Question

Two profit maximizing firms produce complementary goods (video games and game consoles, for instance). The demand functions are given by QA = 32 - 3PA - PB and QB = 32 - PA- 3PB, where QA and QB are the quantities of goods A and B, and PA and PB are the respective prices. The unit cost for each producer is 4.

a) Write an expression for the profits of firm B as a function of the prices PA and PB.

b) Treating prices as the choice variables, what is firm B's best response if firm A chooses the price PA? Illustrate firm B's best response function in a figure with PA and PB on the two axes

c) Find the Nash equilibrium.

d) Suppose that the two firms were to merge and maximize their joint profits. The merged company would charge prices that are lower than the Nash equilibrium prices. You don't need to show this but explain in words why this would be the case.

Answer 1

a) Profit of form A: Total Revenue - Total Cost

_B = PB*QB - 4QB

= PB*(32 - PA - 3PB) - 4(32 - PA - 3PB)

= 32PB - PA*PB - 3PB² -128 + 4PA + 12PB

  _B =   44PB -PA*PB - 3PB² + 4PA - 128 (1)

b) Considering that firm A chooses price PA, the best response for form B is to choose price PB that will maximize its profits:

Maximising  _B in (1) with respect to PB and setting the partial derivative = 0

d _B /dPB = 44 - PA - 6PB = 0

PB = (44 - PA)/6 is Firm B's best response if Firm A chooses price PA. (2)

c) Profit of firm A: Total Revenue - Total Cost

_A = PA*QA - 4QA

= PA*(32 - 3PA - PB) - 4(32 - 3PA - PB)

= 32PA - 3PA² - PAPB - 128 + 12PA + 4PB

_A   = 44PA -3PA² - PA*PB - 128 + 4PB

To find the best response function for firm A, maximise  _A wrt PA assuming firm B chooses the price PB

d _A /dPA = 44 - 6PA - PB = 0

PA = (44 - PB)/6 (3)

Solving (2) and (3)

PA + 6PB = 44

6PA + PB = 44

Solving this gives:

PA = PB

PA^* = PB* = 44/7

Substitute the value of PA^* and PB* in the quantity function to get QA^* and QB^*.

QA^* = 32 - 3PA - PB = 32 - 3*44/7 - 44/7 = 32 - 132/7 - 44/7 = (224 - 132 - 44)/7 = 48/7

QB^* = 32 - PA - 3PB = 32 - 44/7 - 3*44/7 = (224 - 44 - 132)/7 = 48/7

d) The merged company would charge prices lower than Nash equilibrium prices as it can now set its own price and not as a response to the price chosen by another firm.