the Jacobian matrix of first derivatives evaluated locally at equilibrium (Y*, K*), call it A, has determinant:
|A| = a (IY - SY) IK - a (IK - SK) IY, = a (SKIY - IKSY)
where, since IK < 0 and SK, SY, IY > 0 then |A| > 0, thus we have regular (non-saddlepoint) dynamics. To examine local stability, the trace is simply:
tr A = a (IY - SY) + IK
whose sign, obviously, will depend upon the sign of (IY - SY). Now, examine the earlier Figures 3 and 4 again. Notice around the extreme areas, i. e. around YA and YC, the slope of the savings function is greater than the slope of the investment function, i. e. dS/dY > dI/dY or, in other words, IY - SY < 0. In contrast, around the middle areas (around YB) the slope of the savings function is less than the slope of the investment function, thus IY - SY > 0. Thus, assuming Ik is sufficiently small, the trace of the matrix will be positive around the middle area (around YB), thus equilibrium B is locally unstable, whereas around the extremes (YA and YC), the trace will be negative, thus equilibrium A and C are locally stable. This is as we expected from the earlier diagrams.
To obtain the phase diagram in Figure 5, we must obtain the isoclines dY/dt = 0 and dK/dt = 0 by evaluating each differential equation at steady state. When dY/dt = 0, note that a [I (Y, K) - S (Y, K)] = 0, then using the implicit function theorem:
dK/dY|dY/dt = 0 = - (IY - SY) / (IK - SK)
Now, we know from before that Ik < 0 and Sk > 0, thus the denominator (Ik - Sk) < 0 for certain. The shape of the isocline for dY/dt = 0, thus, depends upon the value of (Iy - Sy). As we claimed earlier, for extreme values of Y (around YA and YC), we had (IY - Sy) < 0, thus dK/dY|Y < 0, i. e. the isocline is negatively shaped. However, around middle values of Y (around YB), we had (IY - SY) > 0, thus dK/dY|Y > 0, i. e. the isocline is positively shaped. This is shown in Figure 5.
Fig.5 - Isokine for dY/dt = 0
From Figure 5, we see that at low values of Y (below Y1) and high values of Y (above Y2), the isokine is negatively-sloped - this corresponds to the areas in our earlier diagrams where the savings function was steeper than the investment function (e. g. around YA and YC). However, between Y1 and Y2, the isokine is positively-sloped - which corresponds to the areas where investment is steeper than savings (around YB in our earlier diagrams).
The off-isokine dynamics are easy, namely differentiating the differential equation dY/dt for K:
d (dY/dt) /dK = a [IK - SK] < 0
as IK < 0 and SK > 0. Thus, above the isokine, dY/dt < 0, so output falls whereas below the isokine, dY/dt > 0, so output rises. The directional arrows indicate these tendencies. We can already get a flavor of Kaldor's trade cycle from Figure 6. Remember that our earlier Kaldorian diagrams were drawn for a particular level of capital, K. Thus, as we see in Figure 6, for a given level of K0, we can find the corresponding equilibria (YA, YB, YC) at the intersection between the isokine and the level line K0. However, suppose we start at point C so that K begins to rise: notice that when K0 rises to K2, the points YC and YB begin to move together and finally "merge" at the critical point D (which corresponds to our old B=C) at point Y2. The underlying dynamic (represented by the phase arrows) implies that point D is completely unstable so there will be a catastrophic jump from D to the lower equilibrium E (which is where A moved to as K rose from K0 to K2). Notice that during this catastrophic fall in output is driven solely by the fast multiplier dynamic - the slower-moving capital dynamic is inoperative as, in moving from D to E, capital is constant at K2.
Figure 6 - Dynamics of the Kaldor Cycle
The rest of the story then follows in reverse. At E in Figure 6, we are at a pretty low output level and thus capital decumulates and so K declines from K2 past K0 and then on towards K1. In the meantime, our lower two equilibrium begin to move towards each other and A and B meet and merge at point F at Y1 (equivalent to our old A=B). Note that the stability arrows in Figure 6 are such that now we must have a catastrophic jump in output from F to point G (the high equilibrium to which point C moved to as capital fell from K2 to K1). From G, the process then begins again as capital rises at high output levels from K1 to K0 and onto K2.
The output cycle that can be traced out from the arrows drawn in Figure 6 (from slow movement from G to C to D then catastrophic jump to E then slow movement from E to A to F then catastrophic jump to G, etc) is Kaldor's trade cycle: output thus fluctuates over time between the boundaries imposed by the extreme points E (lower bound) and G (upper bound). Notice that the cycle is completely endogenous: no exogenous shocks, ceilings, floors, structurally unstable parameter values or ratchets are necessary to obtain constant cycles. The non-linearity of the curves, which are economically-justified and not exceptional, is more than sufficient to generate endogenous cycles.
This version of Kaldor's model is derived a bit more formally in Varian (1979) using catastrophe theory. However, we can also use regular non-linear dynamical theory, which makes no assumptions about the relative speeds of the dynamics, to obtain a cycle from the Kaldor model - and this is what Chang and Smyth (1971) do. We have already derived the isokine dY/dt = 0, so to get the full story, we need the dK/dt = 0 isokine. This is simple. dK/dt = I (Y, K), thus at steady-state, dK/dt = 0 = I (Y, K) so the isokine has slope:
dK/dY|K = - IY/IK > 0
as IY < 0 and IK > 0. Thus, the dK/dt = 0 isokine is positive sloped. We have superimposed it on the other isokine in Figure 7. For the off-isokine dynamics, note that:
d (dK/dt) /dY = IY > 0
so that the directional arrows drawn in for the dK/dt = 0 isokine imply that to the left of it, dK/dt < 0 (capital falls) whereas to the right, dK/dt > 0 (capital rises). In the figure below, we have superimposed the isokines for the whole system. As is obvious, the global equilibrium, where dK/dt = dY/dt = 0, is at K* and Y* (point E). Now, as we showed earlier, the trace of the system, tr A, is positive around point E, thus we know that the global equilibrium is locally unstable - as is shown in Figure 7 by the unstable trajectory that emerges when we move slightly off the equilibrium point E. .
Fig.7 - Kaldor's Trade Cycle
However, notice that the system as a whole is "stable": when we draw a "box" around the diagram by imposing upper boundaries Km and Ym in Figure 7 and letting the axis act as lower boundaries, then the directional phase arrows around the boundaries of the box indicate that we do not get out of the confines of this box - any trajectory which enters the "box" will not leave it. Indeed, from the distant boundaries, it almost seems as if we are moving towards the global equilibrium (K*, Y*). Thus, while the equilibrium (K*, Y*) is locally unstable, we are still confined within this "box". Note that we are assuming complex roots to obtain stable and unstable focal dynamics as opposed to simply monotonic ones.
In fact, these three conditions - complex roots, locally unstable equilibrium and a "confining box" - are all that is necessary to fulfill the Poincarй-Bendixson Theorem on the existence of a dynamic "limit cycle". This limit cycle is shown by the thick black circle in Figure 7 (not perfectly drawn) orbiting around the equilibrium. Any trajectory that begins within the circle created by the limit cycle will be explosive and move out towards the cycle. In contrast, any trajectory that begins outside the circle will be dampened and move in towards it. Two such trajectories are shown in Figure 7. Thus, the limit cycle "attracts" all dynamic trajectories to itself and once a trajectory confluences with the limit cycle, it proceeds to follow the orbit of the limit cycle forever (as shown by the directional arrows on the cycle).
In essence, then, all trajectories are "stable", but we are not speaking of a stable path towards a point (such as an equilibrium) but rather of a stable path towards another path, i. e. the limit cycle. This is the limit cycle version of the Kaldor trade cycle. For details on this version of the Kaldor model and more general non-linear dynamics, particularly on Poincarй-Bendixson and, more generally, the Hopf-Bifurcation necessary to yield dynamic limit cycles, consult Chang and Smith (1971), Gabisch and Lorenz (1987), Lorenz (1989) and Rosser (1991).
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