Status competition, endogenous risk attitudes, and the insurance–lottery puzzle

Individuals purchase insurance while buying lottery tickets, contradicting expected utility's prediction of globally concave preferences. This paper provides a structural explanation for the coexistence of risk attitudes commonly associated with insurance purchase and lottery participation. An agent's utility combines a concave consumption component with a contest-based status component derived from the difference-form Tullock–Hirshleifer contest success function. Risk curvature is endogenous: weak agents are strictly risk-loving under re-optimization, while all agents are strictly risk-averse over background risks at fixed status expenditure, so any weak agent simultaneously exhibits both risk attitudes. This coexistence rests on irreversibility: background risks destroy assets that cannot be instantly reallocated, while lottery prizes arrive as liquid cash permitting immediate reallocation toward status expenditure. Indirect utility is globally S-shaped around an endogenously determined reference wealth level that shifts one-for-one with rival expenditure, sharing the S-shaped curvature and endogenous reference point properties of the prospect theory value function, while remaining silent on loss aversion and probability weighting.