Contents

Introduction

1. Endogenous Cycle Models: Kaldor's Non-Linear Cycle

Conclusion

Literature

Introduction

The multiplier-accelerator structures reviewed above have linear dynamic structures. As a result, cycles are generated and maintained only by structurally unstable parameter values (Samuelson) or dampened dynamics with continuous exogenous shocks (Frisch-Slutsky) or exogenously-constrained explosive dynamics (Hicks). As a result, early Keynesian linear multiplier-accelerator fall dangerously close to an "untheoretical" explanation of the cycle - precisely what the original Oxbridge research programme was designed to avoid.

However, linear structures are often adopted because they are simple and the results they yield are simple. But simplicity is sometimes more a vice than a virtue - particularly in the case of macrodynamics and economic fluctuations. It is not only unrealistic to assume linearity, but the very phenomena that we are out to uncover, the formation of cycles and fluctuations, becomes relegated to the "untheory" of exogenous shocks, ceilings, floors, etc. The contention of Lowe (1926) and many Keynesian writers is that theories of fluctuations ought really to explain how fluctuations arise endogenously from a working system otherwise (paraphrasing Lowe's title), how is business cycle theory possible at all?

As a result, many economists have insisted that non-linear structures should be employed instead. Why interest ourselves with non-linear dynamics? As one famous scientist answered, for the same reason we are interested in "non-elephant animals". In short, non-linear dynamical structures are clearly the more general and common case and restricting attention to linear structures not only unrealistically limits the scope of analysis, it also limits the type of dynamics that are possible.

1. Endogenous Cycle Models: Kaldor's Non-Linear Cycle

One of the most interesting theories of business cycles in the Keynesian vein is that expounded in a pioneering article by Nicholas Kaldor (1940). It is distinguishable from most other contemporary treatments since it utilizes non-linear functions, which produce endogenous cycles, rather than the linear multiplier-accelerator kind which rely largely on exogenous factors to maintain regular cycles. We shall follow Kaldor's simple argument and then proceed to analyze Kaldor in the light of the rigorous treatment given to it by Chang and Smyth (1971) and Varian (1979).

What prompted Kaldor's innovation? Besides the influence of Keynes (1936) and Kalecki (1937), in his extremely readable article, Kaldor proposed that the treatment of savings and investment as linear curves simply does not correspond to empirical reality. In (Harrodian version of) Keynesian theory, investment and savings are both positive functions of output (income). The savings relationship is cemented by the income-expenditure theory of Keynes:

S = (1-c) Y

whereas investment is positively related to income via an accelerator-like relationship, (which, in Kaldor, is related to the level rather than the change in income):

I = vY

where v, the Harrod-Kaldor accelerator coefficient, is merely the capital-output ratio. Over these two relationships, Kaldor superimposed Keynes's multiplier theory: namely, that output changes to clear the goods market. Thus, if there is excess goods demand (which translates to saying that investment exceeds savings, I > S), then output rises (dY/dt > 0), whereas if there is excess goods supply (which translates to savings exceeding investment, I < S), then output falls.

The implications of linearity can be visualized in Figure 1, where we draw two positively-sloped linear I and S curves. To economize on space, we place two separate sets of curves in the same diagram. In the left part of Figure 1, the slope of the savings function is larger than that of the investment function. Where they intersect (I = S) is the equilibrium Y*. As we can note, left of Y*, investment is greater than savings (I > S), hence output will increase by the multiplier dynamic. Right of point Y*, savings is greater than investment (I < S), hence output will fall. Thus, the equilibrium point Y* is stable.

In the right side of Figure 1, we see linear S and I functions again, but this time, the slope of the investment curve is greater than that of the savings curve. Where they intersect, Y*, investment equals savings (I = S) and we have equilibrium. However, note that left of the equilibrium Y*, savings are greater than investment (I < S), thus output will contract and we will move away from Y*. In contrast, right of Y*, investment is greater than savings (I > S), so output will increase and move further to the right of Y*. Thus, equilibrium Y* is unstable.

Fig.1 - Savings, Investment and Output Adjustment

Both exclusive cases, complete stability and complete instability, are implied by linear I and S curves in figure 1, are incompatible with the empirical reality of cycles and fluctuations. Hence, Kaldor concluded, it might be sensible to assume that the S and I curves are non-linear. In general, he assumed I = I (Y, K) and S = S (Y, K), where investment and savings are non-linear functions of income and capital as in the Figure 2 below.

We shall focus the relationship with income first. The logic Kaldor (1940) gave for this is quite simple.

The non-linear investment curve, shown in Figure 2 can be explained by simply recognizing that the rate of investment will be quite low at extreme output levels. ............