Nonlinear Pricing Research Articles

Price transmission during food safety incidents: The case of Spanish tuna fraud

Using a two-regime threshold vector autoregressive model, this study investigates the nonlinear price transmission among tuna, salmon and hake during a food safety incident. The tuna fraud in Spain caused a histamine outbreak in 2017, which made 105 people fall ill. To evaluate the degree to which the food safety incident affected price transmission and reflected consumer awareness of the tuna fraud scandal, we have developed a communication index based on the number of Twitter posts and use its first difference to identify the increasing and decreasing regime of the development of the incident. There exists a greater reaction among prices in the increasing regime when communication strengthens on Twitter. Although the impacts of communication are not statistically significant, we find evidence that, in an increasing regime, a change in the communication index Granger causes the price of tuna and hake. The paper concludes with policy implications.

Optimal nonlinear pricing by a monopoly with smooth ambiguity preferences

This paper examines the optimal nonlinear pricing by a monopolist who has smooth ambiguity preferences and sells a single good to two types of buyers with high and low valuation. Ambiguity arises from the monopolist’s uncertainty about which of the subjective beliefs govern the unobserved types of buyers. We show that the prevalence of ambiguity aversion distorts the rent extraction-efficiency tradeoff under asymmetric information in the standard model. Specifically, the ambiguity-averse monopolist acts in a pessimistic manner with the perception that buyers are less likely to have high valuation than the objective beliefs. Compared with the standard model, low valuation buyers consume a less downward distorted quantity and high valuation buyers enjoy a greater information rent. The unit prices paid by all buyers are reduced and the ambiguity-averse monopolist earns a smaller expected profit.

Reconciling business analytics with graphically initialized subspace clustering for optimal nonlinear pricing

The relationship between price and quantity in nonlinear pricing transcends simple proportionality, as conditional rebates and discounts may be contingent upon the quantity of goods or services purchased by consumers. This dynamic introduces significant challenges for both consumers and business operators because incomplete information arises from the inherent uncertainty of consumer behavior. In light of this, the present research elucidates the pursuit of optimal nonlinear pricing strategies by business operators through an innovative data-driven approach. Our contributions encompass two distinctive facets: a novel unsupervised spectral clustering method, termed graphically initialized subspace clustering, and a decision optimization framework. The proposed data-driven method introduces an optimization problem aimed at minimizing subspace partitioning costs, leveraging the efficient utilization of a mixture multivariate skewed t distribution to effectively capture heavy users and to characterize their parametric behavioral patterns. In addition, the decision optimization component builds upon the aforementioned method, employing a convex optimization algorithm to enable seamless modification of attributes in nonlinear pricing, while ensuring revenue consistency during pre- and post-modification. Notably, we substantiate the interpretability and practical applicability of our proposed methodology in the realm of business analytics, through empirical analysis utilizing real-world data obtained from a cellular carrier. The findings of this study confirm the efficacy and viability of our approach in enabling business operators to navigate the complexities of nonlinear pricing optimization with confidence and informed decision-making.

A Novel Non-Ferrous Metals Price Forecast Model Based on LSTM and Multivariate Mode Decomposition

Non-ferrous metals are important bulk commodities and play a significant part in the development of society. Their price forecast is of great reference value for investors and policymakers. However, developing a robust price forecast model is tricky due to the price’s drastic fluctuations. In this work, a novel fusion model based on Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN), Singular Spectrum Analysis (SSA), and Long Short-Term Memory (LSTM) is constructed for non-ferrous metals price forecast. Considering the complexity of their price change, the dual-stage signal preprocessing which combines CEEMDAN and SSA is utilized. Firstly, we use the CEEMDAN algorithm to decompose the original nonlinear price sequence into multiple Intrinsic Mode Functions (IMFs) and a residual. Secondly, the component with maximum sample entropy is decomposed by SSA; this is the so-called Multivariate Mode Decomposition (MMD). A series of experimental results show that the proposed MMD-LSTM method is more stable and robust than the other seven benchmark models, providing a more reasonable scheme for the price forecast of non-ferrous metals.

Cheap Exclusion in Markets with Multiple Complements

We extend the theory of exclusive dealing in first-mover environments to settings where the incumbent seller’s product is used with multiple complements in a distribution chain and the incumbent can sign exclusive dealing contracts with more than one of them. The model is motivated by the market for biosimilar pharmaceuticals, where incumbent sellers that face a threat of entry can sign exclusionary contracts with both providers and insurance carriers prior to entry. We show that when the incumbent’s complementors are vertically related, it can be profitable for the incumbent to sign exclusive contracts with indirect buyers, who operate downstream from the direct buyers of the product. Under linear pricing, such exclusion is profitable if the pass-through rate is sufficiently low, and under nonlinear pricing and symmetric Nash bargaining, it is profitable for all pass-through rates. Complementors face a more severe coordination problem than independent buyers that can make anticompetitive exclusion more likely and especially cheap.

Taxing sugar‐sweetened beverages: A nonlinear pricing approach

AbstractTaxation is frequently implemented to discourage the consumption of sugary beverages. Despite their popularity, little is known about the impacts of taxes when sellers practice price discrimination. To address this issue, we use a standard nonlinear pricing model with one product and two buyer types to study the effects of taxation on (i) consumption, (ii) consumer and producer surpluses, and (iii) the seller's choice of market segmentation scheme. We find that a tax would lead to reductions in consumption, consumer surplus, and expected profit. Additionally, the measure increases the likelihood that the sellers would exclude buyers with low preferences for the beverage to exclusively serve buyers with high willingness to pay for the product.

Pricing energy consumption and residential energy‐efficiency investment: An optimal tax approach

AbstractWe analyze a Pareto optimal income tax problem à la Mirrlees in which households consume three types of goods: energy goods, energy‐efficient investments, and nonenergy goods. The two main ingredients of our normative analysis are: (i) an indirect relationship between energy and the satisfaction of energy needs, as energy‐efficient investments transform energy into services, such as light, heating, and air conditioning; and (ii) imperfect information of the policy designer as regards the level of energy efficiency of households' housing and their labor market productivity. Each household differs with respect to these two latter characteristics, and the government designs a nonlinear income tax combined with energy and energy‐efficient investment nonlinear pricing that maximizes a weighted sum of households' utilities. We show that a benevolent social planner should distort energy prices in a way that depends on the difference between the saturation of energy needs and the complementarity between energy and the level of energy efficiency in the provision of energy services. A sufficient condition for energy consumption to be subsidized is that the rebound effect is small. Second, when individuals can invest in energy efficiency on top of energy consumption, these investments should always be subsidized and the marginal subsidy should always be higher than the one on energy consumption.

Comparing information in general monotone decision problems

I study the value of information in monotone decision problems with potentially multidimensional action spaces. As a criterion for comparing information structures, I develop a condition called monotone quasi-garbling, which involves adding reversely monotone noise to an existing information structure. Specifically, this noise is more likely to return a higher signal in a lower state and a lower signal in a higher state. I show that monotone quasi-garbling is a necessary and sufficient condition for decision makers to obtain a higher ex-ante expected payoff. This new criterion refines the garbling condition by Blackwell (1951, 1953) and is equivalent to the accuracy condition by Lehmann (1988) under the monotone likelihood ratio property. To illustrate, I apply the result to problems in nonlinear monopoly pricing and optimal insurance.

On resource allocation in health care: The case of concierge medicine

Resource allocation generally involves a tension between efficiency and equity, particularly in health care. The growth in exclusive physician arrangements using non-linear prices is leading to consumer segmentation with theoretically ambiguous welfare implications. We study concierge medicine, in which physicians only provide care to patients paying a retainer fee. We find limited evidence of selection based on health and stronger evidence of selection based on income. Using a matching strategy that leverages the staggered adoption of concierge medicine, we find large spending increases and no average mortality effects for patients impacted by the switch to concierge medicine.

Optimal Contracting with Altruistic Agents: Medicare Payments for Dialysis Drugs

We study health-care provider agency and optimal payments, considering an expensive medication for dialysis patients. Using Medicare claims data we estimate a structural model of treatment decisions, in which providers differ in their altruism and marginal costs, and this heterogeneity is unobservable to the government. In a novel application of nonlinear pricing methods, we empirically characterize the optimal contracts in this environment. The optimal contracts eliminate medically excessive dosages and reduce expenditures, resulting in approximately $300 million in annual gains from better contracting. This approach could be applied to a broad class of problems in health-care payment policy. (JEL D64, D86, H51, I11, I13, J33, L21)

Contract design for a closed-loop supply chain considering information asymmetry of remanufacturing process innovation

PurposeAsymmetric cost information exists between a supplier and a manufacturer regarding the manufacturer's process innovation for remanufacturing (PIR), which may hurt the supplier's profit. The authors therefore seek to develop a menu of nonlinear pricing contracts for channel information sharing.Design/methodology/approachBased on principal–agent theory, the supplier, acting as a Stackelberg leader, designs a menu of nonlinear pricing contracts to impel the manufacturer to disclose its private cost information on PIR (i.e. PIR efficiency). In addition, the authors compare the equilibrium outcomes under asymmetric and symmetric information to examine the effects of asymmetric PIR information on the production policies and profits of the supplier and the manufacturer.FindingsThe proposed contract menu encourages th4e manufacturer to spontaneously share PIR efficiency information with the supplier. Asymmetric PIR information may distort the output of new products upward or downward, but the output of remanufactured products may only be distorted downward. In addition, the manufacturer with high PIR efficiency gains information rent, and interestingly, the increase in the probability of low PIR efficiency amplifies its information rent. Finally, an asymmetric information environment may increase the threshold for the manufacturer to enter remanufacturing.Originality/valueThe authors probe the issue of the supplier's contract design by jointly considering remanufacturing, process innovation and information asymmetry. The paper expands the influencing mechanism of process innovation information in the remanufacturing field. The authors also observe new results that may offer guidance to decision makers.

Competitive contract design in a retail supply chain under demand uncertainty

AbstractThis article studies the design of contracts involving a single retailer and multiple competing manufacturers who supply substitutable products. We consider a retail context in which contracts with manufacturers are negotiated relatively infrequently and signed before the demand environment is known, and the retail prices are determined when the demand is known. We develop a Stackelberg model to study the retailer's product selection and pricing decisions and the manufacturers' contract design decisions. We show that it is optimal for each manufacturer to offer a contract with nonlinear prices so that total payments are the total production cost plus a fixed additional cost. In the case of two manufacturers this result allows us to characterize an equilibrium in which the retailer's choice maximizes the supply chain profit, each manufacturer makes a profit equal to its marginal contribution to the supply chain, and the retailer takes the remaining profit. We also find that while increasing demand correlation always benefits the retailer, it benefits the manufacturers only when the production costs are convex. In an extension it is found that our equilibrium continues to hold when the retailer's reservation profit is below a threshold, but the competition dynamics may change when the reservation profit is above the threshold. Finally, we show that the equilibrium results remain true for the case with more than two manufacturers under a submodularity property, which holds in the case of quadratic costs and linear demand.

Optimal Nonlinear Pricing with Data-Sensitive Consumers

We study monopolistic screening when some consumers are data sensitive and incur a privacy cost if their purchase reveals information to the monopolist. The monopolist discriminates between data-sensitive and classical consumers using privacy mechanisms that consist of a direct mechanism and a privacy option. A privacy mechanism is optimal for large privacy costs and leaves classical consumers better off than data-sensitive consumers with the same valuation. When privacy preferences become public information, data-sensitive consumers and the monopolist gain, whereas classical consumers lose. Our results are relevant for policies targeting consumers’ data awareness, such as the European General Data Protection Regulation. (JEL D11, D42, D82, D83, L12)

Bundling and nonlinear pricing in telecommunications

AbstractI develop a multiproduct nonlinear pricing model where a firm sells both discrete and continuous services to consumers with multidimensional heterogeneity. I derive the optimal selling mechanism and provide primitive conditions under which different bundling strategies arise. Exploiting both the firm and the consumer's optimality conditions, I show that the model structure is nonparametrically identified and propose a semiparametric estimation procedure. An application to China Telecom data shows that mixed bundling of internet and landline phone services is more beneficial to both the firm and the consumer relative to component pricing.

Omitted Budget Constraint Bias and Implications for Competitive Pricing

Standard choice-based conjoint models often ignore or insufficiently approximate consumers’ budget constraints, despite the prominent role of budget constraints in economic theory. The authors offer a theoretically motivated improvement to the choice-based conjoint model that is especially appropriate for high-ticket durable goods and develop a Bayesian method for the inference of unobserved budget constraints. The proposed method leverages respondents’ stated budget constraints that suffer from measurement error and respondents’ financial demographic variables as additional information to reduce the dependency on functional form assumptions in the estimation. The authors show that accounting for budget constraints substantially increases model fit and the accuracy of competitive pricing in an industry-grade discrete-choice experiment on consumer preferences for high-end laptops. The proposed model performs better than the canonical linear price benchmark model, which is not flexible enough to approximate budget constraints. In theory, more flexible utility specifications, such as the nonlinear dummy price model, can approximate consumers’ budget constraints. However, they perform poorly when only finite data are available. The authors conclude that applied researchers in industry and academia will benefit from having a better tool for estimating budgets in high-ticket categories.

Electricity price forecasting using hybrid deep learned networks

AbstractThis paper presents a novel hybrid model integrating maximal overlap discrete wavelet transform (MODWT) denoising and empirical mode decomposition (EMD) with sequence‐to‐sequence (seq2seq) long short‐term memory (LSTM) neural networks for day‐ahead electricity price forecasting. The nonstationary and nonlinear time series electricity price data are first denoised using MODWT. The resulting signal is decomposed into several intrinsic mode functions (IMF) with different resolutions by EMD. The extracted IMF is then introduced into seq2seq LSTM to obtain an aggregated predicted value for electricity price. The proposed method is examined using the Nord pool Elspot energy market data. Empirical results show that the proposed model outperformed the other forecasting models like LSTM and stacked LSTM. The performance measures indicate that data denoising can significantly improve the prediction stability and the generalization ability of the LSTM model.

Nonlinear asset pricing in Chinese stock market: A deep learning approach

The redesign of asset pricing models failed to integrate the frequent financial phenomenon that stock markets exhibit a non-linear long- and short-term memory structure. The difficulty lies in developing a nonlinear pricing structure capable of depicting the memory influence of the pricing variable. This paper presents a Long- and Short-Term Memory Neural Network Model (LSTM) to capture the non-linear pricing structure among five elements in the Chinese stock market, including market portfolio return, market capitalisation, book-to-market ratio, earnings factor, and investment factor. The long–short-term memory structure implies that the autocorrelation function of the stock return series decays slowly and has a long-term characteristic. The LSTM model surpasses the standard Fama–French five-factor model in terms of out-of-sample goodness-of-fit and long–short strategy performance. The empirical findings indicate that the LSTM nonlinear model properly represents the nonlinear relationships between the five components.

Household water savings and response to dynamic incentives under nonlinear pricing

We study the response of residential water demand to nonlinear prices by exploiting a natural experiment arising from a water pricing reform in a major Chinese city. The reform introduced an unconventional Increasing Block Tariff featuring prices set according to annual cumulative consumption and a bimonthly billing cycle. Analyzing data from a household survey and administrative water bills, we detect a small effect on low-use households but find large water savings among high-use households. Moreover, we find strong evidence that high-use households respond to future price while current price remains fixed, and that the small share of households who appear myopic do not respond to dynamic incentives.

Entropy martingale optimal transport and nonlinear pricing–hedging duality

The objective of this paper is to develop a duality between a novel entropy martingale optimal transport (EMOT) problem and an associated optimisation problem. In EMOT, we follow the approach taken in the entropy optimal transport (EOT) problem developed in Liero et al. (Invent. Math. 211:969–1117, 2018), but we add the constraint, typical of martingale optimal transport (MOT) theory, that the infimum of the cost functional is taken over martingale probability measures. In the associated problem, the objective functional, related via Fenchel conjugacy to the entropic term in EMOT, is no longer linear as in (martingale) optimal transport. This leads to a novel optimisation problem which also has a clear financial interpretation as a nonlinear subhedging problem. Our theory allows us to establish a nonlinear robust pricing–hedging duality which also covers a wide range of known robust results. We also focus on Wasserstein-induced penalisations and study how the duality is affected by variations in the penalty terms, with a special focus on the convergence of EMOT to the extreme case of MOT.

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)