Inequality and Mobility Under Social Competition

We provide a micro-founded dynamic framework to analyse the effects of inequality on social competition, mobility, and welfare. We consider an infinitely repeated Tullock contest in which players with concave utility allocate resources between consumption and costly effort, and the prizes from the current competition determine the players' endowments in the subsequent period. We characterize the unique pure-strategy Markov Perfect Equilibrium, proving that the highly endowed player exerts more effort and has a higher probability of winning. Social competition is maximized at an intermediate level of inequality, whereas utilitarian social welfare is maximized under full equality. Assuming non-increasing absolute risk aversion preferences, we find that greater inequality monotonically reduces social mobility (a pattern consistent with the Great Gatsby curve) and lowers the welfare of the lowly endowed player. By contrast, the welfare of the highly endowed player is non-monotonic when the discount factor is sufficiently high. Thus, being richer in a more unequal society does not necessarily imply higher individual welfare. For example, under logarithmic utility and a discount factor of 2/3, an individual must control over 88% of total resources to strictly prefer inequality over full equality.