Differential Approximation to the Demand Function

A fi nal approach is based on a direct approximation of the Marshallian demands. Transforming the differentials of the Marshallian demands yields a set of equations that are local fi rst-order approximations to the underlying relationship between quantities, prices, and income. The most common differential demand system is the Rotterdam model (Theil 1965; Barten 1966). More-recent alternatives include the fi rst-differenced linear AIDS (FDLAIDS) (Deaton and Muellbauer 1980a), the National Bureau of Research (NBR) demand system (Neves 1987), and the Central Bureau of Statistics (CBS) demand system (Keller and Van Driel 1985). Barten (1993) showed that these four differential demand systems can be nested into a model referred to as Barten’s synthetic model.

Consider the Rotterdam model of Theil (1965) and Barten (1966). Theil derived the Rotterdam model, beginning with the logarithmic differential of the Marshallian demand for good n, q_n(p₁,…, p_N, M), such that

, (52) where q_n is quantity of good n, p is price, M is total expenditure, and η_nj and η_nM are Marshal- lian price and expenditure elasticities.18 _{Using the Slutsky equation in (11), (52) becomes}

.

18 _{In practice, the assumption is made that the model derived in continuous time can be approximated}

using data measured in discrete time, i.e., d ln p_n ≈ Δ ln p_n = ln p_n,t – ln p_n,t–1, where t is the time period (Theil

1965). This approximation is discussed in greater detail in the application of the differential type models in section 6.

19 _{Denote the true cost-of-living price index as equation (28), i.e.,}

,

where p– is a vector of base-period prices and u is utility. The proportional rate of change in the price index is then

. Footnote 17 implies that

. Hence, for any fi xed utility level u, the price index can be written as

.

This suggests that w_n(p, u) should be replaced with the observed budget shares, w_n. However, as discussed

in section 2.3.4, unless preferences are homothetic, the utility-constant budget shares are not equal to the actual budget shares. In addition, as discussed in footnote 18, prices are not observed continuously, so the preceding equation would have to be approximated by some formula containing fi nite changes (Deaton and Muellbauer 1980b, pp. 174–175).

Multiplying both sides of this equation by the expenditure share for good n, w_n, results in the Rotterdam demand system:

, (53) where d ln Q is a Divisia volume index; that is

(54) or

, the parameters of the system are defi ned as

, (55) , (56) and s_nj is the Slutsky substitution term from equation (10).

It can be shown that the FDLAIDS is a transformation of the Rotterdam model. The AIDS (or the LAIDS) can be expressed in differential form following Deaton and Muellbauer (1980a). Specifi cally, if the logarithmic price terms in the LAIDS are replaced by their logarithmic dif- ferentials and Stone’s price index is replaced with the Divisia price index, the FDLAIDS is19

, (57) .

The right-hand side terms of the Rotterdam and FDLAIDS are similar. The left-hand side terms differ but the Rotterdam model can be transformed to have the same dependent variable as the FDLAIDS. To show this, note that the differential of a budget share, w_n, can be written as

. (58) Also, the logarithmic differential of the budget equation is

(59) . (60) Substituting (60) into (58), . Solving for w_nd ln q_n, , and substituting this term into (53) yields

. Rearranging this equation yields

, (61) where

.

Hence, if β_n = θ_n – w_n and γ_nj = π_nj – w_nw_j + w_nδ_nj, the two models are approximately equivalent (Brown, Lee and Seale 1994).

The CBS and NBR specifi cations are hybrids of the Rotterdam model and FDLAIDS. The CBS model incorporates Working and Leser’s Engel model into the Rotterdam specifi ca- tion (Brown, Lee and Seale 1994). In particular, Working and Leser proposed modeling the expenditure share for good i as

. (62) Multiplying this by M and then differentiating with respect to M yields

. (63) Solving for α_n in (63) and substituting the resulting expression and (56) into (62) yields

. (64) Replacing θ_n in (53) with (64) yields the CBS model, which has Rotterdam price coeffi cients and an FDLAIDS income term:

Similarly, the NBR model can be derived from the FDLAIDS model by letting β_i = θ_i – w_i in (57), such that

,

where the price coeffi cients are the same as the FDLAIDS price coeffi cients and the expendi- ture term is the same as in the Rotterdam model.

By parameterizing the four models to have the same right-hand-side terms, we can consider the differences in the marginal budget shares between models. Rewriting the Rotterdam (R), CBS (C), FDLAIDS (F), and NBR (N) models so that they all have the same right-hand-side terms yields

, (65) , (66) , (67) . (68) The coeffi cient on the income term in the Rotterdam and NBR models (i.e., θ_n) is the marginal budget share and is constant, whereas the marginal budget shares for the FDLAIDS and CBS models (i.e., β_n = θ_n – w_n) vary with the expenditure shares. Conversely, the Slutsky terms are considered to be constants in the Rotterdam and CBS models (i.e., π_nj) but vary with expen- diture shares in the NBR and FDLAIDS models.

Barten (1993) nested the four differential demand system models into the following general model by exploiting the similarities between the models:

, (69) where y_i, i = R, C, N, F is a t × 1 vector of transformed basic endogenous variables; X is a t × k matrix of exogenous price and expenditure variables; and Ω = α_Rω_R + α_Cω_C + α_Fω_F + α_Nω_N and ω_i, i = R, C, N, F comprise a k × 1 vector of coeffi cients. Without loss of generality, the sum of the αs is set to zero and α_R is

. (70) Substituting α_R into (69) and solving for y_R yields

. (71) Unconstrained estimation of the αs is not possible since α_R is a linear combination of α_F, α_C, and α_N. However, (71) can be rewritten using the fact that

, (72) or

Solving (73) for y_R – y_F yields

, and substituting this into (71) gives

, ,

. (74) The nesting coeffi cient δ₁= α_C + α_F measures the difference between the marginal budget shares of the Rotterdam model and the marginal budget shares of the CBS and FDLAIDS models. The nesting coeffi cient δ₂ = α_N + α_F measures the difference between the price coeffi cients of the Rotterdam model and price coeffi cients of the FDLAIDS and NBR models. Substituting (65)–(68) into (74) yields

. (75) Using (58) and (59), the bracketed term multiplied by δ₂ is equivalent to

, and substituting this into (75) yields

. (76) Since the FDLAIDS (Rotterdam) model has the same coeffi cient on the expenditure variable as in the CBS (NBR) model and the FDLAIDS (Rotterdam) model has the same coeffi cients on the price variables as in the NBR (CBS) model, we can rewrite XΩ as

. (77) Hence, by substituting (77) into (76) and rearranging, Barten’s synthetic model takes the form

, (78) where δ₁ and δ₂ are nesting parameters, a_n = δ₁β_n + (1 – δ₁)θ_n and b_nj = δ₂γ_nj + (1 – δ₂)π_nj are expenditure and price coeffi cients to be estimated, δ_ij is the Kronecker delta, w_n is a t × 1 vector

Table 2. Nesting Parameter Values for Differential Demand Systems

Barten’s Synthetic Model

Generalized Ordinary Differential Demand System

δ₁ δ₂ φ₁ φ₂

Rotterdam 0 0 –1 1

FDLAIDS 1 1 0 0

CBS 1 0 0 1

NBR 0 1 –1 0

of expenditure shares for good n, p_j is a t × 1 vector of prices of good j, and Q is a t × 1 vector of Divisia volume indexes (equation (54)). Table 2 lists the values for δ₁ and δ₂ that allow Barten’s synthetic model to collapse into the various nested models. The formulas for the elasticities of demand with respect to expenditure and prices and the adding-up, homogene- ity, and symmetry conditions are listed in Table 1.

Matsuda (2005) showed that, at an individual level, Barten’s synthetic model has the same marginal budget shares as generated by specifi c forms of Engel curves formulated by a Box-Cox transformation. If δ₁ = 0, then the Engel curves are linear. On the other hand, if

δ₁ = 1, then the Engel curves are linear logarithmic.

Eales, Durham, and Wessells (1997) specifi ed an alternative parameterization of Barten’s synthetic model with an FDLAIDS dependent variable for the generalized ordinary differential demand system (GODDS). Instead of solving for α_R in (70), they solved for α_F and substituted

α_F into (69) to yield

. (79) Their alternative specifi cation takes the form

, (80) where c_n = φ₁β_n + (1 – φ₁)θ_n and d_nj = φ₂γ_nj + (1 – φ₂)π_n are expenditure and price coeffi cients to be estimated. Table 2 lists the values for φ₁ and φ₂ that allow the GODDS to collapse into the various nested models. Adding-up, homogeneity, and symmetry restrictions and formulas for expenditure and price elasticities of demand for (80) are summarized in Table 1.