as the inverse demand for commodity k in demand region k in month m. Qjkm is the
quantity of commodity k consumed in demand region j in month m and the parameters
ajkm and bjkm are both assumed to be non-negative.
 Let
 Xijkm = quantity of commodity k shipped from supply region i to demand region j
 in month m
 c3ijkm = per unit transportation cost from supply region I to demand region j for
 commodity k in month m.
With these definitions, the quadratic programming model can be written as

 J K 12 I Li
MAX I [ ajkm Qjkm (1/2)bjkm Q2jkm] Clhi Wli
 J=1 k=l m=1 1=1 li=l
 1 K 12 1 J K 12
 CI c2ikZikm Z I C C3ijkmXijkm
 j=I k=l m=l I=1 j=l k=l m=I

subject to

 Ulikm = dlikm i h= l ,...,Li; i= 1,...,I;k= 1,...,K ; m=l,...,12

 Zikm= Ulikm, i= ...,I; k=l,...,K ; m=1,....,12
 Li
 J
 SXijkm Zikm i=l,...,I; k=1,...,K ; m=1,....,12
 j=i

 SXijkm Qjkm i= ,...,I; k=l,...,K ; m=1,....,12
 i=1


 Qjkm.Wi. Zikm.Uikm, Xijkm 0 for all i,j, k, m, and li

 The optimal solution to this model provides the equilibrium consumption of each
commodity in every month in each demand region (Qjkm), the optimal level of shipments
between each supply area and each demand region (Xijkm), the optimal production of each
cropping system by production area (W1i) and the quantity of each commodity produced
in each supply region by month (Zikm). The optimal dual solution provides market
clearing prices in each demand area by month and commodity.
 This model incorporates a fixed proportions technology to generate supply.
Although the production functions used in the model follow the fixed proportions
assumption, the supply curves generated by the model are not perfectly elastic. The
upward sloping supply curves result, in part, because of increasing transportation costs to
market as production from a particular supply region expands. This is the implicit supply
model as discussed by McCarl and Spreen (1980) and Peters and Spreen (1989). The
other important simplification imposed on the model is that all parameters are assumed to
be non-stochastic.