Numerical simulations of techniques related to utility function and price elasticity estimators

In many economic models the utility function chosen is based on preconceived ideas of the economic state. In order for a utility function to be fit to raw demand data an assumption is made (amongst others) that a preference relation holds within a cycle of data points (Eberhard et al. (2009)). In this paper we assume that errors have occurred in the data collection process which have somewhat corrupted the quantity consumed for the particular commodity price. This results in inconsistences in the preference relation and infeasibility of Afriat inequalities (Eberhard et al. (2009)). We introduce a method that allows the data to shift in order for a Generalised Axiom of Revealed Preference (GARP) to be satisfied by the data enabling a utility to be fitted. This technique is described in section 1. The commonly used Cobb-Douglas utility is defined as (equcation presented) where S^L_i=1 a_i = 1. This function represents the demand of commodities with respect to commodity costs and household income. Here a represents the commodity share of good i in the total household expenditure. Upon solving the utility maximisation problem the consumer spends the entire household budget x_i, p_i= y, and the demand is given as (equcation presented) We run simulations with generated Cobb-Douglas type price-demand data to compare the fit of an “Afriat” type utility. Errors are included in the Cobb-Douglas data to simulate the corruption of GARP and to test the robustness of the error shifting least squares program. Calculating price elasticities of demand is an important part of an economic model, as it quantifies the susceptibility to change in quantity consumed for an associated change in commodity price. The Hicks-Slutsky partition (Dixon et al. (1980)) describes the changes in demand given by a price change in commodity in conjunction with the associated change in demand given by the consumers change in income. We fit a utility to simulated Cobb-Douglas demand to compare our calculated elasticities (constrained to be consistent with the Hicks-Slutsky Partition) with the known Cobb-Douglas elasticities. Cobb-Douglas price elasticity of demand is unitary as demonstrated by the demand relation (2), therefore a 1% increase/decrease in price will lead to a 1% decrease/increase in demand. Cross-price elasticity for this case is zero, since the demand is directly proportional to it's own price. We investigate and compare the elasticities generated by our method. An example of commodities that are considered to be perfect substitutes is tea and coffee. Using the technique described in section 2 we run a simulation of calculated elasticities with Cobb-Douglas simulated data and also calculate elasticities of real price-quantity data from the ABS to determine the substitutability of tea-coffee.