Suppose the demand for steel bars in Example 20.1 fluctuates with the business cycle. During expansions demand is \\[Q=7,000-100 P,\\] and during recessions demand is \\[Q=3,000-100 P.\\] Assume also that expansions and recessions are equally likely and that firms know what the economic conditions are before setting their price. a. What is the lowest value of \(\delta\) that will sustain a trigger price strategy that maintains the appropriate monopoly price during both recessions and expansions? b. If \(\delta\) falls slightly below the value calculated in part (a), how should the trigger price strategies be adjusted to retain profitable tacit collusion?

Short Answer

Expert verified
The monopoly prices during expansions and recessions can be calculated by equating marginal cost (MC) to the marginal revenue (MR) calculated from the demand functions. For expansions, the monopoly price is \(P_1^* = \frac{7,000-c}{200}\), and for recessions, the monopoly price is \(P_2^* = \frac{3,000-c}{200}\). The trigger price strategy involves setting prices at or above these monopoly prices to maximize profits during both economic conditions. 2. If the discount factor, \(\delta\), falls slightly below the value calculated in part (a), how should the trigger price strategies be adjusted? If \(\delta\) falls slightly below the calculated value, firms should adjust their trigger price strategies by raising the trigger price during expansions and/or lowering the trigger price during recessions. This adjustment allows firms to maintain a profitable tacit collusion in the face of a less favorable discount factor.

Step by step solution

01

1. Determine the Monopoly Prices during Expansions and Recessions

First, we need to find the monopoly prices for both economic conditions. For this, we can use the demand function given in the problem: During expansions: \[Q=7,000-100 P\] During recessions: \[Q=3,000-100 P\] Now, we need to find the corresponding marginal revenue (MR) functions for each demand function. To calculate the MR function from the demand function, differentiate total revenue (TR) with respect to quantity (Q). Total revenue is the product of price (P) and quantity (Q). For expansions: \[TR_1=P_1Q=P_1(7,000-100 P_1)\] Differentiate with respect to \(P_1\): \[MR_1 = \frac{d(TR_1)}{d P_1} = 7,000 - 200P_1\] For recessions: \[TR_2=P_2Q=P_2(3,000-100 P_2)\] Differentiate with respect to \(P_2\): \[MR_2 = \frac{d(TR_2)}{d P_2} = 3,000 - 200P_2\]
02

2. Calculate the appropriate monopoly price during both recessions and expansions

For both economic conditions, we need to equate the corresponding marginal cost (MC) to the marginal revenue (MR) we found in step 1. We assume the same MC for both conditions: MC = \(c\). For expansions: \[MC = MR_1\] \[c = 7,000 - 200P_1\] \[P_1^* = \frac{7,000-c}{200}\] For recessions: \[MC = MR_2\] \[c = 3,000 - 200P_2\] \[P_2^* = \frac{3,000-c}{200}\]
03

3. Find the lowest value of \(\delta\) that will sustain a trigger price strategy

The discount factor \(\delta\) is the present value of future profits. In order to find the lowest value of \(\delta\) that will sustain a trigger price strategy, we need to equate the discounted sum of the net profit during expansions and during recessions: \[ \delta ( \pi (P_1^*) - \pi (P_2^*)) = \pi (P_2^*)\] where \(\pi(P)\) is the profit at price \(P\). We can calculate profit by using the demand function and the cost function: During expansions: \[\pi(P_1^*) = (P_1^*-c)(7,000-100 P_1^*)\] During recessions: \[\pi(P_2^*) = (P_2^*-c)(3,000-100 P_2^*)\] Now, solve for \(\delta\). Then, plug in the values of \(P_1^*\) and \(P_2^*\) found in step 2: \[\delta = \frac{\pi (P_2^*)}{ \pi (P_1^*) - \pi (P_2^*)}\]
04

4. Adjusting trigger price strategies if \(\delta\) falls slightly below the value calculated in part (a)

If \(\delta\) falls slightly below the calculated value, it means the present value of future profits decreases. In this case, firms should adjust their trigger price strategies by raising the trigger price during expansions and/or lowering the trigger price during recessions. This adjustment allows firms to maintain a profitable tacit collusion in the face of a less favorable discount factor.

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Most popular questions from this chapter

Suppose firms \(A\) and \(B\) operate under conditions of constant average and marginal cost, but that \(M C_{A}=10, M C_{B}=8 .\) The demand for the firms' output is given by \\[Q_{D}=500-20 P.\\] a. If the firms practice Bertrand competition, what will be the market price under a Nash equilibrium? b. What will the profits be for each firm? c. Will this equilibrium be Pareto efficient?

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In Example 20.5 we showed that the Nash equilibrium in this first-price, sealed bid auction was for each participant to adopt a bidding strategy of \(b(v)=[(n-1) / n] v\). The total revenue a seller might expect to receive from such an auction will obviously be \([(n-1) / n] v^{*}-\) where \(v^{*}\) is the expected value of the highest valuation among the \(n\) auction participants. a. Show that if valuations are uniformly distributed over the interval \([0,1],\) the expected value for \(v^{*}\) is \(n /(n+1) .\) Hence expected revenue from the auction is \((n-1) /(n+1)\) Hint: The expected value of the highest bid is given by \\[ E\left(v^{*}\right)=\int_{0}^{1} v f(v) d v \\] where \(f(v)\) is the probability density function of the probability that any particular \(v\) is the maximum among \(n\) bidders. Here \(f(v)=n v^{n-1}\) b. In a famous 1961 article ("Counterspeculation, Auctions, and Competitive Sealed Tenders," Journal of Finance, March \(1961,\) pp. \(8-37\) ) William Vickrey examined second-price sealed bid auctions. In these auctions the highest bidder wins, but pays the price bid by the second highest bidder. Show that the optimal bidding strategy for any participant in such an auction is to bid his or her true valuation: \(b(v)=v\) c. Show that the expected revenue provided by the second-price auction format is identical to that provided by the first-price auction studied in part a (this is Vickrey's "revenue equivalence theorem"). Hint: The probability that any given valuation will be the second highest among \(n\) bidders is given by \(g(v)=(n-1)(1-v) n v^{n-2}\). That is, the probability is given by the probability that any of \((n-1)\) bidders will have a higher valuation \([(n-1)(1-v)]\) times the probability that any of \(n\) bidders will have a valuation exceeding that of \(n-2\) other bidders \(\left[n v^{n-2}\right]\)

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