First we consider the existence question in Sraffa’s Chapter I dismissed by counting equations and unknowns. A theorem from the theory of Markov processes, applied to distributions not now of probability but of goods to sectors, shows the general existence of non-negative prices satisfying the conditions imposed by the value equation, that value of output equals value of input. The further condition for these to be unique and positive is that the economy be irreducible, or that no independent sub-economy should exist. Sraffa provides a precise formula determining unique prices, he barely escapes imposing too many conditions on them and certainly cannot require more. In the background and giving motive to the enquiry is the Labour Theory of Value, that goes further. It asserts that the value of anything is ultimately equal to the labour that has gone into making it; so it implies the same principle expressed by the value equation, but a further condition has been added about the nature of the unit. Since the value equation alone makes prices fully determined, there is no room for further conditions, so there is an obstacle to the application of the theory. Standing as a canonical text in a revival of interest in the Theory Of Value serving earlier thought and the later concentration of Ricardo, it offers an exercise in labour value arithmetic, where the only fruit is to find the arithmetic is impossible. An extended interdependence, which applies to repeated production, appears as a stability condition for prices in an adjustment process, and so does the existence of what Sraffa calls a standard commodity, one depending on all others for its production. After treating a case where there is a surplus, and joint production, the relation with Leontief and von Neumann is considered.
