102
Finally, using the deÞnition of π
ε
in (76), we obtain
π
ε
=γ
µq
ε
1−σ
µ
σ+
1−ε
ε−γ
¶
(90)
So far, we have disregarded constraints (74) to (78). However, for ε ≥ γ, the values of q
ε
,v
ε
,
and π
ε
in (88), (89), and (90) are well deÞned and satisfy these constraints. Moreover, at ε = γ,
q
ε
=v
ε
=0. Hence, (a) implies q
ε
=v
ε
=π
ε
=0 for ε ≤ γ. The Hamiltonian is strictly concave
with respect to q
ε
,and when the Hamiltonian is evaluated at the optimal choice of q
ε
it is concave
with respect to v
ε
.Therefore, (80) to (82) characterize the unique maximum of the program. ¤
(c) For ω ∈ [0,0.5], the solution to program (72) subject to (73) to (78), is the following:
q
ε
=
µ
ψ
µ
ε
¶
1
σ
and
(91)
v
ε
=
σ
1− σ
µq
ε
(92)
When ω = 0.5, the variable v
ε
cancels in the objective (72) so the problem becomes separable
across types. Ignoring constraints (76) and (77), the Þrst-order conditions of the problem yield
(91). Integrating (73) from 0 to ε and noting v
0
=0, we obtain (92). Using the deÞnition in (76),
we obtain π
ε
=0 for all ε ∈ [0,1]. The values of q
ε
and v
ε
in (91) and (92) satify constraints (76)
and (77), so they solve program (72) subject to (73) to (78). Furthermore, the solution for ω = 0.5
maximizes the expected payoff of the buyer subject to a zero expected payoff for the seller. A
fortiori, given constraint (76), the same solution must apply for ω ∈ [0,0.5), which assigns a lower
weight to the seller in the maximized welfare function (72). ¤
Finally, Proposition 1 results from combining steps (b) and (c) together with (79). ¥
Proof of Proposition 2
Abuyer facing a price schedule γ purchases a quantity that satisÞes the following specialization
of (10)
εψq
−σ
ε
=γψq
−σ
ε
+(1 −γ) µ.
(93)
Solving this equation for q
ε
,we obtain (21). Using (24), the implied payments are given by (22).
Finally, γ taking values in the interval [0,0.5] spans all values of γ attained in (23) for ω ∈ [0,1].
Thus, Proposition 2 follows. ¥
Proof of Proposition 3
Giventhat the equilibriumvariables are recursively determinedby the set of equations described
in the main text. Existence and uniqueness is implied if equations (33) and (38) have a unique
35