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Calculating optimal output and profit

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Two manufacturers, a & b have combined demand given q=105-p there total costs are given by

TCa=5qa+.5qa^2 and TCb=5qb+.5qb^2.

A. If they successfully collude determine the total output and joint profits.

B. If they cannot collude and instead produce where the market price equals marginal cost determine the total output and joint profit.

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Solution Preview

A. If they successfully collude determine the total output and joint profits.

If they collude successfully, cost function is given by
TC=5q+0.5q^2
Marginal cost=MC=dTC/dq=5+q

Where q is combined output.
q=105-p
or p=105-q
Total revenue=TR=p*q=(105-q)*q=105q-q^2
Marginal Revenue=dTR/dq=105-2q

They will set their combined output level such that MR=MC
105-2q=5+q
3q=100
q=100/3=33.33
p=105-q=105-33.33=71.67
Total revenue=TR=p*q=71.67*33.33=2388.76
Total Cost=5*33.33+0.5*33.33^2=722.09
Joint ...

Solution Summary

The solution calculates the optimal outputs and profits of two firms in case they collude and in case if they do not collude.

$2.19