On the Stability of Certain Economic Systems

possess negative real parts. The answer to this question given by James and Belz in [4] is not very useful from a practical point of view for it amounts essentially to the determination of the roots of the characteristic equation themselves. A stability criterion using only the given coefficients and not the roots of the characteristic equation was given by Hayes in [3]. But here, too, the transcendental auxiliary equation x* cot x = c must be solved first. In this paper we present another proof of the stability criterion using the so-called graphical methods of control engineering, i.e., Cauchy's theorem of residues. We thereby arrive at an expression for the criterion which seems to be better adapted to practical purposes (because with given coefficients only the auxiliary equation cos x = c has to be solved). The connection with the other forms of the stability criterion will be clarified and the criterion will be illustrated by an example from economics. For general discussion of these problems and further economics examples, compare Tustin [6].