• 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?

x

U g

grt)1( bb rtx

U)1()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)

Mean-Variance Analysis

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

Mean

=1w1+ 2w2+ 3w3+ 4w4+ … …….+sws+ ...

s

Ss ws

1

Or in other words

Mean

s

Ss ws

1

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( )

Variance

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?

Return

rx s xrs x rs

S

f 1

1 ( ( ) )

rx x s rs x rs

S

f 1

1 ( ) ( )

rx xrm x rf ( )1

Variance

x s xrs x r rxs

S

f2 1 2

1

( ( ) )

x s xr x rs

S

s f2 1 2

1

( ( ) ) xrm x rf( )1

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

Results

Return

Variance:

rx xrm x rf ( )1

x x m2 2 2