# States

## summary

the state of a system is the magnitude of a set of variables which are considered to represent some aspect of reality. The state of my bank account could thus be given by the following magnitudes which my bank reports to me if I log in:

- the balance on my chequing account
- the balance on my savings account

I can predict the immediate future state of my account from the following additional variables:

- the cheques I have written which are due to be presented by my debitors
- the cheques which I have submitted to the bank which have been issued by my creditors
- other payments in transit

If I wish to predict the state of my account in the medium term I need some further information which can be expressed in more variables:

- the contracts I have undertaken which legally oblige me to pay such as my gas bills
- contracts which my creditors are legally obliged to pay such as my wages

If I wish to go still further into the realms of uncertainty I can add such variables as

- the fine I may have to pay if the camera at the junction or Pomp and Circumstance caught me and the police decide to prosecute
- the money I will receive if my lottery ticket pays off

All these variables, taken together, constitute the state of my personal finances at any given time.

Another and simpler example of a state is a ball thrown in the air. At any given time, this is described by

- its position
- its mass
- its speed

A trajectory is a succession of states in time, such that, at any given time, the state of the system is uniquely defined (so, for example, the state of my bank account is given by the number that appears on my screen when I ask my bank how much money I have and not by some approximate number that depends on whether my aunt gives me a christmas gift or even whether the cheque in the post reaches my account tomorrow).

A simulation provides a set of definite rules which describe how a state at any given time can be calculated from a state at a previous time. The result of a simulation is a trajectory.

## definition

The state of a dynamic system is most rigorously defined in state space theory but is applied widely in the physical sciences. In particular it is to be found in mechanics and its interface with quantum mechanics, but is in quite general use in describing electromagnetic and other physical phenomena. The notion of state features widely in economics although it is rarely subjected to critical examination.

In the above contexts, the state of a system refers to the magnitude of a definite and usually finite set of variables at a definite point in time, which are considered to describe or represent some aspect of reality which can be treated, for the purpose of study, as relatively autonomous, or self-contained.

In state space theory these variables are divided into the two general categories of exogenous and endogenous. Loosely speaking the exogenous variables are those whose magnitudes are defined (in a simulation) or caused (in reality) in ways that cannot be accounted for by any set of mathematical relations between the remaining, endogenous variables.

## example

If an experimenter throws a ball in the air, then the trajectory of the ball is uniquely described by two variables, its speed and its height. What happens to the ball after it is released depends only on

- gravitational acceleration
- the initial speed given the ball by the experimenter, that is to say, how 'hard' it is thrown
- the initial direction given the ball by the experimenter

Provided we know these three exogenous variables, and provided we ignore all other possible causes of where the ball might be such as wind, magnetism, or somebody else getting in the way, we can work out where the ball will be at any point in time after it is thrown.

The ball then passes through a series of states, each given by its speed and height.

At any given moment in time, its speed and height constitute its *state*.