Chapter 20

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?

The world's entire supply of kryptonite is controlled by 20 people, with each having 10,000 grams of this potent mineral. The world demand for kryptonite is given by \[Q=10,000-1,000 P,\] where $P$ is the price per gram. a. If all owners could conspire to rig the price of kryptonite, what price would they set, and how much of their supply would they sell? b. Why is the price computed in part (a) an unstable equilibrium? c. Does a price for kryptonite exist that would be a stable equilibrium in the sense that no firm could gain by altering its output from that required to maintain this market price?

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?

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 $[(n-1)/n]v^{\ast }-$ where $v^{\ast }$ 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 $v^{\ast }$ 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 $\left[n{v}^{n-2}\right]$