THE NUMERICAL APPROXIMATION FOR THE INTEGRABILITY PROBLEM AND THE MEASURE OF WELFARE CHANGES, AND ITS APPLICATIONS

Abstract

This dissertation mainly studied on numerical approximation methods as a solution of the integrability problem and the measure of welfare changes, and demonstrated how numerical algorithms can be applied in empirical studies as a solution method. In general, the integrability problem is described as a system of the partial differential equations (PDE) in terms of the expenditure function, and the measure of welfare changes is defined by the difference between the expenditure function at two different time periods. Both problems can be solved using the same method since solutions for these questions mainly relied on how to recover the compensated income (expenditure) from the ordinary demand function. In order to investigate whether numerical approximation methods can be applied to the integrability problem and the measure of welfare changes, first, we studied the integrability problem mainly focusing on how to transform the system of the partial differential equations to the system of the ordinary differential equation since this transform possibility provides a way to solve the integrability problem using the numerical method. Second, several numerical methods were investigated as a possible solution of both problem including the Vartia, the RK-4th order algorithm, and the Adams Fourth-Order Predictor-Corrector algorithm. In addition, the Rotterdam and Almost Ideal demand system were investigated since the demand system played an important role on recovering the expenditure. Two empirical studies are performed. In the first application, using both the U.S consumer expenditure (CE) data and the consumer price index (CPI), the AI and Rotterdam demand system were estimated, and the expenditure was recovered from the estimated demand system using numerical approximation methods. From this, we could demonstrate the power and the applicability of numerical algorithms. In the second application, we paid attention to analyze the welfare effect on the U.S elderly population when prices changed. The burden index and the compensating variation were calculated using the numerical algorithm. From the evaluation, we could confirm that the welfare changes and consumer welfare losses of the elderly population were larger than that of the general U.S population