Question

Rexburg Technologies operates two plants. The demand equation for Rexburg's product is P = 38 – 2.5Q, where Q is in thousands of units. The marginal cost of production in the two plants are MC1 = 2Q1 and MC2 = 4Q2, respectively. To maximize profits, Rexburg should charge a price of:

A. $32.50

B. $23

C. $8

D. $10.75

E. None of the options

Answer 1

The total quantity produced by the firm =Q=Q1+Q2

The demand function of the firm can be written as P= 38-2.5(Q1+Q2)= 38-2.5Q1-2.5Q2

We are given MC for both the plants of firm, we can calculate the total cost in each plant of the firm using integration.

TC1 = MC1 = 2Q1 = Q1²

And, TC2= MC2= 4Q2 = 2Q2²

And total cost to the firm for producing Q(Q1+Q2) units = TC = TC1+TC2 = Q1²+ 2Q2²

Total Revenue, TR = Price* Output

TR= P*Q = (38-2.5Q1-2.5Q2)( Q1+ Q2) = 38Q1-2.5Q1²-2.5Q2Q1 +38Q2-2.5Q2²-2.5Q1Q2

TR= 38Q1-2.5Q1² -5 Q1Q2 +38Q2-2.5Q2²

The profit function for the firm is -

Profit = TR-TC

Max Profit is the goal

Profit= 38Q1-2.5Q1²-5Q1Q2+38Q2-2.5Q2² - Q1²-2Q2²

Profit = 38Q1-3.5Q1²-5Q1Q2+38Q2-4.5Q2²

Using partial differentiation with respect to Q1 and Q2, we get two first order conditions,

1... 38-7Q1-5Q2=0

2.... 38-9Q2-5Q1=0

Solving these two equations-

38-7Q1= 5Q2 => Q2= (38-7Q1)/5 from(1)

Putting in (2)

38= 9(38-7Q1)/5 +5Q1

38*5= 342-63Q1+25Q1

190= 342-38Q1

38Q1= 342-190 => 38Q1= 152

Q1= 152/38 => Q1= 4

Hence, Q2 = (38-7Q1)/5 = 38-7*4/5= 38-28/5=10/5=2

Thus, profit maximizing quantity for firm from both plants is Q= Q1+Q2= 4+2= 6

Putting the profit maximizing quantity in the price function, we get,

P= 38-2.5*Q

P= 38- 2.5*6

P= 38-15

P= 23

Thus, the firm should charge a price equal to 23.

Correct option is B