PS The questions are about Theory of Industrial Organization

PS. The questions are about Theory of Industrial Organization.

1. Suppose two firms compete by choosing quantities and that neither of them sees the other\'s quantity before having to commit to its own production level. Inverse market demand is given by p(Q) = 95-2Q. where Q is industry quantity. Firm 2 has a cost advantage over Firm 1 in small quantities, but Firm 1 is able to produce at lower marginal cost at larger quantities. This is reflected in the following cost technologies: G(a)-3ai and G(a)-2, where qi and g2 are Firm I and Firm 2 quantities, respectively. (a) What are the equilibrium quantities in this market?

Solution

The equilibrium for firm 1 will be at a point where MR=MC

P= 95-2(Q1+Q2), Because Q is the total output for firm 1 and 2.

Then Q =Q1(Production of firm 1) +Q2 (Production of firm 2)

TR = P*Q

TR = 95Q - 2Q1² – 2*Q2*Q1

MR = 95- 4Q1 - 2Q2 (First order derivative of TR with respect to Q1 or dTR/dQ1)

MR = MC(First order derivative of C with respect to Q1 or dC/dQ1 = 3)

95 - 4Q1 - 2Q2 = 3

46 - 2Q1 = Q2

Again the equilibrium for firm 1 will be at a point where MR=MC

TC = 2*Q2²

MC = 4*Q2 (First order derivative of C with respect to Q2 or dC/dQ1 = 4*Q2)

MR = 95 – 4*Q2 – 2*Q1

MR = MC

95 – 4*Q2 – 2*Q1 = 4*Q2

95 – 2*Q1 = 8Q2

46 – 2*Q1 = Q2

95-2*Q1 = 8*(46-2*Q1)

95-2*Q1 = 368 – 16*Q1

14*Q1 = 273

Then

Q1 =19.5

Q2 = 7

(b) The price set by MR and MC will be same for both firms unlike the quantity

P = 95-2*(19.5+7)

P = 42

(c) If the firm 2 closes down and ceases to operate then firm 1 will still produce at MR = MC

MR = 95 -4*Q

MC = 3

95 – 4*Q = 3

92 = 4*Q

Q = 23

P = 95 - 2*23 = 49

Firm 1 = 23 units.

(d)

Firm 1 and 2 had cumulative production of 19.5+7 = 26.5 units.

After shut down by firm 2 firm 1 will produce 23 units.