eu= u(w+x* r g ) + (1- )u(w+x* r b ) last week saw consumer with wealth w, chose to invest an...
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• EU=U(W+x* rg ) + (1-)U(W+x* rb)
Last week saw consumer with wealth W, chose to invest an amount x* when returns were
rg in the good state with probability
and rb in the bad state with probability 1-.
• EU=U(W+x(1-t)rg) + (1-)U(W+x (1-t)rb)
What is the effect of taxation on the amount we choose to invest in the risky investment?
Now only get (1-t)rg in the good state with probability
and (1-t)rb in the bad state with probability 1-.
• EU=U(W+x(1-t)rg) + (1-)U(W+x (1-t)rb)
What are the FOC’s?
0)1()1()1(
bb
gg rt
x
Urt
x
U
0)1(
bb
gg r
x
Ur
x
U
• EU=U(W+x(1-t)rg) + (1-)U(W+x (1-t)rb)
What are the FOC’s?
0)1(
bb
gg r
x
Ur
x
U
Like last case except that there is no guarantee that x with taxes is the same as x*
Why? Payoff in good and bad states different.
• EU=U(W+x(1-t)rg) + (1-)U(W+x (1-t)rb)
What are the FOC’s?
0])1([()1(])1([(^^
bbb
ggg rrtxW
x
UrrtxW
x
U
Like last case except that there is no guarantee that x with taxes is the same as x*
Why? Payoff in good and bad states different.W+x[1-t]rs
• What is the relationship between x* and
Usually think, if return on asset goes down, want less of it. That is, might think x*>
g
b
bb
gg
r
r
rtxWxU
rtxWx
U
)1(
])1([
])1([
^
^
^
x
^
x
To see why this is wrong sett
xx
1
*^
Then MRS between
the two states is:
])1([
])1([
^
^
bb
gg
rtxWxU
rtxWx
U
To see this suppose set t
xx
1
*^
MRS between the two states is:
])1(1
*[
])1(1
*[
bb
gg
rtt
xW
xU
rtt
xW
x
U
To see this suppose set t
xx
1
*^
MRS between the two states is:
g
b
bb
gg
r
r
rxWxU
rxWx
U
)1(
]*[
]*[
but since 0<t<1,
(1-t)<1
*1
*^
xt
xxSo
Why does investment rise?
• When the good state occurs t is a tax.
• However, if losses can be offset against tax then t is a subsidy when rb occurs
rg(1-t)rg(1-t)rb
rb
•So t reduces spread of returns and therefore risk
•Only way can recreate original spread is to invest more (i.e. x up)
Suppose W = £100 and bet £50 on flip of a coin
• EU=0.5U(50) + 0.5 U(100)
Probability
0.5
£50 £150
Outcomes
More outcomes => More complexity
Probability
1/6
£10 £60Outcomes
• E.g W = £30, Bet £30 on throw of dice
• Prizes 1=£10, 2= £20, 3=£30, 4=£40, 5=£50,6=£60,
• each with probability 1/6
£30 £40 £50£20
More outcomes => More complexity
Probability
1/6
£10 £60Outcomes
• EU= 1/6 U(10)+ 1/6 U(20)+ 1/6 U(30)+ 1/6 U(40)+ 1/6 U(50)+ 1/6 U(60)
£30 £40 £50£20
More complex still: Probability of returns on investment follows a
normal distribution
Probability
Returns
• With EU need to consider every possible outcome and probability
• Too complex, need something simpler
Probability of returns on investment follows a normal distribution
Probability
Returns
• If we can use some representative information that would be simpler.
Probability of returns on investment follows a normal distribution
Probability
Returns
Called Mean-Variance Analysis
U=U(,2)
Like returns to be highLike Risk to be low
U=U(,2)
U=U(,)
(measured by Variance 2 or Standard Deviation, )
Risk
Return
U0
U2
The Mean-Variance approach says that consumers preferences
can be captured by using two summary statistics of a
distribution :
•Mean
•Variance
Similarly Variance is
2 (w2-)2 +
………...3 (w3-)2 + 4 (w4-
)2 + … …….+s (ws-)2
+ ...
Or in other words
=1(w1-)2
2 21
s
Ss ws( )
We like return- measured by the Mean
We dislike risk - measured by the Variance
U U
U U
( , )
( , )
2
Suppose we have two assets, one risk-free and one risky, e.g. Stock
The return on the risk-free asset is rf and its variance is f
2=0
The return on the risky asset is rs with probability s , but on average it is rm and its
variance is m2
If we had a portfolio composed of x of the risky asset and (1-x) of
the risk-free asset, what would its properties be?
x s xr x r xrm x rs
S
s f f2 1 1 2
1
( ( ) ( ) )
x s xr xrms
Ss
2 21
( )
x x s r rms
Ss
2 2 21
( )
x x m2 2 2
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