Constant Unit Cost Research Articles

The Multiple Unit Auction with Variable Supply

The theory of multiple unit auctions traditionally assumes that the offered quantity is fixed. I argue that this assumption is not appropriate for many applications because the seller may be able and willing to adjust the supply to the bidding. In this paper I address this shortcoming by analyzing a multi-unit auction game between a monopolistic seller who can produce arbitrary quantities at constant unit cost, and oligopolistic bidders. I establish the existence of a subgame-perfect equilibrium for price discriminating and for uniform price auctions. I also show that bidders have an incentive to misreport their true demand in both auction formats, but they do that in different ways and for different reasons. Furthermore, both auction formats are inefficient, but there is no unambiguous ordering among them. Finally, the more competitive the bidders are, the more likely the seller is to prefer uniform pricing over price discrimination, yet increased competition among bidders may or may not enhance efficiency.

De l'usage optimal de divers types de ressources naturelles

A partial equilibrium analysis shows that a pool of natural resources with constant unit costs of extraction has to be exploited in strict increasing cost order. KEMP and LONG (1980) have shown that it is not necessarily the case in a general equilibrium framework. In a consumption-leisure trade-off model, we show that the optimal order of exploitation is never strict, and that for a set of model parameters, the resources exploitation paths may be inderteminate although the consumption-leisure paths are perfectly determinated.

Mark-up Pricing versus Normal Cost Pricing in Post-Keynesian Models

The intent of the present paper is to show that some of Frederic Lee's recent criticisms against some of the hypotheses imbedded in some post-Keynesian models are exaggerated. In contrast to what is argued by Lee, there is no flagrant contradiction between the work of Andrews and his followers and that of most modern-day Kaleckian and post-Keynesian authors. Some of the simplifications used by the latter, such as constant direct unit costs or simple mark-up pricing, were, in fact, suggested by well-known Andrewsian authors, Lee himself included. In addition, although most well-known post-Keynesian authors have relied on the writings of Kalecki to establish their pricing models, Presumably because of its obvious macroeconomic implications, the structure of their model is in many instances drawing on the Andrewsian normal cost pricing Principe.

Mixed Duopoly under Vertical Differentiation

In this paper we study a vertically differentiated duopoly market with a profit maximizing firm (private firm) and a total surplus maximizing firm (public firm). The technological conditions are assumed identical for both firms and are described by unit costs which are constant with respect to quantity, though increasing in quality. No specific form is given to the relation between unit costs and quality. We prove that the socially optimal solution can be sustained as a market outcome by using a public firm as a market agent. We also provide conditions on the constant unit cost function under which every market outcome is a social optimum.

WELFARE EFFECTS OF STRATEGIC PRICE SETTING IN ANTICIPATION OF PRICE REGULATION

ABSTRACTA single product monopolist with constant unit costs, in a simple two‐period model, is aware that price regulation will be imposed in the second period. The form of the regulation is such that price in period 2 may not exceed 100 L̄; per cent of price in period 1, where 0 < L̄ < 1. The period 1 price will be set higher than it would be in the absence of anticipated regulation. However, it is not always the case that pre‐regulation price will be raised as L̄ falls. The welfare effects are crucially dependent on the form of the demand function. Under constant elasticity of demand a reduction in L̄ will reduce both consumers' and the producer's welfare. Under linear demand, consumers benefit and the producer loses as L̄ is reduced and the resultant effect on aggregate welfare is ambiguous.

Durable Goods Monopoly with Entry of New Consumers

This paper analyzes a model of a dynamic monopolist who produces at constant unit cost. Each period a new cohort of consumers enters the market. Each entering cohort is identical. Consumers within a cohort have different tastes for the good. My main results are: If players are sufficiently patient, any positive average profit less than the maximum feasible level can be attained in a subgame-perfect equilibrium; in the subset of subgame-perfect equilibria in which players use stationary strategies, the seller cannot make sales at prices significantly greater than the lowest willingness to pay when period length goes to zero; and the seller attains the maximum profit when commitment is feasible by charging the same (static monopoly) price in every period. THIS PAPER ANALYZES a model of a dynamic monopolist who operates in a market in which there is a regular flow of new consumers. The seller produces a durable good at constant unit cost. Each period a new cohort of consumers enters the market. Consumers wish to buy exactly one unit of the item, but are willing to wait. Resales are not allowed. Each entering cohort is identical.

An efficient algorithm for the capacitated single item dynamic lot size problem

A dynamic programming based algorithm is developed for the single item lot size problem with concave costs and arbitrary capacities. By making use of the extreme point properties of the problem, first the set of all feasible cumulative production levels that may occur in an optimal solution is generated. In the second stage, a dynamic programming procedure is carried out over this set. The worst case computational effort is equal to that of the standard dynamic programming approach but extensive computational tests with the algorithm indicate that for T period problems the computational effort does not exceed O( T 4). The performance of the algorithm is compared with the performance of the existing procedures in the literature for the general, the constant capacity, and the constant unit cost problems. The computational results demonstrate that our algorithm is at least three times faster than the other procedures for all problem types considered.

Agro-Industrial Processing and Agricultural Pricing Under Uncertainty

Processing domestically available materials for export has intuitive appeal, e.g. if transport costs are reduced. Uncertainty in the supply of the raw material can be a crucial factor. The theoretical model of this paper shows how to choose the optimal processing capacity and, relatedly, the optimal price to pay producers of the input. Issues include: decision makers' preferences; sources and nature of uncertainty; availability of inputs to produce the raw material; and technology and costs of processing. Econometric estimates of the climate-yield relation are used to simulate these decisions for the case of Senegalese groundnut processing. There is an intuitive appeal to industrial strategies based on processing domestically available raw materials for export. For instance, relative to export of the raw material for processing abroad, domestic processing often achieves cost savings based on certain locational advantages, such as the opportunity to decrease the product's weight prior to transport.' In this paper, I analyse the costs and benefits of investing in processing capacity under particular conditions, and the related question of the appropriate domestic pricing of the raw material. The second part of the paper presents an application of the theory to Senegalese groundnut culture and processing. Especially for agricultural inputs, the domestic supply of the raw material is likely to be uncertain. This uncertainty interacts with locational advantages in important ways affecting both the optimal choice of agro-industrial processing capacity and the optimal choice of the price to be offered agricultural producers of the input. From the viewpoint of the processing mills, the locational advantage means that stochastically low availability of the raw material cannot be (fully) compensated by imports. Thus, the problem of idle processing capacity arises. Similarly, harvests of the input above the level that can be processed imply some loss of the locational advantages, assuming that the input cannot be stored costlessly. To illustrate these basic notions, consider the simplest case of a single input with a given world price, pj, transformed at a constant unit cost, r, into an output with a given world price, p. Let the size of the crop also be certain, say of magnitude m. So long as (p -pI) _ r, optimal processing capacity is m. Now consider a situation of uncertainty with two equiprobable states of nature faced

Gaussian Demand and Commodity Bundling

In an important and widely cited essay, Adams and Yellen (1976) examine the use of package selling as a price discrimination device.' They consider a monopolist producing two goods with constant unit costs and facing buyers with diverse tastes. The marginal utility of a second unit of either good is assumed to be zero for all buyers. The two goods are independent in demand for all buyers, so that any buyer's reservation price for the first unit of either good is independent of the market price of the other. Thus the maximum amount any buyer would pay for a bundle consisting of one unit of each good is the sum of the two reservation prices, and buyers are completely described by the values of those two prices.2 Resale markets are assumed away. In order to obtain comparative results for alternative pricing strategies, the distribution of reservation prices in the Adams-Yellen framework is assumed to be bivariate normal. Implications of Gaussian demand are explored. Pure bundling is shown to 'operate by reducing buyer diversity, thus facilitating the capture of consumers' surplus. It generally makes buyers worse off than unbundled sales; it is more profitable if average reservation prices are high enough. Mixed bundling combines advantages of both pure bundling and unbundled sales, and it is generally strictly more profitable than either. Bundling, which treats goods symmetrically, seems most attractive under symmetric reservation price distributions. * I am indebted to Severin Borenstein for superb research assistance, to the Ford Motor Company for research support through a grant to M.I.T., and to the Harvard Economics Department for housing me as a Visiting Scholar while most of this research was performed. Of course, only I can be held responsible for this essay's shortcomings. 1. The first clear recognition of this possibility seems to have been by Stigler (1968), but also see Burstein (1960). 2. Paroush and Peles (1981) relax the assumption that consumers purchase at most one unit of each good, but they consider only two types of buyers, both with linear demand functions. (They retain the assumption that demands are independent for all buyers.) Telser (1979) and Phillips (1981) use different frameworks to analyze bundling policies. Spence (1980) explores the relation between bundling across commodities, as here, and bundling multiple units of the same

Profit-sharing in a collusive industry

We study a model in which collusive duopolists divide up the monopoly profit according to their relative bargaining power. We are particularly interested in how the negotiated profit shares depend on the sizes of the firms. If each can produce at the same constant unit cost up to its capacity, we show that the profit per unit of capacity of the small firm is higher than that of the large one. We also study how the ratio of the negotiated profits depends on the size of demand relative to industry capacity, and how this ratio changes with variations in demand.

Cost-Containment Efforts

To the Editor.— The effects of a voluntary cost-containment effort on the use of high-cost laboratory tests, roentgenograms, medications, and supplies in the emergency department of a community hospital have been described recently by Stephen Karas, Jr, MD (1980;243:1356). He reported a 10.2% reduction in aggregate charges for laboratory and radiology services. For each item in Karas' table, we looked at the percentage change frombaseline. Such values apply equally, of course, to either frequencies or costs, given constant unit cost. Certain items in Karas'study were "specifically emphasized as potential cost-cutting items." The Table shows, for laboratory and radiology items, the results of our analysis comparing these items with the remaining items (the reduced use of which had been stressed but not so strongly). An appreciable overall decrease for specifically emphasized items was countered by an overall increase for other items. Specific emphasis on a limited number of items was

FINANCE AS AN INDUSTRY: A SIMPLE MODEL OF GROWTH*

THE PURPOSE of the dissertation is to integrate financial intermediation into a theoretical model of real economic output and growth. The methodology involves construction of a model in which capital accumulation is based upon an interest-elastic demand for the capital stock. This model is shown to be capable of adjusting to an equilibrium capital intensity and saving share. To this growth model is added a financial intermediary sector to form a two-sector model of an economy. The production sector uses real inputs of capital and labor to produce a homogeneous good which is then allocated to consumption and real capital accumulation. The financial intermediary sector uses real inputs of capital and labor to produce financial intermediary services. Financial intermediary output is described by a production function for the financial intermediary in which real capital and labor inputs are related to the units of real capital placed by the financial intermediary with production sector firms. That is, for each unit of real capital used to produce the homogeneous good, the financial intermediary expends real resources of capital and labor to gather capital from surplus households in order to place this capital with deficit production units. It is assumed that at the firm level the financial production functions exhibit the usual decreasing average product after some point, which leads to increasing average unit costs. At the aggregate sector level the financial production function exhibits constant returns to scale and constant unit costs, as expansion is carried out by the introduction of new and similarly efficient financial intermediaries. The financial intermediary thus employs real capital and labor in profit-maximizing quantities and pays these factors the competitive factor returns. The intermediary pays deposit rates to attract deposit capital which it then lends to production sector units and earns a loan rate equal to the marginal product of capital. The financial intermediary earns unit revenues equal to the difference between the loan rate and deposit rate. This quantity, called the unit cost of finance, is a per period rate per unit of capital. Total revenues of the intermediary are then composed of these per unit charges. When competition prevails in the financial intermediary market for deposit capital the intermediary is forced to pay a high enough deposit rate so that the intermediary merely covers its real factor input costs. The intermediary thus is capable of adjustments in deposit rates and factor inputs; and in a competitive equilibrium situation it produces financial services at minimum average costs. The deposit rate in a competitive equilibrium is then equal to the production sector marginal product of capital (loan rate) less the minimum average unit cost of financial services. The solution for the variables of the system are determined by a simultaneous solution of the sectoral production functions, the capital demand function (saving function), and capital and labor allocation equations (perfect substitutability). The model is solved for the sectoral capital intensities, the distribution of labor and capital between sectors, the real rates in the system (deposit rate and marginal product of capital), as well as aggregate and per capita income, investment and consumption.

The Influence of Monopoly on Product Innovation: Rejoinder

There is little in dispute between Lawrence J. White, in his Note in this issue, and myself, despite his claim to offer a more complete explanation for the existence of sleeping patents. In fact, his explanation is essentially the same as my own. He does not dispute my basic proposition that, under certain conditions, a monopoly controlling entry to the entire range of a group of sub,stitute products would introduce an identical range of products (but at smaller outputs and higher prices) than would be the case if perfectly competitive free entry conditions prevailed in all markets within the group.' In essence, these conditions require equality of the cross-partial derivatives of the price functions for products within the group and constant unit costs. In Case II White considers the situation in which entry is blocked in one market, but entry can take place in the production of another product that is a partial substitute for the original. In this case of incomplete or impure monopoly, naturally there are circumstances in which it would be profitable for an outside firm to enter in an endeavor to break down the monopoly pricing for the original product, while it would not be profitable for the firm with the monopoly in the original market to introduce the second product (except, possibly, as a last attempt to forestall entry). In these circumstances, there can be only one explanation for the desire of the original monopoly to purchase patents for the second product, without intention of use. By buying patents it hopes to forestall entry of outside firms and so preserve intact its original position, which could be endangered by the introduction of a close substitute. This must be the only explanation because, under the conditions specified, entry of any firm into the second market would be unprofitable with existing technology and costs, if competition were to prevail in the original market.

Reactive Programming of Supply and Demand Relations. Applications to Fresh Vegetables

S INCE is a new term, coined to describe to some extent the process employed, we may well spend a few minutes in learning something about the technique. Then we will move on to one of several applications which have been made. We will define Reactive Programming as a means of obtaining the equilibrium flows of a commodity between areas with given transportation cost functions, given demand schedules in each of the several areas of consumption and given supply schedules in each of the several areas of production. Though the technique is by no means limited to the case considered in this paper, we limit ourselves here to the problem of determining the perfect competition equilibrium flows of a commodity between areas with given demand schedules in each of several areas of consumption, fixed supplies in each of several areas of production, and given constant unit transportation costs. We apologize for the necessity of employing a good many symbols with subscripts and superscripts. But, their use is required for an orderly presentation of the technique. First, we let i=l, 2, ? ? , m denote the different producing areas and j =1, 2, ? ? ?, n denote the different consuming areas. Next, we let Pj denote price of the homogeneous product in the jth consuming area, Qij denote quantity purchased in the jth consuming area which was produced in the ith producing area, Tij denote the constant unit cost of transporting the product from the ith producing area to the jth consuming area, Si denote the fixed supply in the ith producing area, and Rij denote the net average revenue per unit for the product produced in the ith producing area and sold in the jth consuming area. Since an iterative procedure is used, we use the superscript (k) to denote the variables Qij being solved for at any stage of the process and the superscript (k -1) to denote the last values obtaine(l for all Qij's prior to solving for the variables in question. As given data for the problem we must have the n demand equations,'

A Different Loss Function for the Choice between Two Populations

SUMMARY In previous discussions of the problem of balancing sampling costs against the cost of choosing the wrong population it has been assumed that the cost of sampling is equal to the sample size multiplied by a constant known unit cost. In the present paper the cost of sampling is assumed to be the cost of an incorrect choice for half of the sample programme (which is divided evenly between the two populations). The resulting loss function is considered when the choice is between two normal populations of known equal variance, the standard of performance being determined by the mean. Wald's (1950) minimax principle is used to determine fixed size and sequential sampling procedures, and the performance of these procedures is examined.