1
. Consider the following functions. Show that each of those has a diminishing MRS,
but that they exhibit constant, increasing, and decreasing marginal utility, respectively.
a.
U(x; y) = xy
b.
U(x; y) = x
2
y
2
c.
U(x; y) = ln(x) + ln(y)
2.
Suppose there a
I
farmers, each of whom has the right to graze cows on the village
common. The amount of milk a cow produces depends on the total number of cows,
N
, grazing on the green.
The revenue produced by
n
i
cows is
n
i
V(N)
for N<
N
, and
V(N)≡0 for N≥
N
, where V(0)>0, v’<0, and v’’≤0.
Each cos cost
c,
and cows are
perfectly divisible.
Suppose v(0)>
c
. Farmers simultaneously decide how many cows
to purchase; all purchased cows will graze on the common.
a.
Write this a game in strategic form.
b.
Find the Nash Equilibrium, and compart it against the social optimum.
c.
Discuss the relationship between this game and the Cournot oligopoly model.
3.
Snoopy is holding an auction to sell an ice cream cone. There are two bidders,
Lucy (bidder i) and Charlie (bidder ii). Snoopy announces the following rules for the
auction.
1. Each bidder is to submit a sealed bid for the ice cream cone.
2. A bid made be one of three values, $0, $1, or $2.
3. The ice cream cone will be awarded to the bidder submitting the highest
bid, but the winner will pay the bid submitted by the loser (i.e., this is a
second-price auction). The loser will pay nothing.
4. Ties in bidding will be broken by flipping a coin (i.e., the winner will be
selected
randomly with equal probability and will pay the bid at which the
tie occurred).
Suppose that Lucy and Charlie are both risk-neutral (i.e., a 50% chance of receiving
surplus of v dollars provides expected utility of 0:5v dollars). Lucy's utility from
consuming the ice cream cone is $1 and Charlie's is $2.
a.
What are the strategy sets, S1 and S2, of each player?
b.
Letting player i (Lucy) control the rows and player ii (Charlie) control the
columns, write down the matrix game representing this auction.
c.
What are player ii's best-responses if: player i bids $0? if she bids $1? If she
bids $2? Does player i have a strictly/weakly dominant strategy? If so, what is
it?