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Finding optimal output and profit level

See the attached file.

Suppose that the technology of manufacturing widgets is that every firm entering the market has the same total cost curve, as follows:

TC=1000+5Q+0.1Q^2
Where : TC=Total cost per unit time
Q=Number of widgets produced per unit time

The market demand for widgets is:
Q=700-8P
Where P=price ($/widget)

a. What is the maximum number of firms that can survive in the market?
b. Suppose that, for whatsoever reason, there are only four firms in the market and, as noted above, they have identical cost curves. Assuming active price competition and no collusion among the firms, find the market supply function.
c. Given the assumptions and market supply function found in part b., above, find (i) the market price; (ii) the total quantity demanded; and (iii) the total cost, total revenue and profit for each firm.
d. Suppose that entry is attempted by a fifth firm, identical to the four firms already active in the market. Assuming that the fifth firm enters, find the new market supply curve.
e. Given the market supply curve of part d., above, find (i) the market price; (ii) the total quantity demanded; and (iii) the total cost, total revenue and profit for each firm.
f. What action could the original four firms take to prevent the entry of the fifth firm?

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a) TC=1000+5Q+0.1Q^2
Marginal Cost=MC=dTC/dQ=5+0.2Q
Average Total Cost=TC/Q=(1000/Q)+5+0.1Q
d(ATC)/dQ =-(1000/Q^2)+0.1

Let us find the output level at which ATC has minimum value.
d(ATC)/dQ=0
-(1000/Q^2)+0.1=0
Q^2=10000
Q=100

Marginal Cost at Q=100 is given by
MC=5+0.2Q=5+0.2*100=$25

In price competitive environment, each firm sets its output level such that MC=P
Price, P=MC=$25

Demand at P=$25 is given ...

Solution Summary

The solution describes the steps to calculate equilibrium market price and profits of firms existing in the market. It also studies the effect on the market and profits of firms if another firm attempts to enter the market.

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