Slutsky Matrix Research Articles

A New Look at the Symmetry of the Slutsky Matrix

A New Look at the Symmetry of the Slutsky Matrix

Non-smooth integrability theory

We study a method for calculating the utility function from a candidate of a demand function that is not differentiable, but is locally Lipschitz. Using this method, we obtain two new necessary and sufficient conditions for a candidate of a demand function to be a demand function. The first concerns the Slutsky matrix, and the second is the existence of a concave solution to a partial differential equation. Moreover, we show that the upper semi-continuous weak order that corresponds to the demand function is unique, and that this weak order is represented by our calculated utility function. We provide applications of these results to econometric theory. First, we show that, under several requirements, if a sequence of demand functions converges to some function with respect to the metric of compact convergence, then the limit is also a demand function. Second, the space of demand functions that have uniform Lipschitz constants on any compact set is compact under the above metric. Third, the mapping from a demand function to the calculated utility function becomes continuous. We also show a similar result on the topology of pointwise convergence.

Random choice and market demand

AbstractThis paper examines a statistical model of choice in which individual decisions have no preferential basis or optimizing behaviour. Individual consumption is chosen at random, while respecting a linear budget set. All the properties of classical demand theory, including symmetry and negative (semi)definiteness of the Slutsky matrix, are shown to be generally satisfied by mean (i.e., market) demands. Despite the fact that individuals are “irrational,” mean demands can be rationalized as the outcome of utility‐maximizing behaviour, even when demands are interior. The severity and frequency of revealed preference violations are unable to reject economic rationality in irrational individuals.

The energy rebound effect in households: Evidence from urban and rural areas in Beijing

With Beijing facing greater pressure for energy conservation, improving energy efficiency becomes more necessary. However, the energy rebound effect may weaken this energy-saving effect. Using the LA-AIDS (Linear Approximation of the Almost Ideal Demand System) model and simulation analysis, this paper evaluates the energy rebound effects of different commodity groups in Beijing’s urban and rural households during 2000–2019. Then, the Slutsky decomposition equation is introduced to decompose the energy rebound caused by the substitution and income effects. Results suggest that: (1) The energy consumption of urban and rural residents shows a partial rebound in most years, and the energy rebound effect of rural residents exceeded that of urban residents after 2010. (2) The income effect from commodity price changes is the main source of the energy rebound effect both in urban and rural households. (3) The commodity groups Residence, Household facilities & Services, Transportation & Communication and Health care & Medical services have the same influencing direction on the energy rebound effect in urban and rural residents, while the commodity groups Food, Tobacco & Alcohol, Clothing and Culture, Education & Entertainment have different influencing direction. Finally, suggestions are provided for alleviating the energy rebound effect of households in Beijing.

Bounded rationality and animal spirits: a fluctuation-response approach to Slutsky matrices

The Slutsky equation, central in consumer choice theory, is derived from the usual hypotheses underlying most standard models in Economics, such as full rationality, homogeneity, and absence of interactions. We present a statistical physics framework that allows us to relax such assumptions. We first derive a general fluctuation-response formula that relates the Slutsky matrix to spontaneous fluctuations of consumption rather than to response to changing prices and budget. We then show that, within our hypotheses, the symmetry of the Slutsky matrix remains valid even when agents are only boundedly rational but non-interacting. We finally propose an ‘animal spirit’ model where agents are influenced by the choice of others, leading to a phase transition beyond which consumption is dominated by herding (or ‘fashion’) effects. In this case, the individual Slutsky matrix is no longer symmetric, even for fully rational agents. The vicinity of the transition features a peak in asymmetry.

Social welfare and the unrepresentative representative consumer

AbstractIf, for all prices, income distribution is optimal for a planner with a social welfare function, then aggregate demand is the same as that of a single “representative consumer” whose preferences over aggregate consumption are the same as the planner's. This paper shows that the converse is false. Aggregate demand may be the demand function of a representative consumer although the income distribution is not optimal for any social welfare function. The representative consumer may be Pareto inconsistent, preferring situation A to B when all the actual consumers prefer B to A. We give conditions under which existence of a representative consumer implies that the income distribution satisfies first order conditions for optimality. Satisfying the first order optimality conditions for an additively separable social welfare function is essentially equivalent to aggregate demand for every pair of consumers having a symmetric Slutsky matrix.

Estimating Parental Demand for Children’s Screen Time in a Model of Family Labor Supply

In a novel approach to model the demand for the children's screen time as the result of a parent's optimal labor-leisure choice, the study used a simple model of parental utility maximization subject to the money and time budget constraints to derive Marshallian parental demand functions for two types of child upbringing activities: time-intensive (violin lesson) and time-saving (video games). After the Slutsky decomposition, parental demand for children's screen time was shown to be similar to a Giffen good. Using the National Longitudinal Survey of Youth 1979 and Adolescent Brain Cognitive Development data, the wage equation was first estimated based on Heckman’s two-step correction procedure. Then, the total effect of an increase in wage rate on the parental demand for screen time was empirically decomposed into the substitution effect and the income effect. The study findings indicate that the substitution effect is positive, the income effect is negative, and the negative income effect dominates the substitution effect. We add to the existing literature by showing that the empirical findings in the public health and psychology literature can be reconciled with the theoretical predictions of the standard economic labor-leisure trade-off paradigm.

THE SLUTSKY EQUATION AND ITS PRACTICAL SIGNIFICANCE THROUGH THE MAPLE SOFTWARE PRODUCT

In this paper, we will consider the Slutsky equation for the function of demand for goods depending on the prices of these goods and the consumer’s income, taking into account compensation. The practical application of this one-factor model for calculating the demand of two interchangeable goods and the economic interpretation of the results obtained are also given. The work will be useful to those who wish to familiarize themselves with the Slutsky model for demand and consider its use in practice.

Slutsky equation in tourism: an empirical study in Vietnam

ABSTRACT The purpose of the paper is to apply the Slutsky equation to the Bass diffusion model to examine the price and income elasticities of hotel demand in different tourist destinations including cultural, commercial, and coastal cities. The sample was a set of 120 points by ex-post data from the Smith Travel Research in Vietnam. Findings indicate the price in a leisure destination in the beginning of the tourist area life cycle is inelastic, whereas the prices in the business and culture destinations at the end of the cycle are elastic. This study identifies the hidden cost when demand increases in the three destinations, ultimately allowing hoteliers to effectively strategize when making price changes within their industry.

Marginal compensated effects in discrete labor supply models

This paper develops analytic results for marginal compensated effects in discrete labor supply models, including a Slutsky equation. The Slutsky equation is aggregate in the sense that it establishes the relationship between the marginal compensated effects of the probability of working and the mean hours of work in terms of the corresponding marginal uncompensated effects. The Slutsky equation differs somewhat from the Slutsky equation in the standard continuous labor supply models. Specifically, the marginal compensated effect of an increase in the wage rate differs from the corresponding effect of a decrease in the wage rate.To illustrate some qualitative properties of the compensated marginal effects we have used an empirical labor supply model to compute numerical compensated (Hicksian) and uncompensated marginal (Marshallian) effects resulting from wage rate changes.

Income and Substitution Effects: A Rejoinder to Professor Joseph Salerno

Professor Joseph Salerno (2019) has commented on my recent reconstruction of the income effect from a causal-realist perspective (Israel, 2018b). In this rejoinder, I clarify my position and show that the main points of criticism in Salerno’s response are unfounded. In particular, I show that my argument does not involve a claim of greater “realism of assumptions” and it by no means contradicts the law of demand. Moreover, I work out in more detail the similarities and differences of my approach to the standard neoclassical decomposition of income and substitution effects. I show that my approach is closer to the Slutsky decomposition as opposed to the Hicks decomposition.

The Becker Fertility Mode: Theory and Critique

This paper is an exploration of the theoretical properties of the Becker fertility model. I demonstrate that the comparative statics of the Becker fertility model with a general budget constraint and its corresponding expenditure model can be expressed in terms of the ordinary consumer expenditure function with a change of variable. When there are no fixed-costs of quality per child, the Becker fertility model is equivalent to the ordinary consumer model with restrictions on the form of the utility function. Solutions to the Becker fertility model are provided with the Cobb-Douglas, CES, and AIDS specifications under this assumption, Becker’s hypothesis that demand for quality per child increases with income is not valid under any Cobb-Douglas or CES specification, but can be tested with a valid estimation of the AIDS model. I also evaluate the role of fixed-costs of quality per child in Becker’s model and show that they introduce a third term into the Slutsky matrix, i.e. in addition to ordinary substitution and income effects, but that these effects are small when the average fixed costs of quality per child is small relative to the marginal cost of quality per child. When the Becker model is expressed in terms of children, quality, and other goods, children must be a substitute for either their quality or for other goods and either quality or other goods must be a luxury good.

Hicksian complementarity and perturbed utility models

This paper studies aggregate complementarity without price or income variation. We show that for a class of utility functions, variation in non-price observables allows one to recover a measure of complementarity similar to Hicksian complementarity. In addition, the entire Slutsky matrix can be recovered up to scale without price variation. We then examine aggregate complementarity in latent utility models used in discrete choice, bundles, and matching. We show that classical linear instrumental variables can recover Hicksian complementarity for the special case of quadratic utility.

Marginal Compensated Effects in Discrete Labor Supply Models

This paper develops analytic results for marginal compensated effects of discrete labor supply models, including Slutsky equations. It matters, when evaluating marginal compensated effects in discrete choice labor supply models, whether one considers wage increase (right marginal effects) or wage decrease (left marginal effects). We show how the results obtained can be used to calculate the marginal cost of public funds in the context of discrete labor supply models. Subsequently, we use the empirical labor supply model of Dagsvik and Strom (2006) to compute numerical compensated (Hicksian) and uncompensated marginal (Marshallian) effects resulting from wage changes. The mean Hicksian labor supply elasticities are larger than the Marshallian, but the difference is small.

Classifying bounded rationality in limited data sets: a Slutsky matrix approach

Given any observed finite sequence of prices, wealth, and demand choices, we propose a way to measure and classify the departures from rationality in a systematic fashion, by connecting violations of the underlying Slutsky matrix properties to the length of revealed demand cycles. The approach complements our previous study (Aguiar and Serrano in J Econ Theory 172:163–201, 2017), which is based on the entire demand function. The methodology can be easily applied in experimental demand data sets.

Rebound effect of transportation considering additional capital costs and input-output relationships: The role of subsistence consumption and unmet demand

The anticipated carbon mitigation through energy-saving technology progress might be partially or even totally offset due to rebound effect. A better understanding of rebound effect, especially the underling mechanisms, has important implications for energy efficiency policies to achieve the goal of sustainability. Previous studies usually ignore additional capital costs for adopting energy-efficient equipment and indirect carbon emissions for providing new consumption bundles. Also, the precise mechanisms of how living standard affects rebound effect are unknown. Using almost ideal demand system (AIDS) model and input-output relationships among sectors, this paper builds a method to estimate the rebound effect that allows for additional capital costs and carbon emission changes of all other goods and services due to backward-forward linkages. We find that rebound effect is over 100% for transportation in rural China even though additional capital costs would indeed reduce the magnitude, and the results are quite robust to alternative settings. As for the mechanisms, rebound effect is divided into substitution and income channels by Slutsky decomposition. Here we show that both of these two channels are dependent on living standard because of subsistence consumption and unmet demand. Carbon emissions of poor households will rebound more through income channel of unmet demand, while rich households usually do not restricted by subsistence consumption, and thus through substitution channel, rebound of carbon emissions would be larger for rich households. The opposite directions of mechanisms dig more information about the nonlinearity of rebound effect to income levels. Furthermore, given the estimating results, this paper gains several insights in terms of welfare benefits, rather than just focuses on carbon reduction.

Surplus measures and luenberger Hicks–Moorsteen productivity indicator

This paper extends to production theory some notions of compensated and equivalent variation analyzed in Luenberger (Econ Theory 7:445–462, 1996) in a consumer context. Along this line the Luenberger–Hicks–Moorsteen productivity indicator introduced in Briec and Kerstens (Econ Theory 23:925–939, 2004) is derived from these concepts for multi-output production technologies. The dual properties of this productivity indicator are analyzed and an aggregate indicator is introduced inspired from the resource function proposed in Luenberger (1996). A connection to a suitable Slutsky matrix is established.

Decomposing the Slutsky Decomposition for the First Time in a Century

<p>The Slutsky decomposition is a mathematical formula which has been used for a very long time in economics to analyze how the demand for a good changes when its price goes up. Such a change in the demand is called the price effect. Most people will expect a decrease in the demand in response to a rise in the price. In other words, they will have a downward sloping demand curve in mind. Indeed they are right in usual cases.In terms of economicsit is said that the price effect is usually negative. But why? The Slutsky decomposition gives a correct answer to this question by decomposing the price effect into the substitution effect and the income effect. It is already known that the substitution effect is always negative, while the income effect is also negative if a good under considerationis “normal.”Since the sum of the two effects equals the price effect, it can be concluded that a demand for the good decreases when its price goes up, or the demand curve is downward sloping, in a “normal” situation. The Slutsky decomposition is so elegant and powerful that there would not be any economist who studies consumer behavior without mentioning it. Then, is there other decomposition than the Slutsky? This paperintroduces a new one which decomposes the price effect into the unit-elasticity effect and the ratio effect. The unit-elasticity effect means by how much the demand decreases in response to a rise in the price with the expenditure on it as fixed. The ratio effect means by how much the demand changes due to a change in the ratio of the expenditure on it to total income. The former effect is always negative, but the latter effect may be positive even in a “normal” situation. It is shown that anew decomposition is obtained by decomposing the Slutsky decomposition.</p>

Decomposing the Slutsky Decomposition for the First Time in a Century

<p>The Slutsky decomposition is a mathematical formula which has been used for a very long time in economics to analyze how the demand for a good changes when its price goes up. Such a change in the demand is called the price effect. Most people will expect a decrease in the demand in response to a rise in the price. In other words, they will have a downward sloping demand curve in mind. Indeed they are right in usual cases.In terms of economicsit is said that the price effect is usually negative. But why? The Slutsky decomposition gives a correct answer to this question by decomposing the price effect into the substitution effect and the income effect. It is already known that the substitution effect is always negative, while the income effect is also negative if a good under considerationis “normal.”Since the sum of the two effects equals the price effect, it can be concluded that a demand for the good decreases when its price goes up, or the demand curve is downward sloping, in a “normal” situation. The Slutsky decomposition is so elegant and powerful that there would not be any economist who studies consumer behavior without mentioning it. Then, is there other decomposition than the Slutsky? This paperintroduces a new one which decomposes the price effect into the unit-elasticity effect and the ratio effect. The unit-elasticity effect means by how much the demand decreases in response to a rise in the price with the expenditure on it as fixed. The ratio effect means by how much the demand changes due to a change in the ratio of the expenditure on it to total income. The former effect is always negative, but the latter effect may be positive even in a “normal” situation. It is shown that anew decomposition is obtained by decomposing the Slutsky decomposition.</p>

Constrained vs unconstrained labor supply: the economics of dual job holding

This paper develops a unified model of dual and unitary job holding based on a Stone-Geary utility function. The model incorporates both constrained and unconstrained labor supply. Panel data methods are adapted to accommodate unobserved heterogeneity and multinomial selection into six mutually exclusive labor supply regimes. We estimate the wage and income elasticities arising from selection and unobserved heterogeneity as well as from the Stone-Geary Slutsky equations. The labor supply model is estimated with data from the British Household Panel Survey 1991–2008. Among dual job holders, our study finds that the Stone-Geary income and wage elasticities are much larger for labor supply to the second job compared with the main job. When the effects of selection and unobserved heterogeneity are taken account of, the magnitudes of these elasticities on the second job tend to be significantly reduced.