Borrowing-proofness of the Lindahl rule in Kolm triangle economies

In the context of a simple model of public good provision, we study the requirement on an allocation rule that it be immune to manipulation by augmenting one's endowment through borrowing from the outside world. We call it open-economy borrowing-proofness (Thomson, 2009). We ask whether the Lindahl rule satisfies the property. The answer is yes on both the domain of quasi-linear economies and on the domain of homothetic economies. However, on the classical domain (when preferences are only required to be continuous, monotone, and convex), the answer is negative. We compare the manipulability of the rule through borrowing and its manipulability through withholding. We also asks whether the rule is immune to manipulation by borrowing from a fellow trader, closed-economy borrowing-proofness. We obtain a parallel set of answers. The negative results hold no matter how small the amount borrowed is constrained to be.