The Karush-Kuhn-Tucker Test of Convexity of Univariate Observations and Certain Economic Applications

The problem of convexity runs deeply in economic theory. For example, increasing returns or upward slopes (convexity) and diminishing returns or downward slopes (concavity) of certain supply, demand, production and utility relations are often assumed in economics. Quite frequently, however, the observations have lost convexity (or concavity) due to errors of the measuring process. We derive the KarushKuhn-Tucker test statistic of convexity, when the convex estimator of the data minimizes the sum of squares of residuals subject to the assumption of non-decreasing returns. Testing convexity is a linear regression problem with linear inequality constraints on the regression coefficients, so generally the work of Gouriéroux, Holly and Monfort (1982) as well as Hartigan (1967) apply. Convex estimation is a highly structured quadratic programming calculation that is solved very efficiently by the Demetriou and Powell (1991) algorithm. Certain applications that test the convexity assumption of real economic data are considered, the results are briefly analyzed and the interpretation capability of the test is demonstrated. Index terms Cobb-Douglas, convexity, concavity, data fitting, diminishing return, divided difference, Gini coefficient, infant mortality, least squares, money demand, quadratic programming, statistical test