The optimal condition is mc q 1 q 2 mr 1 q 1 mr 2 q 2

The optimal condition is
MC ( q
1
+ q
2
) = MR
1
( q
1
) = MR
2
( q
2
) ;
which gives us a system of two equations with two unknowns:
q
1
+ q
2
= 50 ° q
1
q
1
+ q
2
= 100 ° 4 q
2
:
Thus, we have
q
°
1
= q
°
2
=
50
3
:
Plug these quantities into each demand function to get
p
°
1
=
125
3
; p
°
2
=
200
3
:
Demand elasticities are
°
1
=
@q
1
@p
1
p
1
q
1
= ° 2 ±
125
3
±
3
50
= ° 5
°
2
=
@q
2
@p
2
p
2
q
2
= °
1
2
±
200
3
±
3
50
= ° 2 :
(e) As we learned in class, the equilibrium price and quantity are determined by both supply and
demand functions. Assume demand and supply are given as follows:
D
t
= ± + ²P
t
+ ³X
D
t
+ "
t
(1)
S
t
= ´ + µP
t
+ ¶X
S
t
+ u
t
;
where X
D
t
is a demand shifter and X
S
t
is a supply shifter. If you solve these equations for ( Q
t
; P
t
)
by letting D
t
= S
t
= Q
t
; you will °nd that P
t
is a function of "
t
: Thus, P
t
is endogenous in equation
1

(1) because P
t
is correlated with an unobservable variable ( " ). Therefore, if you regress the observed
quantity on the price (and X
D
t
) by OLS, your estimate for ² would not be consistent.
(f) Example 1. The di/erence between the marginal cost of a monopolist and the price it charges to
consumers depends on the price elasticity of demand. If demand is elastic, the degree of monopoly
power would be low. Example 2. If entry is not costly in the industry in question, entry threats
(existence of potential entrants) may prevent incumbents with a high market share from exercising
their market power.
2