ABSTRACT
The function p(t) is a control instrument for the private firms, and g(t), the control instru ment of the government. The result or the “sum” of p and g is the vector u, contained in the so-called “Pontyagin difference of sets. ” It is defined by
(3. 1. 2)
With this notation, we obtain the system
(3. 1. 3)
If the central bank is an independent player from the government with constraint set, we can split g as follows, g = a -b c, and the system becomes
Is this equivalent to
where
Our aim is to control
(3. 1. 4)
by manipulating the control vector u. It is sometimes more convenient to express (3. 1. 3) in the following form
(3. 1. 5)
The controllability problem can be expressed as follows: given any initial sate xo = x ( t0) at time to and any target x \ e E b, is it possible to steer x0 to x \ (= x(£i)) in finite time 1 1? One can consider the target Xi = (y i , R x, k x, L i , p 1, E x),
a national income growth rate of 5% when
interest rate, if jR0 =: 1 0 %,
This may be viewed as frill employment. Then p(t\) = pi (price for zero inflation), E (ti) = 0, zero cumulative trade balance, no deficit, which could mean fixed exchange rate. With x \ so defined as a target, the country wants to hit the target x \ in time t\. Is it possible? The controllability problem is to determine conditions on A and B , and u such that every solution of (3. 1. 5) with initial point x0 = (t/o> E 0) can be driven to x \ in time t \ . Thus using the resultant of government fiscal and monetary policies and the firms’ private initiative (consisting of consumption, investment, autonomous money hold ing), one wishes to attain the “happy” situation of zero deficit, high income growth rate, low interest rate, full employment and low prices and therefore low inflation. To prove this is possible is to solve the controllability problem.
