3.3 Steady-state properties
3.3.1 Comparative static analysis
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立 政 治 大 學
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Of particular note, our main focus is on the examination of the capital tax.
By holding the proportion of the government spending constant, an increase in the capital income tax must be coupled with a reduction in the labor income tax.
Therefore, we follow the literature on the Chamley-Judd result to assume that the labor income tax endogenously adjusts to balance the government budget. This approach has been dubbed “tax shifting” or “tax swap” in the literature.
3.3.1 Comparative static analysis
In this section, we analyze the e¤ects of the capital taxation on the R&D share, the endogenous labor income tax rate, labour supply, and scale-adjusted variables:
^
a; ^k; ^c; and ^y.2
The long run R&D labour share, s, is given by
s =
1
1 (1+ )gA
r gY + 1 + 11 (1+ )gA
: (21a)
It follows from the above equation that, in the steady state, a change in the capital income tax rate (21a) does not a¤ect the R&D labor share (i,e.,@s=@ K = 0). The intuition underlying @s=@ K = 0 can be grasped as follows. The non-arbitrage condition between physical capital and R&D equity reported in (20a) requires that the return of physical capital should be equal to the return of R&D equity.
Given that the return of R&D equity, r = +11 1 gA+ n, is independent of the capital tax rate, it is clear that the capital income tax rate is impotent to a¤ect the return of R&D equity and hence the R&D labor share.
2We solve the dynamic system in Appendix 3.B, and a detailed derivation of the comparative static analysis is presented in Appendix 3.C.
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Based on (21a), we have:
@ L
@ K = (1 s)
1 < 0: (21c)
Under the tax-shifting scheme, an increase in the capital income tax rate must be coupled with a reduction in the labor income tax rate.
Given a constant capital income tax rate K, labor supply in the steady state is given by:
It is straightforward from eq.(22a) to infer the following result:
@l
Equation (22b) indicates that, when the tax shifts form a labor income tax to a capital income tax, a rise in the capital income tax rate leads to an increase in labor supply. The rationale for this result can be understood intuitively. In response to
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wage income, thereby leading to an increase in labor supply. The latter positive e¤ect dominates the former negative e¤ect, and hence a rise in the capital income tax rate is accompanied with an increase in labor supply.
Moreover, scale-adjusted R&D varieties ^a is given by:
^
a = [ &
(1 + )gA
]1=(1 )(sl) =(1 ); (23a) where s and l are reported in eqs (21a) and (22a). With @s=@ K = 0, it is quite easy to derive from eq. (23a) that:
@^a
@ K =
(1 )^a @l
l@ K > 0: (23b)
Equation (23b) indicates that a rise in the capital income tax rate tends to boost scale-adjusted R&D varieties. The intuition behind this result is not hard to understand. Following a rise in the capital income tax rate coupled with a decline in the labor income tax rate, the household is motivated to raise its labor supply.
This in turn increases labor input allocated to the R&D sector (LA = N sl). Then, as reported in eq. (23a), given that scale-adjusted R&D varieties ^a is positively with R&D labor input sN l, ^a will increase in response following a rise in K.
From eqs (20a), (20c), (20d),(23a), and (20e), we can infer that:
^
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Equation (24b) indicates that a rise in the capital income tax rate has ambiguous e¤ect on scale-adjusted output ^y. As exhibited in eq. (24b), two con‡icting e¤ects emerge following a rise in the capital income tax rate. First, a rise in the capital income tax rate shrinks capital investment, which in turn generates a negative impact on output. Second, a rise in the capital income tax rate is accompanied with a fall in the labor income tax rate, which motivates the household to provide more labor supply. This leads more labor input to be allocated to the R&D sector and in turn boosts R&D varieties, thereby contributing to a positive e¤ect on output. If labor supply is exogenous ( = 0), the second positive e¤ect is absent (@l=@ K = 0), and a higher capital income tax rate lowers output. However, if labor supply is endogenous ( > 0), both con‡icting e¤ects are present, and the output e¤ect of capital income taxation depends upon the relative strength between these two e¤ects.
From eqs (20a), (20c), and (20d), we have:
k =^ (1 K)
( + gY) y;^ (25a)
^
c = [(1 ) (1 K) ]^y; (25b)
where ( + +g( +gY)
Y) is a composite parameter.
Based on eqs (25a) and (25b), the e¤ects of K on ^k and ^ccan be expressed as:
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The economic intuition behind eqs (26a) and (26b) can be explained as follows.
It is clear in eq. (25a) that capital income taxation a¤ects scale-adjusted capital ^k through two channels. The …rst channel is the capital-output ratio (^k=^y = (1( +gK)
Y) ), and the second channel is the level of scale-adjusted output ^y. The …rst term after the …rst equality in eq. (26a) indicates the …rst channel de…nitely lowers the level of ^k. Moreover, as shown in eq. (24b), the second channel may either raise or lower the level of ^k since capital taxation leads to an ambiguous e¤ect on ^y. As a consequence, the net e¤ect of capital taxation on the sale-adjusted capital stock
^k is still uncertain. Similarly, as indicated in eq. (25b), capital income taxation a¤ects ^c also through two channels. The …rst channel is the consumption-output ratio (^c=^y = [(1 ) (1 K) ]), and the second channel is the level of scale-adjusted output ^y. As exhibited in eq. (24b), the …rst channel de…nitely boosts the level of ^c, while the second channel may either raise or lower the level of ^c since capital taxation leads to an ambiguous e¤ect on ^y. As a consequence, the net e¤ect of capital taxation on scale-adjusted consumption ^cremains ambiguous.