# anita

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```					“Dynamic Duopoly Competition
with Limited Supply”
Anita van den Berg, Jean-Jacques
Herings and Hans Peters
Aim of the research

   Analyze strategic firm behavior in settings where
firms have a fixed stock to sell over multiple periods,
   Two settings:
- commitment
- no commitment
   Does commitment matter for:
- prices
- welfare
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?
Literature

   Exhaustible Resources:
   Hotelling (1931)
   Lewis & Schmalensee (1980)
   Salo & Tahvonen (2001)
   Capacity constraints:
   Edgeworth (1925)
   Biglaiser & Vettas (2004)
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,
   δ  (0,1], discount factor
   P(Q) = 1 - Q, inverse demand per period
   Competition in quantity
   Two settings: with and without commitment
Commitment

Firm i maximizes
Пi(qi,ri,qj,rj) = qiP(qi + qj) + δriP(ri + rj)
subject to
0 ≤ qi,ri ≤1, qi + ri ≤ Si
where it takes qj and rj as given.

   Nash equilibrium:
Пi(qi*,ri*,qj*,rj*) ≥ Пi(qi,ri,qj*,rj*) for all qi+ri ≤ Si
Commitment - Concentration of firms, δ =
0.5

x/y/z: x: firms active in period 1, y: firms active in period 2,
z: firms with residual afterwards
Commitment - Results

   price increases over time
   an increase in firm i’s initial supply increases
its profit
   an increase in firm i’s initial supply increases
consumer surplus
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
No commitment

 In second period, firm i maximizes:
πi(ri,rj)=riP(ri+rj)
subject to
0 ≤ ri ≤ Si-qi
where it takes rj as given.

 Second period equilibrium:
πi(ri*,rj*) ≥ πi(ri,rj*) for all ri ≤ Si-qi
No commitment – second period

./y/z: y: firms active in period 2, z: firms with residual
afterwards
No commitment – second period
Reduced profit function:
Пi(qi,qj) = qi(1-qi-qj) + δПi2(qi,qj),
where
δ/9                    if Si-qi,Sj-qj > 1/3
Пi2(qi,qj)=   δ(Si-qi)(1-Si+qi)/2     if 1-2(Sj-qj) <Si-qi ≤1/3
δ(1-Si+qi)2/4          if Si-qi < (1-Sj-qj)/2,Sj-qj ≤ 1/3
δ(1-Si-Sj+qi+qj)(Si-qi) if Si-qi ≤ min{(1-Sj-qj)/2,1-2(Sj-
qj)}

Subgame perfect Nash equilibrium:
Пi(qi*,qj*) ≥ Пi(qi,qj*) for all qi ≤ Si
No commitment – Concentration of firms,
δ = 0.5
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 δ = 1, consumer surplus never decreases
when total supply in market increases.
Conclusions
Commitment                   No commitment
For all values equilibrium   For some values no
equilibrium
Equilibrium outcomes in both settings converge when
second period becomes less and less important (δ↓0)
Equilibrium outcomes coincide when Si = Sj.

Price increases with time    Sometimes, price
decreases with time
Profit increases with initial Not “destroy-proof”
supply

```
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 views: 10 posted: 10/28/2011 language: English pages: 17
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