San José State University
Department of Economics

applet-magic.com
Thayer Watkins
Silicon Valley
USA

 Phase Space Analysis of a Macroeconomic Model

Let Y represent the output of an economy; i.e., the Gross Domestic Product (GDP). Y is not only the aggregate output but also the aggregate buying power generated in the economy. The demand for the output of the economy is a function of the aggregate buying power and the real interest rate R; i.e.,

#### YD(Y, R)

If the output of an economy Y is greater than YD, the demand for the output, then output will decrease. Likewise if output is less than demand and output can be increased then it will. This can be represented as:

#### dY/dt = -α(Y - YD(Y,R))

where dY/dt is the rate of change of output and α is a positive parameter.

In the money market if the demand for money, DD(Y,R), is greater than the money supply MS then interest rates will increase. Likewise if the imbalance runs the other way then the interest rate will decrease. This relationship can be represented as:

#### dR/dt = -β(MS - MD(Y,R))

A phase space diagram is a graph that shows the direction of change at each point of the space of state variables, in this case (Y,R). At each point (Y,R) the values of dY/dt and dR/dt are determined and in the diagram the relative values of dY/dt and dR/dt are represented by an arrow. When a significant proportion of these arrows are shown in the diagram they describe a flow of the system toward equilibrium is an equilibrium exists and is stable.

In the ISLM diagram shown below the IS curve represents equilibrium in the product market. This is where dY/dt=0. The LM curve represents equilibrium in the money market; i.e., where dR/dt=0. The intersection of the IS and LM curves is where both markets are in equilibrium. This is where the system is in equilibrium.

In the above diagram any point above the red IS curve (or to right of it) has Y greater than demand so dY/dt is negative. Any point above the green LM has the quantity of money demand being less than the supply of money so the interest rate decreases; i.e., dR/dt is negative. These conditions divide the phase space into four regions as shown in the diagram below.

The arrows can be placed with an appropriate direction in the four regions. A very rough depiction of the flow lines are shown below. The strictly spirally flow lines cannot be right because the flow line arrows must be vertical on the IS curve and horizontal on the LM curve.

The phase space diagrams give a means of conceptualizing the dynamics of an economy.

(To be continued.)