Marginal Pricing Rule Research Articles

Market Failures and Equilibria in Banach Lattices: New Tangent and Normal Cones

In this paper, we consider an economy with infinitely many commodities and market failures such as increasing returns to scale and external effects or other regarding preferences. The commodity space is a Banach lattice possibly without interior points in the positive cone in order to include most of the relevant commodity spaces in economics. We propose a new definition of the marginal pricing rule through a new tangent cone to the production set at a point of its (non-smooth) boundary. The major contribution is the unification of many previous works with convex or non-convex production sets, smooth or non-smooth, for the competitive equilibria and for the marginal pricing equilibria, with or without external effects, in finite-dimensional spaces as well as in infinite-dimensional spaces. In order to prove the existence of a marginal pricing equilibria, we also provide a suitable properness condition on non-convex technologies to deal with the emptiness of the interior of the positive cone.

An Investigation of Plug-In Electric Vehicle Charging Impact on Power Systems Scheduling and Energy Costs

This paper investigates the impact of plug-in electric vehicle (EV) integration on the power systems scheduling and energy cost. An intermediary entity, the EV aggregator, participates in the market on behalf of the EV owners by optimally self-scheduling under the price-taking approach. Through detailed rolling simulations for a year and different EVs' penetration scenarios at a large insular power system, this work highlights the different impact of direct and smart charging on power system scheduling and energy costs, the limitations of the price-taking approach, which is widely used in self-scheduling models, and the difference in system value and market value that smart charging adoption creates in restructured markets under the marginal pricing rule.

The Survival Assumption in Intertemporal Economies

In an economy with a non-convex production sector, we provide an assumption on each individual producer, which implies that the survival assumption holds true at the aggregate level for general pricing rules. For the marginal pricing rule, we derive this assumption from the bounded marginal productivity of inputs. We apply this approach to intertemporal economies and we show how our assumption fits well with the time structure. We obtain a tractable existence result of equilibria for discrete time growth models.

Existence of equilibria with a tight marginal pricing rule

This paper deals with the existence of marginal pricing equilibria when it is defined by using a new and tighter normal cone introduced by B. Cornet and M.O. Czarnecki. The main interest of this new definition of the marginal pricing rule comes from the fact that it is more precise in the sense that the set of prices satisfying the condition is smaller than the one given by the Clarke’s normal cone. The counterpart is that it is not convex valued, which leads to some mathematical difficulties in the existence proof. The result is obtained through an approximation argument under the same assumptions as in the previous existence results.

The marginal pricing rule revisited

The purpose of the paper is to introduce a tighter definition for the marginal (cost) pricing rule. By means of an example, we illustrate the improvements that one gets with the new definition with respect to the former one using Clarke’s normal cone, and we discuss its consequences in terms of the existence of equilibria.

On the Characterization of Efficient Production Vectors

In this paper we study the efficient points of a closed production set with free disposal. We first provide a condition on the boundary of the production set, which is equivalent to the fact that all boundary points are efficient. When the production set is convex, we also give an alternative characterization of efficiency around a given production vector in terms of the profit maximization rule. In the non-convex case, this condition expressed with the marginal pricing rule is sufficient for efficiency. Then we study the Luenberger’s shortage function. We first provide basic properties on it. Then, we prove that the above necessary condition at a production vector implies that the shortage function is locally Lipschitz continuous and the efficient points in a neighborhood are the zeros of it and conversely.

The Marginal Pricing Rule in Economies with Infinitely Many Commodities

Clarke's normal cone appears as the right tool to define the marginal pricing rule in finite dimensional commodity space since it allows to consider in the same framework convex, smooth as well as nonsmooth nonconvex production sets. Furthermore it has nice continuity and convexity properties. But it is not well adapted for economies with infinitely many commodities since it does satisfy minimal continuity properties. In this paper, we propose an alternative definition of the marginal pricing rule.

Existence of marginal pricing equilibria in economies with externalities and non-convexities

We consider a general equilibrium model with externalities and non-convexities in production. The consumption sets, the preferences of the consumers and the production possibilities are represented by set-valued mappings to take into account possibility of external effects. There is no convexity assumption on the correspondences of production. We propose a definition of the marginal pricing rule, which generalizes the one used in the model without externality and, which satisfies a continuity assumption with respect to the external effect. We prove the existence of general equilibria under assumptions which allow us to encompass together the works on economies with externalities and convex conditional production sets, and those on marginal pricing equilibria in economies without externalities. We provide examples to illustrate the definition of the marginal pricing rule and to show the difference with the standard case.

On the Efficiency of Market Equilibrium in Production Economies

This paper introduces the notion of Market Equilibrium With Active Consumers (MEWAC), in order to characterize the efficiency of market outcomes in production economies. We show that, no matter the behaviour followed by the firms, a market equilibrium is efficient if it is a MEWAC. And also that every efficient allocation can be decentralized as a MEWAC in which firms follow the marginal pricing rule.

Existence of equilibria in the presence of increasing returns: A synthesis

In the literature, there are two theories dealing with the existence of equilibria in economies with non-convex production sets. In the first, the producers follow the marginal pricing rule and in the second, they follow general pricing rules with bounded loss. The purpose of this paper is to propose a synthesis of the two approaches in the sense that one deduces the existence of a marginal pricing equilibrium from the existence result for bounded loss pricing rules.

Existence of Marginal Cost Pricing Equilibria: The Nonsmooth Case

In this article we report a general existence theorem of a marginal (cost) pricing equilibrium for an economy which may exhibit increasing returns to scale or more general types of in the production sector. Our model considers an arbitrary number of firms and no smoothness assumption is made on their production sets or on the aggregate production set. In this article we report a general existence theorem of marginal (cost) pricing equilibria for an economy, which may exhibit increasing returns to scale or more general types of nonconvexities in the production sector. In our model, which considers a positive finite numbers 1 of commodities, m, of consumers, and, n, of firms, (a) consumers maximize their preferences subject to their budget constraints, (b) convex producers maximize profits, while nonconvex producers are instructed to follow the marginal pricing rule, i.e., they fulfill the first-order necessary condition for profit maximization in a precise mathematical sense, formalized with Clarke's normal cone. Our treatment of convex and nonconvex producers will in fact be symmetric. The technological possibilities of the jth producer (j = 1, ..., n)

Optimal Pricing and Advertising Policies for New Product Oligopoly Models

In this paper our previous work on monopoly and oligopoly new product models is extended by the addition of pricing as well as advertising control variables. These models contain Bass's demand growth model, and the Vidale-Wolfe and Ozga advertising models, as well as the production learning curve model and an exponential demand function. The problem of characterizing an optimal pricing and advertising policy over time is an important question in the field of marketing as well as in the areas of business policy and competitive economics. These questions are particularly important during the introductory period of a new product, when the effects of the learning curve phenomenon and market saturation are most pronounced. We consider first the monopoly case with linear advertising cost, exponential demand, and three different pricing rules: the optimal variable pricing, the instantaneous marginal pricing, and the optimal constant pricing rules. Several theoretical results are established for these rules including the facts that the instantaneous marginal pricing rule is a myopic version of the optimal pricing rule and the optimal constant pricing rule is a weighted average over time of the instantaneous marginal pricing rule. Another surprising result is that, after the market is at least half saturated, a pulse of advertising must be preceded by a significant drop in price. Numerical solutions of a number of examples are discussed. Oligopolistic models are analyzed as nonzero-sum differential games in the rest of the paper. The state and adjoint equations are easy to write down, but impossible to solve in closed form. Hence we describe how to reformulate these models as discrete differential games, and give a numerical algorithm for finding open loop Nash solutions. The latter was used to solve three triopoly models. In each case it was found that optimal prices and advertising rates start high and steadily decline.