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Abstract
This supplement presents mathematical expressions of the models of the farm supply cooperative described in Part I and the marketing cooperative described in Part II. Price and output solutions are derived for firms that maximize profit, cooperatives that maximize member returns, and cooperatives that handle whatever quantity of products members choose to purchase or deliver. Those solutions are then compared to the solutions for the maximization of economic welfare to determine the conditions under which profit-maximizing firms and cooperatives are efficient in an allocative sense.