1 Sep 2013 - p u β. = the accumulation constraint for tangible capital,. 4 equation (1) in the main text, by. ( ). 1,.
September 2013
Intangible Investment and Ramsey Capital Taxation Supplementary Online Appendix Juan C. Conesa Stony Brook University
Begoña Domínguez University of Queensland
ABSTRACT __________________________________________________________________________________ This Appendix derives the implementability condition presented in the main text, shows the first order conditions of the Ramsey problem and the details of the Proof of Proposition 1. __________________________________________________________________________________
1
1
1. The implementability condition
2
First, plugging dividend/equity distribution into the household’s budget constraint to get ct + bt +1 = (1 − τ tn ) wt nt + (1 + rt b )bt
3
+ (1 − τ td ) (1 − τ tc ) f ( km ,t , ku ,t , nt ) − xu ,t − wt nt + τ tcδ m km ,t − xm ,t .
(A1)
4
Then equation (A1) is multiplied by pt = β t u1,t , the accumulation constraint for tangible capital,
5
equation (1) in the main text, by χ tm = (1 − τ td ) pt
6
χ tu = (1 − τ td )(1 − τ tc ) pt ∞
∑ p c + b t
7
1 . The resulting equations are added together and over time to obtain I1,ut
− (1 − τ tn ) wt nt − (1 − τ td ) (1 − τ tc ) f ( km ,t , ku ,t , nt ) − wt nt − xu ,t + τ tcδ m km,t − xm,t − (1 + rt b )bt
t +1
t
1 , and the one for intangible capital, equation (2), by I1,mt
t=0
∞
∞
+ ∑ χ tm km ,t +1 − (1 − δ m ) km ,t − I m ( xm ,t , em,t ) + ∑ χ tu ku ,t +1 − (1 − δ u ) ku ,t − I u ( xu ,t , eu ,t ) = 0.
(
)
t=0
8 9
(
)
t=0
Substituting pt , χtm , and χ tu into the above equation, using the homogeneity of degree one in the final good’s production function and that wages equal the marginal productivity of labor, we obtain ∞
∑ β u
c − β t u1,t (1 − τ tn ) f3,t nt − u1,t (1 − τ td ) (1 − τ tc ) f1,t k m,t + f 2,t ku ,t − xu ,t + τ tcδ m km,t − xm ,t
t
1,t t
t=0
10
∞
∞
t=0
t=0
+ ∑ β t u1,t bt +1 − (1 + rt b )bt + ∑ (1 − τ td ) u1,t ∞
+ ∑ (1 − τ td )(1 − τ tc ) u1,t t=0
11
1 km,t +1 − (1 − δ m ) km ,t − I m ( xm,t , em,t ) m I1,t
(
)
1 ku ,t +1 − (1 − δ u ) ku ,t − I u ( xu ,t , eu ,t ) = 0. u I1,t
(
)
Rearranging, the above equation becomes Im Iu n d d c u1,t ct − u1,t (1 − τ t ) f3,t nt − u1,t (1 − τ t ) m − xm,t − u1,t (1 − τ t )(1 − τ t ) u − xu ,t t=0 I1,t I1,t ∞ ∞ 1 1 + ∑ β t u1,t bt +1 − (1 + rt b )bt + ∑ β t u1,t (1 − τ td ) m km,t +1 − (1 − τ tc ) ( f1,t − δ m ) + δ m + (1 − δ m ) m km,t I1,t t=0 t=0 I1,t ∞
∑β
12
t
∞ 1 1 + ∑ β t u1,t (1 − τ td )(1 − τ tc ) u ku ,t +1 − f 2,t + (1 − δ u ) u I1,t t=0 I1,t
ku ,t = 0.
2
1
Substituting in the tax rates through the competitive equilibrium conditions (7)-(9), and imposing
2
the first order conditions for government bonds u1,t = β u1,t +1 (1 + rt b+1 ), and for tangible and intangible
3
capital, equations (10)-(11) in the main text, we find
Im Iu I1,mt I1,ut β u1,t ct − u2,t nt − u2,t m − xm,t m − u2,t u − xu ,t u = u1,0 (1 + r0b )b0 + ∑ I 2,t I 2,t t=0 I 2,t I 2,t 1 1 u1,0 (1 − τ 0d ) (1 − τ 0c ) ( f1,0 − δ m ) + δ m + (1 − δ m ) m km ,0 + u1,0 (1 − τ 0d )(1 − τ 0c ) f 2,0 + (1 − δ u ) u ku ,0 . I1,0 I1,0 ∞
t
4
Finally, noting that e j ,t =
5
6
m u2,0 I1,0
(1 − τ ) = u d 0
1,0
7
m I 2,0
, and (1 − τ
d 0
j I j I1,t x j ,t , from the homogeneity of the investment functions, and − I 2,j t I 2,j t
u u2,0 I1,0
)(1 − τ ) = u c 0
1,0
u I 2,0
, from equations (8) and (9), the implementability
condition becomes ∞
∑β
t
u1,t ct − u2,t ( nt + em ,t + eu ,t ) = u1,0 (1 + r0b )b0 +
t=0
8
u m u2,0 m I1,0 I 2,0 u I f 1 km,0 + 2,0 I u f + (1 − δ u ) ) ku ,0 . − + + − δ δ δ ( ) ( ) 1,0 m m m m 1,0 u m u ( 1,0 2,0 I 2,0 I 2,0 I1,0 I 2,0
9 10
2. The first order conditions for the Ramsey problem
11
The first order conditions for the Ramsey problem presented in the main text are as follows:
12
[ ct ] :
β t u1,t + λ ( u1,t + ct u11,t ) = β t µt ,
[ nt ] :
β t u2,t + λ ( u2,t − ( nt + em,t + eu ,t ) u22,t ) = β t f3,t µt +
∂Θ , ∂nt
∂Θ em ,t : β t u2,t + λ u2,t − ( nt + em ,t + eu ,t ) u22,t = β t χ tmG I 2,mt + , ∂em,t
(
)
∂Θ eu ,t : β t u2,t + λ u2,t − ( nt + em ,t + eu ,t ) u22,t = β t χ tuG I 2,u t + , ∂eu ,t
(
)
∂Θ xm,t : β t χ tmG I1,mt + = β t µt , ∂xm,t
3
∂Θ xu ,t : β t χtuG I1,ut + = β t µt , ∂xu ,t
1
∂Θ km ,t +1 : β t χtmG = β t +1µt +1 f1,t +1 + β t +1χ tmG , +1 (1 − δ m ) + ∂k m ,t +1
∂Θ ku ,t +1 : β t χtuG = β t +1µt +1 f 2,t +1 + β t +1 χtuG , +1 (1 − δ u ) + ∂k u ,t +1
2
with λ , µt , χ tmG , and χ tuG denoting the multipliers on the implementability condition, resource
3
constraint, and tangible and intangible capital accumulation constraints, (1) and (2), respectively, where
4
decentralization constraints and the real value of the initial wealth have been condensed in
5
u m u2,0 m I1,0 I 2,0 u b u Θ = − u1,0 (1 + r0 ) b0 + m I1,0 u m ( f1,0 − δ m ) + δ m + (1 − δ m ) km ,0 + 2,0 I f + − k 1 δ ( ) ( ) 1,0 2,0 u u ,0 u I 2,0 I 2,0 I1,0 I 2,0
u ∞ u Iu Im u u + ∑ β t θtm 2,mt − β I1,mt +1 1,u t +1 2,mt +1 ( f1,t +1 − δ m ) + δ m + (1 − δ m ) 2,mt +1 +θtu 2,u t − β ( I1,ut +1 f 2,t +1 + (1 − δ u ) ) 2,u t +1 . I I I 2,t +1 I 2,t +1 t =0 I 2,t 2,t +1 I1,t +1 2,t
6 7
Note that partial derivatives with respect to Θ are zero in the long run because decentralization
8
constraints are not binding in the limit. This is consistent with the results of Proposition 1.
9 10
3. Proof of Proposition 1
11
The proof reproduces the procedure of Albanesi and Armenter (2007) in our setup.1 We show that there
12
exists a welfare improving perturbation to any candidate policy that at steady state features (i) dividend
13
tax rates different from labor tax rates and/or (ii) corporate taxes different from zero. Therefore, by
14
contradiction, the optimal corporate tax is zero and optimal dividend and labor tax rates are equal at
15
steady state.
16
The perturbation, that is considered, is a tax reform that takes place when the economy under the
17
candidate policy is at steady state. Steady-state variables are denoted with superscript ss. Any tax reform
18
to our candidate policy must imply an allocation that is feasible (satisfying resource and capital
1
Albanesi and Armenter (2007) illustrates the general results published in Albanesi and Armenter (2012).
4
1
accumulation constraints) and admissible (satisfying implementability and decentralization conditions).
2
For utility functions that are not logarithmic in consumption, the perturbation involves a tax reform that
3
affects labor reallocation in a given period and does not affect any other periods. For log utilities, since
4
income and substitution effects cancel out, one-period reforms cannot deliver labor reallocation. Then,
5
for this case, the proof considers a tax reform across two periods. Below, the proof for utilities that are
6
non-logarithmic in consumption is presented first and then followed by the one for log utilities.
7 8
3.1. Non-logarithmic utility functions
9
Let’s then consider non-log utilities and a reform that only affects period t. In what follows, we present
10
how the tax reform affects the allocation, through the adjustment in feasibility and admissibility
11
conditions, and then the taxes that decentralize that allocation. The implementability condition requires
12
d ( nt + em,t + eu ,t ) = Ψdct , where Ψ solves d ct u1,t − u2,t ( nt + em,t + eu ,t ) = 0, so that equilibrium is
13
restored within the same period. For our utility function, equation (12) in the main text, Ψ = u1,ss (1 − σ ) .
14
The change in resources is obtained from the derived variation in the capital accumulation constraints
15
(1) and (2), implying I1,mss dxm,t + I 2,mss dem,t = 0 and I1,uss dxu ,t + I 2,u ss deu ,t = 0 as both tangible and intangible
16
capital are not affected. Using the resource constraint ( dct + dxm,t + dxu ,t = f 3, ss dnt ), plugging in the
17
changes in total labor and in resources and using the equilibrium conditions (7)-(9), it follows that
18
u2, ss 1 1 + χ
1 − τ ssn 1 − Ψ f dc = ( 3, ss ) t d 1 − τ ss
1 − τ ssn − 1 f de + − 1 f 3, ss deu ,t . 3, ss m ,t d c τ τ 1 − 1 − ( )( ) ss ss
(
(A2)
)
19
The implied welfare change is β t u1,ss dct − u2, ss d ( nt + em ,t + eu ,t ) = β t ( u1, ss − Ψu2, ss ) dct . It is easy to
20
show that ( u1,ss − Ψu2, ss ) > 0 always, and (1 − Ψf3, ss ) > 0 at the optimum. Therefore, if τ ssn ≠ τ ssd and/or
21
τ ssc ≠ 0, a policy that reallocates the time devoted to labor/effort can potentially increase welfare. 5
1
So far, feasibility and the implementability condition have been imposed. Admissibility also
2
requires the decentralization constraints to hold. As we consider a reform that affects only period t,
3
satisfying the decentralization constraints means that
4
sixth equations in the Ramsey problem stated in the main text) are constant or, in terms of adjustment, u22, ss
5
u2, ss u22, ss
6
7
8 9
u2, ss m I 21, ss
with D1 =
m 1, ss
I
−
u2,t I
m 2,t
and
u2,t I 2,u t
(the left-hand-side of the fifth and
d ( nt + em ,t + eu ,t ) = D1dem,t ,
(A3)
d ( nt + em ,t + eu ,t ) = D2 deu ,t ,
(A4)
m u u I 21, I 22, I 22, ss ss ss and D > 0 = − u 2 u I 2,mss I I 2, ss 1, ss
> 0 constant.
Using (A2)-(A4), a feasible and admissible reform that increases consumption by dct = ε can be written in terms of labor reallocation as deu ,t = Z ε ,
10
dem ,t =
D2 Zε , D1
D u dnt = − 1 − 2, ss D1 2 + 1 Z ε . u22, ss D1
11
with Z =
(1 − Ψf ) 3, ss
1 − τ n 1 − τ ssn D2 ss − 1 f + − 1 f 3, ss 3, ss D1 (1 − τ ssd )(1 − τ ssc ) 1 − τ ssd
. Let’s consider situations close to the
12
optimum so that (1 − Ψf 3, ss ) > 0. If τ ssd > τ ssn and/or τ ssc > 0, then Z > 0. Then the above require an
13
increase in both types of effort, in intangibles and in tangibles, and a decrease in labor. If, instead,
14
τ ssd < τ ssn and/or τ ssc < 0, then Z < 0, and the opposite movement in labor activities is required. Such a
15
reform can be decentralized using (7)-(9) as follows:
16
D2 f33,ss u2,ss 1 n d τ = − D + 1 − D + 1 Z ε , t 3 f3, ss u22, ss 1 D1 (1 − τ ssn )
(A5) 6
1
D2 u 1 d d τ = − Z − 11, ss D4 t d D1 u1, ss (1 − τ ss )
2
u 1 1 dτ td + dτ tc = − D5 Z − 11, ss d c u1, ss (1 − τ ss ) (1 − τ ss )
3
ε ,
(A6)
ε ,
(A7)
m m u u m u where D3 = − u11,ss + Ψ u22,ss , D4 = I12,m ss − I11,mss I 2,mss > 0, and D5 = I12,u ss − I11,u ss I 2,u ss > 0 are constant. The
u1, ss
u2, ss
I1,ss
I1,ss I1, ss
I1, ss
I1,ss I1,ss
4
sign of D3 is ambiguous. Then if τ ssd > τ ssn and/or τ ssc > 0, a welfare improving perturbation involves a
5
decrease in dividend taxes and changes in labor and corporate taxes (of ambiguous sign) according to
6
(A5)-(A7). If, instead, τ ssd < τ ssn and/or τ ssc < 0, then the opposite movement in taxes is required. But the
7
welfare improving reform exists and in turn implies that, at steady state, optimal corporate taxes are zero
8
and the optimal levels of dividend and labor taxes coincide at steady state. Moreover, manipulating the
9
first order conditions of the Ramsey problem, optimal dividend and labor tax rates at steady state are
10
css u11, ss u − ( nss + em , ss + eu , ss ) 22, ss u1, ss u2, ss
λ − τ ssd * = τ ssn* =
u 1 + λ 1 − ( nss + em, ss + eu , ss ) 22, ss u2, ss
,
11
which are clearly positive. Note that λ denotes the multiplier on the implementability condition, which
12
is strictly positive. For our function (12), the tax rate becomes
λ σ +
13
τ ssd * = τ ssn* =
1
χ
1 1 + λ 1 + χ
.
14 15
3.2. Logarithmic utility functions
16
For utility functions that are logarithmic in consumption, it follows that Ψ = 0, which together with the
17
required change in the decentralization constraints (A3)-(A4), imply no labor/effort reallocation
18
dnt = dem ,t = deu ,t = 0. Therefore there is not a welfare improving perturbation that affects only one 7
1
period. Then, policy reforms that involve periods t and t+1 and no other periods are considered. The
2
proof proceeds as in the case of non-log utility functions. We first present the tax reform through the
3
required adjustment in terms of allocation and then show the taxes that decentralize that allocation.
4
u2,t +1
For period t+1,
I
m 2,t +1
=
u2, ss I
m 2, ss
and
u2,t +1 I
u2, ss
=
u 2,t +1
must hold to meet the decentralization constraints at
I 2,u ss
5
t+2. Then conditions (A2)-(A3) are satisfied at period t+1, which together with the implementability
6
condition d ( nt +1 + dem,t +1 + deu ,t +1 ) = 0, imply that dnt +1 = dem,t +1 = deu ,t +1 = 0. For date t, d ( nt + dem,t + deu ,t ) = 0
7
is satisfied, as well as the following decentralization constraints:
8
9
10 11 12
13 14 15 16
m I1,ut +1 I 2,mt +1 u2, ss f = β I1,t +1 u − δ + δ + 1 − δ , ( ) ( ) 1, t + 1 m m m m I 2,mss I 2,t +1 I1,t +1
u2,t I 2,mt u2,t I
u 2,t
= β ( I1,ut +1 f 2,t +1 + (1 − δ u ) )
u2, ss I 2,u ss
(A8)
(A9)
.
Using feasibility (resource and accumulation constraints (1) and (2)) and d ( nt + dem,t + deu ,t ) = 0 , any change affecting period t can be summarized in
dct +
1 I1,mss
dkm,t +1 +
1 I1,uss
Im Iu dku ,t +1 = 2,mss − f3,ss dem,t + 2,u ss − f3,ss deu ,t . I 1, ss I1,ss
As there is no variation in any labor activity, feasibility requires dct +1 = I1,mss f1, ss + (1 − δ m )
1 m 1, ss
I
dkm ,t +1 + I1,uss f 2, ss + (1 − δ u )
1 I1,uss
dku ,t +1 ,
in period t+1. Adding up the changes in consumption in periods t and t+1 and rearranging, it follows dct + β dct +1 = βτ
c ss
(f
1, ss
(1 − τ ssn ) 1 − τ ssn ) ( − δ m ) dk m ,t +1 + − 1 f de + − 1 f de . (A10) (1 − τ ssd ) 3, ss m ,t (1 − τ ssd )(1 − τ ssc ) 3, ss u ,t
17
The implied welfare change is β t u1, ss ( dct + β dct +1 ) . Then, if τ ssn ≠ τ ssd , and/or τ ssc ≠ 0, a policy reform that
18
affects tangible capital and/or reallocates the time devoted to labor/effort could increase welfare. Is there
19
an admissible policy that achieves that?
8
1
We have three taxes in period t and another three in t+1. For period t+1, our policy is similar to
2
(A5)-(A7) but taking into account that effort/labor cannot be reallocated while consumption can still
3
adjust. This implies a variation in labor taxes as in
4
according to
5
I
m 2, t +1 m 1,t +1
I
=
I
u 2,t +1 u 1,t +1
I
1 dτ tn+1 = − D3 dct +1 a change in dividend taxes (1 − τ ssn )
u 1 dτ td+1 = 11, ss dct +1 , and no movement in corporate taxes. The latter means d u1, ss (1 − τ ss )
, which after rearranging becomes dku ,t +1
m m I 21, 1 I11, ss ss m − m m (1 − δ m ) I I1, ss I1, ss = D6 dk m,t +1 , with D6 = 2, ss > 0. u u I 21, ss I11, 1 ss m − u u (1 − δ u ) I 2, ss I1, ss I1, ss
6
That is, tangible and intangible capital move in the same direction. Then the variation in consumption at
7
1 1 t+1 can be written as dct +1 = Ωdkm ,t +1 , with Ω = I1,mss f1,ss + (1 − δ m ) m + I1,u ss f 2,ss + (1 − δ u ) u D6 > 0. I1, ss I1, ss
8
Differentiating the decentralization constraints (A8)-(A9) and rearranging results in
9
m m I 21, I 22, ss ss m − m I I 2, ss 1, ss
dem ,t = D7 dk m ,t +1 ,
(A11)
10
u u I 21, I 22, ss ss u − u I 2, ss I1, ss
deu ,t = D8 dkm,t +1 ,
(A12)
11
Im Im where D7 = β I1,mss (1 − τ ssc ) f12, ss D6 + β I1,mss (1 − τ ssc ) f11, ss − β 11,mss (1 − τ ssc ) ( f1, ss − δ m ) + δ m (1 − δ m ) + m 21, ssm , I1, ss I 2, ss I1, ss
12
u u u u I11, I 21, ss and D8 = β I1, ss f 21, ss + β I1, ss f 22, ss − β u f 2, ss (1 − δ u ) + u ssu I1, ss I 2, ss I1, ss
D6 .
13
Let’s consider a steady state with τ ssd > τ ssn and/or τ ssc > 0. Using (A10)-(A12), a feasible and
14
admissible reform that decreases consumption at period t by dct = − a , increases consumption at period
15
t+1 by dct +1 = 1 ( a + ε ) and therefore raises welfare by β t u1, ss ( dct + β dct +1 ) = β t u1, ssε > 0 , can be written
16
in terms of labor reallocation and changes in tangible capital as
β
9
deu ,t = Zε ,
1 2
dem ,t =
1 1 dnt = − deu ,t − dem ,t = − + dkm ,t +1 , D9 D10
3
4
5
D10 deu ,t , D9
dkm ,t +1 =
with Z =
1 1 (a + ε ). Ωβ
dct + β dct +1 (1 − τ ssn ) (1 − τ ssn ) − 1 D10 βτ ssc ( f1, ss − δ m ) D10 + f − 1 + (1 − τ ssd ) 3, ss D9 (1 − τ ssd )(1 − τ ssc )
m m I 22, 1 I 21, ss ss , D9 = m − m , I 2, ss D7 I1, ss f3, ss
6
u u and D10 = 1 I 21,u ss − I 22,u ss . D9 and D10 have ambiguous sign. While the sign in the required change of I 2, ss D8 I1, ss
7
the allocation is not determined, we expect an increase in both types of effort, a decrease in labor and a
8
raise in tangible capital. Using (7)-(9), such a reform can be decentralized as
9
f 1 1 1 1 1 dτ tn = D3a − 33, ss + (a + ε ), n (1 − τ ss ) f3, ss D9 D10 Ω β m m I 21, u11, s I11, 1 ss d dτ t = − a + m − mss d I u1, ss (1 − τ ss ) 2, ss I1,ss
10
11
1 1 1 D ( a + ε ) − D11 10 Z ε , m D9 I1, ss Ω β
u u I 21, u11, s I11, 1 1 ss d c dτ t + dτ t = − a + u − u ss d c I u1, ss (1 − τ ss ) (1 − τ ss ) 2, ss I1, ss
D6 1 1 ( a + ε ) − D12 Z ε , u I Ω β 1, ss
12
1 1 dτ tn+1 = − D3 ( a + ε ) , n (1 − τ ss ) β
13
u 1 1 dτ td+1 = 11, ss ( a + ε ) , d u1, ss β (1 − τ ss )
14
dτ tc+1 = 0,
15
u u u u m m m m u m where D11 = I12, ss − I 22, ss + I 2, ss I 21, ss − I11, ss > 0, and D12 = I12,u ss − I 22,u ss + I 21,u ss − I11,u ss I 2,u ss > 0. As mentioned m m m m m
16
I1, ss
I 2, ss
I1, ss I 2, ss
I1, ss
I1, ss
I 2, ss
I 2, ss
I1, ss I1, ss
earlier, dividend and labor taxes should decrease and corporate taxes should not change in period t+1 10
1
(as D3 > 0 for Ψ = 0 ). The sign in the change of other taxes is ambiguous. If, instead, τ ssd < τ ssn and/or
2
τ ssc < 0, the reform may require a transfer of resources from t+1 to t, and the sign of the changes in
3
allocation and taxes would be the opposite. In any case, a welfare improving perturbation exists and, in
4
consequence, our candidate policy is not optimal. Therefore, the optimal corporate tax is zero at steady
5
state and the optimal level of dividend and labor tax rates are positive and identical at steady state.
6 7 8 9 10 11
References Albanesi, S., Armenter, R., 2007. Understanding capital taxation in Ramsey models. Unpublished Manuscript. Albanesi, S., Armenter, R., 2012. Intertemporal distortions in the second best. Review of Economic Studies 79, 1271-1307.
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