Dynamic models of the firm. Determining optimal investment, financing and production policies by computer (Q1910356)

The book is devoted to the application of optimal control (OC) theory to the problem mentioned in the title. The basic model of a firm connects the next main variables: capital assets \(K\), equity \(X\), investment rate \(I\), production rate \(Q\), labor input \(L\), debt \(Y\) and dividend rate \(D\). The main equations are: \[ \dot K=I(t)-aK(t),\quad K=X+Y, \] \[ \dot X= (1-f)[S(Q(t))- wL(t)-aK(t)-rY(t)]- D(t). \] In addition, \(Q(t)=k^{-1}K(t)\), \(L(t)=lQ(t)\) and \(I\) and \(D\) are control variables. The function \(S(Q)\) presents the firm turnover. The control possibilities are bounded by conditions: \(0\leq Y(t)\leq bX(t)\), \(I(t)\geq 0\), \(D(t)\leq 0\). The firm aims to maximize \[ \int^z_0 e^{-it}D(t)dt+ e^{-iz}X(z), \] where \(i\) is shareholder's time preference rate. The decision procedure for the formulated OC problem, complicated by intermediate phase constraints, is based on optimality conditions (Lagrange type) and a heuristic ``path coupling procedure''. As its result the set of paths which can complicate an optimal process is obtained. Each path is separate from the others by differences in the set of active constraints. The optimal string of paths is evaluating numerically by means of a time discretization procedure. This approach is realized for the basic firm dynamic model as well as for its following developments: with start-up costs with a variable utilization rate, with a cash balance, with an inventory of finished goods and others.