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

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 equilibrium price level.

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 )

Transcribed Image Text:(1) Two firms produce goods that are imperfect substitutes. If firm 1 charges price pi and firm 2 charges price p2, then their respective demands are 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, pi = 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. BR1(p2) is (b) Notice that for any given price P1, firm l'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?

Question

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

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 equilibrium price level.

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 )

(1) Two firms produce goods that are imperfect substitutes. If firm 1 charges price pi and
firm 2 charges price p2, then their respective demands are
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, pi =
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.
BR1(p2) is
(b) Notice that for any given price P1, firm l'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?

Accepted Answer

The demand function for firm 1 : The demand function for firm 2 : Marginal cost: