An expenditure based estimate of Britain's black economy revisited

The seminal paper by Pissarides and Weber (1989) is one of several previous studies trying to measure the size of the black economy. Pissarides and Weber compared the relationship between food expenditure and income in two groups of workers, self-employed and employees in employment, assuming that employees reported income correctly. For a given level of reported income, the self-employed had a higher food expenditure than employees. Pissarides and Weber concluded that self-employed's actual income was 1.55 times reported income, and that this part of the black economy was about 5.5 percent of GDP in the UK in 1982. Presumably due to a too informal argumentation, Pissarides and Weber's estimators are not entirely correct and alternative estimators have been overlooked. In all, I suggest three different interval estimators for mean under-reporting. The first is obtained by formally solving optimization problems which Pissarides and Weber tried to solve informally. The other two follows from recognizing, and incorporating, parameter restrictions which were not fully appreciated.