You are the manager of a firm that produces output in two plants. The demand for your firm's product is P = 78 - 15Q, where Q = Q1 + Q2. The marginal cost associated with producing in the two plants are MC1 = 3Q1 and MC2 = 2Q2. How much output should be produced in plant 1 in order to maximize profits?

charged to maximize profits is 40.5

Explanation:

given data

P = 78 - 15 Q

Q = Q1 + Q2

MC1 = 3Q1

MC2 = 2Q2

solution

we get here first total revenue and marginal revenue that is

total revenue TR = P × Q   ............................1

total revenue TR = 78Q - 15Q²

so

marginal revenue MR =    

marginal revenue MR = 78 - 30Q

and now we get here

marginal revenue MR = MC1 = MC2

put here value

78 - 30Q1 - 30Q2 = 3 Q1 or 33 Q1 = 78 - 30Q2   ......................................2

78 - 30 Q1 - 30 Q2 = 2 Q2 or Q2 = 78 - 30Q1/32   ...................................3

by equation 2 and 3  we get here

33 Q1 = 78 - 30 (78 -    )

so here Q1 = 1 and

Q2 = 78 -  

Q2 = 1.5

and here Q will be

Q = Q1 + Q2

Q = 1 + 1.5

Q = 2.5

so we get value of P that is

P = 78 - 15 Q

P = 78 - 15 (2.5)  

P = 40.5

so charged to maximize profits is 40.5