IO 2 Static Oligopoly Models

Example: Market demand Q=90-3P. c1=15 c2=10. How much does each firm sell? Market price? Profits?

Solve: 

• find the slope of the demand curve- Q=0 P=30, Q=50 P=13.33, change in p/change in Q = 1/3=b
• Best response functions to find quantity
• -
• -
• -
• Sub into the inverse demand function: P=(90-10-25)/3 = 18.33
• Profit1 = pq-cq = 10(18.33) - 15(10) = 33.33
• Profit2 = 18.33(25) - 10(25) = 208.25

• 2 firms producing homogenous products
• Unique nash equlibirum - p1 = p2 = c Why? at all other prices theres incentive to deviate.
  • p2>p1>c - firm 2 can increase profit by setting p2'E(c,p1)
  • p1=p2>c - each firm can increase profits by slightly undercutting the rival price
  • pz>p2=c - form 2 can increase profit by increasing its price above c and below p1. 

Bertand Paradox 

With 2 firms and intendical marginal costs c1=c2=c.

• Firms set p=mc
• Firms enjoy no market power

Only 2 firms but perfectly competitive outcome.

Example: Market demand Q=90-3P. c1=15 c2=10. How much does each firm sell? Market price? profits?

Solve:

• p=mc of the highest cost firm = 15 
• Q=90-3(15) = 45 q1=22.5 q2-22.5
• profit1 = 15(22.5) - 15(22.5) = 0
• profit2 = 15(22.5) - 10(22.5) = 112.5

At p=c- both firms need enough capacity to serve all consumers. But at p=c firms half the market meaning alot of excess capacity and changes the equilibrium. 

If both firms set p=c they will fill their capacity so raising the price wont loose any consumers because both firms are at capacity so there is no where to go. 

Firms will serve the customers with the highest willingness to pay.

Example: Market demand P=100-Q. c=8. Firms compete in price. Each firm has a capacity constraint of 30. Market price? Quantity? Profits?

Solve:

• Work out whole market capacity. Q=30(2) = 60. 
• Sub in for price P=100-60 = 40.
• Find TR = P(100-P) = 100P -P^2
• Find MR = 100-2P
• Find MR for each firm MR= 50-P.
• Sub in P=40. MR=50-40 = 10. 

2 stage game - stage 1: capacity choice. stage 2: price competition.

• Firms are unlikely to choose sufficient capactiy to serve the whole marker when p=mc
• So capacity of each firm is less than the whole market
• but then no incentive to cut price to mc.

Result is indentical to the cournot competition even though the competition is in price.

• Increasing capacity implies a lower price in the second stage (firms choose price so total quantity demanded = total capacity)
• only expand their capacity so long as MR>MC
• Firms choose q to serve the entire market and such as MR=MC 

Location Example:

2 shops on one street. A customer located a x incurs and a transport cost of:

• tx when buying from shop 1
• t(1-x) when buying from shop 2.

Solve: xm - the consumer enjoys the same surplus buying from either shop.

• V-p1-txm = V -p2-t(1-xm)
• Rearrange to get xm(p1,p2) = (p2-p1+t)/2t
• Demand for firm 1 : D1= N(p2-p1+T)/2t
• Profit for firm 1 = (p1-c)D1 = N(p1-c)(p2-p1+t)/2t
• FOC = N/2t(p2-2p1+t+c)=0
• Rearrange p1=(p2+t+c)/2