Differential demand systems: a further look at Barten's synthesis.

• Author: Matsuda, Toshinobu

1. Introduction

   Theoretically consistent, complete systems of consumer demand equations (demand systems, for short) give the quantity demanded of each commodity as a function of total expenditure, the prices of all commodities, and other variables that affect demand. Providing expenditure elasticities, own- and cross-price elasticities, and other effects of policy interest, demand systems have been extensively used in econometric analysis of consumer behavior ever since the introduction of the pioneering linear expenditure system (LES) of Stone (1954). The literature shows that demand systems cover a wide range of applications from the aggregate to the disaggregate level. Some recent examples are Nicol (1994), Pashardes (1995), and Phipps (1998) in demographic economics; West and Williams (2004) in environmental economics; Cockx and Brasseur (2003) in health economics; Wolak (1996) and Capps, Church, and Love (2003) in industrial organization; Okamura (1996), Hill (2000), and Irwin (2003) in international economics; Lewbel (2003) and Escario and Molina (2004) in law and economics; Fleissig and Swofford (1996) and Fleissig and Serletis (2002) in monetary economics; Borge and Rattso (1995) and Nichele and Robin (1995) in public economics; and Cheshire and Sheppard (2002) in urban economics. It is noteworthy that demand systems have often been applied, especially by agricultural economists, to estimate the demand for different food products. (1) This is partly because demand systems are based on static utility maximization, which is not suitable for durable commodities, and food is considered the most typical nondurable commodity group. (2)

   Among numerous different functional forms of demand systems developed to date, a class of models known as differential demand systems accounts for a substantial proportion of contributions made by demand systems at large, particularly in terms of empirical applications. Originating in and typified by the Rotterdam model (Barten 1964: Theil 1965), this class of demand systems is derived from a first-order approximation to arbitrary Marshallian demand functions, while other well-known demand systems, such as the LES, the translog model (Christensen, Jorgenson, and Lau 1975), and Deaton and Muellbauer's (1980) almost ideal demand (AID) system in its original formulation are derived from the maximization of an explicit indirect utility function or, equivalently, from the minimization of an explicit expenditure/cost function. Differential demand systems are as much based on consumer demand theory as are the LES and the so-called flexible functional forms, such as the translog and the AID, and are as flexible as the flexible functional forms in that they have enough coefficients to attain arbitrary quantities, expenditure elasticities, and own- and cross-price elasticities. Their popularity may not only be due to these features shared with other models but also due to the fact that differential demand systems are linear in coefficients and therefore easy to estimate, and that, required to convert the differential terms to finite changes, in econometric implementation, the process of first-differencing their variables is likely to make them stationary.

   Differential demand systems other than the Rotterdam model include the AID (although, when first published, it was not derived through the differential approach but from expenditure minimization), the (Dutch) Central Bureau of Statistics (CBS) model of Keller and van Driel (1985), the NBR model of Neves (1987), and the model of Barten (1993) that nests all these four differential demand systems within it. It should also be noted that Theil, Chung, and Seale (1989) developed a cross-country levels version of demand system using the differential approach, and that, extending Barten's (1993) approach to the context of inverse demand, Brown, Lee, and Seale (1995) proposed a synthetic model nesting within it the four models--Barren and Bettendorf's (1989) inverse Rotterdam and inverse AID and Laitinen and Theil's (1979) inverse CBS along with the inverse analogue of the NBR. (3)

   Differential demand systems have been successfully applied to an innumerable number of empirical studies including many recent ones: Alston and Chalfant (1993); Lee, Brown, and Seale (1994); Brester and Schroeder (1995); Nelson and Moran (1995); Kinnucan, Xiao, and Hsia (1996); Brown and Lee (1997); Kinnucan et al. (1997): Nelson (1999); Duffy (2001); Angulo el al. (2002); Schmitz and Seale (2002); Capps, Church, and Love (2003): Cockx and Brasseur (2003): Seale, Marchant, and Basso (2003): and others.

   It is often of practical interest to researchers to determine which to choose among available functional forms. If the alternative models have similar theoretical properties, one of the basic criteria for comparing them is their relative explanatory power. In this light, Barten's (1993) model is empirically attractive because, by its synthetic construction, it is useful for testing the adequacy of the competing functional forms of differential demand systems including the Rotterdam and the AID-two of the most popular demand systems in the literature. Due to its artificial way of nesting, however, the economic implications for this synthetic model are not clear. This article provides a further look at the functional form of the synthetic model, showing that, at the individual consumer level, an arbitrary differential demand system has the same demand response to change in total expenditure as that of a specific form of Engel curve. An empirical illustration is given for Japanese consumer demand for nondurable goods and services.

2. Differential Demand Systems

   Let p = ([P.sub.1],..., [P.sub.n]) denote the nominal price vector of n goods, m denote the total expenditure on the goods (expenditure, for short), and [q.sub.i](p, m) denote the Marshallian demand function of good i. The derivation of differential demand systems starts with totally differentiating [q.sub.i](P, m) so that

   (1) d[q.sub.i](p, m) = [partial derivative][q.sub.i](p, m)/[partial derivative]m dm + [summation over (j)] [partial derivative][q.sub.i](p, m)/[partial derivative][p.sub.j] d[p.sub.j], i = 1,...,n,

   where [[SIGMA].sub.j] is an abbreviated notation for [[SIGMA].sup.n.sub.j=1].

   If hi(p, u) is taken to be the Hicksian (compensated) demand function of good i, where u is a reference utility level, the relation between the Marshallian and the Hicksian demand functions is expressed by the Slutsky equation,

   (2) [partial derivative][q.sub.i](p, m)/[partial derivative][p.sub.j] = [partial derivative][h.sub.i](p, u)/[partial derivative][p.sub.j] - [partial derivative][q.sub.i](p, m)/[[partial derivative].sub.m] [q.sub.j](p, m), i,j = 1,...,n.

   The budget constraint or the adding-up condition, on the other hand, is totally differentiated as follows: (4)

   (3) [summation over(i)] [p.sub.i][dq.sub.i] = [d.sub.m] - [summation over(i)] [q.sub.i][dp.sub.i].

   Substituting Equation 2 into Equation 1, using the results in Equation 3, and then multiplying both sides through by [p.sub.i/m] obtains

   (4) [w.sub.i]d log [q.sub.i] = [p.sub.i] [partial derivative][q.sub.i]/[[partial derivative].sub.m] d log Q + [summation over(j)] [p.sub.i][p.sub.j]/m [partial derivative][h.sub.i]/[partial derivative][p.sub.j] d log [p.sub.j], i = 1,...,n,

   where log denotes the natural logarithm, [w.sub.i] [equivalent to] [p.sub.i][q.sub.i/m] denotes the expenditure share (share, for short) of good i, and d log Q [equivalent to] [[SIGMA].sub.i] [w.sub.i]d log [q.sub.i] denotes the Divisia volume index. [p.sub.i][partial derivative][q.sub.i]/[[partial derivative].sub.m] is the marginal budget share of good i, which determines the allocation of additional expenditure to the good, while ([p.sub.i][p.sub.j/m])([partial derivative][h.sub.i]/[partial derivative][p.sub.j]) is the Slutsky term or the ijth element of the Slutsky matrix, which involves the substitution effect of price changes.

   If both the marginal budget share and the Slutsky terms are approximated to be constant, Equation 4 becomes the Rotterdam, the most well-known and heavily used differential demand system,

   (5) [w.sub.i]d log [q.sub.i] = [b.sub.i]d log Q + [summation over(j)] [s.sub.ij]d log [p.sub.j], i = 1,...,n.

   Although criticized for not being consistent with utility maximization without imposing extreme restrictions on its coefficients, the Rotterdam is still considered a pioneering and fundamental contribution to demand system specification especially in light of its unwavering popularity in applied studies, as well as in light of the defenses against the criticism made by Barnett (1979) and Mountain (1988). (5) Barnett (1979) showed that the discrete, that is, first-differenced form Rotterdam at the aggregate level was a Taylor series approximation to a certain demand system, while Mountain (1988) showed that the discrete Rotterdam at the individual consumer level was also a valid approximation.

   By subtracting [w.sub.i]d log Q from both sides of Equation 5 and then defining the parameterization [c.sub.i] [equivalent to] [b.sub.i] [w.sub.i], an alternative specification of a differential demand system is derived as

   (6) [w.sub.i](d log [q.sub.i] - d log Q) = [c.sub.i]d log Q + [summation over(j)] [s.sub.ij]d log [p.sub.j], i = 1,...,n,

   which is known as the CBS model.

   Let d log P [equivalent to] [[summation over].sub.i] [w.sub.i]d log [p.sub.i] = d log m - d log Q denote the Divisia price index and [[delta].sub.ij] denote the Kronecker delta, which is equal to unity if i = j and zero otherwise. Adding [w.sub.i](d log [p.sub.i] - d log P) to both sides of Equation 6, using in the left-hand side the relation

   (7) [w.sub.i](d log [p.sub.i] + d log [q.sub.i] - d log m) = [dw.sub.i],

   and then parameterizing so that [r.sub.ij] [equivalent to] [s.sub.ij] + [w.sub.i]([[delta].sub.ij] - [w.sub.j]) results in

   (8) [dw.sub.i] = [c.sub.i]d log Q + [summation over(j)]...