1. Demand is given by q = 1-p.
a) What is demand elasticity at p = 0? At p = 1?
b) At what price is demand elasticity 1?
c) At which price is revenue maximised?
2. Two players participate in a contest where the winner gets R and the loser gets
P-R, 0 ≤ R ≤ P. The two players have identical convex cost of effort functions C(e).
The chance player i wins is ei/(e1+e2). Show that, at equilibrium, effort increases in
R.
3. A piece of art is being auctioned. Bidders each have a private valuation drawn
independently from a uniform distribution with range $0 to $1million.
a) The auctioneer runs a second-price sealed auction. What is each bidder’s
optimal strategy? How does it depend on the number of bidders?
b) If the auctioneer runs a first price sealed bid auction instead, what is each
bidder’s optimal strategy if there are 2 bidders? If there are n bidders?
c) What is the expected revenue if there are 2 bidders? If there are n bidders?
4. David has an investment opportunity that pays 33 with probability 1/2 and loses 30
with probability 1/2.
a) If his current wealth is 111 and his utility function is U = W1/2, will he make this
investment?
b) Will he make the investment if he has a partner who will share equally in the net
return (positive and negative) from the investment?
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