I took it upon myself today to not only clarify the solution for Class Problem 10.2, but also produce an “expanded” version which appears here.  What I particularly like about the problem is that it manages to tie together a number of concepts that we been studying, including the notion that profit maximizing firms set marginal revenue equal to marginal cost, that market power translates into positive economic profits (over and above the opportunity cost of capital), that firms base their decisions upon what they believe the “best responses” are for their rivals, etc.

 

Anyway, here is the “new and improved” version of Class Problem 10.2:

 

Question: Glyde Air Fresheners is the dominant firm (i.e., “leader”) in the solid room aromatizer industry which has a total market demand given by Q = 80 – 2P.  Glyde has competition from a fringe of four small firms (i.e., “followers”) that produce where their individual marginal costs equal the market price (i.e., P = MCi for the followers, implying that these firms earn zero economic profit).  Each of the followers has total costs given by TCi = 10Qi + 2Qi2.  If Glyde’s total costs are given by TCG = 100 + 6QG, what price should Glyde establish for air fresheners?  What is Glyde’s maximum profit, and what is the total profit for the “fringe” firms?

 

Solution: This is a “Stackelberg” problem, where Glyde is the leader and the followers are the fringe firms.  The starting point is to figure out the reaction function for the followers, since the leader maximizes profit based upon what she believes is the followers’ “best response” to her pricing and output decisions.

 

The marginal cost for each of the followers is MCi = 10+4Qi.  Let QF = 4Qi; i.e., QF represents the total production from all four followers.  Since Q = 80 – 2P, this implies that P = 40 – 0.5Q = 40 – 0.5QF – 0.5QG.  Since P = MCi for the followers, we use this equation to obtain the followers’ reaction function; i.e., P = MCi, which implies that 40 – 0.5QF – 0.5QG  = 10 + QF,which implies that QF = 20 – QG/3. 

 

Next, in order to maximize Glyde’s profit, we set marginal revenue equal to marginal cost for Glyde.  In order to find Glyde’s marginal revenue (MRG), we calculate total revenue (TRG), where TRG = PQG.  Substituting the followers’ reaction function into the price equation, we obtain

 

P = 40 – 0.5QF – 0.5QG = 40 – 0.5(20 – QG/3) – 0.5QG = 30 – QG/3.

 

Thus, TRG = PQG = (30 – QG/3)QG = 30QGQG2/3, which implies that MRG = 30 – (2/3)QG.  Setting MRG = MCG, we solve for QG:

 

MRG = MC

30 – (2/3)QG = 6

QG = 36.

 

Since QG = 36, everything now falls into place:

 

1.    What price should Glyde establish for air fresheners? 

Since Glyde produces 36 air fresheners, the fringe firms will produce QF = 20 – QG/3 = 20 – 12 = 8 air fresheners.  Therefore, total industry output is Q = QF + QG = 8 + 36 = 44, and P = 40 – 0.5Q = 40 – 22 = $18.

2.    What is Glyde’s maximum profit?

pG = TRG TCG = PQG – $100 – $6QG = $18(36) – $100 – $6(36) = $332.

3.    What is the total profit for the “fringe” firms?

Each “fringe” firm will enjoy profit of pi = TRi TCi = PQi – 10Qi – 2Qi2= $18(2) – $10(2) – $2(22) = $36 – $20 – $8 = $8.  Since there are four such firms, the total profit for the group is $32.

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