Precise analysis of equilibrium in asset markets is difficult since few cases can be solved exactly for equilibrium prices and volume. Many analyses assume that markets are complete, implying that equilibrium is efficient and equivalent to some social planner'sproblem. That approach is limited since it ignores transaction costs, taxes, and incompleteness in asset markets. This paper develops bifurcation methods to approximate asset market equilibrium without assuming complete asset markets.
We begin from a trivial deterministic case where all assets have the same safe return and use local approximation methods to compute asset market equilibrium when assets have small risk. We compute Taylorseries expressing equilibrium asset prices and holdings as a function of preference parameters such as absolute risk aversion, and asset return statistics such as mean, variance, and skewness. The formulas completely characterize equilibrium for small risks.
Implementing this approach is straightforward, but involves an enormous amount of algebraic manipulation far beyond the capacity of human hands. Fortunately, desktop computers using symbolic software can execute the necessary algebraic manipula- tionandcompute the series expansions in reasonable time. We use Mathematica, but the computation could be executed by other symbolic languages such as Macsyma and Maple.
The asymptotic expansions tell us about the qualitative properties of equilibrium and can be used to computea numerical approximation to equilibrium of particular problems with a speci Þednonzerorisk. Therefore, the bifurcation approach is computational in two ways: the formulas are qualitative asymptotic approximations derived by computer algebra, and can be used to produce numerical approximations tospeciÞcproblems. This paper focuses on the qualitative asymptotic results and leaves the numerical applications for future study.
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Asymptotic Methods for Asset Market Equilibrium Analysis
