98 S. Dasgupta and R.G. Hansen
ΠA(v∗,v∗)=G(v∗)v∗−PA(v∗)=0. Now from(14), integrating, we get forvv∗
PA(v)=PA(v∗)+∫vv∗yg(y) dy=G(v∗)v∗+∫vv∗ydG(y)=vG(v)− (15)∫vv∗G(y) dy,where in the last step, we used integration by parts.
The expected revenue for the seller from a single bidder is
∫v ̄
0 PA(v)f (v) dv.Again,using integration by parts, this can be written as
E(RiA)=∫v ̄0PA(v)f (v) dv=
∫v ̄v∗PA(v)f (v) dv=
∫v ̄v∗vG(v)f (v) dv−∫v ̄v∗[∫vv∗G(y) dy]
dF=
∫v ̄v∗vG(v)f (v) dv−∫v ̄v∗G(y) dy+∫v ̄v∗F(v)G(v) dv= (16)
∫v ̄v∗[
vf (v )−(
1 −F(v))]
G(v) dv.Given equal treatment of allNbuyers, the expected revenue to the seller is simplyN
times the above expression.
Notice that what we have shown is that all auction forms in the class of auctions
being considered must provide the seller with the same expected revenue if the marginal
bidder is the same. The reserve price will determine the marginal bidder. If no bidder
has a valuation above that of the marginal bidder, the seller keeps the object. Assume
that the seller values the object atv 0. Then for any auction, the seller should choose the
marginal bidder to maximize
∫v ̄
v∗[
vf (v )−(
1 −F(v))]
G(v) dv+F(v∗)v 0.From the first-order condition with respect tov∗, we getv∗=v 0 + (17)1 −F(v∗)
f(v∗).
Since the optimal marginal bidder is the same in all auctions—all auctions in the class
of auctions we are considering provide the seller with the same expectedprofitas well
as revenue. The revenue equivalence result survives when a reserve price is introduced.