Question

1. The market for frogs has inverse demand p = 30 − 1/3Q. Two suppliers only operate.

   Firm 1 has constant marginal cost 15 and firm 2 has constant marginal cost 10.

   1. If the two firms compete by choosing quantities, find the equilibrium price

      and profits.

   2. If they compete in price, find the equilibrium price and profits.

   3. What price and profits would you anticipate if they compete in quantities

      and firm 1 can choose its quantity first?

Answer 1

Answer) p= 30-Q/3

MC1= 15

MC2= 10

Answer 1) p= 30- [ (Q1+Q2)/3 ]

Case 1) Profit maximsation for firm 1

At profit maximisation MR= MC1

MR= ∆TR1/∆Q1

TR1= p×Q1

TR1=[ 30-(Q1+Q2)/3 ] Q1

MR1 = 30 - [ (2Q1+Q2)/3 ]

•MR1= (90-2Q1-Q2)/3

• MC1=15

90-2Q1-Q2= 45

Q1=( 45-Q2)/2 -----------------(A)

Case 2) Profit maximisation for firm 2

At profit maximisation MR2= MC2

MR2=∆TR2/∆Q2

TR2= p × Q2

TR2=[ 30- (Q1+Q2)/3 ] Q2

MR2= 30 - ( Q1+ 2Q2)/3

• MR2 =( 90-Q1-2Q2)/3

• MC2= 10

So,90-Q1-2Q2= 30

(60-Q1)/2 = Q2

Q2= (60-Q1)/2 ------------(B)

Putting the value of Q1= (45-Q2)/2 in equation (B)

Q2= (60-[ (45-Q2)/2] )/ 2

Q2= (120-45+Q2)/4

4Q2= 75+Q2

•Q2*= 15

Q1*=(45-Q2)/2

•Q1*= 15

At Q1=Q2=15

Equilibrium price= p= 30-[ (Q1+Q2)/3 ]

p= 30- [ ( 30)/3]

•p= 20

•Profit of firm 1 = TR1- TC1

TR1= p× Q1

TC1= 15Q1 [ integrating MC with respect to Q1]

TR1= 20*15= 300

TC1= 15*15 = 225

Profit of firm 1= 300-225= 75

• Profit of firm 2 =TR2-TC2

TR2= p× Q2

TC2= 10Q2 ( integrating MC2 with respect to Q2)

TR2= 20* 15 = 300

TC2= 10* 15 = 150

Profit of firm 2 = 300-150= 150

Answer 2) When they compete in price

Answer c) When firm 1 choose Quantity first

p= 30- (Q1+Q2)/3

​​​​​​MC1= 15

MC2= 10

Step 1) Profit maximsation for firm 2 occurs at a point where MR2=MC2

MR2=∆TR2/∆Q2

TR2= p× Q2= [30- (Q1+Q2)/3] Q2

• MR2= 30 - ( Q1+ 2Q2)/3

• MC2= 10

30-( Q1+2Q2)/3= 10

90-Q1-2Q2 = 30

Q2= (60-Q1)/2 -----------(A)

Step 2) Putting the value of Q2 in p

p= 30- (Q1+Q2)/3

p= 30- (Q1 + { (60 -Q1) } /2 ) /3

p= 30 - { 2Q1+60-Q1}/6

p= (120-Q1)/6

Step 3) Maximisation of firm 1 profit occurs at a point where MR1= MC1

TR1= p* Q1

TR1=[ (120-Q1)/6 ] * Q1

MR1= (120-2Q1)/6

MC1= 15

90= 120 -2Q1

Q1= 15

Q2= (60-Q1)/2

Q2= 22.5

p= 30- ( 15+22.5) /3

p= 17.5