IO2 - Static Oligoply Models

Equilibrium - No firm can increae its profits by changing its output level, given that the other firm produces the cournot output level.

Finding Equilirbium

• Find the inverse demand function.
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• Sub into the production level functions:
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• Sub q1 and q2 in to find price
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• Sub in to find profits:

• Find the residual demand function: P=150-Q = P=150-(q1+q2+q3)
• Sub into the profit function
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• Expand the brackets
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• Find the F.O.C
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• Rearange for q1
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• Sub in qc- indentical MC
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• Find qc,Q, P and profits

• 2 firms sell homogenous products
• Firms simultanously choose the prices at which they sell at to maximise profits
• Firm with the lower price attracts all of the market demand. At equal prices the market is shared equally.
• Unique Nash equilibrium - p1=p2=c
  • p2>p1>c - firm 2 can increases profit by setting p2 = p1
  • p1=p2>c - each firm can increase its profits by slightly undercutting the rival price
  • p1>p2=c - firm 2 can increase profits by increasing its price above c and below p1.  

Finding Equilibrium

• Q=90-3P, c1=15 c2=10
• P= MC of the highest cost firm
• P=15
• Q=90-3P
• Q=90-3(15) = 45
• Profit 1 = 15(45/2) - 15(45/2) = 0
• Profit 2 =15(45/2) - 10(45/2) = 112.5

Assumptions:

• • No capacity constraints
  • Product is homogenous
• With 2 firms and indentical marginal costs c1=c2=c. 
• Firms set P=MC
• Firms enjoy no market power

Only 2 firms but a perfectly competitive outcome

• Firms prepare to serve the whole market but at P=MC the two firms split the market leading to huge excess capacity and could change the equilibrium. 
• Example
• P=MC=8, Q=92 ,profit 1=profit 2=0, P=100-Q
• Assume each firm has a capacity constraint of 30. 
• Q=30(2)=60 P=100-60=40
• If firms share the market evenly, they will produce 30 units each. They don’t have an incentive to reduce their price because they are at capacity and cannot serve more consumers. 
• Find the MR for each firm: 
• Q=100-P   TR=P(100-P)  MR=100-2P  MR1=50-P
• MR1=50-40 = 10.
• At this price, MR = 10 > MC = 8, so they don’t have an incentive to increase price. Profits are equal to 960.

• N consumers live equally spread along Main Street with 2 shops at either end.
• Consumers incur there and back transport costs at t per mile. A consumer located at x incurs a cost of: tx when buying from shop 1 and t(1-x) when buying from shop 2.
• The marginal consumer- enjoys the same consumer surplus buying from shop 1 or 2.
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• Rearrange
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• Demand for firm 1 - there are N consumers
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• Profit for firm 1:
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• Find the F.O.C
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• Rearrange to the firms 1 BR function. Symmety. BR1=BR2 equilibrium
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• The more products are differentiated (i.e. the higher t), the higher the price cost-margin of the firms in equilibrium ⇒ The higher firms’ market power

Bertrand appropiate when:

• Unlimited capacity
• Prices more difficult to adjust in the short run
• Production schedules easily changed

Cournot appropiate when:

• Limited Capacity
• For some products e.g. package holidays quantity is hard to adjust.