# Correction for “Collateral premia and risk sharing under limited commitment” [Econ. Theory 46, 475–501 (2011)]

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## Abstract

This note provides a correction for Lemma 4 in Kilenthong (Econ Theory 46:475–501, 2011). As pointed out in Zierhut (Econ Theory Bull, 2013), Assumption 2 in Kilenthong (Econ Theory 46:475–501, 2011) is not sufficient to guarantee that optimal allocations will be full risk sharing as claimed in Lemma 4 in Kilenthong (Econ Theory 46:475–501, 2011). This note shows that the main result still holds for an economy in which risk sharing rule is linear by providing a corrected sufficient condition for full risk sharing.

## Keywords

Collateral premium Collateralized contracts Limited commitment Full risk sharing## JEL Classification

D52 D53Consider a collateral economy as in Kilenthong (2011). There are two periods. There are \(S\) states in the second period. There is no loss of generality to focus only on an economy with consumption and collateralizable capital goods only. Thus, the commodity space is given by \(\mathbb {R}^{S+2}\). Let \(R_s > 0\) be the payoff of the capital good in state \(s = 1, \ldots , S\). Let \(\mathbf {e}^h =\left( e^h_0, e^h_1, \ldots , e^h_S, e^h_\mathrm{coll}\right) \) be the endowments of an agent of type \(h\). The aggregate collateral is denoted by \(\bar{e}_\mathrm{coll} = \sum _h e^h_\mathrm{coll}\).

There is no loss of generality to focus on collateralized Arrow or contingent securities only because the objective is only to find a sufficient condition. Let \(D_{sj}\) be the payoff of the \(j\)th-contingent security in state \(s\). As in Kilenthong (2011), its payoff in state \(s\) is \(D_{ss} = R_s\) and zero in other states.

Let \(\lambda = \left( \lambda ^1, \ldots , \lambda ^H\right) \) be the vector of the Pareto weights such that \(\lambda ^h > 0\) and \(\sum _h \lambda ^h =1\).

### **Lemma 1**

### *Proof*

In conclusion, this lemma is a corrected version of Lemma 4 in Kilenthong (2011). Accordingly, it can be used to prove Proposition 4 in Kilenthong (2011). Lemma 4 and Proposition 4 hold under a sufficient condition (7) for an economy in which risk sharing rule is linear.

## References

- Kilenthong, W.T.: Collateral premia and risk sharing under limited commitment. Econ. Theory
**46**(3), 475–501 (2011)CrossRefGoogle Scholar - Townsend, R.M.: The Medieval Village Economy. Princeton University Press Princeton, NJ (1993)Google Scholar
- Zierhut, M. Comment on “Collateral Premia and Risk Sharing under Limited Commitment” [Econ. Theory 46, 475–501 (2011)]. Econ. Theory Bull. (2013, forthcoming)Google Scholar