A monopolist manufactures and sells two competing products, call them I and II, that cost $42 and $35 per unit, respectively, to produce. The revenue from marketing x units of product I and y units of product II is R(x, y) = 110x + 127y − 0.04xy − 0.1x^2 − 0.2y^2. Find the exact values of x and y that maximize the monopolist's profits.

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topeadeniran2 topeadeniran2 · Answer 1 · 2022-02-08T21:54:01+01:00

The values of x and y that can be used to maximimize the profit of the monopolist will be 300 and 200 respectively.

The cost function will be:

C(x, y) = 42x + 35y

The revenue function will be:

R(x, y) = 110x + 127y − 0.04xy − 0.1x² − 0.2y²

The profit function will be:

= Revenue - Cost

= (110x + 127y − 0.04xy − 0.1x² − 0.2y²) - 42x + 35y

= 68x + 92y - 0.04xy − 0.1x² − 0.2y²

We'll maximize P with respect to x. This will be:

= 68 - 0.04y - 0.2x

We'll maximize P with respect to y. This will be:

= 92 - 0.04x - 0.4y

Equating them to zero will give:

x = 340 - 0.2y

92 - 0.04x - 0.4y = 0

92 - 0.04(340 - 0.2y) - 0.4y = 0

92 - 13.6 + 0.008y - 0.4y = 0

78.4 - 0 392y = 0

y = 78.4/0.392

y = 200.

x = 340 - 0.2y

x = 340 - 0.2(200)

x = 340 - 40

x = 300

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