Let Q_{1} and Q_{2} be the outputs of plants A and B. Solve the demand function for price and find total revenue:
P=100  Q/4
TR = PQ = 100Q  Q²/4
TR = 100(Q_{1}+Q_{2})  (Q_{1}+Q_{2})²/4
Find total cost:
TC = 20Q_{1}+0.5Q_{2}^{2}
Profit = total revenue minus total cost:
Π = TR  TC = 100(Q_{1}+Q_{2})  (Q_{1}+Q_{2})²/4  20Q_{1}  0.5Q_{2}^{2}
Now take the partial derivatives and set them to zero to find the critical point(s):
Π_{1} = 100  (Q_{1}+Q_{2})/2  20 = 0
Π_{2} = 100  (Q_{1}+Q_{2})/2  Q_{2} = 0
Solving these two equations gives only one critical point:
Q_{1}=140, Q_{2}=20.
You can check that this critical point is indeed a maximum of the profit function by checking the 2nd partials:
Π_{11} = 1/2 < 0, Π_{22} = 3/2 < 0, Π_{11}Π_{22
} Π_{12}^{2} = 1/2 > 0.
These are the conditions for a maximum.
Therefore, the total output of the two plants is Q=140+20=160, at a price of P=100160/4=60.
Nov 14

Andre W.