Download Problem Set 8: Decision under Uncertainty 1. Let X be a random

Problem Set 8: Decision under Uncertainty
1. Let X be a random variable taking values x ∈ [0, 1]. Let an agent’s Bernoulli utility function
be u(x) = xβ where 0 < β < 1. Let the probability distribution of X be P r[X ≤ x; α] = F (x; α) =
x1+α where α 6= 0.
i. Characterize the agent’s expected payoff and the certainty equivalent. ii. Show that as β
increases, the agent becomes more risk averse and his expected payoff falls. Show that the certainty
equivalent increases. iii. Show that if α1 > α2 , F (x; α1 ) first-order stochastically dominates
F (x; α2 ). Show that the certainty equivalent is increasing in α and that the agent’s expected payoff
is increasing in α.
i. The agent’s expected payoff is
E[U (X)] =
Z
1
(1 + α)xα+β dx =
0
1+α
1+α+β
and the certainty equivalent is
c(F, u) =
1+α
1+α+β
1β
ii. The Arrow-Pratt coefficient for this agent is:
rA = −
β(β − 1)xβ−2
(1 − β)
=
β−1
βx
x
which is obviously decreasing in β, and goes to zero as β → 1. Now, we know that as the agent
becomes more risk averse, the certainty equivalent falls. More risk aversion is equivalent with
a higher Arrow-Pratt coefficient. Therefore, the certainty equivalent increases as β goes up.
The hard way to show that c(F, u) is increasing in β is to take a derivative:
∂c(F, u)
=
∂β
−1
log
β2
1+α
1+α+β
1+α
11+α+β
−
β 1 + α (1 + α + β)2
−1
log
β
=
or
1+α
1+α+β
1
−
(1 + α + β)
1
1
1
−
β 1 + α + c (1 + α + β)
1
β
1+α
1+α+β
1β
1β
1+α
=
1+α+β
1β
−1
1
1+α
1
1
=
log(1 + α) + log(1 + α + β) −
β
β
(1 + α + β) β 1 + α + β
1β
1
1
1
1+α
=
(log(1 + α + β) − log(1 + α)) −
β
(1 + α + β) β 1 + α + β
1
β
1+α
1+α+β
1β
≥0
where c ∈ (α + β, 1 + α + β). This is positive, since c < β, and the certainty equivalent is
increasing as risk aversion decreases.
iii. If α1 > α2 , we have x1+α1 < x1+α2 for all x ∈ [0, 1], and first-order stochastic dominance
follows. The rest is like work from part ii.
2. There is an agent with a strictly concave Bernoulli utility function u(), trying to decide how
much to invest in a risky asset. Each share of the assets yields rh with probability p and rl with
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probability 1 − p, with rh > 1 > rl . The consumer has wealth w, and the price of a share is q. Let
the number of shares purchased be z. Assume ph rh + pl rl ≥ q, so that the expected return on a
share is greater than its cost.
i. Write out the agent’s expected utility maximization problem.
ii. Characterize necessary and sufficient conditions for a particular portfolio z ∗ to be a solution to
the investment problem. Is z ∗ > 0?
iii. How is the optimal z ∗ varying in p? rh ? w? q? If you can’t derive a clear prediction, explain.
iv. Suppose the agent’s utility function is u(x) = ψ(v(x)), where ψ() and v() are concave functions.
Would an agent with utility function v() invest more or less than the consumer with utility function
u()? If added assumptions are necessary to obtain a prediction, explain their economic content.
i.
max pu(w + (rh − q)z) + (1 − p)u(w + (rl − q)z)
z≥0
ii. The FONC is
pu′ (w + (rh − q)z ∗ )(rh − q) + (1 − p)u′ (w + (rl − q)z ∗ )(rl − q) + µ = 0
and at an interior solution, the SOSC is
pu′′ (w + (rh − q)z ∗ )(rh − q)2 + (1 − p)u′′ (w + (rl − q)z ∗ )(rl − q)2 < 0
which is automatically satisfied. z ∗ = 0 is a solution if
pu′ (w)(rh − q) + (1 − p)u′ (w)(rl − q) < 0
or
prh + (1 − p)rl < q
so that the expected return of a share is less than the price.
iii. Use the implicit function theorem. Without assuming DARA, w is ambiguous,
iv. Substitute into the FONC for u = ψ(v(()) to get
pψ ′ (v(w+(rh −q)zu∗ ))v ′ (w+(rh −q)zu∗ )(rh −q)+(1−p)ψ ′ (v(w+(rl −q)zu∗ ))v ′ (w+(rl −q)zu∗ )(rl −q) = 0
Now divide to get
p
ψ ′ (v(w + (rh − q)zu∗ )) ′
v (w + (rh − q)zu∗ )(rh − q) + (1 − p)v ′ (w + (rl − q)zu∗ )(rl − q) = 0
ψ ′ (v(w + (rl − q)zu∗ ))
{z
}
|
<1
Since rh > q, this implies
pv ′ (w + (rh − q)zu∗ )(rh − q) + (1 − p)v ′ (w + (rl − q)zu∗ )(rl − q) > 0
To get this FONC equal to zero, we have to increase zu∗ to zv∗ , since the SOSC implies the
equation defining the FONC is decreasing. Therefore, zv∗ > zu∗ .
Suppose DARA. Then decreasing w is a concave transformation, and the above work shows
that more agents with more concave utility functions will purchase less of the risky asset. So
under DARA, increasing wealth increases purchases.
3. A risk averse agent with positive, increasing, concave Bernoulli utility function u(x) buys
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insurance against a stochastic loss, L. The probability density function of the loss is f (L) on [a, b]
with b > a > 0, with mean L̄. The price of a unit of insurance is q, and each unit pays out α if a
loss occurs. The probability of a loss occurring is p, and the agent has wealth w. Let x ≥ 0 be the
number of units of insurance the agent purchases.
i. Write out the agent’s expected utility and characterize the profit-maximizing level of insurance. Under what circumstances does the agent buy no insurance?
ii. Let L′ be a mean-preserving spread of L. Does the agent buy more or less insurance at L′ or L?
iii. If the distribution F1 (L) first-order stochastically dominates F2 (L), does the agent buy more
or less insurance under F1 (L) or F2 (L)?
iv. If agent α is “more risk averse” than agent β. Define this term rigorously, and use your definition to show which agent buys more insurance.
v. How does the amount of insurance purchased vary with w?
i. The agent solves
max p
x≥0
Z
b
u(w − L + (α − q)x)f (L)dL + (1 − p)u(w − qx)
a
with FONC
p
Z
b
u′ (w − L + (α − q)x)(α − q)f (L)dL − q(1 − p)u′ (w − qx) + µ = 0
a
and SOSC at an interior solution
Z b
u′′ (w − L + (α − q)x)(α − q)2 f (L)dL + (1 − p)q 2 u′′ (w − qx) < 0
p
a
which is automatically satisfied. x∗ = 0 is a solution if
Z b
u′ (w − L)(α − q)f (L)dL − q(1 − p)u′ (w) ≤ 0
p
a
which doesn’t reduce to something trivial that doesn’t include u(w).
ii. Let L = L′ + ε, where E[ε|L′ ] = 0. Then the FONC can be written
Z Z
p
u′ (w − L′ − ε + (α − q)x)(α − q)f [ε|L′ ]f (L′ )dεdL − q(1 − p)u′ (w − qx) = 0
Now, if u′ () is convex, we have by Jensen’s inequality that
Z Z
0=p
u′ (w − L′ − ε + (α − q)x)(α − q)f [ε|L′ ]f (L′ )dεdL − q(1 − p)u′ (w − qx) ≥
Z
p u′ (w − L′ + (α − q)x)(α − q)f (L′ )dL − q(1 − p)u′ (w − qx),
so that — since the FONC is decreasing — the agent will buy less insurance when faced with
just the risk of L′ rather than L = L′ + ε. If u′ () is concave, Jensen’s inequality would imply
the opposite result. Note that you can’t apply Jensen’s inquality to u′ () without taking a stand
on whether u′ () is concave or convex: Assuming u(x) is concave implies nothing about whether
u′ (x) is concave.
iii. Let F1 (L) ≤ F2 (L) for all L, and consider the FONC:
Z b
u′ (w − L + (α − q)x)(α − q)f1 (L)dL − q(1 − p)u′ (w − qx) = 0
p
a
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Integrate by parts to get
Z b
′′
2
′
u (w − L + (α − q)x)(α − q) (1 − F1 (L))dL −q(1−p)u′ (w−qx) = 0
p u (w − a + (α − q)a) −
a
Now, 1 − F1 (L) ≥ 1 − F2 (L) for all L, so that
Z b
′′
2
′
u (w − L + (α − q)x)(α − q) (1 − F1 (L))dL −q(1−p)u′ (w−qx) ≥
0 = p u (w − a + (α − q)a) −
a
Z b
′′
2
′
u (w − L + (α − q)x)(α − q) (1 − F2 (L))dL −q(1−p)u′ (w−qx)
p u (w − a + (α − q)a) −
a
so that the FONC at F1 (L) is zero, but the FONC at F2 (L) evaluated at the solution for F1 (L)
is negative. Therefore, the agent should buy less insurance at F2 (L) than at F1 (L), since the
FONC is a decreasing function in x.
iv. An agent u is more risk averse than agent v if u(x) = ψ(v(x)), where ψ() is an increasing,
concave function. Consider the FONC for u,
p
Z
b
ψ ′ (v(w−L+(α−q)xu ))v ′ (w−L+(α−q)xu )(α−q)f (L)dL−q(1−p)ψ ′ (v(w−qxu ))v ′ (w−qxu ) = 0
a
Dividing by the ψ ′ (v − w − a + (α − q)xu ) — which is less than all of the other ψ ′ (...) terms—
yields
p
Z
a
b
ψ ′ (v(w − L + (α − q)xu )) ′
v (w − L + (α − q)xu )(α − q)f (L)dL
ψ ′ (v(w − a + (α − q)xu ))
ψ ′ (v(w − qxu ))
− q(1 − p) ′
v ′ (w − qxu ) = 0
ψ (v(w − a + (α − q)xu ))
and throwing away the fractional terms places more weight on the negative terms in the integral,
reducing the value of the FONC, so that
p
Z
b
v ′ (w − L + (α − q)xu )(α − q)f (L)dL − q(1 − p)v ′ (w − qxu ) < 0
a
so that the less risk averse agent v is consuming too much insurance at u’s solution, xu , and
xv ≤ xu .
v. Suppose DARA. Then a decrease in w is a concave transformation, so by iv, the agent
would purchase more insurance with less wealth. Thus, more wealth implies less insurance
under DARA. Basically, more wealth leads to less risk aversion, so agents “self-insure” more.
4. At a first-price auction, the buyers each simultaneously and non-cooperatively submit a
sealed bid. The seller opens all the bids, and the highest bidder wins a unit of the good and pays
the seller his bid.
Consider a first-price auction auction, where a buyer with value v for the good submitting bid
b(v) wins with probability p(b(v)), and receives a monetary payoff of v − b(v) if his bid is the highest
and pays his bid, but gets zero otherwise. The buyer is also risk averse, with increasing, strictly
concave utility function u(x) that satisfies u(0) = 0. Let p(b) be the probability that a bid of b
wins.
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i. If buyers solve
max p(b(v))u(v − b(v)),
b(v)
what are the first- and second-order conditions in b(v)? When does a buyer bid zero? Can you sign
b′ (v)?
ii. Let G(z) = p(b(z)), with G′ (z) = g(z), and consider
max G(z)u(v − b(z))
z
where b(v) is the bidding function from i; this is like telling the seller any value z, and the seller
then bidding for you as if your type was z. Show that
b′ (v) =
u(v − b(v)) g(v)
u′ (v − b(v)) G(v)
Is this positive? If u(x) = x, we then have
(v − γ(v))
g(v)
= γ ′ (v)
G(v)
which would characterize the bid function γ(v) in the risk neutral case.
iii. Show that if a function u(x) is strictly concave and u(0) = 0, then for all x > 0 we have
u(x)/u′ (x) > x. Use this with part ii to show that γ(v) > b(v) implies b′ (v) > γ ′ (v). Show that
b(0) = γ(0) = 0.
iv. Conclude that b(v) > γ(v) for all v > 0.
v. Explain why this proves that at first-price auctions, a set of risk-averse buyers will bid more
than a set of risk-neutral ones (think fundamental theorem of calculus). Is it always true that if
one set of buyers are more risk averse than another set of buyers, they will always bid more?
vi. At a second-price auction, the buyers each independently submit a sealed bid; the highest bidder
wins the good, but pays the second highest bid to the seller. Explain why bidding b(v) = v is a
payoff maximizing strategy, regardless of the level of risk aversion or what the other buyers do.
Which auction format will raise more revenue, the first- or second-price auction? (It is unlikely you
will get the correct answer, so don’t invest too much time in this part. It is just a fun question to
think about.)
i. The FONC for
max p(b(v))u(v − b(v)),
b(v)
is
p′ (b(v))u(v − b(v)) − p(b(v))u′ (v − b(v)) + µ = 0
and the SOSC at an interior solution is
p′′ (b(v))u(v − b(v)) − 2p′ (b(v))u′ (v − b(v)) + p(b(v))u′′ (v − b(v)) < 0
The buyer should bid zero if
p′ (0)u(v) − p(0)u′ (v) < 0
and if p(0) = 0,
p′ (0)u(v) < 0
so that the probability of winning is decreasing in the bid at zero. This is impossible, since if
you have a zero probability of winning, bidding something can’t make you less likely to win, so
everyone always bids something.
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The sign of b′ (v) is the same as
p′ (b(v))u′ (v − b(v)) − p(b(v))u′′ (v − b(v))
which is positive as long as the probability of winning is increasing in the bid.
ii. The FONC for the transformed problem
max G(z)u(v − b(z))
z
is
g(z)u(v − b(z)) − G(z)u′ (v − b(z))b′ (z) = 0
or
b′ (z) =
g(z)u(v − b(z))
G(z)u′ (v − b(z))
This is definitely positive, since g/G > 0, and u/u′ > 0.
iii. Use a Taylor series:
u(0) = u(x) + u′ (x)(0 − x) + u′′ (c)(0 − x)2 /2
and since u(0) = 0,
u′ (x) = u(x)/x + u′′ (x)|x|/2 < u(x)/x
The inequality follows immediately. b(0) = γ(0) = 0 since if an agent with value 0 bids
something strictly positive, she wins with positive probability, but gets a worse payoff than if
she had bid nothing, since u() is increasing.
iii. Now, suppose to the contrary that γ(v) > b(v). Then we have
Z v
Z v
b′ (z)dz
γ ′ (z)dz − b(0) −
γ(v) − b(v) = γ(0) +
0
0
Since γ(v) > b(v), we have b′ (v) > γ ′ (v) from the previous part, so
Z v
γ ′ (z) − b′ (z) dz
=
{z
}
0 |
<0
which leads to a contradiction. Therefore, b(v) > γ(v).
iv. Since the seller’s revenue is
Z
Z
b(v)f(1) (v)dv > γ(v)f(1) (v)dv
where f(1) (x) is the density of the first-order statistic of N draws from the density f (v), the
FPA with risk aversion will raise more revenue than the FPA with no risk aversion.
v. Here, we can pin down the bid at zero and then characterize how the slopes of the two
bid functions compare. Now, if we think about a concave transformation, u(w) = ψ(h(w)), the
FONC becomes
g(z)ψ(h(v − bu (z)))
b′u (z) =
′
G(z)ψ (h(v − bu (z))h′ (v − bu (v))
which doesn’t seem to compare easily to the FONC for h. Now, we rederive the intermediate
step that u(x)/x > u′ (x), using the fact about concavity we just proved by linearizing ψ()
around h(). The result follows that more risk averse bidders will bid more, so the results are
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not just for the case of risk aversion compared with risk neutrality, but any comparison of any
group of bidders against a more risk averse group.
vi. In the second-price auction, it is a weakly dominant strategy to bid honestly. Suppose
you bid b′ > v. Then if you win but would have lost bidding honestly, you get a negative payoff;
if you were going to lose anyway, this doesn’t help. Suppose you bid b′ < v. Then if you lose
but would have won bidding honestly, you trade a positive payoff for zero; if you were to win
anyway, this doesn’t help. Therefore, bidding b = v weakly dominates over- or under-bidding
in every state of the world, so it is payoff-maximizing.
The revenue equivalence principle is that in an independent private values environment with
risk neutrality, any auction in which the highest bidder wins the good and losers pay nothing
gives the same revenue. Consequently, the first- and second-price auctions with risk neutrality
yield the same amount of revenue. If there is more bidding in the FPA with risk aversion
than the FPA with risk neutrality, then, the FPA will raise more revenue, since the bidding in
the SPA doesn’t depend on the risk aversion of the agents. C’mon, how cool is that? If you
want to make bidding easy and stress-free for bidders, you should offer the SPA. If you want
to maximize revenue and you think the bidders are sufficiently capable to determine a bid for
the FPA, then you should choose that. There’s clearly a trade-off in where the computational
burden placed is placed in making the decision: The FPA requires a lot of work by the bidders,
but the SPA requires the auctioneer to do the work. This is all super interesting, right? Right??
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