Cost of Living Index Based on an Estimated Generalized Ces Utility Function Computed Using Multi-step Perturbation

The paper describes and illustrates a multi-step perturbation (MSP) method for accurately and quickly computing changes in the real cost of living, ∆ct, as nominal prices change over observation periods. As prices change, for a given nominal income, a representative consumer moves from one utility-maximizing point to another in combined income and substitution effects. By definition, ∆ct is an adjustment of a consumer's real income, which restricts them to a pure substitution effect, i.e., to be on the same initial indifference curve as prices change. Using MSP to compute ∆ct has two advantages. (1) The MSP method is based on kth-order approximating polynomials, computed in h steps, for each observation period, with resulting accuracy of order h. (2) The MSP method uses only derivatives of a utility function and can compute results even when there are no known explicit closed-form solutions of the utility maximization problem. This is illustrated with the generalized constant elasticity of substitution (GCES) utility function.