Equivalence Scale

Definition

An equivalence scale is a factor that adjusts household income or expenditure to account for differences in household size and composition, enabling welfare comparisons across household types on a common per-equivalent-adult basis. A scale of 1.23, for example, means a household of type h needs 23% more income than the reference household to achieve the same standard of living.

Key Ideas

• Commodity-specific scales $s_{ih} = a_{ih}/a_{ir}$ relate the expenditure on commodity i for household type h to that of a reference type r, holding utility constant
• General equivalence scales aggregate across commodities by equating indirect utility functions across demographic types
• The Extended Linear Expenditure System (ELES; Lluch 1973) provides a tractable demand-system framework for estimating equivalence scales: $y_{ih} = \theta_{ih} + \eta_i \cdot x_h$, where $\theta_{ih}$ are set-specific subsistence quantities and $\eta_i$ are common marginal budget shares across demographic types
• Bayesian estimation via multi-set seemingly unrelated regression (SUR) Gibbs allows pooling of marginal budget share information ($\eta$) across demographic groups while allowing group-specific baseline expenditure ($\Theta_h$) and error covariance ($\Omega_h$)

How It Works

In the ELES framework (Griffiths-Valenzuela 2002):

1. Each household type h forms a set of SUR equations — one per commodity — with the same 11 commodities for each of 8 demographic types (H=8)
2. $\Theta_h$ (set-specific intercepts, the $\theta_{ih}$ parameters) are estimated separately per demographic group
3. $\eta$ (marginal budget shares $\eta_i$) are constrained equal across demographic groups — the cross-set restriction
4. Commodity-specific scales are recovered as $s_{ih} = \hat{a}_{ih}/\hat{a}_{ir}$ (ratio of estimated intercepts to reference type)
5. General scales are computed by equating indirect utility functions across types, yielding a single summary adjustment factor per household type

Griffiths-Valenzuela (2002) estimate that a first child increases the two-adult household budget requirement by approximately 23%, with economies of scale for a second child (scale increment smaller than for the first).

Why It Matters

• Equivalence scales underpin poverty measurement, social benefit design, and inter-household welfare comparisons
• The multi-set SUR structure is natural: demographic groups share the same commodities (same $\eta$) but differ in baseline consumption patterns (different $\Theta_h$) and residual correlations (different $\Omega_h$)
• Bayesian posterior distributions over equivalence scales propagate all estimation uncertainty, enabling credible intervals rather than point estimates alone

Open Questions

• Equivalence scales are not identified from expenditure data alone without additional restrictions on the form of the utility function; ELES restrictions (linear Engel curves) may be rejected by the data
• General scales depend on the base utility level chosen for comparison — different reference points yield different scales
• The assumption of common marginal budget shares ($\eta$) across demographic types is testable but rarely formally tested in applied work

Related

• Griffiths-Valenzuela (2002)
• Seemingly Unrelated Regression
• Gibbs Sampler
• William E. Griffiths
• Ma. Rebecca Valenzuela