Question

Apple and Samsung compete in the market for phones. Apple is the market leader and so they choose how many households to service, and then Samsung responds with their quantity choice. The marginal cost of providing a quality phone to a is $40 for each user. The total market demand for phones is 200 – 0.4Q. According to the Stackelberg model, what is Samsung reaction function to Apple?

Answer 1

According to the stackelberg model, one player will choose their quantity first (Apple) and get a first mover's advantage while the other player (Samsung) will choose decide their supply based on Apple's decision.

Let's assume quantity supplied by Apple = Q_A and the quantity supplied by Samsung = Q_S

The total quantity 'Q' is the sum of the quantity supplied by Apple and Samsung ie;

Q = Q_A + Q_S  

Because of full information in the market, Apple knows that Samsung will base it's strategy on the quantity that Apple supplies. Hence, we will solve this by backward induction. This is because Apple will first predict Samsung's Strategy, and will then set it's own Strategy accordingly. Solving for Samsung:

P = 200 - 0.4Q

P = 200 - 0.4 (Q_A + Q_S) As mentioned above, Q = Q_A + Q_S  

P = 200 - 0.4Q_A -0.4Q_S

In order to maximize it's profit, Samsung will equate its Marginal Revenue to its Marginal Cost. (MR = MC is the profit maximizing condition in economics)

Total Revenue = Quantity * Price  

Hence, Samsung's Total Revenue = Q_S * (200 - 0.4Q_A - 0.4Q_S)

= 200Q_S -0.4Q_AQ_S - 0.4Q_S²

We get Marginal Revenue by Differentiating Total Revenue

Hence, MR for Samsung = 200 - 0.4Q_A - 0.8Q_S and Marginal Cost = $40 (Given in the question)

Equating MR and MC, we get:

200 - 0.4Q_A - 0.8Q_S = 40

160 - 0.4Q_A = 0.8Q_S

Q_S = 200 - 0.5Q_A This is Samsung's reaction function to Apple.

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Extra Information: We can get Apple's response function in a similar way. The only difference now would be that we know that Q_S = 200 - 0.5Q_A

For apple, P = 200 - 0.4Q_A -0.4Q_S

Now that we know what Q_S is equal to, P =  200 - 0.4Q_A -0.4(200 - 0.5Q_A) = 120 - 0.2Q_A . We can now calculate Apple's Quantity supplied by finding TR (P * Q_A), differentiating it w.r.t Q_A to find MR, and then equating it to MC)  

TR_A = 120Q_A - 0.2Q_A²

MR = 120 - 0.4Q_A (By differentiating TR)

when we equate MR and MC we get 120 - 0.4Q_A = 40, hence, Q_A = 200

By putting this in Q_S = 200 - 0.5Q_A,we'll get Q_S= 100

We can also calculate their profits through Total Revenue - Total Cost.