Question

2. Matthew is considering selling a painting in an English auction. There are n bidders whose valuations are distributed independently and uniformly on (VL, Vu]. a) What is his expected payoff from the auction? b) A private buyer is willing to pay (VL + vu)/2. What is the minimum number of bidders required for Matthew to choose the auction rather than sell to the private buyer? (Assume Matthew is risk neutral.) 3. Two firms compete by setting quantities. Firm A has a marginal cost of 10, while Firm B has a marginal cost of 20. The demand function is given by: p = 90 - Q/2, where Q = 9A + 9B a) What are the equilibrium quantities if both firms set quantities simultaneously (Cournot competition)? b) What are the equilibrium quantities if Firm A sets quantity first (Stackelberg competition)?

Answer 1

the bid is offered to one with 2)a) in an english auction highest valuation valuation at cost of toits second highest individuals are Bids of independent and uniformly distributed ft in [ve, vu]. in ura,b] - man is n [b-a] ta ntl - and (n-1} (b-a) ta is mar nt So, expected pagoff for mathew is vu -V] tvi nti a) b) for him auction to still auction, his expected payoff from z seller payment Vu V) + ♡ z Vutui nt 2 =) tv, 11 - 11) CV u V.) tv , 2 vu u - 2 ) Cu",) 2 uurui - n-) ne 2 -) 2n-2 ? nt) >) n-l 2% ntl n23

134) for any q, firm A chooses firm B objective is with (no - (012), -2004 La differentiate w.rit qq and set it to zero 90-91 qr .-20 2 70-9 a =92. -70 For any q, firm B chooses, fim a's objective in man (60 - (4*)]4,- q, -109, wort.9, differentiate wirot q, and set it to zero 90-qa -a i 10 =0 2 a, = 80-92/27 =) q, G in 2 80 70 -9) a -) 9,2 80-35+ 9 , 2) 3a, 45 Z 4 9,2 =) 60 and 2 =50-60 = 40. 2

b) If firm À sets quantity first. It knows that for every 'd, it chooses, firm & will choose 70 - 9, we found it 3(a) a . as & So, its objective is, man 9, + 70-91 90 -) man wort.a 90 qi -35+9, 2 q, 109, u man worota, ( ss . 55 104 9, 4 )a, and set it to zero differentiate wirit to q, 1 55-9 2 -10 =0 q, I : 45 - 9,90 - 2 and 85 70- 'Va = 70-45 - 3/3

If you want derivation for 2(a), please comment.

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