Talented youngsters, like everyone, need to experi ence the milk of human kindness. Because these students are extremely able persons, they are often neglected emotionally—they are often erroneously seen as completely independent individuals. The writers remind educators here that these bright stu dents need emotional and psychological satisfac tions.

AN earlier paper by one of us (Farrell, 1957) developed, inter alia, a method for estimating efficient production functions from observations of the inputs and outputs of a number of individual production units. The method consisted essentially in plotting these observations as points in a space of a suitable number of dimensions, forming the convex closure of this set of points and taking the appropriate part of the surface of this convex closure as the estimate of the efficient production function. In the case where n inputs are used to produce a single output under conditions of constant returns to scale, if output is represented by X and the inputs by xl, x2, ..., Xw then each firm can be represented as a point in n-dimensional space with coordinates (xl/X, x2/X, ..., x./X). This set of points is then augmented by the points at infinity on each of the n axes, and the convex closure of the augmented set is formed. The surface of this convex closure, omitting the facet at infinity, is our estimate of the efficient production function. The earlier paper was primarily concerned with this simple version of the method. Not only was it used for the basic exposition but a large part of the paper was devoted to applying it to observations on American agricultural production. The paper did, however, discuss at a formal level the relaxation of the assumptions of single product and constant returns to scale, and in Section 2.3 showed how the method could be generalized to the case of m outputs X1, X2, ..., Xm by treating each observation as a point in n + m-dimensional space with coordinates (X,, X2, ..., Xm, x1, x2, ..., xn). A broadly similar procedure which would accommodate the possibility of decreasing returns to scale was indicated in Section 2.4. Underlying the use of the method in all these cases is the assumption that the efficient production function to be estimated is convex. In no case does this assumption hold necessarily, but it is one frequently made in economics, and will probably hold in most actual situations. Certainly there is nothing in the nature of multiple outputs or decreasing returns to scale that is inconsistent with the convexity assumption. This happy state of affairs ceases to obtain as soon as we consider a situation with increasing returns to scale, because this essentially involves a non-convex production function; indeed they are in practice probably the most important example of nonconvex production functions. The estimation of efficient production functions under

Previous articleNext article No AccessComments on Non-Convexity: RejoinderM. J. FarrellM. J. Farrell Search for more articles by this author PDFPDF PLUS Add to favoritesDownload CitationTrack CitationsPermissionsReprints Share onFacebookTwitterLinkedInRedditEmail SectionsMoreDetailsFiguresReferencesCited by Journal of Political Economy Volume 69, Number 5Oct., 1961 Article DOIhttps://doi.org/10.1086/258544 Views: 2Total views on this site Citations: 1Citations are reported from Crossref Copyright 1961 The University of ChicagoPDF download Crossref reports the following articles citing this article:Isabella M Weber On the necessity of money in an exchange-constituted economy: the cases of Smith and Marx, Cambridge Journal of Economics 16 (Aug 2019).https://doi.org/10.1093/cje/bez038

Previous articleNext article No AccessThe Convexity Assumption in the Theory of Competitive MarketsM. J. FarrellM. J. Farrell Search for more articles by this author PDFPDF PLUS Add to favoritesDownload CitationTrack CitationsPermissionsReprints Share onFacebookTwitterLinkedInRedditEmail SectionsMoreDetailsFiguresReferencesCited by Journal of Political Economy Volume 67, Number 4Aug., 1959 Article DOIhttps://doi.org/10.1086/258197 Views: 18Total views on this site Citations: 59Citations are reported from Crossref Copyright 1959 The University of ChicagoPDF download Crossref reports the following articles citing this article:Martin Bichler, Johannes Knörr, Felipe Maldonado Pricing in Nonconvex Markets: How to Price Electricity in the Presence of Demand Response, Information Systems Research 86 (Jul 2022).https://doi.org/10.1287/isre.2022.1139Walter Briec, Kristiaan Kerstens, Ignace Van de Woestyne Nonconvexity in Production and Cost Functions: An Exploratory and Selective Review*, (Jun 2022): 721–754.https://doi.org/10.1007/978-981-10-3455-8_15Kristiaan Kerstens, Ignace Van de Woestyne Cost functions are nonconvex in the outputs when the technology is nonconvex: convexification is not harmless, Annals of Operations Research 305, no.1-21-2 (Jun 2021): 81–106.https://doi.org/10.1007/s10479-021-04069-1Magdalena Kapelko, Alfons Oude Lansink, Spiro E. 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Previous articleNext article No AccessBook ReviewsDemand for Automobiles in the United States: A Study in Consumer Durables. Gregory C. Chow M. J. FarrellM. J. Farrell Search for more articles by this author PDFPDF PLUS Add to favoritesDownload CitationTrack CitationsPermissionsReprints Share onFacebookTwitterLinkedInRedditEmailPrint SectionsMoreDetailsFiguresReferencesCited by Journal of Political Economy Volume 67, Number 3Jun., 1959 Article DOIhttps://doi.org/10.1086/258182 Views: 3Total views on this site Copyright 1959 The University of ChicagoPDF download Crossref reports no articles citing this article.