(1) Two firms produce goods that are imperfect substitutes. If firm 1 charges price p₁ and firm 2 charges price p2, then their respective demands are 91 = 122p1 + P2 and 92 = 12+ p1 - 2p2. So this is like Bertrand competition, except that when p₁ > p2, firm 1 still gets a positive demand for its product. Regulation does not allow either firm to charge a price higher than 20. Both firms have a constant marginal cost c = 4. (a) Construct the best reply function BR₁(p2) for firm 1. That is, p₁ = BR₁(p2) is the optimal price for firm 1 if it is known that firm 2 charges a price p2. Construct a Nash equilibrium in pure strategies for this game. Are there any Nash equilibria in mixed strategies? If yes, construct one; if no provide a justification. (b) Notice that for any given price p₁, firm 1's demand increases with p2, so firm 1 is better off when firm 2 charges a high price p2. What is the best reply to p2 = 20? What is the best reply to p2 = 0? (c) What prices for firm 1 are not strictly dominated? What prices would survive two rounds of strict dominance? Provide a reason for each strategy that you eliminate. (d) Challenge question: If you continue the iterative elimination of strictly dominated strategies, what strategies will survive?
Two firms produce goods that are imperfect substitutes. If firm 1 charges
q1 = 12 - 2p1 + p2 and q2 = 12 + p1 - 2p2
So this is like Bertrand competition, except that when p1 > p2, firm 1 still gets a positive demand for its product. Regulation does not allow either firm to charge a price higher than 20. Both firms have a constant marginal cost c = 4.
(a) Construct the best reply function BR1(p2) for firm 1. That is, p1 = BR1(p2) is the optimal price for firm 1 if it is known that firm 2 charges a price p2. Construct a Nash equilibrium in pure strategies for this game. Are there any Nash equilibria in mixed strategies? If yes, construct one; if no provide a justification.
(b) Notice that for any given price p1, firm 1’s demand increases with p2, so firm 1 is better off when firm 2 charges a high price p2. What is the best reply to p2 = 20? What is the best reply to p2 = 0
(c) What prices for firm 1 are not strictly dominated? What prices would survive two rounds of strict dominance? Provide a reason for each strategy that you eliminate.
(d) Challenge question: If you continue the iterative elimination of strictly dominated strategies, what strategies will survive?
Solution for (a) and (b)
Step 1
Demand function for firm 1 : q1 = 12 - 2p1 + p2
Demand function for firm 2 : q2 = 12 + p1 - 2p2
Both firms have constant marginal cost : c = 4
In Bertrand competition both firms compete in prices by optimizing their prices taking other firm's prices as given .
Step 2
For Nash equilibrium strategy we compute best response functions of both the firms by simultaneously optimizing profits taking other firm's prices as given .
For firm 1 :
Optimal condition for production is achieved at the point where rate of change of profits w.r.t to p1 is equal to zero .
Profit of firm 1 => ( p1 -MC )(q1)
=> (p1 - 4 ) (12 - 2p1 + p2 )
d(profit )/dp1 = 0
=> 12 - 4p1 + p2 + 8 = 0
p1 = (20 + p2 )/4 (Best response function of firms 1 )
For Firm 2 :
Optimal condition for production is achieved at the point where rate of change of profits w.r.t to p2 is equal to zero.
Profit of firm 1 => ( p2 -MC )(q2)
= ( p2 - 4 )(12 + p1 - 2p2 )
d(profit )/dp2 = 0
=> 12 + p1 - 4p2 + 8 = 0
p2 = ( 20 + p1)/4 (Best response function of firm 2 )
For pure strategy Nash equilibrium we take value of p2* and put it into Best response function of firm 1 .
4p1* = 20 + ( 4 + p1)/4
16p1* = 84 + p1
p1* = 83/15 = 5.6
Putting this value into BR of firm 2 :
p2* = 5.6
Therefore , p1 = p2 = 5.6 is the pure strategy Nash
A mixed strategy game with two firms can also be set up , with strategies ,
For firm 1 : {set P1 > MC , set P1 = MC }
For firm 2 : { set P2 > MC , set P2 = MC }
B .)
Best reply to p2 = 20
Best response function of firm 1: p1 = (20 + p2 )/4
Putting p2 = 20 , we get :
p1* = 10 is the best response of firm1 for p2 = 20 .
For p2 = 0
Best response : p1 = (20 + 0 ) /4
p1* = 5 (Best response for p2 = 0 )
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