Answers to Odd-Numbered Problems 31From the cost function, C Q^2 /40, we know that MC dC/dQ
Q/20. Setting 3 Q/5 Q/20, we find Q 12. In turn, P 1.80
and so R 21.6. The seller’s profit is R C 21.6 3.6 18. The
buyer’s profit is B R 28.8 21.6 7.2.
c. Acting as a monopolist, the seller quotes a price that leads to the
purchase of too few units (12 units instead of 20). The monopoly
price is the source of the inefficiency.
*11. The buyer’s expected profit is b(vbP)F(P). The buyer determines
the optimal price P that maximizes this expression by setting marginal
profit equal to zero. Therefore, Mbdb/dP (vbP)dF(P)/dP
F(P) 0. This can be rewritten as (vbP)f(P) F(P) 0, where f(p)
dF(P)/dP is the density function of F(P). Solving for P we confirm that
P vbF(P)/f(P).
*13. The value of the target under current management ranges between
$60 and $80 per share, with an expected value of $70 (since all values
are equally likely). What if firm A offered a price of $70? Current
management accepts this price when vTis between $60 and $70.
(Obviously, if vT70, firm T will not sell.) Thus, when its offer is
accepted, the acquisition value to firm A ranges between $60 and $75.
(Remember that vA1.5vT30.) This means that firm A’s expected
acquisition value is $67.5. On average, it obtains a company worth less
than the price it pays! The trick is to realize that companies that
accept its offer are likely to be low-value companies. One can check
that firm A cannot earn a positive profit at anyprice between $60
and $80.Chapter 16
- a. Each buyer should bid bivi. If the buyer bids above her value, it
makes a difference only when she outbids an opponent who bids bjvi,
in which case she obtains the good for a price bjabove her value. In
short, bidding above one’s value makes no sense. If she bids below her
value, she cannot improve the price she pays. (This is fixed at the
second-highest bid.) But she risks losing the item if her bid is below the
second-highest bid, that is, if bibjvi. Bidding below one’s true
value is disadvantageous. Thus, the bidder’s dominant strategy is bivi.
b. In the English auction, the bidding stops at (or just above) the second-
highest value. In the second-price auction, the final price is set at the
second-highest bid (which corresponds to the second-highest value).
c. The absent buyer should report a bid equal to his true value, bivi.
If he wins, he pays only the price required to win the auction, which
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