Контрольная работа: Endogenous Cycle Models
The clincher in Kaldor's system is the phenomenon of capital accumulation at a given point in time. After all, as Kaldor reminds us, investment and savings functions are short term. At a high stable level of output, such as that at point YC in the figure above, if investment is happening, the stock of capital is increasing. As capital stock increases, there are some substantial changes in the I and S curves. In the first instance, as capital stock increases, the return or marginal productivity of capital declines. Thus, it is not unreasonable to assume that investment will fall over time. Thus, it is acceptable that dI/dK < 0, i. e. the I curve falls.
However, as capital goods become more available, a greater proportion of production can be dedicated to the production of consumer goods. As consumer goods themselves increase in number, the prices of consumer goods decline. For the individual consumer, this phenomenon is significant since it implies that less income is required to purchase the same amount of goods as before. Consequently, there will be more income left over to be saved. Thus, it is also not unreasonable to suspect that the savings curve, S, will gradually move upwards, i. e. dS/dK > 0. This is illustrated in Figure 3.
Fig.3 - Capital Accumulation and Gravitation of Investment and Savings Curves
So, we can see the story by visualizing the move from Figure 2 to Figure 3. Starting from our (old) YC , as I (Y, K) moves down and S (Y, K) moves down, point B will gradually move from its original position in the middle towards C (i. e. YB will move right) while point C moves towards B (YC moves left). As shown in Figure 3, as time progresses, and the investment and savings curves continue on their migration induced by capital accumulation, and B and C approximate each other, we will reach a situation where B and C meet at YB = YB and the S and I curves are tangential to each other. Notice that at this point in time, C is no longer stable - left and right of point C, savings exceeds investment, thus output must fall - and indeed will fall catastrophically from YB = YC to the only stable point in the system: namely, point A at YA .
At YA , we are again at a stable, short-run equilibrium. However, as in the earlier case, the S and I curves are not going to remain unchanged. In fact, they will move in the opposite direction. As investment is reigned back, there might not even be enough to cover replacement.
Thus, previous investment projects which were running on existing capital will disappear with depreciation. The usefulness (i. e. productivity) of the projects, however, remains. Thus, the projects reemerge as "new" opportunities. In simplest terms, with capital decumulation, the return to capital increases and hence investment becomes more attractive, so that the I curve will shift upwards (see Figure 4).
Similarly, as capital is decumulated, consumer industries will disappear, prices rise and hence real income (purchasing power) per head declines so that, to keep a given level of real consumption, savings must decline. 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 Kaldor's trade cycle.
W. W. Chang and D. J. Smyth (1971) and Hal Varian (1979) translated Kaldor's 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 < S, dY/dt < 0. Now both savings and investment are positive (non-linear) functions of income and capital, hence I = I (K, Y) and S = S (K, Y) where dI/dY= IY > 0 and dS/dY = SY > 0 while dI/dK = IK < 0 and dS/dK = SK > 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 (SK IY - IK SY )
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: