The choice between owner’s wages and dividends under
the dual income tax: Extended appendix
Diderik Lund∗
Department of Economics, University of Oslo
P.O. Box 1095, Blindern, NO-0317 Oslo, Norway
tel. +47 2285 5129, fax. +47 2285 5035
e-mail: diderik.lund@econ.uio.no
web page: http://folk.uio.no/dilund/
First version March 1999
Slightly revised February 2003
This appendix presents in some detail the system of taxes and regulations which were in
effect in Norway in 1991, which are the basis for the research presented in Fjærli and Lund
(2001). In particular, the appendix concentrates on the systems of equations and inequalities which are necessary in order to calculate the budget sets which describe our observations. There are additional sources available at http://folk.uio.no/dilund/wagediv/,
desribing the computer programs we used.
There are also sections on some anomalies in the tax system, in particular the occurence
of marginal tax rates above one hundred percent in some instances.
1
Regulations of retained earnings
The most central relation is
Y − Tc − G = R = ∆F + Va − Vw + Ea − Ew ,
which follows from equations (1) and (A.6) in Fjærli and Lund (2001).
Some of the notation is:1
• Ea is the allocation to free equity,
• Ew is the withdrawal from free equity,
• Eb is the free equity at the beginning of the year,
∗
1
This is an extended appendix for Fjærli and Lund (2001).
The variable names which are not explained here, are explained in Fjærli and Lund (2001).
1
(1)
• Ee is the free equity at the end of the year,
• ∆F is the allocation to the consolidation fund,
• Va is the allocation to the reserve fund,
• Vw is the withdrawal from the reserve fund,
• Vb is the reserve fund at the beginning of the year,
• Ve is the reserve fund at the end of the year.
While the variables Y, R, and Ea may be positive or negative, the variables Tc , G, ∆F,
Va , Vw , and Ew are assumed to be non-negative.
The stock variables Eb and Ee related to Ea and Ew by the equation Ee − Eb = ∆E ≡
Ea − Ew . One or both stock variables may be negative, which is called an uncovered loss.
The withdrawal Ew plays a role in the tax base definition, so it is useful to define it as a
separate variable and restrict it to be non-negative and not to exceed Eb .
Similarly, Va plays a role in the tax base definition, and is restricted to be non-negative.
Vw is also non-negative, and the stock variables Vb and Ve are related to these by Ve − Vb =
∆V ≡ Va − Vw . There is either an allocation or a withdrawal, so when Vw > 0, then Va = 0,
and when Va > 0, then Vw = 0. Both can also be zero. The stock variables for the reserve
fund cannot be negative.
In our derivation of the budget set, we assume that the variables Π, R∗ , Vb and Eb are
exogenously given, where R∗ is the observed level of R. Also, the elements Te and Y∆
of the tax equations, explained in Fjærli and Lund (2001) p. 117, are exogenous. As we
vary Wg , there will be different levels of Y ≡ Π − Wg (1 + a), and for each of these, we
find the maximum G given the tax system and the exogenous variables. This will allow
us to draw one budget set in the (W, D) diagram for each value of the exogenous vector
(Π, R∗ , Te , Y∆ , Vb , Eb ). In addition to this vector, the budget set also depends on variables
such as the owner’s age, through the tax rules.
In Norway in 1991 there were two regulations in order to promote corporate retention of
earnings. These were funds under the equity section of the balance sheet, called the reserve
fund and the consolidation fund. There was no restriction on what kind of assets the funds
corresponded to. The reserve fund mandated certain retentions, while the consolidation
fund offered tax incentives.2 While the rules are somewhat complicated, perhaps confusing,
and country specific, the topic should be of interest outside Norway as well, since quite a
few countries have similar arrangements.3 Allocations to the funds influence the corporate
tax bases, so they are a necessary part of the analysis.
The consolidation fund rules allowed the corporation to retain a share of profits free
from corporate income taxes, if allocated to this fund. The maximum allocation was each
year limited to
(2)
ϕ max(0, Y − Y∆ − Lm ),
2
The consolidation fund incentives were established in 1984 and abolished in 1992.
According to a Norwegian government report on corporate law, NOU (1992:29) section 13, Denmark
abolished reserve fund regulations in 1991, while Sweden still has something similar. The EU corporate
directive no. 2 allows member states to legislate similar funds.
3
2
i.e., a fraction ϕ = 0.23 of the base for the municipal corporate income tax.4 The allocation
to the fund is ∆F between zero and the maximum given by (2).
The reserve fund rules mandate the corporation to build a reserve fund, V , until it
equals at least ζCe , where Ce is the end-of-year capital stock (book value of shares), and
ζ = 0.2. (We use subscript e to denote end-of-year values, and b to denote beginning-of-year
values.) The intention (cf. Marthinussen and Aarbakke (1986)) is to smoothen dividend
payments, probably regulated as a protection for minority shareholders. We regard Ce as
exogenous.
The fund should also be large enough so that the reserve fund plus Ce + Ae equals
the corporate debt, i.e., the target debt/equity ratio is unity. The intention of this is to
protect debtholders. Here Ae is the end-of-year value of the appreciation fund (“oppskrivningsfond”), a part of equity created from appreciation of assets, considered exogenous. In
relation to the reserve fund target, “debt” includes the year’s taxes and dividends.5 We
assume that apart from these, the corporation has debt Be , which is considered exogenous.
This target becomes Be + Tc + G − Ae − Ce , and the combined target is
Vt ≡ max(ζCe , Be + Tc + G − Ce − Ae ),
(3)
which is endogenous (because of Tc and G) when the second of the two alternatives is
effective.
The requirement is not to establish the fund at once, but to allocate to it in all years
with positive after-tax profits and/or positive dividend payments, until the two targets are
met at the end of the year.
The allocations are mandated to be:
• Ordinary reserve fund allocations,
max{0, κ[Y − ∆F − Tc + min(Eb , 0)]},
(4)
where κ = 0.1,6
• and extraordinary reserve fund allocations,
max{0, G − χ(Cb + Vb + Ab )},
4
(5)
As part of the tax reform in 1992, it was decided that the consolidation fund at the end of 1991 would
never be taxed, and could be converted to free equity in 1992. Because of this, we have assumed that
no corporation made withdrawals from the consolidation fund in 1991. The consolidation fund rules say
that the maximum deductible allocation is 23 percent of taxable profits after deduction for previous years’
losses, see Norwegian Tax Directorate (1991), p. 440. Presumably this refers to tax-deductible losses, Lm ,
or more specifically those losses that are actually deducted in 1991. Since the allocation saves taxes, there
could be an argument for not claiming deductions for previous years’ losses, if one had the possibility of
reclaiming the deduction for losses in a later year. We are not sure whether such a practice would be
allowed. If allowed, it would be particularly interesting in 1991, the last year which allowed allocation
to the consolidation fund. We have ignored this possibility, and assume that tax-deductible losses are
deducted to the extent possible in 1991.
5
See Marthinussen and Aarbakke (1986), p. 439.
6
Even with positive after-tax profits, the reserve fund allocation is not necessarily required. As the
formula shows, there should first be an allocation to cover an “uncovered loss” in the balance sheet
(negative free equity), if any, i.e., min(Eb , 0).
3
where χ = 0.1, while Cb , Vb , and Ab are the beginning-of-year values of the capital
stock, reserve fund, and appreciation fund, respectively.
In a year when some allocation is required to meet one or both targets, but mandated
allocations according to (4) and (5) would lead to an excess above both targets, then it is
sufficient to meet the targets. In such a year the allocation is thus
∆V = Vt − Vb ,
(6)
where Vb is the beginning-of-year value of the fund. This equation has two linear variants,
since Vt has.
Withdrawals from the reserve fund are allowed under two circumstances: If it exceeds
the target at the beginning of the year, a fraction η = 0.2 of the excess may be withdrawn
and used as free equity. If the corporation has negative after-tax profits, Y − ∆F − Tc < 0,
there are rules for how this loss should affect the balance sheet variables.
The rules are as follows: The loss should first be deducted from free equity, Eb , if this is
positive. If the (absolute value of the) loss exceeds free equity, the rest should be deducted
from the reserve fund. If the (absolute value of the) loss exceeds both, i.e., if it exceeds
Eb + Vb , then the rest should be allocated to the balance sheet as a negative entry under
equity, Ee < 0, called “uncovered loss.” Both the second and the third of these possibilities
are of little interest to us, since we are only interested in cases in which dividends can be
paid.7
This means that the required allocation ∆V , if positive, can be either one of the two
expressions related to profits or dividends, or the sum of both, or just sufficient to bring
V = Vt , which again has two different linear expressions, cf. (3). This gives altogether five
alternative linear expressions when ∆V > 0.
Another important restriction is that free equity at the end of the year, Ee = Eb + ∆E,
is not allowed to become negative if dividends are paid. Using (1) this can be written
G ≤ max(0, Y − Tc + Eb − ∆F − ∆V ).
(7)
Observe that we allow the use of previously retained earnings, Eb , to pay dividends. This
is relevant in a detailed and short-run model, although it is often neglected in long-run
models, e.g., explicitly in Sinn (1987), cf. his footnote 3, p. 75. All previous retentions may
be used, cf. equation (21) below.
All budget sets will include one point on the horizontal axis (the W axis) and one point
on the vertical axis (the D axis). The variables W (and Wg ) and D (and G) are defined
to be positive, so the budget set does not extend into other quadrants. On the horizontal
axis, a corporation can be located at a point (observed or hypothetical) in which it cannot
pay any dividend, even if it reduced W marginally. This would happen if
Y − Tc + Eb − ∆F − ∆V < 0,
(8)
cf. (7). At such points, there is no immediate trade-off between W and D, so the points
are not part of our budget set. If a corporation is observed to be at such a point, it will
7
The rules put a restriction on the exogenous variables. Assuming that the rules were followed in the
previous year: If Eb < 0, then Vb = 0.
4
be excluded from our sample, since only an increase in R (from its observed value) would
allow for a positive G. Hypothetically such points would occur if W is increased until D is
zero, and then further. That further increase is not part of our budget set. More generally,
an observation will be excluded if there is no observed dividend payment, and there is no
way for the corporation to pay dividends (by reducing Wg ) while keeping retentions at the
observed level.
2
The sets of equations for retained earnings
A particular feature of our method is the exact calculation of budget sets. The computer
programs doing these calculations are available at the web site8 , and explained in Lund
(2003). These can only be understood in connection with the present section, which details
the sets of equations and inequalities and the numbering of these.
We present how the 24 sets of equations for the allocations are distinguished. There
are six main categories, (a) – (f), distinguished by whether ∆V > 0, whether R > R∗ ,
and whether there is an interior solution for ∆F . Then there are numbered subcategories
distinguished by the exact rule for ∆V . There are 3 possible rules when ∆V ≤ 0, and 5
possible rules when ∆V > 0.
In all cases we require that dividends are positive,
G ≥ 0.
(9)
(When calculating the budget constraint, we in fact assume G > 0, but the point G = 0 is
contiguous to the constraint, since all the relations defining the constraint are continuous.)
(a),(b),(f ). Alternatives with no reserve fund allocation:
These cases have either a deficit (not a positive after-tax profits, at least not sufficient
to cover an uncovered loss (Eb < 0) if any, i.e., Y − ∆F − Tc + min(Eb , 0) < 0), or a
withdrawal from the reserve fund because it exceeds the targets. Although these distinct
alternatives seem rather different, they give rise to the same equations apart from the ∆V
equation.
Considering the deficit cases, we can narrow down our attention significantly. We can
ignore the possibility that a deficit leads to withdrawals from the reserve fund. Such a
situation would not allow for dividend payments.
We therefore concentrate on the deficit cases in which the deficit is covered by a positive
Eb . We require that
−Eb ≤ ∆E
(10)
in all of these cases.
(a). In case (a) there is full allocation to the consolidation fund. This implies that the
remaining three equations are: Either
∆V = 0
(11)
∆V = η(ζCe − Vb )
(12)
(called the (a3) subcase) or
8
http://folk.uio.no/dilund/wagediv/
5
(called (a1)) or
∆V = η(Be + Tc + G − Ce − Ae − Vb )
(13)
(called (a2)), moreover, the following two,
∆F + ∆V + ∆E = R∗ ,
(14)
where R∗ is the observed value of retentions, and finally,
∆F = ϕ max(0, Y − Y∆ − Lm ).
(15)
The latter is seemingly not a linear equation, but it adds no new equation systems, since
its two linear variants correspond to the two alternatives for the municipal tax base.
The linear equation systems (a1) and (a2) require that
∆V < 0,
(16)
Vb > ζCe > Be + Tc + G − Ce − Ae
(17)
Vb > Be + Tc + G − Ce − Ae > ζCe
(18)
and that
(case (a1)), or
(case (a2)).
Case (a3), on the other hand, is the deficit-covered-by-positive-Eb case. In order for
there to be no reserve fund allocation, we need
Y − ∆F − Tc + min(Eb , 0) < 0,
(19)
G < χ(Cb + Vb + Ab ).
(20)
and
We also need to know that there is no withdrawal, so the first inequalities in (17) or in (18)
must not be fulfilled, while the second inequalties should be fulfilled in order to distinguish
cases (a31) or (a32), respectively. Case (a3) is thus an equation system which is implied
by two sets of systems of linear inequalities, (a31) and (a32).
(b). There is a deficit (cf. alternative (a) above) or a withdrawal from the reserve fund,
but less than full allocation to the consolidation fund. Instead ∆F comes out as an interior
solution (somewhere between zero and the legal maximum), determined by the equation
system.
There are three equation system subcases, (b1), (b2), and (b3), just as under (a) above,
distinguished by the three alternative ∆V equations, (12), (13), and (11). Equation (14)
also holds, but (15) is replaced by
(21)
∆E = −Eb .
The relevant inequalities are the same as for case (a), except that (10) need not be
checked. Instead we require
∆F > 0,
(22)
and
R∗ − Va + Vw − ϕ max(0, Y − Y∆ − Lm ) < −Eb .
6
(23)
(f ). In case (f) there is no allocation to the reserve fund, but possibly (in cases (f1)
and (f2)) a withdrawal. There is no allocation to the consolidation fund. Even though ∆E
is at its minimum (while still allowing dividends, ∆E = −Eb ), retentions exceed observed
retentions. We thus keep the three alternative equations (12), (13), and (11) from (a) and
(b), which yield subcases (f1), (f2), and (f3), respectively. We keep both (15) and (21), but
dismiss (14).
The inequalities to check are the same as under the corresponding (b) cases, except
that we should not require (22) and (23).
Instead we should check that
(24)
R∗ < ∆V − Eb .
(c)–(e). Alternatives with reserve fund allocation: For the remaining cases,
there will be allocations to the reserve fund. These allocations can have five different
linear expressions. The first three of these can be valid under two alternative sets of linear
inequalities, giving subcases (x11), (x12), (x21), (x22), (x31), and (x32), where x is one of c,
d, or e. The fourth and fifth can be valid under three alternative sets of linear inequalities,
giving subcases (x41), (x42), (x43), (x51), (x52), and (x53).
For all cases (c)–(e) there is a requirement that
∆V > 0.
(25)
There is then a two-dimensional structure, the cases (c), (d), and (e) should all be
combined with all the five alternative equations for ∆V , in fact with all the twelve inequality
sets leading to these five equations.
Equation subcase (x1):
∆V = κ[Y − ∆F − Tc + min(Eb , 0)].
(26)
This is valid when the following inequalities hold:
κ[Y − ∆F − Tc + min(Eb , 0)] > 0,
(27)
G < χ(Cb + Vb + Ab ),
(28)
Vb + ∆V < Vt .
(29)
and
Since Vt has two alternative expressions, the last of these inequalities has two alternative
linear variants, giving rise to subcases (x11) and (x12). We also need to check which of
the linear variants is valid.
Equation subcase (x2):
∆V = G − χ(Cb + Vb + Ab ).
(30)
G > χ(Cb + Vb + Ab ),
(31)
κ[Y − ∆F − Tc + min(Eb , 0)] < 0,
(32)
This is valid when
7
and
Vb + ∆V < Vt .
(33)
Again there are two linear variants of Vt , and the validity of each must be checked.
Equation subcase (x3):
∆V = κ[Y − ∆F − Tc + min(Eb , 0)] + G − χ(Cb + Vb + Ab ).
(34)
This is valid when
κ[Y − ∆F − Tc + min(Eb , 0)] > 0,
(35)
G > χ(Cb + Vb + Ab ),
(36)
Vb + ∆V < Vt .
(37)
and
Again there are two linear variants of Vt , and the validity of each must be checked.
Equation systems (x4) and (x5):
∆V = Vt − Vb ,
(38)
with one of the two linear variants of Vt , giving rise to the two alternatives (x4) and (x5).
Here some reserve fund allocation is required, but the targets are met by less than those
allocations given for alternatives (x1), (x2) and (x3).
The inequality subcases (x41), (x42), (x43), (x51), (x52), and (x53), reflect which of
the three alternative ∆V expressions would have been in effect if the target had not been
met. For instance, in subcase (x41) the target is
Vt = ζCe ,
(39)
and the hypothetical allocation if the target had not been met is
κ[Y − ∆F − Tc + min(Eb , 0)].
(40)
Case (x41), in which the target is met, thus requires the inequalities
κ[Y − ∆F − Tc + min(Eb , 0)] > 0,
(41)
G < χ(Cb + Vb + Ab ),
(42)
ζCe > Be + Tc + G − Ce − Ae ,
(43)
Vb + κ[Y − ∆F − Tc + min(Eb , 0)] > ζCe .
(44)
and
To work out the other five sets of inequalities, (x42), (x43), (x51), (x52), and (x53), is left
to the reader. The computer code will be included in an appendix.
(c). There is an allocation to the reserve fund, and full allocation to the consolidation
fund. This implies that the remaining three equations are as follows: ∆V is determined
by one of the five possible equations, (26), (30), (34), or one of the two variants of (38).
Furthermore,
(45)
∆F + ∆V + ∆E = R∗ ,
8
and
∆F = ϕ max(0, Y − Y∆ − Lm ).
(46)
Equation system (c) requires
∆E > −Eb .
(47)
(d). There is an allocation to the reserve fund, and less than full allocation to the
consolidation fund. This implies that the remaining three equations are as follows:
∆V is determined by one of the five possible equations, (26), (30), (34), or one of the
two variants of (38). Furthermore,
∆F + ∆V + ∆E = R∗ ,
(48)
∆E = −Eb .
(49)
R∗ − ∆V − ϕ max(0, Y − Y∆ − Lm ) < −Eb ,
(50)
∆F > 0.
(51)
and
Equation system (d) requires
and
(e). There is an allocation to the reserve fund, which exceeds the observed retentions.
Nothing is allocated to the consolidation fund. This implies that the remaining three
equations are as follows: ∆V is determined by one of the five possible equations, (26),
(30), (34), or one of the two variants of (38). Furthermore,
∆F = 0,
(52)
∆E = −Eb .
(53)
−Eb > R∗ − ∆V.
(54)
and
Equation system (e) requires
3
Maximum number of segments
Figure 1 shows that budget set which, within our data set, had the maximum number of
segments, 13.
4
The possible values for dG/dY
The 288 possible expressions for dG/dY are given in a separate document.9 The 57 unique
numerical values which are obtained in the data, are given below.10 This illustrates strong
9
To be found at http://folk.uio.no/dilund/wagediv/
Remember that the variation in dG/dY is only part of the source of variation in the slopes of budget
sets in the W, D diagram. The other source is progressive personal taxation.
10
9
variation, but still much fewer distinct values (57) than the number of equation system
(288). This is partly due to coinciding theoretical values for some of the equation systems,
and partly to coinciding numerical values due to particular data values, such as the value
zero for the firm’s received dividends.
These are the observed dG/dY values:
-0.022161
-0.011080
0.181440
0.253057
0.270000
0.285714
0.306648
0.307479
0.340720
0.346500
0.362881
10
0.366374
0.369668
0.371357
0.385000
0.400000
0.419280
0.450000
0.450600
0.490474
0.500000
0.514774
0.516807
0.536383
0.540000
0.540720
0.544900
0.590557
0.601041
0.603333
0.613296
0.614958
0.617729
0.626016
0.632030
0.658145
0.677701
0.679992
0.681440
0.693000
0.702471
0.722000
0.724000
0.735255
0.750813
0.754709
0.769160
0.770000
0.822900
0.833333
0.843269
0.866631
0.900000
0.922992
0.958973
0.962923
11
1.000000
The negative values are commented below.
5
Reserve fund allocations exceeding observed retentions plus Eb
This is an anomaly which, strictly speaking, means that our budget set definition is not
applicable. It may happen in cases denoted (e) and (f) in section 2.
Starting at the observed point, we move north west by reducing W and increasing
D. In many cases ∆V will increase continuously during this process.11 In order to keep
retentions fixed, ∆E + ∆F will be decreased.
But when these reach their minimum value,12 −Eb (with ∆F = 0), the process will
change its character. When W is reduced (and Y is increased) further, one can only
continue paying dividends by letting R increase. This additional R is added to D to give
the dashed curve in the diagram.
Consider first whether the point exists: In our considerations of dG/dY , we will first
calculate this magnitude for values of Y all the way to plus infinity. It is clear that when
Y becomes large enough, the required reserve fund allocation becomes large also, and
eventually it will exceed R∗ + Eb . Therefore, all our sequences of segments with different
dG/dY values will end with segments where the observed retentions and Eb are insufficient
to pay the reserve fund allocations.
On the other hand, all of these sequences will be cut off at the point where we leave
the first quadrant in the (W, D) plane. Intervals with ∆V > R∗ + Eb will be irrelevant if
R∗ + Eb is high enough to cover required reserve fund allocations when W is reduced to
zero.
Consider now the question: What does the budget set look like to the north west of the
point, for those corporation-owner pairs for which the point exists in the first quadrant?
Two extreme formulations are possible:
• We could assume that the owner is indifferent between dividends and reserve fund
retentions, or that the owner paid in more capital to the firm in the form of new
shares, being indifferent between this and saving by other means. Under one of
these assumptions the budget set could be extended to the north west almost as if
no reserve fund allocations had to be made. This will be called the “indifference
alternative.” The reason for the “almost,” is that allocations to the reserve fund do
affect what amount is available for allocation to the consolidation fund, which again
affects taxes.
• We could assume that the reserve fund had no value to the owner, just as a tax
payment. This would define a budget set which would be restricted compared to
11
This can happen as a negative ∆V is reduced in absolute value, or as a positive ∆V is increased in
absolute value, but it will not happen at those segments where ∆V is zero.
12
In order to pay dividends, the corporation should not have a negative Ee ≡ Eb + ∆E. −Eb is not the
minimum value of ∆E, but the minimum value which allows the payment of dividends, which is the point
of interest to us.
12
the indifference alternative. As shown in the figure, it would have a lower slope (in
absolute value) in the leftmost part where reserve fund allocations were required.
This will be called the “zero-value alternative.”
Between these there is at least one more realistic alternative, namely, that one krone
in the reserve fund has some value, but less than one krone paid out as dividends.
The reserve fund can only be distributed to shareholders to the extent that it (in some
future years) exceeds the two targets, and then in each year only by twenty percent of the
excess amount. This is an argument why its shadow value should be lower than unity. On
the other hand, from 1992 there is no taxation of dividends at the personal level, so an
owner who faces a fixed interest rate and is not credit constrained should not have any
strong reason to withdraw funds from the corporation. Actually, the dividends paid out of
1991 profits were the last dividends to be subject to personal taxes, given that the agents
believe that the tax system will not later be changed in this respect.13
The extension of the budget constraint to the north west of the point (where the required
reserve fund allocation exhausts R∗ + Eb ) may not be very important for our regressions.
When R is not kept constant at R∗ , there are actually more than the two variables W
and D which are changing simultaneously. This means that we ought to model the owner’s
preferences over more variables than W and D. We could choose to keep retentions in free
equity, R − ∆V = ∆F + ∆E, fixed, and let the resulting reserve fund allocation ∆V be a
separate argument in the owner’s utility function. This creates a three-dimensional budget
set, and the econometric problem becomes much less tractable.
In the paper we have simply chosen the indifference alternative.
6
A case with G decreasing in Y
When the owner of the corporation decides to reduce own wages, and the corporation is
in a position where it pays dividends, it is almost always the case that the reduced wages
allow for increased dividends. The immediate consequence of reduced wages is that (both
taxable and book) profits are increased. We have found, however, that there were cases
in Norway in 1991 for which the increased profits resulted in tax payments increased by a
larger amount. With a given amount of retentions, this would imply lower dividends, G.
We point out immediately that the statutory tax rate did not exceed one hundred
percent. However, the effective marginal rate did, in the sense defined by our model. This
rests on our assumptions that the owner keeps the retained earnings constant and minimizes
taxes. In the case to be described, there will be an increase in the required allocation to
the reserve fund, which will require an equally large reduction in the allocation to the
consolidation fund. This is part of the reason for the increased taxes.
Empirically, of the 228 observations with non-empty budget sets, there were 3 who had
a segment of their budget constraints with negative dG/dY .
13
This raises the question why one would want to pay anything out at all in 1991, since both wages and
dividends were subject to personal taxes, while retentions could be distributed later with no taxation. We
want to avoid this question by considering the amount to be retained as exogenous.
13
The case to be considered is called (d5) in 2. We now give the five equations determining
G, Tc , ∆V, ∆E, and ∆F as functions of Y in this case. After presenting the equations, we
shall show their solution.
Y − Tc − G = R∗ ,
(55)
which is just the payout budget equation, with R∗ as the observed value of the retentions.
R∗ = ∆F + ∆V + ∆E,
(56)
which is the other side of the same equation, specifying that the three parts of the retentions
should sum to R∗ .
∆E = −Eb ,
(57)
which means that ∆E is at its minimum value, so that there will be no free equity at the
end of the year.
∆V = Be + Tc + G − Ce − Ae − Vb ,
(58)
where Be + Tc + G − Ce − Ae is known as one of the target values of the reserve fund.
Case (d5) occurs when the target is exactly met in 1991, by an allocation which is less
than those which would be mandated if the target had not been met. Thus the allocation
to the fund is determined by the difference between the target and the fund’s size at the
beginning of 1991.
Of course, one feature of this regulation which contributes to the effective marginal tax
rate above one hundred percent, is that the target itself depends on the two endogenous
variables Tc + G.
The equations given above should be combined with a tax equation. Of the possible
tax cases, there are more than one which would here lead to dG/dY < 0. One of these is
Tc = Te + cm (Y − Y∆ − Lm − ∆F ) + cn (Y − Y∆ − Ln − ∆F − G + Gr ).
(59)
(Another occurs if we have +∆V in the base for the national tax instead of −(∆F + G).)
Observe that there is no equation stating that ∆F is equal to its maximum value,
ϕ(Y − Y∆ − Lm ). The precondition for the case to be valid is that the maximum value is
positive, and that ∆F is a positive number less than it. This interior solution for ∆F is
relevant because we have assumed that retentions are fixed, ∆V is given by rules, and ∆E
is at its minimum.
Instead of presenting the complete solution to the five equations, we shall differentiate
them, and derive the solution for the total derivatives with respect to Y with some care.
We observe that ∆E is equal to an exogenous variable, so its derivative is zero, and we are
left with four equations, which we differentiate with respect to Y, G, ∆V, Tc , and ∆F .
From (55) we observe that
(60)
dY = dTc + dG.
From (58) we observe that
d∆V = dTc + dG = dY.
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(61)
Then it is clear from (56) that
d∆F = −d∆V = −dY.
(62)
Plugging these into (55) and (59) gives
dY − dG = dTc = (cn + cm )dY − (cn + cm )d∆F + cn dG = 2(cn + cm )dY + cn dG.
(63)
This yields the result
1 − 2(cn + cm )
dG
,
=
dY
1 − cn
which is negative, equal to -0.022 with the 1991 rates. It follows that
cn + 2cm
dTc
=
,
dY
1 − cn
(64)
(65)
equal to 1.022 with the 1991 rates.
7
Why include W m + Dm?
If the owner was only interested in minimizing taxes for a given payout from the corporation, then the wage would simply be W m . This indicates that it might be reasonable to
run regressions without W m + Dm as an explanatory variable. The reason why we nevertheless include it in most regressions, is that there should be a credible H0 to test. When
we test whether the coefficient of W m is zero, this would be rather incredible if there was
no other variable included in the regression which reflected the size of the payout budget.
Whatever determines the wage payout, it must be affected by the size of the total payout,
in particular in a regression with the large heterogeneity which is found here.
8
8.1
More robust regressions
Endogeneity
There are reasons to suspect that some of the explanatory variables may be correlated
with the error terms. One reason is that they are decision variables, which may be affected
by almost any unobserved heterogeneity. This is true for the maximum total payout,
W m + Dm . For instance, an owner may have a high desire for a wage payout because of
some unobserved variable, such as a high valuation of pension points because of a high
probability of becoming disabled. This means that this owner has a high error term.
But it may also cause her to give up good investments opportunities in the corporation
and withdraw more cash than otherwise. This means that the total payout is positively
correlated with the error term. The payout-maximizing net wage, W m , is correlated with
total payout, and may thus also be correlated with the error term.
It cannot be ruled out a priori that the owner’s wage income from other sources, W e ,
is also correlated with the error term. It is a decision variable, and if some unobserved
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variable causes a high desire for wage payout from the corporation we observe, then it may
also cause a high desire for W e .
Because most of the variables behind this endogeneity are unobservables, we cannot
run regressions based on a more elaborate model with several equations. We are left with
the instrumental variable technique to overcome the endogeneity. Because of a lack of
good instruments, and because of only a weak suspicion of its endogeneity, we have not
tried to instrument for W e . The variables W m + Dm and W m are not included among
the instruments, but two other variables substitute for these. These two instruments are
the total assets of the firm at the beginning of the year, for which we have a book value,
reflecting the size of the firm. Next we have the year’s profits before wages to the owner,
Π. When the W m variable is included in the regression as W m χm
i , these are excluded as
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instead.
instruments, while we include Πχm
i
The instrumental variable estimator is known as the two-stage least squares (2SLS)
estimator. When X is the n × k matrix of explanatory variables, W is the n × matrix
of instruments, and y is the n × 1 vector of dependent variables, the coefficient vector
estimator is
β̂2SLS = (X W (W W )−1 W X)−1 X W (W W )−1 W y.
The estimator of the variance of the residual is
1
s2 = (y − X β̂2SLS )(y − X β̂2SLS ),
n
and the estimator of the covariance matrix of the coefficient estimator is
V̂ (β̂2SLS ) = s2 (X W (W W )−1 W X)−1 ,
(66)
(67)
(68)
cf. Davidson and MacKinnon (1993) pp. 216ff.
8.2
Heteroskedasticity
There may well be heteroskedasticity, as is quite common in cross-sectional studies. The
form of the heteroskedasticity is not known. We have used a method which is robust under
heteroskedasticity, known as H2SLS in Davidson and MacKinnon (1993). This estimator
also takes care of the endogeneity problem. Unfortunately, we know little about its smallsample properties. The coefficient estimator is
β̂H2SLS = (X W (W ΩW )−1 W X)−1 X W (W ΩW )−1 W y,
(69)
where Ω is a matrix with the squared residuals from the 2SLS regression along the diagonal,
and zeroes off the diagonal, cf. Davidson and MacKinnon (1993) p. 599. The estimator for
the variance of this is
V̂ (β̂H2SLS ) = (X W (W ΩW )−1 W X)−1 ,
(70)
cf. Davidson and MacKinnon (1993) pp. 612. We only report t ratios computed from the
diagonal of this.
14
There is an inconsistency in the use of the variable name χ, which is a retention regulation parameter
in the previous section, but a general name for indicator variables in the present context. These uses
are distinguished, however, by super- and subscripts, which are always and only used for the indicator
variables.
16
References
Davidson, Russel, and James G. MacKinnon (1993), Estimation and Inference in Econometrics, Oxford University Press, Oxford, UK.
Fjærli, Erik, and Diderik Lund (2001), “The choice between owner’s wages and dividends
under the dual income tax,” Finnish Economic Papers, vol. 14, no. 2, pp. 104–119, available at http://www.taloustieteellinenseura.fi/fep/articles/f2001_2c.pdf
Lund, Diderik (2003), “Exact nonlinear budget constraints determined by systems of equations and inequalities,” unpublished, University of Oslo, latest revision January 2003,
available at http://folk.uio.no/dilund/wagediv/wadvcomp.pdf
Marthinussen, Hans Fredrik, and Magnus Aarbakke (1986), Aksjeloven med kommentarer,
(Corporations Act with comments), Aschehoug, Oslo, Norway.
Norwegian Tax Directorate (1991), Lignings-ABC, (Tax Assessment ABC), Gyldendal,
Oslo, Norway.
NOU (1992:29), Lov om aksjeselskap, (Corporations’ Act), Governmental Report, The
Ministry of Justice of Norway, Oslo.
Sinn, Hans-Werner (1987), Capital Income Taxation and Resource Allocation, North-Holland, Amsterdam, The Netherlands.
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