Abstract
Ambiguous assets are characterized as assets where objective and subjective probabilities of tomorrow’s asset-returns are ill-defined or may not exist, e.g., bitcoin, volatility indices or any IPO. Investors may choose to diversify their portfolios of fiat money, stocks and bonds by investing in ambiguous assets, a fourth asset class, to hedge the uncertainties of future returns that are not risks.
(IR)rational probabilities are computable alternative descriptions of the distribution of returns for ambiguous assets. (IR)rational probabilities can be used to define an investor’s (IR)rational expected utility function in the class of non-expected utilities. Investment advisors use revealed preference analysis to elicit the investor’s composite preferences for risk tolerance, ambiguity aversion and optimism.
Investors rationalize (IR)rational expected utilities over portfolios of fiat money, stocks, bonds and ambiguous assets by choosing their optimal portfolio investments with (IR)rational expected utilities. Subsequently, investors can hedge future losses of their optimal portfolios by purchasing minimum-cost portfolio insurance.
Recently Cherchye et al. (2011) reformulated the Walrasian equilibrium inequalities, introduced by Brown and Matzkin (1996), as an integer programming problem and proved that solving the Walrasian equilibrium inequalities is NP-hard. Brown and Shannon (2002) derived an equivalent system of equilibrium inequalities, i.e., the dual Walrasian equilibrium inequalities. That is, the Walrasian equilibrium inequalities are solvable iff the dual Walrasian equilibrium inequalities are solvable.
We show that solving the dual Walrasian equilibrium inequalities is equivalent to solving a NP-hard minimization problem. Approximation theorems are polynomial time algorithms for computing approximate solutions of NP-hard minimization problems. The primary contribution of this paper is an approximation theorem for the equivalent NP-hard minimization problem. In this theorem, we derive explicit bounds, where the degree of approximation is determined by observable market data.
Recently Cherchye et al. (2011) reformulated the Walrasian equilibrium inequalities, introduced by Brown and Matzkin (1996), as an integer programming problem and proved that solving the Walrasian equilibrium inequalities is NP-hard. Following Brown and Shannon (2000), we reformulate the Walrasian equilibrium inequalities as the dual Walrasian equilibrium inequalities.
Brown and Shannon proved that the Walrasian equilibrium inequalities are solvable iff the dual Walrasian equilibrium inequalities are solvable. We show that solving the dual Walrasian equilibrium inequalities is equivalent to solving a NP-hard minimization problem. Approximation theorems are polynomial time algorithms for computing approximate solutions of NP-hard minimization problems.
The primary contribution of this paper is an approximation theorem for the equivalent NP-hard minimization problem. In this theorem, we propose a polynomial time algorithm for computing an approximate solution to the dual Walrasian equilibrium inequalities, where the marginal utilities of income are uniformly bounded. We derive explicit bounds on the degree of approximation from observable market data.
The second contribution is the derivation of the Gorman polar form equilibrium inequalities for an exchange economy, where each consumer is endowed with an indirect utility function in Gorman polar form. If the marginal utilities of income are uniformly bounded then we prove a similar approximation theorem for the Gorman polar form equilibrium inequalities.
Recently Cherchye et al. (2011) reformulated the Walrasian equilibrium inequalities, introduced by Brown and Matzkin (1996), as an integer programming problem and proved that solving the Walrasian equilibrium inequalities is NP-hard. Brown and Shannon (2002) derived an equivalent system of equilibrium inequalities, i.e., the dual Walrasian equilibrium inequalities. That is, the Walrasian equilibrium inequalities are solvable iff the dual Walrasian equilibrium inequalities are solvable.
We show that solving the dual Walrsian equilibrium inequalities is equivalent to solving a NP-hard minimization problem. Approximation theorems are polynomial time algorithms for computing approximate solutions of NP-hard minimization problems. The primary contribution of this paper is an approximation theorem for the equivalent NP-hard minimization problem. In this theorem, we derive explicit bounds, where the degree of approximation is determined by observable market data.
Recently Cherchye et al. (2011) reformulated the Walrasian equilibrium inequalities, introduced by Brown and Matzkin (1996), as an integer programming problem and proved that solving the Walrasian equilibrium inequalities is NP-hard. Following Brown and Shannon (2000), we reformulate the Walrasian equilibrium inequalities as the Hicksian equilibrium inequalities.
Brown and Shannon proved that the Walrasian equilibrium inequalities are solvable iff the Hicksian equilibrium inequalities are solvable. We show that solving the Hicksian equilibrium inequalities is equivalent to solving an NP-hard minimization problem. Approximation theorems are polynomial time algorithms for computing approximate solutions of NP-hard minimization problems.
The contribution of this paper is an approximation theorem for the NP-hard minimization, over indirect utility functions of consumers, of the maximum distance, over observations, between social endowments and aggregate Marshallian demands. In this theorem, we propose a polynomial time algorithm for computing an approximate solution to the Walrasian equilibrium inequalities, where explicit bounds on the degree of approximation are determined by observable market data.
This paper is an exposition of an experiment on revealed preferences, where we posite a novel discrete binary choice model. To estimate this model, we use general estimating equations or GEE. This is a methodology originating in biostatistics for estimating regression models with correlated data. In this paper, we focus on the motivation for our approach, the logic and intuition underlying our analysis and a summary of our findings. The missing technical details are in the working paper by Bunn, et al. (2013).
The experimental data is available from the corresponding author: donald.brown@yale.edu. The recruiting poster and informed consent form are attached as appendices.
This paper is a revision of my paper, CFDP 1865. The principal innovation is an equivalent reformulation of the decision problem for weak feasibility of the GE inequalities, using polynomial time ellipsoid methods, as a semidefinite optimization problem, using polynomial time interior point methods. We minimize the maximum of the Euclidean distances between the aggregate endowment and the Minkowski sum of the sets of consumer’s Marshallian demands in each observation. We show that this is an instance of the generic semidefinite optimization problem: infx in Kf(x) ≡ Opt(K,f), the optimal value of the program,where the convex feasible set K and the convex objective function f(x) have semidefinite representations. This problem can be approximately solved in polynomial time. That is, if p(K,x) is a convex measure of infeasibilty, where for all x, p(K,x) ≥ 0 and p(K,z) = 0 iff z in K, then for every ε > 0 there exists an epsilon-optimal y such that p(K,y) ≤ ε and f(y) ≤ ε + Opt(K,f) where y is computable in polynomial time using interior point methods.
We replicate the essentials of the Huettel et al. (2006) experiment on choice under uncertainty with 30 Yale undergraduates, where subjects make 200 pair-wise choices between risky and ambiguous lotteries. Inferences about the independence of economic preferences for risk and ambiguity are derived from estimation of a mixed logit model, where the choice probabilities are functions of two random effects: the proxies for risk-aversion and ambiguity-aversion.
[Our principal empirical finding is that we cannot reject the null hypothesis that risk and ambiguity are independent in economic choice under uncertainty. This finding is consistent with the hypothesized independence of the neural mechanisms governing economic choices under risk and ambiguity, suggested by the double dissociation-fMRI study reported in Huettel et al.
We propose two algorithms for deciding if the Walrasian equilibrium inequalities are solvable. These algorithms may serve as nonparametric tests for multiple calibration of applied general equilibrium models or they can be used to compute counterfactual equilibria in applied general equilibrium models defined by the Walrasian equilibrium inequalities.
Keywords: Applied general equilibrium analysis, Walrasian equilibrium inequalities, Calibration
Changes in total surplus and deadweight loss are traditional measures of economic welfare. We propose necessary and sufficient conditions for rationalizing consumer demand data with a quasilinear utility function. Under these conditions, consumer surplus is a valid measure of consumer welfare. For nonmarketed goods, we propose necessary and sufficient conditions on market data for efficient production , i.e., production at minimum cost. Under these conditions we derive a cost function for the nonmarketed good, where producer surplus is the area above the marginal cost curve.
We propose two algorithms for deciding if systems of Walrasian inequalities are solvable. These algorithms may serve as nonparametric tests for multiple calibration of applied general equilibrium models or they can be used to compute counterfactual equilibria in applied general equilibrium models defined by systems of Walrasian inequalities.
Keywords: Applied general equilibrium analysis, Walrasian inequalities, Calibration
An applied general equilibrium analysis of monopoly power is proposed as an alternative to the partial equilibrium analyses of monopoly pricing current in antitrust economics. This analysis introduces a new notion of market equilibrium where firms with monopoly power are cost-minimizing price-takers in competitive factor markets and make supracompetitive profits in equilibrium, i.e., the monopoly price exceeds the marginal cost of production.
We assume that the primary goals of antitrust policy are the promotion of competition and the enhancement of consumer welfare. To that end, we use Debreu’s coefficient of resource utilization to determine the counterfactual competitive price levels in monopolized markets and then impute the economic costs of monopolization.
Keywords: Monopoly power, Antitrust economics, Applied general equilibrium analysis
A general equilibrium analysis of monopoly power is proposed as an alternative to the partial equilibrium analyses of monopolization common to most antitrust texts. This analysis introduces the notion of a cost minimizing market equilibrium. The empirical implications of this equilibrium concept for antitrust policy is derived in terms of a family of equilibrium inequalities over market data from observations on a market economy with competitive factor markets. The social cost of monopoly power is measured using Debreu’s coefficient of resource utilization. That is, we propose Pareto optimality as the ultimate objective of antitrust policy.
Keywords: Monopoly power, Antitrust economics, Applied general equilibrium analysis
We propose a nonparametric test for multiple calibration of numerical general equilibrium models, and we present an effective algorithm for computing counterfactual equilibria in homothetic Walrasian economies, where counterfactual equilibria are solutions to the Walrasian inequalities.
Keywords: Applied general equilibrium analysis, Walrasian inequalities, Calibration