Phase Diagrams

Purpose

To exhibit visually a dynamic economy.
Analogous to supply and demand diagrams.
Exhibit movement of "state" and "control" variables.
Multiple equilibria and stability can often be seen.

Main examples

Consumption vs. Capital in the basic growth model.
Labor vs. Capital. (This extension of the basic model not common, but very revealing.)
Tobin's q. (Price q of capital vs. capital.)
Exchange rate and price level changes. (See Rudiger Dornbusch)

Main components

Two axes, say x (horizontal axis) and y (vertical axis).

These axes represent two variables.

Ex. 1: x = capital (a state variable), y = consumption (a control var.)

Ex. 2: x = capital (state variable), y = price of capital (a co-state variable)

Two curves: Define dx/dt = f(x,y), dy/dt = g(x,y). Curve (1) f(x,y) = 0, curve (2) g(x,y) = 0

Crossing point or points of the two curves. A crossing point is a steady state.

Direction arrows. At points off of the two curves dx/dt and dy/dt are not zero. The arrows indicate whether they are positive or negative. (Right arrow for dx/dt > 0, left arrow for dx/dt <0, up arrow for dy/dt >0, down arrow for dy/dt < 0.

An equilibrium path or paths. The equilibrium path is "what happens."

The equilibrium path follows the arrows. (The purpose of the arrows is to find the direction of the equilibrium path.)

When there is one steady state and it is a "saddle point," the equilibrium path is a "saddle path."

When there are multiple crossing points, there are multiple steady state equilibria. Often some are stable and others are unstable. Stable points have arrows pointing toward them, unstable have the arrows pointing away. Saddle points have some of each, some pointing in, others pointing out.

In physics, unstable means unlikely to occur in reality.

In economics, saddle paths are likely to occur in reality, despite being unstable (or at least half unstable). Saddle paths have a uniqueness property, which is desirable if you want to say with certainty what happens (in the model).

Typical analysis

First, the dx/dt = f(x,y) and dy/dt = g(x,y) conditions are obtained. In the standard growth model (x = capital, y = consumption), dx/dt comes from the givens of the economy. dy/dt comes from the optimization problem.

The curves representing dx/dt = 0 and dy/dt = 0 are drawn.

The steady states are computed (solve: dx/dt = 0, dy/dt = 0).

The arrows are drawn.

The equilibrium path is sketched.

Some shock is invoked. Technology is raised or lowered, capital is increased or decreased.

The new curves (dx/dt = 0 and dy/dt = 0) are drawn.

The new equilibrium path is drawn.

Movement from the old situation to the new situation is described.

When there are multiple equilibria, some analysis is needed in deciding which are most likely to arise.

Strengths, limitations

The phase diagram is not in time-space. Time is not on the horizontal axis. How long things take cannot be seen immediately from the phase diagram.

The phase diagram indicates co-movements among variables. In Tobin's q, the main result is that when the capital stock is high the price of capital is low and vice versa (a result that may seem trivial, but one which does not occur in the basic growth model).

Adjustments speeds can be inferred by the equilibrium paths, even if they cannot be read directly. A steep saddle path in the basic growth model indicates rapid convergence, whereas a flat curve represents slow adjustment.

An example: the basic growth model

Here, x = capital (k), y = consumption (c).
dk/dt = f(k,A) - delta k(t) - c(t)
dc/dt = (c(t)/gamma)(f'(k(t) - delta - rho)
delta = 0.08 (depreciation rate)
rho = 0.02 (time rate of preference)
gamma = 1.5 u(c) = (c^(1-gamma))/(1-gamma).
alpha = 0.25. f(k,A) = A k^alpha.
The picture shows the equilibrium movement as A changes from A = 10 to A = 20.
Saddle path (A = 10 to 20, gamma = 1.5)