∑ (∆*P*s + ∆*P*r )j + ∆*P*he

j

i

is identical for more than one consumer.

2. If the consumers have different values of ∆*P*cvm,i such that the maximum value

of the quantity

∑ (∆*P*s + ∆*P*r )j + ∆*P*he + ∆*P*cvm,i

j

i

is identical for more than one consumer.

Under either of these two conditions, the corresponding constraints from the eq

4-5 set for the consumers with identical values as described above will be active.

However, under neither condition will more than one be critical, since the deletion

of any additional constraints from the problem will not make the objective un-

bounded. This is true because the constraints from the set of eq 4-2 directed as

described earlier will bound the ∆*P*cv,i of all the consumers but one in the proper sense.

What is not immediately apparent is why all (*n * 1) of the constraints from the set

of eq 4-2 must be critical for some ∆*P*cv,i objective variable. To illustrate why this is

so, consider the case where for an arbitrary consumer *k *when we let ∆*P*cv,k be

bounded below by the constraint from the set of eq 4-5, thus ∆*P*cv,k = ∆*P*cvm,k. Now

suppose that

∑ (

∆*P*s + ∆*P*r )j + ∆*P*he =

∑ (

∆*P*s + ∆*P*r )j + ∆*P*he

j

k

j

i

and that ∆*P*cvm,k > ∆*P*cvm,i, where consumer *i *is the consumer with the minimum

value of ∆*P*cv,i. The constraint from the set of eq 4-2 that states that

∑ (∆*P*s + ∆*P*r )j + ∆*P*cv + ∆*P*he -

∑ (∆*P*s + ∆*P*r )j + ∆*P*cv + ∆*P*he = 0

j

i

j

k

would be violated. Notice that this constraint has not been directed and shown to be

active, yet it still must not be violated by any feasible solution. Thus, this is not a

feasible solution and we see that only one of the constraints from the set of eq 4-5 may

be critical, since (*n * 1) of the critical constraints must come from the set of eq 4-2.

Thus, our general result is that we must have (*n * 1) of the constraints from the set

of eq 4-2 active and critical for (*n * 1) of the ∆*P*cv,i values. Thus, we have (*n * 1) critical

constraints of the form

∑ (∆*P*s + ∆*P*r )j + ∆*P*cv,k + ∆*P*he,k -

∑ (∆*P*s + ∆*P*r )j + ∆*P*cv,i + ∆*P*he,i ≡ < 0

j

j

k

i

(5-5)

Where the consumer index *i *does not equal the consumer index *k *and the monoto-

nicities are

(∆*P*cv,i , ∆*P*c+v,k )

we also have one critical constraint on the remaining ∆*P*cv,i not bounded below by

one of the (*n * 1) constraints of eq 5-5 of the form

48

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