11
obtain the optimal choices for output and stringency of controls represented by y*(ω,β,a)
and 5 * (ω, β, a), respectively. We are interested in signing the following: dy*∕~ω , ds∕θω ,
dy */ ∂s */ dУ */ and ∂s */
The first four of these can be signed under reasonable
∕∂β, ∕∂β, ∕∂a, and ∕∂a
assumptions. Stronger assumptions are needed, however, to sign the last two derivatives.
We next show that the ability of consumers to perceive quality increases the optimal
level of stringency of the QAS and is likely to decrease the output rate. Differentiating the
system of equations (3) and (4) with respect to ω (at an interior solution) and using the chain
rule, we get (after some rearrangement and omitting arguments for brevity)
|
l∖ ∂2π |
∂2π |
Ï |
∖ ∂y * ^ |
∖ | |
|
(5) |
ll ∂y2 |
∂y∂s |
ll∂ω |
II=ll | |
|
l∂2π |
∂2π ∂2m 2+ πβ 2 |
ɪ |
l∂s * |
I | |
|
^∂y ∂s |
1 -βm)II |
к ∂ω I |
к |
πβ
(1 - βm )2
∂2 m z, r, ∂ ,> ∂m ∂m ∣
-----(1 - βm ) + β--I
∂s∂ω v ∂s ∂ω I
<√
Because the parameter ω enters by itself only in equation (4) of the necessary
conditions, Samuelson’s conjugate pairs theorem immediately asserts that
Г /.2 - -
∂ss*/ ∖ ∏β ∖ ∂ m (λ n ∖. omm ∂m ∣ r, ...
sgn(ds∕dω) = sgn ;------ʒ-l (l-βm)+— I . Recalling that
' dω' Ц1 -βm) ^∂s∂ω ∂s ∂ωI II
m(s,ω)=λ(s)+(1-λ(s))(1-ω), the previous expression simplifies to
sgn (ds >∂ω )=sgn ∣(1Z⅛ ⅛ (1 - β )]
>0. As consumers become more able to discern
quality, processors will find it optimal to adopt more stringent controls. This result is similar in
a sense to one of the findings of Darby and Karni. These authors argued that it is very likely
(albeit not necessarily true) that as consumers become more knowledgeable, the optimal