Question

Two firms operate in an industry with inverse demand given by p = 12 – q. each firm operates with constant marginal cost equal to 0 and fixed cost equal to 4. Firms compete by setting the quantity to sell in the market . A) Determine the best reply function of each firm. b) Determine what are in equilibrium the quantities offered by each firm, the market price and the profits obtained by each firm. Assume now N firms operate in this market c) Determine how many firms can profitably operate in this market .

Answer 1

Given that 2 firms have:

o Inversed demand function: p = 12 –q, MC =0 and FC or TC = 4. There is no veritable cost given. Let q1 = Quantity produced by Firm 1 and q2 = quantity produced by firm 2. Thus the market demand function can be written as:

p = 12 – (q1 + q2)

o Marginal Cost (MC)=0 and Fixed Cost (FC)= 4. There is no variable cost given so the Total Cost (TC) = 4

A. Given that the firms compete by setting quantities to sell in the market. Let’s assume, the firms set their quantities simultaneously. Thus, to find the best response function of each firm we need to find the profits firm and then equate the partial derivative of the profit function (respect to firm’s quantities) equal to zero.

Total Revenue of Firm 1(TR1) = P.q1 = (12 –q)q1

= (12 – q1 – q2)q1

= 12q1 – q12 – q1q2

Total Revenue of Firm 2(TR2) = P.q2 = (12 –q)q2

= (12 – q1 – q2)q2

= 12q2 – q1q2 – q22

TC1 = TC2 = 4

Profits of Firm 1 (PR1) = TR1 – TC1 = 12q1 – q12 – q1q2 – 4

Profits of Firm 2 (PR2) = TR2 – TC2 = 12q2 – q1q2 – q22 - 4

Partially Differentiating Profit Function of Firm 1 with respect to q2 and equating it with zero, we get the best response function of firm1:

dPR1/dq1 = 12 – 2q1 – q2

if dPR1/dq1 = 0

12 – 2q1 – q2 =0

12 – q2 = 2q1

6 – 0.5q2 = q1   {Best Response Function of Firm 1}

Partially Differentiating Profit Function of Firm 2 with respect to q2 and equating it with zero, we get the best response function of firm2:

dPR1/dq1 = 12 – q1 – 2q2

dPR2/dq2 = 0

12 – q1 – 2q2 = 0

12 – q1 = 2q2

6 – 0.5q1 = q2    {Best Response Function of Firm 1}

B) To determine the equilibrium quantities of each firm we first put the best response function of firm 2 in firm 1.

q1 = 6 – 0.5q2

q1 = 6 – 0.5(6 – 0.5q1)

q1 = 6 – 3 + 0.25q1

q1 – 0.25q1 = 3

0.75q1 = 3

q1 = 3/0.75

q1 = 4

Now using the value of q1 = 4 in the best response function of firm 2 we get q2:

q2 = 6 – 0.5q2

q2 = 6 –0.5(4)

q2 = 6 – 2

q2 = 4

Thus, Market price: p = 12 – (q1 + q2)

= 12 – 4 – 4

= 12 – 8

= 4

Profits of Firm 1 (PR1) = 12q1 – q12 – q1q2 – 4

= 12(4) – (4)2 – 4.4 - 4

= 48 – 16 – 16 - 4

= 48 – 36

= 12

Profits of Firm 2 (PR2) = TR2 – TC2 = 12q2 – q1q2 – q22 - 4

= 12(4) – 4.4 – (4)2 – 4

= 48 – 36

= 12

C) Now if we assume that there are N firms in the industry, say from 1, 2, 3 , ,,,,,,,,,,N.

Let qi = Quantity Supplied by Each Firm

(PR)i = Profits earned by each firm “I”

q = Market Supply = Sum of all qi = q1 + q2 + q3 + ….qN

q(-i) = Industry Supply excluding the quantity supplied by firm i

Thus, q = q(-i) + qi          ---------------- equation 1

We are given the market demand p = 12 – q, where q = q1 + q2 + q3 + ……qn and the TC for each firm is 4

Thus the profit function for each firm “i“ will be :

PR(i) = TR(i) – TC

PR(i) = (12 – q)qi – 4

PR(i) = (12 - q(-i) + qi)qi – 4     {From equation 1}

PR(i) = 12qi - q(-i)qi – qi2 – 4   

We maximize the profit function with respect to qi and find the first order derivative:

PR(i) = 12qi - q(-i)qi – qi2 – 4   

dPR(i)/dqi = 12 - q(-i) – 2qi

Let, dPR(i)/dqi = 0

12 - q(-i) – 2qi = 0

12 - q(-i) = 2qi

6 – 0.5q(-i) = qi ------ This is the best response function of each firm in the market

As we know all the firms in this market face identical demand and cost structure, so their best response functions would be identical too. Since market demand is given by :

q = q1 + q2 + ……..qN

Thus, qi = q/N

Or N = q /qi , which gives the number of firms that can operate in the market .

To verify this, as in this market, qi (1,2) = 4 and total market demand was q1 + q2 = 8

So N = 8/4 = 2