On trees and logs

In this paper we critically examine the main workhorse model in asset pricing theory, the Lucas (1978) tree model (LT-Model), extended to include heterogeneous agents and multiple goods, and contrast it to the benchmark model in ...nancial equilibrium theory, the real assets model (RA-Model). Households in the LT-Model trade goods together with claims to Lucas trees (exogenous stochastic dividend streams speci...ed in terms of a particular good) and longlived, zero-net-supply real bonds, and are endowed with share portfolios. The RA-Model is quite similar to the LT-Model except that the only claims traded there are zero-net-supply assets paying out in terms of commodity bundles (real assets) and households’ endowments are in terms of commodity bundles as well. At the outset, one would expect the two models to deliver similar implications since the LT-Model can be transformed into a special case of the RA-Model. We demonstrate that this is simply not correct: results obtained in the context of the LT-Model can be strikingly di¤erent from those in the RA-Model. Indeed, specializing households’ preferences to be additively separable (over time) as well as log-linear, we show that for a large set of initial portfolios the LT-Model – even with potentially complete ...nancial markets – admits a peculiar ...nancial equilibrium (PFE) in which there is no trade on the bond market after the initial period, while the stock market is completely degenerate, in the sense that all stocks o¤er exactly the same investment opportunity – and yet, allocation is Pareto optimal. We then thoroughly investigate why the LT-Model is so much at variance with the RA-Model, and also completely characterize the properties of the set of PFE, which turn out to be the only kind of equilibria occurring in this model. We also ...nd that when a PFE exists, either (i) it is unique, or (ii) there is a continuum of equilibria: in fact, every Pareto optimal allocation is supported as a PFE. Finally, we consider to what extent our results continue to hold true in the presence of various types of restrictions on transactions in ...nancial markets. While our analysis is carried out in the framework of the traditional two-period Arrow-DebreuMcKenzie pure exchange model with uncertainty (encompassing, in particular, many types of contingent commodities), we show that similar results hold for the analogous continuous-time martingale model of asset pricing. ¤We are very grateful to Suleyman Basak for numerous conversations throughout the project, and in particular for sharing insights into solving for equilibrium in the presence of portfolio constraints; to Svetlana Boyarchenko for providing the elegant, general proof of Proposition 4, as well as helping to uncover the precise possibilities for nonuniqueness; and to Paolo Siconol... for many illuminating conversations, especially regarding the implications of adding seemingly irrelevant portfolio constraints.