1. In a market with annual demand Q-100-p , there are two firms, and B, that mak
ID: 1141740 • Letter: 1
Question
1. In a market with annual demand Q-100-p , there are two firms, and B, that make identical products. Because their products are identical, if one charges a lower price than the other, all consumers will want to buy from the lower-priced firm. If they charge the same price, consumers are indifferent and end up splitting their purchases about evenly between the firms. Marginal cost is 10 which is constant and there are no capacity constraints.* (a) What are the single-period Nash equilibrium prices, p. and p (b) What prices would maximize the two firms' joint profits? (c) If the interest rate (r) is 10%, s one repeated-game Nash equilibrium for both firms to charge the price you found in part (b)? What if the interest rate is 110%? What is the highest interest rate at which the joint profit-maximizing price is sustainable? (Note: we use a formula for the discount rate such as 1+7 Assume that one firm cannot observe the other's price until after it has set its own price for the year. Assume further that both firms know that if one undercuts the other, they will revert forever to the non-cooperative behavior you described in (a).- (d) Describe qualitatively how your answer to (c) would change if neither firm was certain that it would be able to detect changes in its rival's price. In particular, what if a price change is detected with a probability of 0.7 each period after it occurs? (Note: Do not try to calculate the new equilibria.)' Return to the situation in part (c), with an interest rate of 10%. But now suppose that the market for this good is declining. The demand is Q-A- p with A 100 in the current period, but the value of A is expected to decline by 10% each year (ie, to 90 next year, then 81 the following year, etc.). (e) Now is it a repeated-game Nash Equilibrium for both firms to charge the monopoly price from part (b)? (Note: Do not try to calculate the new equilibria)-Explanation / Answer
Answer :-
a) The single period nash equilibrium will be the price of each firms that would be equal to their marginal cost.
Therefore it will be equal to $10.
b) The firms can maximize the joint profit if they set MR ( marginal revenue ) equal to their MC ( marginal cost ).
MC of both firms is 10
MR would be dP*Q/dQ
we have Q = 100-P or P = 100-Q
thus TR = P*Q = 100Q - Q2
and hence MR = dTR/dQ = 100 - 2Q
equate it with MC
100 - 2Q = 10
Q =90/2 = 45
and hence P = 100 - 45 = 55
therefore, each will produce 45/2 = 22.5 quanities at price of $55.
c) When interest rate is equal to 10%, then the discount rate, would be =1/(1+r) = 1/(1+.10) = 0.909.
And when nash equilibrum is played, then the monopoly profit would become half because of charging of monopoloy price of $10. thus the profit would be (55-10)45/2 = 1012.5
therefore, the present value of an infinite stream of of half of monopoly profit will be equal to 1/(1-0.909) * 1012.5 = 11137.50.
Alternatively if one cheated this period they will receive (54.99-10)45 = 2025. but in future they will receive 0.
Since 11137.50 > 2025, the monopoly price will be sustainable. This implies that 1- < 1012.5/2025
or > 1 - 91012.5/2025) = .5
Since =1/(1+r) this implies that 1/(1+r)>0.5 or 1+r<2 which in turn implies that as long as the interest rate is 100% or less, the monopoly price will be sustainable.
d) Here the probability of detection is less than 1, thus if one firm cheats it can cheat for more than one period before being caught. This will make cheating more attractive and sustain collusion less possible.
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