Supplement to “Foundations of Intrinsic Habit Formation”

Lemma 23. ∀ T ∈ N, x̄ = (x0, x1, . . . , xT ) ∈ R, ∃ hx̄,T ∈ H such that (x0, x1, . . . , xT , 0, 0, . . .) ∈ C∗ hx̄,T+1 . Proof. For arbitrary h, define c by c0 = x0+φ(h), c h t = xt+φ(hc h 0c h 1 · · · ct−1) for all 1 ≤ t ≤ T , and ct = φ(hc0c1 · · · ct−1) for t > T . φ is strictly increasing, so we may choose hx̄,T ∈ H sufficiently large so that (c hx̄,T 0 , c hx̄,T 1 , . . . c hx̄,T T ) is nonnegative. But if (c hx̄,T 0 , c hx̄,T 1 , . . . c hx̄,T T ) is nonnegative, then so is cx̄,T . Moreover, the stream is ultimately weakly decreasing. Therefore cx̄,T ∈ C.