We study endogenous leverage in a general equilibrium model with incomplete markets. We prove that in any binary tree leverage emerges in equilibrium at the maximum level such that VaR = 0, so there is no default in equilibrium, provided that agents get no utility from holding the collateral. When the collateral does aﬀect utility (as with housing) or when agents have suﬀiciently heterogenous beliefs over three or more states, VaR = 0 fails to hold in equilibrium. We study commonly used examples: an economy in which investors have heterogenous beliefs and a CAPM economy consisting of investors with diﬀerent risk aversion. We ﬁnd two main departures from VaR = 0. First, both examples show that with enough heterogeneity among the investors, equilibrium default is normal. Second, we ﬁnd that more than one contract is actively traded in equilibrium on the same collateral, that is, the same asset is bought at diﬀerent margin requirements by diﬀerent agents. Finally, we study the relationship between leverage and asset prices. We provide an example that shows that as the regulatory authority gradually relaxes leverage restrictions from low levels and permits leverage to rise, asset prices start to rise, but eventually increased leverage paradoxically tends to reduce asset prices because the risky bonds become substitutes for the asset used as collateral.
Fostel, Ana and Geanakoplos, John, "Endogenous Leverage: VaR and Beyond" (2011). Cowles Foundation Discussion Papers. 2144.