When does specification or aggregation across consumers matter for economic impact analysis models? An investigation into demand systems

Abstract

Economic impact analysis simulation models frequently rely on some kind of representation of consumption behavior. However, the sensitivity of such results with respect to the choices of the specification and the level of aggregation across consumers has not yet been thoroughly examined. We exploit a unique dataset to simulate various stereotypical scenarios and investigate the influence of the choice between six demand system specifications and household-level versus national-level models on several outcome measures. We find that both choices have a large influence on simulation results and thus on policies deduced therefrom. Our results point to pragmatic recommendations for various settings.

Appendices

Appendices

A: The quadratic almost ideal demand system (QUAIDS)

The QUAIDS was introduced by Banks et al. (1997) and is an extension of the AIDS due to Muellbauer(1980b, see Sect. 3.5). It is defined by the budget share Eq. (37), the price indices a(p) (38) and b(p) (39), and restrictions to impose adding up (40), homogeneity (41), and symmetry (42). The income and price elasticities are defined by Eqs. (43) and (44)

$$\begin{aligned} w_i&= \alpha _i + \sum _{j \in I} \gamma _{ij} \ln p_j + \beta _i \ln \left( \frac{M}{a(p)} \right) + \frac{\lambda _i}{b(p)} \left( \ln \left( \frac{M}{a(p)} \right) \right) ^2 \quad \forall i \in I \end{aligned}$$

(37)

$$\begin{aligned} \ln a(p)&= \alpha _0 + \sum _{i \in I} \alpha _i \ln p_i + \frac{1}{2} \sum _{i \in I} \sum _{j \in I} \gamma _{ij} \ln p_i \ln p_j \end{aligned}$$

(38)

$$\begin{aligned} b(p)&= \prod _{i \in I} p_i^{\beta _i} \end{aligned}$$

(39)

$$\begin{aligned} \sum _{i \in I} \alpha _i&= 1 ,\quad \sum _{i \in I} \beta _i = 0 ,\quad \sum _{i \in I} \lambda _i = 0 ,\quad \sum _{i \in I} \gamma _{ij} = 0 \quad \quad \forall j \in I \end{aligned}$$

(40)

$$\begin{aligned} \sum _{j \in I} \gamma _{ij}&= 0 \quad \quad \forall i \in I \end{aligned}$$

(41)

$$\begin{aligned} \gamma _{ij}&= \gamma _{ji} \quad \quad \forall i,j \in \{I | i<j\} \end{aligned}$$

(42)

$$\begin{aligned} \eta _i&= 1 + \frac{1}{w_i} \left[ \beta _i + \frac{2 \lambda _i}{b(p)} \ln \left( \frac{M}{a(p)} \right) \right] \quad \quad \forall i \in I \end{aligned}$$

(43)

$$\begin{aligned} \varepsilon _{ij}&= -\delta _{ij} + \frac{1}{w_i} \left[ \gamma _{ij} - \left( \beta _i + \frac{2 \lambda _i}{b(p)} \ln \left( \frac{M}{a(p)} \right) \right) \right. \left( \alpha _j + \sum _{k \in I} \gamma _{jk} \ln p_k \right) \nonumber \\&\quad -\, \left. \frac{\lambda _i \beta _j}{b(p)} \left( \ln \left( \frac{M}{a(p)} \right) \right) ^2 \right] \quad \forall i,j \in I \end{aligned}$$

(44)

with Kronecker $\delta _{ij}$ (see Sect. 3.3). The QUAIDS has $(n^2 + 3n)/2 - 1$ independent parameters and is thus flexible. In comparison with the rank two AIDS that restricts budget shares to be linear in the log of income, the QUAIDS is of rank three and adds flexibility to the Engel curves representable by adding a term quadratic in the log of income. As a consequence, the QUAIDS allows income elasticities to vary with the level of income such that an item might be a luxury for low and a necessity for high income levels. The QUAIDS specification does not guarantee compliance with the monotonicity or curvature properties and thus might not be regular even for the price-income points of the sample observations. McLaren and Yang (2016) state that its regularity properties are not better than those of the AIDS. The QUAIDS does not impose any bounds on the value ranges of the income and price elasticities.

B: Generalized cross-entropy estimation model

The goal of this estimation is to generate a theoretically consistent set of parameters for a specific demand system which precisely replicates the given budget shares $w_i=\frac{p_i q_i}{M}$ ($q_i$ denotes the quantity demanded) at a given income (M) and a vector of prices $p_i$ while fitting a given set of prior income $\overline{\eta }_i$ and price elasticities $\overline{\varepsilon }_{ij}$ as closely as possible, where $i,j \in I$ and I denotes the set of consumption items.

Equations (45) to (50) form the core model of the GCE estimation, which then needs to be amended with the specific constraints of the particular demand system model:

$$\begin{aligned} \min H(\pi )= & {} \sum _{i \in I} w_i \sum _{l \in L} \left( \pi ^\eta _{il} \ln \dfrac{\pi ^\eta _{il}}{\overline{\pi }^\eta _{il}} + \pi ^\varepsilon _{iil} \ln \dfrac{\pi ^\varepsilon _{iil}}{\overline{\pi }^\varepsilon _{iil}} \right) \nonumber \\&+ \left( \sum _{i \in I} \sum _{j \in \{I|j \ne i\}} w_i w_j \sum _{l \in L} \pi ^\varepsilon _{ijl} \ln \dfrac{\pi ^\varepsilon _{ijl}}{\overline{\pi }^\varepsilon _{ijl}} \right) \end{aligned}$$

(45)

subject to

$$\begin{aligned} \eta _i&= \sum _{l \in L} \pi _{il}^\eta z_{il}^\eta \quad \forall i \in I \end{aligned}$$

(46)

$$\begin{aligned} \varepsilon _{ij}&= \sum _{l \in L} \pi _{ijl}^\varepsilon z_{ijl}^\varepsilon \quad \forall i,j \in I \end{aligned}$$

(47)

$$\begin{aligned} \sum _{l \in L} \pi _{il}^\eta&= 1 \quad \forall i \in I \end{aligned}$$

(48)

$$\begin{aligned} \sum _{l \in L} \pi _{ijl}^\varepsilon&= 1 \quad \forall i,j \in I \end{aligned}$$

(49)

$$\begin{aligned} \pi _{il}^\eta ,\, \pi _{ijl}^\varepsilon&\ge 0 \quad \forall i,j \in I, \, l \in L \end{aligned}$$

(50)

The cross-entropy objective function (45) minimizes the distance—as measured by the Kullback-Leibler measure (see, e.g., Golan et al. 1996) of information divergence—of the estimated probability distribution $\pi _{i(j)l}^s$ from their prior probabilities $\overline{\pi }^s_{i(j)l}$, where $s \in S=\{\eta , \varepsilon \}$ indicates parameters associated with the income and price elasticities, respectively. The prior probabilities are chosen such that their implied expected value is equal to the prior elasticity. The importance of each item’s elasticity with respect to the entire demand system is assumed to be proportional to the corresponding budget share $w_i$ which enters the objective function as a weight. Since the number of cross-price elasticities is quadratically related to the number of items, these are weighted by the product of the budget shares of the two items involved $w_i w_j$.²¹ Equations (46) and (47) link the income and price elasticity variables $\eta _i$ and $\varepsilon _{ij}$ to their expected values, defined as the sum over the support points $z_{i(j)l}^s$ (indexed by $l \in L$) multiplied by their respective probabilities $\pi _{i(j)l}^s$. Equations (48) and (49) ensure that probabilities sum to one. In addition, restrictions (50) confine probability variables to the nonnegative domain.

The vector of support points associated with a specific elasticity defines the support range in which the estimated elasticity is allowed to fall. Since the estimated elasticity, for example, $\eta _i$, is calculated as the sum of all support points multiplied by their associated probabilities (Eq. 46), the prior elasticity is chosen as the midpoint of the support point vector $z_{il}$ and the support points extend equidistantly from each other and symmetrically around the midpoint. For example, if the prior income elasticity is $\eta _i=1.2$, a support space of $|L|=5$ points is chosen and it is assumed that the error around the elasticity is uniformly distributed (the prior distribution) $\pi _i^\eta = \overline{\pi }_i^\eta = (0.2,0.2,0.2,0.2,0.2)$, then $z_i^\eta = (0.2,0.7,1.2,1.7,2.2)$ so that Eq. (46) initially evaluates to the prior income elasticity $\overline{\eta }_{i}$. Alternatively, the error might be assumed to be normally distributed around the prior elasticity, which can be accomplished by assigning different probabilities to $\overline{\pi }_i^\eta$ which approximate a normal distribution function. This attributes a higher probability to a support point the closer it is to the midpoint of the support range and thus assumes there is information about the distribution of the error around the prior elasticity.

To complete the GCE demand system estimation model, the core GCE model needs to be complemented by equations which define the demand system (budget share equations and parameter domain definitions) and its income and price elasticities.