Felix Kubler

Borrowing Costs and the Demand for Equity over the Life Cycle

We analyze consumption and portfolio behavior in a life-cycle model with realistic borrowing costs and income processes. We show that even a small wedge between borrowing costs and the risk-free return dramatically shrinks the demand for equity. When the cost of borrowing equals or exceeds the expected return on equity the relevant case according to the data households hold little or no equity during much of the life cycle. The model also implies that the correlation between consumption growth and equity returns is low at all ages, and that risk aversion estimates based on the standard excess return formulation of the consumption Euler Equation are greatly upward biased. The demand for equity in the model is non-monotonic in borrowing costs and risk aversion, and the standard deviation of marginal utility growth is an order of magnitude smaller than the Sharpe ratio.

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

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.

Computing equilibrium in OLG models with stochastic production

Abstract In this paper we develop a projection algorithm to approximate equilibria in overlapping generations economies with a large number of generations and stochastic aggregate production. In these types of economies the state space includes the distribution of wealth across generations. We use Smolyaks algorithm to approximate policy functions, which map the current state into agents’ optimal choices, by linear combinations of polynomials. This allows us to compute equilibria for models where agents live for 20–30 periods. We also provide examples which demonstrate that approximating the cross-sectional wealth distribution only by its first moment (and thereby reducing the dimension of the state space to two continuous state variable, independently of the number of agents) often leads to very high relative errors in agents’ Euler equations.

Asset Trading Volume with Dynamically Complete Markets and Heterogeneous Agents

Trading volume of infinitely lived securities, such as equity, is generically zero in Lucas asset pricing models with heterogeneous agents. More generally, the end-of-period portfolio of all securities is constant over time and states in the generic economy. General equilibrium restrictions rule out trading of equity after an initial period. This result contrasts the prediction of portfolio allocation analyses that portfolio rebalancing motives produce nontrivial trade volume. Therefore, other causes of trade must be present in asset markets with large trading volume.

Intergenerational Risk-Sharing via Social Security when Financial Markets Are Incomplete

This paper develops an overlapping generations model with stochastic production and incomplete markets to assess whether the introduction of an unfunded social security system can lead to a Pareto improvement, even if the initial equilibrium is neither production-ine±cient in the spirit of Diamond (1965) nor dynamically ine±cient in the spirit of Samuelson (1957). When returns to capital and wages are imperfectly correlated and subject to aggregate shocks, then the consumption variance of all generations can be reduced if private markets or government policies enable them to pool their labor and capital incomes. A social security system that endows retired households with a claim to labor income may serve as an e®ective tool to share aggregate risk between gener

RECURSIVE EQUILIBRIA IN ECONOMIES WITH INCOMPLETE MARKETS

We examine minimal sufficient state spaces for equilibria in a Lucas asset pricing model with heterogeneous agents and incomplete markets. It is clear that even if all fundamentals of the economy follow a first-order Markov process, equilibrium prices and allocations generally will depend not only on the current exogenous shock but also on the distribution of wealth among the heterogeneous agents. The main contribution of this paper is to give an example of an infinite-horizon economy with Markovian fundamentals, where the joint process of equilibrium asset holdings and exogenous shocks does not constitute a sufficient state space either.

Regulating collateral-requirements when markets are incomplete☆

In this paper we examine the effects of default and collateral on risk sharing. We assume that there is a large set of assets which all promise a risk less payoff but which distinguish themselves by their collateral requirements. In equilibrium agents default, the assets have different payoffs, and there are as many linearly independent assets available for trade as there are states of the world. We derive necessary and sufficient conditions for equilibria to be Pareto-efficient in the presence of uncertainty. We explore some examples for which the collateral equilibrium allocation is identical to the Arrow–Debreu allocation, either when agents have a high preference for the durable good, or when the endowment distribution of the durable good is relatively homogeneous. We examine a series of examples to understand which collateral-levels prevail in equilibrium and under which conditions there is scope for regulating margin-requirements, that is, restricting the sets of tradable assets through government intervention. In these examples equilibrium is always sub-optimal but regulation never leads to a Pareto-improvement. While the competitive equilibria are constrained efficient, there do exist regulations which make large groups of agents in the economy better off. These regulations typically restrict all trades to take place in the low-collateral loans and benefit the poor and the rich agents in the economy through their effects on the equilibrium interest rate and the equilibrium prices of the durable goods.

Competitive Equilibria in Semi-Algebraic Economies

This paper examines the equilibrium correspondence in Arrow-Debreu exchange economies with semi-algebraic preferences. We show that a generic semi-algebraic exchange economy gives rise to a square system of polynomial equations with finitely many solutions. The competitive equilibria form a subset of the solution set and can be identified by verifying finitely many polynomial inequalities. We apply methods from computational algebraic geometry to obtain an equivalent polynomial system of equations that essentially reduces the computation of all equilibria to finding all roots of a univariate polynomial. This polynomial can be used to determine an upper bound on the number of equilibria and to approximate all equilibria numerically. We illustrate our results and computational method with several examples. In particular, we show that in economies with two commodities and two agents with CES utility, the number of competitive equilibria is never larger than three and that multiplicity of equilibria is rare in that it only occurs for a very small fraction of individual endowments and preference parameters.

Tackling Multiplicity of Equilibria with Gröbner Bases

Multiplicity of equilibria is a prevalent problem in many economic models. Often equilibria are characterized as solutions to a system of polynomial equations. This paper gives an introduction to the application of Grobner bases for finding all solutions of a polynomial system. The Shape Lemma, a key result from algebraic geometry, states under mild assumptions that a given equilibrium system has the same solution set as a much simpler triangular system. Essentially, the computation of all solutions then reduces to finding all roots of a single polynomial in a single unknown. The software package Singular computes the equivalent simple system. If all coefficients in the original equilibrium equations are rational numbers or parameters, then the Grobner basis computations of Singular are exact. Thus, Grobner basis methods cannot only be used for a numerical approximation of equilibria, but in fact may allow the proof of theoretical results for the underlying economic model. Three economic applications illustrate that without much prior knowledge of algebraic geometry, Grobner basis methods can be easily applied to gain interesting insights into many modern economic models.

Observable restrictions of general equilibrium models with financial markets

This paper examines whether general equilibrium models of exchange economies with incomplete financial markets impose restrictions on prices of commodities and assets given the stochastic processes of dividends and aggregate endowments. We show that the assumption of time-separable expected utility implies restriction on the cross-section of asset prices as well as on spot commodity prices. However, a relaxation of the assumption of time separability will generally destroy these restriction.