¹⁸This limiting result requires that we impose the $FHWC (Γ < R )$ , because the perfect foresight consumption function is not deﬁned if $Γ ≥ R$ . Informally, the proof is as follows. Deﬁne $c ≡ (m − 1)κ = cu(m )$ as the consumption function for the unemployed consumer who will receive no future labor income. Then $c(m) < c(m ) ≤ ¯c(m )$ , and so $1 < c(m)∕c(m) < ¯c(m )∕c(m )$ . In the limit as $m ↑ ∞$ , however, human wealth accounts for an arbitrarily small proportion $κh ∕κ(h + m − 1)$ of consumption, so $limm ↑∞ ce(m)∕cu(m) = 1$ so the precautionary motive captured by $∇$ vanishes.