In Nonparametric Multivariate Regression Subject to Constraint (with Paul Ruud) we review Hildreth's algorithm for computing the least squares regression subject to inequality constraints and subsequent generalizations, notably Dykstra's. We provide a geometric proof of convergence and a modification to the algorithm which improves computation speed and provides some robustness w.r.t. rounding error.

On the Nonconvexity of the Set of Utility-based Demand Functions(with Paul Ruud) examines a fundamental question of economic theory: when will a given collection of economics data be consistent with a given theory. For demand analysis, this problem is elegantly solved and elaborated in Afriat (S. Afriat, "On a System of Inequalities in Demand Analysis: An Extension of the Classical Method," International Economic Review, 14, 1973, 460-472) and Varian (H. Varian, "The Nonparametric Approach to Demand Analysis," Econometrica, 50, 1982, 945-973). The empirical version of this query asks what is the ''best'' estimate of economic behavior satisfying some theory with respect to a given collection of observations. This estimation problem is customarily addressed through regression analysis but an alternative procedure involving nonparametric analysis has recently experienced renewed interest. This technique, originates with Hildreth (C. Hildreth, "Point Estimates of Ordinates of Concave Functions," Journal of the American Statistical Association, 49 ,1954, 598-619) and is described in Varian (H. Varian, "Nonparametric Analysis of Optimizing Behavior with Measurement Error, Journal of Econometrics, 30, 1985, 445-458) and Goldman and Ruud ("Nonparametric Multivariate Regression Subject to Constraint," Working Paper No. 93-213, Department of Economics, University of California, revised 1995). We wish to address this question of estimating an individual demand function without imposing any restrictions as to functional form except those directly implied by revealed preference. The problem will be described by the attempt to identify a least squares solution to the distance between the actual observations and those obtained from a demand function derivable from utility maximization. In order for a unique solution to exist to this problem, the allowable set of demand functions in general must be closed and convex. In the note below, we will demonstrate that the set of utility based demand functions fails to be convex and that, consequently, unique estimation is problematic.

The analysis in The SIS Model of Infectious Disease with Treatment (with James Lightwood) examines the steady state solutions to rational treatment patterns of SIS diseases. Individual behavior is shown to lead to the possibility of significant undertreatment through the presence of unrewarded external benefits. The main conclusion of the analysis notes that the individually rational and socially optimal level of disease will usually differ, with the socially optimal level of disease being no larger than the individually rational level.

In Cost Optimization in the SIS Model of Disease with Treatment (with James Lightwood) we consider the intertemporal social optimization problem of minimizing the present value of the costs incurred from both disease and treatment Our analysis allows for a robust collection of treatment cost functions and so extends the previous analysis by Sanders who examines only the case of constant marginal treatment cost. Though this extension of the analysis is substantially complicated by the resulting failure of concavity , we are able to characterize both the long run equilibria and the adjustment paths. Sanders results generalize is the following manner: The "bang-bang" treatment pattern discerned by Sanders where, below some level of infection, the entire population of infecteds is treated while, above that level, treatment is completely discontinued is here replaced by a montonically decreasing level of treatment with respect to the level of disease incidence. The socially optimal program is then compared to individually rational behavior and the inefficiencies in private behavior from the infection externality are shown to cause potentially large increases in the equilibrium rate of infection. The negative monotonic relationship (described above) is shown to hold in both the individual and social cases and is a direct consequence of the increased likelihood of reinfection at higher levels of disease incidence which renders the benefits less worthwhile.