Optimal Price Schedule Research Articles

Nonlinear Pricing with Underutilization: A Theory of Multi-part Tariffs

We study the nonlinear pricing of goods whose usage generates revenue for the seller and of which buyers can freely dispose. The optimal price schedule is a multi-part tariff, featuring tiers within which buyers pay a marginal price of zero. We apply our model to digital goods, for which advertising, data generation, and network effects make usage valuable, but monitoring legitimate usage is infeasible. Our results rationalize common pricing schemes including free products, free trials, and unlimited subscriptions. The possibility of free disposal harms producer and consumer welfare and makes both less sensitive to changes in usage-based revenue and demand. (JEL D11, D21, D42, L86, M37)

Nonlinear Pricing with Average-Price Bias

Empirical evidence suggests that consumers facing complex nonlinear prices often make choices based on average (not marginal) prices. Given such behavior, we characterize a monopolist’s optimal nonlinear price schedule. In contrast to the textbook setting, nonlinear prices designed for “ average-price bias” distort consumption downward for consumers with the highest marginal utility and typically feature quantity premia rather than quantity discounts. These properties arise because the bias replaces consumer information rents with “curvature rents.” Whether or not a monopolist prefers consumers with average-price bias depends upon underlying preferences and costs. (JEL D11, D21, D42, L12)

Optimal multi-unit mechanisms with private demands

A seller can produce multiple units of a single good. The buyer has constant marginal value for each unit she receives up to a demand, and zero marginal value for units beyond the demand. The marginal value and the demand are drawn from a distribution and are privately known to the buyer. We show that under natural regularity conditions on the distribution, the optimal (revenue-maximizing) selling mechanism is deterministic. It is a price schedule that specifies the payment based on the number of units purchased. Further, under the same conditions, the revenue as a function of the price schedule is concave, which in turn implies that the optimal price schedule can be found in polynomial time. We give a more detailed characterization of the optimal prices when there are only two possible demands.

Nonlinear Pricing of Software with Local Demand Inelasticity

Nonlinear usage-based pricing is applied extensively to price software products. Different from other products, software customers often cannot vary their required usage volume, a property we label local demand inelasticity. For instance, a client firm that needs a salesforce automation software either buys one user license for every salesperson in its organization or does not buy at all. It is unlikely to buy licenses for some salespersons, but not the others. This demand feature violates a critical assumption of the standard nonlinear pricing literature that consumers are flexible with their usage volume, and the utility that consumers derive from using the product changes smoothly with their usage volume. Consequently, standard pricing solutions are inapplicable to many software products. This paper studies the optimal nonlinear usage-based pricing of software when customers demand is locally inelastic. We demonstrate that this unique demand feature necessitates a fundamental reformulation of the traditional nonlinear pricing problem. We provide this reformulation and characterize the solution to a complicated nonlinear pricing problem with discontinuous and inelastic individual demand functions, virtually no restriction on demand distribution, and no single-crossing restriction on utility functions. We show that under a weak ordering condition of customer types, this complex pricing problem can be decomposed into a set of much simpler pricing problems with known solutions. Our pricing solution is applicable to a broad range of demand systems including the very popular normal and exponential distributions in prior literature. Managerially, our solution is based on parameters that are easily measurable in practice: the required consumption volume and value from this consumption for each customer. In contrast, typical nonlinear pricing problems require one to specify a complete utility function for every customer drawn from a continuum of customer types. Therefore, our solution likely makes empirical demand estimation and the real-world use of the analytical solution more viable. We discuss the characteristics of the optimal nonlinear pricing schedule and its welfare implications at the end.

Spatial Nonlinear Pricing with Per-Trip versus Per-Unit Transportation Costs

In this paper, we introduce a nonlinear pricing model with transportation costs that do not vary with the quantity purchased. In contrast to the quantity discounts seen in a model with per-unit transportation charges, we find that the optimal nonlinear pricing schedule takes the form of either a two-part tariff or package pricing, which appears to be consistent with observed pricing structures. We compare efficiency measures between optimal nonlinear pricing and the single price monopoly result, finding that per trip transportation costs can lead to qualitatively different results for pricing structure and efficiency measures.

Many-to-many matching and price discrimination

We study centralized many-to-many matching in markets where agents have private information about (vertical) characteristics that determine match values. Our analysis reveals how matching patterns reflect cross-subsidization between sides. Agents are endogenously partitioned into consumers and inputs. At the optimum, the costs of procuring agents-inputs are compensated by the gains from agents-consumers. We show how such cross-subsidization can be achieved through matching rules that have a simple threshold structure, and deliver testable predictions relating the optimal price schedules to the distribution of the agents' characteristics. The analysis sheds light on the practice of large matching intermediaries, such as media and business-to-business platforms, advertising exchanges, and commercial lobbying firms.

Pricing storage of outbound containers in container terminals

In container yard of container terminals, a storage charge is imposed to encourage customers to limit the space required for their containers. This study addresses the storage price scheduling problem for the storage of outbound containers. The price schedule consists of the free-time limit, which is the maximum duration for a container to stay in the yard without any charge, and storage charge per day for storing a container past the free-time-limit. Empirical data suggests that the efficiency of loading operations significantly depends on the space utilized by a vessel’s outbound containers. Mathematical models are developed to determine the optimal storage price schedule in such a manner that the terminal’s total profit is maximized or the total system’s cost is minimized. Both single and multi-vessel cases are considered. Properties of the optimal solution are derived from the mathematical models and numerical experiments are conducted to validate solutions.

Regulating a Manager-controlled Natural Monopoly with Unknown Costs

We study the regulation of a manager-controlled natural monopoly with unknown costs, borrowing from Baron and Myerson (BM) (1982), where the monopoly is controlled by the owner. We consider the case where the regulator can tax the owner as well as the case where she cannot. We find that the optimal price schedule generally lies below the one in the BM model and that it can be as low as the marginal cost if the compensation parameter is sufficiently small and the regulator cannot tax the monopoly owner. We also identify the cases where the monopoly owner prefers to delegate the control to a manager. Copyright © 2016 John Wiley & Sons, Ltd.

A Note on Second-Degree Price Discrimination: The Optimal Quantity-Discount Pricing Schedule

This paper examines a monopolist's optimal multi-tier quantity-discount pricing scheme. In the linear demand case, it derives a new formula for determining the optimal total output for any number of tiers. The formula yields a convenient graphical representation of the optimal pricing scheme. The paper shows that all tiers' outputs are equal, the last tier's price is always higher than the marginal cost, and the larger is the number of tiers, the larger are the total output, profits, and social welfare but the lower is the consumer surplus. In the non-linear demand case, the paper shows how the individual tier's output structure is affected by the curvature of the demand curve.

On the optimality of pure bundling for a monopolist

This paper considers a monopolist selling two objects to a single buyer with privately observed valuations. We prove that if the virtual valuation for each object is non-negative for all possible types, then the optimal price schedule is such that the objects are sold only in a bundle. Under an additional regularity condition, pure bundling is the optimal sale mechanism among all individually rational and incentive compatible mechanisms.

The Value of Free Water: Analyzing South Africa's Free Basic Water Policy

This paper analyzes South Africa's Free Basic Water Policy, under which households receive a free water allowance equal to the World Health Organization's recommended minimum of 6 kiloliters per month. I structurally estimate residential water demand, evaluate the welfare eects of free water, and provide optimal price schedules derived from a social planner's problem. I use a unique dataset of monthly metered billing data for 60,000 households for 2002-2008 from a particularly disadvantaged suburb of Pretoria. The dataset contains rich price variation across 18 dierent nonlinear tarischedules, and includes a policy experiment which removed the free allowance. I …nd that without government subsidy, the mean monthly consumption would decrease from 12.6 to 5.6 kiloliters, which is below the clean water consumption recommended by the WHO. However, it is possible to reallocate the current subsidy to form an optimal tari¤ without a free allowance, which would increase welfare while leaving the water provider's revenue unchanged. This optimal tariwould also reduce the number of households consuming below the WHO-recommended level.

Storage pricing strategies for import container terminals under stochastic conditions

This paper presents a model for determining the optimal storage pricing schedule for import containers. A generic schedule which is characterized by a flat rate and a storage time charge is adopted. The model considers analytically the stochastic behavior of the storage yard, as input/output flows are random variables, and includes the migration to an off-dock warehouse. Two objective functions are proposed: maximizing terminal operator profits and minimizing total integrated cost of the system. Some numerical experiments are provided and a sensitivity analysis is performed to investigate the effect of main variables and approaches on the optimal solution.

Proper Treatment of Buyer Power in Merger Review

In this paper, we assess the appropriate treatment of buyer power in merger review. We conclude that, for changes in bargaining outcomes due to a buyer merger to create efficiencies, it must be the case that, post-merger, the parties are able to arrive at a more optimal price schedule, perhaps due to reduced transactions costs. Empirical tests will be important to the evaluation of such efficiencies. We further conclude that, under certain conditions, powerful buyers may be able to prevent higher prices from a merger of suppliers. Once again, empirical tests should guide the evaluation of this merger defense.

Determining optimal selling price and lot size with process reliability and partial backlogging considerations

In this article, we extend the classical economic production quantity (EPQ) model by proposing imperfect production processes and quality-dependent unit production cost. The demand rate is described by any convex decreasing function of the selling price. In addition, we allow for shortages and a time-proportional backlogging rate. For any given selling price, we first prove that the optimal production schedule not only exists but also is unique. Next, we show that the total profit per unit time is a concave function of price when the production schedule is given. We then provide a simple algorithm to find the optimal selling price and production schedule for the proposed model. Finally, we use a couple of numerical examples to illustrate the algorithm and conclude this article with suggestions for possible future research.

Evaluating Pricing Strategies for Storage in Import Container Terminals

In most container terminals around the world, storage space is scarce, and pricing policies are needed to increase performance. Specifically, congestion when inbound containers are temporarily stored in terminal yards leads to high operational costs. This paper focuses on the introduction of a yard storage tariff to encourage early pickup of containers. Different from previous approaches, the price schedule introduced has a nonzero flat rate. Both demand reactions and changes in pickup decisions are considered in the analysis. A model is developed for the objective function (profit of the terminal operator). A numerical example illustrates an optimal price schedule, shows how sensitive the profit is to the basic constituents of the model, and provides general pricing rules.

Inventory and pricing strategies for deteriorating items with limited capacity and time-proportional backlogging rate

In the present study, a two-warehouse deterministic inventory model for deteriorating items with price-sensitive demand is developed. A rented warehouse (RW) is used to store the excess units over the fixed capacity of the own warehouse (OW). In addition, shortages are allowed in the OW and it is assumed that the backlogging demand rate is dependent on the duration of the stock-out. The objective is to find the optimal inventory and pricing strategies maximising the total average profit over the infinite planning horizon. For any given selling price, we first prove that the optimal replenishment schedule not only exists but is unique. Next, we show that the total average profit is a concave function of price when the replenishment schedule is given. The optimal selling price and replenishment schedule is also calculated for the proposed model. Finally, the results have been validated with the help of a numerical example.

Learning in nonlinear pricing with unknown utility functions

Optimal screening is one of the basic models of contracting under incomplete information, and we study the problem in a quality pricing application. We present a simple numerical method for solving the pricing problem when the firm has limited information about the buyers’ utility functions. In the method, the firm learns the optimal price schedule as the demand data is collected. We examine what the firm can learn about the preferences by observing the sales, and how the revealed information can be used in adjusting the quality-price bundles to increase the profit. We analyze the properties of the solution and derive the first-order optimality conditions under different assumptions. We show that the problem can be solved by making use of these optimality conditions together with the buyers’ marginal valuations. The firm can estimate the marginal valuations either by offering linear tariffs or by selling test bundles near the current solution.

The Economic Characteristics of Registries and Their Policy Implications

The discipline of economics supplies principles that may contribute to the discourse about investments in trauma registries and the role of the public sector, as well as the optimal use of the datasets those investments create. Principles from production economics, information economics, and public economics are employed to explore the reasons that trauma registries may be prone to underfunding, relative to their value, and to describe a threat to value maximization. The typical production activities and cost structures of registries are analyzed, along with the way registries generate social benefits. Assuming that the purpose of a trauma registry is to maximize the value or social good it creates, a number of investment, governance, and pricing principles are then proposed. Trauma registries are multiproduct enterprises. They are generally characterized by large and indivisible fixed, joint costs, and relatively low marginal costs. This implies that registries are subject to strong economies of scale and scope. Additionally, because registry data are not depleted by use, the registry's output is, technically, nonrival in consumption. The value created by registries may be maximized when a marginal-cost pricing policy is adopted. This means that the optimal price schedule for access to trauma registry data are likely to be zero, or close to zero, for some users. The economic principles presented here complement the contributions from other disciplines in discussions about the establishment of new registries or about improving some aspects of existing registries.

Optimal pricing and replenishment schedule for deteriorating items over a finite planning horizon

This paper considers a monopolistic retailer who stocks a single product that is subject to continuous decay, faces a price-dependent and time-varying demand, and has the objective of determining replenishment schedule and selling price to maximise the total profit stream over multi-period planning horizon. We propose both coordinated and decentralised decision-making policies that drive the solution of the model. The numerical result shows that the solution generated by the coordinated policy outperforms that by the decentralised policy in maximising profit and many other quantifiable measures such as minimising inventory investment and storage capacity.

Optimal price schedules for storage of inbound containers

This paper discusses a method of determining the optimal price schedule for storing inbound containers in a container yard. The price schedule in this study is characterized by the free-time-limit during which a container can be stored without any charge, and by the storage price per unit time for the storage beyond the free-time-limit. The profit or cost models for optimal price schedule are developed from the viewpoint of a public terminal operator as well as a private terminal operator. The probability distribution of delivery times is expressed by a continuous probability function. Various characteristics of the optimal solution are analyzed by numerical experiments.