Phase Diagrams

BIBLIOGRAPHY

Phase diagrams display the characteristics of dynamical systems. They include fixed points (equilibrium points); isoclines, which subdivide the phase space into different vector forces; trajectories, which show particular paths of the system over time; and information on the stability/ instability properties of the dynamical system. Phase diagrams have been employed in a number of areas in the social sciences. In economics, for example, they have been used in environmental economics, growth theory, and macroeconomics—most recently in structural macroeconomics. They can be one-dimensional (called a phase line ) or two-dimensional, involving a two-dimensional phase space. Phase diagrams can also be used to display discrete systems of difference equations or continuous systems of differential equations—although the latter is the most common.

A deterministic dynamical system has three elements: (1) a set of equations showing the direction of motion; (2) a set of parameters; and (3) a set of initial conditions. Given these elements, the trajectory of the system is defined and can be plotted in a phase diagram. To illustrate these concepts, consider the simple continuous dynamical system in Figure 1.

ẋ (t ) and ẏ (t ) define the system’s motion for the variables x and y, which are a continuous function of time, t. The system has a set of parameters a, b, c, d, e, and f. The initial condition is x (0) = x ₀ and y (0) = y ₀. This system is autonomous since it does not involve time as a separate variable. A nonautonomous system, for example, might have an equation such as ẋ (t ) = a + bx (t ) + cy (t ) + he^gt. This is important because the mathematical properties of dynamical systems apply largely to autonomous systems.

One of the first considerations of a dynamical system is whether it has a fixed point: an equilibrium point. A fixed point for a two-dimensional system is where x (t ) = x ^*and y (t ) = y ^* for all t. In other words, the system remains at this position. When systems represent economic or social systems, such fixed points identify equilibrium states: They denote a balance of forces and so remain in

that position unless something exogenous, such as a shock, disturbs the system. Fixed points are identified by the conditions ẋ (t ) = 0 and ẏ (t ) = 0. In the example in Figure 1, therefore, the equilibrium is where x ^* = 6 and y ^* = 9. Because this example is linear, there is only one equilibrium point.

But far more meaning can be given to the equations in the example when they refer to economic or social systems. For example, the first equation may refer to the goods market in a macroeconomic model of the economy, and the second equation may refer to the money market. Each of these markets can be in equilibrium separately or simultaneously. The first is in equilibrium when ẋ (t ) = 0 for all t, while the second is in equilibrium when ẏ (t ) = 0 for all t.

Such conditions for equilibrium in each market separately identify isoclines. Isoclines represent partitions in the phase plane. Each side of the isoclines represents a disequilibrium state. More importantly, each side of the isocline represents vector forces pushing one of the variables in a particular direction. In the example in Figure 1, there are two isoclines. Along the ẋ (t ) = 0 -isocline market, ẋ is in equilibrium; along the ẏ (t ) = 0 -isocline market, y is in equilibrium. The system is in total equilibrium when x and ẏ are simultaneously in equilibrium. This must be where the two isoclines intersect. Figure 2(a) illustrates this example, and point E denotes the fixed point of the system. This figure also shows a typical phase diagram in two-dimensional space. The initial point is denoted A. The differential equations indicate the conditions by which x and y will increase or decrease over time. But exactly whether x is increasing (decreasing) or y is increasing (decreasing) will depend on the sign of ẋ (t ) and ẏ (t ) at some time t. For this reason we need to identify the vector forces in each of the quadrants.

The isoclines in this example divide the phase plane into four quadrants (areas). In each quadrant the market for x (or y ) is in disequilibrium. When the market is in disequilibrium, it will change over time according to the nature of the differential equations (for continuous systems) or the difference equations (for discrete systems). The vector forces are shown by the arrows in the diagrams. They supply qualitative information about the movement of the system over time. For example, the vector forces in Figure 2(a) show that the system will move in a counterclockwise direction over time. Although the system will move between the vector forces in any quadrant, only a detailed specification of the equations can indicate whether the horizontal force or the vertical force is dominant. With autonomous systems there is only one trajectory through point A, and this is the path AE.

Consider another example illustrating a different application. Europeans discovered the Polynesian civilization

of Easter Island in 1722 with a population of about three thousand and a set of enormous statues (moai ). The island itself was first settled around 400 CE, when there was a large palm forest, although the island was virtually treeless in 1772. The palm was a natural resource used for a variety of purposes, and it sustained the population. However, the island’s rising population, which at its height built the enormous statues, led to deforestation and, along with violent conflict, to the eventual collapse of the Easter Island civilization. Based on information about population and resources, the rise and fall of this civilization has been modeled, and is illustrated in the phase diagram in Figure 3(c) and the time profile in Figure 3(d).

The trajectory illustrates that as the population expanded, the resource base declined, with the population reaching a peak of about ten thousand around 1400 CE. Analytically, we have a stable clockwise spiral. Sociologically, what was so particular about Easter Island? A major characteristic is that the forest involved a slow-growing palm (the Jubea palm), which grew nowhere else in Polynesia. The other Polynesian islands grew the coconut palm and the Fiji fan palm, which have a much shorter fruit-growing age—and none of these islands exhibited the same rise and fall noted on Easter Island.

A feature of dynamical social systems is asymmetric behavior. If, for example, the variable on the x -axis adjusts more quickly than that on the vertical axis, then the system will move in a more horizontal direction in any given quadrant. Another feature illustrated by the vector forces is the possible existence of saddle-path solutions. Systems can generate unstable paths except for a saddle-path, which directs the system to the equilibrium. Such saddle-paths are a particular feature of economic systems that postulate rational expectations. In example (b) of Figure 2, the saddle-path for the Ramsey growth model denotes a balanced growth path. Saddle-path solutions, however, highlight an unsatisfactory feature of such modeling. There is no reason for the system to initially be on the saddle-path, and if it is not, there needs to be some mechanism to move the system to the one and only path that will take the system to equilibrium. A third feature that dynamical systems highlight is the possibility that systems will follow orbital paths around the equilibrium, never actually moving away from or toward the equilibrium. A major consideration, therefore, of dynamical systems is whether they are stable or not. Many of the features just mentioned, however, are more likely to occur in systems that involve nonlinear isoclines, such as the Ramsey growth model.

SEE ALSO Comparative Dynamics; Cumulative Causation; Differential Equations; Hume Process; North-South Models; Stability in Economics; Taylor, Lance

BIBLIOGRAPHY

Azariadis, Costas. 1993. Intertemporal Macroeconomics. Oxford: Blackwell.

Brander, James A., and M. Scott Taylor. 1998. The Simple Economics of Easter Island: A Ricardo-Malthus Model of Renewable Resource Use. American Economic Review 88 (1): 119–138.

Lynch, Stephen. 2001. Dynamical Systems with Applications Using MAPLE. Boston: Birkhäuser.

Sandifur, James T. 1990. Discrete Dynamical Systems: Theory and Applications. Oxford: Clarendon.

Shone, Ronald. 2002. Economic Dynamics: Phase Diagrams and Their Economic Application. 2nd ed. Cambridge, U.K.: Cambridge University Press.

Taylor, Lance. 1983. Structuralist Macroeconomics: Applicable Models for the Third World. New York: Basic Books.

Ronald Shone