John Riley
6 December 2010
ANSWERS TO EVEN NUMBERED EXERCISES IN CHAPTER 7
Section 7.1
Exercise 7.1-2: Allais Paradox
(a) Draw a tree diagram showing that the prospect C can be represented as a compound
gamble between A and (0,0,1). Draw another tree diagram showing that the prospect D
can be represented as a compound gamble between B and (0,0,1), where the probability
of (0,0,1) is the same.
(b) Show that the ranking of A and B must be the same as the ranking of C and D. Hence
establish the original version of the Allais Paradox., that is C ~ D ⇔ A ~ E
ANSWER
(a)
D = (ab, 0,1- ab)
C = (0, b,1- b)
p
A = (0,1, 0)
1-p
Z = (0, 0,1)
p
B = (a, 0,1- a)
1-p
Z = (0, 0,1)
(b) Then (0, b,1 − b) = p (0,1, 0) + (1 − p)(0, 0,1) if and only if p = b . Then
(ab, 0,1 − ab) = p(a, 0,1 − a) + (1 − p)(0, 0,1)
If A ∼ B ; then for any prospect Z and probabilities p1 , p2 > 0 and p1 + p2 = 1 ,
C = ( p1 , p2 ; A, Z ) ∼ ( p1 , p2 ; B, Z ) = D
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John Riley
6 December 2010
Exercise 7.1-4: Rabin paradox
You are offered either $100 for sure or one of the following risky alternatives. Which
would you accept?
Payoff if
Number of green balls (out of 100)
Payoff if
green ball
red ball
200
52
55
60
66
70
0
1100
13
20
33
48
57
0
5100
7
18
33
48
57
0
25100
7
18
33
48
57
0
Compare your answers to the answers you gave above in the discussion of the Rabin
Paradox. Are the differences consistent with expected utility maximization?
ANSWER
Define ŵ = w + g where g = 100 . Then the choice is between a final wealth of ŵ and an
uncertain payoff of ŵ + ng or ŵ − g depending on the color of the ball drawn from the
urn, where n = 1,10,50, 250 . Note that this problem is identical in structure to the one
analyzed above so the answers should be exactly the same.
Section 7.2
Exercise 7.2-2: Relative risk aversion
The degree of relative risk aversion R ( x ) = − xv′′( x ) / v′( x ) .
(a) Show that if an individual exhibits constant relative risk aversion, his marginal rate of
substitution M ( x1 , x2 ) is constant along a ray from the origin. Assuming that he can
trade in claims to each state, show that the risk he will take on rises proportionally with
his wealth.
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John Riley
6 December 2010
(b) Show that if v′( x) = x −1/σ , σ > 0 , the individual exhibits constant relative risk
aversion. Hence solve for the constant relative risk aversion utility functions.
(c) It is usually argued that individuals exhibit increasing relative risk aversion and
constant
absolute risk aversion. What does this imply about the shape of wealth expansion paths?
ANSWER
(a)
D
B
c2
Certainty line
A
c1
We must show that starting from a point A above the certainty line, the marginal rate of
substitution increases along the ray AD under the assumption of increasing risk aversion.
A point on the ray is x(θ ) = (1 + θ )( x1 , x2 ) . Then
M ( x(1 + θ )) =
π 1v′( x1 (1 + θ ))
π 1v′( x2 (1 + θ ))
Taking the logarithm and differentiating,
1 ∂M
M ∂θ
=
θ =0
x1v′′( x1 ) x2 v′′( x2 )
−
= R ( x2 ) − R ( x1 ) .
v′( x1 )
v′( x2 )
Since c2 > c1 it follows that the marginal rate of substitution increases along the ray AD.
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John Riley
6 December 2010
Hence as wealth increases, the consumer chooses a state claims bundle lying below the
ray. It is therefore less risky relative to his wealth.
(b) Differentiating, v′′( x) = −
1
σ
x
1
− −1
σ
. Therefore R ( x) = −
xv′′( x) 1
= . Thus the CES
v′( x) σ
family of VNM utility functions exhibit constant relative risk aversion. If σ = 1 ,
1−
1
x
σ
1−
1
v′( x) = x −1 and so v( x) = ln x . If σ ≠ 1 v( x) =
.
σ
(c) An almost identical argument to that made in (a) establishes that the marginal rate of
substitution decreases along a line parallel to the certainty line. Therefore the
consumption bundles with a marginal rate of substitution equal to that at A lie in the
shaded region. Hence as wealth increases, the consumer chooses a state claims bundle
with greater absolute risk but less relative risk.
*Exercise 7.2-4 Wealth effects
An individual with wealth W must decide how much to invest in a riskless asset with
yield 1 + r1 and a risky asset with yield 1 + r2 where E{r2 ) > r1 .
(a) If the individual exhibits constant absolute risk aversion, show that his investment in
the risky asset is independent of his wealth.
*(b) If the individual exhibits decreasing absolute risk aversion, show that he will invest
more as his wealth increases.
ANSWER
(a) From section 7.2, if an individual invests x in the risky asset his marginal expected
utility is
S
U ′( x) = ∑ π sθ s v′( w + xθ s ) , where θ s = r2 s − r1 and w = (1 + r1 )W
s −1
We then ask what is the effect of an increase in wealth on the marginal utility of investing
in the risky asset.
S
d
U ′( x) = ∑ π sθ s v′′( w + xθ s )
dw
s −1
.
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John Riley
6 December 2010
Let x* be the optimal investment. If
d
U ′( x* ) > 0 , increasing wealth raises the marginal
dw
utility of investing in the risky asset. Then, at the higher wealth level, the individual
invests more in the risky asset.
(a) If absolute risk aversion is constant so that v′′(cs ) = − Av′(cs ) . Substitute this into the
above equation,
S
d
U ′( x) = − A∑ π sθ s v′( w + xθ s )
dw
s −1
(D.7-1)
At the optimum it follows from the first order condition that the right hand side is zero.
Thus
d
U ′( x* ) = 0
dw
and so a change in wealth has no effect on the optimal investment.
(b) The key is to make use of our assumption that absolute risk aversion is diminishing.
We index states so that the yield on the risky asset is lower in higher states. We also
define state t to be the lowest state for which θ s < 0 .
Substituting A(c) = −
v′′(c)
into (D.7-1),
v′(c)
S
d
U ′( x) = ∑ π sθ s v′′( w + xθ s )
dw
s −1
t −1
S
s =1
s =t
= ∑ π s (−θ s ) A( w + xθ s )v′( w + xθ s ) − ∑ π sθ s A( w + xθ s )v′( w + xθ s )
If θ s > 0, A( w + xθ s ) < A( w) . Hence
S
∑
s =t
S
π sθ s A( w + xθ s )v′( w + xθ s ) < ∑ π sθ s A( w)v′( w + xθ s )
s =t
Also, if θ s < 0, A( w + xθ s ) > A( w) . Hence
t −1
∑
s =1
t −1
π s (−θ s ) A( w + xθ s )v′( w + xθ s ) > ∑ π s (−θ s ) A( w)v′( w + xθ s )
s =1
Combining these results, it follows that
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John Riley
6 December 2010
S
d
U ′( x) > ∑ − π sθ s A( w)v′( w + xθ s )
dw
s =1
Since A(w) is independent of the state, we can take it outside the summation. Then
d
U ′( x) > − A( w)U ′( x)
dw
In particular, at x* ,
d
U ′( x* ) > − A( w)U ′( x* ) = 0 .
dw
Thus investment in the risky asset rises with wealth.
Exercise 7.2.6: Concave function
A function u (c) where c ∈
is strictly concave if for any c1 and c3 > c1
c2 = (1 − λ )c1 + λ c3 , 0 < λ < 1 ⇒ u (c2 ) > (1 − λ )u (c1 ) + λu (c3 )
(a) Rearrange these two expressions and hence show that u (c) is concave if
λ (c3 − c2 ) − (1 − λ )(c2 − c1 ), 0 < λ < 1 ⇒ λ (u (c3 ) − u (c2 )) − (1 − λ )(u (c2 ) − u (c1 )) < 0
(b) Hence show that the concavity of u (c) is equivalent to the statement that for all c1
and c3 > c1 and c2 ∈ (c1 , c3 )
c2 − c1 u (c2 ) − u (c1 )
u (c2 ) − u (c1 ) u (c3 ) − u (c2 )
>
<
.
. That is,
c3 − c2 u (c3 ) − u (c2 )
c2 − c1
c3 − c2
ANSWER
(a) c2 = (1 − λ )c2 + λ c2 = (1 − λ )c1 + λ c3 and
u (c2 ) = (1 − λ )u (c2 ) + λu (c2 ) > (1 − λ )u (c1 ) + λ u (c3 ) .
Rearranging these statement yields the desired result.
(b) Hence
c2 − c1
u (c2 ) − u (c1 )
u (c2 ) − u (c1 ) u (c3 ) − u (c2 )
λ
and so
>
.
=
<
c3 − c2 1 − λ u (c3 ) − u (c2 )
c2 − c1
c3 − c2
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John Riley
6 December 2010
Section 7.3
Exercise 7.3-2: First order stochastic dominance
(a) For S = 2 and S = 3 confirm the following identity.
S
∑
s =1
S −1
t
t =1
s =1
π s us = Π S uS − ∑ Π t (ut +1 − ut ) where Π t ≡ ∑ π s .
(4.7-2)
Prove by induction that the identity holds for all S.
Consider the two consequences (c, π A ), (c, π B ) where cs +1 > cs . The second probability
distribution stochastically dominates the first there is less weight in the left tail, that is
Π sB ≤ Π sA , s = 1,..., S .
(b) If v(⋅) is an increasing function, show that the individual prefers B over A.
s
(c) Define H s = ∑ Π t and Δut = ut +1 − ut . Appeal to the identity in (a) to show that
t =1
S
∑
t =1
S
Π t Δut = H S −1 − ∑ ( H S −1 (Δut +1 − Δut ) .
s =1
Substitute this expression into (4.7-2), hence prove Proposition 7.3-2.
ANSWER
(a)
π 1u1 + π 2u2 = π 1u1 + u2 (Π 2 − π 1 ) = Π 2u2 − π 1 (u2 − u1 ) = Π 2u2 − Π1 (u2 − u1 )
Suppose that the Proposition is true with S states. We must show that the proposition
must also be true with S + 1 states.
Let (π 1 ,...., π S , π S +1 ) be the probability vector with S + 1 states. Then
π
π
S
1
(
,....,
)
1 − π S +1
1 − π S +1
is a probability vector over S states and so
π u
ΠS
1 S −1
π 1u1
+ .... + S S =
uS −
∑ Π t (ut +1 − ut ) .
1 − π S +1
1 − π S +1 1 − π S +1
1 − π S +1 t =1
Therefore
S −1
π 1u1 + .... + π S uS = uS − ∑ Π t (ut +1 − ut ) .
t =1
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John Riley
6 December 2010
Hence
S −1
π 1u1 + .... + π S uS + π S +1uS +1 = π S +1uS +1 + Π S uS − ∑ Π t (ut +1 − ut )
t =1
S −1
= uS +1 (Π S +1 − Π s ) + Π s uS − ∑ Π t (ut +1 − ut )
t =1
S
= Π S +1uS +1 − ∑ Π t (ut +1 − ut ) .
t =1
(b) Note that Π SA = Π SB = 1 . Therefore
S
∑
s =1
S
S −1
S −1
s =1
t =1
t =1
π sBu (cs ) − ∑ π sAu (cs ) = −∑ Π tB (ut +1 − ut ) + ∑ Π tA (ut +1 − ut )
S −1
= ∑ (Π tA − Π tB )(u (ct +1 ) − u (ct )) > 0 .
t =1
(c) From the identity,
S −1
∑
t =1
S −2
Π t Δut = H S −1ΔuS −1 − ∑ H t (Δut +1 − Δut ) .
t =1
Therefore
S
∑
s =1
S −1
S −2
t =1
t =1
π s us = uS − ∑ Π t (ut +1 − ut ) = us − H S −1ΔuS −1 + ∑ H t (Δut +1 − Δut ) .
If ct − ct −1 = δ > 0 and u is concave then Δut +1 < Δut . Therefore if
( H1B ,..., H SB−1 ) < ( H1A ,..., H SA−1 ) for probability vectors π B and π A , it follows that
S
∑
s =1
S
π sBus > ∑ π sAus .
s =1
Exercise 7.3.4 Equivalent definition of conditional stochastic dominance
(a) If the probability distribution π exhibits conditional strict stochastic dominance over
πˆ , show that
1−
ˆ
Π t −1
Π
> 1 − t −1
ˆ
Πt
Π
t
Hence establish the “only if” part of the following proposition
Answers to Chapter 7 page
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John Riley
6 December 2010
Proposition 7.3-5: Conditional strict stochastic dominance
The probability distribution π exhibits strict conditional stochastic dominance over πˆ if
πs
and only if
Πs
>
πt
Πt
, ∀t and ∀s > t .
(4.7-3)
(b) Show that if (4.7-3) holds then
ˆ
Π s −1 Π
< s −1 . Use this to establish strict conditional
ˆ
Πs
Π
s
stochastic dominance.
ANSWER
(a) ONLY IF
If π exhibits conditional stochastic dominance over πˆ than for all s < t
s
ˆ
Πs Π
≤ s where Π s ≡ ∑ π s .
ˆ
Πt Π
θ =1
t
In particular,
ˆ
ˆ
Π t −1 Π
π
Π
Π
πˆ
≤ t −1 hence t = 1 − t −1 ≥ 1 − t −1 = t .
ˆ
ˆ
ˆ
Πt
Πt
Πt
Π
Π
Π
t
t
t
(b) IF
πt
Πt
≥
πˆt
, t = 1,..., S . Then
ˆ
Π
t
π
πˆ
1 − t ≤ 1 − t , t = 1,..., S ,
ˆ
Πt
Π
t
Suppose that
Hence
ˆ
Π t −1 Π
≤ t −1 , t = 1,..., S .
ˆ
Πt
Π
t
Therefore
t
t
ˆ
ˆ
Πs
Π
Π
Π
t −1
= × τ −1 ≤ ×
= s.
ˆ
ˆ
Π t τ = s +1 Πτ τ = s +1 Π
Π
t
t
Section 7.4
Exercise 7.4-2: Pareto efficiency and monotonicity with more than one binding
incentive constraint
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John Riley
6 December 2010
Extend the analysis to show that as long as all the binding constraints are associated with
less costly actions, the efficient wage is higher in states where output is higher.
ANSWER
Suppose that the set of binding constraints is indexed by B. That is constraint i is binding
if i ∈ B . By hypothesis, xi < x* , i ∈ B . The new maximization problem is
Max{U P | U A ( x* , w) ≥ U A , U A ( x* , w) ≥ U A ( xi , w), for all i ∈ B} .
w
The Lagrangian of this optimization problem is
L = U P + λ (U A ( x* , w) − U A ) + ∑ μi (U A ( x* , w) − U A ( xi , w))
i∈B
= U P + (λ + ∑ μi )(U A ( x* , w) − ∑ μi ) μiU A ( x, w)) + a constant
i∈B
i∈B
S
S
S
s =1
s =1
= ∑ π s ( x* )( ys − ws ) + (λ + ∑ μi ∑ π s ( x* )v( ws ) − μ ∑ π s ( xi )v( ws )
s =1
i∈B
plus constant terms.
The first order conditions are therefore
∂L
= −π s ( x* ) + (λ + ∑ μiπ s ( x* )v′( ws* ) − ∑ μiπ s ( xi )v′( ws* ) = 0 , s = 1,..., S
∂ws
i∈B
i∈B
Hence
π (x )
1
= λ + ∑ μi − ∑ μi s *i where xi < x*
*
v′( ws )
π s (x )
i∈B
i∈B
By hypothesis, the likelihood ratio
π s ( x* )
is increasing in the output state s. Thus the
π s ( x)
right hand side of this expression increases with s. Therefore, since v(⋅) is concave, ws is
increasing in s. That is, for efficiency, the higher the output, the higher is the payment to
the agent.
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