Question

GAME THEORY: Please show all work and explanation so i understand.

Two firms, A and B, compete by each choosing a price. Demand for good Q is Q = 13 – P, where P denotes price. If one firm offers a lower price than the other firm, the firm with the lower price meets the entire demand at his price. If the two firms set the same price, then they equally split demand at that price. Consider prices P = 4, 8 & 10. Use revenues for the firms’ payoffs. There are no costs!

(a.) Draw the normal form game.

(b.) If firm B chooses PB = 8, what are the three possible payoffs he can earn?

(c.) If firm B chooses PB = 8, which price choice for A would give her the highest payoff?

Answer 1

A) Normal form of the game is show below. Given that payoffs are revenue we find that

• When both set a price of 4, the quantity demanded is Q = 13 - 4 = 9. Each of them sells half of this which is 4.5 and therefore earns 4.5*4 = $18
• When both set a price of 8, the quantity demanded is Q = 13 - 8 = 5. Each of them sells half of this which is 2.5 and therefore earns 2.5*8 = $20
• When both set a price of 10, the quantity demanded is Q = 13 - 10 = 3. Each of them sells half of this which is 1.5 and therefore earns 1.5*10 = $15
• When one of them charges $4 and the other charges $8 or $10, it caters the entire demand of Q = 13 - 4 = 9 and earns $4*9 = $36. The other firm earns zero
• When one of them charges $8 and the other charges $10, it caters the entire demand of Q = 13 - 8 = 5 and earns $8*5 = $40. The other firm earns zero

b) If firm B chooses PB = 8, the three possible payoffs he can earn are $0 (when PA is $4), $20 (when PA is $8) and $40 (when PA is $10).

(c.) If firm B chooses PB = 8, firm A observes that it cannot charge $10, because this gives it no profit. The price that would give her the highest payoff is PA = 4. At this price, her payoff is $36 and is maximum (when PB is 8).