In this appendix, we present the details of the extended model with debt in our paper "Transitional Dynamics of Dividend and Capital Gains Tax Cuts." Section 1 presents the extended model and results. Section 2 presents the numerical algorithm to solve this model. Section 3 presents an extensiongure for the simulation conducted in Section 3 of our original paper.additional

Francois Gourio

Jianjun Miao

English

Research Papers in Economics

Technical Appendix for \Transitional Dynamics of
Dividend and Capital Gains Tax Cuts"
Fran» cois Gourio¤ and Jianjun Miaoy
June 24, 2010
In this appendix, we present the details of the extended model with debt in our paper
\Transitional Dynamics of Dividend and Capital Gains Tax Cuts." Section 1 presents the
extended model and results. Section 2 presents the numerical algorithm to solve this model.
Section 3 presents an additional ¯gure for the simulation conducted in Section 3 of our original
paper.
1 The Extended Model with Debt
We extend the baseline model to incorporate debt ¯nancing. To keep the model tractable, we
consider risk-free debt and ignore the issue of default. Debt has a tax advantage in that interest
payments are tax deductible. But debt is limited by a collateral constraint, as in Kiyotaki and
Moore (1997) and Hennessy and Whited (2005). Suppose a ¯rm issues debt bt with interest
rate rt. We interpret the case with bt < 0 as saving. The collateral constraint is given by:
(1 + rt)bt · ´kt; b0 given, (1)
where ´ > 0: The ¯rm's °ow of funds constraint becomes:
xt +
Ãx2
t
2kt
+ dt + (1 + rt)bt = (1 ¡ ¿c)¼ (kt;zt;wt) + ¿c (±kt + rtbt) + st + bt+1: (2)
Its decision problem is to choose fbt+1;kt+1;st;dt;xtg so as to maximize
Vt = Et
1 X
j=0
1
Rt;t+j
Ã
1 ¡ ¿d
t+j
1 ¡ ¿
g
t+j
dt+j ¡
¡
1 + ¸1st+j>0
¢
st+j
!
;
¤Department of Economics, Boston University, and NBER. Address: 270 Bay State Road, Boston MA 02215.
Tel.: 617-353-4534. Email: fgourio@bu.edu.
yDepartment of Economics, Boston University, 270 Bay State Road, Boston MA 02215, USA, and Zhongnan
University of Economics and Law. Email: miaoj@bu.edu. Tel: (617) 353 6675.
1subject to (2), (1), and
dt ¸ 0; (3)
st ¸ ¡¹ s: (4)
In this case, there are three state variables (kt;bt;zt) in the ¯rm's dynamic programming
problem. As a result, ¯rms can be di®erentiated by these three characteristics. In the cross
section, there is a distribution ¹t of ¯rms over (kt;bt;zt): We use this distribution to conduct
aggregation. We can then de¯ne a competitive equilibrium as in Section 2.4.
We set ´ = 0:3; which is within the range of estimates of capital resale discounts in Ramey
and Shapiro (2001). Our results are robust to changes in this parameter value. In addition, this
value implies that the ratio of debt to ¯rm value is 0.14, which is within the range of empirical
estimates. We take all other parameter values as in Table 1 in the original paper. Based on
these parameter values, we solve the model numerically and compare the solution with that in
the baseline model.
1.1 Steady State
We start with the steady-state properties of the extended model with debt. We conduct the
policy experiment as in Section 3.2, in which dividend and capital gains tax rates are cut from
0.25 and 0.20, respectively, to the same 0.15 permanently. These tax cuts are unexpected
and implemented in the initial period 1. Table 1 presents the pre-tax-cut and post-tax-cut
steady states for both the baseline and extended models. We ¯nd that the impacts of the tax
cuts on the economy in the two models are qualitatively similar, though there are quantitative
di®erences. The tax cuts have a smaller e®ect (in percentage term) in the extended model
with debt. For instance, the capital stock increases by 3.12% following the reform, whereas
it is 4.05% in the baseline model without debt. This re°ects that the proportional reduction
in the user cost is smaller, since the user cost is smaller in the model with debt. Moreover,
the productivity e®ect of relaxing ¯nancial frictions is smaller, since debt alleviates ¯nancial
frictions somewhat.
Compared to the baseline model without debt, the °exibility of using debt and equity ¯-
nancing allows ¯rms to reduce the cost of capital and thus bene¯ts the economy. In particular,
the steady-state aggregate real quantities such as investment, capital stock, consumption, em-
ployment, and output are all higher in the extended model than in the baseline model. TFP
is also higher since ¯rms' expansions can be ¯nanced with debt. However, aggregate dividends
2and new equity issuance are smaller in the extended model than in the baseline model. This
is because ¯rms must use part of earnings to pay interests of debt instead of distributing div-
idends in the extended model. In addition, in the extended model, ¯rms can raise debt to
¯nance investment and distribute dividends and thus may reduce equity issuance.
[Insert Table 1 Here.]
1.2 Transitional Dynamics
Now, we study transitional dynamics for the policy experiments considered in Sections 3.2 and
3.3. Figures 1-4 present the results. These ¯gures reveal that the transitional dynamics of real
quantities in the baseline model and in the extended model are similar. The main di®erence
between the two models' predictions is re°ected in the ¯nancial quantities. In the extended
model with debt, ¯rms can borrow or save to transfer cash from the future to the present
or from the present to the future. This °exibility allows ¯rms to conduct intertemporal tax
arbitrage so that they can take advantage of low dividend taxes. In the baseline model without
debt, in order to take advantage of low dividend taxes, the only way to pay more dividends for
¯rms is to cut back investment, ceteris paribus.
Figure 2 reveals that, in response to the unexpected and permanent tax cut, aggregate
debts rise over time. This is because the collateral constraints are gradually relaxed as ¯rms
build up capital stock over time (see Figure 1). Because ¯rms can borrow against their future
earnings, they can distribute more dividends initially to take advantage of the dividend tax cut
immediately, as revealed in the top left panel of Figure 2.
Figures 3-4 show that, when the dividend and capital gains tax cuts are unexpected and
last from periods 1-8 temporarily, the economy will stay in the same steady state as that before
the tax cuts in the long run. But investment decreases during periods 1-8 and jumps up in
period 9. From periods 1-8 ¯rms raise more debt. Firms use the funds raised by debt to
distribute more dividends, rather than to make more investment. As in the baseline model
without debt, ¯rms still cut back investment to pay more dividends. In period 9, the dividend
tax rate reverts back to the original higher level. Anticipating this policy, ¯rms reduce dividend
payments and make more investment in period 9. In addition, ¯rms borrow more in period
8 and repay debts in period 9. Overall, the transitional dynamics of real quantities are very
similar in the models with and without debt, but the dividends and equity issuance are more
volatile in the extended model with debt. In particular, dividend payments rise by about 35
3and 30 percent, respectively, in periods 1 and 8 (compared to about 15 and 25 percent in the
model without debt), and decrease by about 20 percent in period 9 (compared to 10 percent
in the model without debt).
2 Numerical Method
The solution method is similar to that for the model without debt. However, because we now
have three state variables (capital, debt, and productivity), we need to modify the previous
algorithm to make the computation faster. Our algorithm solves the value function on a
relatively coarse grid, but allows the ¯rm's choices for future capital and debt to lie on a
thinner grid.
2.1 Steady State
We use the same three general steps as in the case without debt. However, the details di®er.
Step 1. Starting with a guess of wage w, solve the ¯rm's dynamic programming problem
by value function iteration on a grid. We use a coarse grid with nk = 25 points for capital,
and nb = 15 points for debt. The choice of capital tomorrow and debt tomorrow has to lie in a
di®erent (thinner) grid, with n0
k = 180 and n0
b = 100 points. To ¯nd the value outside the grid
points, we use spline interpolation. We keep the same grid for z as in the case without debt.
Step 2. After obtaining the value function in step 1, we solve for the optimal decision
rules on the thin grid by solving the dynamic programing problem once. Call these policy
functions k0 = g(k;b;z) and b0 = h(k;b;z): Next, we solve for the stationary distribution of
¯rms ¹¤(k;b;z;w) by simply iterating on the following equation:
¹t+1 (A £ B £ C) =
Z
1g(k;b;z)2A1h(k;b;z)2BQ(z;C)¹t (dk;db;dz);
starting from a uniform distribution over (k;b;z): This equation is similar to equation (12) in
the main text but is adapted to also allow for a debt choice.
Step 3. As in the case without debt, we obtain the aggregate labor demand Ld(w) =
P
k;b;z ¹¤(k;b;z;w)l(k;b;z;w); and then check whether the labor market clears, i.e. whether
the equation ¡U2(C;Ld (w))=U1(C;Ld (w)) = (1 ¡ ¿i)w holds, where aggregate consumption
C is deduced from the resource constraint and the stationary distribution. If the equilibrium
condition is not satis¯ed, we use the bisection method to update the wage rate and go back to
step 1.
42.2 Transitional Dynamics
Assume that the economy starts in the steady state associated with the tax rates
¡
¿
g
0;¿d
0
¢
:
Assume that for t ¸ T, the economy reaches a new steady state with constant tax rates
¡
¿
g
T;¿d
T
¢
: We can then solve the transitional dynamics implied by a sequence of tax rates
©
¿d
t ;¿
g
t
ªT
t=0 ; from periods 0 to T; as follows.
Steps 1-2. As in the model without debt, we compute the initial and ¯nal steady-states
and guess a path for the interest rate frtg
T
t=1 and a path for the wage rate fwtg
T
t=1.
Step 3. Given fwt;rtg, solve the ¯rm's dynamic programing problem by ¯nite backward
induction, assuming that VT(k;b;z) is the new steady-state value function V ¤(k;b;z): As we
do for the steady-state, we use a coarse grid for k;b and a thin grid for future choices k0;b0: We
obtain the policy functions for each date by linear interpolation, so that we generate the policy
functions kt+1 = gt(k;b;z) and bt+1 = ht(k;b;z):
Step 4. Given these policy functions, we compute the evolution of the cross-sectional
distribution for any time t. However, since the policy functions are interpolated, they may not
fall in the (thin) grid. We proceed as follows: for any t;k;b;z; we ¯nd the unique index i such
that ki < gt(k;b;z) < ki+1; where fkig is the thin grid, and we then assume that gt(k;b;z) = ki
with probability
ki+1¡gt(k;b;z)
ki+1¡ki ; and gt(k;b;z) = ki+1 with probability
gt(k;b;z)¡ki
ki+1¡ki : (This method is
suggested by Rios-Rull (2000). Alternatively, we can use simulations to ¯nd the cross-sectional
distribution. In practice, our method seems to work better for our problem.) We can thus
¯nd the distribution ¹t(k;b;z), for any t; given ¹0; on a discrete support n0
k £ n0
b £ nz: Then,
we deduce the aggregates Yt;Nt;Ct, for t = 1;:::;T ¡ 1; using aggregation and the resource
constraints.
Step 5: As in the model without debt, we update the interest rate and wage paths. We
set ½ = 0:94:
3 An Additional Figure
Figure 5 presents the simulated series of aggregate investment, output, consumption, employ-
ment and dividends analyzed in Section 3 of our original paper. This ¯gure complements Figure
7 in the original paper.
5Table 1. Steady-state results for the baseline and extended models
Baseline model Extended model
pre-tax cut post-tax cut pre-tax cut post-tax cut
Investment 0:085 0:088 (4:05) 0:092 0:095 (3:12)
Capital 0:892 0:929 (4:05) 0:971 1:001 (3:12)
Consumption 0:467 0:472 (1:11) 0:485 0:489 (0:86)
Labor 0:298 0:300 (0:37) 0:301 0:302 (0:36)
Output 0:564 0:574 (1:86) 0:594 0:603 (1:50)
Dividends 0:053 0:061 (15:23) 0:047 0:052 (10:80)
Equity issuance 0:012 0:019 (55:01) 0:010 0:018 (72:45)
Notes: This table presents the pre-tax-cut and post-tax-cut steady states for both the baseline
and extended models. The numbers in the brackets give the percentage changes after the
tax cuts. Dividends and capital gains taxes are reduced from 25 percent and 20 percent,
respectively, to the same 15 percent. The tax cuts are unexpected and permanent.
60 10 20 30
0
0.5
1
1.5
2
2.5
3
K
0 10 20 30
0
0.5
1
1.5
Y
0 10 20 30
−0.4
−0.2
0
0.2
0.4
0.6
0.8
C
0 10 20 30
0
0.5
1
1.5
2
2.5
3
3.5
I
0 10 20 30
0
0.1
0.2
0.3
0.4
0.5
N
0 10 20 30
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
TFP
Figure 1: Impact of unexpected permanent dividend and capital gains tax cuts in
the extended model with debt. The economy before period 1 is at the initial steady state
with parameter values given in Table 1. The ¯gure plots the responses of capital (K), output
(Y ), consumption (C), labor (N), investment (I), and TFP to the unexpected permanent cuts
of the dividend tax rate from 0.25 to 0.15 and of the capital gains tax rate from 0.20 to 0.15. In
each panel, the horizontal axis measures time period, and the vertical axis measures percentage
deviation from the initial steady state before the tax cuts.
70 10 20 30
0
5
10
15
20
25
30
Dividends
0 10 20 30
0
5
10
15
20
25
30
35
Equity issuance
0 10 20 30
0
1
2
3
4
5
6
Capital gains
0 10 20 30
0
5
10
15
Debt
Figure 2: Impact of unexpected permanent dividend and capital gains tax cuts in
the extended model with debt. The economy before period 1 is at the initial steady state
with parameter values given in Table 1. The ¯gure plots the responses of dividends, equity
issuance, the ratio of capital gains to equity value, and ¯nance regimes to the unexpected
permanent cuts of the dividend tax rate from 0.25 to 0.15 and of the capital gains tax rate
from 0.20 to 0.15. In each panel, the horizontal axis measures the time period. In the top two
panels, the vertical axes measure the percentage deviation from the initial steady state before
the tax cuts. The bottom left panel, the vertical axis measures the percentage of the rate of
capital gains. In the bottom right panel, the vertical axis measures the share of ¯rms in each
¯nance regime.
80 10 20 30
−2
−1.5
−1
−0.5
0
K
0 10 20 30
−0.6
−0.4
−0.2
0
0.2
Y
0 10 20 30
−0.5
0
0.5
1
C
0 10 20 30
−10
−8
−6
−4
−2
0
I
0 10 20 30
−0.8
−0.6
−0.4
−0.2
0
0.2
N
0 10 20 30
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
TFP
Figure 3: Impact of unexpected temporary dividend and capital gains tax cuts in
the extended model with debt. The economy before period 1 is at the initial steady state
with parameter values given in Table 1. The ¯gure plots the responses of capital (K), output
(Y ), consumption (C), labor (N), investment (I), and TFP to the unexpected temporary cuts
of the dividend tax rate from 0.25 to 0.15 and of the capital gains tax rate from 0.20 to 0.15.
The tax cuts last from periods 1-8. In each panel, the horizontal axis measures time period,
and the vertical axis measures percentage deviation from the initial steady state before the tax
cuts.
90 10 20 30
−10
0
10
20
30
Dividends
0 10 20 30
0
5
10
15
20
25
30
35
Equity issuance
0 10 20 30
−2
−1
0
1
2
3
Capital gains
0 10 20 30
0
5
10
15
Debt
Figure 4: Impact of unexpected temporary dividend and capital gains tax cuts in the
extended model with debt. The economy before period 1 is at the initial steady state with
parameter values given in Table 1. The ¯gure plots the responses of dividends, equity issuance,
the ratio of capital gains to equity value, and ¯nance regimes to the unexpected temporary
cuts of the dividend tax rate from 0.25 to 0.15 and of the capital gains tax rate from 0.20 to
0.15. The tax cuts last from periods 1-8. In each panel, the horizontal axis measures the time
period. In the top two panels, the vertical axes measure the percentage deviation from the
initial steady state before the tax cuts. The bottom left panel, the vertical axis measures the
percentage of the rate of capital gains. In the bottom right panel, the vertical axis measures
the share of ¯rms in each ¯nance regime.
102005 2010 2015
0.8
0.85
0.9
0.95
1
1.05
Investment
 
 
2005 2010 2015
0.96
0.98
1
1.02
Consumption
2005 2010 2015
0.88
0.9
0.92
0.94
0.96
0.98
1
1.02
Employment
2005 2010 2015
0.9
0.92
0.94
0.96
0.98
1
1.02
Output
2005 2010 2015
0.8
0.9
1
1.1
1.2
1.3
Dividends
Data
Model: Perm
Model: Temp

Online Appendix to "Transitional Dynamics of Dividend and Capital Gains Tax Cuts"

http://www.EconomicDynamics.org/appendix/08/08-187.pdf

Online Appendix to "Transitional Dynamics of Dividend and Capital Gains Tax Cuts"

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