Suppose that an agent has preferences over a finite time horizon, and that, in addition, marginal rates of substitution between adjacent time periods are independent of the level of consumption in other time periods. This means, for example, that marginal rates of substitution between commodities in periods one and two are independent of consumption in period three, and that marginal rates of substitution between commodities in periods two and three are independent of consumption in period one. Then, by the overlapping theorem, the utility function must be additive in the consumption of the three periods. A proof along these lines is presented in Sect. 4 of ’The structure of utility functions’ (Ch. 12); this paper presents an alternative proof of the same theorem. John Whitaker at the Oxford Mathematical Economics Seminar posed the problem in 1964, and Gorman submitted the paper to Econometrica in July 1965 and it was published in Econometrica 36 (1968).
Keywords: additive separability, additive utility functions, additivity, consumption, marginal rates of substitution, overlapping theorem, separability, time periods