288 research outputs found
On Computability of Equilibria in Markets with Production
Although production is an integral part of the Arrow-Debreu market model,
most of the work in theoretical computer science has so far concentrated on
markets without production, i.e., the exchange economy. This paper takes a
significant step towards understanding computational aspects of markets with
production.
We first define the notion of separable, piecewise-linear concave (SPLC)
production by analogy with SPLC utility functions. We then obtain a linear
complementarity problem (LCP) formulation that captures exactly the set of
equilibria for Arrow-Debreu markets with SPLC utilities and SPLC production,
and we give a complementary pivot algorithm for finding an equilibrium. This
settles a question asked by Eaves in 1975 of extending his complementary pivot
algorithm to markets with production.
Since this is a path-following algorithm, we obtain a proof of membership of
this problem in PPAD, using Todd, 1976. We also obtain an elementary proof of
existence of equilibrium (i.e., without using a fixed point theorem),
rationality, and oddness of the number of equilibria. We further give a proof
of PPAD-hardness for this problem and also for its restriction to markets with
linear utilities and SPLC production. Experiments show that our algorithm runs
fast on randomly chosen examples, and unlike previous approaches, it does not
suffer from issues of numerical instability. Additionally, it is strongly
polynomial when the number of goods or the number of agents and firms is
constant. This extends the result of Devanur and Kannan (2008) to markets with
production.
Finally, we show that an LCP-based approach cannot be extended to PLC
(non-separable) production, by constructing an example which has only
irrational equilibria.Comment: An extended abstract will appear in SODA 201
Modelling vintage structures with DDEs : principles and applications
A comprehensive study of the linkages between demographic and economic variables should not only account for vintage specificity but also incorporate the relevant economic and demographic decisions in a complete optimal control set-up. In this paper, a methodological set-up allowing to reach the objectives is described. In this framework, time is continuous but agents take discrete timing decisions. The mixture of continuous and discrete time yields differential-difference equations (DDEs). This paper shows clearly that the approach allows for a relatively complete and rigorous analytical exploration in some special cases (mainly linear or quasi linear models), and for an easy computational appraisal in the general case.Demography, Economic growth, Vintage structures, Optimal control, Differential-difference equations, State-dependence
Market Equilibrium in Exchange Economies with Some Families of Concave Utility Functions
We present explicit convex programs which characterize the equilibrium for certain additively separable utility functions and CES functions. These include some CES utility functions that do not satisfy weak gross substitutability.Exchange economy, computation of equilibria, convex feasibility problem
The Homotopy Continuation Method and the Walrasian General Equilibrium Theory
The homotopy continuation method becomes a. powerful tool for comparative statics in the large. It is also, by its nature, to be used for showing the existence of zeroes of a map. In this paper, we first present some existence theorems concerning set-valued maps. These are obtained from a fundamental theorem on a homotopy continuation method and theorems on continuos selections. We then apply them to the Walrasian general equilibrium models and show the existence of equilibria. Subsequently, we consider a regular economy, and get some insights into the number of equilibria due to the homotopy invariance theorem and the norm-coerciveness theorem. Although these problems have been investigated by many economists, the homotopy continuation method clarifies in a clear and systematic manner why our results have been obtained
An Interior-Point Path-Following Method to Compute Stationary Equilibria in Stochastic Games
Subgame perfect equilibrium in stationary strategies (SSPE) is the most important solution concept used in applications of stochastic games, which makes it imperative to develop efficient numerical methods to compute an SSPE. For this purpose, this paper develops an interior-point path-following method (IPM), which remedies a number of issues with the existing method called stochastic linear tracing procedure (SLTP). The homotopy system of IPM is derived from the optimality conditions of an artificial barrier game, whose objective function is a combination of the original payoff function and a logarithmic term. Unlike SLTP, the starting stationary strategy profile can be arbitrarily chosen and IPM does not need switching between different systems of equations. The use of a perturbation term makes IPM applicable to all stochastic games, whereas SLTP only works for a generic stochastic game. A transformation of variables reduces the number of equations and variables of by roughly one half. Numerical results show that our method is more than three times as efficient as SLTP
Computing equilibria in general equilibrium models via interior-point method
In this paper we study new computational methods to find equilibria in general equilibrium
models. We first survey the algorithms to compute equilibria that can be found in the literature on
computational economics and we indicate how these algorithms can be improved from the computational
point of view. We also provide alternative algorithms that are able to compute the equilibria
in an efficient manner even for large-scale models, based on interior-point methods. We illustrate the
proposed methods with some examples taken from the literature on general equilibrium modelsPublicad
Algorithms for Competitive Division of Chores
We study the problem of allocating divisible bads (chores) among multiple
agents with additive utilities, when money transfers are not allowed. The
competitive rule is known to be the best mechanism for goods with additive
utilities and was recently extended to chores by Bogomolnaia et al (2017). For
both goods and chores, the rule produces Pareto optimal and envy-free
allocations. In the case of goods, the outcome of the competitive rule can be
easily computed. Competitive allocations solve the Eisenberg-Gale convex
program; hence the outcome is unique and can be approximately found by standard
gradient methods. An exact algorithm that runs in polynomial time in the number
of agents and goods was given by Orlin.
In the case of chores, the competitive rule does not solve any convex
optimization problem; instead, competitive allocations correspond to local
minima, local maxima, and saddle points of the Nash Social Welfare on the
Pareto frontier of the set of feasible utilities. The rule becomes multivalued
and none of the standard methods can be applied to compute its outcome.
In this paper, we show that all the outcomes of the competitive rule for
chores can be computed in strongly polynomial time if either the number of
agents or the number of chores is fixed. The approach is based on a combination
of three ideas: all consumption graphs of Pareto optimal allocations can be
listed in polynomial time; for a given consumption graph, a candidate for a
competitive allocation can be constructed via explicit formula; and a given
allocation can be checked for being competitive using a maximum flow
computation as in Devanur et al (2002).
Our algorithm immediately gives an approximately-fair allocation of
indivisible chores by the rounding technique of Barman and Krishnamurthy
(2018).Comment: 38 pages, 4 figure
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