45


where the are the undetermined lagrange multipliers. Differentiation
 1
of equation (35) with respect to C., X.. and x. yields the N(n + 2)
 1 3frst-o r c i
first-order conditions


aCi


(35a)


 N aP
 k
P. + C = 0
k=1 aCi k 1


 af.
 -r. + *. 1 0
 +j 1 aX i



fi (Xli' Xni) Ci


i 1, ..., N


i = 1, ..., N
j = 1, ..., n


= 0

i =1, ..., N


From equation (35a) it can be immediately seen that X. = P. +


i = 1, .
 aP.
+
 3a. Ci
term on

revenue

output.

(MRi).


N aP
al k Ck'
k=l i


.., N. This expression for A. can be rewritten as X. = (P.
 1 1 i
 aP
) + z k C. In this form, it can be seen that the first
 kfi aCi k.

the right-hand side of the equality is precisely the change in
 th th
 in the i region with respect to variations in the i region's
 th
 Thus, this term is equal to the i region's marginal revenue

 Now, substituting for Xi in (35b) yields


 af. af.
(MR. + E 1 C) =- r.
 1 kti Ci k aXji


j = l, ..., n
i = 1, ..., N


Rearranging terms in equation (36) results in the expression for the ith

region


aL
aX..
31


aL
ax.


(35b)


(35c)


(36)