程序代写代做代考 game C Bertrand Price Competition

Bertrand Price Competition
Bertrand Price Competition with Homogenous Products
Consider a market with two firms, Firm 1 and Firm 2. Both firms produce homogenous (identical) products at a unit cost c = 0 (for simplicity).
Two firms are competing by simultaneously setting prices of an identical product to place on the market.
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Price Competition

Bertrand Price Competition
Firms’ products are viewed identically to consumers — all consumers buy from the firm with a lower price.
When the firms charge the same price, the firms split the market and each firm captures exactly half of the market demand.
Saltuk Ozerturk (SMU)
Price Competition

Bertrand Price Competition
Suppose firm i sets price pi 2 [0, 1) when the rival firm j sets a price pj 2 [0, 1). Then the demand qi for Firm i’s product is given by
8 1p if p

pj
Saltuk Ozerturk (SMU)
Price Competition

Bertrand Price Competition
Strategic (Normal) Form of the game: Players: Two Firms N = {1, 2}
Strategies: Firm i 2 N chooses price pi 2 [0, +1) .
Saltuk Ozerturk (SMU)
Price Competition

Bertrand Price Competition
Payo↵s of the firms:
8><(p1c)(1p1) if p1: (p1c)(1p1) if p1=p2 0 2 ifp1>p2
8><(p2c)(1p2) if p2: (p2c)(1p2) if p1=p2 0 2 ifp2>p1
Saltuk Ozerturk (SMU)
Price Competition

Bertrand Price Competition
Is there any Nash Equilibria with p1 6= p2?
Suppose, without loss of generality, there is a NE in which
firm1setsp1 =aandp2 =bwhere
c0
2
Hence we cannot have a NE in which firm 1 (or firm 2) sets a higher price than its rival.
Saltuk Ozerturk (SMU)
Price Competition

Bertrand Price Competition
Is there any Nash Equilibria with p1 = p2 > c?
Suppose, there is a NE in which firms set p1 = p2 = a where
c0 2
Saltuk Ozerturk (SMU)
Price Competition

Bertrand Price Competition
But note that, instead of setting p1 = a, F11 can set p1 =a”where”>0
In this case, F1 would get the whole market and receive
⇡1(p1 =a”,p2 =b)=(a”c)(1a+”)> (ac)(1a)
2
Hence no NE in which two firms both set a price strictly higher than marginal cost c
Saltuk Ozerturk (SMU)
Price Competition

Bertrand Price Competition
Is p1 = p2 = c a Nash Equilibrium?
If both firms set a price equal to their marginal cost, they
share the market but receive a zero profit. ⇡1(p1=c,p2=c) = (p1c)(1p1)=0
2
⇡2(p1=c,p2=c) = (p2c)(1p1)=0 2
Saltuk Ozerturk (SMU)
Price Competition

Bertrand Price Competition
Is p1 = p2 = c a Nash Equilibrium?
Is there a profitable devitation for any of the two firms. The
answer is no.
If a firm sets a lower price than c, this firm will sell each unit a loss and make negative profits. If the same firm deviates and sets a higher price than c, it will not be able to sell anything and continue to receive zero.
Therefore p1 = p2 = c is the unique Nash Equilibrium.
Saltuk Ozerturk (SMU)
Price Competition

Price Competition
Price Competition with Di↵erentiated Products
Suppose two firms produce di↵erentiated products at a unit cost c = 0.
The firms are competing by simultaneously setting prices
Firms’ products are viewed as imperfect substitutes by consumers.
Saltuk Ozerturk (SMU)
Price Competition

Price Competition
Suppose firm i sets price pi 2 [0, 1) when the rival firm j sets a price pj 2 [0, 1). Then the demand qi for Firm i’s product is given by
which implies
qi (pi,pj)=10↵pi +pj q1 (p1, p2) = 10 ↵p1 + p2
q2 (p1, p2) = 10 ↵p2 + p1
We assume that ↵ > 1 so that own-price e↵ect is larger than
the cross-price e↵ect.
Saltuk Ozerturk (SMU)
Price Competition

Price Competition
Strategic (Normal) Form of the game:
Players: Two Firms N = {1, 2}
Strategies: Firm i 2 N chooses price pi 2 [0, +1) . Payo↵s of the firms:
⇡1(p1,p2)=p1q1(p1,p2)=p1(10↵p1 +p2) ⇡2(p1,p2)=p2q2(p1,p2)=p1(10↵p2 +p1)
Both firms want to maximize profits.
Saltuk Ozerturk (SMU)
Price Competition

Price Competition
Deriving the Best Response Function of Firm 1
Given any p2 by Firm 2, Firm 1 chooses p1 to maximize ⇡1(p1,p2) = p1(10↵p1+p2)
) ⇡1(p1,p2)=10p1↵p12+p2p1 First order derivative with respect to p1 yields the first order
condition
102↵p1+p2 = 0
) p 1⇤ ( p 2 ) = 5 + 1 p 2 ↵ 2↵
That is, Firm 1 sets a higher price as the rival firm’s price p2 increases and lower price as ↵ increases.
Saltuk Ozerturk (SMU)
Price Competition

Price Competition
Deriving the Best Response Function of Firm 2
Given any p1 by Firm 1, Firm 2 chooses p2 to maximize ⇡2(p1,p2) = p2(10↵p2+p1)
) ⇡2(p1,p2)=10p2 ↵p2 +p1p2 First order derivative with respect to p2 yields the first order
condition
102↵p2+p1 = 0
) p 2⇤ ( p 1 ) = 5 + 1 p 1 ↵ 2↵
That is, Firm 2 sets a higher price as the rival firm’s price p1 increases and lower price as ↵ increases.
Saltuk Ozerturk (SMU)
Price Competition

Price Competition
The Nash equilibrium pair (p1⇤,p2⇤) solves the system of equations described by the best responses.
p 1⇤ ( p 2⇤ ) = 5 + 1 p 2⇤ ↵ 2↵
p 2⇤ ( p 1⇤ ) = 5 + 1 p 1⇤ ↵ 2↵
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Price Competition

Price Competition
) p 1⇤ = 5 + 5 + ✓ 1 ◆ p 1⇤ ↵ 2↵2 4↵2
) ✓ 4↵2 1 ◆ p1⇤ = 10↵ + 5 4↵2 2↵2
) (2↵1)(2↵+1)p1⇤ = 5(2↵+1) 4↵2 2↵2
)p1⇤=p2⇤= 10 2↵1
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Price Competition

Price Competition
Note that the Nash Equilibrium Price pair p1⇤=p2⇤= 10
2↵1
is drastically di↵erent than the unique Nash Equilibrium of the
Bertrand duopoly with identical products.
Saltuk Ozerturk (SMU)
Price Competition

Price Competition
With identical products, the NE was p1⇤ = p2⇤ = c which would imply here that
p 1⇤ = p 2⇤ = 0 since we assumed that c = 0.
Note that with
p1⇤=p2⇤= 10 2↵1
as ↵ approaches to infinity we again have p1⇤ = p2⇤ = 0.Why?
Saltuk Ozerturk (SMU)
Price Competition