# The choice between owner’s wages and dividends under dG/dY

```The choice between owner’s wages and dividends under
the dual income tax: Formulae for dG/dY
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: [email protected]
web page: http://folk.uio.no/dilund/
First version March 1999
Slightly revised February 2003
This is a documentation of the 288 formulae for the dG/dY which follow from the
definition of the corporate budget set in the model of Fjærli and Lund (2001), explained
in Lund (2003).1
The tax alternatives which are defined below, are indexed by two numbers, (i, j). i ∈
{0, 1} denotes which of the two linear versions of the municipal income tax formula is used,
cf. Fjærli and Lund (2001), eq. (A.3). j ∈ {0, 1, 2, 3, 4, 5} denotes which of the six linear
versions of the national income tax formula is used, cf. Fjærli and Lund (2001), eq. (A.4).
These are the 24×12 expressions for dG/dY :
Case (a1), tax alternative (0,0):
dG
= 1.
(1)
dY
Case (a1), tax alternative (0,1):
dG
= 1.
(2)
dY
Case (a1), tax alternative (0,2):
dG
= 1.
(3)
dY
∗
This is an extended appendix for Fjærli and Lund (2001).
Unfortunately, we have changed notation for a few of the variables during the project. The published
version, Fjærli and Lund (2001), defines two tax rates on corporate income, a municipal rate, cm , and a
national rate, cn . Originally we wanted to use the more general definitions of marginal corporate income
tax rates found in King (1977), but it turns out that the relation between the two statutory tax rates and
King’s rates varies from case to case (e.g., being in or out of tax position), so we decided that we needed
the statutory rates as our basic variables. In the computer programs, and in this paper, we use instead
the rates cu and cd , which here should be taken to be defined simply as cu = cn + cm and cd = cm .
1
1
Case (a1), tax alternative (0,3):
dG
= 1.
dY
(4)
dG
= 1.
dY
(5)
dG
= 1.
dY
(6)
dG
= 1 − cd + cd ϕ.
dY
(7)
dG
= 1 − cd + cd ϕ.
dY
(8)
dG
−1 + cu − cd ϕ
=
.
dY
−1 − cd + cu
(9)
−1 + cu − cd ϕ
dG
=
.
dY
−1 − cd + cu
(10)
−1 + cu − cd ϕ
dG
=
.
dY
−1 − cd + cu
(11)
1 − cu + c u ϕ
dG
=
.
dY
1 + cd − cu
(12)
dG
= 1.
dY
(13)
dG
= 1.
dY
(14)
dG
= 1.
dY
(15)
Case (a1), tax alternative (0,4):
Case (a1), tax alternative (0,5):
Case (a1), tax alternative (1,0):
Case (a1), tax alternative (1,1):
Case (a1), tax alternative (1,2):
Case (a1), tax alternative (1,3):
Case (a1), tax alternative (1,4):
Case (a1), tax alternative (1,5):
Case (a2), tax alternative (0,0):
Case (a2), tax alternative (0,1):
Case (a2), tax alternative (0,2):
2
Case (a2), tax alternative (0,3):
dG
(cd − cu ) η
.
=1−
dY
−1 − cd + cu
(16)
dG
(cd − cu ) η
.
=1−
dY
−1 − cd + cu
(17)
dG
= 1.
dY
(18)
dG
= 1 − cd + cd ϕ.
dY
(19)
dG
= 1 − cd + cd ϕ.
dY
(20)
−1 + cu − cd ϕ
dG
.
=
dY
−1 − cd + cu
(21)
dG
−1 + cu − cd η + cu η − cd ϕ
.
=
dY
−1 − cd + cu
(22)
Case (a2), tax alternative (0,4):
Case (a2), tax alternative (0,5):
Case (a2), tax alternative (1,0):
Case (a2), tax alternative (1,1):
Case (a2), tax alternative (1,2):
Case (a2), tax alternative (1,3):
Case (a2), tax alternative (1,4):
dG
−1 + cu − cd η + cu η − cd ϕ
=
.
dY
−1 − cd + cu
(23)
Case (a2), tax alternative (1,5):
dG
1 − cu + c u ϕ
.
=
dY
1 + cd − cu
(24)
dG
= 1.
dY
(25)
dG
= 1.
dY
(26)
dG
= 1.
dY
(27)
Case (a3), tax alternative (0,0):
Case (a3), tax alternative (0,1):
Case (a3), tax alternative (0,2):
3
Case (a3), tax alternative (0,3):
dG
= 1.
dY
(28)
dG
= 1.
dY
(29)
dG
= 1.
dY
(30)
dG
= 1 − cd + cd ϕ.
dY
(31)
dG
= 1 − cd + cd ϕ.
dY
(32)
dG
−1 + cu − cd ϕ
=
.
dY
−1 − cd + cu
(33)
−1 + cu − cd ϕ
dG
=
.
dY
−1 − cd + cu
(34)
−1 + cu − cd ϕ
dG
=
.
dY
−1 − cd + cu
(35)
1 − cu + c u ϕ
dG
=
.
dY
1 + cd − cu
(36)
dG
= 1.
dY
(37)
dG
= 1.
dY
(38)
dG
= 1.
dY
(39)
Case (a3), tax alternative (0,4):
Case (a3), tax alternative (0,5):
Case (a3), tax alternative (1,0):
Case (a3), tax alternative (1,1):
Case (a3), tax alternative (1,2):
Case (a3), tax alternative (1,3):
Case (a3), tax alternative (1,4):
Case (a3), tax alternative (1,5):
Case (b1), tax alternative (0,0):
Case (b1), tax alternative (0,1):
Case (b1), tax alternative (0,2):
4
Case (b1), tax alternative (0,3):
dG
= 1.
dY
(40)
dG
= 1.
dY
(41)
dG
= 1.
dY
(42)
dG
= 1 − cd .
dY
(43)
dG
= 1 − cd .
dY
(44)
dG
cd
.
=1−
dY
1 + cd − cu
(45)
dG
cd
.
=1−
dY
1 + cd − cu
(46)
dG
cd
.
=1−
dY
1 + cd − cu
(47)
dG
1 − cu
=
.
dY
1 + cd − cu
(48)
dG
= 1.
dY
(49)
dG
= 1.
dY
(50)
dG
= 1.
dY
(51)
dG
(cd − cu ) η
=1−
.
dY
−1 − cd + cu
(52)
Case (b1), tax alternative (0,4):
Case (b1), tax alternative (0,5):
Case (b1), tax alternative (1,0):
Case (b1), tax alternative (1,1):
Case (b1), tax alternative (1,2):
Case (b1), tax alternative (1,3):
Case (b1), tax alternative (1,4):
Case (b1), tax alternative (1,5):
Case (b2), tax alternative (0,0):
Case (b2), tax alternative (0,1):
Case (b2), tax alternative (0,2):
Case (b2), tax alternative (0,3):
5
Case (b2), tax alternative (0,4):
dG
(cd − cu ) η
.
=1−
dY
−1 − cd + cu
(53)
1 + cd − cu + cd η − c u η
dG
.
=
dY
1 + cd − cu
(54)
Case (b2), tax alternative (0,5):
Case (b2), tax alternative (1,0):
dG
= 1 − cd − cd η.
dY
(55)
dG
= 1 − cd − cd η.
dY
(56)
dG
−1 + cu + cd η
.
=
dY
−1 − cd + cu
(57)
−1 + cu + cu η
dG
.
=
dY
−1 − cd + cu
(58)
−1 + cu + cu η
dG
=
.
dY
−1 − cd + cu
(59)
−1 + cu + cu η
dG
.
=
dY
−1 − cd + cu
(60)
dG
= 1.
dY
(61)
dG
= 1.
dY
(62)
dG
= 1.
dY
(63)
dG
= 1.
dY
(64)
Case (b2), tax alternative (1,1):
Case (b2), tax alternative (1,2):
Case (b2), tax alternative (1,3):
Case (b2), tax alternative (1,4):
Case (b2), tax alternative (1,5):
Case (b3), tax alternative (0,0):
Case (b3), tax alternative (0,1):
Case (b3), tax alternative (0,2):
Case (b3), tax alternative (0,3):
6
Case (b3), tax alternative (0,4):
dG
= 1.
dY
(65)
dG
= 1.
dY
(66)
dG
= 1 − cd .
dY
(67)
dG
= 1 − cd .
dY
(68)
dG
cd
=1−
.
dY
1 + cd − cu
(69)
dG
cd
.
=1−
dY
1 + cd − cu
(70)
dG
cd
.
=1−
dY
1 + cd − cu
(71)
1 − cu
dG
.
=
dY
1 + cd − cu
(72)
dG
= 1.
dY
(73)
dG
= 1.
dY
(74)
dG
= 1.
dY
(75)
1 + cd − cu
dG
=
.
dY
1 + c d − cu − cd κ + c u κ
(76)
Case (b3), tax alternative (0,5):
Case (b3), tax alternative (1,0):
Case (b3), tax alternative (1,1):
Case (b3), tax alternative (1,2):
Case (b3), tax alternative (1,3):
Case (b3), tax alternative (1,4):
Case (b3), tax alternative (1,5):
Case (c1), tax alternative (0,0):
Case (c1), tax alternative (0,1):
Case (c1), tax alternative (0,2):
Case (c1), tax alternative (0,3):
Case (c1), tax alternative (0,4):
dG
1 + cd − cu
=
.
dY
1 + c d − cu − cd κ + c u κ
7
(77)
Case (c1), tax alternative (0,5):
dG
= 1.
dY
(78)
dG
= 1 − cd + cd ϕ.
dY
(79)
dG
= 1 − cd + cd ϕ.
dY
(80)
dG
−1 + cu − cd ϕ
.
=
dY
−1 − cd + cu
(81)
dG
1 − cu + cd ϕ − cd κ ϕ + c u κ ϕ
=
.
dY
1 + c d − cu − cd κ + c u κ
(82)
Case (c1), tax alternative (1,0):
Case (c1), tax alternative (1,1):
Case (c1), tax alternative (1,2):
Case (c1), tax alternative (1,3):
Case (c1), tax alternative (1,4):
dG
1 − cu + cd ϕ − cd κ ϕ + c u κ ϕ
=
.
dY
1 + c d − cu − cd κ + c u κ
(83)
Case (c1), tax alternative (1,5):
dG
−1 + cu − cu ϕ
.
=
dY
−1 − cd + cu
(84)
dG
= 1.
dY
(85)
dG
= 1.
dY
(86)
dG
= 1.
dY
(87)
dG
= 1 + cd − cu .
dY
(88)
dG
= 1 + cd − cu .
dY
(89)
Case (c2), tax alternative (0,0):
Case (c2), tax alternative (0,1):
Case (c2), tax alternative (0,2):
Case (c2), tax alternative (0,3):
Case (c2), tax alternative (0,4):
8
Case (c2), tax alternative (0,5):
dG
= 1.
dY
(90)
dG
= 1 − cd + cd ϕ.
dY
(91)
dG
= 1 − cd + cd ϕ.
dY
(92)
dG
−1 + cu − cd ϕ
.
=
dY
−1 − cd + cu
(93)
dG
= 1 − cu + cd ϕ.
dY
(94)
dG
= 1 − cu + cd ϕ.
dY
(95)
dG
1 − cu + c u ϕ
.
=
dY
1 + cd − cu
(96)
dG
= 1.
dY
(97)
dG
= 1.
dY
(98)
dG
= 1.
dY
(99)
dG
1 + cd − cu
=
.
dY
1 − cd κ + c u κ
(100)
dG
1 + cd − cu
=
.
dY
1 − cd κ + c u κ
(101)
Case (c2), tax alternative (1,0):
Case (c2), tax alternative (1,1):
Case (c2), tax alternative (1,2):
Case (c2), tax alternative (1,3):
Case (c2), tax alternative (1,4):
Case (c2), tax alternative (1,5):
Case (c3), tax alternative (0,0):
Case (c3), tax alternative (0,1):
Case (c3), tax alternative (0,2):
Case (c3), tax alternative (0,3):
Case (c3), tax alternative (0,4):
9
Case (c3), tax alternative (0,5):
dG
= 1.
dY
(102)
dG
= 1 − cd + cd ϕ.
dY
(103)
dG
= 1 − cd + cd ϕ.
dY
(104)
dG
−1 + cu − cd ϕ
=
.
dY
−1 − cd + cu
(105)
1 − cu + cd ϕ − cd κ ϕ + c u κ ϕ
dG
=
.
dY
1 − cd κ + c u κ
(106)
Case (c3), tax alternative (1,0):
Case (c3), tax alternative (1,1):
Case (c3), tax alternative (1,2):
Case (c3), tax alternative (1,3):
Case (c3), tax alternative (1,4):
dG
1 − cu + cd ϕ − cd κ ϕ + c u κ ϕ
=
.
dY
1 − cd κ + c u κ
(107)
Case (c3), tax alternative (1,5):
dG
1 − cu + c u ϕ
=
.
dY
1 + cd − cu
(108)
dG
= 1.
dY
(109)
dG
= 1.
dY
(110)
dG
= 1.
dY
(111)
dG
= 1.
dY
(112)
dG
= 1.
dY
(113)
Case (c4), tax alternative (0,0):
Case (c4), tax alternative (0,1):
Case (c4), tax alternative (0,2):
Case (c4), tax alternative (0,3):
Case (c4), tax alternative (0,4):
10
Case (c4), tax alternative (0,5):
dG
= 1.
dY
(114)
dG
= 1 − cd + cd ϕ.
dY
(115)
dG
= 1 − cd + cd ϕ.
dY
(116)
dG
−1 + cu − cd ϕ
.
=
dY
−1 − cd + cu
(117)
−1 + cu − cd ϕ
dG
=
.
dY
−1 − cd + cu
(118)
−1 + cu − cd ϕ
dG
=
.
dY
−1 − cd + cu
(119)
dG
1 − cu + c u ϕ
.
=
dY
1 + cd − cu
(120)
dG
= 1.
dY
(121)
dG
= 1.
dY
(122)
dG
= 1.
dY
(123)
−1 − 2 cd + 2 cu
dG
.
=
dY
−1 − cd + cu
(124)
−1 − 2 cd + 2 cu
dG
=
.
dY
−1 − cd + cu
(125)
Case (c4), tax alternative (1,0):
Case (c4), tax alternative (1,1):
Case (c4), tax alternative (1,2):
Case (c4), tax alternative (1,3):
Case (c4), tax alternative (1,4):
Case (c4), tax alternative (1,5):
Case (c5), tax alternative (0,0):
Case (c5), tax alternative (0,1):
Case (c5), tax alternative (0,2):
Case (c5), tax alternative (0,3):
Case (c5), tax alternative (0,4):
11
Case (c5), tax alternative (0,5):
dG
= 1.
dY
(126)
dG
= 1 − cd + cd ϕ.
dY
(127)
dG
= 1 − cd + cd ϕ.
dY
(128)
−1 + cu − cd ϕ
dG
.
=
dY
−1 − cd + cu
(129)
cu − cd ϕ
dG
.
=1+
dY
−1 − cd + cu
(130)
cu − cd ϕ
dG
.
=1+
dY
−1 − cd + cu
(131)
dG
1 − cu + c u ϕ
.
=
dY
1 + cd − cu
(132)
dG
= 1.
dY
(133)
dG
= 1.
dY
(134)
dG
= 1.
dY
(135)
dG
(cd − cu ) κ
=1+
.
dY
1 + cd − cu − κ − 2 cd κ + 2 c u κ
(136)
Case (c5), tax alternative (1,0):
Case (c5), tax alternative (1,1):
Case (c5), tax alternative (1,2):
Case (c5), tax alternative (1,3):
Case (c5), tax alternative (1,4):
Case (c5), tax alternative (1,5):
Case (d1), tax alternative (0,0):
Case (d1), tax alternative (0,1):
Case (d1), tax alternative (0,2):
Case (d1), tax alternative (0,3):
Case (d1), tax alternative (0,4):
dG
(cd − cu ) κ
=1+
.
dY
1 + cd − cu − κ − 2 cd κ + 2 c u κ
12
(137)
Case (d1), tax alternative (0,5):
dG
(−1 − cd + cu ) (−1 + κ)
=
.
dY
1 + cd − cu − κ − 2 cd κ + 2 c u κ
(138)
Case (d1), tax alternative (1,0):
dG
(−1 + cd ) (−1 + κ)
=
.
dY
1 − κ + cd κ
(139)
Case (d1), tax alternative (1,1):
dG
(−1 + cd ) (−1 + κ)
=
.
dY
1 − κ + cd κ
(140)
Case (d1), tax alternative (1,2):
dG
cd
=1+
.
dY
−1 − cd + cu + κ − cu κ
(141)
Case (d1), tax alternative (1,3):
dG
(1 − cu ) (−1 + κ)
=
.
dY
−1 − cd + cu + κ + cd κ − 2 cu κ
(142)
Case (d1), tax alternative (1,4):
dG
(1 − cu ) (−1 + κ)
=
.
dY
−1 − cd + cu + κ + cd κ − 2 cu κ
(143)
Case (d1), tax alternative (1,5):
dG
(1 − cu ) (−1 + κ)
=
.
dY
−1 − cd + cu + κ + cd κ − 2 cu κ
(144)
Case (d2), tax alternative (0,0):
dG
= 1.
dY
(145)
dG
= 1.
dY
(146)
dG
= 1.
dY
(147)
dG
= 1 + cd − cu .
dY
(148)
Case (d2), tax alternative (0,1):
Case (d2), tax alternative (0,2):
Case (d2), tax alternative (0,3):
13
Case (d2), tax alternative (0,4):
dG
= 1 + cd − cu .
dY
(149)
dG
= 1 + cd − cu .
dY
(150)
1 − cd
dG
.
=
dY
1 + cd
(151)
dG
1 − cd
.
=
dY
1 + cd
(152)
−1 + cu
dG
.
=
dY
−1 − 2 cd + cu
(153)
1 − cu
dG
.
=
dY
1 + cd
(154)
dG
1 − cu
=
.
dY
1 + cd
(155)
dG
1 − cu
.
=
dY
1 + cd
(156)
dG
= 1.
dY
(157)
dG
= 1.
dY
(158)
dG
= 1.
dY
(159)
(−1 − cd + cu ) (−1 + κ)
dG
=
.
dY
1 − κ − 2 cd κ + 2 cu κ
(160)
Case (d2), tax alternative (0,5):
Case (d2), tax alternative (1,0):
Case (d2), tax alternative (1,1):
Case (d2), tax alternative (1,2):
Case (d2), tax alternative (1,3):
Case (d2), tax alternative (1,4):
Case (d2), tax alternative (1,5):
Case (d3), tax alternative (0,0):
Case (d3), tax alternative (0,1):
Case (d3), tax alternative (0,2):
Case (d3), tax alternative (0,3):
Case (d3), tax alternative (0,4):
dG
(−1 − cd + cu ) (−1 + κ)
=
.
dY
1 − κ − 2 cd κ + 2 cu κ
14
(161)
Case (d3), tax alternative (0,5):
dG
(−1 − cd + cu ) (1 − κ)
=
.
dY
−1 + κ + 2 cd κ − 2 cu κ
(162)
Case (d3), tax alternative (1,0):
dG
(−1 + cd ) (−1 + κ)
=
.
dY
1 + cd − κ + cd κ
(163)
Case (d3), tax alternative (1,1):
(−1 + cd ) (−1 + κ)
dG
=
.
dY
1 + cd − κ + cd κ
(164)
Case (d3), tax alternative (1,2):
dG
(−1 + cu ) (−1 + κ)
=
.
dY
1 + 2 cd − cu − κ + c u κ
(165)
Case (d3), tax alternative (1,3):
dG
(−1 + cu ) (−1 + κ)
=
.
dY
1 + cd − κ − cd κ + 2 c u κ
(166)
Case (d3), tax alternative (1,4):
(−1 + cu ) (−1 + κ)
dG
=
.
dY
1 + cd − κ − cd κ + 2 c u κ
(167)
Case (d3), tax alternative (1,5):
dG
(1 − cu ) (−1 + κ)
=
.
dY
−1 − cd + κ + cd κ − 2 cu κ
(168)
Case (d4), tax alternative (0,0):
dG
= 1.
dY
(169)
dG
= 1.
dY
(170)
dG
= 1.
dY
(171)
dG
= 1.
dY
(172)
dG
= 1.
dY
(173)
Case (d4), tax alternative (0,1):
Case (d4), tax alternative (0,2):
Case (d4), tax alternative (0,3):
Case (d4), tax alternative (0,4):
15
Case (d4), tax alternative (0,5):
dG
= 1.
dY
(174)
dG
= 1 − cd .
dY
(175)
dG
= 1 − cd .
dY
(176)
dG
cd
.
=1−
dY
1 + cd − cu
(177)
dG
cd
.
=1−
dY
1 + cd − cu
(178)
dG
cd
.
=1−
dY
1 + cd − cu
(179)
dG
1 − cu
.
=
dY
1 + cd − cu
(180)
dG
= 1.
dY
(181)
dG
= 1.
dY
(182)
dG
= 1.
dY
(183)
dG
−1 − 2 cd + 2 cu
.
=
dY
−1 − cd + cu
(184)
dG
−1 − 2 cd + 2 cu
.
=
dY
−1 − cd + cu
(185)
Case (d4), tax alternative (1,0):
Case (d4), tax alternative (1,1):
Case (d4), tax alternative (1,2):
Case (d4), tax alternative (1,3):
Case (d4), tax alternative (1,4):
Case (d4), tax alternative (1,5):
Case (d5), tax alternative (0,0):
Case (d5), tax alternative (0,1):
Case (d5), tax alternative (0,2):
Case (d5), tax alternative (0,3):
Case (d5), tax alternative (0,4):
16
Case (d5), tax alternative (0,5):
dG
1 + 2 cd − 2 cu
.
=
dY
1 + cd − cu
(186)
dG
= 1 − 2 cd .
dY
(187)
dG
= 1 − 2 cd .
dY
(188)
dG
2 cd
.
=1+
dY
−1 − cd + cu
(189)
−1 + 2 cu
dG
.
=
dY
−1 − cd + cu
(190)
−1 + 2 cu
dG
.
=
dY
−1 − cd + cu
(191)
−1 + 2 cu
dG
.
=
dY
−1 − cd + cu
(192)
dG
= 1 − κ.
dY
(193)
dG
= 1 − κ.
dY
(194)
dG
= 1 − κ.
dY
(195)
(−1 − cd + cu ) (−1 + κ)
dG
=
.
dY
1 + cd − cu − cd κ + c u κ
(196)
Case (d5), tax alternative (1,0):
Case (d5), tax alternative (1,1):
Case (d5), tax alternative (1,2):
Case (d5), tax alternative (1,3):
Case (d5), tax alternative (1,4):
Case (d5), tax alternative (1,5):
Case (e1), tax alternative (0,0):
Case (e1), tax alternative (0,1):
Case (e1), tax alternative (0,2):
Case (e1), tax alternative (0,3):
Case (e1), tax alternative (0,4):
(−1 − cd + cu ) (−1 + κ)
dG
=
.
dY
1 + cd − cu − cd κ + c u κ
17
(197)
Case (e1), tax alternative (0,5):
dG
(−1 − cd + cu ) (−1 + κ)
=
.
dY
1 + cd − cu − cd κ + c u κ
(198)
Case (e1), tax alternative (1,0):
dG
= 1 − cd − κ + cd κ.
dY
(199)
dG
= 1 − cd − κ + cd κ.
dY
(200)
dG
(−1 + cu ) (−1 + κ)
=
.
dY
1 + cd − cu
(201)
Case (e1), tax alternative (1,1):
Case (e1), tax alternative (1,2):
Case (e1), tax alternative (1,3):
dG
(−1 + cu ) (−1 + κ)
=
.
dY
1 + c d − cu − cd κ + c u κ
(202)
Case (e1), tax alternative (1,4):
dG
(−1 + cu ) (−1 + κ)
=
.
dY
1 + c d − cu − cd κ + c u κ
(203)
Case (e1), tax alternative (1,5):
dG
(−1 + cu ) (−1 + κ)
=
.
dY
1 + c d − cu − cd κ + c u κ
(204)
Case (e2), tax alternative (0,0):
1
dG
= .
dY
2
(205)
dG
1
= .
dY
2
(206)
1
dG
= .
dY
2
(207)
−1 − cd + cu
dG
=
.
dY
−2 − cd + cu
(208)
Case (e2), tax alternative (0,1):
Case (e2), tax alternative (0,2):
Case (e2), tax alternative (0,3):
18
Case (e2), tax alternative (0,4):
dG
−1 − cd + cu
=
.
dY
−2 − cd + cu
(209)
dG
1 + cd − cu
.
=
dY
2 + cd − cu
(210)
dG
1 − cd
=
.
dY
2
(211)
dG
1 − cd
=
.
dY
2
(212)
dG
1
cd
= −
.
dY
2 2 (1 + cd − cu )
(213)
dG
1 − cu
.
=
dY
2 + cd − cu
(214)
dG
1 − cu
.
=
dY
2 + cd − cu
(215)
dG
1 − cu
.
=
dY
2 + cd − cu
(216)
dG
1−κ
=
.
dY
2
(217)
dG
1−κ
=
.
dY
2
(218)
dG
1−κ
=
.
dY
2
(219)
dG
(−1 − cd + cu ) (−1 + κ)
=
.
dY
2 + cd − cu − cd κ + c u κ
(220)
Case (e2), tax alternative (0,5):
Case (e2), tax alternative (1,0):
Case (e2), tax alternative (1,1):
Case (e2), tax alternative (1,2):
Case (e2), tax alternative (1,3):
Case (e2), tax alternative (1,4):
Case (e2), tax alternative (1,5):
Case (e3), tax alternative (0,0):
Case (e3), tax alternative (0,1):
Case (e3), tax alternative (0,2):
Case (e3), tax alternative (0,3):
19
Case (e3), tax alternative (0,4):
dG
(−1 − cd + cu ) (−1 + κ)
=
.
dY
2 + cd − cu − cd κ + c u κ
(221)
Case (e3), tax alternative (0,5):
dG
(−1 − cd + cu ) (−1 + κ)
=
.
dY
2 + cd − cu − cd κ + c u κ
(222)
Case (e3), tax alternative (1,0):
dG
(−1 + cd ) (−1 + κ)
=
.
dY
2
(223)
dG
(−1 + cd ) (−1 + κ)
=
.
dY
2
(224)
(−1 + cu ) (−1 + κ)
dG
=
.
dY
2 (1 + cd − cu )
(225)
Case (e3), tax alternative (1,1):
Case (e3), tax alternative (1,2):
Case (e3), tax alternative (1,3):
dG
(−1 + cu ) (−1 + κ)
=
.
dY
2 + c d − cu − cd κ + c u κ
(226)
Case (e3), tax alternative (1,4):
dG
(−1 + cu ) (−1 + κ)
=
.
dY
2 + c d − cu − cd κ + c u κ
(227)
Case (e3), tax alternative (1,5):
dG
−1 + cu + κ − cu κ
=
.
dY
−2 − cd + cu + cd κ − cu κ
(228)
Case (e4), tax alternative (0,0):
dG
= 1.
dY
(229)
dG
= 1.
dY
(230)
dG
= 1.
dY
(231)
Case (e4), tax alternative (0,1):
Case (e4), tax alternative (0,2):
20
Case (e4), tax alternative (0,3):
dG
= 1.
dY
(232)
dG
= 1.
dY
(233)
dG
= 1.
dY
(234)
dG
= 1 − cd .
dY
(235)
dG
= 1 − cd .
dY
(236)
cd
dG
.
=1−
dY
1 + cd − cu
(237)
cd
dG
=1−
.
dY
1 + cd − cu
(238)
dG
cd
.
=1−
dY
1 + cd − cu
(239)
1 − cu
dG
.
=
dY
1 + cd − cu
(240)
1
dG
= .
dY
2
(241)
dG
1
= .
dY
2
(242)
dG
1
= .
dY
2
(243)
dG
−1 − 2 cd + 2 cu
=
.
dY
2 (−1 − cd + cu )
(244)
Case (e4), tax alternative (0,4):
Case (e4), tax alternative (0,5):
Case (e4), tax alternative (1,0):
Case (e4), tax alternative (1,1):
Case (e4), tax alternative (1,2):
Case (e4), tax alternative (1,3):
Case (e4), tax alternative (1,4):
Case (e4), tax alternative (1,5):
Case (e5), tax alternative (0,0):
Case (e5), tax alternative (0,1):
Case (e5), tax alternative (0,2):
Case (e5), tax alternative (0,3):
21
Case (e5), tax alternative (0,4):
−1 − 2 cd + 2 cu
dG
=
.
dY
2 (−1 − cd + cu )
(245)
dG
1 + 2 cd − 2 cu
=
.
dY
2 (1 + cd − cu )
(246)
1
dG
= − cd .
dY
2
(247)
dG
1
= − cd .
dY
2
(248)
dG
1
cd
.
= −
dY
2 1 + cd − cu
(249)
dG
−1 + 2 cu
=
.
dY
2 (−1 − cd + cu )
(250)
dG
−1 + 2 cu
=
.
dY
2 (−1 − cd + cu )
(251)
dG
1 − 2 cu
=
.
dY
2 (1 + cd − cu )
(252)
dG
= 1.
dY
(253)
dG
= 1.
dY
(254)
dG
= 1.
dY
(255)
dG
= 1.
dY
(256)
Case (e5), tax alternative (0,5):
Case (e5), tax alternative (1,0):
Case (e5), tax alternative (1,1):
Case (e5), tax alternative (1,2):
Case (e5), tax alternative (1,3):
Case (e5), tax alternative (1,4):
Case (e5), tax alternative (1,5):
Case (f1), tax alternative (0,0):
Case (f1), tax alternative (0,1):
Case (f1), tax alternative (0,2):
Case (f1), tax alternative (0,3):
22
Case (f1), tax alternative (0,4):
dG
= 1.
dY
(257)
dG
= 1.
dY
(258)
dG
= 1 − cd .
dY
(259)
dG
= 1 − cd .
dY
(260)
cd
dG
.
=1−
dY
1 + cd − cu
(261)
dG
cd
.
=1−
dY
1 + cd − cu
(262)
dG
cd
.
=1−
dY
1 + cd − cu
(263)
dG
1 − cu
.
=
dY
1 + cd − cu
(264)
dG
1
=
.
dY
1+η
(265)
dG
1
=
.
dY
1+η
(266)
dG
1
=
.
dY
1+η
(267)
dG
−1 − cd + cu − cd η + cu η
=
.
dY
(−1 − cd + cu ) (1 + η)
(268)
Case (f1), tax alternative (0,5):
Case (f1), tax alternative (1,0):
Case (f1), tax alternative (1,1):
Case (f1), tax alternative (1,2):
Case (f1), tax alternative (1,3):
Case (f1), tax alternative (1,4):
Case (f1), tax alternative (1,5):
Case (f2), tax alternative (0,0):
Case (f2), tax alternative (0,1):
Case (f2), tax alternative (0,2):
Case (f2), tax alternative (0,3):
23
Case (f2), tax alternative (0,4):
−1 − cd + cu − cd η + cu η
dG
=
.
dY
(−1 − cd + cu ) (1 + η)
(269)
Case (f2), tax alternative (0,5):
dG
−1 − cd + cu − cd η + cu η
=
.
dY
(−1 − cd + cu ) (1 + η)
(270)
Case (f2), tax alternative (1,0):
dG
1
= −cd +
.
dY
1+η
(271)
1
dG
= −cd +
.
dY
1+η
(272)
dG
1
cd
+
=
.
dY
−1 − cd + cu 1 + η
(273)
Case (f2), tax alternative (1,1):
Case (f2), tax alternative (1,2):
Case (f2), tax alternative (1,3):
dG
−1 + cu + cu η
=
.
dY
(−1 − cd + cu ) (1 + η)
(274)
Case (f2), tax alternative (1,4):
dG
−1 + cu + cu η
=
.
dY
(−1 − cd + cu ) (1 + η)
(275)
Case (f2), tax alternative (1,5):
dG
−1 + cu + cu η
=
.
dY
(−1 − cd + cu ) (1 + η)
(276)
Case (f3), tax alternative (0,0):
dG
= 1.
dY
(277)
dG
= 1.
dY
(278)
dG
= 1.
dY
(279)
Case (f3), tax alternative (0,1):
Case (f3), tax alternative (0,2):
24
Case (f3), tax alternative (0,3):
dG
= 1.
dY
(280)
dG
= 1.
dY
(281)
dG
= 1.
dY
(282)
dG
= 1 − cd .
dY
(283)
dG
= 1 − cd .
dY
(284)
dG
cd
.
=1−
dY
1 + cd − cu
(285)
dG
cd
.
=1−
dY
1 + cd − cu
(286)
dG
cd
.
=1−
dY
1 + cd − cu
(287)
dG
1 − cu
.
=
dY
1 + cd − cu
(288)
Case (f3), tax alternative (0,4):
Case (f3), tax alternative (0,5):
Case (f3), tax alternative (1,0):
Case (f3), tax alternative (1,1):
Case (f3), tax alternative (1,2):
Case (f3), tax alternative (1,3):
Case (f3), tax alternative (1,4):
Case (f3), tax alternative (1,5):
25
References
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,