A positional game for an overlapping generation economy

We develop a model with intra-generational consumption externalities, based on the overlapping generation version of Diamond (1965) model. More specifically, we consider a two-period lived overlapping generation economy, assuming that the utility of each consumer depends also on the average level of consumption by other consumers in the same generation. In this way we obtain a positional game embedded in an overlapping generation economy. We characterize the consumption and saving choices for the two periods in the Nash equilibrium path and we determine a dynamic equation for capital accumulation coherent with the agents' choices in the Nash equilibrium. Hence, also the behavior, both static and dynamic, described by the equation for the capital accumulation is coherent with the Nash equilibrium. For the associated dynamical system we find a unique positive steady state for capital, which is globally stable. Its position is decreasing with respect to positive variations in the degree of interaction in the first period, while the opposite relation holds in the second period. We then compare the steady states for capital with and without social interaction. In this respect we show that the steady state with social interaction is larger than the steady state in the absence of social interaction if and only if the degree of interaction in the second period exceeds the degree of interaction in the first period. In particular, if the degrees of interaction in the two periods coincide, the dynamical system is equivalent to the one without social interaction.