Problem Set 4 Solutions
ECON105 Industrial Organization and Firm Strategy
Professor Michael Noel
University of California San Diego
1.
Claudia operates a fruit stand and sells bananas and coconuts. Marginal cost is zero as
Claudia just picks up the fruit as it falls from the trees onto her land. There are two
consumers, Bavanee and Cecilia. Cecilia values a banana at $4 and a coconut at $9. For
Bavanee, her valuations are $6 and $3 respectively. Show that Cecilia could make more
money by bundling one banana and one coconut into a fruit basket and selling the basket
rather than selling the fruits separately.
Separate: p
banana
= 8, p
coconut
= 9, TR
separate
= 17.
Bundled: p
basket
=9, TR
basket
= 18.
2.
Two firms compete Cournot. The market demand curve is Q = 16/p and each firm faces a
cost function C(qi) = cqi.
a.
Find the equilibrium firm quantities, industry quantity, price and firm profits.
Let us do maximize the profits of either firm (i) by choosing this firm’s quantity (that is
what makes the game ―Cournot‖) as a function of the opponent’s quantity (j):
0
2
16
0
2
0
)
(
)
(
0
)
(
:
*
)
(
*
)
(
max
2
2
2
2
2
2
cQ
j
Q
iQ
j
c
cQ
i
Qj
cQ
j
Q
j
c
cQ
i
Qi
c
c
FOC
cQ
i
C
P
Qi
This last equation is still just the firstorder condition which implicitly defines the optimal
quantity for firm i as a function of the quantity of firm j and the parameter c. If we wanted
to, we could express this relationship explicitly by solving this quadratic equation for
Qi(Qj) which would give us the bestresponse function of firm i: Ri(Qj;c)=Qi(Qj). But
what we are looking for ultimately is Qi and Qj only as functions of the parameter c and
not each other. We could find those optimal quantities by solving a system of two
equations, where the two equations are just the two FOCs for the two firms. One FOC is
just the one we found above for firm i, and another one is just like it but with switched
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View Full Documentsubscripts everywhere which would come out if we resolved the above problem for firm j
instead:
0
2
16
:
0
2
:
2
2
2
2
cQi
Qi Qj
c
cQj
Qi
FOCj
cQj
c
cQi
Qj
FOCi
However, since the two firms are symmetrical (have the same MC=c), we know that there
must be a symmetric equilibrium (i.e., equal quantities), so we can replace one of those
two equations with the much simpler equation: Qi=Qj.
cQ
j
Q
iQ
j
c
cQ
i
0
2
2
2
→
0
2
16
2
2
cQi
Qi Qi
c
→
0
2
2
2
2
c
→
0
4
2
cQ
i
→
0
)
4
(
4
When both firms are producing at Q1=Q2=4/c, neither has any incentive to change its
quantity so these quantities must be a Nash Equilibrium. (Note that it is not an
equilibrium for both firms to produce zero, because either one would have the incentive
to behave like a monopolist if they thought their opponent dropped out.)
Total quantity is Q=Q1+Q2=8/c. Prices are P=16/Q=2c.
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 Spring '10
 Game Theory, Supply And Demand, Firm

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