A Karush-Kuhn-Tucker test of convexity for univariate observations

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 Karush-Kuhn-Tucker test statistic of convexity, when the estimator of the data minimizes the sum of squares of residuals subject to the assumption of non-decreasing returns. It is a highly structured quadratic programming problem that allows a very efficient calculation of the test statistic. Certain applications that test the convexity assumption of real economic data are considered and the interpretation capability of the test is demonstrated. Some numerical results illustrate the computation and present the efficacy of the test in small, medium and large data sets. They suggest that the test is suitable when the number of observations is very large