  Two critical questions about intergenerational outcomes are: one, whether
significant barriers or traps exist between different social or economic
strata; and two, the extent to which intergenerational outcomes do (or can be
used to) affect individual investment and consumption decisions. We develop a
model to explicitly relate these two questions, and prove the first such `rat
race' theorem, showing that a fundamental relationship exists between high
levels of individual investment and the existence of a wealth trap, which traps
otherwise identical agents at a lower level of wealth. Our simple model of
intergenerational wealth dynamics involves agents which balance current
consumption with investment in a single descendant. Investments then determine
descendant wealth via a potentially nonlinear and discontinuous competitiveness
function about which we do not make concavity assumptions. From this model we
demonstrate how to infer such a competitiveness function from investments,
along with geometric criteria to determine individual decisions. Additionally
we investigate the stability of a wealth distribution, both to local
perturbations and to the introduction of new agents with no wealth.
