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Unformatted text preview: ky equation in question is
@x1
@xh
1
=
@p2
@p2 1 x2 @ x1
:
@y Given the results from (a) and (b) and that p1 = 1=2, we have
@x1
=
@p2 164 y
;
(164 =2 + p2 )2 @xh
1
= 0;
@p2 x2 = y
;
4 =2 + p
16
2 @x1
164
=4
:
@y
16 =2 + p2 Obviously, the Slutsky equation does hold here.
Solution 2 [All students get full credit for this question as it is outside the coverage.]
See notes 11, pp. 611.
Solution 3 (a) Given Roy’ identity, the group1’ Marshallian demand for good 1 is
s
s
x1 =
1
x2 =
1 @ v1 (p; y) =@p1
=
@v1 (p; y) =@y
@ v2 (p; y) =@p1
=
@v2 (p; y) =@y 1=p1
y
=
2=y
2p1
2
2
y =p1
y
=
2y=p1
2p1 Then the aggregate demand is
Qd = 40 200
8000
100
+ 60
=
:
2p1
2p1
p1 (b) The …rm’ pro…tmaximization problem is given by
s
max p1 q
q c(q ) = p1 q q2: The …rstorder condition is
p1 2q = 0: Solving the above gives the output supply function q (p2 ) = p1 =2 and the pro…t function is
p2 =4.
1
To solve for the price, use the marketclearing condition:
8000
p1
= Qd = Qs = 160 :
p1
2
Thus, the price is p1 = 10.
(c,i) The equilibrium price should be p1 = 2q . This is a direct result of the theorem that
says an equilibrium in the Bertrand competition is when at least two …rms charge the same
price equal to the marginal cost.
(c,ii) Given p1 = 4000, we have q = p1 =2 = 2000. The marketclearing condition is
8000
8000
=
= Qd = Qs = 2000n:
p1
4000
2 Thus n = 1=1000. 3...
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This document was uploaded on 03/26/2014 for the course ECON 310 at Queens University.
 Winter '13
 Utility

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