Microeconomics I
c
Leopold S¨ogner
Department of Economics and Finance
Institute for Advanced Studies
Stumpergasse 56
1060 Wien
Tel: +43-1-59991 182
soegner@ihs.ac.at
http://www.ihs.ac.at/∼soegner
March, 2014
Course Outline (1)
Applied Micro
Learning Objectives:
• This course covers key concepts of microeconomic theory. The
main goal of this course is to provide students with both, a basic
understanding and analytical traceability of these concepts.
• The main concepts are discussed in detail during the lectures. In
addition students have to work through the textbooks and have
to solve problems to improve their understanding and to acquire
skills to apply these tools to related problems.
1
Course Outline (2)
Applied Micro
Literature:
• Andreu Mas-Colell, A., Whinston, M.D., Green, J.R.,
Microeconomic Theory, Oxford University Press, 1995.
Supplementary Literature:
• Gilboa, I., Theroy of Decision under Uncertainty, Cambridge University Press,
2009.
• Gollier C., The Economics of Risk and Time, Mit Press, 2004.
• Jehle G.A. and P. J. Reny, Advanced Microeconomic Theory, Addison-Wesley
Series in Economics, Longman, Amsterdam, 2000.
• Ritzberger, K., Foundations of Non-Cooperative Game Theory, Oxford
University Press, 2002.
2
Course Outline (3)
Applied Micro
• Decision Theory, Decisions under Uncertainty: von
Neumann-Morgenstern expected utility theory, risk-aversion,
stochastic dominance, states and state dependent utility.
Mas-Colell. Chapter 6; Ritzberger Chapter 2.
• Partial Equilibrium Theory: Pareto optimality and competitive
equilibrium, welfare theorems, the competitive model, (bilaterial
externality, public goods, second best, monopoly, Cournot model,
Betrand model, competitive limit.) Mas-Colell, Chapter 10 A-D,
F, ;11 A,B,C,E; 12 A,B,C,F.
3
Course Outline (4)
Applied Micro
• Summer Term 2014 in Brno
– First block: March 13-14: 11:00-12:30 and 14:00-15:30 on
Thursday, 10:00-11:30 and 13:00-14:30 on Friday.
– Second block: April, 10-11: 11:00-12:30 and 14:00-15:30 on
Thursday, 10:00-11:30 and 13:00-14:30 on Friday.
– Third block: May, 15-16: 11:00-12:30 and 14:00-15:30 on
Thursday, 10:00-11:30 and 13:00-14:30 on Friday.
• Practice session will be organized by Rostislav Stanek.
4
Course Outline (5)
Applied Micro
Some more comments on homework and grading:
• Final test (80%), homework and class-room participation (20%).
• Final test: tba
• Reset Test: tba
5
Expected Utility
Uncertainty (1)
Applied Micro
• Preferences and Lotteries.
• Von Neumann-Morgenstern Expected Utility Theorem.
• Attitudes towards risk.
• State Dependent Utility, Subjective Utility
MasColell Chapter 6.
6
Expected Utility
Lotteries (1)
Applied Micro
• A risky alternative results in one of a number of different events
or states of the world, ωi.
• The events are associated with consequences or outcomes, zn.
Each zn involves no uncertainty.
• Outcomes can be money prices, wealth levels, consumption
bundles, etc.
• Assume that the set of outcomes is finite. Then
Z = {z1, . . . , zN}.
• E.g. flip a coin: Events {H, T} and outcomes Z = {−1, 1}, with
head H or tail T.
7
Expected Utility
Lotteries (2)
Applied Micro
• Definition - Simple Gamble/Simple Lottery: [D 6.B.1] With
the consequences {z1, . . . , zN} ⊆ Z and N finite. A simple
gamble assigns a probability pn to each outcome zn. pn ≥ 0 and
N
n=1 pn = 1.
• Notation: L = (p1 ◦ z1, . . . , pN ◦ zN) or L = (p1, . . . , pN)
• Let us fix the set of outcomes Z: Different lotteries correspond
to a different set of probabilities.
• Definition - Set of Simple Gambles: The set of simple
gambles on Z is given by
LS = {(p1◦z1, . . . , pN◦zN)|pn ≥ 0 ,
N
n=1
pn = 1} = {L|pn ≥ 0 ,
N
N=1
pn = 1}
8
Expected Utility
Lotteries (3)
Applied Micro
• Definition - Degenerated Lottery:
˜Ln
= (0 ◦ z1, . . . , 1 ◦ zn, . . . , 0 ◦ zN).
• ’Z ⊆ LS’, since ˜Ln
= (0 ◦ z1, . . . , 1 ◦ zn, . . . , 0 ◦ zN) for all i;
• If z1 is the smallest element and zn the largest one, then also
(α ◦ z1, 0 ◦ z2, . . . , 0 ◦ zN−1, (1 − α) ◦ zN) ∈ LS.
• Remark: In terms of probability theory, the elements of Z where
p > 0 provide the support of the distribution of a random
variable z. I.e. a lottery L is a probability distribution.
9
Expected Utility
Lotteries (4)
Applied Micro
• With N consequences, every simple lottery can be represented by
a point in a N − 1 dimensional simplex
∆(N−1)
= {p ∈ RN
+ | pn = 1} .
• At each corner n we have the degenerated case that pn = 1.
• With interior points pn > 0 for all i.
• See Ritzberger, p. 36,37, Figures 2.1 and 2.2 or Figure 6.B.1,
page 169.
• Equivalent to Machina’s triangle; with N = 3;
{(p1, p3) ∈ [0, 1]2
|0 ≤ 1 − p1 − p3 ≤ 1}.
10
Expected Utility
Lotteries (5)
Applied Micro
• The consequences of a lottery need not be a z ∈ Z but can also
be further lottery.
• Definition - Compound Lottery:[D 6.B.2] Given K simple
lotteries Lk and probabilities αk ≥ 0 and αk = 1, the
compound lottery LC = (α1 ◦ L1, . . . , αk ◦ Lk, . . . , αK ◦ LK). It
is the risky alternative that yields the simple lottery Lk with
probability αk.
• The support of the compound lottery is the union of the supports
generating this lotteries.
11
Expected Utility
Lotteries (6)
Applied Micro
• Definition - Reduced Lottery: For any compound lottery LC
we can construct a reduced lottery/simple gamble L ∈ LS.
With the probabilities pk
for each Lk
we get p = αkpk
, such
that probabilities for each zn ∈ Z are pn = αkpk
n.
• Examples: Example 2.5, Ritzberger p. 37
• A reduced lottery can be expressed by a convex combination of
elements of compound lotteries (see Ritzberger, Figure 2.3, page
38). I.e. αpl1
+ (1 − α)pl2
= plreduced
.
• Remark: This linear structure carries over to von
Neumann-Morgenstern decision theory.
12
Expected Utility
von Neumann-Morgenstern Utility (1)
Applied Micro
• Here we assume that any decision problem can be expressed by
means of a lottery (simple gamble).
• Only the outcomes matter.
• Consumers are able to perform calculations like in probability
theory, gambles with the same probability distribution on Z are
equivalent.
13
Expected Utility
von Neumann-Morgenstern Utility (2)
Applied Micro
• Axiom vNM1 - Completeness: For two gambles L1 and L2 in
LS either L1 L2, L2 L1 or both.
• Here we assume that a consumer is able to rank also risky
alternatives. I.e. Axiom vNM1 is stronger than Axiom 1 under
certainty.
• Axiom vNM2 - Transitivity: For three gambles L1, L2 and L3:
L1 L2 and L2 L3 implies L1 L3.
14
Expected Utility
von Neumann-Morgenstern Utility (3)
Applied Micro
• Axiom vNM3 - Continuity: [D 6.B.3] The preference relation
on the space of simple lotteries is continuous if for any L1, L2, L3
the sets {α ∈ [0, 1]|αL1 + (1 − α)L2 L3} ⊂ [0, 1] and
{α ∈ [0, 1]|L3 αL1 + (1 − α)L2} ⊂ [0, 1] are closed.
• Later we show: for any gambles L ∈ LS, there exists some
probability α such that L ∼ α¯L + (1 − α)L .
• This assumption rules out a lexicographical ordering of
preferences (safety first preferences).
• Small changes in the probabilities do not change the ordering of
the lotteries.
15
Expected Utility
von Neumann-Morgenstern Utility (4)
Applied Micro
• Consider the outcomes Z = {1000, 10, death}, where
1000 10 death. L1 gives 10 with certainty.
• If vNM3 holds then L1 can be expressed by means of a linear
combination of 1000 and death. If there is no α ∈ [0, 1] fulfilling
this requirement vNM3 does not hold.
• vNM3 will rule out Bernoulli utility levels of ±∞.
16
Expected Utility
von Neumann-Morgenstern Utility (5)
Applied Micro
• Axiom - Monotonicity: For all probabilities α, β ∈ [0, 1],
α¯L + (1 − α)L β ¯L + (1 − β)L
if and only if α ≥ β.
• Counterexample where this assumption is not met: Safari hunter
who prefers an alternative with the bad outcome.
17
Expected Utility
von Neumann-Morgenstern Utility (6)
Applied Micro
• Axiom vNM4 - Independence, Substitution: For all
probabilities L1, L2 and L3 in LS and α ∈ [0, 1]:
L1 L2 ⇔ αL1 + (1 − α)L3 αL2 + (1 − α)L3 .
• This axiom implies that the preference orderings of the mixtures
are independent of the third lottery.
• This axiom has no parallel in consumer theory under certainty.
18
Expected Utility
von Neumann-Morgenstern Utility (7)
Applied Micro
• Example: consider a bundle x1
consisting of 1 cake and 1 bottle
of wine, x2
= (3, 0); x3
= (3, 3). Assume that x1
x2
.
Axiom vNM4 requires that αx1
+ (1 − α)x3
αx2
+ (1 − α)x3
;
here α > 0.
19
Expected Utility
von Neumann-Morgenstern Utility (8)
Applied Micro
• Lemma - vNM1-4 imply monotonicity: Moreover, if L1 L2
then αL1 + (1 − α)L2 βL1 + (1 − β)L2 for arbitrary
α, β ∈ [0, 1] where α ≥ β. There is unique γ such that
γL1 + (1 − γ)L2 ∼ L.
• See steps 2-3 of the vNM existence proof.
20
Expected Utility
von Neumann-Morgenstern Utility (9)
Applied Micro
• Definition - von Neumann Morgenstern Expected Utility
Function: [D 6.B.5] A real valued function U : LS → R has
expected utility form if there is an assignment of numbers
(u1, . . . , uN) (with un = u(zn)) such that for every lottery
L ∈ LS we have U(L) = zn∈Z p(zn)u(zn). A function of this
structure is said to satisfy the expected utility property- it is
called von Neumann-Morgenstern (expected) utility function.
• Note that this function is linear in the probabilities pn.
• u(zn) is called Bernoulli utility function.
21
Expected Utility
von Neumann-Morgenstern Utility (10)
Applied Micro
• Proposition - Linearity of the von Neumann Morgenstern
Expect Utility Function: [P 6.B.1] A utility function has
expected utility form if and only if it is linear. That is to say:
U
K
k=1
αkLk =
K
k=1
αkU(Lk)
22
Expected Utility
von Neumann-Morgenstern Utility (11)
Applied Micro
Proof:
• Suppose that U(
K
k=1 αkLk) =
K
k=1 αkU(Lk) holds. We have
to show that U has expected utility form, i.e. if
U( k αkLk) = k αkU(Lk) then U(L) = pnu(zn).
• If U is linear then we can express any lottery L by means of a
compound lottery with probabilities αn = pn and degenerated
lotteries ˜Ln
. I.e. L = pn
˜Ln
. By linearity we get
U(L) = U( pn
˜Ln
) = pnU(˜Ln
).
• Define u(zn) = U(˜Ln
). Then
U(L) = U( pn
˜Ln
) = pnU(˜Ln
) = pnu(zn). Therefore
U(.) has expected utility form.
23
Expected Utility
von Neumann-Morgenstern Utility (12)
Applied Micro
Proof:
• Suppose that U(L) =
N
n=1 pnu(zn) holds. We have to show
that utility is linear, i.e. if U(L) = pnu(zn) then
U( k αkLk) = k αkU(Lk)
• Consider a compound lottery (L1, . . . , LK, α1, . . . , αK). Its
reduced lottery is L = k αkLk.
• Then U( k αkLk) = n k αkpk
n u(zn) =
k αk n pk
nu(zn) = k αkU(Lk).
24
Expected Utility
von Neumann-Morgenstern Utility (13)
Applied Micro
• Proposition - Existence of a von Neumann Morgenstern
Expect Utility Function: [P 6.B.3] If the Axioms vNM 1-4 are
satisfied for a preference ordering on LS. Then admits an
expected utility representation. I.e. there exists a real valued
function u(.) on Z which assigns a real number to each outcome.
For any pair of lotteries we get
L1 L2 ⇔ U(L1) =
N
n=1
pl1(zn)u(zn) ≥ U(L2) =
N
n=1
pl2(zn)u(zn) .
25
Expected Utility
von Neumann-Morgenstern Utility (14)
Applied Micro
Proof:
• Suppose that there is a best and a worst lottery. With a finite set
of outcomes this can be easily shown by means of the
independence axiom. In addition ¯L L.
• By the definition of ¯L and L we get: ¯L Lc L, ¯L L1 L
and ¯L L2 L.
• We have to show that (i) u(zn) exists and (ii) that for any
compound lottery Lc = βL1 + (1 − β)L2 we have
U(βL1 + (1 − β)L2) = βU(L1) + (1 − β)U(L2) (expected utility
structure).
26
Expected Utility
von Neumann-Morgenstern Utility (15)
Applied Micro
Proof:
• Step 1: By the independence Axiom vNM4 we get if L1 L2
and α ∈ (0, 1) then L1 αL1 + (1 − α)L2 L2.
• This follows directly from the independence axiom.
L1 ∼ αL1+(1−α)L1 αL1+(1−α)L2 αL2+(1−α)L2 = L2
27
Expected Utility
von Neumann-Morgenstern Utility (16)
Applied Micro
Proof:
• Step 2: Assume β > α , then (by monotonicity)
β ¯L + (1 − β)L α¯L + (1 − α)L and vice versa.
• Define γ = (β − α)/(1 − α); the assumptions imply γ ∈ [0, 1].
28
Expected Utility
von Neumann-Morgenstern Utility (17)
Applied Micro
Proof:
• Then
β ¯L + (1 − β)L = γ ¯L + (1 − γ)(α¯L + (1 − α)L)
γ(α¯L + (1 − α)L) + (1 − γ)(α¯L + (1 − α)L)
∼ α¯L + (1 − α)L
29
Expected Utility
von Neumann-Morgenstern Utility (18)
Applied Micro
Proof:
• Step 2: For the converse we have to show that
β ¯L + (1 − β)L α¯L + (1 − α)L results in β > α. We show this
by means of the contrapositive: If β > α then
β ¯L + (1 − β)L α¯L + (1 − α)L.
• Thus assume β ≤ α, then α¯L + (1 − α)L β ¯L + (1 − β)L
follows in the same way as above. If α = β indifference follows.
30
Expected Utility
von Neumann-Morgenstern Utility (19)
Applied Micro
Proof:
• Step 3: There is a unique αL such that L ∼ αL
¯L + (1 − αL)L.
• Existence follows from ¯L L and the continuity axiom.
Uniqueness follows from step 2.
• Ad existence: define the sets {α ∈ [0, 1]|α¯L + (1 − α)L L}
and {α ∈ [0, 1]|L α¯L + (1 − α)L}. Both sets are closed. Any α
belongs to at least one of these two sets. Both sets are nonempty.
Their complements are open and disjoint. The set [0, 1] is
connected ⇒ there is at least one α belonging to both sets.
31
Expected Utility
Connected Sets
Applied Micro
• Definition: Let X be a topological space. A separation of X is
a pair U, V of disjoint nonempty open subsets of X whose union
is X. The space is said to be connected, if there does not exist
a separation of X. (see e.g. Munkres, J. Topology, page 148)
• Example: The rationals are not connected.
• Example: [−1, 1] is connected, [−1, 0] and (0, 1] are disjoint and
cover X. The first set is not open. Alternatively, if
X = [−1, 0) ∪ (0, 1] we would get a separation.
32
Expected Utility
von Neumann-Morgenstern Utility (20)
Applied Micro
Proof:
• Step 4: The function U(L) = αL represents the preference
relations .
• Consider L1, L2 ∈ LS: If L1 L2 then α1 ≥ α2. If α1 ≥ α2 then
L1 L2 by steps 2-3.
• It remains to show that this utility function has expected utility
form.
33
Expected Utility
von Neumann-Morgenstern Utility (21)
Applied Micro
Proof:
• Step 5: U(L) is has expected utility form.
• We show that the linear structure also holds for the compound
lottery Lc = βL1 + (1 − β)L2.
• By using the independence we get:
βL1 + (1 − β)L2 ∼ β(α1
¯L + (1 − α1)L) + (1 − β)L2
∼ β(α1
¯L + (1 − α1)L) + (1 − β)(α2
¯L + (1 − α2)L)
∼ (βα1 + (1 − β)α2)¯L + (β(1 − α1) + (1 − β)(1 − α2))L
• By the rule developed in step 4, this shows that
U(Lc) = U(βL1 + (1 − β)L2) = βU(L1) + (1 − β)U(L2).
34
Expected Utility
von Neumann-Morgenstern Utility (22)
Applied Micro
• Proposition - von Neumann Morgenstern Expect Utility
Function are unique up to Positive Affine Transformations:
[P 6.B.2] If U(.) represents the preference ordering , then V
represents the same preference ordering if and only if
V = α + βU, where β > 0.
35
Expected Utility
von Neumann-Morgenstern Utility (23)
Applied Micro
Proof:
• Note that if V (L) = α + βU(L), V (L) fulfills the expected
utility property (see also MWG p. 174).
• We have to show that if U and V represent preferences, then V
has to be an affine linear transformation of U.
• If U is constant on LS, then V has to be constant. Both
functions can only differ by a constant α.
36
Expected Utility
von Neumann-Morgenstern Utility (24)
Applied Micro
Proof:
• Alternatively, for any L ∈ LS and ¯L L, we get
f1 :=
U(L) − U(L)
U(¯L) − U(L)
and
f2 :=
V (L) − V (L)
V (¯L) − V (L)
.
• f1 and f2 are linear transformations of U and V that satisfy the
expected utility property.
• fi(L) = 0 and fi(¯L) = 1, for i = 1, 2.
37
Expected Utility
von Neumann-Morgenstern Utility (25)
Applied Micro
Proof:
• L ∼ L then f1 = f2 = 0; if L ∼ ¯L then f1 = f2 = 1.
• By expected utility U(L) = γU(¯L) + (1 − γ)U(L) and
V (L) = γV (¯L) + (1 − γ)V (L).
• If ¯L L L then there has to exist a unique γ, such that
L L ∼ γ ¯L + (1 − γ)L ¯L. Therefore
γ =
U(L) − U(L)
U(¯L) − U(L)
=
V (L) − V (L)
V (¯L) − V (L)
38
Expected Utility
von Neumann-Morgenstern Utility (26)
Applied Micro
Proof:
• Then V (L) = α + βU(L) where
α = V (L) − U(L)
V (¯L) − V (L)
U(¯L) − U(L)
and
β =
V (¯L) − V (L)
U(¯L) − U(L)
.
39
Expected Utility
von Neumann-Morgenstern Utility (27)
Applied Micro
• The idea of expected utility can be extended to a set of
distributions F(x) where the expectation of u(x) exists, i.e.
u(x)dF(x) < ∞.
• For technical details see e.g. Robert (1994), The Bayesian Choice
and DeGroot, Optimal Statistical Decisions.
• Note that expected utility is a probability weighted combination
of Bernoulli utility functions. I.e. the properties of the random
variable z, described by the lottery l(z), are separated from the
attitudes towards risk.
40
Expected Utility
VNM Indifference Curves (1)
Applied Micro
• Indifferences curves are straight lines; see Ritzberger, Figure 2.4,
page 41.
• Consider a VNM utility function and two indifferent lotteries L1
and L2. It has to hold that U(L1) = U(L2).
• By the expected utility theorem
U(αL1 + (1 − α)L2) = αU(L1) + (1 − α)U(L2).
• If U(L1) = U(L2) then U(αL1 + (1 − α)L2) = U(L1) = U(L2)
has to hold and the indifferent lotteries is linear combinations of
L1 and L2.
41
Expected Utility
VNM Indifference Curves (2)
Applied Micro
• Indifference curves are parallel; see Ritzberger, Figure 2.5, 2.6,
page 42.
• Consider L1 ∼ L2 and a further lottery L3 L1 (w.l.g.).
• From βL1 + (1 − β)L3 and βL2 + (1 − β)L3 we have received
two compound lotteries.
• By construction these lotteries are on a line parallel to the line
connecting L1 and L2.
42
Expected Utility
VNM Indifference Curves (3)
Applied Micro
• The independence axiom vNM4 implies that
βL1 + (1 − β)L3 ∼ βL2 + (1 − β)L3 for β ∈ [0, 1].
• Therefore the line connecting the points βL1 + (1 − β)L3 and
βL2 + (1 − β)L3 is an indifference curve.
• The new indifference curve is a parallel shift of the old curve; by
the linear structure of the expected utility function no other
indifference curves are possible.
43
Expected Utility
Allais Paradoxon (1)
Applied Micro
Lottery 0 1-10 11-99
pz 1/100 10/100 89/100
La 50 50 50
Lb 0 250 50
Ma 50 50 0
Mb 0 250 0
44
Expected Utility
Allais Paradoxon (2)
Applied Micro
• Most people prefer La to Lb and Mb to Ma.
• This is a contradiction to the independence axiom G5.
• Allais paradoxon in the Machina triangle, Gollier, Figure 1.2,
page 8.
45
Expected Utility
Allais Paradoxon (3)
Applied Micro
• Expected utility theory avoids problems of time inconsistency.
• Agents violating the independence axiom are subject to Dutch
book outcomes (violate no money pump assumption).
46
Expected Utility
Allais Paradoxon (4)
Applied Micro
• Three lotteries: La Lb and La Lc.
• But Ld = 0.5Lb + 0.5Lc La.
• Gambler is willing to pay some fee to replace La by Ld.
47
Expected Utility
Allais Paradoxon (5)
Applied Micro
• After nature moves: Lb or Lc with Ld.
• Now the agents is once again willing to pay a positive amount for
receiving La
• Gambler starting with La and holding at the end La has paid two
fees!
• Dynamically inconsistent/Time inconsistent.
• Dicuss Figure 1.3, Gollier, page 12.
48
Expected Utility
Risk Attitude (1)
Applied Micro
• For the proof of the VNM-utility function we did not place any
assumptions on the Bernoulli utility function u(z).
• For applications often a Bernoulli utility function has to be
specified.
• In the following we consider z ∈ RN
and u (z) > 0; abbreviate
lotteries with money amounts l ∈ LS.
• There are interesting interdependences between the Bernoulli
utility function and an agent’s attitude towards risk.
49
Expected Utility
Risk Attitude (2)
Applied Micro
• Consider a nondegenerated lottery l ∈ LS and a degenerated
lottery ˜l. Assume that E(z) = z˜l holds. I.e. the degenerated
lottery pays the expectation of l for sure.
• Definition - Risk Aversion: A consumer is risk averse if ˜l is at
least of good as l; ˜l is preferred to l in a stronger version.
• Definition - Risk Neutrality: A consumer is risk neutral if ˜l ∼ l.
• Definition - Risk Loving: A consumer is risk loving if l is at
least as good as ˜l.
50
Expected Utility
Risk Attitude (3)
Applied Micro
• By the definition of risk aversion we see that u(E(z)) ≥ E(u(z)).
• To attain such a relationship Jensen’s inequality has to hold: If
f(z) is a concave function and z ∼ F(z) then
f(z)dF(z) ≤ f( zdF(z)) .
• For sums this implies:
pzf(z) ≤ f( pzz) .
For strictly concave function, < has to hold, for convex functions
we get ≥; for strictly convex functions >. 51
Expected Utility
Risk Attitude (4)
Applied Micro
• For a lottery l where E(u(z)) < ∞ and E(z) < ∞ we can
calculate the amount C where a consumer is indifferent between
receiving C for sure and the lottery l. I.e. l ∼ C and
E(u(z)) = u(C) hold.
• In addition we are able to calculate the maximum amount π an
agent is willing to pay for receiving the fixed amount E(z) for
sure instead of the lottery l. I.e. l ∼ E(z) − π or
E(u(z)) = u(E(z) − π).
52
Expected Utility
Risk Attitude (5)
Applied Micro
• Definition - Certainty Equivalent [D 6.C.2]: The fixed amount
C where a consumer is indifferent between C an a gamble l is
called certainty equivalent.
• Definition - Risk Premium: The maximum amount π a
consumer is willing to pay to exchange the gamble l for a sure
event with outcome E(z) is called risk premium.
• Note that C and π depend on the properties of the random
variable (described by l) and the attitude towards risk (described
by u).
53
Expected Utility
Risk Attitude (6)
Applied Micro
• Remark: the same analysis can also be performed with risk
neutral and risk loving agents.
• Remark: MWG defines a probability premium, which is
abbreviated by π in the textbook. Given a degenerated lottery
and some ε > 0. The probability-premium πR
is defined as
u(˜lz) = (1
2 + πR
)u(z + ε) + (1
2 − πR
)u(z − ε). I.e.
mean-preserving spreads are considered here.
54
Expected Utility
Risk Attitude (7)
Applied Micro
• Proposition - Risk Aversion and Bernoulli Utility: Consider
an expected utility maximizer with Bernoulli utility function u(.).
The following statements are equivalent:
– The agent is risk averse.
– u(.) is a (strictly) concave function.
– C ≤ E(z). (< with strict version)
– π ≥ 0. (> with strict version)
55
Expected Utility
Risk Attitude (8)
Applied Micro
Proof: (sketch)
• By the definition of risk aversion: for a lottery l where E(z) = z˜l,
a risk avers agent ˜l l.
• I.e. E(u(z)) ≤ u(z˜l) = u(E(z)) for a VNM utility maximizer.
• (ii) follows from Jensen’s inequality.
• (iii) If u(.) is (strictly) concave then E(u(z)) = u(C) ≤ u(E(z))
can only be matched with C ≤ E(z).
• (iv) With a strictly concave u(.),
E(u(z)) = u(E(z) − π) ≤ u(E(z)) can only be matched with
π ≥ 0.
56
Expected Utility
Arrow Pratt Coefficients (1)
Applied Micro
• Using simply the second derivative u (z) causes problems with
affine linear transformations.
• Definition - Arrow-Pratt Coefficient of Absolute Risk
Aversion: [D 6.C.3] Given a twice differentiable Bernoulli utility
function u(.), the coefficient of absolute risk aversion is defined
by A(z) = −u (z)/u (z).
• Definition - Arrow-Pratt Coefficient of Relative Risk
Aversion: [D 6.C.5] Given a twice differentiable Bernoulli utility
function u(.), the coefficient of relative risk aversion is defined by
R(z) = −zu (z)/u (z).
57
Expected Utility
Comparative Analysis (1)
Applied Micro
• Consider two agents with Bernoulli utility functions u1 and u2.
We want to compare their attitudes towards risk.
• Definition - More Risk Averse: Agent 1 is more risk averse
than agent 2: Whenever agent 1 finds a lottery F at least good
as a riskless outcome ˜x, then agent 2 finds F at least good as ˜x.
I.e. if F 1
˜L˜x then F 2
˜L˜x.
In terms of a VNM-ultility maximizer: If
EF (u1(z)) = u1(z)dF(z) ≥ u1(˜x) then
EF (u2(z)) = u2(z)dF(z) ≥ u2(˜x) for any F(.) and ˜x.
58
Expected Utility
Comparative Analysis (2)
Applied Micro
• Define a function φ(x) = u1(u−1
2 (x)). Since u2(.) is an
increasing function this expression is well defined. We, in
addition, assume that the first and the second derivatives exist.
• By construction with x = u2(z) we get:
φ(x) = u1(u−1
2 (x)) = u1(u−1
2 (u2(z))) = u1(z). I.e. φ(x)
transforms u2 into u1, such that u1(z) = φ(u2(z)).
• In the following we assume that ui and φ are differentiable. In the
following theorem we shall observe that φ > 0 for u1 and u2 > 0.
59
Expected Utility
Comparative Analysis (3)
Applied Micro
• Proposition - More Risk Averse Agents [P 6.C.3]: Assume
that the first and second derivatives of the Bernoulli utility
functions u1 and u2 exist (u > 0 and u < 0). Then the
following statements are equivalent:
– Agent 1 is (strictly) more risk averse than agent 2.
– u1 is a (strictly) concave transformation of u2.
– A1(z) ≥ A2(z) (> for strict) for all z.
– C1 ≤ C2 and π1 ≥ π2; (<> for strict).
60
Expected Utility
Comparative Analysis (4)
Applied Micro
Proof:
• Step 1: (i) follows from (ii): We have to show that if φ is concave, then if
EF (u1(z)) = u1(z)dF (z) ≥ u1(˜x) ⇒
EF (u2(z)) = u2(z)dF (z) ≥ u2(˜x) has to follow.
• Suppose that for some lottery F the inequality
EF (u1(z)) = u1(z)dF (z) ≥ u1(˜x) holds. This implies
EF (u1(z)) = u1(z)dF (z) ≥ u1(˜x) = φ(u2(˜x)).
• By means of Jensen’s inequality we get for a concave φ(.); (with strict
concave we get <) E(u1(z)) = E(φ(u2(z)) ≤ φ(E(u2(z))).
• Then φ(E(u2(z))) ≥ E(u1(z)) and E(u1(z)) ≥ u1(˜x) = φ(u2(˜x))
implies φ(E(u2(z))) ≥ φ(u2(˜x)).
• Since φ is increasing this implies E(u2(z)) ≥ u2(˜x).
61
Expected Utility
Comparative Analysis (5)
Applied Micro
Proof:
• (ii) follows from (i): Suppose that
EF (u1(z)) = u1(z)dF(z) ≥ u1(˜x) ⇒
EF (u2(z)) = u2(z)dF(z) ≥ u2(˜x) for any F(.) and ˜x holds
and φ is not concave.
• Then EF (u1(z)) = u1(CF 1) has to hold as well with ˜x = CF 1.
This implies EF (u1(z)) = EF (φ(u2(z))) = φ(u2(CF 1)) for
lottery F.
• Since φ is not concave, there exits a lottery where
φ(EF (u2(z))) < EF (φ(u2(z))) = φ(u2(CF 1)). This yields
EF (u2(z)) < u2(CF 1). Contradiction!
62
Expected Utility
Comparative Analysis (6)
Applied Micro
Proof:
• Step 2 (iii)∼ (ii): By the definition of φ and our assumptions we
get
u1(z) =
dφ((u2(z)))
dz
= φ (u2(z))u2(z) .
(since u1, u2 > 0 ⇒ φ > 0) and
u1(z) = φ (u2(z))u2(z) + φ (u2(z))(u2(z))2
.
63
Expected Utility
Comparative Analysis (7)
Applied Micro
Proof:
• Divide both sides by −u1(z) < 0 and using u1(z) = ... yields:
−
u1(z)
u1(z)
= A1(z) = A2(z) −
φ (u2(z))
φ (u2(z))
u2(z) .
• Since A1, A2 > 0 due to risk aversion, φ > 0 and φ ≤ 0 (<)
due to its concave shape we get A1(z) ≥ A2(z) (>) for all z.
64
Expected Utility
Comparative Analysis (8)
Applied Micro
Proof:
• Step 3 (iv)∼ (ii): Jensen’s inequality yields (with strictly concave
φ)
u1(C1) = E(u1(z)) = E(φ(u2(z)) < φ(E(u2(z))) = φ(u2(C2)) = u1(C2)
• Since u1 > 0 we get C1 < C2.
• π1 > π2 works in the same way.
• The above considerations also work in both directions, therefore
(ii) and (iv) are equivalent.
65
Expected Utility
Comparative Analysis (9)
Applied Micro
Proof:
• Step 4 (vi)∼ (ii): Jensen’s inequality yields (with strictly concave
φ)
u1(E(z)−π1) = E(u1(z)) = E(φ(u2(z)) < φ(E(u2(z))) = φ(u2(E(z)−π2)) = u1(E(z)−π2)
• Since u1 > 0 we get π1 > π2.
66
Expected Utility
Stochastic Dominance (1)
Applied Micro
• In an application, do we have to specify the Bernoulli utility
function?
• Are there some lotteries (distributions) such that F(z) is
(strictly) preferred to G(z)?
• E.g. if X(ω) > Y (ω) a.s.?
• YES ⇒ Concept of stochastic dominance.
• Mascollel, Figure 6.D.1., page 196.
67
Expected Utility
Stochastic Dominance (2)
Applied Micro
• Definition - First Order Stochastic Dominance: [D 6.D.1] A
distribution F(z) first order dominates the distribution G(z) if
for every nondecreasing function u : R → R we have
∞
−∞
u(z)dF(z) ≥
∞
−∞
u(z)dG(z).
• Definition - Second Order Stochastic Dominance: [D 6.D.2]
A distribution F(z) second order dominates the distribution G(z)
if EF (z) = EG(z) and for every nondecreasing concave function
u : R+ → R the inequality
∞
0
u(z)dF(z) ≥
∞
0
u(z)dG(z)
holds.
68
Expected Utility
Stochastic Dominance (3)
Applied Micro
• Proposition - First Order Stochastic Dominance: [P 6.D.1]
F(z) first order dominates the distribution G(z) if and only if
F(z) ≤ G(z).
• Proposition - Second Order Stochastic Dominance: [D
6.D.2] F(z) second order dominates the distribution G(z) if and
only if
¯z
0
F(z)dz ≤
¯z
0
G(z)dz for all ¯z in R+
.
• Remark: I.e. if we can show stochastic dominance we do not
have to specify any Bernoulli utility function!
69
Expected Utility
Stochastic Dominance (4)
Applied Micro
Proof:
• Assume that u is differentiable and u ≥ 0
• Step 1: First order, if part: If F(z) ≤ G(z) integration by parts
yields:
∞
−∞
u(z)dF (z) −
∞
−∞
u(z)dG(z)
= u(z)(F (z) − G(z))|
∞
−∞ −
∞
−∞
u (z)(F (z) − G(z))dz
= −
∞
−∞
u (z)(F (z) − G(z))dz ≥ 0 .
• The above inequality holds since the terms inside the integral
(F(z) − G(z)) ≤ 0 a.s..
70
Expected Utility
Stochastic Dominance (5)
Applied Micro
Proof:
• Step 2: First order, only if part: If FOSD then F(z) ≤ G(z)
holds. Proof by means of contradiction.
• Assume there is a ¯z such that F(¯z) > G(¯z). ¯z > −∞ by
construction. Set u(z) = 0 for z ≤ ¯z and u(z) = 1 for z > ¯z.
Here we get
∞
−∞
u(z)dF(z) −
∞
−∞
u(z)dG(z)
= (1 − F(¯z)) − (1 − G(¯z)) = −F(¯z) + G(¯z) < 0
71
Expected Utility
Stochastic Dominance (6)
Applied Micro
Proof:
• Second Order SD: Assume that u is twice continuously
differentiable, such that u (z) ≤ 0, w.l.g. u(0) = 0.
• Remark: The equality of means implies:
0 =
∞
0
zdF(z) −
∞
0
zdG(z)
= z(F(z) − G(z))|∞
0 −
∞
0
(F(z) − G(z))dz
= −
∞
0
(F(z) − G(z))dz .
72
Expected Utility
Stochastic Dominance (7)
Applied Micro
Proof:
• Step 3: Second order, if part: Integration by parts yields:
∞
0
u(z)dF (z) −
∞
0
u(z)dG(z)
= u(z)(F (z) − G(z))|
∞
0 −
∞
0
u (z)(F (z) − G(z))dz
= −
∞
0
u (z)(F (z) − G(z))dz
= −u (z)
z
0
(F (x) − G(x))dx|
∞
0 −
∞
0
−u (z)
z
0
(F (x) − G(x))dx dz
=
∞
0
u (z)
z
0
(F (x) − G(x))dx dz ≥ 0
• Note that u ≤ 0 by assumption.
73
Expected Utility
Stochastic Dominance (8)
Applied Micro
Proof:
• Step 4: Second order, only if part: Consider a ¯z such that
u(z) = ¯z for all z > ¯z and u(z) = z for all z ≤ ¯z. This yields:
∞
0
u(z)dF(z) −
∞
0
u(z)dG(z)
=
¯z
0
zdF(z) −
¯z
0
zdG(z) + ¯z ((1 − F(¯z)) − (1 − G(¯z)))
= z (F(z) − G(z)) |¯z
0 −
¯z
0
(F(z) − G(z)) dz − ¯z (F(¯z) − G(¯z))
= −
¯z
0
(F(z) − G(z)) dz < 0 .
74
Expected Utility
Stochastic Dominance (9)
Applied Micro
• Definiton - Monotone Likelihood Ratio Property: The
distributions F(z) and G(z) fulfill, the monotone likelihood rate
property if G(z)/F(z) is non-increasing in z.
• For x → ∞ G(z)/F(z) = 1 has to hold. This and the fact that
G(z)/F(z) is non-increasing implies G(z)/F(z) ≥ 1 for all z.
• Proposition - First Order Stochastic Dominance follows
from MLP: MLP results in F(z) ≤ G(z).
• Remark: If F(z) and G(z) have Lebesgue-densities f(z) and
g(z), then F(z) ≤ G(z) if the ratio of the densities g(z)/f(z) is
non-increasing. More on the topic - see Lehmann (1986).
75
Expected Utility
Arrow-Pratt Approximation (1)
Applied Micro
• By means of the Arrow-Pratt approximation we can express the
risk premium π in terms of the Arrow-Pratt measures of risk.
• Assume that z = w + kx, where w is a fixed constant (e.g.
wealth), x is a mean zero random variable and k ≥ 0. By this
assumption the variance of z is given by
V(z) = k2
V(x) = k2
E(x2
).
• Proposition - Arrow-Pratt Risk Premium with respect to
Additive risk: If risk is additive, i.e. z = w + kx, then the risk
premium π is approximately equal to 0.5A(w)V(z).
76
Expected Utility
Arrow-Pratt Approximation (2)
Applied Micro
Proof:
• By the definition of the risk premium we have
E(u(z)) = E(u(w + kx)) = u(w − π(k)).
• For k = 0 we get π(k) = 0. For risk averse agents dπ(k)/dk ≥ 0.
• Use the definition of the risk premium and take the first derivate
with respect to k on both sides:
E(xu (w + kx)) = −π (k)u (w − π(k)) .
77
Expected Utility
Arrow-Pratt Approximation (3)
Applied Micro
Proof:
• For the left hand side we get at k = 0:
E(xu (w + kx)) = u (w)E(x) = 0 since E(x) = 0 by assumption.
• Matching LHS with RHS results in π (k) = 0 at k = 0.
78
Expected Utility
Arrow-Pratt Approximation (4)
Applied Micro
Proof:
• Taking the second derivative with respect to k yields:
E(x2
u (w + kx)) = (π (k))2
u (w − π(k)) − π (k)u (w − π(k))
• At k = 0 this results in (note that π (0) = 0):
π (0) = −
u (w)
u (w)
E(x2
)
79
Expected Utility
Arrow-Pratt Approximation (5)
Applied Micro
• A second order Taylor expansion of π(k) around k = 0 results in
π(k) ≈ π(0) + π (0)k +
π (0)
2
k2
• Thus
π(k) ≈ 0.5A(w)E(x2
)k2
• Since E(x) = 0 by assumption, the risk premium is proportional
to the variance of x.
80
Expected Utility
Arrow-Pratt Approximation (6)
Applied Micro
• For multiplicative risk we can proceed as follows: z = w(1 + kx)
where E(x) = 0.
• Proceeding the same way results in:
π(k)
w
≈ −
wu (w)
u (w)
k2
E(x2
) = 0.5R(w)E(x2
)k2
• Proposition - Arrow-Pratt Relative Risk Premium with
respect to Multiplicative risk: If risk is multiplicative, i.e.
z = w(1 + kx), then the relative risk premium π/w is
approximately equal to 0.5R(w)k2
V(x).
• Interpretation: Risk premium per monetary unit of wealth.
81
Expected Utility
Decreasing Absolute Risk Aversion (1)
Applied Micro
• It is widely believed that the more wealthy an agent, the smaller
his/her willingness to pay to escape a given additive risk.
• Definition - Decreasing Absolute Risk Aversion: Given
additive risk z = w + x, x is a random variable with mean 0. The
risk premium is a decreasing function in wealth w.
82
Expected Utility
Decreasing Absolute Risk Aversion (2)
Applied Micro
• Proposition - Decreasing Absolute Risk Aversion: [P 6.C.3]
The following statements are equivalent
– The risk premium is a decreasing function in wealth w.
– Absolute risk aversion A(w) is decreasing in wealth.
– −u (z) is a concave transformation of u. I.e. u is sufficiently
convex.
83
Expected Utility
Decreasing Absolute Risk Aversion (3)
Applied Micro
Proof: (sketch)
• Step 1, (i) ∼ (iii): Consider additive risk and the definition of
the risk premium. Treat π as a function of wealth:
E(u(w + kx)) = u(w − π(w)) .
• Taking the first derivative yields:
E(1u (w + kx)) = (1 − π (w))u (w − π(w)) .
84
Expected Utility
Decreasing Absolute Risk Aversion (4)
Applied Micro
Proof: (sketch)
• This yields:
π (w) = −
E(1u (w + kx)) − u (w − π(w))
u (w − π(w))
.
• π (w) decreases if E(1u (w + kx)) − u (w − π(w)) ≥ 0.
• Note that we have proven that if E(u2(z)) = u2(z − π2) then
E(u1(z)) ≤ u1(z − π2) if agent 1 were more risk averse.
85
Expected Utility
Decreasing Absolute Risk Aversion (5)
Applied Micro
Proof: (sketch)
• Here we have the same mathematical structure (see slides on
Comparative Analysis): set z = w + kx, u1 = −u and u2 = u.
• ⇒ −u is more concave than u such that −u is a concave
transformation of u.
86
Expected Utility
Decreasing Absolute Risk Aversion (6)
Applied Micro
Proof: (sketch)
• Step 2, (iii) ∼ (ii): Next define P(w) := −u
u which is often
called degree of absolute prudence.
• From our former theorems we get: P(w) ≥ A(w) has to be
fulfilled (see A1 and A2).
• Take the first derivative of the Arrow-Pratt measure yields:
A (w) = −
1
(u (w))2
(u (w)u (w) − (u (w))
2
)
= −
u (w)
(u (w))
(u (w)/u (w) − u (w)/u (w))
=
u (w)
(u (w))
(P (w) − A(w))
87
Expected Utility
Decreasing Absolute Risk Aversion (7)
Applied Micro
Proof: (sketch)
• A decreases in wealth if A (w) ≤ 0.
• We get A (w) ≤ 0 if P(w) ≥ A(w).
88
Expected Utility
HARA Utility (1)
Applied Micro
• Definition - Harmonic Absolute Risk Aversion: A Bernoulli
utility function exhibits HARA if its absolute risk tolerance (=
inverse of absolute risk aversion) T(z) := 1/A(z) is linear in
wealth w.
• I.e. T(z) = −u (z)/u (z) is linear in z
• These functions have the form u(z) = ζ (η + z/γ)
1−γ
.
• Given the domain of z, η + z/γ > 0 has to hold.
89
Expected Utility
HARA Utility (2)
Applied Micro
• Taking derivatives results in:
u (z) = ζ
1 − γ
γ
(η + z/γ)
−γ
u (z) = −ζ
1 − γ
γ
(η + z/γ)
−γ−1
u (z) = ζ
(1 − γ)(γ + 1)
γ2
(η + z/γ)
−γ−2
90
Expected Utility
HARA Utility (3)
Applied Micro
• Risk aversion: A(z) = (η + z/γ)
−1
• Risk Tolerance is linear in z: T(z) = η + z/γ
• Absolute Prudence: P(z) = γ+1
γ (η + z/γ)
−1
• Relative Risk Aversion: R(z) = z (η + z/γ)
−1
91
Expected Utility
HARA Utility (4)
Applied Micro
• With η = 0, R(z) = γ: Constant Relative Risk Aversion
Utility Function: u(z) = log(z) for γ = 1 and u(z) = z1−γ
1−γ for
γ = 1.
• This function exhibits DARA; A (z) = −γ2
/z2
< 0.
92
Expected Utility
HARA Utility (5)
Applied Micro
• With γ → ∞: Constant Absolute Risk Aversion Utility
Function: A(z) = 1/η.
• Since u (z) = Au (z) we get u(z) = − exp(−Az)/A.
• This function exhibits increasing relative risk aversion.
93
Expected Utility
HARA Utility (6)
Applied Micro
• With γ = −1: Quadratic Utility Function:
• This functions requires z < η, since it is decreasing over η.
• Increasing absolute risk aversion.
94
Expected Utility
State Dependent Utility (1)
Applied Micro
• With von Neumann Morgenstern utility theory only the
consequences and their corresponding probabilities matter.
• I.e. the underlying cause of the consequence does not play any
role.
• If the cause is one’s state of health this assumption is unlikely to
be fulfilled.
• Example car insurance: Consider fair full cover insurance. Under
VNM utility U(l) = pu(w − P) + (1 − p)u(w − P), etc. If
however it plays a role whether we have a wealth of w − P in the
case of no accident or getting compensated by the insurance
company such the wealth is w − P, the agent’s preferences
depend on the states accident and no accident.
95
Expected Utility
State Dependent Utility (2)
Applied Micro
• Definition - States: Events ω ∈ Ω causing the consequences
z ∈ Z are called states of the world/states of nature. Ω is called
set of states (sample space).
• For these states we assume that they
– Leave no relevant aspect undescribed.
– Mutually exclusive. At most one state can be obtained.
– Collectively exhaustive, ω = Ω.
– ω does not depend on the choice of the decision maker.
96
Expected Utility
State Dependent Utility (3)
Applied Micro
• Definition - Uncertainty with State Dependent Utility: To
formulate uncertainty consider the following parts:
– Set of consequences Z.
– Set of states Ω.
– Probability measure π on (Ω, F).
97
Expected Utility
State Dependent Utility (4)
Applied Micro
• Remark: Note that this construction corresponds to the idea of a
random variable.
• A function g : Ω → Z will be called random variable. With the
sigma field F generated by this random variable we get the
probability measure π. An event is a subset of Ω. If Z ⊆ RN
it is
a real valued random variable.
• A random variable assigns to each state ω a consequence z ∈ Z,
the preimage is g−1
(z) = ω.
98
Expected Utility
States (1)
Applied Micro
• A random variable f mapping from the set of states into
consequences gives rise to a lottery
(π1 ◦ z1, . . . , πn ◦ zn)
for finite Ω.
• There is a loss of information when going from the random
variable to the lottery/distribution representation. We do not
know which state gave rise to a particular consequence.
99
Expected Utility
States (2)
Applied Micro
• A random variable z is called measurable if f−1
(z) = ω ∈ F. I.e.
the preimage has to be contained in the sigma field.
• With finitely many states we can define the set
P = {f−1
(¯z)}¯z=z∈Z with f−1
(¯z) := {ω ∈ Ω|f(ω) = ¯z}. By
construction P is a partition.
• If f−1
(¯z1) ∩ f−1
(¯z2) = ∅ then z1 = z2 , i f−1
(zi) = Ω
f−1
(zi) = ∅ by construction.
• Within f−1
(¯z1) the function f(ω) is constant. f(ω) = ¯z1 for
ω ∈ f−1
(¯z1).
100
Expected Utility
States (3)
Applied Micro
• Example - Asset Price: Assume the price of an asset is
permitted to move upwards (by 1 + ut) for downwards (1 − dt)
with probability p and 1 − p. The initial price S0 = 1. We
consider two periods. To keep the analysis simple assume that
(1 + u1)(1 + d2) = (1 + d1)(1 + u2).
• Then ω1 corresponds to the consequence (1 + u1)(1 + u2), ω2 to
(1 + u1)(1 − d2), ω3 to (1 − d1)(1 + u2) and ω4 to
(1 − d1)(1 − d2). The sigma field generated by this random
variable consists of all subsets of Ω.
101
Expected Utility
States (4)
Applied Micro
• At t = 2 the partition P2 is given by the sets ω1, . . . , ω4. For
each consequence the preimage f−1
(zi) ∈ F or P2.
• At t = 1 only the subsets (ω1, ω2) and (ω3, ω4) are measurable
with respect to F1. For t = 0 only the constant S0 is measurable
with respect to the trivial sigma field F0 = {∅, Ω}.
• P1 = {(ω1, ω2), (ω3, ω4)}.
102
Expected Utility
States (5)
Applied Micro
• I.e. we get the filtration F0 ⊆ F1 ⊆ F2.
• The corresponding partitions are P0 and P1. P2 is finer than P1
and P1 is finer than P0.
103
Expected Utility
States (6)
Applied Micro
• The corresponding partitions are P0 and P1. P2 is finer than P1
and P1 is finer than P0.
• The subsets of P2 are f−1
2 (zi) = ωi, i = 1, . . . , 4. For P1 we get
the subset f−1
1 (¯zi) = (ω1, ω2) for i = 1, 2 and f−1
1 (¯zi) = (ω3, ω4)
for i = 3, 4 . While for P0 we get Ω.
• Note that f−1
2 (¯zi) ⊆ f−1
1 (¯zi) but not vice versa.
104
Expected Utility
States (7)
Applied Micro
• Example - Signals: Assume that a random variable f maps
from Ω to a set of reports/signal R, r are the elements of R.
• Hf is the partition generated by f−1
(r), i.e. Hf = {f−1
(¯r)}r∈R.
• For two random variables f and g, the events
f−1
(¯r1) ∩ g−1
(¯r2) = {ω ∈ Ω|f(ω) = ¯r1 and g(ω) = ¯r2} also
partition the state space.
• If for every r1 it happens that f−1
(¯r1) ⊆ g−1
(¯r2) for some ¯r2,
then the addition of g does not result in further information.
105
Expected Utility
States (8)
Applied Micro
• Definition - Information Partition: A partition on the state
space Ω is called information partition, the subsets of this
partition are h. For every state ω ∈ Ω: The event/function h(ω)
assigning an element of H to each ω ∈ Ω is called information
set containing ω (possibility set).
• Note that if H = {h1, . . . , hm} then by h(ω) we are looking for
the hi where ω is contained. I.e. h(ω) : Ω → H or h(ω) → hi.
• This assignment satisfies: ω ∈ h(ω) for all ω ∈ Ω. If ω = ω and
ω ∈ h(ω) then h(ω) = h(ω ).
106
Expected Utility
States (9)
Applied Micro
• Definition - Knowledge: An event E ∈ Ω is known at the state
ω ∈ Ω if h(ω) ⊆ E.
• I.e. E is known if anything possible implies it. What is known to
the decision maker depends on the state ω.
• See Ritzberger, page 63, Example 2.10.
107
Expected Utility
States (10)
Applied Micro
• When a decision maker observes realizations of a random variable
she will update her probability assignments on z.
• Call π prior beliefs, and the ˜π posterior beliefs.
• A decision maker regards states outside h(ω) is impossible if
˜π(h(ω)) = 1.
• Only ω ∈ h(ω) are assigned with a positive probability.
• The posterior probability of a set E given h(ω) is then given by
the Bayes theorem: For π(h(ω)) > 0)
π(E|h(ω)) =
π(h(ω) ∩ E)
π(h(ω))
108
Expected Utility
States (11)
Applied Micro
• Note that π(E|h(ω)) depends on ω and is therefore a random
variable.
• For a finite probability space with z ∈ Z we get:
π(f−1
(z)|h(ω)) =
π(h(ω)|f−1
(z))π(f−1
(z))
z ∈Z π(h(ω)|f−1(z ))π(f−1(z ))
• Note that π(f−1
(z)|h(ω)) = π(z|h(ω)) by construction; the
denominator above is different from zero.
• For an infinite probability space see textbooks on Probability
theory.
109
Expected Utility
State Dependent Utility (1)
Applied Micro
• With VNM utility theory we have considered the set of simple
lotteries LS over the set of consequences Z. Each lottery li
corresponds to a probability distribution on Z.
• Assume that Ω has finite states. Define a random variable f
mapping from Ω into LS. Then f(ω) = lω for all ω of Ω. I.e. f
assigns a simple lottery to each state ω.
• If the probabilities of the states are given by π(ω), we arrive at
the compound lotteries lSDU = π(ω)lω.
• I.e. we have calculated probabilities of compound lotteries.
110
Expected Utility
State Dependent Utility (2)
Applied Micro
• The set of lSDU will be called LSDU. Such lotteries are also
called horse lotteries.
• Note that also convex combinations of lSDU are ∈ LSDU.
• Definition - Extended Independence Axiom: The preference
relation satisfies extended independence if for all
l1
SDU, l2
SDU, lSDU ∈ LSDU and α ∈ (0, 1) we have l1
SDU lSDU
if and only if αl1
SDU + (1 − α)l2
SDU αlSDU + (1 − α)l2
SDU.
111
Expected Utility
State Dependent Utility (3)
Applied Micro
• Proposition - Extended Expected Utility/State Dependent
Utility: Suppose that Ω is finite and the preference relation
satisfies continuity and in independence on LSDU. Then there
exists a real valued function u : Z × Ω → R such that
l1
SDU l2
SDU
if and only if
ω∈Ω
π(ω)
z∈supp(l1
SDU
(ω))
pl1(z|ω)u(z, ω) ≥
ω∈Ω
π(ω)
z∈supp(l2
SDU
(ω))
pl2(z|ω)u(z, ω) .
112
Expected Utility
State Dependent Utility (4)
Applied Micro
• u is unique up to positive linear transformations.
• Proof: see Ritzberger, page 73.
• If only consequences matter such that u(z, ω) = u(z) then state
dependent utility is equal to VNM utility.
113
Expected Utility
Subjective Utility (1)
Applied Micro
• In the above settings we have assumed that π(ω) are objective
probabilities.
• In some applications the likelihood of an event is more or less a
subjective estimate.
• With subjective probability theory π(ω) are subjective beliefs.
• Here the probability of an event depends on the agent’s
preferences.
114
Expected Utility
Subjective Utility (2)
Applied Micro
• Consider an extended expected utility formulation where u(z, ω)
and π(ω) depend on preferences.
• Here we need some way to disentangle the Bernoulli utility
function from the probabilities. This requires a further axiom.
• Definition - State Preferences: Consider the set of simple
lotteries LS (with ω fixed): L1 ω L2 if and only if
pl1(ω)u(z, ω) ≥ pl2(ω)u(z, ω) .
• Axiom - State Uniform Preferences: ω= ω for all ω and ω
in Ω.
115
Expected Utility
Subjective Utility (3)
Applied Micro
• Claim: With state uniform preferences we get
u(z, ω) = π(ω)u(z) + β(ω).
• L1 ω L2 has to be fulfilled for all ω. Therefore
pl1(ω)u(z, ω) ≥ pl2(ω)u(z, ω) has to hold for each ω.
• This can only be the case if pl1(ω)u(z, ω) is a positive affine
of pl1(ω )u(z, ω ) for arbitrary pairs ω, ω (transformation
properties of VNM utility functions).
• For notational issues and w.l.g. let us consider degenerated
lotteries, here u(z, ω) is PAT of u(z, ω )
116
Expected Utility
Subjective Utility (4)
Applied Micro
• Thus, a(ω)u(z, ω) + b(ω) = a(ω )u(z, ω ) + b(ω )
• W.l.g. use ω1 as benchmark, Then
a(ω)u(z, ω) + b(ω) = u(z, ω1) = u(z).
• ⇒ u(z, ω) = (u(z) − b(ω))/a(ω). For all ω, a(ω1) = 1 and
b(ω1) = 0.
• Thus u(z, ω) = π(ω)u(z) + β(ω) with π(ω) = 1/a(ω) and
β(ω) = −b(ω)/a(ω).
117
Expected Utility
Subjective Utility (6)
Applied Micro
• u(z, ω) ≥ u(z , ω) for all ω holds if ω u(z, ω) ≥ ω u(z , ω)
holds and vice versa with u(z, ω) PAT of u(z, ω ).
• Plug in (π(ω)u(z) + β(ω)) results in
ω u(z, ω) = ω π(ω)u(z) + β(ω)
• The same preferences are represented if we divide all a and b by
the same constant.
• Choose this constant such that ω w(ω) = 1, then
u(z, ω) = w(ω)v(z, ω).
118
Expected Utility
Subjective Utility (7)
Applied Micro
• These weights have to correspond to the subjective probabilities
to result in an extended expected utility function.
• Proposition - Subjective Expected Utility: Suppose that the
preference relation satisfies continuity and in independence on
LSDU. Assume that these preferences are state uniform. Then
there exists subjective probabilities and an extended expected
utility function representing these preferences.
• Limitations see e.g. the Ellsberg Paradoxon.
119
Expected Utility
Knight Uncertainty (1)
Applied Micro
• Knight distinguished between risk and uncertainty.
• For risk the probabilities are objectively given, for uncertainty not.
• With subjective probability theory uncertainty can be once again
expressed by probabilities.
• Non - vNM approaches see e.g Gilboa
120
Expected Utility
Capital Asset Pricing Model (1)
Applied Micro
• To derive the Capital Asset Pricing Model (CAPM) we choose
a representative agent model with CARA preferences; the returns
are normally distributed. There are also other ways to get the
CAPM, see e.g. ? and ?.
• The preference of the representative agent are described by
E(u(z)) = E(− exp(−ρz)). I.e. we consider a von
Neumann-Morgenstern utility maximizer with Bernoulli utility
function u(z) = − exp(−ρz). The absolute Arrow-Pratt measure
is ρ, therefore the expression constant absolute risk aversion
(CARA).
• The CAPM is an equilibrium model.
121
Expected Utility
Capital Asset Pricing Model (2)
Applied Micro
• Definition - Returns: Consider the prices of asset i at time t,
then rit =
pit−pi,t−1
pi,t−1
is called return. Rit = 1 + rit is called
gross-return. E(ri,t+1) =
E(pi,t+1)−pt
pt
is called expected return.
• Definition - Portfolio: Given the assets 1, . . . , n with prices pt,
a portfolio is given by a vector qt = (qt1, . . . , qtn) ∈ Rn
, qti is
the number of assets i held by some investor at t. The value of
the portfolio is wt = pt · qt =
n
i=1 pitqit.
• The money amounts invested in the assets are yit = pitqit, where
yt = (y1t, . . . , ynt) .
• The relative amounts are ωit = yit
wt
.
122
Expected Utility
Capital Asset Pricing Model (3)
Applied Micro
• Definition - Portfolio Returns: The returns of the portfolio qt
are rpt =
wpt−wp,t−1
wp,t−1
.
• rpt can be written as follows:
rpt =
qt−1 · (pt − pt−1)
qt−1 · pt−1
or
rpt =
n
i=1
ωi,t−1
pit − pi,t−1
pi,t−1
.
123
Expected Utility
Capital Asset Pricing Model (4)
Applied Micro
• Equipped with this terminology we consider a two period
economy; consumption takes place in the second period, q is
bought in the initial period. A risk free asset is assumed to exist.
The return is rf; the supply of this asset is perfectly elastic.
• Since we are only considering a two period model we can skip the
time indixes for the returns. There is only one expected return for
asset i, abbreviated by E(ri) and an expected return of the
portfolio q. For some portfolio we get
E(rp,t+1) = E(rp) =
n
i=1
ωiE(ri) =
n
i=1
ωtiE(ri,t+1).
124
Expected Utility
Capital Asset Pricing Model (5)
Applied Micro
• yf
, yr
= (y1, . . . , yn) are the amounts of the risk-free asset and
the risky assets held by the representative investor.
y = (yf
, yr
) .
• The value of the portfolio in the second period is a random
variable, it is given by w = yf
Rf + yr
· Rr. Rf = 1 + rf, Rr is
the vector of gross-returns of the n risky assets. Rr and
rr = 1 − Rr are n dimensional vectors of returns.
• We assume that the returns are normally distributed.
125
Expected Utility
Capital Asset Pricing Model (6)
Applied Micro
• The preference of the representative agent are described by
E(u(z)) = E(− exp(−ρz)). I.e. we consider a von
Neumann-Morgenstern utility maximizer with Bernoulli utility
function u(z) = − exp(−ρz).
• The CAPM is an equilibrium model. The amounts of the risky
assets available are a = (a1, . . . , an). ai is measured in monetary
units (like yi).
• For the gross-returns we observe E(Rp,t+1) = 1 + E(rp,t+1) and
V(rr) = V(Rr).
126
Expected Utility
Capital Asset Pricing Model (7)
Applied Micro
• The expected wealth is (w0 = yf
+
n
i=1 yr
i is the initial wealth)
E(w) =
n+1
i=1
yi(1 + E(ri)) = yf
Rf + yr
· E(Rr).
• The variance of the wealth is yr
V(rt)yr
. yr
is a n dimensional
vector, V(rt) is the n × n covariance matrix of the wealth. The
variances are in the main diagonal of this matrix.
• Since Rf and rf are a constants, the variance and the
covariances with the other returns are zero. This also implies
yr
V(rt)yr
= ((yf
, yr
) ) V((rf, rt ) )(yf
, yr
) .
127
Expected Utility
Capital Asset Pricing Model (8)
Applied Micro
• If yr
= a we say that the investor holds the market portfolio;
wy =
n
i=1 yi, wa =
n
i=1 ai.
• Here E(wM) = yf
Rf + a · E(Rr),
E(rM) = ω · E(rr) = 1
wa
a · E(rr) and V(wM) = a E(Rr)a.
• σ2
M = V(rM) = 1
w2
a
· a V(rr)a = ω V(rr)ω, where
ω = (ωr
1, . . . , ωr
n) and
n
i=1 ωr
i = 1; here ωr
i = ai
wa
.
128
CAPM - Mathematical Note (1)
Applied Micro
• Given the above notation and assumptions we obtain
E(− exp(−ρw)) = E − exp(−ρ[(w0 − yi)Rf + ρyr
· E(Rr)]) .
• If X ∈ R1
is normally distributed with mean vector µ and
covariance matrix Σ, then
E(exp(t · X)) = exp(t · µ + t2
/2 · Σ)
Laplace transform/ Moment generating function of a normal
random variable; t ∈ R is called convolution parameter.
129
CAPM - Mathematical Note (2)
Applied Micro
• In our case w ∈ R1
is normally distributed with mean vector
(w0 − yi)Rf + yr
· E(Rr) and variance yr
V(Rr)yr
. The
convolution parameter is −ρ. This yields
E(− exp(−ρw)) = − exp −ρ(w0 − yi)Rf − ρy
r
· E(Rr) +
ρ2
2
y
r
V(Rr)y
r
.
130
Expected Utility
Capital Asset Pricing Model (9)
Applied Micro
• Consider
E(− exp(−ρw)) = − exp −ρ(w0 − yi)Rf − ρy
r
· E(Rr) +
ρ2
2
y
r
V(Rr)y
r
.
• By taking first derivatives with respect to yi, i = 1, . . . , n we
obtain the vector of optimal amounts invested. I.e.
Rf − E(Rr) + ρV(Rr)yr
= 0
such that
yr
= 1
ρV(Rr)−1
(E(Rr) − Rf) = 1
ρV(rr)−1
(E(rr) − rf) .
• Note that the optimal yr
does not depend on the initial wealth
w0. 131
Expected Utility
Capital Asset Pricing Model (10)
Applied Micro
• We consider an equilibrium model, therefore yr
= a. This yields
E(rr) = rf + ρV(Rr)a = 0,
and
a =
1
ρ
V(Rr)−1
(E(rr) − rf) .
• Note that V(Rr)a is equal to the vector of covariances
Cov(Rr, RM) between the returns of the assets, i = 1, . . . , n,
with the return of the market portfolio (weighted by ωr
i = ai/wa)
times the market capitalization wa. To see this calculate
E(Rr(a · Rr) ) = E(RrRr )ωwa = . . .
132
Expected Utility
Capital Asset Pricing Model (11)
Applied Micro
• From the equilibrium condition applied to the market portfolio
we get
E(rr) − rf = ρV(Rr)a = ρCov(Rr, RM)wa = ρCov(rr, rM)wa
• Left-multiply both sides by ω , then we get
ω (E(rr) − rf) = ρω V(Rr)ωwa and
E(rM) − rf = ρV(rM)wa such that
ρ =
1
σ2
Mwa
(E(rM) − rf).
133
Expected Utility
Capital Asset Pricing Model (12)
Applied Micro
• Finally from E(rr) − rf = ρCov(rr, rM)wa and
ρ = 1
σ2
M
wa
(E(rM) − rf) we obtain the equilibrium returns:
E(ri) = rf +
Cov(ri, rM)
σ2
M
(E(rM) − rf) .
• Cov(ri,rM)
σ2
M
will be called β-factor.
• The model can also be derived with heterogeneous agents
i = 1, . . . , I with CARA utility (different ρi), in this case the
equilibrium condition is given by
I
i=1 yr
i = a.
134