Exact Aggregation--A Discussion of Miller's Theorem

T OM MILLER, in an article in this journal,1 has shown that the optimal responses of different farms to a given set of relative product prices will be proportional if two conditions are met. The conditions are that the farms have homogeneous activity vectors and that the same activities appear in the linear programming solution vector for each farm. Miller's paper and this discussion of it are concerned with the aggregation problem, which is essentially one of grouping farms that respond alike in linear programming models of agricultural supply. Miller starts by observing that levels of management and practices are assumed given or are specified in most research projects so the input-output matrices for large groups of farms are identical. He further observes that all farm managers can be assumed to behave in a consistent, objective manner; thus, an acceptable procedure is to assume all farmers have the same net return expectations. These two observations allow Miller to focus his concern on Richard Day's condition of proportionality of resource vectors among all farms in the aggregate.2 Miller sets forth the following theorem: