The decomposition of a matrix multiplier derived from a social accounting matrix (SAM) by Pyatt and Round [(1979). Accounting and Fixed Price Multipliers in a Social Accounting Matrix Framework. <italic>Economic Journal</italic>, 89, 850--873] has prompted a number of subsequent applications. In one of the earliest examples Stone [(1985). The Disaggregation of the Household Sector in the National Accounts, Chapter 8. In: G. Pyatt and J.I. Round (eds.) <italic>Social Accounting Matrices: A Basis for Planning</italic>. Washington, DC, The World Bank, 145--185] made the intriguing observation that the higher order (circular) effects of an exogenous change in final demand on the distribution of income and the structure of production were more or less independent of the sectoral composition of the initial injection. Our initial objective in this article is to explore this phenomenon of distributional invariance and to derive sufficient conditions for it. We then argue that these conditions have important implications for the design of SAMs, for the taxonomies they adopt and for levels of disaggregation, all of which strongly condition the quality of results that can be generated <italic>via</italic> subsequent modelling.
