Journal
of Public
Economics
15 (1981) 1633184. North-Holland
RISK TAKING
AND
Publishing
Company
TAXATION
An alternative perspective
SM. KANBUR*
Glare College, Cambridge
Received
University,
April 1978, revised version
Cambridge CBZ 1 TL, UK
received August
1980
The analysis of risk taking and taxation
has almost invariably
been in a portfolio
choice
framework. This paper presents the alternative perspective of an occupational
choice framework,
where risk taking involves the additional
element of discrete choice between safe and risky
activities. It is shown that the specification
of equilibrium
must of necessity have a general
equilibrium character. In this setting the paper develops rules for government
intervention
in the
market equilibrium, and analyses the effects of taxation on risk taking.
1. Introduction
The study of the effects of taxation on risk taking has a long tradition in
economic analysis. The ‘yield versus risk’ analysis of Domar and Musgrave
(1944) pioneered the field in first drawing attention
to the effect of taxation
on the choice between safe and risky assets in terms of the government
sharing some of the risk with investors, with the result that in some cases
investment
in risky assets would increase with taxation. The ‘mean-variance’
approach
of Tobin (1958) was applied by Richter (1960) but the more
general expected utility approach
of Mossin (1968) and Stiglitz (1969) has
now become standard
[see, for example, Ahsan (1974) Cowell (1975) and
Sandmo (1977)].
A distinctive
feature of this whole literature
is that the analysis is in a
portfolio choice framework,
where the agent controls
continuous
choice
variables - the proportions
of wealth invested in safe and risky assets - to
maximise his utility function by adjustments
at the margin. I would argue
that this framework
cannot capture all aspects of risk taking in a society.
Thus, for example, we observe in the real world that people make discrete,
non-marginal choices between alternative
activities which differ in their risk
characteristics.
Moreover,
entrepreneurial
risk taking cannot,
I think, be
viewed simply and solely in terms of adjustments
in portfolio at the margin,
or indeed engaging ‘a little bit’ more in entrepreneurial
activity.’
*I am indebted to Jim Mirrlees, Kevin Roberts, Joe Stiglitz and participants
at the SSRC
Workshop
on Public Economics, University of Warwick, 1978, for helpful discussions.
‘See Kanbur (1980b) for a discussion of the writings of Knight, Schumpeter
and Cantillon on
the entrepreneurial
function.
0047-2727/81/000&0000/$02.50
0
North-Holland
Publishing
Company
S.M. Kunbur,
164
Risk taking
and taxation
This paper can, therefore, be seen as an attempt to extend the literature on
risk taking and taxation in a new direction by modelling risk taking in a
manner where the choice of a continuous
control variable is combined with a
discrete choice between safety and risk. I call this an occupational choice
framework
of analysis, though it should be noted that models [such as that
of Cowell (1978)] where an individual
allocates time between alternative
activities through adjustments
at the margin will not fall in this category;
they would belong, rather, in a portfolio
choice framework
as I have
characterised
it.
Another important
feature of the model of risk taking put forward in this
paper is that there is imperfect insurance.
In fact, the model is one of the
polar case where there is no insurance whatsoever, either in terms of market
insurance of risky entrepreneurial
incomes, or in terms of mutual insurance
between entrepreneurial
and safe activities.2 It is in such a setting that I want
to investigate the role of public policy in altering market equilibrium
so as to
increase welfare, and to investigate
the effects of this lack of insurance
on
public policy. Moreover, it will be shown that in the discrete choice setting
the specification
of equilibrium
must of necessity have a general equilibrium
character, and an analysis of the effect of taxation on risk taking and welfare
must take these general equilibrium
feedbacks into account.
The plan of the paper is as follows. Section 2 presents the basic model, of
choice between entrepreneurship
and wage labour, and characterises
the
equilibrium
of the economy. Section 3 introduces the major issues of interest
for public policy in a mixed economy by considering
the opposite extreme of
an economy with an all powerful government.
Section 4 compares market
equilibrium
to that of such a mixed economy which controls access to the
occupations.
Section 5 develops rules for the introduction
of an occupational
tax-subsidy
scheme, while Section 6 does the same for the introduction
of
progressive
income
taxation.
Section
7 presents
some
concluding
observations.
An Appendix
to the paper develops some technical
results
which are used in the main text.
2. The basic model
The model of risk
(1977). It is a single
under risk. There
function which uses
F(L, ‘J),
taking developed in this section is introduced
in Kanbur
period general equilibrium
model of occupational
choice
are two occupations
centred
on a risky production
labour to produce a homogeneous
output
(1)
*It might be questioned
why insurance
is not explicitly incorporated
into the model and
shown in equilibrium
to be imperfect, rather than assuming imperfection
and taking the polar
case. This is, in fact, done in Kanbur
(1980a, appendix
to ch. 3) and it is shown that the
endogenous
determination
of market insurance introduces intractabilities
without disturbing
the
essential character of the model.
S.M. Kanbur,
Risk taking and taxation
165
where L is the labour employed and 6 is a random variable defined on the
density function g(0). The production
range [e mln,Qmax] with a probability
function satisfies the following properties:
F,>O,
F,,<O,
F,>O,
F,,>O,
F(0,0)=0,
(2)
where, as throughout
the paper, subscripts denote partial derivatives.
The agent has two alternatives
open to him. He can either become a
‘labourer’ in which case he supplies a unit of labour
of uniform quality and
receives the safe competitive
wage. Or he can become an ‘entrepreneur’,
which in this model is simply the management
of a production
function (1)
by employing
labour at a guaranteed
wage while bearing the production
(and hence income) risk represented by 0. One interpretation
of 8 is in terms
of ability
risk. It is assumed
that agents
do not know
their own
entrepreneurial
ability 0 but take g(Q), the density of 0 in the population,
as
the relevant risk. (The interpretation
of 8 as an ability index also makes
intuitive sense of the assumptions
F,>O, F,,>O in (2).)
The risk faced by the prospective entrepreneur
is that he has to make his
labour hiring decision before he discovers his entrepreneurial
ability 8. The
income of an entrepreneur
who hires L units of labour and then discovers his
ability to be 0 is given by
y=F(L,O)-wL.
(3)
Agents are assumed to have a common von NeumannMorgenstern
function U(y) with the usual properties:
and prospective
entrepreneurs
income, which gives
choose
L to maximise
V(w)=maxE(U(F(L,o)-wL)),
expected
utility
utility
of
(5)
L
where E is the expectation
operator with respect to the density of 0. The first
and second order conditions
from the problem (5) are, respectively,4
E(U,[F,-w])=O,
(6)
E(U,,CFL-w12)+E(UyFLL)<0,
(7)
3Problems of labour-leisure
choice are abstracted
from in order to focus on the essentials of
risk taking choice.
4Throughout
the paper, round brackets are used to enclose arguments
of functions, whereas
square brackets represent the usual bracket operation.
S.M. Kanbur,
166
Risk taking and tuxation
and it is seen from the properties of F and U, given in (2) and (4) that (7) is
satisfied. We assume the existence of an interior solution to (6).
How is market equilibrium
characterised
in this model? Firstly, the labour
market must clear: those who enter the entrepreneurial
activity must demand
just enough labour to employ those who enter wage labour. Secondly, to
sustain equilibrium
when all agents have a common utility function it must
be the case that neither activity is preferred to the other. Along with the
continuous
variable choice problem
(5) agents also face a discrete
choice
between the entrepreneurial
activity, which gives an expected utility I/(w),
and wage labour, which gives a utility U(w). If either of these is strictly
greater than the other then all agents will enter one activity or the other, and
the labour market will not clear. Thus we require that
V(w)=U(w).
(8)
When this holds5 the distribution
of population
between the two activities is
determined
by the full employment
condition.
Denoting x as the proportion
of population
(a representative
subsample of the whole) that enters the risky
occupation,
and normalising
population
size at unity, full employment
requires that
xL=l-X
or
x=
l/[l +L],
where
e=e(w)
is the optimised
entrepreneur,
derived from (5).
labour
demand
of
a
prospective
51 have assumed here that a value of w satisfying (8) exists. The existence question is discussed
further in Kanbur (1980a). Since U(w) is a monotonic
function of w that takes on all values
between -x,
and + zc as w varies from y,,, to 00, and since, as shown in (22), V(w) is a
monotonic
decreasing
function of W, a solution to (8) exists if V(y,,,) is finite. In this case
equilibrium is unique as well.
There is, however, a possibility that equilibrium
may not exist. This happens if the V(w)
relationship
is such that at the wage y,,, entrepreneurs
cannot avoid ruin, so that V(y,,,,p)
is
_ CC. To rule out such a case it is sufficient that it be possible for the entrepreneur
to choose his
labour demand so as to avoid ruin even in the worst possible realisation
of 0=0,,,.
If it is
possible to avoid ruin the optimisation
problem will ensure that it is in fact avoided. Thus we
can ask: if the entrepreneur
chooses L to maximise his income when O=O,,,, is it possible for
him to get an income greater than y,,, when the ruling wage is y,,,? It can be shown that this is
so if cc(L, O,,,) < L/Cl + L], where cc(L, 0) = LF,(L, 0)/F (L, 0) is the labour elasticity of output. The
above condition is sufficient to rule out non-existence.
It is not necessary, however. If the density
then the integral V(y,,,) will have a finite value
of 0 tends to zero sufficiently fast as B-O,,,,
and equilibrium will exist.
In what follows I shall assume that non-existence
is ruled out.
KM.
Kanbur,
Risk taking
and taxation
167
Given this equilibrium,
we are now in a position to analyse the role of
government
intervention
in the market so as to improve welfare. In the next
section I study the case of a fully controlled economy and then move on to
an analysis of the mixed economy case.
3. The fully controlled economy
The policy to be followed by an all powerful government
in the context of
the model of section 2 is trivially simple. Given that entrepreneurial
input is
required to operate a production
function, while labour is required as an
input
into the production
function
brought
into being by a unit of
entrepreneurial
input, and that the total size of population
is fixed (at unity),
national income is maximised by choosing the numbers of entrepreneurs
to
(10)
which has the first order condition
1
‘=1+L=
E(F,)
E(F)
(11)
The government,
if it is all powerful, then simply distributes
this maximised
national income equally, and this situation clearly leads to maximum welfare
for the (identical) individuals
of the economy modelled in section 2.
Of course, not only is the above result trivially simple, it is also of little
interest in a mixed economy, where the government
has to interact
with the
market through a system of instruments
that only give it imperfect control of
agents’ choices. Among others, these instruments
are mainly of the ‘tax and
subsidy’ variety, though sometimes a system of ‘licensing’ is also available. I
shall therefore focus on three sets of instruments
that a government
can use
in the context of occupational
choice. First, I consider government
control of
access to occupations
through a system of licensing, but without control over
the market wage or the entrepreneur’s
choice of labour input. Second, I
suppose that the government
can identify the occupational
origin of incomes
and can tax these differentially.
Third, I assume that the government
implements
a self-linancing
linear tax regime on all incomes,
whether
entrepreneurial
or labour.
There is an important
sense in which the analysis of public policy with
market instruments
may turn out to be uninteresting,
and it is this: with a
sufficient number of market instruments,
and with a sufficiently intensive and
severe use of them, the ‘mixed economy’ becomes one that is controlled
completely
by the government,
and the first best given by (10) can be
168
KM. Kunbur, Risk taking and taxation
achieved. In the analysis of this paper, therefore, I shall restrict attention
to
the three instruments
one at a time, and furthermore
I shall restrict the
severity
of their application
by only considering
their impact
in a
‘neighbourhood’
of the market equilibrium.
Such a restriction is in any case
dictated by the fact that the analysis of the instruments
in combination
and
away from equilibrium
poses intractabilities
that are hard to resolve.
4. Control of access to occupations
In this section I will examine the response to market equilibrium
of a
government
which has quantitative
control of the occupational
distribution
of population,
but which does not control the labour hiring decision of
entrepreneurs.
Thus, in the model of section 2, the government
chooses x, a
random sample of the population,
to enter the entrepreneurial
occupation,
and the market wage then adjusts to clear the labour market. Hence for
given x the wage adjusts so that
XL(w)= 1 -x,
where t(w)
is the prospective
demand, as given by the solution
w=tm’
entrepreneur’s
optimising
ex ante
to (2). Rewriting the above,
labour
l-x
(--I,
(12)
X
Under these circumstances,
therefore, would the government
wish to direct
more or less people into entrepreneurship
than are to be found in the market
equilibrium?
The answer clearly depends
on the government’s
objective
function.
I assume this to be given by the sum of (ex ante or ex post)
utilities:
S=xE(U(F(L,O)-wE))+[l-x]CJ(w).
Then, differentiating
+[l
(13)
(13) with respect to x,
-x,$r;,,-ci(w).
Using the basic first order condition
(2) and (12), this can be written
as
(14)
S.M. Kanbur,
Risk taking and tazution
169
An intuitive account of (14) can be given as follows. Shifting one person from
a labouring to an entrepreneurial
activity (increasing x) has two effects. First,
it changes his utility by EU(y)U(w). Secondly, it alters the market wage
from (12), and this has further consequences.
The utility of workers now
changes by U,. The expected utility of entrepreneurs
as a whole changes by
the number of workers each employs times the total number of entrepreneurs
times E(U,). But this is simply the total numbers of workers times E(U,).
Hence we get the first term in (14).
I wish to analyse the policy of a government
with a limited control of
access to occupations,
when faced with the market equilibrium.
I thus
evaluate (14) at the market equilibrium
(ME) to give
dS
-
dx
=[W~[U,-EW,)l;
(15)
ME
where I have used the fact that at market equilibrium
expected utilities in the
two occupations
are equal [see (S)]. The expression
(15) illuminates
the
possible sub-optimality
of market equilibrium.
Because of the discrete choice
element in our modelling
of risk taking, equilibrium
requires that expected
total utility in the two activities be equal. But from the point of view of
policy it is the difference in expected marginal utility between the two
activities which is relevant, and this magnitude
need not necessarily be equal
to zero.
Notice, of course, that when the utility function is linear,
dS
dx
=o
ME
so we have the only too obvious
result that in a risk neutral
society
government
control of access to occupations
cannot improve upon market
equilibrium.
But now consider the more interesting
case of a risk averse
society. Then, from (12)
1
dw
dx
(16)
-v’
Define the Arrow-Pratt
follows:
A(y)=
measures
-+,
and relative
risk aversion
as
(17)
Y
R(Y)=-
of absolute
yu,,
u
.
Y
170
SM. Kanbur,
In the Appendix
Risk taking and taxation
it is shown that
(19)
so that, from (16)
dw
A,sO=>->O.
(20)
dx
When dw/dx >O, (16) tells us that
dS
dx
The expression
follows :
Given
$OoE(U,)~U,.
(21)
ME
(21) provides
non-increasing
us with a characterisation
absolute
risk
direct more people
market equilibrium
(less people)
if the expected
than (greater
the marginal
than)
aversion,
the
result that I state as
government
would
into the risky activity than present
marginal utility in the risky activity
utility
wish
to
in the
is less
in the sufe activity.
The interpretation
of this proposition
is of interest, since it may seem
surprising at first sight - the government
is directing more people into the
activity with lower expected marginal
utility. But this result is a natural
consequence
of the general equilibrium
nature
of the problem.
Welfare
improvement
requires
a narrowing
in the expected
marginal
utility
differential between the two activities. But, by differentiation
of (5),
if,.=
-ME,
(22)
and with non-increasing
absolute risk aversion dwldx >O, so that directing
more people into the activity with the lower expected marginal utility will
lower its expected total utility and precisely for that reason, raise expected
marginal utility in that activity. Hence the result.
We can relate (21) to certain conditions
on preferences. Define Cp as the
function
(23)
S.M. Kanbur,
Then, using the equilibrium
171
Risk taking and taxation
condition
(8), it follows that
u,-E(U,)=~(U(w))-E(~(U(y)))
=~(E(U(Y)))-E(~(U(Y)))
~O~c#J”“~O,
o
A,$O.
(24)
Thus with respect to the objective
function
chosen and the instrument
considered, it follows that if absolute risk aversion is strictly decreasing the
market equilibrium has too many entrepreneurs.
Notice of course that this
proposition
only states a sufficient condition
- the property of decreasing
absolute risk aversion is not necessary for market equilibrium
to have an
excessive number of entrepreneurs.
In the other case it is possible that &,>O,
which together with (24) leads to dS/dx<O.
5. Occupational
taxation
In this section I move away from the instrument
of direct control of access
to occupations
and assume that the government
has only tax and subsidy
instruments
to hand. Specifically, I assume that the government
can identify
the occupational
origin of incomes,
and can tax occupational
incomes
differentially.
When
faced
with the market
equilibrium,
should
the
government
tax or subsidise the risky activity? This section develops rules
which provide an answer to this question, at least in the neighbourhood
of
market equilibrium.
Let there be a propor: \q ’ ‘1x rate t on incomes from the risky activity,
wage income (of course t and s can be
and a proportional
subs.
negative, which is a tax on ,
-ome to subsidise risky incomes). Then
ivities ‘are given by
post tax incomes in the risky ar __
i
i’..
(25)
w*=[l+s]w.
The prospective
post-tax income,
(26)
entrepreneur’s
which gives
problem
V(t,w)=maxEU([l-t][F(L,8)-wL]).
L
IPE
B
is to maximise
expected
utility
of
(27)
S.M. Kanhur, Risk taking
172
The first order condition
and taxation
is
W,(Y*)C~,-WI)=Q
which defines labour
(28)
demand
L(b w).
(29 I
Market equilibrium
is specified,
distribution
of population
as before,
by a labour
market
clearing
1
(30)
X=1+&w)’
and the condition
that agents be indifferent
between
the two activities:
V(t,w)=U([l+s]w).
(31)
disbursed for given t and s depends on
when this regime is imposed. Net tax
The tax revenue raised and subsidy
the market equilibrium
which attains
revenue is given by
T=xtE(y)and the self-financing
s=t
x
[l -x]sw,
condition
thus requires
WY)
w)’
E(Y)
1-X
As before, we assume
post-tax utilities:
w
-
(32)
(33)
wL(t,
social
welfare
,. ,
“!
1i
_ii:.,‘
tea
L
131 ’
I’,
i;
;y the ex post
sum of
.I
S=xv(t,w)+[1-x]U([1-ts]w)
=V(t,w)=U([l+s]w),
using (31). The effect on social welfare of a small change
thus given by
dS
-=u,
dt
(34)
in the tax rate t is
(35)
S.M. Kanbur,
where dw/dt and ds/dt are jointly
condition
(31) and the self-financing
(33) we get, respectively,
determined
condition
by the market equilibrium
(33). Differentiating
(31) and
[l-t]L(t,w)E(U,)+U,[l+s]
ds
173
Risk taking and taxation
+E(yU,)+wU,.;=O,
_Eo?)+t. dIW)W(t>
dt-WI&W)
w)l
(37)
’
dt
(36)
A combination
of (35), (36) and (37) will give us a general characterisation
of the direction of welfare change consequent
upon a tax change. It should
be clear that this general case presents us with intractabilities
of considerable
severity. However, we can restrict attention
to government
policy in the
neighbourhood
of market equilibrium.
Mathematically,
this means that we
evaluate the derivatives at t = 0, s = 0:
ds
z
E(Y)
(38)
,++=z
-dw
dt t=,,s=o=
-
E(@,)+
CEb)lLlU,
LE(U,)+
U,
’
=EOI)W,)-E(YU,).
LE(U,)+
With decreasing marginal
correlated, so that
utility
U,
income
(39)
u
(40)
w
and marginal
utility
are negatively
and
dS
> 0.
dt t=o,s=o
Hence it follows that starting from a position of market equilibrium,
the
government should tax the risky occupation and subsidise the safe occupation.
S.M. Kanbur,
174
Risk taking and taxation
What are the implications
of the above policy for the distribution
population
between safe and risky activities? Differentiating
(30)
of
(41)
In the Appendix
it is shown that
R,$O*
Thus, since L,<O
L,$Oo.
(42)
with non-increasing
absolute
risk aversion
and
dw
CO,
dt r=o,s=o
we have the result that
dx
t0
if R,hO
and A,SO.
dt t=o,s=o
(43)
Thus with non-increasing
absolute risk aversion and non-decreasing
relative
risk aversion, a government imposing a cross-occupational
tax-subsidy scheme
to increase welfare will do so to reduce entry into the risky activity.
It should be noted, of course, that the above proposition
is specific to the
proportional
tax subsidy scheme ~ not all cross-occupational
tax-subsidy
regimes that reduce entry into the risky activity will increase welfare, even
given the assumptions
of non-increasing
absolute risk aversion and nondecreasing
relative risk aversion. To see this, consider the case where the
government
imposes an entry tax (or ‘license fee’) on the entrepreneurial
occupation,
and uses the revenue to subsidise wage labour. Let the entry tax
be A and the entry subsidy to the safe occupation
be B. Then the prospective
entrepreneur’s
problem is to
V(A,w)=maxE(U(F(L,O)-wL-A)),
(44)
L
leading
to a labour
demand
L= L(A, w).
Following
earlier arguments
V(A,w)=U(w+B)
(45)
equilibrium
is given by
(46)
S.M. Kanbur,
Risk
taking
175
and taxation
and
1
(47 )
x= 1 +L,(A,w)’
with the government’s
self-financing
condition
as
(1 -x)B.
xA=
Then, differentiating
(48 )
(47)
(49)
The response dw/dA is given by the interactions
of the market equilibrium
condition (46) and self-financing condition (48). Differentiating
these we get
-E(U,)-gu,
dw
-zz
dA
U,+LE(U,)
’
(51)
so that
-E(U,)-;
dw
dA A=O=
Now differentiating
L,=
U,
< 0.
(52)
u, + WU,)
the first order condition
WU,,{~,-w~)
for problem
10
E(U,,{F,-W}~)+E(U~FLL)-
using the basic lemma proved
constant absolute risk aversion,
dx
dA *=,,
using (52) and (19).
<o
in the Appendix.
(44) we get
if A,,50
Thus,
(53)
for example,
with
176
S.M. Kanbur,
Risk taking and taxution
Thus the imposition
of this cross-occupational
tax-subsidy
scheme reduces
entry into the risky activity. But does it increase welfare? We know that with
a utilitarian
objective social welfare is a monotonic
increasing function of the
post tax wage
w*=w+L?.
(55)
Then
(56)
from (50) and (51). Thus, relative to this instrument,
market equilibrium
has
neither an excessive number nor too few entrepreneurs.
Yet its operation will
in general lead to a change in this number (a decrease if there is constant
absolute risk aversion).
6. Income taxation
In this section I focus on the use of income taxation to modify market
equilibrium
in such a way as to increase welfare. In the context of risk taking
as modelled in the discrete choice framework
of this paper, should selffinancing income taxation be progressive
or regressive? I will look for an
answer to this question in the framework of a linear tax regime
y*=a+by,
(57)
where y is pre-tax and y* is post-tax
and the average tax rate is given by
income.
The marginal
tax rate is [ 1 -b]
-;+[l-b].
Hence the regime is progressive (regressive), in the sense that the average tax
rate rises (falls) with income, if a>0 (CO).
Prospective
entrepreneurs
will now choose their labour demand so as to
maximise the expected utility of post-tax income (57), which gives
V(a,b,w)=maxE(U(a+b[F(L,0)-wLI)).
(58)
L
The first order
condition
defines
labour
demand
as a function
of the wage
S.M. Kunbur, Risk taking and taxation
177
and the tax parameters:
L(a, b, w).
Equilibrium
(59)
is again specified by
x=l/(l+L(a,b,w))
from the labour
market
(60)
clearing
condition,
and
V(a,b,w)=U(w*),
(61)
w*=a+bw
(62)
is the post-tax
that
wage. If tax revenue
is 17;then the self-financing
requirement
is
T=xE(y-y*)+[l-x][w-w*]
=-a+[l-b]Y(a,b,w)
=o,
(63)
where
y(u
b
3
is national income
As throughout
utilitarian
sum
,w
$WWAw),Q))
l+L(a,b,w)
gross of tax.
the paper, social
welfare
is again
assumed
to
be the
S=xI’(u,b,w)+[l-x]U(w*)
=V’(u,b,w)=U(w*).
(65)
I parametrise
tax regime/changes
by changes in b, and the object is to
examine the effect on welfare of a change in the marginal tax rate subject to
the self-financing
condition
(63), and the response of market equilibrium
from (60) and (61). From (65) we see that social welfare is a monotonic
S.M. Kanbur,
178
function
of the post-tax
dw*
da
db
db
Risk taking and taxation
wage w*. Thus differentiating
(62) with respect to b:
dw
-=-+bdh+w.
(66)
The derivatives da/db and dwldb are jointly determined
by the self-tinancing
condition
and the response of market equilibrium.
Differentiating
(61) and
(63) we get, respectively, that
=wU,(w*)-WYU,),
da
p=
db
Solving
(67)
[l-b][Y,+Y,[dw/db]]-Y
(68)
l-[l-b]Y,
.
(67) and (68) simultaneously,
dw
p=
db
CY-Cl-blr,lCE(U,)-U,(w*)l
wU,(w*)-E(yU,)+
l-[l-b]Y,
C~~C’,~-~,~~*~ll~-~lY,_bLLE~U,~+u,(w~~l
l-[l-b]Y,
(69)
As can be seen, the effect on welfare of a revenue compensated
change in
the tax schedule is somewhat complicated
in the general case. An explicit
analysis of this general case encounters
severe problems
of intractability.
to focus attention
on a
However,
as in previous
sections, we propose
neighbourhood
of market
equilibrium.
This means
that we evaluate
derivatives at b = 1 (a= 0):
da
db b=l=
dw
wU,-E(yU,)+
db b=l=Substituting
(70)
-r,
YCWJ,)-
WU,)+
Uwl
U,
(71)
(70) and (71) into (68) gives us the simple expression
dw*
db
b=l=
W’UJ-WMU,)
LE(U,)+
U,
’
(72)
S.M. Kanbur,
Since F and U, are negatively
dw*
db
b=l
Risk
taking
correlated,
and taxation
179
(72) is negative:
<o.
Notice that under self-financing
b > 1 (aa ~0) represents
a regressive regime
and b < 1 (=>a>O) represents a progressive
regime. This leads to the next
proposition:
If the (marginal)
imposition
of a self-naming
linear tax regime
is being
considered,
the regime
introduced
should
be a progressive
one.
As in the previous section, a question of interest is whether or not the
welfare increasing
policy is consistent
with a reduction
of numbers in the
risky activity. But in this case the analysis does not lead to definite results
and is relegated
to the Appendix.
In general, however, a self-financing
progressive tax may either increase or decrease the number of entrepreneurs,
so that, relative to this instrument,
market equilibrium
may in general have
an excessive number or too few entrepreneurs.
7. Conclusion
The recognition
of occupational
choice as a major vehicle for risk taking
goes back at least as far as Adam Smith (1776). Yet the analysis of taxation
and risk taking in the literature
since the pioneering
work of Domar and
Musgrave (1944) has almost invariably
been in a portfolio choice framework.
The first object of this paper has been to construct a model in which some
old questions
of taxation
and risk taking
can be posed in terms of
occupational
choice. The distinctive feature of the model is that risk taking
combines an element of discrete choice with an element of continuous
choice.
The discrete choice element forces us to specify general equilibrium
in the
model, with the result that in the analysis of government
policy, we have to
take account of the general equilibrium
feedbacks. In this framework
this
paper has developed
rules for government
intervention
in the market
equilibrium
with different types of instruments.
It is shown that market
equilibrium
is not necessarily optimal, since in each case we find conditions
under which intervention
would be desirable.
The paper has also asked the following question: does market equilibrium
have too many or too few entrepreneurs?
It is argued that this question can
only be answered relative to a specific instrument
under consideration
(and a
specific objective
function),
and found that, for certain
restrictions
on
market
equilibrium
does
have
an excessive
number
of
preferences,
entrepreneurs
relative to a system of occupational
licensing and a system of
cross-occupational
taxation.
180
S.M. Kanbur,
Risk
taking
and taxation
Appendix
In this Appendix I examine the effect of the linear tax regime (57) on
I consider
the effect of a revenue
entrepreneurship.
In particular,
compensated
increase
in the marginal
tax rate on the occupational
distribution
of population.
In doing so I shall also obtain the result used in
deriving (42). Total differentiation
of (60) with respect to b gives us
da
dw
Lqg+Lb+Lw~
1
2
(73)
where
1
XL=
(74)
-m+o.
But it is seen that in order to sign (73) we have five other terms to consider,
each of which represents
a separate aspect of the interrelated
equilibrium
system modelled in section 6. L,, L,, and L, reflect the optimisation
problem
in (58) while da/db and dw/db are jointly determined by (68) and (69).
We can obtain general expressions for L,, L, and L, by differentiating
the
first order condition for the maximisation
problem (58):
(75)
-E(U,,CF,-wlCF-wL1)
Lb=bE(UgylFL-w]2)+E(UyFI.I.)1
L
=
WU,)+bWU,,[I%-WI)
w bE(U,,CF,-wl')+E(U,F,,)'
(76)
(77)
From the second order conditions
the denominator
of (75), (76) and (77) is
negative.
We start by considering the sign of E(U,,[F,
- w]). Define 6 such that
F,(L,B)-w$Oo8$&
From the properties of the production
unique.
Moreover
notice that since
increasing function of 0. Thus, recalling
(78)
function given in (2), e^ exists and is
F, >O from (2), y=F - WL is an
that A(y) is the measure of absolute
S.M. Kanbur,
Risk taking and taxation
181
risk aversion,
A,503
A(y(Q))~A(y(B))
if Q>O,
A(y(Q))=A(y(@)
if O=&
A(y(8))2A(y(d))
if O-C&
(79)
Using (78) and (79)
A,~O~A(y(B))CF,-~l~A(y(~))CF~-wlV8.
From
the definition
of A(y),
(80)
in (17) it follows that
A,~O~U,,[F,-w]~A(y(O^))U,[F,-w].
Taking
expectations
and using the first order condition,
we have
A,~O+E(U,,[F,-w])zO.
A symmetrical
argument
(81)
for A,10
would provide
the general
result
A,$OoE(U,,[F,-w])zO.
(82)
A,~O=L,~O,
(83)
A,sOs-L,<O.
(84)
Hence
Turning
now to the measure
R,zO+
From
of relative
risk aversion,
R(y(O))ZR(y(B))
if Q>O,
R(y(B))=R(y(@)
if Q=8,
i R(y(8))5R(y(@)
if 0~8.
(85)
(78) and (85),
R,~O~R(y(~))CF,-wl~R(y(B))[F,-w]V8.
Taking
expectations
in (86) and using the definition
(86)
of R(y)
in (18) and the
S.M. Kunbur, Risk taking and taxation
182
first order condition,
R,lO+E(U,,[FL-w][F-wL])Z
When a=0
-;E(U,,Ck.,w]).
(87)
and b=l,
Using a symmetrical
development
R,$O-=-Lblb=l
for R,sO,
it follows that
$0,
(89)
which establishes (42), identifying b with 1 - t.
To show the ambiguities
that arise in the sign of (73), let me first of all
consider the sign of dw/db evaluated at the market equilibrium.
To sign (71),
let us define IL as the function
Then, using the equilibrium
condition
(61),
We can now attempt to sign (73) under
aversion. Consider first of all a utility function
various assumptions
of the type
on
risk
(91)
which displays constant
relative risk aversion and decreasing
aversion. Then, using (90) and (24), we know that
dw
db b=l
<O.
absolute
risk
From
the analysis
L,>O,
S.M. Kanbur,
Risk taking and taxation
of the Appendix
it is also clear that
183
L,<O,
L,=O,
while (70) gives us
da
db b=l
<o.
Using these signs in (73), we see that
dw
da
L#&-4
L,pO,
so that the sign of (73) is ambiguous
function
U(y)=
the utility
-_Boe-B*y,
which displays constant
aversion. In this case
L,=O,
in this case. Now consider
absolute
Lb-CO,
risk aversion
and
increasing
relative
risk
L,<O,
which again leaves the sign of (73) ambiguous.
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