

S.-H. Chen

E0 k0

1

)1

= A (1

˜ 2(1

) [˜

(1

)] ,

RHS > 0

(26)

LHS > 0 4

) ˜ (1 ) > 0. where 1 (2 Fig. 2 illustrates that the left-hand side of (26) is an upward-sloping straight line with a positive vertical intercept

(= (1

)2

( ) ); in addition, the right-hand side of (26) is a convex curve that starts from the origin with a negative slope and is E0 k0

upward-sloped in the positive quadrant.5 As depicted, there exists a unique ˜ . I then derive that an increase in the tax progressivity E0 shifts the locus of LHS upwards (see (26)), thereby giving rise to a higher growth-rate maximizing marginal income tax rate ˜ :

˜ [˜ (1

˜ = E0

(1 )] ) E0 2

1

> 0,

(27) 6

+ 2(1 )[1 (2 ) ˜] > 0 . The result in (27) follows from the fact that the marginal income tax rate τ where 2 (2 and the tax progressivity E0 (which determines the exemption level Et) exert opposite effects on taxes paid (= (yt Et ) ) and hence the macroeconomic performance. With regard to the growth-rate effect of the tax progressivity E0, by taking a total differentiation on (13)–(36), I derive that: ) ˜2

y

E0

(1

=

) (1 (1

) A (x *) f ) k0

0 when

f

1.

(28)

Therefore, raising the tax progressivity suppresses (enhances) long-run output growth when the economy is located at the high- (low) growth BGP equilibrium. For the intuition behind the result in (28), I illustrate in Fig. 2 that a higher E0 shifts the locus of f(x*) downwards, because:

f (x *) = E0

k0

< 0.

(29)

If the economy starts at the high-growth BGP equilibrium associated with x 2*, then a higher E0 reduces x 2*. The consequential decline in the net rate of return on capital suppresses capital investment and output growth, giving rise to a negative growth effect of tax progressivity

(

y (x1*)

E0

)

(

y (x2*)

E0

)

< 0 . Conversely, the BGP’s output growth is positively related with the tax progressivity on the low-growth BGP

> 0 . It is noticeable that the effect on x* of a rise in E0 is opposite to that of a rise in τ. This again follows from the fact that

the marginal income tax rate τ and the tax progressivity E0 exert opposite effects on tax paid. I summarize the above results in the following proposition. Proposition 2. Under linearly progressive income taxation:

(i) an increase in the marginal income tax rate reduces long-run output growth along the low-growth BGP; (ii) along the high-growth BGP, there exists an interior positive marginal income tax rate that maximizes output growth; this growth ratemaximizing tax rate is higher than the output elasticity of government spending, 1 ; (iii) an increase in the tax progressivity reduces (raises) the output growth rate along the high- (low-) growth BGP. The results in Proposition 2(i) and (ii) are different from those in Barro (1990) which shows under a flat income tax schedule that the model displays a unique BGP where there exists an interior growth rate-maximizing marginal income tax rate that is equal to the output elasticity of public expenditures. In addition, in a closely-related work, Chen and Guo (2013) show that the endogenously growing economy of Barro (1990) under Guo and Lansing's (1998) non-linear fiscal formulation possesses a unique BGP that displays: (i) a negative growth-rate effect of tax progressivity; (ii) saddle-path stability when the long-run average tax rate is lower than the output elasticity of government spending, and the degree of tax progressivity is below a critical value; and (iii) equilibrium indeterminacy otherwise. Alternatively, this section’s analysis shows that Barro (1990) model under the linear fiscal formulation á la Dromel and Pintus (2007) exhibits: (i) dual BGPs; (ii) equilibrium indeterminacy with a low-growth trap; and (iii) an ambiguous growth rate-effect of tax progressivity that is linked to the local stability properties of a BGP. 4. Automatic stabilizers Because the implementation of a linearly progressive income tax leads to an undesirable economic situation of multiple BGPs with a low-growth trap and endogenous growth fluctuations, it is worthwhile to find instruments that help to recover equilibrium determinacy. This section proposes that a lump-sum tax, a consumption tax, and investment subsidies are able to achieve this goal and 4 5

)2 > 0 . , I derive 1 > (1 Since ˜ > 1 For the locus of the left-hand side of (26), I derive: (i)

LHS ˜

= (2

)

( ) E0 k0

> 0 ; (ii) LHS (˜ = 1 (2

= RHS· LHS (˜ ) . For the locus of the right-hand side of (26), I derive: (i) RHS ˜ ) (iv) RHS (˜ . 6 ) = (1 )2 > 0 and 2 = 2(2 )[˜ (1 )] > 0 for all ˜ > 1 Since 2 (˜ = 1 384

) ˜ 2(1 )2 ˜ [˜ (1 )]

0, if ˜

) = (1 2(1 2

)2

)2

( ) E0 k0

> 0 ; and (iii)

; (ii) RHS (˜ = 1

, I have Δ2 > 0 for all ˜ > 1

.

) = 0 ; and

Journal of Macroeconomics 60 (2019) 378–395

S.-H. Chen

Fig. 2. Unique Optimal Marginal Income Tax Rate at High-Growth BGP.

Fig. 3. Tax Progressivity.

hence operate as automatic stabilizers. Moreover, by eliminating the indeterminate low-growth BGP, these policy instruments help to ensure the existence of a unique BGP that displays high output growth. Under a balanced budget rule that comprises constant wage and capital income tax rates and no endogenous growth, Guo and Harrison (2004) show that government consumption and lump-sum transfers/taxes display the same stabilizing effect that guarantees 385

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S.-H. Chen

saddle-path stability of the model. It is thus interesting to find out whether or not government consumption operates as an automatic stabilizer like lump-sum taxes do within this paper’s circumstance. As it turns out, the answer is no. In the presence of lump-sum taxes, a consumption tax, investment subsidies, and government consumption, the budget constraints of the representative household and the government, i.e. (2) and (8), are respectively modified as:

ct + it = yt

(yt

Et )

c ct

+ sit

(30)

Tt ,

and

gt + gtc + sit = (yt

Et ) +

c ct

(31)

+ Tt ,

where τc ∈ (0, 1) is the constant tax rate levied on consumption purchases, and s is the investment subsidy rate; in addition, Tt and gtc respectively denote lump-sum taxes and government consumption, both of which are postulated to be proportional to the level of capital stock within the sustained economic growth setting: Tt = kt and gtc = ckt . Other model features presented in Section 2 are maintained. It is straightforward to show that the dynamical system for this section’s model is described by a pair of differential equations comprising:

z t = xt

(1 1

1

) s

Axt1

+ zt

+

c

zt ,

+

c]

(32)

and

xt =

Et kt

( c + s) zt 1

Ax t1

[

s

(1

+ xt + z t + )(

s ) Axt

,

(33)

=( (1 s )(xt + + ( c + s) zt + . where It follows immediately from the dynamical system (32) and (33) that the BGP’s consumption-to-capital ratio and government spending-to-capital ratio are: Et kt

s ) Axt1

z* =

x* =

1

(

(1 1

c)

) s

A (x *)1

s ) A (x *)1 1

x* +

c,

E0 / k 0 + ( c + s ) z * + s

(34) c,

(35)

respectively; in addition, the BGP expression of the balanced growth rate is:

=

(1 1

) A (x *)1 s

.

(36)

In what follows, I systematically show the mechanisms that lump-sum taxes, a consumption tax, and investment subsidies stabilize the economy by eliminating the indeterminate low-growth BGP. I then demonstrate why government consumption fails to achieve this goal. 4.1. Lump-sum taxes I first examine the case of lump-sum taxes; hence, I let ϕ > 0 and

c

=

c = E

s = 0 . Fig. 4 plots the locus of the right-hand side of

0 + , along with the 45-degree line. As depicted, the (35) under this parameter configuration, denoted as fl(x*) ≡ k0 7 locus of fl(x*) has the same shape as that of f(x*) shown in Fig. 1. Nevertheless, while f(x*) has a negative vertical intercept, the vertical intercept of fl(x*) can be positive, zero, or negative, depending on the value of ϕ. In particular:

A (x *)1

fl (0) =

E0 0 k0

when

E0 . k0

(37)

In terms of the local stability properties of a BGP, I compute the Jacobian matrix of the dynamical system (32) and (33) under ϕ > 0 and c = c = s = 0 and evaluate it at (z*, x*). I derive that the determinant of the resulting Jacobian matrix, denoted as Detl, 1 ) A] . I further derive that Trl = Tr has the same expression as that given by (20). Thus, Detl = Det 0, when x * x^ = [(1 . Let ^ ^ denote the critical level of the lump-sum taxes-to-capital ratio at which f (x *; = ) is tangent to the 45-degree line, and the l

corresponding x* is x^ . Fig. 1 illustrates that, when ^ < < , the locus of fl (x *; ^ < < ) has a negative vertical intercept. In this case, the model possesses two interior BGPs, wherein the low-growth one associated with x1* < x^ is a locally indeterminate sink (Detl < 0), and the high-growth one associated with x 2* > x^ exhibits equilibrium determinacy (Detl > 0 and Trl > 0).8 When 7

It is straightforward to show that fl(x*) has the same first and second derivatives as those of f(x*). Under a wide range of parameter values, Trl (x 2*) can be negative only when ϕ takes on high values such that z2* is too low to be empirically implausible. For example, under the calibrated parameter values used in Section 5, Trl (x 2*) becomes negative when z2* < 0.0075. 8

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Fig. 4. Eliminating Low-Growth Trap via Lump-Sum Taxes.

ϕ < ϕ < 1 such that fl(x*; ϕ < ϕ < 1) has a positive vertical intercept, the indeterminate low-growth BGP is eliminated. In this case, the economy possesses a unique determinate BGP that displays high output growth. I therefore derive the following proposition. Proposition 3. A sufficiently high lump-sum taxes-to-capital ratio (ϕ > ϕ) eliminates the indeterminate low-growth BGP, thereby ensuring equilibrium uniqueness and high output growth of the economy. 4.2. Consumption tax In the case of consumption taxation, I let τc > 0 and = c = s = 0 . Under this parameter configuration and with (12), the rightE0 1 + c , where c + c > 0 . Fig. 5 depicts that: (i) fc(x*) is hand side of (35) can be expressed as: fc (x *) c A (x *) c x* k0 a concave curve that slopes upwards (downwards) under lower (higher) levels of the government spending-to-capital ratio:

) A (x *)

f c = (1 fc

fc =

x*

c

when x *

0

(1

)

cA

1

,

(38)

c

< 0,

(39)

and, (ii) the vertical intercept of fc(x*) can be positive, zero, or negative, depending on the value of τc:

fc (0) = Let c x^c =

^cc (1

E0 0 k0

c

when

c

c

E0 . k0

denote the consumption tax rate such that fc (x *; ) cA c

1 + ^c

1

(40) c

=

^cc )

is tangent to the 45-degree line and the corresponding x* equals

c . Panels (a) and (b) of Fig. 5 illustrate that, under low consumption tax rates (^c <

c

<

c)

such that fc(x*) has a

negative vertical intercept, the model exhibits two BGPs. Panel (c) in turn shows that when the consumption tax rate is sufficiently high (τc < τc < 1), the locus of fc(x*; τc < τc < 1) has a positive vertical intercept, therefore the low-growth BGP is eliminated. In this case, the model economy possesses a unique BGP that exhibits high long-run output growth.9 In terms of the local stability properties of a BGP, I compute the Jacobian matrix of the dynamical system (32) and (33) under τc > 0 and = c = s = 0 and evaluate it at (z*, x*). I derive that the determinant and trace of the resulting Jacobian matrix Jc are: 9

It can be derived that an increase in τc shifts the locus of fc(x*) upward: 387

fc (x *) c

= z* > 0 .

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Fig. 5. Eliminating Low-Growth Trap via Consumption Tax.

Detc =

Trc =

) (1 ) A (x *) z *E0 0 when (1 ) A (x *) ] k 0

(1 [1

+

x * x^ = [(1

x *k 0 Detc +1 . E0 z*

1

) A] ,

(41)

(42)

When x * < x^ , Eq. (41) shows that the model’s Jacobian matrix possesses a negative determinant (Detc < 0), indicating that the two eigenvalues have opposite signs. Hence, such BGPs associated with x * < x^ exhibit equilibrium indeterminacy and belief-driven growth fluctuations. When x * > x^ , the BGP displays Detc > 0, which in turn guarantees Trc > 0 according to (42). Indeterminacy and sunspots therefore do not arise under these BGPs associated with x * > x^. Let ^c denote the consumption tax rate such that x^ = fc (x^ ; c = ^c ) . Fig. 5 illustrates that x^ divides the regions labeled as indet c c (Detc < 0) and deter (Detc > 0, Trc > 0). The figure also shows that x^ is lower than x^c at which fc (x *; c = ^c ) is tangent to the 45c 10 ^ degree line. Note that the low-growth BGP associated with x1* must be below xc and can be higher or lower than x^ . The low-growth c BGP therefore exhibits equilibrium (in)determinacy if x1* < (> ) x^ . On the other hand, since x 2* must be higher than x^c ( >x^), the highgrowth BGP associated with x 2* must display saddle-path stability. 10

Since (1

) A

(1

) cA c 1 + ^c

=

(1

c )2^c (1 c 1 + ^c

)A

c < 0, I have x^ < x^c .

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Based on the above information, Fig. 5 illustrates how the number and the local stability properties of the BGPs change as the consumption tax rate rises. The results are summarized in the following proposition. Proposition 5. In the presence of a consumption tax: c

(i) when the consumption tax rate is low such that ^c < c < ^c , the model possesses two BGPs, both of which display equilibrium determinacy; (ii) when the government levies a mild consumption tax rate such that ^c < c < c , the model possesses two BGPs, wherein the low-growth one displays equilibrium indeterminacy and the high-growth one exhibits saddle-path stability. (ii) when the consumption tax rate is high such that τc < τc < 1, the model possesses a unique determinate BGP that exhibits high output growth. The above proposition indicates that, in order to guarantee the existence of a unique BGP that displays saddle path stability without the possibility of endogenous growth fluctuations, the consumption tax rate needs to be set at a value higher than τc. 4.3. Investment subsidies For this case, I set s > 0 and 1 s A (x *)

sx * 1 s

=

c

=

c

s (1

E0 / k 0 + s

= 0 . With (12), the right-hand side of (35) can be expressed as: )

0, when s s˜ fs (x *) , where s 1 s can be positive or negative, depending on the investment subsidy rate: fs (0) =

s 1

E0 / k 0 0 s

when

s s

E0 . k0

(1

) +

( > ) . It follows that the vertical intercept of fs(x*)

(43)

Under empirically plausible parameter values, s˜ < s (see Fig. 7(d)). I then derive that the slope and curvature of fs(x*) are respectively:

fs =

(1

)

s A (x *)

1

s

s

and f s =

(1

) 1

s A (x *)

s

1

.

(44)

Panels (a) and (b) of Fig. 6 illustrate that, when s < s˜ (Πs > 0), the locus of fs(x*) has a negative vertical intercept and exhibits the c c same shape as that of fc(x*). Hence, the model possesses two (no) BGP(s) if s > ( <) s^ , where s^ denotes the investment subsidy rate c ^ such that fs (x *; s = s ) is tangent to the 45-degree line. Panel (c) of Fig. 6 illustrates that, when s > s˜ (Πs < 0), the locus of fs(x*) is a negatively-sloped convex curve ( f s < 0 and f s > 0 ) whose vertical intercept is positive (negative), if s > ( < )s. Hence, as depicted, the model possesses two (no) BGP(s) if s > ( < )s. In terms of the local stability properties of a BGP, I derive that the determinant and trace of the resulting Jacobian matrix Js under this parameter configuration are:

Dets =

Trs =

(1

+

(1 s )[1

(1

s

) (1 (1

) A (x *) z *E0 0, )( s ) A (x *) ] k 0

(45)

s ) x *k 0 Dets +1 . E0 z*

(46)

Since s˜ > , in the case where s > s˜ (i.e. Firgure 6(c)), the model’s Jacobian matrix must display Dets > 0 and Trs > 0; therefore, belief-driven growth fluctuations will not arise. By contrast, when s < s˜ (i.e. panels (a) and (b) of Fig. 6), the determinant of Js is 1

(1 )( s) A positive (negative), if x * > ( < ) x^s . Let s^ denote the investment subsidy rate such that x^s = fs (x^s ; s = s^) . Panels (a) 1 s and (b) of Fig. 6 illustrate that x^ divides the regions labeled as indet (Dets < 0) and deter (Dets > 0, Trs > 0). Based on the above information, I establish the following proposition that states how the number and local stability properties of the BGPs change as the investment subsidy rate rises, as depicted in Fig. 6.

Proposition 6. In the presence of investment subsidies: c (i) when the investment subsidy rate is low such that s^ < s < s^, the model possesses two BGPs, both of which display equilibrium determinacy; (ii) when s^ < s < s˜, the model possesses two BGPs, wherein the low-growth one displays equilibrium indeterminacy and the high-growth one exhibits saddle-path stability. (iii) when s˜ < s < s , no BGP equilibrium exists. (iv) when the investment subsidy rate is high such that s < s < 1, the model possesses a unique BGP that exhibits equilibrium determinacy.

According to the above proposition, the economy will exhibit a unique BGP equilibrium and be immune to sunspot-driven growth when the investment subsidy rate is set at a value higher than s. 389

Journal of Macroeconomics 60 (2019) 378–395

S.-H. Chen

Fig. 6. Eliminating Low-Growth Trap via Investment Subsidies.

4.4. Government consumption c . It is obvious A (x *)1 E0 / k 0 By setting ϕc > 0 and = c = s = 0, the right-hand side of (35) is expressed as fg (x *) that fg(x*) has the same shape as f(x*) shown in Fig. 1. Since the vertical intercept of fg(x*) must be negative c < 0 ( fg (0) = E0/ k 0 ), the low-growth BGP cannot be eliminated using government consumption. Furthermore, it is straightforward to show that the inclusion of government spending on goods and services does not change the local stability properties of a BGP presented in Section 3. I therefore present the following proposition.

Proposition 4. Government consumption is unable to remove equilibrium multiplicity and the low-growth trap emerging under the linearly progressive income tax scheme.

5. Quantitative analysis The preceding section analytically demonstrates that, under a given level of the marginal income tax rate, sufficiently high lumpsum taxes-to-capital ratio (ϕ > ϕ), consumption tax rate (τc > τc), and investment subsidy rate (s > s) effectively remove the indeterminate low-growth trap emerging under the linearly-progressive income tax scheme. This section quantitatively investigates how the optimal marginal income tax rate changes with key policy parameters, and on how effective ϕ, τc, and s help to recover 390

Journal of Macroeconomics 60 (2019) 378–395

S.-H. Chen

equilibrium uniqueness when the marginal income tax rate is optimally set. Before carrying out the quantitative analysis, recall from Section 3 that: (i) output growth is inversely related with the marginal income tax rate τ along the low-growth BGP; and (ii) on the high-growth BGP, there exists an interior positive ˜ that maximizes longrun output growth. These results remain true when incorporating lump-sum taxes, a consumption tax, and/or investment subsidies, indexed respectively by j= l, c, s. In particular, I derive from (36) that the growth-rate effect of τ on the BGP is: j (x *)

(1

=

(1

)(1

)

yt*

f j x*

s) 1

Et* kt*

1

A (x *) 1 s

. (47)

Since f j (x1*) > 1 and f j (x 2*) < 1 for j= l, c, s, the above equation indicates a negative growth-rate effect on the low-growth BGP

(

j (x1*)

)

< 0 and the existence of an interior positive marginal income tax rate that attains maximum output growth on the highj (x *)

2 = 0. growth BGP: For the criterion of the optimal marginal income tax rate on the high-growth BGP, I follow Barro (1990) and Palivos and Yip (1995) in considering a benevolent government whose objective is to maximize the utility attained by the representative households. Note that there are no transitional dynamics on the determinate high-growth BGP. By integrating the right-hand side of (1), I obtain that the lifetime utility is a function of the balanced growth rate θj and the initial level of consumption

c0, j =

(1

s )[(1

Uj =

) j+ + ]

log c0, j

+

j , 2

+

E0 k0

k0 : 1+ c

j = y, l , c , s.

(48)

where j = y represents the case where there is no lump-sum taxes, consumption tax, and investment subsidies. Differentiation of Uj with respect to the marginal income tax rate yields:

Uj

1 (1

=

s )(1 ) k0 1 + (1 + c ) c0, j

j

E0 . (1 + c ) c0, j

+

(49)

Evaluating (49) at the marginal income tax rate that attains the maximum rate of output growth yields:

Uj j

= =0

E0 (1 + c ) c0, j

0 if

E0

0.

(50)

The above equation implies that, under a constant income tax rate (E0 = 0 ), maximizing output growth is equivalent to maximizing social welfare. This is what Barro (1990) model derives. However, when a progressive income tax is implemented (E0 > 0), the welfare-maximizing marginal income tax rate is higher than the rate that attains the highest output growth. This result is also different from that obtained in Futagami et al.'s (1993) setting under a flat income tax schedule and a stock of public capital, wherein the welfare-maximizing income tax rate is lower than the rate that maximizes output growth. I thus derive the following proposition. Proposition 5. Under linearly progressive income taxation, the welfare-maximizing marginal income tax rate is higher than the rate that attains the maximum rate of output growth, whether or not there are lump-sum taxes, a consumption tax, or investment subsidies. For the quantitative analysis, I am particularly interested in studying a calibrated version of the model economy where the combination of parameter values is consistent with post-war U.S. data. The benchmark parameterization for the quantitative analysis is as follows. The time unit is assumed to be one year. As is common in the literature of endogenous growth, the rates of time preference and capital depreciation are fixed at = 0.03 and = 0.1, respectively. The existing empirical estimates of the output elasticity of government spending, 1 , exhibit a wide range from 0.03 (Eberts, 1986) to 0.39 (Aschauer, 1989). I adopt Li and = 0.25 as a “consensus” benchmark. It follows that the elasticity of output with respect to the Sarte's (2004) calibration and set 1 broad concept of capital is = 0.75. The average marginal income tax rate is chosen to be = 25% . The steady-state ratio of public expenditures to GDP (or the government size) is set equal to

gt* yt*

=

(x *) A

1

= 0.2, which implies x * = (0.2A) . Substituting this re-

) A (x *)1 , the scale parameter A in the production function is chosen such that lationship into the expression of = (1 the output growth rate θ is equal to 2%; hence, A = 0.555 . I normalize the value of the initial stock of capital at k 0 = 1. Both the lumpsum taxes-to-capital ratio and the consumption tax rate are set at 0 as a benchmark. It follows that the calibrated initial tax prox* c z* k 0 = 0.0533. gressivity is E0 = A (x *)1 Under the calibrated values of ρ, δ, α, A, k0, and ϕ, Fig. 7(a) presents two loci of the relationship between the tax progressivity E0 and the marginal income tax rate τ when parameters other than E0 are fixed at the calibrated values. The solid blue locus illustrates the growth rate-maximizing τ which is in consistent with Section 3’s prediction: (i) equal to the output elasticity of government spending (= 0.25) when the income tax schedule is flat (E0 = 0 ); and (ii) increasing in the tax progressivity E0. On the other hand, the dashed blue locus illustrates the welfare-maximizing τ like (50) predicts: (i) equal to the rate that maximizes output growth when the income tax schedule is flat (E0 = 0 ); and (ii) strictly higher than the growth rate-maximizing τ when the income tax schedule is progressive (E0 > 0). It is also noticeable that when the degree of tax progressivity increases, the wedge between the two rates expands. 391

Journal of Macroeconomics 60 (2019) 378–395

S.-H. Chen

Fig. 7. Optimal Marginal Income Tax Rate at High-Growth BGP.

Likewise, with parameters other than ϕ fixed at the calibrated values, the solid blue locus in Fig. 7(b) illustrates the inverse nexus between the growth rate-maximizing τ and the lump-sum taxes-to-capital ratio ϕ, and the dashed blue locus shows that the welfaremaximizing τ is negatively related with ϕ. It is clear that both rates are higher (lower) than the output elasticity of government = 0.25, provided that ϕ is set at a lower (higher) value. spending, 1 E The upward-sloped red solid locus in Fig. 7(b) plots the threshold value of = k 0 . Hence, points located to the right (left) of the 0 red solid locus are associated with ϕ > ( < )ϕ and hence correspond to the case where the model possesses a unique (two) BGPs. Fig. 7(b) clearly shows that, when ϕ is set at a value higher than ϕ such that the indeterminate low-growth trap is eliminated, both the = 0.25. For example, when growth rate- and the welfare-maximizing marginal income tax rates are strictly smaller than 1 = 0.02, the growth rate- and the welfare-maximizing marginal income tax rates are 16.83% and 19.21%, respectively; both points E are located to the right of the locus of = k 0 . 0 Fig. 7(c) presents the case of consumption taxation. The solid blue locus and the dashed blue locus likewise illustrate that both the growth rate- and the welfare-maximizing τ decline when the consumption tax rate τc rises. The upward-sloped red solid locus E illustrates that the threshold value of c = k0 above which the low-growth BGP is eliminated increases with the marginal income tax 0 rate τ. It is clear that, when using consumption tax to recover equilibrium uniqueness, both the growth rate- and the welfare= 0.25. maximizing τ are strictly smaller than the output elasticity of government spending, 1 Fig. 7(d) finally presents the case of investment subsidies. The upward-sloped red solid locus illustrates the requisite investment E subsidy rate s = k0 such that there exists a unique BGP equilibrium as depicted in Fig. 6(d). The upward-sloped green solid locus 0

plots the level of s˜ =

. Therefore, points located to the left of s˜ correspond to the case where the model possesses two BGPs, (1 ) + i.e. Fig. 6(a) and (b). Points located between the loci of s˜ and s correspond to fs (˜s < s < s ) in Fig. 6(c) where no BGP exists. The solid blue locus and the dashed blue locus in Fig. 7(d) show that both the growth rate- and the welfare-maximizing τ: (i) 392

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S.-H. Chen

increase when the investment subsidy rate s rises; and (ii) are always higher than the output elasticity of government spending, 1 = 0.25. Noticeably, both of these two loci are above the locus of s˜ . This indicates that, when the marginal income tax rate is set optimally, investment subsidies are unable to eliminate the low-growth trap. For example, when s = 0.1, the growth rate- and the welfare-maximizing marginal income tax rates are 30.92% and 36.06%, respectively; both points are located above the loci of s˜ and s, indicating that there exist two BGPs. I summarize the above results in the following propositions. Proposition 8. Under linearly progressive income taxation, both the welfare-maximizing and the growth rate-maximizing marginal income tax rates: (i) increase with tax progressivity and the investment subsidy rate; (ii) decrease with the lump-sum taxes-to-capital ratio and the consumption tax rate. Proposition 9. When using either lump-sum taxes or a consumption tax to eliminate the low-growth trap, both the welfare- and the growth rate-maximizing marginal income tax rates are strictly lower than the elasticity of output with respect to government spending. When the marginal income tax is set to attain either maximum welfare or maximum output growth, investment subsidies are unable to eliminate the lowgrowth trap. 6. Non-linear tax scheme Under a non-linear progressive income tax formulation á la Guo and Lansing (1998) that displays continuously increasing average and marginal tax rates, Chen and Guo (2013) analytically demonstrate that a sufficiently progressive income tax scheme destabilizes the economy of Barro (1990), and that a less progressive tax schedule can maintain saddle-path stability of the model. This section reexamines the (de)stabilization effect of such a non-linear fiscal policy rule within the macroeconomy described in Section 2. The budget constraint faced by the representative household is now changed to:

ct + it = yt

t (yt

Et ) ,

E0

0 given,

(51)

Tax Paid

where yt Et is the household’s taxable income, and τt is the tax rate taking on the functional form which is continuously increasing and differentiable in the taxable income:

=

t

yt yt*

+

Et Et*

,

0<

< 1,

0<

< 1.

(52)

In (52), yt* Et* denotes the benchmark level of taxable income that is taken as given by the representative household, where Et* = Et = E0 e n t for all t. In this paper’s environment with endogenous growth, yt* is set equal to the level of per capita income on the economy’s BGP whereby

yt* yt*

=

n

> 0 for all t.11 I follow Dromel and Pintus (2007, p.27) in defining tax progressivity as

=(m a )/(1 a ), where τm and τa respectively denote the marginal and the average tax rates. I then derive that ( + ) E0 / y0 + = 1 ( + )(1 (0, 1), and hence the tax progressivity π increases with both the exemption threshold level E0 and the slope E /y ) 0

0

parameter of the tax scheme φ: Let

mt

E0

(1 )( + ) [( + ) E0 / y0 + ] y0

=

> 0 and

=

+ (1 + ) ut denote the marginal tax rate, where ut

=

( + ) E0 / y0 + yt Et is a yt* Et*

> 0. function of (xt; τ, η, φ, E0). It is straightforward to

show that (i) the equilibrium conditions for this specification can be represented by the following autonomous dynamical system with no given initial condition:

z t = {x t xt =

1

[1 xt (1

(1

mt )

1 mt Ax t

)

mt Ax t

] Axt1 [

+ zt Ax t1

n

(53)

} zt , + xt + z t + ];

(54)

and (ii) the existence and number of the economy’s interior balanced growth path(s) are governed by u* = 1 and:

z * = [1

(1

m)

x * = ( + ) A (x *)1 where

] A (x *)1

(55)

x* + ,

( + ) E0 / k 0

(56)

h (x *),

+ (1 + ) is the marginal tax rate on a BGP. With (56), the balanced growth rate is expressed as:

m

=

n

= (1

m)

A (x *)1

(57)

.

Fig. 8 plots the locus of h(x*) defined in (56) together with the 45-degree line. It is clear that h(x*) has a similar shape as f(x*) 11 To ensure the existence of a balanced-growth equilibrium in the subsequent analyses, the household’s taxable income yt needs to grow at the same rate as the baseline level of taxable income yt* Et* .

393

Et in equilibrium

Journal of Macroeconomics 60 (2019) 378–395

S.-H. Chen

Fig. 8. Non-linear Tax Scheme. c shown in Fig. 1.12Fig. 8(b) shows that the locus of h (x *; E0 = E^0 ) is tangent to the 45-degree line when the exemption threshold level c c 1 [( )(1 ) A] . equals E^ and the corresponding government spending to capital ratio is x^ n

0

m

In terms of the local stability properties of a BGP, I derive that the determinant and trace of the Jacobian matrix Jn of the dynamical system (53) and (54) are respectively:

( + ) A (x *) z *

Detn = Trn =

2

,

(58)

1

( + ){(

y

+ )

m (1

) A (x *)

+ }

,

(59)

1

where 1

m [1

2

a2 2

2

h (x *)]

+ a1

0 when x * x^n

+ a 0 0 when

˜

[

m (1

a1 +

a12

1

) A] ,

2a 0 a2

a2

(60)

.

(61)

2( + ) < 0. 0, and a2 )(1 (1 ) In the above equation, a0 ( + )(1 m ) E0 / k 0 > 0, a1 Fig. 9 graphically proves the existence and uniqueness of ˜ .13 c c Let E^0 denote the exemption threshold level such that x^n = h (x^n ; E0 = E^0), where E^0 < E^0 and x^n > x^n . Fig. 8 illustrates how the number and the local stability properties of the BGPs change as the exemption threshold level rises. The results are summarized in the following proposition. A (x *)1

A (x *)1

Proposition 10. Under (51) and (52), dual balanced growth equilibria emerge wherein: (i) the low-growth one is a locally indeterminate sink; (ii) the high-growth one exhibits saddle-path stability when the exemption threshold level is lower than E^0 and the slope parameter of the income tax schedule is below ˜ ; otherwise, the high-growth BGP exhibits equilibrium indeterminacy. Proof. First, Eqs. (59) and (60) indicate that Trn < 0 if Ω1 < 0 ( x * < x^n ) and that Trn > 0 if Ω1 > 0 ( x * > x^n ). Next, when x * < x^n (Ω1 < 0 and Trn < 0), Eqs. (58) and (61) reveal that Detn ⋛0 if Ω2 ⋚0 ( ˜ ), indicating the presence of one (two) eigenvalues with negative real parts. Thus, for x * < x^n , the Jacobian Jn displays one (two) degree (degrees) of indeterminacy if > (< ) ˜ . Similarly, 12

Specifically, h(x*) has a negative vertical intercept h (0) =

h (x *) = ( + )(1 13

) A (x *)

> 0 and h (x *) =

h (x *) x*

( + ) E0/ k 0 < 0, and is a positively-sloped concave curve in the positive quadrant:

< 0.

The locus of Ω2 is a concave curve ( 2 = a2 < 0 ) that has a positive vertical intercept ( 2 (0) = a0 > 0 ). Fig. 9 plots the case where a1 < 0 and a1 hence 2 = a2 + a1 < 0, for all φ. When a1 > 0, 2 0, if . Although in this case Ω2 is upward-sloping (downward-sloping) at lower (higher) a2 values of φ, the locus of Ω2 still intersects the horizontal axis once, and hence there exists a unique ˜ such that 2 = 0 . 394

Journal of Macroeconomics 60 (2019) 378–395

S.-H. Chen

Fig. 9. Identifying the Sign of Ω2.

when x * > x^n (Ω1 > 0 and Trn > 0), I have Detn⋛0 if Ω2⋛0 ( (one degree of indeterminacy) if < (> ) ˜ . □

˜ ). Thus, for x * > x^n , the Jacobian Jn exhibits saddle-path stability

It is clear by comparing the results in Propositions 2 and 10 that the non-linear progressive formulation of the income tax schedule expands the scope of indeterminacy. The result in Propositions 10(ii) implies that, to guarantee saddle-path stability of the highgrowth BGP, the income tax scheme cannot be too progressive, in the sense that E0 < E^0 and < ˜ are needed. This result is qualitatively consistent with that obtained in Chen and Guo (2013). Thus, Proposition 10 demonstrates the robustness of the result in Section 3 as well as that in Chen and Guo (2013). 7. Conclusion This paper shows that a linearly progressive income tax scheme destabilizes Barro's (1990) endogenously growing economy under a flow of productive government spending by giving rise to multiple BGPs with an indeterminate low-growth trap. I demonstrate that both lump-sum taxes and a consumption tax effectively eliminate the low-growth balanced growth equilibrium, thereby ensuring the existence of a unique BGP that displays saddle-path stability and high output growth. In these cases, both the growth rate- and the welfare-maximizing marginal income tax rates are lower than the elasticity of output with respect to government expenditures. This paper can be extended in several directions. For example, it would be worthwhile to incorporate variable labor supply (Palivos et al., 2003), the stock of public capital (Futagami et al., 1993, 2006), or national debt (Greiner, 2007) into the analytical framework. These possible extensions will allow me to examine the robustness of this paper’s theoretical results and policy implications, as well as further enhance the understanding of the relationship between linearly progressive taxation, economic growth, and macroeconomic (in)stability in an endogenously growing setting with public production services. I plan to pursue these research projects in the near future. References Aschauer, D.A., 1989. Is public expenditure productive? J. Monet. Econ. 23, 177–200. Barro, R.J., 1990. Government spending in a simple model of endogenous growth. J. Polit. Econ. 98, S103–S125. Benhabib, J., Farmer, R.E.A., 1994. Indeterminacy and increasing returns. J Econ Theory 63, 19–41. Chen, S.H., Guo, J.T., 2013. On indeterminacy and growth under progressive taxation and productive government spending. Canadian J. Econ. 46, 865–880. Chen, S.H., Guo, J.T., 2016. Progressive taxation, endogenous growth, and macroeconomic (in)stability. Bull. Econ. Res. 68 (S1), 20–27. Chen, S.H., Guo, J.T., 2018. On indeterminacy and growth under progressive taxation and utility-generating government spending. Pacif. Econ. Rev. 23, 533–543. Chen, S.H., Guo, J.T. Progressive taxation as an automatic destabilizer under endogenous growth. J. Econ. (2018b) forthcoming (accepted). Christiano, L.J., Harrison, S.G., 1999. Chaos, sunspots, and automatic stabilizers in a business cycle model. J. Monet. Econ. 44, 3–31. Dromel, N.L., Pintus, P.A., 2007. Linearly progressive income taxes and stabilization. Res. Econ. 61, 25–29. Eberts, R.W., 1986. Estimating the Contribution of Urban Public Infrastructure to Regional Growth. Federal Reserve Bank of Cleveland Working paper no. 8610. Futagami, K., Morita, Y., Shibata, A., 1993. Dynamic analysis of an endogenous growth model with public capital. Scand. J. Econ. 95, 607–625. Greiner, A., 2006. Progressive taxation, public capital, and endogenous growth. FinazArchiv 62, 353–366. Greiner, A., 2007. An endogenous growth model with public capital and sustainable government debt. Jpn. Econ. Rev. 58, 345–361. Guo, J.-T., Lansing, K.J., 1998. Indeterminacy and stabilization policy. J. Econ. Theory 82, 481–490. Guo, J.-T., Harrison, S.G., 2004. Balanced-budget rules and macroeconomic (in)stability. J. Econ. Theory 119, 357–363. Li, W., Sarte, P.-D., 2004. Progressive taxation and long-run growth. Am. Econ. Rev. 94, 1705–1716. Palivos, T., Yip, C.K., 1995. Government expenditure financing in an endogenous growth model: a comparison. J. Money Credit Bank. 27, 1159–1178. Palivos, T., Yip, C.Y., Zhang, J., 2003. Transitional dynamics and indeterminacy of equilibria in an endogenous growth model with a public input. Rev. Devel. Econ. 7, 86–98. Yamarik, S., 2001. Nonlinear tax structures and endogenous growth. Manchester School 69, 16–30.

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