Agricultural Household Model

Separability (or recursivity) means the household can solve its production problem independently of its consumption problem. Under complete markets, the farm is run to maximise profit at given prices, and that profit then funds consumption like any other income; production decisions do not depend on the household's tastes or size. When a market fails — typically for labour — this independence breaks, a shadow price replaces the market price, and production and consumption must be solved jointly. Separability is therefore the dividing line between treating the farm as a standalone firm and treating it as inseparable from the consuming household.

Benjamin's test exploits a clean implication: under separation, farm labour demand (and other production choices) should be determined by prices and fixed factors, not by household demographic composition, because the household can adjust labour through the market. So you regress on-farm labour use on household composition while controlling for farm size and prices. If composition has no significant effect, separation is supported and the recursive model is valid; if larger or differently structured households systematically farm more intensively, separation is rejected, signalling a labour-market failure and the need for a non-separable model.

Because the household is both producer and consumer of the good. A higher price for, say, rice raises the profit from growing rice (a production incentive to supply more) but also raises the cost of the rice the household eats and increases its real income as a net seller (income and substitution effects on the consumption side). These effects can partly offset, so the net change in marketed surplus is theoretically ambiguous and can even be negative for a net-consuming household. Capturing this interaction is precisely what the agricultural household model adds over a pure firm or pure consumer analysis.