**Contents:**

Under the assumptions of utility maximization and preference independence additive preferences , mathematical relationships between income elasticity values and the uncompensated own and cross price elasticity of demand are here derived using the differential approach to demand analysis. Key parameters are: the elasticity of the marginal utility of income, and the average budget share.

The proposed method can be used to forecast the direct and indirect impact of price changes and of financial instruments of policy using available estimates of the income elasticity of demand. This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: All relevant data are within the paper and its Supporting Information files. The funder provided support in the form of salary for the author [LS], but did not have any additional role in the study design, data collection and analysis, decision to publish or preparation of the manuscript. The author received no additional funding, from any other source, for this work.

Part of the results contained in the manuscript are being used by GLOBMOD to develop a software application to support charity and commercial activities.

There are no further patents, products in development, or marketed products to declare. This does not alter the author's adherence to all the PLOS ONE policies on sharing data and materials, as detailed online in the guide for authors. A change in the price of a market good determines a change in the purchasing power of consumers income effect , and a change in the relative price of goods substitution effect.

The aggregate consumer responsiveness to changes in price and in income is measured using the own and cross price elasticity and the income elasticity of demand, respectively. Knowing the uncompensated own and cross price elasticity of demand is essential to anticipate the impact of price changes, and of financial instruments of policy such as subsidies, cost sharing schemes, and taxation, nonetheless forecasting it requires data that are not always readily available. Contingency studies, e.

Unlike the price elasticity, the income elasticity of demand can often be estimated from routinely collected data e. Nonetheless, it does not contain in and of itself enough information to infer the consequences of changing prices. The mathematical relationship between demand and price can be modeled using the neoclassical consumer theory, assuming a representative economic agent with preferences over consumption goods captured by a utility function [ 3 ].

The Rotterdam model, first proposed by Barten [ 4 ] and Theil [ 5 ], builds on this approach, allowing for the estimation of substitutes and complements, and the separability of preferences. The Rotterdam model produces constant marginal shares, a problem that can be avoided using a demand function called the almost ideal demand system AIDS model [ 6 ], which was subsequently extended using the differential approach of the Rotterdam model by Theil, Chung, and Seale [ 7 ][ 8 ].

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They added a non-linear substitution term to the basic linear function, which allows for separability and has fewer parameters to be estimated than in the AIDS model, creating the Florida model [ 7 ][ 8 ]. If separability holds, total expenditure can be partitioned into groups or bundles of goods, making it possible to analyze the preferences for one group independently of other groups.

In that case, the mathematical relationship between price and demand becomes amenable to analytical calculations. In the present study, mathematical relationships that allow the estimation of the uncompensated own price elasticity and of the cross price elasticity of demand for independent bundles of goods are obtained following the differential approach used to derive the Florida model.

The proposed equations require three inputs: the income elasticity of demand, the elasticity of the marginal utility of income, and the mean budget share allocated to the bundle of goods of interest.

The definitions used throughout this paper are reported in Table 1. The following assumptions are made:. To take into account the effect of parametric uncertainty on model estimates, credible intervals for the estimates of the price elasticity of demand are calculated via Monte-Carlo simulation, drawing random model parameter values from normal N and uniform U distributions: 3.

The proof of Eqs 1 and 2 uses Lagrange multipliers and differential equations, and is based on the fact that any change in the price of a good determines a change in the purchasing power of the consumer income effect , and a change in the relative price of goods substitution effect.

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The substitution effect depends on two elements: a the deflationary impact that a change in the price of a single good has on all market goods; and b the relative importance of different goods to the consumer. The proof follows three steps:. The following vector notation is used:. Assumption I implies that first and second order derivatives of the utility function u exist, and that the Hessian matrix of u is symmetric negative.

The first term on the right of Eq 8 is the real income term of demand, which results from the change in money income and the income effect of the price change. The second term on the right of Eq 8 is the substitution term. Under preference independence assumption III , the term containing the Frisch deflated price of good i is the only non-zero term in the substitution term. Therefore Eq 8 becomes: Differentiating the budget constraint Eq 4 yields: 11 where: 12 If prices are kept constant, the following equation holds: As an example, the derivation of Eq 1 for a hypothetical market, trading only in two independent bundles of goods, is outlined in the S1 Text.

The average budget share was drawn from a uniform distribution ranging from 0.

The sensitivity of predictions to model parameter values and especially to the value of the elasticity of the marginal utility of income increases with the income-elasticity of demand, and therefore so does the width of credible intervals associated with predictions. The cross price elasticity is negative, null or positive, depending on whether the income elasticity of B is smaller of, equal to, or larger of the absolute value of the elasticity of the marginal utility of income.

The average budget share is equal to 0. In this example, the income elasticity of the bundle of goods B is equal to 0. Eq 1 can be used to forecast the impact of a change in the own price of bundle A on the demand for A, while Eq 2 can be used to assess how the demand for a bundle A changes if the price of an independent bundle B changes.

As such, they provide a theoretical basis for estimating the potential impact of financial instruments of policy, e. The face validity of this approach was tested using publicly available data. Model predictions from Eq 1 were compared with published estimates of income and price elasticity for different bundles of goods and services that had been calculated fitting the Florida model to real world data collected in national surveys of consumption [ 8 ]. Specifically, the bundles considered were: clothing and footwear, education, healthcare, and recreation.

Average elasticity values for low-, middle-, and high-income countries are displayed in Fig 4. Country-specific data can be found in S2 Fig. The estimates colored circles in the three panels refer to low-income, middle-income, and high-income countries and were obtained fitting the Florida model to country survey data source: Seale JL, Regmi A, Bernstein J.

International evidence on food consumption patterns. Nevertheless, there are several assumptions and caveats that need careful consideration. The accuracy of model estimates may be limited by the assumption of instantaneous maximization of consumer utility, which in practice requires maximization of utility to be performed in a relatively short time.

The proposed approach relies on strong separability assumptions which are often not met in real markets. In addition, preferences exhibit non-satiation, i. Whilst these assumptions allow a useful simplification of the calculations involved, they might also be sources of bias in the results. Nonetheless, similar caveats do also apply to other models [ 7 ][ 8 ][ 12 ][ 13 ]. In fact, the mathematical framework here described builds on approaches already used in those studies, but with different objectives.

For instance, Barnett and Serletis [ 13 ] used the differential approach to demand analysis, and then implemented the Rotterdam parameterization to move from a model based on infinitesimal instantaneous changes to a discrete model, where prices, income, and demand change over finite time intervals days, months, years. Subsequently they fitted the discrete model to time series of prices and income, to calculate the parameter values of the demand system. Brown and Lee [ 12 ] followed an approach often used when modeling advertising effects in the Rotterdam model, and introduced preference variables in the utility function.

They then studied the effect of imposing restrictions on preference variables.

Income elasticity of demand

Nevertheless, to date no published study has investigated the relationship between income elasticity and price elasticity of demand. When compared with the results of studies that have concurrently estimated price and income elasticity of demand [ 8 ][ 14 ][ 15 ][ 16 ], the predictions of the model here presented appear consistent with the available data see Fig 4 , S1 and S2 Figs. In conclusion, based on theoretical considerations and on the available evidence, the estimates of price-elasticity of demand obtained using the analytical relationships here proposed are comparable with those generated using other models based on the differential approach to demand analysis.

If used to infer the price elasticity from available estimates of the income elasticity of demand, under conditions of additive preferences, the proposed model provides an effective shortcut to forecast the impact of price changes on consumption patterns. The red rectangles refer to specific empirical studies conducted in the US.

The white rectangles refer to global estimates, based on survey data, produced by the US Department of Agriculture Seale et al. This indicates the good is not a necessity like food, and would be considered a relative luxury for this individual. When YED is negative , the good is classified as inferior.

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The negative sign means that the good is inferior, and, because the coefficient is less than one, demand for the good does not respond significantly to a change in income. The sign and the number provide different information about the relationship between income and demand. Income elasticity of demand can also be illustrated by Engel curves. A firm can forecast the impact of a change in income on sales volume Q , and sales revenue P x Q.

It can now predict the impact of this change. The same producer sells a small car with a YED of - 5. Knowing YED helps the firm decide whether to raise or lower price following a change in consumer incomes. If incomes are falling and YED is positive, a reduction in price might help compensate for the reduction in demand. Firms can diversify and offer a range of goods with different YEDs to spread the risks associated with changes in the level of national income.

For example, a car manufacturer may produce cars with a range of YED values, so that sales are stabilised as the economy grows and declines. Changes in real national income tend to be cyclical.