ECONOMICE HARD QUESTION

2007-05-06 9:31 pm

I TRIED...BUT QUITE HARD
THANKS FOR ANYONE HELP

1)A monopolistic producer of two goods has a joint total cost function

TC=10Q(1)+Q(1)Q(2)+10Q(2)

where Q(1) and Q(2) denote the quantities of Good 1 and Good 2, respectively. If P(1)and P(2) denote the corresponding prices then the demand equations are

P(1) = 50-Q(1)+Q(2)
P(2) = 30+2Q(1) - Q(2)

Find the maximum profit if the firm is contracted to produce a total of 15 goods of either type.

回答 (1)

仲泉

2007-05-06 10:48 pm

✔ 最佳答案

Produce a total of 15 goods of either type, so:
Q(2)=15-Q(1)

TC=10Q(1)+Q(1)(15-Q(1))+10(15-Q(1))
=10Q(1)+15Q(1)-Q(1)^2+150-10Q(1)
=150+15Q(1)-Q(1)^2

P(1)=50-Q(1)+15-Q(1)
=65-2Q(1)

P(2)=30+2Q(1) - (15-Q(1))
=30+2Q(1) - 15+Q(1)
=15+3Q(1)

TR=P(1)Q(1)+P(2)Q(2)
=(65-2Q(1))Q(1)+(15+3Q(1))(15-Q(1))
=65Q(1)-2Q(1)^2+225+30Q(1)-3Q(1)^2
=225+95Q(1)-5Q(1)^2

Profit=TR-TC
=225+95Q(1)-5Q(1)^2-(150+15Q(1)-Q(1)^2)
=75+80Q(1)-4Q(1)^2

d(Profit)/dQ(1)=80-8Q(1)

To maximize profit, make d(Profit)/dQ(1)=0,so:
80-8Q(1)=0
8Q(1)=80
Q(1)=10
so, Q(2)=15-10=5
P(1)=65-20=45
P(2)=15+30=45
TR=45*10+45*5=675
TC=10*10+10*5+10*5=200

Therefore,
Maximum Profit=675-200=475

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