A note on a closed input-output model with finite life-times and gestation lags

Abstract

The presentation of the dynamic linear input-output model by Lange [4] in 1957 has recently been followed by some studies investigating the properties of such models. See, e.g., Brady [ 1) and Johansen [2]. In Lange’s model production requirements stem from current use of commodities as well as from growth of stocks. Brady [ 1 ] in his closed input-output model used disaggregated capital-output coefficients (as did Lange); mathematically, the model is posed as an eigenequation. Recently Johansen [3] has generalized Brady’s model by assuming gestation lags and finite life-time of capital equipment. (The concept of gestation lags in an input-output framework may also be found in an article by Lange [S] written in 1959.) The purpose of this note is to present an alternative proof and a solution algorithm in a slightly generalized Johansen model. In our model the productive capacity of investment changes by age, but in both models investment may require more than one time period for its construction, and capital goods have different finite life-times. By a mathematically appropriate formulation of the model, the existence of a unique solution, characterized by balanced growth and full capacity utilization, is easily proved. Furthermore, the mathematical formulation of the non-linear system of equations suggests a simple solution algorithm.