Trouble Understanding Cornout Competition?

Question

Find the quantity of units produced (Q) for each company (A+B) given the price equation p=5,000-Q(a)-Q(b) and the total cost functions of each company

TC(A)=500,000+100Q(a)+.5Q(a)^2

TC(B)=200,000+200Q(b)+Q(b)^2

Answer 1

Find the profit function:

Pr(A)=Q(A)(5000-Q(A)-Q(B))-500000-100Q...

Take the derivative, then find the function in terms of Q(B)

Pr(A)'=5000-2Q(A)-Q(B)-100-Q(A)

3Q(A)=4900-Q(B)

Q(A)=4900/3-Q(B)/3

Find the profit function:

Pr(B)=Q(B)(5000-Q(A)-Q(B))-200000-200Q...

Take the derivative, then find the function in terms of Q(A)

Pr(B)'=5000-Q(A)-2Q(B)-200-2Q(B)=0

4Q(B)=4800-Q(A)

Q(B)=1200-Q(A)/4

Combine the two profit functions and solve for Q(A)

Q(A)=4900/3-(1200-Q(A)/4)/3

Q(A)=4900/3-400+Q(A)/12

11Q(A)/12=3700/3

Q(A)=1345

Q(B)=1200-(1345/4)=864