Bródy’s Capital

  • Thijs ten Raa


The dynamic input-output model reads (Leontief, 1970)
$$x = Ax + B\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} + y$$
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, , 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.


Material Balance Material Input Output Coefficient Input Flow Convolution Product 
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  1. BRÓDY, A. (1974) Production, Prices and Planning, North-Holland, Amsterdam.Google Scholar
  2. LEONTIEF, W. (1970) ‘The Dynamic Inverse’, in A. P. Carter and A. Bródy (eds), Contributions to Input-Output Analysis, North-Holland, Amsterdam, pp. 17–46.Google Scholar
  3. MINC, H. (1988) Nonnegatieve Matrices, Wiley, New York, 1988.Google Scholar
  4. VON NEUMANN, J. (1945) ‘A Model of General Economic Equilibrium’, The Review of Economic Studies, 13, pp. 1–9.CrossRefGoogle Scholar
  5. TEN RAA, Th. (1986a) ‘Dynamic Input-Output Analysis with Distributed Activities’, The Review of Economics and Statistics 68, 2, pp. 300–10.CrossRefGoogle Scholar
  6. TEN RAA, Th. (1986b) ‘Applied Dynamic Input-Output with Distributed Activities’, European Economic Review 30, 4, pp. 805–31.CrossRefGoogle Scholar

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© András Simonovits and Albert E. Steenge 1997

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  • Thijs ten Raa

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