Dynamic Duopoly Competition with Limited Supply

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Dynamic Duopoly Competition with Limited Supply

• Anita van den Berg, Jean-Jacques Herings and Hans
  Peters

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Aim of the research

• Analyze strategic firm behavior in settings where
  firms have a fixed stock to sell over multiple
  periods, instead of per-period.
• Two settings
• - commitment
• - no commitment
• Does commitment matter for
• - prices
• - welfare

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Example

• exhaustible resources
• Limited supply fashionable products
• otherwise limited
• For instance
• Two firms
• A finite set of identical televisions
• A new type every two periods, old type no longer
  produced
• Sell how much of the stock per period?

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Literature

• Exhaustible Resources
• Hotelling (1931)
• Lewis Schmalensee (1980)
• Salo Tahvonen (2001)
• Capacity constraints
• Edgeworth (1925)
• Biglaiser Vettas (2004)

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The model

• i ?1,2, two firms
• Si, the finite amount owned by firm i
• Two periods
• qi, quantity firm i sells in period 1,
• ri, quantity firm i sells in period 2,
• d ? (0,1, discount factor
• P(Q) 1 - Q, inverse demand per period
• Competition in quantity
• Two settings with and without commitment

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Commitment

• Firm i maximizes
• ?i(qi,ri,qj,rj) qiP(qi qj) driP(ri
  rj)
• subject to
• 0 qi,ri 1, qi ri Si
• where it takes qj and rj as given.
• Nash equilibrium
• ?i(qi,ri,qj,rj) ?i(qi,ri,qj,rj) for all
  qiri Si

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Commitment - Concentration of firms, d 0.5

• x/y/z x firms active in period 1, y firms
  active in period 2,
• z firms with residual afterwards

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Commitment - Results

• price increases over time
• an increase in firm is initial supply increases
  its profit
• an increase in firm is initial supply increases
  consumer surplus

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No Commitment

• Firm i observes qj before second period second
  period is subgame
• Strategy firm i is (qi,ri(qi,qj))
• ri(qi,qj) second period subgame perfect Nash
  equilibrium output
• Subgame perfect N.E. found by backwards induction

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No commitment

• In second period, firm i maximizes
• pi(ri,rj)riP(rirj)
• subject to
• 0 ri Si-qi
• where it takes rj as given.
• Second period equilibrium
• pi(ri,rj) pi(ri,rj) for all ri Si-qi

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No commitment second period

• ./y/z y firms active in period 2, z firms
  with residual afterwards

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No commitment second period

• Reduced profit function
• ?i(qi,qj) qi(1-qi-qj) d?i2(qi,qj),
• where
• d/9 if Si-qi,Sj-qj gt 1/3
• ?i2(qi,qj) d(Si-qi)(1-Siqi)/2 if
  1-2(Sj-qj) ltSi-qi 1/3
• d(1-Siqi)2/4 if Si-qi lt
  (1-Sj-qj)/2,Sj-qj 1/3
• d(1-Si-Sjqiqj)(Si-qi) if Si-qi
  min(1-Sj-qj)/2,1-2(Sj-qj)
• Subgame perfect Nash equilibrium
• ?i(qi,qj) ?i(qi,qj) for all qi Si

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No commitment Concentration of firms, d 0.5
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No commitment - results

• Not always an equilibrium
• Price can decrease over time
• An increase in the supply of firm i can decrease
  its profit
• For d 1, consumer surplus never decreases when
  total supply in market increases.

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Conclusions