2. Basic principles

16 October 1999

To Chapter 1

  1. State spaces
  2. Exercises.
  3. References.

2.1 State spaces

Part 1 of Economic Circular Flows described how to build and use static models. Static models show the situation, state, of the economy at a certain time instant. It shows what are the possible combinations of flows in the economy.

Dynamic models can show what state can follow after a previous state. A starting point has to be defined, then successive states can be calculated for later times. The values of a minimum number of variables have to be known in order to define the state of the economy. They are called the state variables of the system. State variables can be chosen in different ways but their number is always the same and equal to the degrees of freedom of the system.

Take a system with two state variables y1 and y2 . It is fully defined at time t by a point p(t) in a two-dimensional space. See figure below:

Figure 2.1:1 A point in the state space.

There may be many other variables x1, x2, … , xn that are functions of y1 and y2. They are called dependent variables and not state variables. If we have two states, state a and state b, then we can show them as two points in the state diagram:

Figure 2.1:2 Two different states a and b.

The two points pa and pb represent two different states of the system but we do not know anything about the time of each point. We can calculate the difference between the dependent variables (x1, x2, … , xn)b and (x1, x2, … , xn)a but we do not know when and how the change took place since we have no time dimension of the problem.

In order to study the dynamic behavior of the system we must introduce trajectories in the state space. The trajectories have time coordinates ta and tb along the trajectory curve. The figure below shows two different trajectories. They both pass the point p(ta).Two possibilities for the point p(tb) are shown, one on each curve.

Figure 2.1:3 Trajectory curves in the state space.

Knowing the points p(tb) and p(ta), we can calculate the velocity between the two points. The velocities often correspond to flows in our economic models. But how do we know which trajectory the system will follow, the solid curve or the dashed curve or any other curve? For that we need expressions of the derivatives d y1/dt and d y2/dt or the differences y1(tb) - y1(ta) and y2(tb) - y2(ta). These are the dynamic equations of the system.

1.6 Exercises


1.7 References

  1. Ref.1

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