The Keynesian concept in economic of endogenous cycles seems to require non-linear structures. One of the theories of business cycles in the Keynesian vein is that expounded in a pioneering article by N. Kaldor. Non-linear dynamics of the Kaldors cycle.
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So, the S curve falls. Ultimately, as time progresses and the curves keep shifting, as shown in Figure 4, until we will reach another tangency between S and I analogous to the one before. Here, points B and A merge at YA = YB and the system becomes unstable so that the only stable point left is C. Hence, there will be a catastrophic rise in production from YA to YC. Fig.4 - Capital Decumulation and Gravitation Thus, we can begin to see some cyclical phenomenon in action. YA and YC are both short-term equilibrium levels of output. However, neither of them, in the long-term, is stable. Consequently, as time progresses, we will be alternating between output levels near the lower end (around YA) and output levels near the higher end (around YC). Moving from YA to YC and back to YA and so on is an inexorable phenomenon. In simplest terms, it is Kaldors trade cycle. W. W. Chang and D. J. Smyth (1971) and Hal Varian (1979) translated Kaldors trade cycle model into more rigorous context: the former into a limit cycle and the latter into catastrophe theory. Output, as we saw via the theory of the multiplier, responds to the difference between savings and investment. Thus: dY/dt = a (I - S) where a is the speed by which output responds to excess investment. If I > S, dY/dt > 0. If I 0 and dS/dY = SY > 0 while dI/dK = IK 0, for the reasons explained before. At any of the three intersection points, YA, YB and YC, savings are equal to investment (I - S = 0). We are faced basically with two differential equations: dY/dt = a [I (K,Y) - S (K,Y)], dK/dt = I (K, Y) To examine the local dynamics, let us linearize these equations around an equilibrium (Y*, K*) and restate them in a matrix system: dY/dt = a (IY - SY) a (IK - SK) Y dK/dt IY IK [Y*, K*] K 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 then |A| > 0, thus we have regular (non-saddlepoint) dynamics.