Egbert Dierker

Topological methods in Walrasian economics

1. The Economic Framework.- 2. Introduction to the Mathematics.- 3. Differentiable Manifolds and Mappings, Tangents, Vectorfields.- 4. Regular Equilibria. A First Approach.- 5. Scarfs Example.- 6. Excess Demand Functions.- 7. Debreus Theorem on the Finiteness of the Number of Equilibria of an Economy.- 8. Continuity of the Walras Correspondence for C Demand Functions.- 9. Density of Transversal Intersection.- 10. Regular Economies.- 11. Stability Questions and the Number of Equilibria.- 12. Large Economies.- Some Standard Notation.- References.

General Equilibrium When Some Firms Follow Special Pricing Rules

IN THIS PAPER, we study the existence of a general equilibrium in an economy in which some of the firms behave competitively, whereas others follow special pricing rules. An even casual observation of economic reality confirms the need for the inclusion of price setting firms in the study of general equilibrium models. And, in fact, this has been emphasized in the economic literature for some time. We consider this a sufficient justification to present a general equilibrium model which allows for both types of behavior-price setting as well as price taking behavior. The model to be presented will account for a wide range of price setting rules. We, however, find it appropriate to point out at the start that not all economically relevant price setting rules are covered. Although our model permits the drawing of useful conclusions for the general equilibrium theory of oligopolistic competition through prices, it is not especially designed for that purpose. In particular, our results are not applicable to the general equilibrium analysis with price making monopolistic firms, a subject on which there exists some literature.2 Our motivation as well as our modelling options and assumptions are directed to two fields where an increased interest in the theoretical foundations recently could be noticed.

Gains and losses at core allocations

Abstract The search for a coalition which can possibly improve upon a given allocation and the redistribution of endowments within such a coalition are conducted through the use of prices. Prices permit the expression of how much every agent gains or loses in the allocation. With any feasible allocation one can associate a price system such that either the total loss of all losers does not exceed a certain bound independent of the number of agents or the losers can improve. The definition of gains and losses that we use implies that the total gain is also bounded in core allocations. Our theorem is closely related to that of Vind (1965).

Nonexistence of Constrained Efficient Equilibria When Markets are Incomplete

We consider economies with incomplete markets, production, an a given distribution of initial endowments. The main purpose of the paper is to present a robust example of an economy with only one firm and one good per state in which no production decision entails a constrained efficient outcome. In particular, the unique Dreze equilibrium is dominated by every other production decision.

When does marginal cost pricing lead to Pareto efficiency

It is well-known that Pareto-efficiency requires, at least in many cases, that prices be set equal to marginal costs. For this reason economists often argue in favor of the marginal cost pricing rule as long as no overly large deficit results. If strongly increasing returns to scale cause a deficit which is considered too large, the marginal cost pricing rule is appropriately modified. To cover the deficit partly or fully, necessary conditions for optimal departures from marginal cost pricing are derived and their implementation is recommended. Normative considerations of this kind lead to what is called Ramsey pricing or pricing ~t la Boiteux. The normative character of Boiteux-Ramsey pricing is the starting point of a large part of the industrial organization literature and the basis for many specific policy recommendations, a typical example being proposals for the structure of telephone rates. Even if one is aware of the fact that quite often first best optimality requires marginal cost pricing and second best optimality requires Boiteux-Ramsey pricing, a further thought leads to the question of what the final outcome will actually be if such pricing rules ate adopted. Leaving distributional aspects aside, although they can be of great relevance, and focusing on efficiency only, o n e

THE OPTIMALITY OF BOITEUX-RAMSEY PRICING

Consider an economy with one public enterprise that is subject to a budgetary constraint and charges prices according to the first-order necessary conditions suggested by M. Boiteux (1956) or F. P. Ramsey (1927). The authors state conditions sufficient to ensure that the resulting allocation is second best. The proof builds on arguments used to demonstrate the Pareto efficiency of marginal cost pricing under appropriate assumptions. Copyright 1991 by The Econometric Society.

Price-dispersed preferences and C1 mean demand

In this paper we introduce the concept of price-dispersed preferences. Moreover we state conditions under which economies with price-dispersed preference distributions have a continuously differentiable mean demand function.

Tax systems and direct mechanisms in large finite economies

The equivalence of anonymous direct mechanisms and tax systems for continuum economies, which has been demonstrated by Hammond (1979) and Guesnerie (1981), is shown to have its counterparts in large finite economies.

Continuous mean demand functions derived from non-convex preferences

Abstract In this paper we show that for a large subset of utility functions in the space of all C1 utility functions and for all prices the mean demand of those consumers whose taste is represented by a given utility function in that subset is uniquely determined. This implies that for a large set of economies mean demand is a continuous function. Our analysis uses derivatives of first and of higher order. The result is essentially a consequence of the multijet transversality theorem.

Product differentiation and market power

Assuming asymmetry across firms and constant unit costs Perloff and Salop (1985) show: If product differentiation increases, the prices rise in a symmetric equilibrium. This raise the question of whether, in general, more product differentiation leads to higher market prices. Giving up the symmetry and the constant unit costs assumptions we present examples in which at least one firm lowers its equilibrium price when product differentiation increases. We formulate a model of product differentiation and state and discuss, within the theory of supermodular games, conditions ensuring that all firms raise their prices in a Nash equilibrium if product differentiation increases.