Estimation of Complete Demand Systems

Demand Analysis

2.5. Estimation of Complete Demand Systems

Estimation of single demand functions either from time series data following the pragmatic approach or from price variations across clusters in household surveys creates the problem that the quantity projections obtained may not satisfy the requirements of demand theory, particularly the budget constraint. Such predictions are consequently inadequate for use in complete models such as multimarkets (Chapter 11) and CGEs (Chapter 12). For this purpose, complete systems of demand equations which are able to take into account consistently the mutual interdependence of large numbers of commodities in the choices made by consumers need to be specified and estimated.

Three demand systems have received considerable attention because of their relative empirical expediency. They are the Linear Expenditure System (LES) developed by Stone

(1954), the Almost Ideal Demand System (AIDS) developed by Deaton and Muellbauer (1980), and the combination of these two systems into a Generalized Almost Ideal Demand System (GAIDS) proposed by Bollino (1990). Other complete demand systems found in the literature but not as widely used are the Rotterdam model of Theil (1976) and Barten (1969) and the translog model of Christensen, Jorgenson, and Lau (1975).

2.5.1. The Linear Expenditure System

The Linear Expenditure System (LES) is the most frequently used system in empirical analyses of demand. It derives from the Stone-Geary utility function, which is pointwise separable:

u (q_ic_i)^bi

i1



n ^{or u b}iln(q_ic_i)

i1



n ^with

0bi 1 b_i



i ^1

q_ic_i0





 .

The c’s are usually interpreted as minimum subsistence or “committed” quantities below which consumption cannot fall. The demand functions derived from maximization of this utility function under a budget constraint constitute the LES:



p_iq_ic_ip_ib_i y

j ^cjp_j



 

, i1,,n.

This shows that the b’s are the marginal budget shares, ∂pq/∂y, which tell how expenditure on each commodity changes as income changes. Since b_i > 0, this system does not allow for inferior goods. c_jp_j is the subsistence expenditure and the term (yc_jp_j⁾ is generally interpreted as “uncommitted” or “supernumerary” income which is spent in fixed proportions b_i between the commodities.

An important drawback of this system is that it implies linear Engel functions, a specification not supported by empiricism and that can at best be true only over a short range of variation of y. If the equations are to be used for predictions, only short-term predictions can consequently be made. The price and income elasticities in these equations are (see derivation in the Appendix):

Eii 1(1bi) c_i

q_i, E_ij b_ic_jp_j

p_iq_i , _i b_i w_i ,

where w_i is, as before, the budget share of commodity i. The flexibility of money can also be measured as

  y y c_j

j

 ^p_j

Like all pointwise-separable models, the LES model is better applied to large categories of expenditure than to individual commodities, since it does not allow for inferior goods and implies that all goods are gross complements (E_ij < 0). Estimation of the linear expenditure system is difficult due to nonlinearity in the coefficients b and c, which enter in multiplicative form. Two iterative approaches have been followed to overcome this difficulty.

The most common and relatively less sophisticated technique is a two-stage iterative procedure. It exploits the fact that, for given b, the LES is linear in c:

(6) piqibiyci(pibipi)

ji



cj(bipj).

Similarly, for given c, the LES is linear in b:

(7) p_iq_ic_ip_i b_i y

j ^cjp_j



 

.

The iterative estimation sequence is as follows. Start with an initial value of b and estimate the c in (6) using OLS regression without intercept. Then, given this estimate of the c, estimate b in (7), again by OLS regression without intercept. The iteration continues until the sequence converges to stable estimates of b and c. An improved estimation method that treats (6) and (7) as a system of equations has been proposed by Parks (1969).

The other approach which has been used to estimate the LES is based on the technique of full information–maximum likelihood. It requires a computer algorithm that solves for a nonlinear system of equations.

2.5.2. The Almost Ideal Demand System

The Almost Ideal Demand System (AIDS) derives from a utility function specified as a second-order approximation to any utility function. The demand functions are derived in budget share form as:

p_iq_i

y w_i a_i



j ^bij lnp_jc_i ln y P,

where w_iis the budget share, P is a price index defined as:

lnPa_o



k ^aklnp_k 1 2



_j



k ^bjklnp_k lnp_j, and the parameters are subject to the following restrictions:



i ^ai 1,



i ^bij0,



i ^ci 0,



j ^bij 0, b_ij b_ji.

Deaton and Muellbauer (1980) suggest approximating the price index P by the Stone geometric price index:

lnP^*



i w_iln p_i.

This linear approximation is all the better if there is collinearity in prices over time. The equation to be estimated is thus:

wi ai* 



j bij ln pj ci ln y P^*,

where and P =  P^ is the approximation to P. The linear-approximate AIDS should be estimated as a system of equations with the above-mentioned restrictions on the parameter estimates. The price and income elasticities can be derived from the parameter estimates as (see derivation in the Appendix):

a_i^* a_ic_iln

E_ii 1 b_ii

w_i c_i, E_ij b_ij w_i  c_i

w_i w_j, _i1 c_i w_i .

The AIDS implies a money flexibility value of minus one (Blanciforti, Green, and King, 1986).

2.5.3. The Generalized Almost Ideal Demand System

The Generalized Almost Ideal Demand System (GAIDS) combines the LES and AIDS models, preserving some of the interesting features of the LES (the concept of committed and supernumerary expenditures) while adding flexibility to the estimated elasticities (Bollino, 1990). The basic idea is to replace in the LES the fixed proportions (b_i) in the allocation of supernumerary expenditures (y^S) by an AIDS specification that makes this allocation a function of income and prices. The equation to be estimated is:



q_i c_i  1

p_i _i  _ijlnp_j_ilny^S

j P

 

 

y

j ^cjp_j



 

, i1,,n, where y^S y c_jp_j

j , under the constraints



i ^i 1,



i ^ij 0,



i ^i 0,



j ^ij0, _ij _ji.

Calling AIDS_i the square bracket in the GAIDS equation and w^S = y^S/y the supernumerary share, the price and income elasticities are:

E_ii 1



1AIDS_i_i



^c_qⁱ

i

w^S

w_i



_ii_iw_i



^,

E_ij c_jp_j

p_iq_i



AIDS_i_i



^w_w^S

i

_ij_iw_j

 

^{, and}

_i AIDS_i_i w_i .

These expressions simplify to the LES elasticities for AIDS_i b_iand_ij _i0.

2.5.4. Estimation Problems*

There are three econometric problems which deserve caution in estimating the AIDS or GAIDS systems.

2.5.4.1. Probit Analysis of Decision to Consume

During the survey period some of the goods may not have been consumed by some of the households, implying zero values for the corresponding observations of the endogenous variable in the regression equations. The dependent variable is thus truncated, creating a bias in the OLS estimates, since the assumption of zero correlation between independent variables and error term is violated. This problem is solved by using a two-stage estimation procedure proposed by Tobin (1958) that combines a probit analysis with a standard OLS (see Hein and Wessells, 1990). In the first stage, the decision by an individual (k) to consume the particular item (i) or not is modeled as a probit as:

Prob



q_ik^ 1



^{ Prob}



^fik



p_k,y_k,z_k



^u_ik0



where is equal to one if the kth household consumes the ith commodity and zero otherwise, p_k, y_k, and z_k are the prices, income, and characteristics that apply to that household, and u_ik an error term. The probit estimation gives the inverse Mill’s ratio:

q_ik^

_ik 

 

f_ik /

 

f_ik ,

where is the probability density function and  the cumulative density function of the standard normal distribution. In the second stage, this ratio is used as an additional exogenous variable in the OLS estimation of the demand equation in order to correct for the bias created by use of a limited dependent variable. (See also Maddala, 1983, Chapter 8.

These regression options are programmed in most econometric packages.) 2.5.4.2. Seemingly Unrelated Regressions

Demand equations appear to be unrelated, since none of the endogenous quantities or budget shares appear on the right-hand side of the equations. This is not the case, however, since error terms across equations are correlated by the fact that the dependent variables need satisfy the budget constraint (e.g., the budget shares in AIDS and GAIDS sum to one). While an OLS estimate of these equations would be consistent and unbiased, the estimation method developed by Zellner (1962) for Seemingly Unrelated Regressions (SUR) provides estimates

that are more efficient. In a first stage, OLS is used to estimate the variance-covariance matrix among residuals; in a second stage this estimated matrix is used in a generalized least squares estimation. Since the covariance matrix among residuals is singular because the residuals satisfy the budget constraint, the typical procedure consists in deleting one of the equations of the demand system. The parameters from the deleted equation can be calculated from the parameters of the other equations through the restrictions on parameters. Barten (1969) has suggested an Iterated Seemingly Unrelated Regression (ITSUR) routine which produces results that are invariant to the equation deleted.

2.5.4.3. Imposition of Inequality Restrictions

Demand parameters need to satisfy a number of exact restrictions, and these must be imposed on the estimators. Equality constraints are imposed by using a restricted least squares approach. Imposition of inequality restrictions is more demanding. Bayesian estimation methods have been developed for this purpose by Geweke (1986). In the Bayesian approach, prior beliefs are combined with sample information into a posterior distribution from which point estimates of the parameters and confidence intervals are then derived.

Here, the demand system is first estimated without the inequality constraints, yielding a vector of unconstrained parameter estimates and their distribution. Prior information is then introduced by truncating this distribution. The expected value of this truncated distribution then becomes the parameter vector to be used in the calculation of the elasticity estimates.

This method was applied to an AIDS for meat in Canada by Chalfant, Gray, and White (1991) and by Rose (1992) to a GAIDS with a Mexican household consumption panel.

2.5.5. Effects of Household Characteristics*

Households differ by a set of characteristics (z_k,k1,...,s) such as age and sex composition, race and religion, and urbanization status that affect the pattern of demand.

From a policy standpoint, it is important to estimate the impact of these characteristics on demand to establish the determinants of observed household-specific consumption levels, help target government programs such as food aid on particular classes of households, and determine the amount of assistance needed to bring the malnourished to acceptable consumption standards. Since the budget constraint needs to be satisfied, any increase in expenditure on some commodities due to a change in z_k must be compensated by a corresponding decline in the consumption of the other commodities. Hence, the z_k parameters are constrained to satisfy the s constraints:

p_iq_i

z_k

i1



n ^{0 for all k.}

Several approaches have been followed to incorporate household characteristics in the estimation of complete demand systems. The idea is to introduce additional parameters into the original system and postulate that household characteristics affect demand only through these parameters (Goungetas and Johnson, 1992).

2.5.5.1 Translating Approach

Using the same idea as in the LES, we postulate that characteristics affect demand only through “translation” parameters (c_i) that represent subsistence or “necessary” levels of demand. Hence, quantities are decomposed into:

q_i c_i q _i(p,y c_j



j ^pj), where c_i a_ik

k ^zk,

which contains ns parameters to be estimated (Pollak and Wales, 1978).

2.5.5.2. Scaling Approach

The idea here is to use scaling parameters (m_i) to reflect the number of “equivalent persons” in the household measured in a scale specific to each commodity i. This corresponds to replacing, in the standard demand model, quantities q_i by scaled quantities q_i /m_i and prices p_i by scaled prices p_im_i. The demand equations to be estimated are then:

, where m

q_i m_iq_i(p1m1, ...,p_nm_n,y) _i1 b_ik

k ^zk, adding again ns parameters to be estimated (Barten, 1964).

These two methods can also be combined as done by Gorman (1976). Results obtained under the different approaches were compared by Pollak and Wales (1981).