Poverty and Self-control

Proof of Proposition 1. See the Online Appendix. Let H(A) and L(A) be the maximum and minimum values of the equilibrium value correspondence V . Because the graph of V is closed, H is usc and L is lsc. We now describe the supremum of continuation values that a deviating individual can obtain. Let x be a feasible choice of continuation asset at A. Consider all limits of sequences of the form {L(xn)}, where xn ∈ [B,α(1 − υ)A] for all n and xn → x. Each limit is an equilibrium value at x, because V has closed graph. Moreover, the collection of all such limits at x (given A) is compact, so a largest value M(x,A) exists for A ≥ B and x ∈ [B,α(1− υ)A]. LEMMA 2. For given A, M(x,A) is usc in x, nondecreasing in A, and independent ofA as long as x < α(1− υ)A.