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A firm has the following (long-run) cost function: TC(Q)=Q^3-10Q^2+100Q. Suppose the market demand is given by...

A firm has the following (long-run) cost function: TC(Q)=Q^3-10Q^2+100Q. Suppose the market demand is given by Q=200-P, and all firms have identical cost functions. How many firms are present in the market.

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Answer #1

The long-run price is equal to the minimum average total cost.

The minimum of average total cost output is found by differentiating the ATC function and equating to the zero.

ATC=TC/Q=Q^2-10Q+100

dATC/dQ=2Q-10

equating to zero

2Q-10=0

2Q=10

Q=5

P=ATC=5^2-10*5+100

=75

The price is $75

the quantity demanded by the market at a price is found by the demand curve

Q=200-75=125

Number of firms =market quantity/individual firms quantity

=125/5

=25

25  firms are present in the market.

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