Econ 203C
Problem Set # 6 — Answer Key
Problem 1.
For arbitrary (possibly infinite) C, let L (C) denote the set of all lotteries
with finite support. Generalize the vNM Theorem demonstrated in class to
arbitrary infinite sets of consequences C.
Suppose that  on L (C) satisfies the vNM axioms. We will derive the
existence of an EU representation with u : C → R. The other parts of the
vNM theorem generalize trivially.
Fix any c, c0 ∈ C such that δc  δc0 . For any finite subset D ⊆ C,
let L (D) denote the set of lotteries with support in D. Define a utility
function u : C → R as follows. For any x ∈ C, consider the restriction of
the preference relation  to L ({c, c0, x}). This restriction obviously satisfies
the vNM axioms; hence there exists a vNM utility function representing Â
on L ({c, c0, x}) vx . By invariance of the representation to positive affine
transformations, vx can be chosen such that vx (c) = 1 and vx (c0 ) = 0. Set
u(x) := vx (x) .
We need to show that u represents  on L (C) . To see this, take any two
finite lotteries L = [pi , xi ] and L0 = [p0i , xi ] such that L Â L0 ; let D denote
the union of their supports.
By the vNM Theorem, Â on L (D) has a (unique) EU representation
with vNM utility function vD satisfying vD (c) = 1 and vD (c0 ) = 0. In
particular,
X
X
pi vD (xi ) >
p0i vD (xi ) .
i
i
Since L (D) ⊇ L ({c, c0, x}) for all x ∈ D, clearly (clearly?) vD = u; hence
X
X
pi u (xi ) >
p0i u (xi ) ,
i
i
verifying the claim since L and L0 were arbitrary.
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Problem 2.
Assume C to be finite. A lottery is binary iff it assigns positive probability to
at most two distinct outcomes; let L2 (C) denote the set of all binary lotteries
on C. Prove the analogue to the von Neumann-Morgenstern theorem. That is
prove in particular that if a preference relation on L2 (C) satisfies Weak Order,
Independence, and the Archimedean axiom, Â has an EU representation.
1. (a) Verify that Steps 1 and 2 in our proof can be used literally, so you
don’t need to redo them
(b) Why can’t you always apply Step 3-4 literally?
(c) However, you can apply Step 3-4 sometimes; this is helpful in obtaining a workable proof in the general case.
b) The difficulty is that the substitution of certain outcomes by bestworst lotteries may lead to non-binary lotteries.
c) I will write binary lotteries as αδx + (1 − α) δy . Let x− and x+ denote
the % −worst and -best outcomes, respectively, and write α for αδx+ +
(1 − α) δx− .
For any x ∈ X, let u (x) be the unique α such that δx ∼ α ; uniqueness
and existence follows from Steps 1 and 2 in our proof of the von NeumannMorgenstern theorem (part a).
Let αδx + (1 − α) δy denote any binary lottery. We need to show that
αδx + (1 − α) δy ∼
αu(x)+(1−α)u(y) .
(1)
Consider first the special case that y = x− .
By Independence,
αδx + (1 − α) δx− ∼ α (u (x) δx+ + (1 − u (x)) δx− ) + (1 − α) δx− =
αu(x) ,
(2)
verifying (1).
Consider now the general case. By (2), δy ∼ βδx + (1 − β) δx− , where
u(y)
. Hence by Independence,
β = u(x)
αδx +(1 − α) δy ∼ αδx +(1 − α) (βδx + (1 − β) δx− ) = (α + (1 − α) β) δx +(1 − α − (1 − α) β) δx− .
By (2) again, the latter is indifferent to
as was to be shown.
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2
(α+(1−α)β)u(x)
=
αu(x)+(1−α)u(y) ,
Problem 3.
c) Let p∗ denote your answer to b). If you are an EU maximizer with
Bernouilli utility function u, u satisfies
p∗ u(∞) + (1 − p∗ )u(1 − q) = u(1),
1
. By this
where q = 0.1. With CRRA with ρ > 1, write u(x) = − xρ−1
normalization, u(1) = −1 and u(∞) = 0, and thus
(1 − p∗ )
= 1,
(1 − q)ρ−1
whence
log(1 − p∗ ) = (ρ − 1) log(1 − q)
and thus
ρ=
For smallish q and p∗ , thus
log(1 − p∗ )
+ 1.
log(1 − q)
ρ≈
p∗
+ 1.
q
e)
p(q)u(∞) + (1 − p(q))u(1 − q) = u(1)
Hence we get for ρ = 2
p(q)0 + (1 − p(q))
1
≈ 1, and thus p(q) ≈ q.
1−q
f) Two such axioms are the following (each is necessary and sufficient
for boundedness of the utility function on its own).
AXIOM 1. For all x, y ∈ R+ such that y > x there exists α ∈ (0, 1)
such that, for all z ∈ R+ , δy % [α, z; (1 − α), x].
AXIOM 2.
For all x ∈ R+ and all α ∈ (0, 1) there exists y ∈ R+ such that, for all
z ∈ R+ , δy % [α, z; (1 − α), x].
Note the different order of quantifiers in each case; they make the content
of the axiom quite different.
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