Competitive General Equilibrium with network externalities

We identify sufficient conditions for existence of competitive equilibrium with network externalities and indivisibilities. Such combination of externalities and indivisibilities is present in many goods and services with network effects, and it makes existence of equilibrium non-trivial. We provide an existence theorem in a model with a measure space of consumers. Key conditions for existence are anonymity of network effects and dispersion in the economy’s income distribution.     

Some results on general assumptions about the existence of economic equilibrium

In the present paper we maximally use the possibilities provided by the Nash approach and the Kakutani fixed point theorem for proving the existence of an economic equilibrium. We obtain a general existence theorem which does not require a special form for income distribution functions and producer's objectives, independence of consumers' tastes, ordered preference and zero degree homogeneous price dependence. The role of the non-satiation assumption becomes more clear.     

Pure strategy equilibria in games with countable actions

In this paper, we extend the Radner–Rosenthal theorem with finite action spaces on the existence of a pure strategy equilibrium for a finite game to the case that the action space is countable and complete. We also prove the existence of a pure strategy equilibrium for a game with a continuum of players of finite types and with a countable and complete action space. To work with the countably infinite action spaces, we prove some regularity properties on the set of distributions induced by the measurable selections of a correspondence with a countable range by using the Bollobás–Varopoulos extension of the marriage lemma.     

Existence of equilibrium configurations of competitive firms on an infinite two-dimensional space

This paper analytically examines the existence of equilibrium configurations of competitive firms on an infinite two-dimensional space. It is shown that the Löschian configuration (a regular-hexagonal lattice) and a square lattice are in global equilibrium; the Löschian configuration is in the strongest global equilibrium among the regular lattices (in this sense, spatial competition leads to the social optimum); and the so-called back-to-back configuration is not in equilibrium. These results are in marked contrast to those obtained from a one-dimensional model, implying that spatial equilibrium configurations in a two-dimensional space should not be inferred only from one-dimensional models frequently employed in spatial economics.     

Sequential decisions under uncertainty and the maximum theorem

For sequential decision problems in which the decision-maker observes a process of state variables and chooses an adapted process of action variables, the paper defines a topology on the space of measures of processes of state variables which ensures the applicability of Berge's maximum theorem to the decision-maker's optimal behavior. The topology controls for the information available to the decision-maker at each decision date. The paper also discusses the implications of the analysis for the dynamic-programming approach to sequential decision-making under uncertainty, and for equilibrium existence proof strategies in sequential-market models and games.     

Equilibria in a large production economy with an infinite dimensional commodity space and price dependent preferences

We prove the existence of a competitive equilibrium in a production economy with infinitely many commodities and a measure space of agents whose preferences are price dependent. We employ a saturated measure space for the set of agents and apply recent results for an infinite dimensional separable Banach space such as Lyapunov’s convexity theorem and an exact Fatou’s lemma to obtain the result.     

Timing Equilibria in an Urban Model with Congestion

All spatial urban models with congestion assume that the departure (arrival) time of commutes is exogenously determined and assume that travel speed at a given time and location depends upon the traffic density at that point in time and space. This paper presents a framework that encompasses such models, but allows workers to choose the time at which they leave home (arrive at work). This paper then proves that in general only one equilibrium exists for urban models with traditional congestion technology: a commuting pattern in which commuting is continuous and the rush hour never ends, which is unrepresentative of traffic patterns anywhere. This paper concludes by discussing alternative general equilibrium urban models with congestion that may have more realistic equilibria.     

The joint continuity of the expected payoff functions

Glicksberg [Glicksberg, I.L., 1952. A further generalization of the Kakutani fixed point theorem, with applications to Nash equilibrium points. In: Proceedings of the American Mathematical Society 3, pp. 170–174] generalized the Kakutani fixed point theorem to the setting of locally convex spaces and used it to prove that every k-person strategic game with action sets convex compact subsets of locally convex spaces and continuous payoff functions has a Nash equilibrium. He subsequently used this result to establish the following fundamental theorem of game theory: Every k-person strategic game with action sets metrizable compact topological spaces and continuous payoff functions has a mixed strategies equilibrium. However, in his proof of the latter result, Glicksberg did not show that the expected payoff functions were jointly continuous, something that was required for the existence of a mixed strategies equilibrium.     

On the existence of walrasian equilibrium in large-square economies

An existence theorem for Walrasian equilibrium is demonstrated for an economy with a continuum of consumers and an infinite-dimensional commodity space, such as l₁ or c₀, having an ‘order-compatible’ basis.     

Equilibrium in economies without ordered preferences or free disposal

Recent work in competitive equilibrium analysis, pioneered by Mas-Colell (1974), ‘An equil- brium existence theorem without complete or transitive preferences’, has established existence of equilibrium in an economy in which preferences are not orderings. The purpose of this note is to show that the assumption of free disposal is not necessary for existence.     

Pure and randomized equilibria in the stochastic von Neumann–Gale model

The paper examines the problem of the existence of equilibrium for the stochastic analogue of the von Neumann–Gale model of economic growth. The mathematical framework of the model is a theory of set-valued random dynamical systems defined by positive stochastic operators with certain properties of convexity and homogeneity. Existence theorems for equilibria in such systems may be regarded as generalizations of the Perron–Frobenius theorem on eigenvalues and eigenvectors of positive matrices. The known results of this kind are obtained under rather restrictive assumptions. We show that these assumptions can be substantially relaxed if one allows for randomization. The main result of the paper is an existence theorem for randomized equilibria. Some special cases (models defined by positive matrices) are considered in which the existence of pure equilibria can be established.     

On the existence,uniqueness and stability of spatial equilibrium in an open city with externalities

This paper deals with the existence, uniqueness and stability of a spatial equilibrium in an open city with external diseconomies like air pollution generated by manufacturing activities. First, assuming that both production functions and utility functions are Cobb-Douglas, we prove the existence of a spatial equilibrium under some reasonable assumptions. It is shown, however, that the uniqueness and stability of the equilibrium may not be obtained, unless the degree of externality is sufficiently small. In fact, none of the equilibria will be stable for a certain set of speeds of adjustment in the labor market and the land market, if the degree of externality is relatively large. Finally, some implications of our results are derived in regard to the application of the hedonic price concept within the context of a spatial equilibrium model.     

Spatial externalities and agglomeration in a competitive industry

We introduce spatial spillovers as an externality in the production function of competitive firms operating within a finite spatial domain under adjustment costs. Spillovers may attenuate with distance and the overall externality could contain positive and negative components with the overall effect being positive. We show that when the spatial externality is not internalized by firms, spatial agglomerations may emerge endogenously in a competitive equilibrium. The result does not require increasing returns at the private or the social level, increasing marginal productivity of private capital with respect to the externality, or location advantages. In fact agglomerations may emerge with decreasing returns to scale, declining marginal productivity of private capital with respect to the externality, and no location advantage. The result depends on the interactions between the structures of production technology and spatial effects as shown in the paper. No agglomerations emerge at the social optimum when spillovers are internalized and diminishing returns both from the private and the social point of view prevail. Numerical experiments with Cobb–Douglas and CES technologies and an isoelastic demand confirm our theoretical predictions.     

On existence and stability of spatial equilibria and steady-states

This note discusses the existence and stability of two equilibrium concepts for a spatial economy in which the utility of an agent depends on the overall distribution of agents over space.     

从分工视角解读城市产业空间结构的演变机理

城市产业空间结构是城市经济结构的表现形式,是城市空间结构最为核心的内容,它的演变历程是城市成长变迁的历程,其演变机理也折射出城市发展的内在动力。以分工为切入点,对产业成长和生产组织结构的演变以及其在空间上的表现———城市经济结构的变迁进行研究。撇开各行业的特殊性,将各个产业的区位选择联系起来,在一个统一的框架内研究城市各产业的区位选择及变迁,揭示城市产业空间结构演变的本质和机理,并分析其所体现出的普遍性规律。认为在分工深化的过程中,随着企业功能的不断分化,不同功能之间的联系成本决定了功能空间的结合或分离,联系成本和区位成本共同构成了企业的空间成本,而企业的区位决策即是要寻求空间成本的最小化。     

Solvability of perturbation solutions in DSGE models

We prove that the undetermined Taylor series coefficients of local approximations to the policy function of arbitrary order in a wide class of discrete time dynamic stochastic general equilibrium (DSGE) models are solvable by standard DSGE perturbation methods under regularity and saddle point stability assumptions on first order approximations. Extending the approach to nonstationary models, we provide necessary and sufficient conditions for solvability, as well as an example in the neoclassical growth model where solvability fails. Finally, we eliminate the assumption of solvability needed for the local existence theorem of perturbation solutions, complete the proof that the policy function is invariant to first order changes in risk, and attribute the loss of numerical accuracy in progressively higher order terms to the compounding of errors from the first order transition matrix.     

A note on the equilibrium theory of economies with asymmetric information

In this note, we give an equilibrium existence theorem for exchange economies with asymmetric information and with an infinite dimensional commodity space. In our model, we assume that preferences are represented by well behaved utility functions, the positive cone has a non empty interior and the individual rational utility set is compact. Our result complements the corresponding one in Podczeck and Yannelis (2008), in the sense that is applicable to commodity spaces in which the order intervals are (possibly) not compact with respect to any Hausdorff linear topology.     

Thick market externalities in a spatial model

It is natural to think of thick market externalities as spatial phenomena. When agents are in close physical proximity, potential trading partners are more numerous and less costly to reach. Counteracting such agglomeration benefits is the dispersion force due to land being an essential input in production. The distribution of economic activities over space is an outcome of how decisions on location, land demand, and the search strategy of agents interact in spatial equilibrium. More desirable locations are those that allow their occupants more abundant and less costly access to potential trading partners. In spatial equilibrium, these are the densest locations, the occupants of which benefit from the strongest thick market externalities.     

Tiebout's tale in spatial economies: Entrepreneurship, self-selection, and efficiency

This paper establishes the existence and efficiency of equilibrium in a local public goods economy with spatial structures by formalizing Hamilton's [Hamilton, B.W., 1975. Zoning and property taxation in a system of local governments Urban Studies 12, 205–211] elaboration of Tiebout's [Tiebout, C., 1956. A pure theory of local public expenditures. Journal of Political Economy 64, 416–424] tale. We use a well-known equilibrium concept from Rothschild and Stiglitz [Rothschild, M., Stiglitz, J.E., 1976. Equilibrium in competitive insurance markets: an essay on the economics of imperfect information. Quarterly Journal of Economics 40, 629–649] in a market with asymmetric information, and show that Hamilton's zoning policy plays an essential role in proving the existence and efficiency of equilibrium. We use an idealized large economy following Ellickson, Grodal, Scotchmer and Zame [Ellickson, B., Grodal, B., Scotchmer, S., Zame, W.R., 1999. Clubs and the market, Econometrica 67, 1185–1217] and Allouch, Conley and Wooders [Allouch, N., Conley, J.P., Wooders, M.H., The Tiebout Hypothesis: On the Existence of Pareto Efficient Competitive Equilibria, (2004), mimeograph]. Our theorem is directly applicable to the existence and efficiency of a discrete spatial approximation of mono- or multi-centric city equilibria in an urban economy with commuting time costs, even if we allow the existence of multiple qualities of (collective) residences, when externalities due to traffic congestion are not present.     

An equilibrium existence theorem without complete or transitive preferences

The Walrasian equilibrium existence theorem is reproved without the assumptions of complete of transitive preferences.