Question

StarPoint manufactures engine parts for Ford using two plants, one in Austin, TX and one in Fairfax, VA.

The Inverse Demand functions (demand curves) for the two StarPoint plants are:

Austin: P_A= 30 - 0.125Q_A

Fairfax: P_F= 35 - 0.25Q_F

Where Q_A and Q_F are outputs of engine parts (in units) from the Austin and Fairfax plants, respectively.
The marginal cost of production in Austin (MC_A) is 10 and the marginal cost of production in Fairfax (MC_F) is 5.

If StarPoint has a total produce limit of 110 units of engine parts, how many of the engine parts should be produced in Fairfax (Q_A)?

Answer 1

Given: P(A)=30-0.125Q(A) ------------------(1)

P(F)=35-0.25Q(F) --------------------(2)

MC(A)=10, MC(F)=5

So, multiplying equation (1) with Q(A) and equation (2) with Q(F) to get total revenue function of (A) and (F),

TR(A)=30Q(A)-0.125Q(A)^2 ---------------(3), TR(F)=35Q(F)-0.25Q(F)^2 ---------------------(4)

Differentiating eq. (3) w.r.t. Q(A) to get marginal revenue of (A) and eq. (4) w.r.t. Q(F) to get marginal revenue of (F),

MR(A)=30-0.25Q(A), MR(F)=35-0.5Q(F)

Equating both equations to their marginal cost function to get values of Q(A) and Q(F),

MR(A)=MC(A) MR(F)=MC(F)

30-0.25Q(A)=10 35-0.5Q(F)=5

Q(A)=80 Q(F)=60

Inputting values of Q(A)=80 and Q(F)=60 into equation (1) and equation (2) to get values of P(A) and P(F)

P(A)=30-0.125Q(A) P(F)=35-0.25Q(F)

P(A)=30-0.125(80) P(F)=35-0.25(60)

P(A)=20 P(F)=20

At this point the question is confusing as it asks about the quantity of Fairfax produced with the total limit being 110 but in brackets mentions Q(A) which is the quantity for Austin. Assuming, quantity of Fairfax is the one that is required:

Now, we notice that price of production for each plant is the same. However, the quantity is different that is Q(A)>Q(F). So, we will assume that the Austin plant produces at full capacity i.e, Q(A)=80 because it is more profitable. Thus, quantity to be produced in Fairfax is:

Q(A)+Q(F)=Q(Total)

80+Q(F)=110

Q(F)=30

Thus, the Fairfax plant should produce 30 engine parts