IMPROVING AGGREGATION VALIDITY by John E. Lee, Jr.* In recent years, increasing concern has been expressed in the literature of our profession over the problems of aggregation in 1l models of agricultural demand and supply ( 1, 3, 4, 5, 7, 8, 12) • 1 Most frequently the concern has been with the validity of aggregated supply estimates when micro supply situations are simulated in linear programming models • The aggregation problem stems from the fact that while most economic activity originates with individual firms and households, economists, with increasing frequency, want to say something about the behavior, in the aggregate, of all or a group of such units. One obvious approach is to build up to the aggregate, firm by firm. This is not feasible in an atomistic economy such as agriculture. A second approach is to ignore the individual units by working with observed or postulated relationships among economic aggregates. A third alternative, especially for an atomistic or purely competitive economy, is to approach the aggregate through "representative" units or through the analysis of subaggregates. The latter approach has been particularly popular in agricultural supply research. The so-called Regional Adjustment Studies (NC-54, S-42, etc.) have used linear programming models of representative farms to derive regional estimates of supply schedules for the major commodities. The basic units of analysis for a model of national agricultural production developed by the Economic Research Service are representative resource situation subaggregates. Linear programming models of these units are used to estimate aggregate supply response to economic stimuli. These are but two examples of the micro-oriented approach to estimation of aggregate behavior--an approach that has almost become conventional in farm management and production economics research. *Agricultural Economics, Production Adjustments Branch, Economic Research Service. This version of the paper has benefited greatly from the suggestions of Lee M. Day. The author also appreciates the comments of Fred Abel, Glen T. Barton, and W. Neill Schaller. The views expressed are the author's and do not necessarily reflect the views of the U.S. Department of Agriculture. 11 Numbers in parentheses refer to Literature Cited at end of paper. -341- The concern for the validity of aggregate estimates of supply derived from the approach typified in these models is justified. There are at least three sources of invalidity in that approach: 1. The failure to be consistent with respect to time in the formulation and specification of supply models; 2. The failure to recognize that constraints on economic activity at higher levels of aggregation may be different than at micro levels; and 3. The failure to systematically group farms and construct units of analysis so as to recognize the interfarm variation in forces that shape economic behavior. These failures are interrelated; thus, the distinctions between them are somewhat arbitrary. But the breakdown provides some "handles" on the aggregation problem and an organization framework for the discussion which follows. I will work from conceptual and theoretical benchmarks, but the emphasis in this paper will be on practical guidelines for improving aggregation validity in models of agricultural supply. It seems appropriate to begin with a look at the aspect of the aggregation problem which has been discussed most frequently. The problem is that of exact aggregation; or simply, the problem of grouping farms so that analyses of aggregate behavior will be free from bias or error attributable to the grouping process itself. Grouping Microunits for Aggregation Validity Recent Conceptual Developments Richard H. Day (3) has shown that conditions sufficient for exact aggregation are proportional variations of resources and behavioral "bounds," proportional variation of net return expectations among all farms in the aggregate, and finally, common technical coefficients which appear in the constraints on the farm's dec is ions • Tom Miller (7) , searching for less restrictive conditions for exact aggregation, has shown conceptually that the responses of different farms to a given set of relative product prices will be proportional if the farms have homogeneous activity vectors and if the s arne activities appear in the linear programming solution vector for each farm. In other words, farms that have the same enterprises in the optimum enterprise mix can be grouped without bias or error. While Miller's conditions are less restrictive than Day's (groupings of farms under Day's conditions -342- would be subsets of groupings under Miller's conditions), they are of limited practical value because they are defined as requirements of the solutions to the individual farm problems, rather than as observable characteristics of the farms themselves. In another recent paper, Sheehy and McAlexander {13) hypothesized that if farms could be sorted into groups having the same absolute re­ striction (limiting resource) on output, benchmark farms based on the averaging of resource levels within each group would give unbiased aggregate results. They proceeded to apply these grouping criteria only to single-product firms and not to the more general multiple­ product, multiple-resource case where variations in relative· prices must be recognized. Thus, several writers have advanced sufficient conditions for exact aggregation. However, these have been rather restrictive con­ ditions or have applied only to special cases. None of these efforts answered Lee Day's (1) call for research to determine the necessary conditions for exact aggregation and to determine the magnitude and direction of error at various prices associated with different levels of variance in the proportionality of resources. We turn to that task now. The Necessary Conditions for Exact Aggregation In an extension of Miller's analysis, this writer (6) showed how the "qualitatively homogeneous solution vector" conditions could be translated into observable characteristics of the farms themselves. This was done by solving the dual to Miller's primal problem. In so doing it was revealed that the shadow prices (marginal value products) of the resources were the same for all farms which Miller showed could be grouped without aggregation error. Further, these shadow prices were constant over the range of resource ratios represented by the ·aggregated farms. The task remaining was the practical one of determining the exact ranges of resource ratios over which the marginal value product was constant. In the article to which I just referred, simple graphics were used to demonstrate that, in effect, the activity vectors in a linear programming situation represent marginal value product "borders." All farms with combinations of resources bounded by the same marginal value product •• borders" or activity vectors have the s arne shadow prices in the dual, have the same activities in the primal solution vector, and can be aggregated without error. The number of cells or bounded areas represents the maximum number of groups of farms needed for exact aggregation. -343- Working with the dual counterpart to Miller's theorem rather than with the primal was a matter of convenience rather than necessity. All farms whose resource coordinates lie between the same two feasible activity vectors will maximize revenue with some combination of those same vectors, thus meeting the "qualitatively homogeneous solution vector" conditions of Miller's primal theorem. From this exercise it was apparent that the key to determination of the resource ratios relevant to error-free grouping of farms is the relationship between the ratios in which resources are required by alternative activities and the ratios in which resources are available to farms. This observation has been used to develop an exact aggre­ gation algorithm for multiple-product farms in a completely general price situation. The difficulty with multiple-product farms is that the resources available to a given commodity (activity) are subject to change as prices (and thus the relative profitability of the activities) change. The answer to that difficulty, of course, is to recognize all possible orderings of the relative profitableness of activities (that is, to recognize all the possible orders in which activities could come into the linear programming solution) . This can be done in the following way. First, all the relevant combinations of two resources are deter­ mined. For example, if one or more activities use all three of resources A, B, and C, the relevant combinations of two resources would be AB, BC, and AC. Then the critical ratios of each of these combinations of two resources are determined from the technical coefficients of the activity vectors . In other words, the boundaries to the critical ranges of each resource ratio are the ratios in which the various activities use these resources. Then farms are grouped according to the critical resource ratio range in which each combination of two resources falls. Sequential sorting on the basis of all possible relevant combinations of two resources (assuming that prices of all outputs could vary from zero to infinity) produces a number of groups of farms free of aggregation error. Since the same conditions must be met for each relevant resource ratio by the farms in each group, the order of the sequential sorting process will not affect the composition of the final groups of farms. Because for each farm in each group the output of each commodity will be limited by the same restraint at any given price, the shadow prices for the resources of each firm are identical and the conditions for zero errors are met. In fact, this "critical resource ratios" grouping algorithm provides the minimum sufficient conditions (and therefore the necessary conditions) -344- for exact aggregation in a completely general model. If any farm in any one of the groups derived meets anything less than these condi­ tions (that is, if any of its resource ratios fall outside the critical ranges which delineate that group) there will be aggregation error at some combination of prices.