Introduction to Game Theory

1 - Decision Theory

Helena Perrone

Universitat Pompeu Fabra

 Decision Theory Introduction

Introduction
Risk and Uncertainty
 Decision Theory Introduction

Uncertainty

I Until now, consumers and rms knew the exact

consequences/outcomes of their actions
Example a monopolist knows exactly its prot if it xes price at a
certain level
I But usually theres UNCERTAINTY about the payos from
choices
Example
- A farmer chooses to cultivate either apples or oranges
- when he makes the decision, hes uncertain about the prots that hell
obtain
- best choice depends on rain conditions, future world prices, plagues...
I Gap between actions and consequences - Decision Theory
shows us how to deal with this gap
 Decision Theory Introduction

Example
Uncertainty

I youll open a shop in a very touristic town

I you can open an ice-cream parlor or a bookshop specialized in
travel guides and maps
I prots are uncertain: they depend on a variable that you
dont control

Table: Actions, States, Consequences

1 2
ice-cream 6 0
bookshop 2 2
 Decision Theory Introduction

Example
Risk

I Suppose is the weather: if its sunny, youll sell a lot of

ice-cream; if it rains, youll sell nothing
I the prot from the bookshop doesnt depend on the weather
I the realizations of the variable follows a probability
distribution

Sun Rain
ice-cream shop 6 0
bookshop 2 2

Fundamental problem of Decision Theory: how to use the

information on possible payos and probabilities to choose between
risky alternatives
(rst topic of the IGT course)
 Decision Theory Introduction

Example
Strategic Interaction

I Suppose is the decision of a neighbor that may open a

tobacco shop or an ice-cream parlor
I your potential benets depend on the decision of the neighbor
Tabacco ice_cream
I ice-cream 6 0
bookshop 2 2
I the neighbor wont choose what to do randomly (following a
probability distribution); hell choose whats best for him
I Fundamental problem of Game theory: determine the decision
of a rational player in a context where his decisions depend on
the decision of other players (and aect their decisions)
(Topics 2-8)
 Decision Theory Taking Decisions

A formal approach to taking decisions

I set of actions A = fa1 , a2 , ..., al g

I set of possible realizations of the state of nature
= f 1 , 2 , ..., m g
I set of consequences (outcomes) C = fc11 , c12 , ..., clm g

Table: Example with l = 2 and m = 3

1 2 3
a1 c11 c12 c13
a2 c21 c22 c23
 Decision Theory Taking Decisions

I realizations of follow a probability distribution

I it can take m dierent values 2 f 1 , 2 , ..., m g
I with probabilities p = fp1, p2 , ..., pm g where pi > 0 and
m
pi = 1
i =1

in a risky situation, we dont know what will happen but we know

what could happen (the dierent values of the state of nature) and
its likelihoods (the probabilities)

Example: we dont know what the weather will be but we know it

can be sunny or rainy and we know the probabilities
 Decision Theory Taking Decisions

Decision Theory
exploits the information given by probabilities and the information
about agentspreferences in order to make good decisions - choose
favorite lottery
 Decision Theory Taking Decisions

Expected Utility Theory

 Decision Theory Taking Decisions

The farmers example

Table: The farmers decision

Cold Warm
tangerines 0 100
apples 20 100
oranges -10 140
cherries 40 40

1
Suppose the probability of cold weather is p = 2
 Decision Theory Taking Decisions

set of actions A = f_______g

set of possible realizations of state of nature = f_____g

probabilities p = f_____g

set of consequences C = f________________g

 Decision Theory Taking Decisions

Lotteries

I representation of risky alternatives

I a set of consequences (in this case, monetary payos), each
with an assigned probability pi
L = hp1, p2 , ..., pN jx1 , x2 , ..., xN i

Lotteries for Farmers Example:

Lt = _____________________
La = _____________________
Lo = _____________________
Lc = _____________________

Tree representation
 Decision Theory Taking Decisions

Rational Behavior

I an individual is rational if he has well dened objectives

(preferences) and makes decisions according to them

I its convenient to work with numerical indicators of

preferences (utilities) so that rational behavior can be
expressed as a maximizing behavior

I how to assign utility numbers to lotteries that represent

individual preferences so we can choose the best/preferred
lottery?
 Expected Value

the expected value of a lottery?

I if $$$ is the only objective of the decision maker, then the

expected value of the lotteries (i.e., the expected prots)
could be a good utility function (i.e., a good representation of
the decision makers preferences)

I the expected value of Lj = hp1 , ..., pm j x1 , ..., xm i is the sum

of all its possible consequences weighted by its respective
probabilities:
m
E (Lj ) = pi cji
i =1
 Expected Value

Back to the farmers example

Expected Value of the farmers lotteries:

E (Lt ) = ________________
E (La ) = _______________
E (Lo ) = _______________
E (Lc ) = _______________

If the farmer is an "expected value maximizer", then his preference

ranking is ______________________

Spoiler : an expected value maximizer is also called risk neutral

 Expected Value

I do real people use expected value to choose between lotteries?

I is expected value a good criterion to choose between lotteries?

 Expected Value

An experiment

choose a lottery
 Expected Value

E (L) = 11000 > 10000 = E (L0 )

I so expected utility maximizers would choose lottery L

I but if people dislike risk (risk averse), theyll probably choose
L0
I the expected value doesnt seem to be a good criterion for
people that dislike risk
I there is considerable empirical evidence that many people are
not expected utility maximizers and therefore the expected
value is not an appropriate utility representation of their
preferences over risky alternatives
 Expected Value

The Saint Petersburg Paradox

I consider a game in which you pay a xed fee to play and then
you toss a coin repeatedly until it turns up heads
I you get $2 if the coin comes up heads in the 1st toss
I $4 if heads in the 2nd toss
I $8 if heads in the 3rd toss
I etc.

I how much would you pay to play?

 Expected Value

Expected Value of the game

1 1 1 1
E (L) = (0) + (2) + (4) + ... + n +1 2n ... =
2 4 8 2
1
= (1 + 1 + 1 + 1 + ...) =
2
=
 Expected Utility

Expected Utility Theory

I individuals dont care only about the monetary values of the

lotteries
I in general expected value, isnt a correct representation of
preferences
I If preferences over risky alternatives satisfy certain consistency
conditions, then there exists a utility function u dened on
consequences such that for all L = hp1 , ..., pm j x1 , ..., xm i and
L0 = hp10 , ..., pm
0 j x 0 , ..., x 0 i
1 n

m n
L L0 , pi u (xi ) > pi0 u xi0
i =1 i =1
 Expected Utility von Neumann-Morgenstern utility function

The von Neumann-Morgenstern utility function

I utility function u dened on consequences

I allows the construction of a utility function U dened on
lotteries
I U (the expected utility) is linear with respect to probabilities
and given by
m
U (L) = pi u (xi )
i =1

Economic agents choose the lottery with highest expected utility

 Expected Utility von Neumann-Morgenstern utility function

NOTE the expected value is a special case of the von

Neumann-Morgenstern utility function

Indeed, if u (xi ) = xi , then expected utility equals expected value

m m
U (L) = pi u (xi ) = pi xi = E (L)
i =1 i =1
 Expected Utility von Neumann-Morgenstern utility function

Example
I Suppose an economic agent has preferences that can be
represented
p by the von Neuman-Morgenstern utility function
u (x ) = x.
I Which lottery will this agent choose?

Calculate and compare expected utilities:

U (L1 ) =
U (L2 ) =
Hell choose lottery ____
 Expected Utility von Neumann-Morgenstern utility function

What if expected utility maximizer?

E (L1 ) =

E (L2 ) =

He would have chosen lottery ____

 Expected Utility von Neumann-Morgenstern utility function

Preference Representations
 Expected Utility von Neumann-Morgenstern utility function

Preference representations are not unique

I if u represents preferences < over lotteries and

v (x ) = u (x ) + with > 0 then v represents the same
preferences

Note: a function of the form f (x ) = x + , with > 0, is called

a positive a ne transformation
 Expected Utility von Neumann-Morgenstern utility function

Proof.

L < L0
m
n n
pi u (xi ) pi0 u (xi0 )
i =1 i =1
m
n n
pi u (xi ) pi0 u (xi0 )
i =1 i =1
m
n n
pi v (xi ) pi0 v (xi0 )
i =1 i =1
 Expected Utility The VNM: Application

Application: freedom to choose the origin and unit of

measure

Suppose we have a function u representing some preferences:

1
u (x ) = 10 x + 1 . Wed like to have a utility representation such
that w (0) = 0 and w (10) = 1.

I Check with u:

1
u (0) = 10 = 9.
0+1

! u isnt satisfactory.
 Expected Utility The VNM: Application

I Construct v subtracting 9 from u:

1 x
v (x ) = u (x ) 9=1 = .
x +1 x +1

I The new function represents the same preferences and

satises our 1st condition since v (0) = 0 +0 1 = 0

I 10
But it doesnt satisfy the 2nd condition since v (10) = 11 . So
v is not satisfactory.
 Expected Utility The VNM: Application

I 10
Construct a new w dividing v by 11 :

11 11x
w (x ) = v (x ) = .
10 10(x + 1)

I The function obtained is satisfactory since

w (0) = 10 (00+1 ) = 0 and w (10) = 1011(10 +
10
1)
=1
 Expected Utility The VNM: Application

Estimating the Utility Function

 Expected Utility The Investor Example

Estimation of the Utility Function

The Investor Example

An investor has to choose between two risky projects. You are a

consultant. You have information about the prot levels in four
dierent scenarios. You obtain an estimation of the probabilities of
every outcome:

Retail store Oil well

Probability Wealth Probability Wealth
0,021 200 0,830 200
0,410 160 0,005 160
0,559 130 0,001 130
0,010 60 0,164 60
 Expected Utility The Investor Example

Should the expected prots be used?

We have to choose between two lotteries

whose expected values (expected prots) are

E (L) = 0, 021 200 + 0, 41 160 + 0, 559 130 + 0, 01 60 = 143, 07

0
E (L ) = 0, 83 200 + 0, 005 160 + 0, 001 130 + 0, 164 60 = 176, 77

The oil well L0 should be chosen if the investor is risk neutral. Yet,
you should check his attitude toward risk before giving any advice.
 Expected Utility The Investor Example

How to estimate the Utility Function?

I The set of consequences is X = f60, 130, 160, 200g.
I We have to assign a utility number to each outcome.
I Fix the utility of the best and worst consequences (can we do
this?):

u (60) = 0
u (200) = 1
I Now we assess the utility of the remaining consequences by
means of the reference lottery:

that has the best and worse outcomes as possible consequences.

 Expected Utility The Investor Example

Assessing the utility of 160e

I Ask your client: what is the probability p that makes 160e
with certainty indierent to the reference lottery?

I Suppose that the answer is p = 0.95

I In that case, we know that both lotteries are indierent so L
and Lp must have the same expected utility, that is

u (160) = 0, 05u (60) + 0, 95u (200) = 0, 05 0 + 0, 95 1 = 0, 95

and from here we get that the utility of 160e is 0,95

Of course, dierent people could have dierent preferences.

Utility is subjective.
 Expected Utility The Investor Example

Assessing the utility of 130e

I Now we ask what is the probability p that makes 130e with

certainty indierent to the reference lottery.

I Suppose that the answer is p = 0.85

I In that case we know that both lotteries are indierent so L

and Lp must have the same expected utility, that is

u (130) = 0, 15u (60) + 0, 85u (200) = 0, 05 0 + 0, 85 1 = 0, 85

and from here we get that the utility of 130e is 0,85

 Expected Utility The Investor Example

The estimated VNM utility function

Euros 60 120 160 200

Utility 0 0.85 0.95 1
 Expected Utility The Investor Example

The optimal investment decision

The best investment opportunity is the one with higher expected

utility. Using the utility function obtained we can compute the
expected utilities of both lotteries

U (L) = 0, 021u (200) + 0, 41u (160) + +0, 559u (130) = 0, 88565

U (L0 ) = 0, 83u (200) + 0, 005u (160) + 0, 001u (130) = 0, 8356

,! The retail store investment is better than the oil well.

 Expected Utility The Investor Example

Summary of decision making recipe

1. Identify set of actions, states of nature, and outcomes (no

model can exactly reproduce the real world, so we need to
make simplifying assumptions to reduce the problem to whats
really important)
2. uncover preferences over outcomes and translate them into
utilities
3. assess probabilities of each state of nature
4. calculate expected utilities of each action and choose the
action that maximizes expected utility
 Expected Utility Graphical Analysis

Graphical Representation of the Utility

Function
 Expected Utility Graphical Analysis

Expected Utility in a Graph

L = h, 1 j x1 , x2 i
remember: 0 1 ( is a probability)
 Expected Utility Graphical Analysis

Steps to determine graphically the expected utility

I Represent the lottery in the horizontal axis, that is, the

outcomes of the lottery, x1 and x2 , and its expected value,
E (L).
I Draw a vertical line from the two outcomes, x1 and x2 until
you hit the utility function (points A and B)
I Draw the linear segment from A to B.
I From the point E (L) in the horizontal axis draw a vertical line
The height of this line is the
until it hits the line segment AB.
expected utility.
 Expected Utility Example of a graphical analysis

Example of a graphical analysis

Initial Wealth and Investment

I an investment project can yield a prot of 40 if things go well

and a loss of 20 if things go wrong
I a long experience with projects of this type show that they fail
one out of three times
I initial wealth of the investor = 30
I utility function is
 Expected Utility Example of a graphical analysis

I compare 2 lotteries: invest (L) and not invest (H )

I remember to add the initial wealth to the outcomes of each
lottery (level of nal wealth may be important to investor)
 Expected Utility Example of a graphical analysis

Investor should _____________

 Expected Utility Example of a graphical analysis

Exercise same preferences, richer investor = 60

Attitude towards risk

Risk Aversion
 Attitude towards risk

Actuarially Equivalent Lotteries

I two lotteries are actuarially equivalent if they have the same

expected value
I given a lottery L, dene L as the degenerate lottery that
gives with certainty (probability 1) the expected value of L

E (L) = pg + (1 p )b = E (L )
risk averse/neutral/loving - based on the reaction when comparing
2 lotteries with the same expected value: a risky lottery L and an
actuarially equivalent degenerate lottery L
 Attitude towards risk

Review Parentheses: Concave and Convex Functions

 Attitude towards risk

Risk Aversion

I a decision maker is said to be risk averse if he always prefers

the certain lottery L instead of the actuarially equivalent
uncertain lottery L
I the decision maker is risk averse if and only if his utility u is
strictly concave

Proof.
 Attitude towards risk

Compare the expected utilities of L and L

U (L ) = u (pg + (1 p ) b ) >? pu (g ) + (1 b ) u (b )

Graph
 Attitude towards risk

Risk Loving

I a decision maker is said to be risk loving if he always prefers

the uncertain lottery L instead of the actuarially equivalent
certain lottery L
I the decision maker is risk loving if and only if his von
Neumann-Morgenstern utility u is strictly convex

Proof.
Homework

Graph
 Attitude towards risk

Risk Neutral

I a decision maker is said to be risk neutral if hes always

indierent between the uncertain lottery L and the actuarially
equivalent certain lottery L
I the decision maker is risk neutral if and only if his von
Neumann-Morgenstern utility u is linear

Proof.
Homework

Graph
 Attitude towards risk

Summary of Attitudes towards Risk

risk aversion , L L , u strictly concave

risk neutrality , L L , u linear
risk loving , L L , u strictly convex

If u is dierentiable, then concavity/convexity depends on the sign

of the second derivative:

u 00 (x ) < 0 , risk _____

u 00 (x ) = 0 , risk _____
u 00 (x ) > 0 , risk _____
 Attitude towards risk Certainty Equivalent

Certainty Equivalent
 Attitude towards risk Certainty Equivalent

Certainty Equivalent

The certainty equivalent of a lottery L is the amount of money

C (L) such that the agent is indierent between having C (L) and
participating in L.

u [C (L)] =p u (x ) + (1 p ) u (x 0 )
C (L) =u 1
[p u (x ) + (1 p ) u (x 0 )]
 Attitude towards risk Certainty Equivalent

Graphical Representation of the Certainty Equivalent

u [C (L)] =p u (x ) + (1 p ) u (x 0 )
C (L) =u 1
[p u (x ) + (1 p ) u (x 0 )]
 Attitude towards risk Certainty Equivalent

Certainty Equivalent and Risk Aversion

The relation between the certainty equivalent and the expected
value of a lottery characterizes the attitude towards risk

Risk averse , C (L) < E (L) for all L

Risk loving , C (L) > E (L) for all L
Risk neutral , C (L) = E (L) for all L

Proof.

p u (x ) + (1 p ) u (x 0 ) < u p x + (1 p) x 0
or, equivalently
u 1
[pu (x ) + (1 p )u (x 0 )] < px + (1 p )x 0
Applications

Applications
 Applications

Applications

1 - Buying and selling risky assets

2 - Insurance Contracts (Homework)

3 - Optimal Portfolio (Homework)

 Applications Buying and Selling Risky Assets

1. Buying and Selling Risky Assets

Setup

I A lottery ticket (a risky asset) yields 12 with 25% probability

and 0 otherwise.
p
I An individual has utility u (x ) = x and wealth = 4.

2 questions:

1. If the individual owns the risky asset, at which price hed be

willing to sell it?
2. If he doesnt own the asset, at which price hed be willing to
buy it?
 Applications Buying and Selling Risky Assets

Determining the selling price

I actions: sell or not

I lotteries:

where x is the selling price

 Applications Buying and Selling Risky Assets

Solution of the selling problem

I He will sell if U (L2 ) U (L1 )
I The expected utility of not selling is
U (L1 ) = ______________
I and of selling is
U (L2 ) = ______________
I the minimum selling x price hell accept gives him
U (L2 ) _____
I Hence, the minimum price to make him sell is the solution to

______________________
) x = ____

I The minimum price at which he is willing to sell is ____e

 Applications Buying and Selling Risky Assets

Determining the buying price

I Assume the individual does not own the risky asset and is
considering whether to buy it
I compare two lotteries: not buying, L4 , and buying at a price
y , L3 .

Compute the expected utilities

U (L3 ) = _______________________
U (L4 ) = _______________________
 Applications Buying and Selling Risky Assets

Solution to the buying problem

I The individual will be willing to buy as long as the price y is

such that U (L3 ) U (L4 )
I We obtain the maximum buying price solving the equation

__________________________
) y = ____

I The buying price is y = ____, much less then the selling

price.
 Applications Insurance Contract

2 - Insurance Contracts

I an insurance contract is a contract in which the insured pays

a relatively small and certain fee (premium) to the insurer in
exchange for the insurers promise to compensate the insured
in case of an uncertain event that causes nancial/personal
loss (accident; robbery, re, death etc.)
I the contract establishes the premium the insurer has to pay
and the compensation x paid by the insurer
I the insurance contract is a contract of transference of risk:
the insurance company accepts the insureds risk in exchange
for the premium. when the contract includes a excess
(franquicia), part of the risk remains in the hands of the
insurer
I the insurance mechanism doesnt change the risk probabilities
or consequences
 Applications Insurance Contract

Insurance: Notation

0 initial wealth
0 d loss
p probability of re
x d compensation in case of re
s price of a unit of insurance
sx insurance premium
 Applications Insurance Contract

Full and partial insurance

I We have a family of insurance contracts, Lx , one for each

level of coverage x
I If x = 0 we get the no insurance as represented by lottery L0
I If x = d we get full insurance as represented by a riskless
lottery L1 : in case of re you are fully compensated and you
always get the same payo s d.
 Applications Insurance Contract

The optimal insurance contract

I The expected utility of the lottery Ld is a function x (all other

variables are given)

U (Ld ) = f (x ) = pu ( d sx + x ) + (1 p ) u ( sx )
 Applications Insurance Contract

The optimal insurance contract

I The economic agent has to choose the level of insurance x

that maximizes expected utility

max f (x ) = p u ( d sx + x ) + (1 p ) u ( sx )
x
s.t. 0 x d

I FOC (for an interior solution 0 < x < d):

p (1 s ) u 0 ( d sx + x ) s (1 p ) u 0 ( sx ) = 0
 Applications Insurance Contract

NOTE An interior solution 0 < x < d is a partial insurance.

Often you will get a corner solution with x = 0 (no insurance) or
x = d (full insurance).
 Applications Insurance Contract

Insurance Contracts: A Numerical Example

An industrial plant is worth 1.000.000e . The probability of a re

is 1/1000. If there is a re, the plant has a residual value of
400.000e . The attitude towards risk of the owner is summarized
by a von Neumann-Morgenstern utility function
10.000.000
u (x ) = 100
x
Two questions:
1. Would the entrepreneur accept to pay 1.200e for a full
insurance contract to cover the risk of re?
2. Is that contract attractive for a risk neutral insurance
company?
 Applications Insurance Contract

Insurance contracts example: the two alternatives

I For both, the entrepreneur and the insurance rm, we have to
compare the initial situation with the situation with an
insurance contract.
I The initial situation for the plant owner is the lottery L0 , and
for the insurance company its the lottery B0 .

I If they agree on an insurance contract, the situation for the

plant owner is L1 and for the insurance company, B1
 Applications Insurance Contract

Insurance contracts example: solution

I The insurance company is risk neutral and therefore
maximizes the expected value.

E (B0 ) = 0
E (B1 ) = 600

I The expected prot is higher with the insurance contract. The

plant owner has to decide between L0 and L1 . Computing the
expected utilities we get

U (L0 ) = 0, 999 u (1.000.000) + 0, 001 u (400.000) = 89, 985

U (L1 ) = u (998.800) = 89, 988
I The expected prot is higher with the insurance contract for
both agents.
 Applications Insurance Contract

Insurance markets

I As we have seen, the possibility of mutually advantageous

contracts is clear when there are dierences in the attitude
toward risk. In the previous example the insurance company
was happy to take the risk of the plant owner in exchange for
a certain amount of money.
I Is it possible to have an insurance market in a society where
all agents are identical and risk averse?
I We shall show that the answer is yes: identical risk averse
individuals have an incentive to insure each other and share
their risk.
 Applications Insurance Contract

Example: a hunters community

I Suppose that in a small island there are two hunters with
identical utility function

30x
u (x ) =
x +4
where x represents the number of deer.
I It is easily veried that the second derivative is negative
I
d 2u 240
= <0
dx (x + 4)3
I so the utility function utility function is strictly concave,
meaning that both hunters are risk averse.
I When an individual goes hunting he gets 2 units of deer.
I From time to time (with probability 20%) a hunter gets ill and
cannot go hunting. In that case he gets nothing.
 Applications Insurance Contract

Solving the huntersinsurance problem

The lottery L0 represents the situations of a hunter without any
insurance contract. Suppose now that they decide to insure
themselves and share the risk: if one of them gets ill, they will
share what the other gets. The lottery L1 represents this situation:
there are 3 possible outcomes:
1. With probability 0.64 both are healthy and each gets 2 deer.
2. With probability 0.04 both are ill and both get 0 deer.
3. With probability 0.32 one of them is ill and the other is
healthy. In that case they consume 1 deer each.
 Applications Insurance Contract

Solution the huntersinsurance problem

Computing the expected utilities we see that it is much better for

both to insure.
30 2
U (L0 ) = 0.8 =8
6
U (LI ) = 0.32 6 + 0.64 10 = 8, 32
 Applications Optimal Portfolio

3 - Optimal portfolio

I We have saved an amount of = 100. We consider making

an investment.
I There is a riskless asset, H, that gives zero returns.
I There is a risky asset, L. If things go well (prob 35 ) we obtain
40% of prots. If things go wrong (prob 25 ) we lose 50%.
I Preferences can be represented by a von
Neumann-Morgenstern utility function u = ln x.

Should we invest in the risky asset?

 Applications Optimal Portfolio

Comparing the two assets

I Let H and L be the two lotteries that result from investing all
the money in one asset.

I The expected value of the risky asset is

E (L1 ) = 0.6 140 + 0.4 50 = 104. So the expected return
of the risky asset is 4%.
I The best asset is the one giving a higher utility:
U (H ) = ln(100) = 4.6051702
U (L) = 0.6 ln(140) + 0.4 ln(50) = 4.5297947
I It is better not to invest in the risky asset.
 Applications Optimal Portfolio

I In any case we could also decide to invest only a part of our

wealth in the risky asset. There is a whole range of possible
portfolios. If z is the proportion of our wealth that we decide
to invest in the risky asset, then let Lz denote the lottery we
are confronted with.

I The payment for each state of nature consists of two parts.

For instance, for the most favorable state of nature the rst
part equals 100(1 z ) which represents the returns of the
riskless asset, and the second part equals 100z (1 + 0.3) which
represents the returns of the risky asset.
 Applications Optimal Portfolio

Solving the optimal portfolio problem

I We obtain the optimal portfolio by solving the following

optimization problem:
3 2
maxf (z ) = U (Lz ) = u (100(1 + 0.3z )) + u (100(1 0.4z ))
5 5
s.t.0 z 1

I The rst order condition is

0.24 0.2
f 0 (z ) = =0
0.4z + 1 1 0.5z
I and solving the system we get z = 0.2, that is, we should
invest 20% of our wealth in the risky asset.
 Applications Optimal Portfolio

Solving the optimal portfolio problem

Comparing the three possibilities:

spezializing in the riskless asset U (H ) = 4.6051702

spezializing in the risky asset U (L) = 4.5297947
optimal portfolio: diversication U (L0.2 ) = 4.6092026

we see that the optimal diversication provides higher utility than

specialization.