A monopolist manufactures and sells two competing products, call them I and II,

Question

A monopolist manufactures and sells two competing products, call them I and II, that cost $35 and $22 per unit, respectively, to produce. The revenue from marketing x units of product I and y units of product II is 103x + 114y - .04xy - .1x2 - .2y2. Find the values of x and y that maximize the monopolist's profits.

I am confused about this - I tried to determine f(x,y) and then determine the constraint, but I ended up getting myself confused. I know that at some point I will need to set the first derivative equal to 0, if you could walk me through trying to solve for one variable and then substitute back in, I would be really grateful - as my algebra skills are weak. Thank you! Any help would be greatly appreciated - thank you.

Explanation / Answer

cost marketing x units of product I and y units of product II is C(x)=35x+22y

The revenue from marketing x units of product I and y units of product II is R(x)=103x + 114y - 0.04xy - 0.1x2 -0.2y2.

profit from marketing x units of product I and y units of product II is P(x)=R(x)-C(x)

P(x)=103x + 114y - 0.04xy - 0.1x2 -0.2y2-35x-22y

P(x)=68x +92y - 0.04xy - 0.1x2 -0.2y2

for maximum profit Px=0, Py =0

Px=68 - 0.04y - 0.2x =0

=>x=[68-0.04y]/0.2

Py=92 - 0.04x - 0.4y =0

92 - 0.04[68-0.04y]/0.2 - 0.4y =0

=>y=200

x=[68-0.04y]/0.2

x=[68- 0.04*200]/0.2

x=300

Pxx=-0.2 ,Pyy=-0.4 ,Pxy=-0.04

D=PxxPyy-Pxy2

D=(-0.2 *-0.4)-(-0.04)2

D=0.0784 >0 ,Pxx<0

so

values of x=300 and y=200 that maximize the monopolist's profits.

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