Compatible with economic theory

The quest for the development of a demand equation that is compatible with economic theory lead to the development of almost idea demand model. This model developed by Deaton and muebullear . Prior to the development of the AIDS model, economics scholars such as stone, barten and theil have been able to model the demand for individual commodity ; running a separate demand equation for each of the of the commodity. That is, the earlier equations only consider only a fraction/ proportion of the total expenditure devoted to that particular commodity. According to (.......), even though the modelling separate demand equation for each commodity was considered to be flexible and of immense advantage, but from the theoretical point of view, it only allows the homogeneity restriction to be imposed. The adding up restriction irrelevant since only a fraction of the outlay was considered in the demand equation and the symmetry assumption can also be tested, the reason being that consumption of other food items are not included. It does not allow for the imposition of other basic characteristics of demand equation.

Demand concepts

Consumer demand can be defined as the quantity of commodity that an individual consumer is willing and able to purchase at a various possible prices over a period of time. This amount of commodity depends on price of the commodity, price of other commodities, consumer's money income and taste (shcam series). It then follows that the concept of consumer demand relates to how the quantity of a commodity that consumer would buy would changes as the price of that particular commodity price change, price of other commodity and his/her money income changes. Varying the own price of the commodity while keeping other factors constant produces the individual demand schedule and the graphical representation of the demand schedule gives us the consumer's demand curve.

In mathematical form, It can be expressed as follows.

Q = f(p, y, p*,z)

Where, Q = quantity of a commodity demanded, p =own price of the commodity, p* = price of other commodities and z = other factors (time, population, seasons, taste, etc)

The law of demand suggests an inverse relationship between the quantities of a commodity demanded and its own price, ceteris paribus. This implies that as price of the commodity falls, the quantity demanded increases and as price increases, the quantity demanded will fall with other factors remaining constant. The implication of this is that the slope of demand curve is negative.

Also demand theory suggests that prices of other commodities are crucial factors influencing demand. Other commodities can either substitute or complement the main commodity of concerns. This is in principle determined by the sign of the price variable of these other commodities in relation to the quantity of the main commodity. This regarded as the cross price-effect. If the price of the other commodity is negative, then, the other commodity is regarded as substitute to the main commodity but if it is positive, is a complement. Commodity that is not related to the main commodity is said to have a zero price effect.

The extent to which the commodity respondent to changes in own price or changes in price of other commodities is explained by the concept of elasticity of demand. The degree of responsiveness of quantity demanded of a commodity to its own price is the own price elasticity while the degree of responsiveness to the price of other commodities is referred to as cross- price elasticity. This value often assumes a negative value but in order to avoid working with negative value, a minus sign is often introduced into the formula for computing elasticity (shcuam series, page 39). Elasticity value can be greater than, less than or equal to unity. The value of the price elasticity has implication of the demand schedule and consequently the demand curve facing an individual consumer.

If the elasticity is less value is greater than zero but less that one, the demand is inelastic; meaning that the consumer responds to price change by buying commodity lower in proportion compared to the accompanied price change. In case of unity elasticity, the consumer responds to price change by purchasing goods in the same proportion compared to the accompanied price change. When the elasticity is greater than one, consumer responds to price change by purchasing goods in higher proportion relative to the accompanied price change.

Apart from price playing a significant role in consumer demand, we know that demand is only effective if the consumer is able to back up his/her willingness by purchasing power (money income). Economic theory suggests a direct relationship between the consumer's income and the quantity demanded. The type of relationship, either negative or positive provides an initial insight into the nature of the commodity, relative to the income level of the consumer. A negative relationship suggests that the commodity is an inferior commodity while a positive sign denote normal goods. The income elasticity of demand which is an indication of the degree of responsiveness of quantity of commodity demanded to change in income provides valuable information as to whether the income elastic or not. It also tells us whether a commodity is a necessity or a luxury. Acording to ................), a commodity can be a luxury at low level of income, necessity at intermediate level and can also become an inferior commodity at high level of income. The same commodity may be perceived differently as his or her income status changes.

Utility and Demand Analysis

Economists are interested in how consumers make choices among alternative goods and services within available resources in order to maximize utility. Utility here is defined as the satisfaction the consumer derives from allocation of income among several alternative products. Given that consumers who make choices between different commodities are also faced with several commodity prices consumer, the question facing the consumer now is how to allocate his/her resources (income) among different available products. The concept of scarcity suggests that consumers are always income constrained while trying to allocate resources to satisfy their wants. This therefore, make consumers to want to obtain maximum utility within the limit imposed by his/her income.

The subject of demand analysis premise therefore, on how consumer would maximize utility subject to a given level of income. This is also referred to as outlay or budget constraint.

This can be expressed as:

Maximize u = v (q) where q = q1, q2,q3 ....qn

Subject to ?pkqk = x

Where u is a utility function of the quantities of goods consumed, x is total outlay, and p and q are vector of commodity prices and quantities, respectively. Solving this maximization problem by setting up the Langrangean function will lead to a set of Marshallian demand that express the quantity demanded for each good as a function of the price and total expenditure. Showing how the optimum quantity would as change prices and total outlay changes.

qi = gi(x,P), where P is a vector of prices.

The logarithmic function is given by log qi = eilogx + ?eijlogPj

By taking the derivative of the logarithmic function with respect to income and prices, we

Obtain the expenditure elasticity and the uncompensated price elasticities

Where ei is the income elasticity and eij are the uncompensated price elasticities.

The consumer's demand problem can also be taken to be a situation where he intends to minimize total expenditures or costs subject to a given level of utility.

Minimize x = ?pkqk

Subject to v (q) = u; where q = q1, q2,q3 ....qn

Solving this minimization problem by setting up the Langrangean function will lead to a set of Hicksian demand that express the quantity demanded for each good as a function of the price and with utility determined in advance. This utility level is the optimal utility level possible.

The Hicksian demand function is given by


The Hicksian demand function is also referred to as the compensated demand function. Since utility maximization also entails cost minimization, the optimal commodity choice from the two processes is the same. Therefore the Marshalling demand function derived from the utility maximization is equal to the Hicksian demand function derived from cost minimization.

In terms of elasticity, the price elasticity from the Hicksian demand also called compensated or Slusky price elasticity is equal to the Marshalling price elasticity plus the product of income elasticity and budget share. The mathematical relationship can be expressed as:

eij= ij + ?i wj

Where eij is the Hicksian or Slutsky price elasticity, i, is the Marshalling elasticity, wj is the budget share and ?i is the income elasticity

Restrictions on demand equations

If the system of equation must be regarded as demand system, they must satisfy some theoretical properties of demand function. Otherwise, they are more or less structural equations which can be estimated separately. In order for these systems of equations to be treated as demand function, the properties of demand function are imposed on the parameter estimates of the systems of equation. These imposed properties are known as restrictions on demand equations. These restrictions are adding-up, homogeneity, symmetry, and negativity restrictions. Estimating the demand equations separately will not satisfy all these restriction except for the adding up. Apart from the adding- up restriction, other restrictions are obtained through the substitution matrix. Another dimension introduced to demand studies through the Slutsky matrix is that it helps us establish the relationship between the Marshallian demand and Hicksian demand.

Properties of demand function

Adding up :

The almost idea demand system came up to solve the problem of aggregation of comer preferences which previous demand model failed to tackled. It possesses the properties of the Rotterdam model and that of the translog model but completely different from the two models. At the same time much better than them in modelling consumer demand. ............... (1980). One of the unique features of AIDS model is that it possesses a superior property of treating market preferences as if it emanates from the preference of an individual consumer without violating the associated demand properties. According to ...............(1980), AIDS model can be regarded as a first order approximation to any demand system and has been considered to be straight forward to estimstes. It is a demand model which evolves from a cost function that represents a second order approximation to an unknown arbitrary function (Holt and Goodwin, 2009). Exploy more of ppty

AIDS model permits exact exacts aggregation of consumer's choices and allow elaticities to change with levels of income a property that also makes it to be superior to the earlier demand systems. The model was advancement over the previous demand system which has attempted to estimate consumer demand through the imposition of some theoretical restriction. It allows the restrictions that economic theory places on demand model to be easily tested by placing a linear restriction on the parameters of the demand model (1980).

AID model began from the recognition of a unique class of preferences which allows exact aggregation over consumers, a problem that the previous demand systems fail to solve. They were able to present the market (aggregate demands) as though it emerge from the decision of a rational representative consumer. While considering the aggregation problem, they assumed that all consumers in the market are face with the same price situation (set) but their incomes are different from one another. Given that Engel curve measure the relationship between the quantities of commodity purchased by individual consumer at different level of income, therefore, the shape of the Engel curve under market demand should also be considered, if average demands are to be related to average outlay. This they did by placing some restriction on the shape of the Engel curve.

The model also satisfies the exact nonlinear aggregation because it is based on the cost function that satisfies the underlining preference of the price independent generalized logarithm. This function is represent the minimum level of expenditure that will help attain a given level of utility at a given price sets.

The PIGLOG class of the cost function is given by

log c(u,p) = (l-u) log a(p) + u log b(p)

Where, c(u,p) = the expenditure function

p = the vector of prices

u = the utility lines between 0 and 1

a(p) = the cost of subsistence when u =0

b(p) = the cost of bliss when u =1

log a(p) = a0 + ? ak logpk + 1/2 ? ? ?k* logpk logpj

log b(p) = loga (p) + o ?pk k

Substituting equation ................ to.............

The AIDS model can be specified as log c(u,p) = a0 + ? ak logpk + 1/2 ? ? ?k* logpk logpj + uo ?pk k

Where u = utility level, pk and pj are prices of commodity k and j respectively; ak, ?k*, o are parameters to be estimated.

This cost function is homogenous of degree one in prices such that if we double the prices of commodity, total expenditure must also be double for the consumer to maintain same level of utility. It is concave in prices, increasing in u, non-decreasing in price and increasing in at least one price. The cost function is also continuous in prices and twice differentiable with respect to prices at every point where price is not zero. Taking the partial derivatives of this function with respect to price leads us in to a system of demand functions in its budget share form.

?logc(u,p) = piqi = wi

?logpi c(u,p)

Where, wi is the budget share of good i. Therefore the logarithmic differentiation of equation.....

results in budget shares as a function of prices and utility.

Wi = ai + ? ?kijlogpj + iuo ?pk k

Where, ?ij = (?ij* + ?ji*)

In the function above, the determining variable is utility level and prices. Given that utility maximization and to cost minimization implies the same choice for a utility maximizing consumer; this equality can be rearranged in such a way that gives a system of demand equation in its the budget share form as a function of price and real income. This referred to as the Almost I deal Demand function ...................). It is represented as thus:

Wi = ai + ? ?ijlogpj + ilog(x/P)

Where,P = price index defined by.

log P = ? + ? ak logpk + 1/2 ? ? ?k logpk logpj

?ij = (?ij*+ ?ji*) = ?ji

The price index is nonlinear and generally poses a lot of challenges in the empirical situation, Deaton resolved this problem by approximating the price index to the linear stone price index; therefore the AIDS model is now been referred to as the Linear approximate AIDS model.

The stone index is given by.

log P = ? wak logpk

So as to preserve the budget constraint restriction, symmetry as well as the homogeneity condition of the cost function, the following restriction were place on the demand equations. These are:

? ai = 1 , ? i = 0 , Adding up restriction

i =1 i =1

??ij= 0, homogeneity restriction i =1

?ij = ?ji symmetry restriction , for i?j

The elasticity obtained from AIDS model is the set of Marhallian uncompensated elasticities and the Hicksian compensated elasticity can be obtained through the Slutsky substitution matrix.

The price elasticities from the AIDS model are given as:

?log qi = wj + ?ij/wi - dij


Where d is the Kronecker delta term which takes up the values of zero if i=j and 1 for i?j

The income elasticity is expressed as: ?log qi = 1 + i /wi


The suggestion of Stone linear price index as deflator in the AIDS model instead of the original translog price index (in the quadratic term) has initiated the linear approximate AIDS model into several criticisms. Many (....) the scholars argued that the LAAIDS is too restrictive and that the of the accuracy of the elasticity obtained from LA/AIDS model is questionable. The invariant of the stone index on the units of measurements was also pointed out by (Moschini, 1995). Consequently, there has been an advocating of the more flexible full AIDS model as well as other flexible demand system to the study of consumer demand (William and Apostolos, 2008)

Apart from the factors affecting quantity demanded as suggested by economics theory, there are other factors bring about change in demand. These factors are shifter factors. These factors are carefully incorporated into the AIDS model through interaction with the intercept term and by placing restriction on the parameters of the shifter (variables) in such a way as to maintain the adding up restriction (Julian, James and Nicholas, 2001)

The intercept function is expressed as:

ai = ?i + sik Sk

Where ? is the new intercept, s is the coefficient of the shifter variables and S represents the identifiable demand shifter variables (Alston, Chalfant and Piggott, 2001) which are also referred to as demographic characteristics of consumers.

The restrictions on the intercept function with the view to preserving the budget constraint restriction of the AIDS model are:

n n

? ?i = 1 , ?si= 0

i =1 i =1

The AIDS model, with the shifter variables can then be re-specified as: Wi = ?i + ?sik Sk + ? ?ijlogpj + ilogx - i ( a0 + ?sik Sk logpk + 1/2 ? ? ?kj logpk logpj ) k

A major problem associated with the transformation of ai in this manner is that the economic effect (elasticity and other welfare measures) obtained now varies with the unit of measurement of price as well as the quantities of commodity (Alston, Chalfant and Piggott, 2001). Alston et al 2001 proposed a solution to this problem through what they called Closure Under Unit Scaling (CUUS) which suggests that the intercept terms of both the price index (deflator) and that of the AIDS model should be made a linear function of the shifter variables the same time. The resulting model can be expressed as:

Wi = ?i + ?sik Sk +??ijlogpj + ilogx - i (?0 + ?sik Sk + (?i +?sik Sk logpk)) + 1/2???kj logpk logpj k

This model is not immune from estimation problem because the tranlog price index of the AIDS model is non linear in parameters .This in practice makes it very difficult to estimate the parameters especially in the case of many price vectors (deaton m1980), hence interacting the shifter variables with the constant term of the price index may not be possible; thereby making it difficult to estimate the AIDS model with CUUS approach.

A practical approach suggested by Lewbel (1985), Alston et al 2001 is to employ the general version of the AIDS model derived from the expenditure function similar to the LES but different from the expenditure function. The expenditure function is given by:

c(u,p) = pi?i + c*(p, u)

where the c*(p, u) is the cost function of the supernumerary expenditure which varies with the utility level. ?i is the subsistent quantity which is price and expenditure independent. This special characteristic allows shifter variables to be integrated into the generalized model without distortion of the unique characteristic of a demand system. In the generalized AIDS model, the subsistent quantities are expresses as the linear function of the shifter variables. Since the subsistent quantities are price independence and expenditure independents, the Generalized model does overcome the problems associated with the scaling (unit of measurement) problems.

The resulting demand function in its budget share form is expressed as:

wi = (pi?i)/X + i X*/X

where X* (X - pi?i) is the supernumerary expenditure, i is the coefficient of the share of the supernumerary expenditure expressed as wi* (X*, p)= ai + ? ?ijlogpj + ilog(X*/P)

Quadratic Almost Ideal demand system.

It was Bank, Blundell and Lewbel (1997) that extended version of the AIDS model They argued that Engel consumption curve predicts a non linear relationship between the expenditure share and income as the level of consumer income. They found out though a nonparametric analysis of consumer consumption habits quadratic terms in the logarithm of expenditure needed to be incorporated in the Engel curves in order to better predict consumption pattern. This information is lacking in the popular Translog or the Almost Ideal Demand Systems in which expenditure share Engel curves are linear in the logarithm of total expenditure. Bank et al therefore modified the AIDS model by incorporating the quadratic logarithmic income component into the model. The utility function underlining the QUAIDS model is:

Log(U) = {[ logX-loga(P)bP ]-1 1 + ?(p)}-1

Where the first term on the right hand side of the equation is the indirect utility function of a linear approximate AIDS model and the second term is term is a differentiable, homogeneous function of degree zero of prices p. The function ?(p) =??ilogpi, where ??i = 0 ( Barnett and Serletis, 2008, 12). By applying Roy Identity, the QUIADS model is given by:

Wi = ai + ? ?ijlogpj + ilog(X/a(p)) + ?i b(P) {log(X/a(p)}-2

The simplicity of the AIDS model as a result of the linearization through stone price index has brought up a lot of debates particularly on the capability of the LAAIDS model to produce the correct elasticity values. These have motivated researchers (Bernett and Seck, 2008) to compare different LAAIDS model and the non-linearized full AIDS model to see whether elasticities of LA/AIDS model compare well with the full model.

Linear expenditure system.

It was Stone that first attempted to estimate a system of demand equations which has properties that is consistent with traditional theory of consumer demand. Literature has it that Stone was the first person who attempted to use economic theory with empirical data. (.............).The starting point of the model was the specification of consumption as a linear function of price and outlay.

Piqi = iX + ? ipj

Stone explained further that consumer's spending can be divided into two main categories. The first is the consumption of some certain commodities which he referred to as basic consumption. In other word, consumers have some basic consumption obligation to make out of his money income before spending on other items. The remaining part of his income constitutes the second category which is spent on other items on fixed proportion. This is referred to as the supernumerary income. Since income is treated as the total expenditure, the supernumerary income is the total expenditure less the committed expenditure. According to him, the supernumerary income is only available for spending if the committed expenditure is not above the total expenditure and; it is premised upon this assumption that the demand function holds. The properties of demand function- adding up, homogeneity and symmetry were then imposed on demand function to produce linear expenditure systems. The resulting linear expenditure function is given by:

Piqi = pi?i + i (x - ?pk ?k)

Where, ?i is the basic or subsistence quantity, x = total income (outlay), i = proportion in which super numeracy incomes is allocated. Since ?i is considered to be basic or fixed, it meant therefore that the utility level varies mainly on the supernumerary income. Keeping price constant and varying the income level, we are back to the Engel curve and the traditional demand relationship if income is kept constant while varying prices. From the equation above, it follows that the super -numerary income is positive and i could assume values ranging between zero and 1 and sum to unity for all commodity. The implication of this is that the values of i will always be positive permitting no commodity to be inferior and cross price elasticity will always be positive, allowing no goods to be complement even though these commodities may complement each other (stone). The demand equations are homogenous of degree zero in prices and income, satisfy adding up and Slutsky symmetry condition (....lai bai......) but the substitution matrix is not negative semi definite.

For a utility maximizing consumer, the expenditure function above becomes the minimum cost required to attain a given level of utility and can be represented as:

C (u,P) =?pk ?k + u?pk k

Where, pk ?k is the subsistence expenditure and ?pk k permit utility to be acquired a constant price (........deatons..................).

The direct utility function (utility as a function of quantity vector) underlining the demand (expenditure function) is given by U(q) = ?(qk - ?k ) k while the indirect utility function (maximum utility level s a function of outlay and prices) is ?(x,p) = (x - ?pk ?k)/ ?pk k


This demand equation is homogenous of degree zero is prices as well as income. It also satisfies the adding up restriction and the symmetry restriction (lee and bai, 1974). The

Rotterdam model

The Rotterdam model was developed by Theil (1965) and Barten (1966) through the introduction of method of differentiation to demand analysis. Theil (1965) demand model approached demand from the probabilistic point of view. The main striking component was the values share which he regarded as probability becauses it is nonnegative and adds up to unity for all commodities. The model was able to introduce substitution matrix into the demand model and as a result be able to categorize commodities into complements or substitutes, a claim that is very impossible in the LES. The model has its root on the traditional assumption that utility maximising quantities are obtained as consumers strives to maximize his/her utility subject to the available budget constraint. The model therefore, derives from the optimal solutions of the resulting first order derivatives obtained from a constrained utility maximization problem.

Supposed the resulting demand function is

qi = q(x, P)

Total derivative of the function is

dqi = ?qi?xidx + ?qi?pjdpj

qidlogqi =x?qi?xidlogx + pj?qi?pjdlogpj

dlogqi = eilogx + eijdlogpj

The Slutsky condition suggest the relationship between Marhallian and Hicksian demand in terms of elasticity as eij = ij - eiwj.

where eij is uncompensated cross- price elasticity, ij is the compensated cross price elasticity and ei is the income elasticity. Substituting the for the Marshallian elasticity, equation................, can be written as.

dlogqi = eilogx + eijdlogpj

dlogqi = ei (logx - ?wkdlogpk) + ? ijdlogpj

Since the budget share wi = (piqi/x) and dm = mdlogm; therefore, following the multiplication of equation above by pi/x, the logarithmic form of the demand equation can be represented by.

widLog qi= PiqiX ei(dlogx -?wkdlogpk) + ?PiqiX uijdlogpj

widLog qi= pi?qi?xi (dlogx -?wkdlogpk) + ?PiPjX ?qi?pjdlogpj

Where dlogP = ? wkdlogpk which is referred to as Divisia (1925) price index, by Barnett and Serletis (2008). Equation................. can be written as

widLog qi= ?i(dlog(x /P) + Fijdlogpj.

Where, ?i is the marginal propensity to consume, x /P is the real income and Fi is the price elasticities, obtainable through the Slutsky substitution matrix.

It was barten (1967) the carried out homogeneity and symmetry test on the result obtained from the Dutch post war data. The homogeneity restriction was rejected.

Translog model

The translog model was developed by...................... (1975). they argued that the assumption of constant elasticity of substitution among pairs of commodity was too restrictive. In addition the assumption of constancy of proportion of all expenditure at different level of incomes may not be ideal consumer behaviour. Hence, the quest to develop a demand system that stems from a utility function which is non linear in the logarithm of quantity and better predict consumer behaviour. They began with utility function that is quadratic in the logarithm of quantity consumed. By so doing, the resulting utility function now relax the assumption of constant elasticity of substitution and then allow elasticity's of substitution to vary among commodities and the degree of homogeneity to vary across commodities at different level of income. The demand model also employed the duality between prices and quantity and as such established an indirect utility function and the aftermath direct demand function and vice versa. Besides the demand restrictions, the tested the assumption of additively and homotheticity on the direct and indirect utility function.

The indirect translog utility function derives from a logarithmic utility function in which each of the commodity prices in the utility function is normalized by diving them through by the total expenditure. The logarithmic utility function is given below.

Log (W) = log W( P1X, P2X , P3X........................, PkX)

The logarithmic utility function which has the quadratic term of logarithm of the normalized prices is regarded as the transcendental logarithmic utility function. The indirect transcendental logarithmic function yields a system of direct demand function in which the budget share is influenced by expenditure normalized commodity price consequent upon the application of Roy's identity.

...................... (1975) in order to preserve the symmetry with respect to the direct utility function approximates the translog utility function as:

Log ? (p,x) = a0 +? ailog PiX, + 1/2? ?jilog PiXlog PjX k =1 k =1 j =1

Where ai is vectors of parameter and ij is a symmetry matrix of parameters. Arising from the application of Roy's identity to the equation above, the budget share (demand) equation is given by:

Wi = -( ?Log?(.)?logpj)/ ?Log?(.)?logX

Where, ?Log?(.)?logpj = aj +? jilog PiX

- ?Log?(.)?logX = ? (ak +? kilog PiX)

  • The budget constraint or the adding up restriction suggest that ak sums up to unity, therefore, ?ak = -1 and the homogeneity restriction implies that ? ki = 0 for all k = 1, 2, 3..............n. If homogeneity holds, then the translog model is homothetic.
  • Barnett, William A. & Serletis, Apostolos 2008. "Consumer preferences and demand systems," Journal of Econometrics, Elsevier, vol. 147(2), pages 210-224, December.
  • Deaton, Angus and Muellbauer, John (1980). "An Almost Ideal Demand System,"The American Economic Review, Vol. 70, No. 3 pp. 312-326.
  • Barnett , William A. and Seck, Ousmane 2008 "Rotterdam Model versus Almost Ideal Demand System: Will the Best Specification Please Stand Up?", Journal of Applied Econometrics, Vol. 23, No. 6, pp. 795-824.
  • Liu, Kang E. 2006. A Quadratic Generalization of the Almost Ideal and Translog Demand Systems: An Application to Food Demand in Urban China. Selected Paper prepared for presentation at the American Agricultural Economics Association Annual Meeting, Long Beach, California, July 23-26, 2006.
  • Holt, Matthew T. and Goodwin, Barry K (2009). The Almost Ideal and Translog Demand Systems. Munich Personal RePEc Archive (MPRA) Paper No. 15092. Online at
  • Christensen, Laurits R ., Jorgenson, Dale W. and Lau, Lawrence J . 1975. "Transcendental Logarithmic Utility Functions" American Economic Review, vol. 65, issue 3, pages 367-83
  • Theil, H. (1965). "Information approach to demand analysis" Econometrica, vol. 33, No. 1. Pp 67 - 87
  • Stone, R. (1954). "Linear expenditure systems and demand analysis: An application to the pattern of British demand" The Economic Journal LXIV, 255 (1954): 511-527.
  • Banks, J., Blundell, R. and Lewbel, A. (1997): "Quadratic Engel curves and consumer demand" The Review of Economics and Statistics Vol. LXXIX November 1997 Number 4, pp 527- 542
  • Alston, J. M., Chalfant, James A. and Piggott Nicholas E. (2001). "Incorporating demand shifters in the almost ideal demand system", Economic letters" 70 (2001) 73- 78

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