I have been rereading Kahn’s classic 1931 paper The Relation of Home Investment to Unemployment for my coming work on a general theory of prices. The paper lays out the argument for the existence of a Keynesian multiplier for the first time. There are many interesting aspects to this paper and it is well worth a read as it is very dense, but here I want to focus on a singular point: the structure of the multiplier equation as laid out by Kahn.
First of all, let us take a look at the standard, modern-day Keynesian multiplier equation as laid out in textbooks. It goes like this. First we lay out a consumption function as we have in Equation 1.1 below:
Here, Co is autonomous consumption, and cYt-1 is the amount by which consumption rises due an initial rise in income. Finally Ct is the amount of consumption at the end of the period.
We then go on to lay out an equation for income, Y, which is the familiar GDP equation as shown in Equation 1.2 below:
We then substitute the import term and the consumption term to take account of how an initial rise in income will affect total income after the multiplier has taken effect. This is shown in Equation 1.3 below (note that the import multiplier works in the same way as the consumption multiplier laid out above):
What we see here is that an increase in income will lead to a multiplied increase in consumption but we also see a rise in imports due to the rise in income, this is represented by the import multiplier mYt-1 which is included together with the autonomous import variable M0. The rise in consumption has a positive effect on total end period income, Yt, while the rise in imports has a negative effect on total end period income.
Kahn’s presentation is altogether different. He breaks down the components of the multiplier into wage and profit incomes. This can be seen in Equation 1.4 below:
Where we have a multiplier relation for each variable, n for the profit generated by an additional man employed and b for wage of that man, which is the amount of expenditure spent on home produced goods. We then have the wage and profit variables, W and P respectively; again, per man employed. Kahn then lays out a more complete equation that can be seen in Equation 1.5 below:
Here we see that Kahn has included a term for the amount of imports that are bought as a result of a one man rise in employment, R, together with a variable for the increase in employment, k. It should be noted that Kahn is also more interested in the rise in imports coming from an increase in raw materials and unfinished goods than he is the rise in imports coming from consumption.
Kahn’s multiplier then tells us how many additional men will be employed, k, at each level of the multiplier. As can be seen there are separate multipliers for the wage consumption, b, and the profit consumption, n. This shows clearly the breakdown between the two and how each adds to employment via separate consumption multipliers for each term. Thus, in the case of a high wage multiplier and a low profit multiplier, for example, most of the income effects will be shown to be generated by the increase in wages; this could lead to the conclusion that a redistribution of income will lead to higher national income.
Although I still prefer the standard textbook method of laying out the multiplier and its effects on national income, I nevertheless find Kahn’s approach interesting for two broad reasons.
First of all, it lays out quite explicitly the distributional aspect of the multiplier process.
Secondly, it stresses not the effects that the multiplier has on national income, but rather the effects it has on employment.
It appears to me that Kahn’s multiplier, which is derived so far as I can tell from Keynes’ equations in his Treatise on Money, is more so along the lines of what we would today refer to as a Kaleckian distributional approach rather than a Keynesian aggregate income approach. Just another interesting note in the history of economic thought.
Update: I have laid out how to transform the Kahn multiplier into a truly macroeconomic multiplier for wages and profits in a new post here.
Fixing the Economists 
A rise in income can only come about if there is new credit creation to enable a rise in demand. The multiplier is a credit multiplier. New credit supplied by a bank becomes new deposits when the credit is spent, and the new deposits can be spent or lent, and the frequency of circulation of the new money can obviously be more than one. The new credit creation does not have to be for real investment; it can be for anything.
The standard ‘Keynesian’ model has things the wrong way around.
I am not surprised that Kahn did not get this; he was one of the most unpleasant academics I ever had contact with, and the way he prejudiced the financial security of Keynes’ widow was monstrous, and a breach of trust as an executor should not favour the ultimate beneficiary at the expense of the life tenant. It was a shameful episode in the long history of Kings College.
Kahn does take account of this in the paper:
“It is, however, important to realise that the intelligent
co-operation of the banking system is being taken for granted.
It is supposed that the object of the Central Bank is to achieve
the maximum of employment that is- consistent with remaining
on the gold standard. If the increased circulation of notes and
the increased demand for working capital that may result from
increased employment are made the occasion for a restriction of
credit, then any attempt to increase employment-whether it by way of road-building or by any other means, or, indeed, by
awaiting the return of world prosperity-may be rendered
nugatory.” (Pp174-175)