Computing general equilibrium with incomplete markets and default

General economic equilibrium with incomplete markets and default represents an important tool in competitive economic theory, and has led to fundamental insights into the behavior of real and financial markets. The numerical computation of these models is important for understanding the behavior of the real world economy, and may lead to insights on the effects of regulation and welfare. However, the current approach to the computation of general equilibrium, based on homotopy methods, does not scale well to complex economic models such as general equilibrium with default. The computation is technically difficult because the utility function and constraints may not be smooth on the entire domain when considering default penalties and collateral models. We consider a two-period exchange economy with default for two classes of models. In the first (collateral) model, promises have to be backed by a durable good, held by the borrower as collateral. In the second (default penalties) model, an agent incurs a loss in utility when she defaults, the loss increasing proportionately with the amount of default. In Chapter 1 we compute general equilibrium with incomplete markets, collateral and default penalties. The computation of general equilibrium is treated as a nonlinear programming problem and solved by an optimization procedure for large computation ALGENCAN an Augmented Lagrangian Method for general nonlinear programming problems. We illustrate the proposed method by computing equilibria for some examples, showing its robustness. In Chapter 2 (based on collaboration with Alóısio Araújo and Felix Kubler) we examine the effects of default and scarcity of collateralizable durable goods on risk-sharing. We assume that there is a large set of assets, but which distinguish themselves by the collateral requirement. There are at least as many assets available for trade as there are states of the world. In the example 1, if there is an abundance of commodities that can be used as collateral and if each agent owns a large fraction of these commodities, markets are complete and competitive equilibrium allocations Pareto optimal. If, on the other hand, the collateralizable durable good is scarce or if some agents do not own enough of the collateralizable durable good in the first period, markets can be endogenously incomplete, not all of the available assets are traded in the competitive equilibrium and allocations are not Pareto optimal. We give examples that show that welfare losses can be quantitatively large and examine the scope for government intervention. We also show that if the borrower owns almost no durable goods, the only asset traded in equilibrium is the one with the lowest possible collateral that can be interpreted as a subprime loan. In Chapter 3 (based on collaboration with Alóısio Araújo) we examine, through numerical examples, when the equilibria allocations can approach the Pareto frontier by the use of a default mechanism. As in the Chapter 2 we assume that there are at least as many assets available for trade as there are states of the world. Our main focus is on the extent that the equilibria allocations approximate the Pareto frontier, and we exhibit how this quantitative problem is quite sensitive to qualitative features of the endowment distribution. In our examples, if the endowment distribution displays only heterogeneity between periods (e.g., one agent is the richest in the first period and another is the richest in all states of nature of the second period), some collateral equilibria are Pareto optimal. If the heterogeneity of the endowments is also manifest between states of nature, default penalties equilibria are often Pareto superior with respect to collateral equilibria.