The TR curve of a monopoly firm is given by the equation
$T R = - q^{2} + 12 q - 20$ , where TR is in millions of dollars.

What is the maximum revenue of the firm?

$ 12 million

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$ 16 million

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$ 20 million

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$ 36 million

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Solution

The correct option is B
$ 16 million

The TR equation is given as $T R = - q^{2} + 12 q - 20$
This is the equation of an inverted parabola. We can rearrange the equation in the following manner.

$T R = - q^{2} + 12 q - 20 T R = (- q^{2} + 12 q - 36) + 36 - 20 (add and subtract 36) T R = - (q^{2} - 12 q + 36) + 36 - 20 T R = - (q - 6)^{2} + 16$

The term $- (q - 6)^{2}$ is always less than or equal to zero.
Hence, TR is maximum when this term is zero i.e. when q=6.
At this output level, TR = $16 million.

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