Only part C. Make sure to thoroughly explain how you got equations step by step
ID: 1147315 • Letter: O
Question
Only part C. Make sure to thoroughly explain how you got equations step by step and explain clearly if there's a derivative process, or if there's another way. Thank You!
Suppose that two firms produce differentiated products and compete in prices. As in class, the two firms are located at two ends of a line one mile apart. Consumers are evenly distributed along the line The firms have identical marginal cost, $60. Firm B produces a product with value $110 to consumers. Firm A (located at 0 on the unit line) produces a higher quality product with value $120 to consumers The cost of travel are directly related to the distance a consumer travels to purchase a good. If a consumer has to travel a mile to purchase a good, they incur a cost of S20. If they have to travel x fraction of a mile, they incur a cost of $20x (a) Write down the expressions for how much a consumer at location d would value the products sold by firms A and B, if they set prices Pa and Pb? (b) Based on your expressions in (a), how much will be demanded from each firm if prices Pa and Pb are set? (c) What are the Nash equilibrium prices?Explanation / Answer
Answer:
a) Surplus from Firm A = $120 - $20d - Pa
Surplus from B = $110 - $20(1-d) - Pb
b) We will set surplus A = surplus B, to solve for d
Thus, $120 - $20d - Pa = $110 - $20(1-d) - Pb
It gives: d* = (30 + pb - pa) / 40
Because the consumers are evenly distributed along the unit interval
Qa = (30 + pb - pa) / 40;
Qb = (10 + pa - pb) / 40
c) Writing profits for firm A, we get A = Qa (Pa - 60)
When we sSubstitute Qa into the above equation and takes the derivative to Pa, then:
p*a = (90+pb) / 2
Similarly, p*b = (70+pb) / 2
Solving it we get nash equibrium pa* = 83.33; pb* = 76.66
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