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It has been conjectured that canonical Bewley--Huggett--Aiyagariheterogeneous-agent models cannot explain the joint distribution of income andwealth. The results stated below verify this conjecture and clarify itsimplications under very general conditions. We show in particular that if (i)agents are infinitely-lived, (ii) saving is risk-free, and (iii) agents haveconstant discount factors, then the wealth distribution inherits the tailbehavior of income shocks (e.g., light-tailedness or the Pareto exponent). Ourrestrictions on utility require only that relative risk aversion is bounded,and a large variety of income processes are admitted. Our results showconclusively that it is necessary to go beyond standard models to explain theempirical fact that wealth is heavier-tailed than income. We demonstratethrough examples that relaxing any of the above three conditions can generatePareto tails.

An Impossibility Theorem for Wealth in Heterogeneous-agent Models with Limited Heterogeneity