One of the components of a Keynesian strategy is the concept of the investment multiplier first introduced by R.F. Kahn in 1931( there were also several other economists in this era who developed a comparable concept around the same time as Kahn see Robert Dimand’s, The Origins of the Keynesian Revolution) as part of a paper that refuted the Treasury view that a deficit financed investment by government would have little or no effect on net employment creation because of the equivalent loss of jobs in the private sector.”The Relation of Home Investment to Unemployment”,  Economic Journal, June 1931 by Kahn demonstrated how this was not true and that in fact a given amount of new investment in the first round would lead to multiple rounds of subsequent spending and job creation in later rounds. He called his new invention the multiplier. It was derived from the notion that recipients of income typically have a marginal propensity to consume that is  much greater than zero and somewhat less than one. Keynes described it in the General Theory as follows: Kahn’s argument” in this article depended on the fundamental notion that, if the propensity to consume …is taken as given and we conceive the monetary or other public authority to take steps to stimulate …investment, the change in the amount of employment will be a function of the net change in the amount of investment;and it aimed at laying down general principles by which to estimate the actual quantitative relationship between an increment of net investment and the increment of aggregate employment which will be associated with it.” (113-114 G.T.)

So for example if the amount of new deficit financed spending on infrastructure is 1 billion dollars and those who receive the money in the first round of spending then spend 70 % of it they will have a marginal propensity to consume of .7 and a marginal propensity to save of .3.  So in the next round (time t +1) .7 x 1 billion =7oo million is then spent. In the next round (t+2).7x 700=490 is then spent.   In the next round (t+3) .7 x 490=343 is spent and in the next(t+4) .7 x 343=240 is spent. We continue this spending circuit so that eventually the new spending approaches the limit of zero. It is clear that the initial stimulus spending will lead to a total contribution to new spending that is larger than the initial amount by a multiple. It is also clear from the above that we have an infinite geometric progression where the formula for its solution 1/1-mpc or 1/1-.7 = 1/.3 =3.334. So the multiplier in this case is 3.334 . The formula is 1/marginal propensity to save.(Mankiw has a neat proof of this in his macro text as follows: let z be the multiplier and x the marginal propensity to consume             a) z=1 + x+x2 + x3 +…..

b) xz=x+x(2) + x(3)+….. [ where x(2)=x.x and x3= x.x.x etc]

c)  a-b= z-xz=1

d) z(1-x) = 1

e) z = 1/(1-x) or 1/1-m.p.c.(See the more formal general  proof  in Raymond Brink, A First Year of College Mathematics, pp.150-151. Here lim Sn= a/1-x as n goes to  ∞)

Now anti Keynesian critics of this approach claim that ever since Milton Friedman  and Franco Modigliani  and James Duesenberry developed the permanent income hypothesis we should not presume that sudden windfalls of income or transitory income will be spent as opposed to saved for later as part of the tendency for people to adjust their spending habits to what they think is their permanent  average income.

In such circumstances it is argued that the marginal propensity to consume stimulus money will be very small and may even approach zero. If this were the case than the multiplier would be drastically reduced perhaps even to zero. But there is  little evidence that Friedman and his allies are correct in this, particularly when the income recipients are moderate to low income households although some economists argue otherwise, ( see Bodkin’s essay in Seccareccia below).There is a fair bit of econometric evidence that the consumption function in the U.S. in the post war period was such that the mpc varied from .5 to .7 Ackley,p.236 )

In addition Friedman defines durable goods consumption as a form of saving rather than as consumption and so distorts the analysis. (See Paul Davidson, Post Keynesian Macro Theory, p.43 ; Mario Seccareccia , the Determinants of Consumption and Saving from a Heterodox Perspective in H.Bougrine and M.Seccareccia, eds. Introducing macroeconomic analysis, pp.79-88. & also Ronald Bodkin, Consumption theory to the Turn of the Millenium(and shortly beyond) pp.65-78 in Bougrine &Seccareccia; Gardner Ackley, Macroeconomic theory pp.208ff)

So if the multiplier is considered to be somewhere between 1.4 and 1.5 in assessments of current expenditures on infrastructure this seems to be a quite reasonable and conservative estimation. Alvin Hansen pointed out both Kahn and Keynes knew there would be leakages from the spending process. ”part of the increment of income is used to pay off debts; part is saved in the form of idle bank deposits; part is invested in securities purchased from others who in turn fail to spend the proceeds; part is spent on imports, which does not help home employment (unless the trading partner is stimulated by this to import more from you); part  of the purchases is supplied from excess stocks of consumer goods, which may not be replaced.” (Alvin Hansen, A Guide to Keynes,N.Y. McGraw-Hill  p89-90) So because of these leakages the total employment impact of a stimulus is somewhat weakened and comes to an end after a period of time even though its net effect is larger than the initial amount spent. As Hansen argues in the first stages of recovery from a deep depression the multiplier will be lower as people use the new funds to pay off some of their debts , ”at moderately high levels of income it would tend to rise and and finally very high levels of income it may again decline.” (p.102)

(For a very anti Keynesian pessimistic assessment of the multiplier that relies on rational expectations, wage and price rigidities and crowding out assumptions in its modelling and presumes wrongly that near zero interest rates could not be maintained by the Fed without near term inflationary consequences see John F.Cogan, Tobias Cwik, John Taylor and Volker Wieland, New Keynesian versus Old Keynesian Government Spending Multipliers.  See also my later post on the multiplier, impair savings and the work of N.Johanssen( Feb.10 2013 this blog) and the link to a post which explores the value of the multiplier in the 1940 s war period.)