Friday, August 27, 2010

Why Income Inequality Leads To Recessionary Conditions

1.0 Introduction
Apparently, some have been discussing whether the gross increased inequality in the USA is connected with the depressionary conditions we are in. So I thought I would climb on my bicycle and do some arithmetic.

I take it as a stylized fact that an increase in inequality is associated with an increase in the average and marginal propensity to save.

There's something called the Harrod-Domar model of growth. I'm not sure I've ever read Domar. I've certainly read more of Harrod than I have of Domar. So in the sequel, I refer exclusively to Harrod.

Harrod defined three rates of growth: the actual rate, the warranted rate, and the natural rate. Increased inequality can result in the warranted rate exceeding the natural rate. Since the warranted rate is unstable and the actual rate cannot long exceed the natural rate, increased inequality is likely to lead to the actual rate of growth falling below and away from the warranted rate, that is, to depressions.

2.0 Harrod's Model
Harrod's model is fairly simple, but it raises deep questions.

2.1 The Actual Rate
Along a steady state growth path, the ratio, v, of the value of capital to the value of net income is constant:
v = K/Y,
where K is the value of the capital stock, and Y is the value of net income. v is known as the capital-output ratio. Thus:
dY/dt = (1/v) dK/dt
Investment, I, is defined to be the change in the value of capital with time. Hence,
(1/Y) dY/dt = (1/v) (I/Y)
The left-hand-side of of the above equation is, by definition, the rate of growth, g, of the economy. The equality of investment and savings is an accounting definition in a model with no foreign trade and no government. Therefore,
g = (1/v) (S/Y)
Define the savings rate, s:
s = S/Y
Then, a steady state growth ratio is the ratio of the savings rate to the capital-output ratio:
g = s/v
That is, the (actual) rate of growth is the quotient of the savings rate and the capital-output ratio.

2.2 The Warranted Rate
Suppose the savings rate and the capital-output ratio are as desired by income recipients (consumers) and firms, respectively. This defines Harrod's warranted rate of growth:
gw = sd/vd
where the subscripts on the right hand side stand for "desired". The warranted rate of growth is being achieved when expectations are being realized and current actions are not setting up forces to disturb current expectations.

The warranted rate of growth extends Keynes' analysis to the long period. Consider the stability of a warranted growth path. If the actual rate of growth exceeds the warranted rate, capacity will be utilized at a greater rate than firms expected. They will increase investment faster than the warranted rate, and the rate of growth will deviate from the warranted rate even more. Likewise, if the actual rate falls below the warranted rate, firms will cut back on investment since the plans upon which their investment was made are not being realized. Hence, the warranted rate is unstable.

Harrod suggested that this instability of the warranted rate is more like an inverted flat-bottomed bowl than a knife-edge.

2.3 The Natural Rate
Suppose the labor force is initially fully employed. Let n be the rate of growth of the labor force:
n = (dL/dt)/L
Define the value of output produced per employed worker:
f = Y/L
Harrod-neutral technical change occurs when the value of output per worker grows at a constant rate, m, while the rate of profit stays unchanged:
m = (1/f) df/dt
Harrod-neutral technical progress implies that the productivity of labor is growing at the same rate in all industries.

Anyways, the following equation follows:
dY/dt = f dL/dt + L df/dt
Some algebra yields:
(1/Y) (dY/dt) = ( 1/L) (dL/dt) + (1/f) (df/dt)
The left hand side of the above equation is the rate of growth that keeps the labor force fully employed (or a constant percentage unemployed). Harrod calls this the natural rate of growth. Hence, assuming Harrod-neutral technological progress, the natural rate of growth is the sum of the rate of growth of the labor force and the rate of growth of labor productivity.
gn = n + m

3.0 Conclusions
Notice that the determinants of the warranted rate of growth - the savings rate and the desired capital-output ratio - are taken as exogeneous constants. The determinants of the natural rate of growth - the growth of the labor force and Harrod-neutral technological progress - are also given. Hence, the warranted and natural rates can only be equal by a fluke.

Solow, following up on some work by Pivlin, suggested that the desired equality between the warranted and natural rates can be brought about by considering the capital-output ratio as a well-behaved function of the rate of interest. Divergences between the two rates can be corrected by variations in the distribution of income. This approach of neoclassical macroeconomics is exemplified in Solow's eponymous growth model, but it has been shown to be not well-founded in the Cambridge Capital Controversy.

If the warranted rate is below the natural rate, a moderate increase in the saving rate is desirable if the economy is exhibiting boom-like conditions. This would bring the warranted rate towards the actual rate of growth while still keeping it below the natural rate of growth.

Notice that when the warranted rate exceeds the natural rate, the economy must sometime fall below the warranted rate. The natural rate sets a limit which the economy cannot long exceed. Because of the instability of the warranted rate, such an economy will experience frequent and perhaps prolonged recessionary conditions. Since increased savings intensify the discrepancies between the warranted and natural growth rates under these conditions, increased savings intensify the frequency and severity of recessions. That is, increased inequality can intensify the frequency and severity of recessions.

  • A. Asimakopulos (1991) Keynes's General Theory and Accumulation, Cambridge.
  • Roy F. Harrod (1948) Towards a Dynamic Economics, Macmillan.
  • Joan Robinson (1962) Essays in the Theory of Economic Growth, Macmillan.


Magpie said...


I'm not sure I quite follow the whole argument, as I'm not very good at differential equations.

However, I think there might be a typo in the title:

"Why income EQUALITY leads to recessionary conditions"

Robert Vienneau said...

So there was. It's now fixed. Thanks.

Laci the Chinese Crested said...

I like this, but you realise that this is far to complicated for the tea party crowd to grasp.

Anonymous said...

dY/dt = f dL/dt + L df/dt

I think this is wrong... ¿Shouldn´t it be this other way:

dY/dt = (f + df/dt) dL/dt + L df/dt?

Robert Vienneau said...

By definition, I have f = Y/L. Rearrange to get Y = f L. Apply the chain rule and you get the derivative in the post.

Anonymous said...

Yeah, applying the chain rule I get the same result as you, buy it seems quite contraintuitive to me.

If you think of Y as the product of L times f, you can see that dY/dt = (L + dL/dt)(f + df/dt) - Lf => dY/dt = (dL/dt)(df/dt) + L(df/dt) + (dL/dt)f + Lf - Lf => dY/dt = f(dL/dt) + L(df/dt) + (dL/dt)(df/dt)...

Where´s my mistake?

Anonymous said...

I mean, counter-intuitive*.

Emil Bakkum said...

@ Robert Vienneau
I like your explanation, since she is commendably clear, and agrees with the account of Eckhard Hein. According to Hein it is Domar, who first studied the effects of partial capacity utilization. Domar described the conditions necessary for growth under a full capacity utilization and an equilibrium on the commodity markets (I=S). Harrod studied dynamics on the short and long term. Incidentally, Hein prefers to call g the rate of the capital accumulation. Just like you Hein and also Jan Pen point to the shocking consequence, that the economy would in fact be highly unstable. Slight devations from the knife-edge would provoke the producers into a self-destructing behaviour. It creates either excess demand and inflation, or excess supply and deflation. In addition there may be structural unemployment. Hein criticizes the assumed constancy of the capital coefficient and the savings rate. The constancy of the former means that Harrod ignores changes in the capacity utilization, which would dampen the unstable behaviour. The constancy of the savings rate means that Harrod ignores the changes in the income distribution between workers and capitalists. These will also dampen the instability. So his theory is in fact far from perfect. Note that in the mean time Kalecki was studying variations in the capacity utilization and Kaldor studied variations in the income distribution. Incidentally, I have some trouble in interpreting your conclusion unambiguously (perhaps I have read the wrong books). When the warranted rate exceeds the natural rate, there will not be a full capacity utilization. I wonder why the warranted rate itself would be unstable, it is simply not practicable. Apparently the savings rate is too large under these conditions, and this may indicate a large inequality. But why would savings increase?

Anonymous said...

(I´m the previous anonymous.)

Sorry, I may be fully wrong, but, well..., I would like to know where´s my mistake. Can anybody help me, please?

Emil Bakkum said...

This is true but it raises many questions. A lack of saving leads to insufficient investments, and thus to an unsatisfactory growth rate. In fact, Mankiw in his text book on macro-economics is certain that the USA should save MORE.

Another point concerns the Harrod model itself. The warranted rate gw is steady, which implies in particular that the capital-to-output ratio vw is constant. However, according to Harrod other growth rates are also in the form g=s/v. So in those cases the system is still or again steady.

A different v implies a different utility of productive capacity. Harrod suggests that this troubles the firms. It is not clear why, since the growth path is again steady.

It may be that firms prefer a utility of 100%. But Harrod does not say this (at least in his paper of 1939). The later text books seem to take this more obvious interpretation. They do not talk about different c-to-o ratios. In fact and surprisingly Harrod is now generally known as the man who believes in fixed c-to-o ratios.

In his paper of 1939 Harrod states that gw itself could move. In this case it becomes unclear why this growth path is called warranted. For apparently the growth rate causes sufficient dissatisfaction to change it. Harrods paper of 1939confuses me. Perhaps I should read his book?

Robert Vienneau said...

For months I've been thinking I should rewrite this post with illustrations. It's been years since I've read Harrod's book, but, as I recall, he has lots of discussion that isn't captured in the few equations. Some might have related to variations in capacity utilization.

Anyways, I also liked Asimakopulos's Keynes's General Theory and Accumulation, which concludes with an attempt to compare and contrast Harrod's model with Robinson's models of metallic ages.

I think the work of economists like Harrod, Kahn, Kaldor, and Robinson offer lots of insights into empirical reality while still leaving much to think about.