Empirical modelling of the aggregation error in the representative consumer model

Abstract

This paper examines different approaches to modelling the aggregation error associated with the representative consumer model. Each approach is based on an analytical framework intended for modelling aggregate time series data on quantities and prices with potential additional measures of income distribution. Simple functions that track aggregation error over time are found to perform better than more complex and theoretically sophisticated models. An explanation is given based on typical time series characteristics of economic data.

Notes

Lewbel (1990) shows that quadratic generalizations of the PIGL and PIGLOG systems attain Gorman's (1981) standard of generality and that these generalizations are analogous to the quadratic expenditure system developed by Howe et al. (1979) and perfected by van Daal and Merkies (1989).

See Anderson and Blundell (1983), Duffy (2001), Alston et al. (2002), De Mello et al. (2002).

This proposition can be extended to obtain an analogous result for the case of heterogeneous preferences (see the Appendix).

The relevance of an exact price index rather than the Stone index in the AI model is discussed by Deaton and Muellbauer (1980) and Pashardes (1993). However, both an exact price index and an exact aggregate income index are required for integrability conditions to hold with aggregate data.

In this model, Σ _i f ^k (p _t , M _it )/n is approximated by f ^k ( p _t , M _t ) + ∂f ^k (p _t , M _it )/∂M _it | _M [Σ _i (M _it  − M _t )/n] + ∂² f ^k (p _t , M _it )/

| _M [Σ _i (M _it  −M _t )²/2n] where specifically for the AI model ∂f ^k ( p _t , M _it )/∂M _it  = A( p _t )/p _kt  + γ _k lnM _it /p _kt  + γ _k /p _kt and ∂² f  ^k (p _t , M _it )/∂M _it ² = γ _k (1/M _it )(1/p _kt ). Model 3 follows upon noting that Σ _i  (M _it  − M _t )/n = 0.

The lognormal assumption is the most common for modelling income distributions in empirical work. Under lognormality (see Jorgenson, 1997, p. 327),

where ln M _t  = Σ_i lnM _it /n, in which one can further substitute

and

(see Greene, 1993, p. 60). Models 1 and 5 are thus equivalent under lognormality.

With heterogeneous tastes, the term Σ _i  γ _ik  M _it  ln M _it /n in the AI model is alternatively represented by sample covariance as (Σ _i γ _ik /n)(Σ _i M _it lnM _it /n) + Cov(γ _ik , Σ _i M _it lnM _it ). Thus, correlation between tastes and income in the AI model is equivalent to time varying intercepts. Thus, polynomial trends may reflect existence of heterogeneous tastes.

For US consumption, the PSID only reports disaggregate data on food expenditures. For the distribution of total expenditures, the Consumer Expenditure Survey (CES) only has comparable data after 1985 (see DeJuan and Seater, 1999). Nevertheless, a broad branch of applications of the AI model considers food consumption, for which weak separability is consistent with some empirical evidence and is widely assumed.

In the AI model, the income elasticity of the kth good is 1 +(γ _k /s _k ) where s _k is the share of the good. Therefore, if γ _k is −0.09 and s _k is 0.1, then the income elasticity is 0.01.

Detailed tables such as Tables 3 and 5 for cases with γ _k  = −0.07 (γ _k  = −0.03) for the income inelastic goods (food, housing, medical care) and γ _k  = 0.03 (γ _k  = 0.07) for the income elastic goods (clothing, transportation, and entertainment) are available upon request.

Again, the simulation results reported here are based on specific assumptions of elasticities but other simulation results too voluminous to report here verify considerable robustness. When increasing or decreasing γ _k from –0.03 to −0.07 for the income inelastic goods and from 0.03 to 0.07 for the income elastic goods, the ranking of models by aggregation bias was almost unaffected. For personal income data, {Model 6 or Model 7, Order 1} where best followed by {Model 7, Order 2, or Model 2}; Model 7, Order 2; {Model 7, Order 3, or Model 5}; Model 4; and Model 3, in that order (brackets reflect the only cases where orders were reversed depending on elasticities). For food expenditure data, Model 7 was best in every case although the dominant order was either 2 or 3, followed by Model 6, Model 3, Model 5, and {Model 4 or Model 2}.