Linear Utility Research Articles

Risk Aversion and a Calculus for Finitely Generated Piecewise Linear Functions: A Calculus that Economists Ought to Develop?

In this note we propose a calculus for piece-wise linear functions, in order to obtain derivatives and second derivatives at points where the function is not differentiable. Such derivatives can be used to calculate coefficients of risk aversion at initial wealth for piece-wise linear utility functions for gains, which display loss aversion-and hence non differentiability at zero gains.

Extending the Multiple Discrete Continuous (MDC) modelling framework to consider complementarity, substitution, and an unobserved budget

Many decisions can be represented as interrelated discrete and continuous choices, i.e. what and how much to choose from a set of finite alternatives (incidence and quantity of consumption). In the last twenty years, several models of Karush–Kuhn–Tucker demand systems have been developed and used to study these kinds of decisions. While strongly grounded in economic theory, most of these models have two limitations: they require specifying a budget, and usually omit any complementarity effects. In this paper, we propose two extensions to the Multiple Discrete Continuous (MDC) modelling framework: (i) an MDC model including explicit complementarity and substitution effects, and (ii) an MDC model with complementarity, substitution that requires no budget definition. Model (ii) relies on the hypothesis that total expenditure on the alternatives under consideration is small compared to the overall budget. This allows using a linear utility function for the numeraire good, leading to a likelihood function without the budget or numeraire good in it. The lack of a budget is specially useful when forecasting, as it avoids cascading errors due to an inaccurate budget specifications. The inclusion of complementarity and substitution effects enriches the interpretability of the models, while the resulting functional form avoids theoretical issues present in previous formulations. Alongside the derivation of the models, we discuss their main properties and propose an efficient forecasting algorithm for (ii). We also report four applications to datasets about time use, household expenditure, supermarket scanner data, and trip generation. Free estimation code for both models is made available online.

What residents of potential Olympic cities want: using conjoint analysis to deal with dominant and heterogeneous preferences

ABSTRACT Understanding residents’ preferences for mega-sports events is a hot topic in tourism and event research. Thus far, most studies have assumed homogeneous preferences, and linear utility additivity of all event attributes when measuring residents’ support. This study uses an adaptive choice-based conjoint analysis combined with hierarchical Bayesian estimation. In contrast to previous research, the method considers the entire spectrum of residents’ preferences at the individual level and identifies non-substitutable dominant preferences (must-haves and unacceptables). A survey with 687 residents was conducted in Tyrol, an Austrian state with a remarkable Olympic tradition. The findings of this study extend the current literature by showing that the usually applied assumptions (homogeneous and linear additive substitutable preferences) are violated with substantial consequences. A simulation allows transforming the survey results into hypothetical referendum support rates for different feasible concepts of the Olympic Games.

Correcting for discounting and loss aversion in composite time trade-off.

Time trade‐off utilities have been suggested to be biased upwards. This bias is a result of the method being applied assuming linear utility of life duration, which is violated when individuals discount future life years or are loss averse for health. Applying a “corrective approach”, that is, measuring individuals' discount function and loss aversion and correcting time trade‐off utilities for these individual characteristics, may reduce this bias in utilities. Earlier work has developed this approach for time trade‐off in a student sample. In this study, the corrective approach was extended to composite time trade‐off (cTTO) methodology, which enabled correcting utilities for health states worse than dead. In digital interviews a sample of 150 members of the general public completed cTTO tasks for six health states, and afterward they completed measurements of loss aversion and discounting. cTTO utilities were corrected using these measurements under multiple specifications. Respondents were also asked to reflect on and adjust their cTTO utilities directly. Our results show considerable loss aversion and both positive and negative discounting were prevalent. As predicted, correction generally resulted in lower utilities. This was in accordance with the direction of adjustments made by respondents themselves.

Mobility and sustainable transportation in higher education: evidence from Monterrey Metropolitan Area in Mexico

PurposeThis paper aims to analyze the demand for mobility in higher education to understand the critical elements of students' mobility and the potential impact of accessing sustainable alternatives. The demand for different means of transportation accounts for socioeconomic variables such as household income and size, gender, age, among other travel factors such as distance, time, speed and direct cost.Design/methodology/approachThis research uses a unique data set encompassing a representative sample of households across gender and municipalities in the Monterrey Metropolitan Area in Mexico. This paper uses the value of transportation time savings approach to estimate the derived demand for mobility in higher education across different means. The empirical strategy uses a linear utility framework in a multinomial non-ordered logit model to analyze the conditional selection of transportation as a function of sociodemographics, direct monetary costs and travel determinants such as travel time, distance and speed.FindingsMen students are more likely to use an unsustained transportation mode such as a car or cab than women. Income is a crucial determinant for transportation demand. There is a negative relationship between household income and walking or riding the bus but positive with driving a car. An increase in the time of the trip decreases the probability of walking while increasing the likelihood of using a car or riding the bus to school. Students who feel safe while riding a bus are more likely to use the bus or walk as transportation than using a car. Finally, having access to better quality sidewalks significantly incentive walking over other transportation options.Practical implicationsResults show that individuals would almost always choose private vehicles because of the personal comforts and time savings.Social implicationsUnderstanding the demand for higher education and access to sustainable transportation might provide new insights into campus planning and development. Access to public transportation options can reduce the travel costs for students with lower incomes and enlarge options for students who have to commute longer distances by car. Sustainable transportation access plays a role in promoting equality and economic development.Originality/valueThis study’s results bring new light on how transportation becomes an essential component of the opportunity cost of college education and could define its role in promoting equality or increasing inequality. This study’s finding is of particular interest for cities where housing location seems to be independent of access to school, work or other amenities.

A Strongly Polynomial Algorithm for Linear Exchange Markets

We present a strongly polynomial algorithm for computing an equilibrium in Arrow-Debreu exchange markets with linear utilities. Our algorithm is based on a variant of the weakly polynomial Duan–Mehlhorn (DM) algorithm. We use the DM algorithm as a subroutine to identify revealed edges—that is, pairs of agents and goods that must correspond to the best bang-per-buck transactions in every equilibrium solution. Every time a new revealed edge is found, we use another subroutine that decides if there is an optimal solution using the current set of revealed edges or, if none exists, finds the solution that approximately minimizes the violation of the demand and supply constraints. This task can be reduced to solving a linear program (LP). Even though we are unable to solve this LP in strongly polynomial time, we show that it can be approximated by a simpler LP with two variables per inequality that is solvable in strongly polynomial time. Funding: Financial support from the Division of Computing and Communication Foundations, National Science Foundation (NSF) [Grants 1755619 and 1942321] and from European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme [Grant ScaleOpt-757481] is gratefully acknowledged.

Solving some stochastic partial differential equations driven by Lévy noise using two SDEs*

ABSTRACT The method of characteristics is a powerful tool to solve some nonlinear second-order stochastic PDEs like those satisfied by a consistent dynamic utilities, see [N. El Karoui and M. Mrad, An exact connection between two solvable SDEs and a non linear utility stochastic PDEs, SIAM J. Financ. Math. 4(1) (2013), pp. 737–783; A. Matoussi and M. Mrad, Dynamic utility and related nonlinear SPDE driven by Lévy noise, preprint (2020), submitted for publication. Available at https://hal.archives-ouvertes.fr/hal-03025475]. In this situation the solution is theoretically of the form where and are solutions of a system of two SDEs, is the inverse flow of and is the initial condition. Unfortunately this representation is not explicit except in simple cases where and are solutions of linear equations. The objective of this work is to take advantage of this representation to establish a numerical scheme approximating the solution V using Euler approximations and of X and ξ. This allows us to avoid a complicated discretization in time and space of the SPDE for which it seems really difficult to obtain error estimates. We place ourselves in the framework of SDEs driven by Lévy noise and we establish at first a strong convergence result, in -norms, of the compound approximation to the compound variable , in terms of the approximations of X and Y which are solutions of two SDEs with jumps. We then apply this result to Utility-SPDEs of HJB type after inverting monotonic stochastic flows.

A comparison of various aggregation functions in multi-criteria decision analysis for drug benefit-risk assessment.

Multi-criteria decision analysis is a quantitative approach to the drug benefit–risk assessment which allows for consistent comparisons by summarising all benefits and risks in a single score. The multi-criteria decision analysis consists of several components, one of which is the utility (or loss) score function that defines how benefits and risks are aggregated into a single quantity. While a linear utility score is one of the most widely used approach in benefit–risk assessment, it is recognised that it can result in counter-intuitive decisions, for example, recommending a treatment with extremely low benefits or high risks. To overcome this problem, alternative approaches to the scores construction, namely, product, multi-linear and Scale Loss Score models, were suggested. However, to date, the majority of arguments concerning the differences implied by these models are heuristic. In this work, we consider four models to calculate the aggregated utility/loss scores and compared their performance in an extensive simulation study over many different scenarios, and in a case study. It is found that the product and Scale Loss Score models provide more intuitive treatment recommendation decisions in the majority of scenarios compared to the linear and multi-linear models, and are more robust to the correlation in the criteria.

A multiple discrete continuous extreme value choice (MDCEV) model with a linear utility profile for the outside good recognizing positive consumption constraints

A variant of the traditional multiple discrete-continuous extreme value (MDCEV) model that obviates the need to have budget information, labeled as the Lγ-profile MDCEV model, has been proposed recently. This new model structure breaks the strong linkage between the discrete and continuous choice dimensions of decision-making. But recent studies show that this Lγ-profile model may not work well in situations when, even if the budget is unobserved, the budget is known to be finite and small in magnitude. The reason is that the formulation, while ensuring the positivity of consumptions of the inside goods (that may or may not be consumed), does not guarantee, within the model formulation and estimation itself, the positivity of the consumption of the essential outside good. In this paper, we develop a formulation based on a reverse Gumbel structure for the stochastic terms in the utility functions of alternatives that develops a closed-form probability expression, while also accommodating the positivity requirement for the outside good. The ability of our proposed Budget-based Reverse Generalized Lγ-profile model (labeled the BR-GLγ-profile model) to recover true underlying model parameters is assessed. Our results clearly point to the benefit of employing the proposed model (relative to extant linear outside utility profile models in the literature) in empirical contexts when there is reason to believe that a finite ceiling applies to the budget (even if the budget is unobserved) or if the budget is actually available. In the latter case when the budget is available, our proposed model is a serious contender to the traditional γ-profile-MDCEV model and will generally outperform the traditional γ-profile-MDCEV when the consumption share of the outside good is high.

On two-player games with pure strategies on intervals $ [a, \; b] $ and comparisons with the two-player, two-strategy matrix case

<p style='text-indent:20px;'>We consider games of two-players with utility functions which are not necessarily linear on the product of convex and compact intervals of <inline-formula><tex-math id="M2">\begin{document}$ \mathcal{R}^2 $\end{document}</tex-math></inline-formula>. An issue is how far an analogy can be drawn with two-player, two-strategy matrix games with linear utility functions, where [0, 1] registers probabilities and equilibria are at the intersection of reaction functions. Now, the idea of <inline-formula><tex-math id="M3">\begin{document}$ \delta $\end{document}</tex-math></inline-formula> functions is exploited to construct mixed strategies to look for Nash equilibria (NE). "Reaction" functions are constructed and results are obtained graphically. They are related to topological theorems on NE. The games chosen make specific points in relation to existence conditions and properties of solutions. It is a distinguishing feature that an interval [a, b] now registers both pure and mixed strategies. For NE a choice has to be justified. Also "reaction" functions are more complicated and their intersection does not guarantee an equilibrium.</p>

General Deep Multinomial Logit Model

Multinomial logit model (MNL) is by far the most widely used discrete choice model that is widely used to explain or predict a choice from a set of two or more discrete alternatives. MNL operates within a framework of the random utility model (RUM) in which the utility of an alternative perceived by an individual consists of two components: systematic component and random component. The systematic component is usually defined as a linear function. However, practical decision processes involve complex considerations regarding various aspects of the alternatives and individual which cannot be adequately represented by simple linear models. To overcome the weakness of linear utility model and improve the performance of MNL, in this paper, we propose a general deep multinomial logit model (GDMNL) that takes advantage of both traditional MNL and deep learning. In this model, deep neural networks are applied to extend MNL by learning different nonlinear utility functions of various alternatives. The empirical study in the domain of transit route choice analysis demonstrates the validity and superiority of the proposed model.

Multiple discrete-continuous choice models with additively separable utility functions and linear utility on outside good: Model properties and characterization of demand functions

In this paper, we enhance the current understanding of the properties of multiple discrete-continuous (MDC) choice models with additively separable (AS), independent and identically distributed (IID) utility functions, and linear utility form on the essential outside good. First, we highlight that the prior implementations of this model in the literature ignore primal feasibility conditions related to the budget constraint and the essential nature of outside good in formulating the model likelihood function. Second, we evaluate the suitability and performance of the model for alternative consumption patterns relative to the budget. In addition, we provide a systematic comparison of the performance of MDC choice models with the linear outside good utility form (i.e., the Lγ-profile model) and those with the non-linear outside good utility form (i.e., the NLγ-profile model). Third, for the Lγ-profile model with infinite budgets (i.e., when a very small proportion of the budget is allocated to inside goods), we derive the distributions of the resulting optimal demand functions and analytic expressions for the corresponding first and second moments, and identify a property that makes it easy to estimate the utility function parameters of an inside alternative even when consumption data is not available for other alternatives. In addition, perhaps for the first time in the literature, we show how an independent system of Tobit models can be derived as a restricted version of the utility-theoretic Lγ-profile MDC model structure. Finally, we apply the model for an empirical analysis of expenditure patterns of leisure trips from a domestic tourism survey sample of households in India.

Exchange markets

The exchange market is a basic model of an economy, where agents bring resources that they own to the market in order to exchange them for other goods that they need. There is a rich literature on the equilibrium properties of such markets starting with the work of Arrow and Debreu. In this note we survey recent results on proportional response dynamics in exchange markets with linear utilities and suggest several directions for future work.

Funding public projects: A case for the Nash product rule

We study a mechanism design problem where a community of agents wishes to fund public projects via voluntary monetary contributions by the community members. This serves as a model for public expenditure without an exogenously available budget, such as participatory budgeting or voluntary tax programs, as well as donor coordination when interpreting charities as public projects and donations as contributions. Our aim is to identify a mutually beneficial distribution of the individual contributions. In the preference aggregation problem that we study, agents with linear utility functions over projects report the amount of their contribution, and the mechanism determines a socially optimal distribution of the money. We identify a specific mechanism—the Nash product rule—which picks the distribution that maximizes the product of the agents’ utilities. This rule is Pareto efficient and incentivizes agents to contribute their entire budget while spending each agent’s contribution only on projects the agent finds acceptable.

Determination of Break-Even Point through Consumer Preference Relation

Social choice theory beliefs about how the consumers function to chose their interested goods and services. Preference relation with affine indifference curves that has a concave representation has a linear utility representation. This study asks how individual preference relations might be combined to give a single ordering which captures the overall wishes of the group of individuals. There are certain properties that one would like such a utility rule, utility have thus become a more abstract concept that is not necessarily solely based on the satisfaction or pleasure received. Concept of cardinal utility is studied in three different situations Debreu (1958) gave quite different approach. This study maintains link between mathematical theory and financial concept to determine break-even point through the consumers’ preference relation.

Ergodicity-breaking reveals time optimal decision making in humans.

Ergodicity describes an equivalence between the expectation value and the time average of observables. Applied to human behaviour, ergodic theories of decision-making reveal how individuals should tolerate risk in different environments. To optimize wealth over time, agents should adapt their utility function according to the dynamical setting they face. Linear utility is optimal for additive dynamics, whereas logarithmic utility is optimal for multiplicative dynamics. Whether humans approximate time optimal behavior across different dynamics is unknown. Here we compare the effects of additive versus multiplicative gamble dynamics on risky choice. We show that utility functions are modulated by gamble dynamics in ways not explained by prevailing decision theories. Instead, as predicted by time optimality, risk aversion increases under multiplicative dynamics, distributing close to the values that maximize the time average growth of in-game wealth. We suggest that our findings motivate a need for explicitly grounding theories of decision-making on ergodic considerations.

Input-dependent noise can explain magnitude-sensitivity in optimal value-based decision-making

AbstractRecent work has derived the optimal policy for two-alternative value-based decisions, in which decision-makers compare the subjective expected reward of two alternatives. Under specific task assumptions — such as linear utility, linear cost of time and constant processing noise — the optimal policy is implemented by a diffusion process in which parallel decision thresholds collapse over time as a function of prior knowledge about average reward across trials. This policy predicts that the decision dynamics of each trial are dominated by the difference in value between alternatives and are insensitive to the magnitude of the alternatives (i.e., their summed values). This prediction clashes with empirical evidence showing magnitude-sensitivity even in the case of equal alternatives, and with ecologically plausible accounts of decision making. Previous work has shown that relaxing assumptions about linear utility or linear time cost can give rise to optimal magnitude-sensitive policies. Here we question the assumption of constant processing noise, in favour of input-dependent noise. The neurally plausible assumption of input-dependent noise during evidence accumulation has received strong support from previous experimental and modelling work. We show that including input-dependent noise in the evidence accumulation process results in a magnitude-sensitive optimal policy for value-based decision-making, even in the case of a linear utility function and a linear cost of time, for both single (i.e., isolated) choices and sequences of choices in which decision-makers maximise reward rate. Compared to explanations that rely on non-linear utility functions and/or non-linear cost of time, our proposed account of magnitude-sensitive optimal decision-making provides a parsimonious explanation that bridges the gap between various task assumptions and between various types of decision making.

Multiple discrete-continuous choice models with bounds on consumptions

This paper derives a multiple discrete–continuous (MDC) choice model formulation with constraints that specify upper bounds on consumption. To do so, considering the conventional utility maximization problem of a consumer, the Karush-Kuhn-Tucker (KKT) conditions are laid out for the MDC model with a general set of linear constraints that include inequalities. Subsequently, we derive a model with constraints that accommodate upper bounds on consumptions and an additive utility structure that accommodates lower bounds on consumptions. The likelihood expression for the proposed model takes a closed form. Furthermore, we extend the formulation to impose bounds on an MDC choice model with activity episode-level choice alternatives that accommodates a logical ordering among different episodes of an activity. The proposed models are derived for two different specifications of the outside good utility – (1) nonlinear utility with respect to consumption and (2) linear utility with respect to consumption. The proposed models are applied to an empirical context to analyze activity-level as well as episode-level activity participation and time allocation while considering bounds on time allocations. Empirical results suggest that the models that consider upper bounds on consumption offer a better fit to data, avoid predictions of unrealistically large time allocations, and result in overall better predictions than those from models without bounds. The proposed models are useful in situations, such as microsimulation models of travel demand, where it is crucial to avoid unrealistically large predictions.

Infinite-Dimensional Fisher Markets: Equilibrium, Duality and Optimization

This paper considers a linear Fisher market with n buyers and a continuum of items. In order to compute market equilibria, we introduce (infinite-dimensional) convex programs over Banach spaces, thereby generalizing the Eisenberg-Gale convex program and its dual. Regarding the new convex programs, we establish existence of optimal solutions, KKT conditions, as well as strong duality. All these properties are established via non-standard arguments, which circumvent the limitations of duality theory in optimization over infinite-dimensional vector spaces. Furthermore, we show that there exists a pure equilibrium allocation, i.e., a division of the item space. Similar to the finite-dimensional case, a market equilibrium under the infinite-dimensional Fisher market is Pareto optimal, envy-free and proportional. We also show how to obtain the (a.e. unique) equilibrium prices and a pure equilibrium allocation from the (unique) equilibrium utility prices. When the item space is the unit interval [0,1] and buyers have piecewise linear utilities, we show that approximate equilibrium prices can be computed in polynomial time. This is achieved by solving a finite-dimensional convex program using the ellipsoid method. To this end, we give nontrivial and efficient subgradient and separation oracles. For general buyer valuations, we propose computing market equilibrium using stochastic dual averaging, which finds approximate equilibrium prices with high probability.

Rational Bubbles and the S&P 500. An empirical approach

We analyze if the dynamics of the S&P500 resemble those of a rational bubble. We find positive evidence in this question by applying the Kalman Filter to a suitable asset pricing model proposed and our conclusion is robust to three different stochastic discount factors SDFs considered: Linear Utility, Log Utility and CRRA utility. We also find evidence of a relationship between the type of SDF and the size of a bubble in the S&P500 case.