Nonlinear Price Schedules Research Articles

Securities Markets in Which Some Investors Receive Information About Cash Flow Betas

We analyze a single-factor setting in which there is private information regarding cash flows as well as their betas. Private information about betas, together with market makers’ risk aversion and mean betas’ nonnegativity, implies a nonlinear price schedule whose stochastic slope covaries positively with order flow when the expected factor payoff is positive and vice versa. We predict a negative relation between the covariance and expected returns and an attenuation of the beta anomaly in asset returns after accounting for this relation. Empirical tests confirm these predictions. This paper was accepted by Lukas Schmid, finance. Funding: L. Yang acknowledges the Social Sciences and Humanities Research Council of Canada [Grant 435-2021-0040] and Bank of Canada Fellowship for financial support. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2022.02061 .

Empirical analysis of network effects in nonlinear pricing data

Network effects, i.e., an agent's utility may depend on other agents' choices, appear in many contracting situations. Empirically assessing them faces two challenges: an endogeneity problem in contract choice and a reflection problem in network effects. This paper proposes a nonparametric approach to tackle both challenges by exploiting restriction conditions from both demand and supply sides. We illustrate our methodology in the yellow pages advertising industry. Using advertising purchases and nonlinear price schedules from seven directories in Toronto, we find positive network effects, which account for a substantial portion of the publisher's profit and businesses' surpluses. We finally conduct counterfactuals to assess the overall and distributional welfare effects of the nonlinear pricing scheme relative to an alternative linear pricing scheme with and without network effects.

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)

Increasing the energy cognizance of electricity consumers in Mexico: Results from a field experiment

We report on the results of a field experiment in Puebla, Mexico that informed randomly-selected households facing a nonlinear price schedule about how different electricity-consuming actions might change their electricity bills. Households that received this 20-minute, in-person intervention reduced their electricity use, with much of this reduction driven by those that paid the highest marginal price for electricity. The estimated impacts were durable with no observed rebound for at least a year. Households with less educational attainment reduced use the most, consistent with the conclusion that the intervention imparted new knowledge to consumers that led to this observed behavior change. The high rate at which customers accepted the intervention, the resulting consumption decrease, and low implementation costs make this intervention cost-effective relative to several previous energy conservation campaigns in Mexico.

Promoting Energy Efficiency in Emerging Economies Through Consumer Education: Results From a Field Experiment in Mexico

We report on the results of a field experiment in Puebla, Mexico that informed randomly-selected households facing a nonlinear price schedule about how different electricity-consuming actions might change their electricity bills. Households that received this 20-minute, in-person intervention reduced their electricity use, with much of this reduction driven by those that paid the highest marginal price for electricity. The estimated impacts were durable with no observed rebound for at least a year. Households with less educational attainment reduced use the most, consistent with the conclusion that the intervention imparted new knowledge to consumers that led to this observed behavior change. The high rate at which customers accepted the intervention, the resulting consumption decrease, and low implementation costs make this intervention cost-effective relative to several previous energy conservation campaigns in Mexico.

Nonlinear Pricing: Evidence of Price Discrimination in the Fluid Milk Market

Firms may use nonlinear price schedules, possibly to the detriment of some consumers, to price discriminate and increase profits. However, it is possible that these nonlinear price schedules may also increase welfare from the consumer's perspective if they are able to serve portions of a market that firms may otherwise ignore. The more extensive theoretical literature on price discrimination shows how firms may use nonlinear price schedules to price discriminate by distorting product characteristics from their efficient levels given heterogeneous preferences among consumers. Additionally, the theoretical literature finds that whether or not this practice is welfare decreasing or increasing from the consumer's perspective depends on a number of factors, thus implying that the welfare implications of price discrimination from the consumer's perspective is an empirical question. Subsequently, some recent empirical papers have found this form of second-degree price discrimination to be welfare decreasing from the consumer's perspective whereas other papers, depending on the analyzed market, have found it to be welfare increasing. In this paper, I study the consumer welfare implications of nonlinear pricing in the market of fluid milk. Variation in the ratio between marginal price per ounce and marginal cost per ounce over the menu of options is evidence that firms are using nonlinear pricing schedules to price discriminate. Once this evidence is shown, I use counterfactual analysis to determine the welfare implications of this practice within the market for fluid milk from the consumer's perspective. I find that firms within this industry are using nonlinear pricing to price discriminate for private label products, of which conservative estimates show that approximately 7.20% of the markup differential between sizes can be explained by price discrimination. However, I find that very little of the markup on Regional branded products, approximately 0.32%, can be attributed to price discrimination. In my most compelling set of counterfactual analyses, I show that consumers are worse off due to price discrimination and my counterfactual analysis shows that households are on average anywhere between $0.0249 to $.0555 to worse off per shopping trip under a forced linear pricing, which corresponds to a total yearly welfare difference between $63 million to $140 million per year within my sample region due to the nonlinear pricing strategies used by firms in the marketplace. Additionally, I estimate and utilize models similar to the status quo in the literature on price discrimination thus far that do not have household demographic characteristics included and compare them my preferred specification which does, showing that ignoring the demographic makeup of the marketplace's individuals and actual consumer decisions leads to potentially biased parameter estimates.

China's medical savings accounts: an analysis of the price elasticity of demand for health care.

Although medical savings accounts (MSAs) have drawn intensive attention across the world for their potential in cost control, there is limited evidence of their impact on the demand for health care. This paper is intended to fill that gap. First, we built up a dynamic model of a consumer's problem of utility maximization in the presence of a nonlinear price schedule embedded in an MSA. Second, the model was implemented using data from a 2-year MSA pilot program in China. The estimated price elasticity under MSAs was between -0.42 and -0.58, i.e., higher than that reported in the literature. The relatively high price elasticity suggests that MSAs as an insurance feature may help control costs. However, the long-term effect of MSAs on health costs is subject to further analysis.

Heterogeneous Effects of a Nonlinear Price Schedule for Outpatient Care

Nonlinear price schedules generally have heterogeneous effects on health-care demand. We develop and apply a finite mixture bivariate probit model to analyze whether there are heterogeneous reactions to the introduction of a nonlinear price schedule in the German statutory health insurance system. In administrative insurance claims data from the largest German health insurance plan, we find that some individuals strongly react to the new price schedule while a second group of individuals does not react. Post-estimation analyses reveal that the group of the individuals who do not react to the reform includes the relatively sick. These results are in line with forward-looking behavior: Individuals who are already sick expect that they will hit the kink in the price schedule and thus are less sensitive to the co-payment. Copyright © 2016 John Wiley & Sons, Ltd.

THE RESPONSE OF DRUG EXPENDITURE TO NON-LINEAR CONTRACT DESIGN: EVIDENCE FROM MEDICARE PART D.

We study the demand response to non-linear price schedules using data on insurance contracts and prescription drug purchases in Medicare Part D. We exploit the kink in individuals' budget set created by the famous "donut hole," where insurance becomes discontinuously much less generous on the margin, to provide descriptive evidence of the drug purchase response to a price increase. We then specify and estimate a simple dynamic model of drug use that allows us to quantify the spending response along the entire non-linear budget set. We use the model for counterfactual analysis of the increase in spending from "filling" the donut hole, as will be required by 2020 under the Affordable Care Act. In our baseline model, which considers spending decisions within a single year, we estimate that "filling" the donut hole will increase annual drug spending by about $150, or about 8 percent. About one-quarter of this spending increase reflects "anticipatory" behavior, coming from beneficiaries whose spending prior to the policy change would leave them short of reaching the donut hole. We also present descriptive evidence of cross-year substitution of spending by individuals who reach the kink, which motivates a simple extension to our baseline model that allows - in a highly stylized way - for individuals to engage in such cross year substitution. Our estimates from this extension suggest that a large share of the $150 drug spending increase could be attributed to cross-year substitution, and the net increase could be as little as $45 per year.

Estimating demand elasticities using nonlinear pricing

Nonlinear pricing is prevalent in industries such as health care, public utilities, and telecommunications. However, this pricing scheme introduces bias into estimating elasticities for welfare analysis or policy changes. I develop a local elasticity estimation method that uses nonlinear price schedules to isolate consumers' expenditure choices from selection and simultaneity biases. This method improves over previous approaches by using commonly-available observational data and requiring only a single general monotonicity assumption. Using claims-level data on health insurance with two nonlinearities, I am able to measure two separate elasticities, and find that elasticity declines from −0.26 to−0.09 by the second nonlinearity. These estimates are then used to calculate moral hazard deadweight loss. This method enables estimation of many policies with nonlinear pricing which previous tools could not address.

Nonlinear pricing in drug benefits and medication use: the case of statin compliance in Medicare Part D.

To examine how enrollees' statin compliance responds to expected prices in Medicare Part D, which features a nonlinear price schedule due to a coverage gap. Prescription Drug Event data for a 5 percent random sample of Medicare Advantage Prescription Drug Plan enrollees in 2008 who did not receive a low-income subsidy. We analyze statin compliance prior to the coverage gap, where the "effective price" is higher than the actual copayment for drugs because consumers anticipate that more spending will make them more likely to reach the gap. We construct each enrollee's effective price as her expected price at the end of the year, which is the weighted average between pre-gap and in-gap copayments with the weight being the predicted probability of hitting the gap. Compliance is defined as at least 80 percent of days covered. Part D enrollees' pre-gap statin compliance decreases by 3.7-4.7 percentage points for a $10 increase in the effective price. The presence of a coverage gap decreases statin compliance prior to the gap, suggesting that incorporating expected future prices is important to assess the full impact of cost sharing on drug compliance under nonlinear price schedules.

Demand for prescription drugs under non-linear pricing in Medicare Part D

We estimate the price elasticity of prescription drug use in Medicare Part D, which features a non-linear price schedule due to a coverage gap. We analyze patterns of drug utilization prior to the coverage gap, where the "effective price" is higher than the actual copayment for drugs because consumers anticipate that more spending will make them more likely to reach the gap. We find that enrollees' total pre-gap drug spending is sensitive to their effective prices: the estimated price elasticity of drug spending ranges between [Formula: see text]0.14 and [Formula: see text]0.36. This finding suggests that filling in the coverage gap, as mandated by the health care reform legislation passed in 2010, will influence drug utilization prior to the gap. A simulation analysis indicates that closing the gap could increase Part D spending by a larger amount than projected, with additional pre-gap costs among those who do not hit the gap.

THE JANUARY EFFECT: MEDICATION REINITIATION AMONG MEDICARE PART D BENEFICIARIES

The Medicare prescription drug program (Part D) standard benefit includes deductible, initial coverage, coverage gap and catastrophic coverage phases. As beneficiaries enter each phase, their out-of-pocket medication costs change discontinuously. The benefit cycle restarts on 1 January of the next year. Taking advantage of variation in drug coverage, we study how individuals reinitiate discontinued medications in response to the non-linear price schedule. Because some beneficiaries who receive low-income subsidies (LIS) have zero or fixed small copayments throughout the year, we perform a difference-in-difference analysis by using the LIS group as a comparison. We find that individuals delay reinitiating important medications in December and are significantly more likely to reinitiate in January than in other months. Although we find some evidence that reinitiation is lower in the final months of the year, it is mostly driven by those who face higher prices due to the coverage gap. Our study suggests that individuals respond more to the current price of medications and do not anticipate future prices as well as theory would suggest.

PER‐PERIOD CO‐PAYMENTS AND THE DEMAND FOR HEALTH CARE: EVIDENCE FROM SURVEY AND CLAIMS DATA

When health insurance reforms involve non-linear price schedules tied to payment periods (for example, fees levied by quarter or year), the empirical analysis of its effects has to take the within-period time structure of incentives into account. The analysis is further complicated when demand data are obtained from a survey in which the reporting period does not coincide with the payment period. We illustrate these issues using as an example a health care reform in Germany that imposed a per-quarter fee of €10 for doctor visits and additionally set an out-of-pocket maximum. This co-payment structure results in an effective 'spot' price for a doctor visit that decreases over time within each payment period. Taking this variation into account, we find a substantial reform effect-especially so for young adults. Overall, the number of doctor visits decreased by around 9% in the young population. The probability of visiting a physician in any given quarter decreased by around 4 to 8 percentage points.

Estimating Demand Elasticities Using Nonlinear Pricing

Nonlinear pricing is prevalent in industries such as health care, public utilities, and telecommunications. However, nonlinear pricing in these sectors causes bias when estimating elasticities for welfare analysis or policy changes. This paper develops a local elasticity estimation method that takes advantage of nonlinear price schedules to isolate consumers' expenditure choices from selection and simultaneity biases. This method improves over previous approaches because it uses commonly-available observational data and requires a single general monotonicity assumption. Using detailed claims-level health insurance data with two nonlinearities, this paper measures declining elasticities from -0.26 to -0.09 along initial expenditure ranges, in contrast to previous work which aggregates over a skewed spending distribution. These estimates are then used to calculate moral hazard deadweight loss. This method enables estimation on many policies with nonlinear pricing which previous tools could not address.

Heterogeneous Effects of a Nonlinear Price Schedule for Outpatient Care

Theoretical considerations suggest that nonlinear health care price schedules have heterogeneous effects on health care demand. In this paper, we develop and apply a finite mixture bivariate probit model to analyze whether there are heterogeneous reactions to the introduction of a nonlinear price schedule in the German statutory health insurance system. In administrative insurance claims data from the largest German health insurance plan, we find that some individuals strongly react to the new price schedule while a second group of individuals does not react. Post-estimation analyses reveal that the group of the individuals who do not react to the reform includes the relatively sick. These results are in line with forward-looking behavior: Individuals who are already sick expect that they will hit the kink in the price schedule and thus are less sensitive to the co-payment.

The adverse selection problem in imperfectly competitive financial markets

We show that two horizontally differentiated banks can implement separating equilibria in markets for bank loans by using non-linear price schedules. The optimal strategies of the banks induce 'high-risk' borrowers to patronize their preferred, that is closer, bank. 'Low-risk' borrowers accept offers from their less preferred bank, which offers lower interest rates. The revelation mechanism is sensitive to the degree of competition between the banks, and credit rationing will be minimized at an intermediate level of competition. Copyright 2013 Oxford University Press 2012 All rights reserved, Oxford University Press.

Do Consumers Respond to Marginal or Average Price? Evidence from Nonlinear Electricity Pricing

Nonlinear pricing and taxation complicate economic decisions by creating multiple marginal prices for the same good. This paper provides a framework to uncover consumers’ perceived price of nonlinear price schedules. I exploit price variation at spatial discontinuities in electricity service areas, where households in the same city experience substantially different nonlinear pricing. Using household-level panel data from administrative records, I find strong evidence that consumers respond to average price rather than marginal or expected marginal price. This suboptimizing behavior makes nonlinear pricing unsuccessful in achieving its policy goal of energy conservation and critically changes the welfare implications of nonlinear pricing. (JEL D12, L11, L94, L98, Q41)

Estimating demand for cellular phone service under nonlinear pricing

Cellular phone carriers typically offer complicated nonlinear tariffs. Consumers make a discrete choice among several rate plans. Each plan has a nonlinear price schedule, and price is usually lower for in-network calls. I present an empirical framework to estimate demand under such nonlinear pricing schemes by using parsimonious carrier-level data and apply the estimation method to analyze the market in Taiwan. I evaluate the impacts of termination-based pricing schemes on the market structure by counterfactual simulations. There is no evidence showing that the network effect resulting from termination-based pricing has significant effects on market structure.

NON-LINEAR PRICING AND EQUALITY OF OPPORTUNITY

This paper investigates non-linear pricing schedules that are based on Roemer's equality of opportunity (EOp) criterion, and compare them with the maximin and the utilitarian non-linear pricing schedules. The main results suggest that the EOp policy offers a reasonable compromise between the maximin and the utilitarian policies in the sense that: (1) the consumption for each individual is highest under the utilitarian policy and lowest under the maximin policy, and (2) the individuals in the extreme positions rank the EOp policies between the maximin and the utilitarian policy in terms of utility level.