Balanced Growth

In the von Neumann balanced growth model, an input of goods this period produces an output of goods next period. The inputs and outputs are the same goods, and there is no consumption of goods. Each period all inputs and outputs grow at the same rate. There exists a maximum rate of balanced growth, and furthermore this balanced growth is attainable as a production equilibrium. The technology is activity analysis, with constant returns to scale. The total input of goods is Ax, and the total output is Bx, in which x denotes the vector of activity levels. We analyze the saddle-point problem $$\min _{\boldsymbol {y}^{\ast }}\left ( \max _{\boldsymbol {x}}\frac {\left \langle \boldsymbol {y}^{\ast },{\textit {{\sf { {B}}}}}\boldsymbol {x}\right \rangle }{\left \langle \boldsymbol {y}^{\ast },{\textit {{\sf { {A}}}}}\boldsymbol {x}\right \rangle }\right ) \!.$$ The ratio is the output value divided by the input value. The numerator and the denominator are both positive. We establish that a saddle point $$\left \langle \boldsymbol {x},\boldsymbol {y}^{\ast }\right \rangle$$ exists. Activityx attains the maximum feasible balanced growth, and the saddle value is the growth factor. Activity x is a production equilibrium for price y*, in which the interest factor equals the growth factor. If the profit of an activity is negative, then the activity level is zero. If a good is in excess supply, then its price is zero. We analyze the saddle-point problem as two standard fractional programs. Together, the two zero-conjugate conditions amount to finding a Nash equilibrium for a two-person zero-sum game.