The market for widgets consists of two firms that produce identical products. Co

Question

The market for widgets consists of two firms that produce identical products. Competition in the market is such that each of the firms independently produces a quantity of output, and these quantities are then sold in the market at a price that is determined by the total amount produced by the two firms. Firm 2 is known to have a cost advantage over Firm 1. A recent study found that the (inverse) market demand curve faced by the two firms is P-280-2(0, + and costs are c,(a)-30, GC,)-242 and a. Determine the marginal revenue for each firm. b. Determine the reaction function for each firm. c. How much output will each firm produce in equilibrium? d. What are the equilibrium profits for each firm?

Explanation / Answer

Consider the given problem, here the market demand curve be, “P=280-2*Q”, where Q=Q1+Q2.

So, the MR for both the firm is given below.

TR1=P*Q1 = (280-2*Q)*Q1 = 280*Q1 – 2 *Q1^2 – 2*Q1*Q2.

=> MR1 = 280 – 4*Q1 – 2*Q2.

Now, for firm2.

TR2=P*Q2 = (280-2*Q)*Q2 = 280*Q2 – 2 *Q2^2 – 2*Q1*Q2.

=> MR2 = 280 – 4*Q2 – 2*Q2.

b).

So, here the Reaction function for “Firm1” be, “MR1=MC1, => 280 – 4*Q1 – 2*Q2 = 3”, since here “MC1=3”.

=> 280 – 4*Q1 – 2*Q2 = 3, => 4*Q1 + 2*Q2 = 277……….(1), be the Reaction Function for firm1.

Now, the Reaction function for “Firm2” be, “MR2=MC2, => 280 – 2*Q1 – 4*Q2 = 2”, since here “MC2=2”.

=> 280 – 4*Q2 – 2*Q1 = 2, => 4*Q2 + 2*Q1 = 278……….(2), be the Reaction Function for firm2.

c).

So, the equilibrium quantity will be determined by the intersection of the 2 curves.

So, if we multiply (1), by 2, then we have,

=> 8*Q1 + 4*Q2 = 554, => 4*Q2 = 554 – 8*Q1………(3), now by substituting this into (1), we get, “554 – 8*Q1 + 2*Q1 = 278, => 276 = 6*Q1, => Q1=276/6=46, Q1=46”.

Now, form (3), we get “4*Q2 = 554 – 8*Q1 = 186, => Q2=186/4=46.5.

So, “Q2=46.5”.

So, the firm1 will produce “Q1=46” and “Q2=46.5”.

d).

So, Q=46+46.5=92.5, so the equilibrium price be, “P=280-2*Q=280-2*92.5=95.

So, the profit earn by the firm1 be, “P*Q1 – C1, =95*46 – 3*46=4232.

So, the profit earn by the firm2 be, “P*Q2 – C2, =95*46.5 – 2*46.5=4324.5.

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