Stationary Equilibria in Asset-Pricing Models with Incomplete Markets and Collateral

Abstract

We consider an infinite-horizon exchange economy with incomplete markets and collateral constraints. As in the two-period model of Geanakoplos and Zame (2002), households can default on their liabilities at any time, and financial securities are only traded if the promises associated with these securities are backed by collateral. We examine an economy with a single perishable consumption good, where the only collateral available consists of productive assets. In this model, competitive equilibria always exist and we show that, under the assumption that all exogenous variables follow a Markov chain, there also exist stationary equilibria. These equilibria can be characterized by a mapping from the exogenous shock and the current distribution of financial wealth to prices and portfolio choices. We develop an algorithm to approximate this mapping numerically and discuss ways to implement the algorithm in practice. A computational example demonstrates the performance of the algorithm and shows some quantitative features of equilibria in a model with collateral and default.

Which recursive equilibrium?

“…In particular, h (k) is a RE i¤ conditions (i), (ii), and (iii.a-iii.c) hold. The key conditions that will be a problem to check for correspondence based recursive methods (e.g., Phelan and Stacchetti [46], Kubler and Schmedders [26]. Miao and Santos [34] and Feng, et.…”

“…Give the subcollections USC(X) and C(X) their relative topologies and partial orders to the space B f (X). 26 In our …rst lemma, mention the order completeness properties of subsets of B f under pointwise partial orders:…”

“…In doing this, we also show the Miao-Santos procedure equilibrium correspondence fails to verify RE. Finally, we propose a new correspondence-based approach, based upon "interval iterations" of interval operators valued in function spaces, that integrate the function-based policy-iteration approach of Coleman [10], with the correspondence-based continuation method advocated in Kubler and Schmedders [26] and Miao and Santos [34]. For this interval mapping, we prove the existence of continuous recursive equilibrium for our class of economies, and prove a simple method for computing it approximately.…”

“…One important class of such methods are known as "monotone continuation methods". 11 To date, two types of monotone continuation approaches have been proposed: (i) continuation methods de…ned in spaces of correspondences (i.e., so-called correspondence-based method …rst proposed in Kydland and Prescott [27], but extended in numerous papers including Kubler and Schmedders [26], or Miao and Santos [34]), among others), or (ii) continuation methods de…ned in spaces of functions (i.e., so-called function-based continuation method, or socalled monotone-map methods …rst proposed in Coleman [10][12], Greenwood and Hu¤man [23], Datta, Mirman, and Re¤ett [14] or Mirman, Morand, and Re¤ett [35]). …”

A Theory of Housing Collateral, Consumption Insurance and Risk Premia

“…In our model, all promises are backed by all collateral assets. Geanakoplos and Zame (2000) and Kubler and Schmedders (2003) consider a different environment in which individual assets collateralize individual promises in an incomplete markets economy. Our paper is the first to argue that the housing collateral mechanism can quantitatively replicate the variation in conditional moments of asset prices observed in the data.…”