The equations representing the economy and its control
 

The controlled economy as treated for the basic model of the present discussion has three variables: price, inventory - or stocks of goods and services for sale - and money in circulation.  Consequently three equations are required for its solution.  All quantities are treated as proportional changes from some target value.
 
The first equation is the statement of the control, as given in the first part of  "The Plan", linked below.  The control specifies that the rate of increase of the money in circulation is made equal to constants multiplied by price deviation, stocks deviation, price rate of change and stocks rate of change.  The values of the constants have to be evaluated during the optimization.
 
The second equation says that the stocks available for sale decreases when there is excess money in circulation and increases when the price is high.

The third equation represents the observation that prices are raised when stocks are short and prices are reduced when stocks are excessive. 

 
The solution to the equations

 
The three equations are linearised and solved to give a third order differential equation.  The writer found it convenient to work in terms of price.
 
The standard solution for such equations allows the price to be given as the sum of three exponentials in time, with the coefficients open to choice to fit the initial conditions.  The powers in the three exponentials may be found as the three roots of a cubic equation.
 
The above derivation gives the solution for any prescribed control operating in any free economy and for any initial conditions.  

 
The optimum control and the optimum economy


 
In order to find the optimum control and economy it is postulated by definition that with the optimum control the economy recovers from a disturbance in the shortest possible time.
 
The solution for the controlled economy obtained as above is a system with three natural modes of response.  Such a system will respond to an initial disturbance by moving in some combination of all its natural modes.  In general the modes will vary in their decay rate, so that the more rapidly decaying modes will tend to die away first, leaving the more slowly decaying modes still giving a disturbance.  Provided, then, that there is no reason why the initial disturbance should be such that the most slowly decaying mode should be scarcely excited - and the economy seems to satisfy that proviso - then, in overall terms, the total response of the system will die out more quickly the more quickly the most slowly decaying mode decays.  The economy would therefore have its shortest possible response, and therefore it's best possible response, and it's fastest recovery from the disturbance, when its most slowly decaying mode has as fast a decay rate as possible.  The corresponding condition for the solution as indicated above is a condition on the cubic equation for the powers in the three exponentials.  The condition is that in that equation the root with the smallest decay component should have that component as large as possible.
 
Now a standard condition which is readily demonstrated for any cubic equation is that the sum of the three roots is equal to minus the coefficient in the second order term in the equation.  A continuation of the present discussion then gives that for the optimum the sum of the three decay components are all equal.  Further discussion, considering qualitatively the undesirable effects of the oscillatory components' disturbances, gives that the optimum responses are non-oscillatory.  The remainder of the solution for the optimum control then follows, and leads to the discussion under the heading: "Significance of critical damping and time lags" in "The Plan", linked below.