Finnish Economic Papers – Volume 20 – Number 1 – Spring 2007
THE ACHILLES HEEL OF THE DUAL INCOME TAX:
THE NORWEGIAN CASE*
ANNETTE ALSTADSÆTER
University of Oslo, Statistics Norway, and CESifo.
Institute of Health Management and Health Economics, University of Oslo,
P.b. 1089 Blindern, N-0317 Oslo. E-mail: [email protected]
The dual income tax provides the self-employed individual with large incentives
to participate in tax minimizing income shifting. The present paper analyses the
income shifting incentives under the Norwegian split model when real capital
investments are risky. It concludes that high-income self-employed individuals can
incorporate and use the legal form of a widely held corporation as a tax shelter.
In addition, real capital investments with a low risk profile are means to shift income from the labor income tax base to the capital income tax base for the highincome self-employed. (JEL: H24, H25, H32)
1. Introduction
In contrast to the global income tax, which levies one tax schedule on the sum of income from
all sources, the dual income tax levies separate
tax schedules on income from labor and capital.
It combines a low proportional tax rate on capital income with a progressive tax rate on other
income, mostly labor income. The dual income
tax was introduced in the Nordic countries in
the early 1990ies.1 Later Austria, Belgium, Ice* I thank the editors of this journal, Panu Poutvaara and
Andreas Wagener, as well as an anonymous referee and Erling Holmøy for most appreciated comments. And I thank
Søren Bo Nielsen, Michael Devereux, Agnar Sandmo, Dirk
Schindler, Guttorm Schjelderup, Joel Slemrod, and Knut
Reidar Wangen for inspiring discussions and comments on
previous versions of this paper. I benefited from stays at
Office of Tax Policy Research, University of Michigan, and
at CES, University of Munich. Grant 158143/510 from the
Research Council of Norway is gratefully acknowledged.
1
The dual income tax was introduced in Sweden in 1991,
Norway 1992, and Finland 1993. The idea originated in
Denmark, and was implemented in their 1985 tax reform.
Later they introduced a hybrid system, mostly due to redis-
land, Italy, Israel, Japan, and Portugal have introduced tax systems that are similar to a dual
income tax, with separate tax schedules for labor income and capital income. As stated by
Sørensen (1994), the taxation of small businesses is the Achilles Heel of the dual income
tax.2 For medium and high income classes, there
is a large difference in the marginal tax rates on
capital and labor income, providing great incentives for income shifting from labor income to
capital income in order to minimize tax payments. Thus a system for imputing the return to
capital and labor in small businesses is required
to counteract this kind of income shifting and
erosion of the tax base. The challenge is which
imputation system to choose, and in particular,
tributive concerns. See Sørensen (1994, 1998) for more on
the dual income tax.
2
But, as Boadway (2004) states, the problem of dealing
with personal business income is one that plagues virtually
every tax system.
Finnish Economic Papers 1/2007 – Annette Alstadsæter
how to set the imputation rate. This has previously been analyzed under two related tax systems. Bond and Devereux (1995, 2003) show
that under a tax system with allowance for corporate equity, the imputed rate of return to capital that ensures investment neutrality is the risk
free nominal interest rate on government bonds.
Panteghini (2001), on the other hand, shows that
the neutral imputation rate under the dual income tax system must be higher when investments are irreversible, which is an argument in
favor of additional risk compensation in the imputation rate.
The Nordic countries have implemented different income splitting systems to impute the
return to labor and capital for small businesses.
The present paper uses the Norwegian imputation system as an example and analyzes how the
dual income tax induces the sole proprietor to
participate in tax minimizing income shifting in
the presence of technology risk. It concludes
that the widely held corporation serves as a tax
shelter for high-income self-employed individuals. The higher the business income of the selfemployed and the higher the difference between
the marginal tax rates on labor and capital, the
larger is the tax-minimizing incentive to incorporate. Real capital investments also serve as
means to shift income from the labor income
tax base to the capital income tax base for the
high-income self-employed. The risk compensation under the split model is an investment tax
subsidy, and it counteracts the investment disincentives of the technology risk and increases
the investment level of the self-employed. The
latter result formalizes the verbal analysis of
Hagen and Sørensen (1998), who argued that
owners of sole proprietorships and closely held
corporations cannot diversify risk to the same
extent that owners of widely held corporations,
and that they are risk averse. Hence sole proprietors and closely held corporations might under-invest in risky capital compared with the
social optima.3 Even though the Norwegian
variant of the dual income tax and the imputation system are used as an example, the results
and the discussion have relevance for other
3
Sandmo (1985), Apel and Sødersten (1999), and
Weisbach (2004) discuss in detail how taxes distort investments in risky assets.
countries as well, both the Nordic countries and
other countries contemplating introducing the
dual income tax.
This issue has been studied by authors in the
other Nordic countries as well. Kari (1999) analyzes the Finnish income splitting model and
shows that it provides entrepreneurs who face
high marginal tax rates with high investment
incentives. Lindhe, Södersten and Öberg (2004)
present all the Nordic income splitting models
and analyze their effects on the cost of capital
of different types of firms in the absence of risk.
They concluded that the cost of capital is approximately the same for closely and widely
held corporations. The exception is debt financed investments of closely held Norwegian
corporations, where the split model represents a
tax subsidy. Lindhe, Södersten and Öberg nevertheless ignore an important aspect in their
analysis, namely the endogeneity of a firm’s tax
system: by changing organizational form the
firm can experience a shift in the taxes it faces.
Most of the literature on tax effects on the
choice of organizational form is based on U.S.
data, where taxes discourage incorporation due
to the double taxation of dividends. Most papers
conclude that even if the tax code provides incentives for smaller corporations with high revenue to shift out of the corporate form, this effect is rather small. Gordon and MacKie-Mason
(1994), Ayers, Cloyd, and Robinson (1996), and
MacKie-Mason and Gordon (1997) all conclude
that non-tax factors seem to dominate the choice
of organizational form. Goolsbee (2004), on the
other hand, reports large negative effects on incorporation by the U.S. tax code. Similarly,
Gordon and Slemrod (2000) document substantial income shifting from the corporate to the
personal tax base following the 1986 U.S. tax
reform.
There is little empirical work on the effects
of the dual income tax on income shifting between tax bases. Two recent exceptions are Romanov (2006) and Pirttilä and Selin (2006).
Romanov explores two recent Israeli tax increases on wage and self-employed income. The
number of Israeli corporations increased by 5%
during this period, and Romanov identifies
many of these new corporations as tax shelters
for high-income professionals. He concludes
Finnish Economic Papers 1/2007 – Annette Alstadsæter
that high-income individuals seem to have responded to these tax increases by incorporating
in order to receive their income as tax favored
dividends instead of wages. Pirttilä and Selin
evaluate the effect on taxable income by the
Finnish 1993 dual tax reform and its introduction of a split model of business taxation. They
document an increase in taxable capital income
of the self-employed individuals after the reform. This they interpret as the result of tax
minimizing income shifting from the personal
to the capital income tax base by the self-employed individuals. However, they do not explicitly consider the effects of tax induced incorporation of the self-employed. In addition,
Fjærli and Lund (2001) analyze how owners of
corporations choose to pay wages and dividends
during a transition period into the dual income
tax in Norway. They conclude that owners pay
themselves more wages than optimal from a
short term tax minimizing view, and suggest
that this can be optimal from a long-term view,
as wage payments are the basis for future pension benefits. The effects of the dual income tax
on related issues as taxable income, demand for
debt and tax progressivity are studied by Aarbu
and Thoresen (2001), Fjærli (2004), and Thoresen (2004) on Norwegian data. Similar studies
are conducted on Swedish data by Selén (2002)
and Hansson (2004).
The outline of the paper is as follows. Section 2 describes the tax system and provides
empirical motivation. In section 3, the basic
model is presented, and the choice of organizational form is limited to two forms. The entrepreneur can organize either as a self-employed
or as a widely held corporation. Section 4 and
5 analyze tax effects on the entrepreneur’s investment behavior in the presence of technology
risk under the two organizational forms, and
section 6 analyzes the tax effects on his incentives to incorporate. Section 7 concludes.
2. Background
2.1 The Norwegian imputation system
– The split model
The Norwegian split model applies to sole proprietorships and closely held corporations. A
corporation is defined as closely held if 2/3 or
more of the shares are held by active owners,
where an owner is characterized as active if he
works more than 300 hours annually in the firm
and passive otherwise. Spouses or under-aged
children of active owners are not recognized as
passive owners. A corporation is defined as
widely held if more than 1/3 of the shares are
held by passive owners, and it is then taxed according to corporate tax rules. Employer’s social security contributions apply to all wage
payments made by the corporation.
Under the split model, an imputed return to
the capital invested in the firm is calculated by
multiplying the value of the capital assets by a
fixed rate of return on capital, which is set annually by the Parliament on the basis of the average rate of return on government bonds plus
a risk premium.4 These assets include physical
business capital, acquired good-will and other
intangible assets, business inventories, and
credit extended to customers net of debt to the
firm’s suppliers. The imputed return to capital
is taxed at the corporate rate, which equals the
capital income tax rate at the individual level.
Business profit net of imputed return to capital
is the imputed return to labor, which is taxed as
labor income independent of whether the wages
are actually paid to the owner or not.5 Employers’ social security contribution does nevertheless not apply to the imputed return to labor. If
imputed labor income is negative, the loss does
not offset other income, but may be carried forward to be deducted against future imputed labor income in the firm. The table in figure 3 in
the appendix summarizes the most important
parameters of the Norwegian tax system.
4
The Norwegian split model is similar to the Finnish
split model, as evaluated by Pirttilä and Selin (2006). The
major difference is that while the imputed return to capital
under the Norwegian model is calculated on the firm’s assets, it is under the Finnish model calculated on the firm’s
assets net of debt. Also, under the Finnish system only corporations listed on the stock exchange are not taxed according to the split model.
5
If the firm has employees in addition to the owner(s),
a salary deduction of a given percentage of the wage bill
from taxable wage payments applies before the return to the
owner’s labor effort is imputed. If the imputed labor income
exceeds a given threshold, the remainder is taxed as capital
income.
Finnish Economic Papers 1/2007 – Annette Alstadsæter
Figure 1. Number of sole proprietors and corporations over time.
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this group increased its real capital by 124 perAn extensive income shifting through increased
investments and changes of organizational form
seem to have taken place in the years after the
1992 introduction of the Norwegian dual in
come tax. The total number of self-employed
individuals in Norway decreased by 14 percent
from 1993 to 2002, while the number of small
corporations with four or less employees increased by 16 percent. In 1994, 55 percent of
new small corporations were widely held, while
this share had risen to 75 percent four years
later. It is striking that nearly the whole growth
in the number of corporations comes as widely
held corporations, as is seen in figure 1. All this
indicates that there might have been a change in
preferred
organizational form as a result of
creative
tax-minimizing
re-labelling
activity,
a
point emphasized by Gordon and Slemrod
(2000). Unfortunately, the available data does
not allow us to trace the business entities as they
change organizational forms.
There also seem to have been a response to
the investment incentives under the split model.
The self-employed individuals with the 10 percent highest business income have increased
their firm specific capital dramatically during
cent from 1992 to 2002, while the corresponding
increase for all self-employed was 68 percent.
Another indication of this income shifting is
that while aggregated wages increased by 44
percent from 1993 to 2002, aggregated business
income of the self-employed only increased by
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For simplicity, the following analysis abstracts
from many of the details discussed above. Consider a utility maximizing entrepreneurial individual
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periods and who is
about to start a business. He needs to decide
how much to invest in real capital in the firm,
which has a stochastic second period return, as
well as which organizational form to choose.
There are many non-tax factors that affect the
choice of organizational form, as discussed by
Ayers et. al (1996) and MacKie-Mason and
Gordon (1997), but in this analysis we simplify
by only including risk and taxes as motives be-
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Figure 2. The value of firm specific capital of Norwegian self-employed individuals from 1992 to 2002, NOK, in 2002prices.
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poration. Individuals differ in their preferences utilities of both first period consumption, C1,
of which is the preferred organizational form. and second period consumption, C2, such that
Here consider the marginal entrepreneur who u’(C1) > 0, u’’(C1) < 0, v’(C2) > 0, and v’’(C2)
initially has no intrinsic value of either of the < 0. Assume that the individual has a precautwo organizational forms, self-employed and tionary saving motive, such that v’’’(C2, s) > 0.
widely held corporation, and who chooses the See Sandmo (1970) and Kimball (1990) for
organizational form that maximizes his utility. more on precautionary saving. The risk averse
The individual has a given time endowment individual chooses the investment portfolio and
in both periods, which he spends working in his organizational form that maximize his expected
firm and enjoying leisure. In order to study the lifetime utility.
Finnish Economic Papers 1/2007 – Annette Alstadsæter
Investments and income In the first period he
has initial wealth Y, which he allocates to investing in risky real capital K in the firm, and
saving B in the financial market. Investments in
the financial market yield the exogenously given safe real rate of return r. Savings may be
negative, and then the individual borrows in the
financial market. Loans are repaid in full in the
second period. The gross return to real capital
investments is the sales income. The net of taxes sales income depends on the tax regime and
thus on the chosen organizational form. It will
be specified separately for each organizational
form in the two following sections, as will the
expressions for first and second period consumption. Real capital investments are in principle reversible, and are realized at the end of
the second period. The sales value of the real
capital is the original value K net of the real
capital depreciation, which is represented by the
shock-related depreciation rate gy and discussed
more closely below. A high depreciation rate
increases the degree of irreversibility in the real
capital investments and may work as a disincentive towards investments in the firm. This is
discussed more closely by Panteghini (2001).
The entrepreneur is the only person employed
in the firm, and thus labor as a production factor
is fixed. The firm produces one product, which
is sold in the second period at a given price set
to unity, p = 1. The production level X varies
according to the amount of capital invested in
the firm, and sales income is thus given by the
production function X = F(K), which has a positive and decreasing marginal product of capital;
FK > 0 and FKK < 0.
Risk The individual invests in real capital in the
first period, and he realizes all his capital in the
second period. The second period sales value of
the capital stock of the firm depends on the depreciation rate, which is given by the stochastic
parameter gy . There will always be some depreciation, and the maximum loss through depreciation is the initial value of the real capital,
such that 0 < gy < 1. The expected value of depreciation is positive and given by the ordinary
y ] = d > 0. The individdepreciation rate d : E[ g ual demands a risk premium in order to invest
in real capital in the firm. Define Q as the rate
10
of return to real capital required to compensate
the individual for the relative expected second
period marginal utility reduction caused by the
depreciation. The size of Q depends on two factors; the individual’s preferences regarding risk,
as well as the probability of a technology shock
that reduces the value of the existing real capital
dramatically:
E[v’(C2) · gy ]
(2) Q ≡ = d + l,
E[v’(C2)]
where l ≡ cov [v’(C2), gy ] / E[v’(C2)] is the risk
premium. A higher probability of a technology
shock increases the expected depreciation rate
d. Also, the real capital depreciation reduces the
second period consumption. The more risk
averse the individual is, the larger is the utility
loss from the drop in second period consumption. Thus the risk premium is positive and
higher the more risk averse the individual is.
The risk premium also varies across different
consumption levels for a given individual, such
that it is not constant at individual level.
Taxes Let tw be the proportional tax rate on labor income and tk the proportional tax rate on
capital income. We simplify by assuming that
the tax on labor income is proportional, when
in fact it is progressive in most countries, including the countries with a dual income tax.
But one might think of this tax as the top marginal tax rate on labor income. The progressive
labor income tax schedule is then in fact ”flat
on the top”. Assume that the tax rate on labor
income is higher than that on capital income,
tw > tk. Total tax payments are given by T. No
wealth tax is present in the model.
4. The self-employed
Let the subscript ”s” denote the previously described variables when the entrepreneur is a
self-employed. First period consumption is
given as the initial wealth net of investments:
(3) C1, s = Y – Ks – Bs.
The self-employed owns the firm and has full
disposal over total sales income. His gross sec-
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Finnish Economic Papers 1/2007 – Annette Alstadsæter
defined as the variable ls ≡ cov [v’(C2, s), gy ] /
E[v’(C2, s)]. As we discussed earlier, the risk
premium is not given on individual level, it depends on both the level of taxes and other variables. The higher a risk premium the individual
requires, the less he invests in real capital when
the degree of irreversibility increases. This effect is counteracted by the risk compensation
factor under the split model, m, which isolated
considered works as a government subsidy on
real capital investments. The total risk compensation under the split model is the relative after
t –t
w k
tax risk compensation rate, · m, which de1– t
effect of the tax increase is positive. Therefore,
the increased tax on labor income induces the
individual to increase his investments in firm
specific real capital if the substitution effect
dominates the income effect. As labor supply is
given in this model, there are no effects on labor
supply from tax changes.
Proposition 3 An increase in the tax rate on
capital income reduces the after-tax risk compensation under the split model and induces the
self-employed to reduce his investments in real
capital.
w
pends positively on the difference in the marginal tax rates on labor and capital income. As
is shown in the proof of proposition 1 in the
appendix, an increase in the risk compensation
rate, m, will induce the self-employed individual to invest more in firm specific real capital if
(1–tk) · r · (tw–tk) · K < E[v’(C2, s)]. That means
if the expected marginal utility of second period
consumption is greater than the after tax return
to the tax savings under the split model from
investing in real capital, and the likelihood for
this is greater the lower the initial level of real
capital in the firm. This is an equivalent to the
Domar-Musgrave (1944)-effect, where the selfemployed entrepreneur invests more in real
capital than he would in the absence of taxes.
Proposition 2 Increased tax on labor income
increases the risk compensation under the split
model and can induce the self-employed to increase his investments in real capital.
The proof of proposition 2 is provided in the
appendix.
A higher tax rate on labor income reduces the
net-of-taxes labor income, which reduces the
incentives to invest in real capital such that the
income effect is negative. At the same time the
higher tax rate on labor income means that the
private return to shifting income from the labor
income tax base to the capital income tax base
increases. It also increases the relative after tax
t –t
1– tw
w k
risk compensation rate · m, making the in-
dividual more willing to invest in risky firm
specific real capital, such that the substitution
12
The proof of proposition 3 is provided in the
appendix.
When the capital income tax rate increases,
the incentive to participate in any kind
of income shifting decreases, since the difference between the two tax rates, tw–tk, decreases,
as does the private gain from income shifting.
Also, a higher tax rate on capital income means
a decreased net risk compensation rate under
t –t
1– tw
w k
the split model, · m. Both factors induce the
self-employed to invest less in risky real capital
in the firm. The higher the self-employed’s
capital income is, the larger share of his total
income is affected by the tax increase, and the
more is his net income reduced.
5. The widely held corporation
The entrepreneur only incorporates in order to
reduce tax payments and wants to keep as much
as possible of the business income. Thus assume that he holds the maximum allowed
amount of shares as a widely held corporation,
such that b ≈ 2/3. The full share capital is spent
on acquiring real capital in the firm, such that
the active owner invests the share b of total real
capital, and the passive shareholders invest the
rest. The passive shareholder may diversify his
investments to a larger extent than the active
shareholder, who invests both his capital and his
labor effort in the firm. Assume therefore that
the passive shareholder does not require a higher risk premium than the active shareholder in
order to invest in the firm.
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Finnish Economic Papers 1/2007 – Annette Alstadsæter
owners to be classified as a widely held corporation, the more of firm specific profits taxed as
capital income may the self-employed keep, and
this increases his incentive to incorporate. This
effect is greater the lower the tax rate on capital
income and the higher the net business income.
Conclusion
The above analysis concluded that the split
model, and in particular the risk compensation
factor in the imputation rate, can induce the
self-employed to increase his investments in
real capital in the firm. Capital becomes a means
to shift income from the labor income tax base
to the capital income tax base. The higher the
value of his real capital, the greater is the imputed return to capital, and the more of his business income is taxed as capital income. The
incentive to participate in this kind of income
shifting is stronger the higher the difference between the two marginal tax rates and the higher
the risk compensation rate under the split model. All types of real capital have the same imputation rate regardless of actual risk, and also
regardless of whether the risk is systematic or
unsystematic. It is thus to be expected that the
self-employed canalizes this tax induced investment into less risky types of real capital in order
to minimize his risk exposure. If the risk that
the self-employed faces is fully unsystematic,
he ought in principle be able do diversify away
from it. Diversification restrictions due to liquidity constraints is an argument in favour of
such a risk compensation via the tax system.
The same is the irreversibility of the investments that the technology risk represents. On
the other hand, if the risk is fully systematic,
there is in principle no such justification for a
tax subsidy to investments of the self-employed,
since all agents in the economy are exposed to
the same risk.
In addition to increasing the capital stock
there are several ways to increase the book value of the capital in the firm, for instance by
shifting from leased to owned premises and machinery, by increasing stocks at the end of the
year, by increasing and extending customers’
16
trade receivables at the end of the year, and by
financing private durable goods in the firm. It
can even be profitable to borrow in the financial
market to invest in business capital. Such debts
are private and entitle the borrower to tax allowances against other capital income. High-income self-employed individuals are subject to
the top marginal tax rate on imputed return to
labor, and these are in particular expected to
take advantage of the income shifting possibilities through increasing their capital stock. And
as we saw in figure 2, there are indications that
this development has taken place.
Only to a certain extent can the self-employed
use real capital investments to shift income to
the capital income tax base. The high-income
entrepreneur thus has incentives to avoid the
split-model completely by becoming a widely
held corporation. The analysis showed that this
tax minimizing incentive to incorporate is
stronger the higher the difference between the
marginal tax rates on labor income and capital
income is, since this increases the return to income shifting. But the larger the required share
of passive investors in order to be classified as
widely held corporation is, the weaker are the
incentives to incorporate. When the active owner substitutes dividend payments for wage payments to himself, it means that he shares any
inframarginal returns to his labor effort with the
passive owners. One way to make the shareholders compensate for their share of the inframarginal returns is initially to demand higher
prices for the shares. He can also avoid this
profit sharing with passive owners in both legal
and illegal ways. One example of legal tax
avoidance is to have grown children or grandchildren as passive owners, which is allowed
under the corporate tax code. Then any dividend
payment to passive owners is in fact a tax free
intergenerational transfer. There are numerous
examples of creative tax evasion in order to
avoid the split model. For instance, two dentists
may be passive owners in each other’s widely
held corporations. They both pay the required
one third of dividends to passive owners, but
since they have about the same income potential, the first dentist receives the same in dividends from the second dentist as she herself
pays him. Both are then left with their full busi-
Finnish Economic Papers 1/2007 – Annette Alstadsæter
ness income taxed at the capital income tax rate.
If detected, this is ruled illegal.
Partly as a response to this income shifting
activity of small businesses, the Norwegian dual
income tax system was remodelled through the
2006 tax reform, as described in Sørensen
(2005). The main feature of this reform is the
introduction of a tax on income from shares on
the individual level, where an imputed normal
return to the shares is tax-exempt. This has removed the need for the split model for closely
held corporations, and all corporations are now
taxed according to the corporate tax code. The
imputation rate under the income splitting model for self-employed is reduced trough the removal of the risk compensation element, such
that the imputation rate now is the before-tax
risk free rate of return.
References
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Apel, M., and J. Södersten (1999). ”Personal Taxation and
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Boadway, R. (2004). ”The Dual Income Tax System – An
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17
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9. Appendix
7D[ SDUDPHWHUV LQ 1RUZD\ Finnish Economic Papers 1/2007 – Annette Alstadsæter
Figure 3. Tax parameters in Norway, 1992–2003.
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mization problem (7) yields
(17)
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Proof
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static analysis
first
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# ofthe
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proposition
4 Comparative
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89787:1;176
analysis of the
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D @?=J A2CE @7 E96
H9:=6 2== :?4@>6 3J E96 @H?6C @7 E96 H:56=J 96=5 4@CA@C2E:@? :D 2Q64E65 E96 E2I 3FC56?
:?4@>6 @7 E96 D6=76>A=@J65 :?5:G:5F2= :D 2Q64E65 3J E96 :?4C62D65 E2I @? 42A:E2= :?4@>6
@7 E96 @H?6C @7 E96 H:56=J 96=5 4@CA@C2E:@? :?4C62D6D >@C6 3J E9:D E2I :?4C62D6
H9:=6 2== :?4@>6 3J E96 @H?6C @7 E96 H:56=J 96=5 4@CA@C2E:@? :D 2Q64E65 E96 E2I 3FC56?
@7 E96 @H?6C @7 E96 H:56=J 96=5 4@CA@C2E:@? :?4C62D6D >@C6 3J E9:D E2I :?4C62D6
$977. 7. 89787:1;176 J 5:Q6C6?E:2E:?8
2?5 C62CC2?8:?8
3J 2AA=J:?8
E96byLCDE
@C56C the
4@?5:E:@?
7C@>
E96
Proof
proposition
8 By
differentiating
(13) and
rearranging
applying
first order
condi$977.of7.
89787:1;176
>2I:>:K2E:@?
AC@3=6> problem
2D H6== (11)
2D 6BF2E:@?
H6 86E(12)
E92Ewe get that
tion
from the maximization
as well as
equation
J 5:Q6C6?E:2E:?8
2?5
C62CC2?8:?8
3J
2AA=J:?8
E96
LCDE @C56C 4@?5:E:@? 7C@> E96
>2I:>:K2E:@?
2D H6== 2D 6BF2E:@? H6 86E E92E
AC@3=6>
$ "
(23) $ " ,96? :7 /6 <?@H E92E " $ J AA=J:?8 E9:D 4@?5:E:@?
E@
>62?D
E92EE96
H9:49
C65F46D
:7
/6 <?@H E92E " $ ,96?
2G6C286 C6EFC? E@ 42A:E2= :? @AE:>F> :D 9:896C E92? E96
>2C8:?2=
C6EFC?
E@
42A:E2=
?5
J AA=J:?8 E9:D 4@?5:E:@? C65F46D E@ H9:49 >62?D E92E E96
E9:D
9@=5D 2D 2?5 2G6C286 C6EFC? E@ 42A:E2= :? @AE:>F> :D 9:896C E92? E96 >2C8:?2= C6EFC? E@ 42A:E2=
?5
E9:D 9@=5D 2D 2?5 22