Bródy’s Capital

Abstract

The dynamic input-output model reads (Leontief, 1970)

$$x = Ax + B\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} + y$$

(1)

The left-hand side features the state variable of the economy, the vector of sectoral capacities, as measured by the output levels. The right-hand side lists the material inputs, investment, and household demand, respectively. The structure of the economy is given by two matrices of technical coefficients. A is the matrix of input flow coefficients and B is the matrix of input stock coefficients. Input flows, for example electricity, are fully consumed, but input stocks, like housing, carry over. Material inputs are, therefore, proportional to the output levels, but investment is proportional to the new capacity, ẋ, where the dot denotes the time derivative. Output x and household demand y are functions of time, but the technical coefficients are constant in the absence of structural change. Implicit in the dynamic input-output model is the assumption that productive activity is instantaneous. If you have the commodity vectors a.₁ and b.₁ (the first columns of technical coefficients matrices A and B), then you get instantaneously the commodity vectors e₁ and b.₁, where e₁ is the first unit vector, representing the output flow, and b.₁ is the carry-over stock. In ten Raa (1986a) I have dropped this assumption, redefining an input flow coefficient as a time profile on the past.