Nonlinear Pricing Model Research Articles

Multi-Dimensional Mechanism Design with Limited Information

We analyze a nonlinear pricing model with limited information. Each buyer can purchase a large variety, d, of goods. His preference for each good is represented by a scalar and his preference over d goods is represented by a d-dimensional vector. The type space of each buyer is given by a compact subset of Rd with a continuum of possible types. By contrast, the seller is limited to offer a finite number M of d-dimensional choices.We provide necessary conditions that the optimal finite menu of the social welfare maximizing problem has to satisfy. We establish an underlying connection to the theory of quantization and provide an estimate of the welfare loss resulting from the usage of the d-dimensional M-class menu. We show that the welfare loss converges to zero at a rate proportional to d/M^{2/d}.We show that in higher dimensions, a significant reduction in the welfare loss arises from an optimal partition of the d-dimensional type space that takes advantage of the correlation among the d parameters.

Nonlinear Pricing with Product Customization in Mobile Service Industry

This paper proposes to incorporate product customization in the Maskin and Riley (1984) nonlinear pricing model in order to capture major features of mobile service data. In particular, consumers are characterized by a two-dimensional type. One dimension is observed by the provider and integrates product customization, while the other is a standard parameter of adverse selection, which is unobserved by the provider and makes it necessary for the provider to discriminate among consumers with dierent tastes through nonlinear pricing. We then propose a novel method to aggregate the multiple-dimensional voice consumption into one-dimensional index. We show that the model structure is identified under the following conditions: The marginal utility function is multiplicatively separable in consumers’ tastes, and consumers’ observed and unobserved heterogeneity are independent. Empirical results show that both dimensions of heterogeneity are important. Due to asymmetric information, 50% of the second-best social welfare is left on the table in order to screen heterogeneous consumers. Moreover, if costly product customization does not aect subscribers’ utility, 20% of subscribers would not be served.

Dynamic Pricing Model for the Operation of Closed-Loop Supply Chain System

A class of closed-loop supply chain system consisting of one manufacturer and one supplier is designed, in which re-distribution, remanufacturing and reuse are considered synthetically. The manufacturer is in charge of recollecting and re-disposal the used products. Demands of ultimate products and collecting quantity of used products are described as the function of prices and reference prices. A non-linear dynamic pricing model for this closed-loop supply chain is established. A numerical example is designed, and the results of this example verified the model’s validity to price for the operation of closed-loop supply chain system.

Brand-Specific Tastes for Quality

This paper develops a model of nonlinear pricing with competition. The novel element is that each consumer's willingness to pay for quality is private information and is allowed to differ across brands. The consumer's preferences are represented by a multidimensional type containing the marginal value of quality for different products. Buyers with high willingness to pay for quality also display strong preferences for particular brands, and require higher discounts in order to switch away from their favorite product. Therefore, competition is fiercer for buyers with lower tastes for quality, and hence more elastic demands. This is in sharp contrast to earlier models in which competition is fiercer for higher-taste, more valuable buyers. In equilibrium, firms either compete intensively for the entire market (providing strictly positive rents to all consumers) or shut down the least profitable segment of the market. Quality levels are distorted downwards for all buyers, except for those with the highest type. Finally, the number of competing firms and the degree of correlation across brand preferences enhance the efficiency of the allocation.

Brand-specific tastes for quality

This paper develops a model of nonlinear pricing with competition. The novel element is that each consumer's willingness to pay for quality is private information and is allowed to differ across brands. The consumer's preferences are represented by a multidimensional type containing the marginal value of quality for different products. Buyers with high willingness to pay for quality also display strong preferences for particular brands, and require higher discounts in order to switch away from their favorite product. Therefore, competition is fiercer for buyers with lower tastes for quality, and hence more elastic demands. This is in sharp contrast to earlier models in which competition is fiercer for higher-taste, more valuable buyers. In equilibrium, firms either compete intensively for the entire market (providing strictly positive rents to all consumers) or shut down the least profitable segment of the market. Quality levels are distorted downwards for all buyers, except for those with the highest type. The number of competing firms and the degree of correlation across brand preferences enhance the efficiency of the allocation.

Mechanism Design with Limited Information: The Case of Nonlinear Pricing

We analyze the canonical nonlinear pricing model with limited information. A seller offers a menu with a finite number of choices to a continuum of buyers with a continuum of possible valuations. By revealing an underlying connection to quantization theory, we derive the optimal finite menu for the socially efficient and the revenue-maximizing mechanism. In both cases, we provide an estimate of the loss resulting from the usage of a finite n-class menu. We show that the losses converge to zero at a rate proportional to 1/n^2 as n becomes large.

Financial Rogue Waves

We analytically give the financial rogue waves in the nonlinear option pricing model due to Ivancevic, which is nonlinear wave alternative of the Black—Scholes model. These rogue wave solutions may he used to describe the possible physical mechanisms for rogue wave phenomenon in financial markets and related fields.

A consistent stable numerical scheme for a nonlinear option pricing model in illiquid markets

Markets liquidity is an issue of very high concern in financial risk management. In a perfect liquid market the option pricing model becomes the well-known linear Black–Scholes problem. Nonlinear models appear when transaction costs or illiquid market effects are taken into account. This paper deals with the numerical analysis of nonlinear Black–Scholes equations modeling illiquid markets when price impact in the underlying asset market affects the replication of a European contingent claim. Numerical analysis of a nonlinear model is necessary because disregarded computations may waste a good mathematical model. In this paper we propose a finite-difference numerical scheme that guarantees positivity of the solution as well as stability and consistency.

Pricing options in illiquid markets: Optimal systems, symmetry reductions and exact solutions

We study a class of nonlinear pricing models which involves the feedback effect from the dynamic hedging strategies on the price of asset introduced by Sircar and Papanicolaou. We are first to study the case of a nonlinear demand function involved in the model. Using a Lie group analysis we investigate the symmetry properties of these nonlinear diffusion equations. We provide the optimal systems of subalgebras and the complete set of non-equivalent reductions of studied PDEs to ODEs. In most cases we obtain families of exact solutions or derive particular solutions to the equations.

A SPOT PRICING FRAMEWORK FOR PRICING INTRA-DOMAIN ASSURED BANDWIDTH SERVICES

The Internet today offers primarily a best-effort service. Research and technology development efforts are currently underway to allow the provisioning of better than best-effort quality of service assurances. In this article, we develop a spot pricing framework for expected bandwidth guaranteed Internet service contracts on a single domain. A nonlinear pricing model that maximizes the total surplus with cost constraints forms the basis for the framework. The spot price process is obtained for different demand profiles of the customer-base and demand arrival characteristics. Simulation modeling and analysis is employed to implement the pricing framework and analyze the price process under different market conditions and changes due to network failures. The framework when implemented at access or exchange points of different service provider domains will provide assured bandwidth for inter-domain traffic.

Optimal Income Taxation, Public-Goods Provision and Public-Sector Pricing: A Contribution to the Foundations of Public Economics

The paper develops an integrated model of optimal nonlinear income taxation, public-goods provision and pricing in a large economy. With asymmetric information about labour productivities and publicgoods preferences, the multidimensional mechanism design problem becomes tractable by requiring renegotiation proofness of the final allocation of private goods and admission tickets for excludable public goods. Under an affiliation assumption on the underlying distribution, optimal income taxation, public-goods provision and admission fees have the same qualitative properties as in unidimensional models. These properties are obtained for utilitarian welfare maximization and for a Ramsey-Boiteux formulation with interim participation constraints.

Network Effects, Nonlinear Pricing and Entry Deterrence

A number of products that display positive network effects are used in variable quantities by heterogeneous customers. Examples include corporate operating systems, infrastructure software, web services and networking equipment. In many of these contexts, the magnitude of network effects are influ- enced by gross consumption, rather than simply by user base. Moreover, the value an individual customer derives on account of these network effects may be related to the extent of their individual consumption, and therefore, the network effects may be heterogeneous across customers. This paper presents a model of nonlinear pricing in the presence of such network effects, under incomplete information, and with the threat of competitive entry. Both homogeneous and heterogeneous network effects are modeled. Conditions under which a fulfilled-expectations contract exists and is unique are established. While network effects generally raise prices, it is shown that accompanying changes in consumption depend on the nature of the network effects — in some cases, it is optimal for the monopolist to induce no changes in usage across customers, while in others cases, network effects raise the usage of all market participants. Optimal pricing is shown to include quantity discounts that increase with usage, and may also involve a nonlinear two-part tariff. These results highlight the impact of network effects on the standard trade-off between price discrimination and value creation, and have important implications for pricing policy. The threat of entry generally lowers profits for the monopolist, and increases customer surplus. When network effects are homogeneous across customers, the resulting entry-deterring monopoly contract is a fixed fee and results in the socially optimal outcome. However, when the magnitude of heterogeneous network effects is relatively high, there are no changes in total surplus induced by the entry threat, and the price changes merely cause a transfer of value from the seller to its customers. The presence of network effects, and of a credible entry threat, are also shown to increase distributional efficiency by reducing the disparity in relative value captured by different customer types. Regulatory and policy implications of these results are discussed.

Implicit solution of uncertain volatility/transaction cost option pricing models with discretely observed barriers

Option pricing models with uncertain volatility/transaction costs give rise to a nonlinear PDE. Previous work has focused on explicit methods. However, pricing discretely observed barrier options requires a very small grid spacing near the barrier, and as a result, the maximum stable timestep for an explicit method is impractically small. A fully implicit method is developed for nonlinear option pricing models, and applied to arithmetic step options, where the option loses a fraction of its value for every day over the barrier.

Modelling chaotic behaviour in agricultural pricesusing a discrete deterministic nonlinear price model

In economic modelling, the generally used deterministic equilibrium models cannotdescribe the “random‐looking” oscillations and irregular motions often observed in realtime series. Recently, many publications have dealt with chaos in economic processes, butthe majority of them used rather difficult nonlinear mathematical functions. Furthermore,chaotic behaviour emerged most often in parameter ranges that are difficult to interpret aseconomically meaningful values. The present paper analyses the behaviour of a discretedeterministic nonlinear model of supply and demand of a single product with many producerson the market. Market supply is determined by the producers' price expectations, the actualprice is the market clearing price. This linear cobweb model is made a piecewise linear oneby putting a lower and an upper limit on the expected prices and the real market price. Thesensitivity of the model to a wide range of negative values of the price elasticity coefficientis tested. As this value increases, the model produces various types of steady‐state behaviour,such as equilibrium point, periodic behaviour with increasing periods, “quasiperiodic‐like”behaviour, period‐3 cycle, and chaotic behaviour. The model showed that a deterministicprice model without any stochastic component is fully capable of producing irregular oscillations,fluctuations often observed in real price series. Furthermore, it was possible to achievethis behaviour with a simple, piecewise linear cobweb model, with parameters that havestraightforward economic meaning, within realistic parameter ranges. A simple method oflinear coupling is applied to show that price stabilisation can always be achieved by usingtwo control parameters, which can be attributed a sound economic meaning.

Modelling chaotic behaviour in agricultural prices using a discrete deterministic nonlinear price model

In economic modelling, a frequent problem is that the generally used deterministic equilibrium models cannot describe the ‘random-looking’ oscillations and irregular motions often observed in real time series. Nonlinear chaotic models offer a way to produce this behaviour without the introduction of stochastic elements. Recently, many publications have dealt with chaos in economic processes, but the majority of them used rather difficult mathematical functions to model these processes. Furthermore, chaotic behaviour emerged most often in parameter ranges that are difficult to interpret as economically meaningful values. The present paper analyses the behaviour of a discrete deterministic non-linear model of supply and demand of a single product with many producers on the market. N producers operate in the market of a single product; their production technology is described by a quadratic cost function. They produce the amount which maximises their gain considering an expected market price in the next time period. The demand function is d(t) = D∗p(t) + d, a linear function of the p( t) real market price, and the real market price is the value which balances the total supply from the N producers and the total demand in the market. This linear model is made a piecewise linear one by putting a lower and an upper limit on the expected prices and the real market price. The sensitivity of the model to a wide range of negative values of D is tested. As the value of D increases, the model produces various types of steady state behaviour, such as equilibrium point, periodic behaviour with increasing periods, ‘quasiperiodic-like’ behaviour, period-3 cycle, and chaotic behaviour. The above model showed that a deterministic price model without any stochastic component is fully capable of producing irregular oscillations, fluctuations often observed in real price series. Furthermore, it was possible to achieve this behaviour with a simple, piecewise linear cobweb model, with parameters that have straightforward economic meaning, in realistic parameter ranges. If parameters have sound meaningful values, then chaotic models can direct the attempts made at price stabilization by describing clearly which is the parameter range the decision maker has to avoid in order to reach a stable or regular behaviour.

Qualitative analysis of a nonlinear stock price model

Attention is focused on stock price models with no closed-form solution, the most likely occurrence, when realistic assumptions are made about the sources of dividends. It is argued that these models can supply valuable information for both financial analysis and financial management, provided that they are treated as dynamic systems and tackled accordingly by means of a qualitative analysis. As an example, the profitability of a firm is taken as a state variable within the celebrated model by Gordon and Shapiro. A one parameter bifurcation analysis is performed on the ensuing nonlinear dynamic system. Novel results about the price-earnings ratio are obtained and financially interpreted.

A note on the marginal tax rate in a finite economy

The marginal tax rate is shown to be non-negative in Guesnerie and Seade's model of nonlinear pricing in a finite economy under assumptions that are as mild as those customarily adopted in nonlinear tax models with a one-dimensional continuum population.

Hospital profit planning under Medicare reimbursement.

The federal Medicare regulations reimburse hospitals on a pro rata share of the hospital's cost. Hence, to meet its financial requirements, a hospital is forced to shift more of the financial burdens onto its private patients. This procedure has contributed to double digit inflation in hospital prices and to proposed federal regulation to control the rate of increase in hospital revenues. In this regulatory environment, we develop nonlinear programming pricing and cost allocation models to aid hospital administrators in meeting their profit maximizing and profit satisfying goals. The model enables administrators to explore tactical issues such as: (i) studying the relationship between a voluntary or legislated cap on a hospital's total revenues and the hospital's profitability, (ii) identifying those departments within the hospital that are the most attractive candidates for cost reduction or cost containment efforts, and (iii) isolating those services that should be singled out by the hospital manager for renegotiation of the prospective or "customary and reasonable" cap. Finally the modeling approach is helpful in explaining the departmental cross subsidies observed in practice, and can be of aid to federal administrators in assessing the impacts of proposed changes in the Medicare reimbursement formula.