John Maynard Keynes,The General Theory (BN Publishing, 2008), pp. 113–131.

The Marginal Propensity to Consume and the Multiplier

Keynes, in this chapter, borrows from earlier work of his (Arthmar and Brady [2011]) and from R.F. Kahn’s “The Relation of Home Investment to Unemployment,” and I think is integral to understanding Keynesian capital theory (i.e. the relationship between consumption and investment). As the chapter’s title suggests, Keynes will take us through the logic of the multiplier, which is a ratio between income and investment. Therefore, allowing for some abstractions, it is also a ratio between total employment and primary employment (that directly employed by investment). No less, it establishes a precise relationship, given the propensity to consume, between aggregate employment and income and the rate of investment.

He offers a brief description of the argument offered by R.F. Kahn, who is usually credited as the originator of the idea. Assume the marginal propensity to consume to be given; employment will be a function of the net change in investment. This chapter is dedicated to elucidating this idea (or an application of it, with some subtle alterations), and providing a foundation by defining terms.

I.

Remember that fluctuations in income come from changes in employment (a given level of employment assumes a certain level of output and a certain level of nominal income). If we assume diminishing marginal returns with increases in output, this means that wages will rise both nominally and in real terms. Keynes holds that real wages and nominal wages will move in the same direction (this is also established in chapter 2), which allows him to posit (since measuring real wages is difficult) that changes in income can be measured in wage-units, Y w , and used as an index. (Keynes, though, argues that nominal wages might fall or rise in greater proportion than real income!)

Knowing that increases in Y w outstrip increases in consumption (C w ), the following relationship holds true: Y w > C w . This allows us to define the marginal propensity to consume (MPC) as dC w /dY w .

The MPC, in turn, tells us how increases in income will have to divided between consumption and investment.

∆Y w = ∆C w + ∆I w (i.e. the change in income is equal to the sum of the change in consumption and the change in investment).

∆Y w = k∆I w , where 1 – 1/k is the MPC. k is therefore the “investment multiplier.”

II.

Kahn’s multiplier — that Keynes defines as k’ and calls the “employment multiplier” — measures the ratio between the increase in total employment associated with increases in primary employment in the investment industries. Formally, this would suggest the following: if ∆I w leads to change in employment ∆N 2 , the increase in total employment is ∆N = k’∆N 2 .

k does not equal k‘, since the supply functions between different industries differ; i.e k’ is properly applied to a single industry (or supply function), where as k is aggregated across industries (implied by the fact that it relates to total income Y w ). Despite this reality, it is easier, writes Keynes, to just assume k = k‘ (the model can be changed to show the more realistic possibility of a divergence between ∆Y w /∆N and ∆I w /∆N 2 .

What the multiplier suggests is that if society consumes 9/10 of their income then multiplier k is 10. So, for example, if the government were to build a pyramid that employed h workers (primary employment), then total employment (everything else being equal) would be 10h. This assumes, of course, that ∆C w is positive as Y w rises. Here, I think, we see a very important rationale in Keynes’ model that reflects on his vision of the relationship between consumption, savings, and investment. If I am reading pp. 117–118 correctly, the temporal sequence is as such: ↑I ⇨ ↑Y ⇨ ↑C,↑S; but, were the ∆S w is enough to cover the previous ↑I. In other words, in a two time period model, I t would be funded by S t+1 . The “secondary employment,” by the way, is that which is stimulated in the consumer industries thanks to a rise in income.

What this means is, according to Keynes, assuming a high MPC a minor negative changes in investment can lead to broad decreases in unemployment, but minor positive changes in investment can cause broad rises in employment (towards full employment). Comparatively, a low MPC would mean smaller changes in employment. Where MPC is measured between 0 and 1, it therefore seems better to increase consumption, so that a relatively minor increase in investment can have greater effects on ” secondary employment.”

III.

All of what we have discussed so far assumes a net increase in investment. In reality, though, the economy is complex. For example, if we apply the multiplier to a public works, then we have to assume that this project didn’t cause some offset (a decrease in investment) elsewhere and it didn’t change the MPC. What are some factors that need consideration?

Depending on how public works are financed, it may raise the rate of interest and therefore dissuade private investment (crowding out?) — assuming monetary policy doesn’t try to assuage this consequence. No less, related inflation will increase the cost of capital goods and reduce their marginal efficiency, requiring a fall in the rate of interest; Given the prevailing “confused psychology” (calculation chaos?), government spending can reduce confidence, which can in turn increase liquidity preference and/or decrease the marginal efficiency of capital; If part of increased income is spent on foreign goods, then part of the multiplier will benefit these other countries. But Keynes is insightful enough that to the extent that this stimulates economic activity there, it may actually be beneficial to us.

Depending on the volume of public works, we also have to account for changes in the MPC. Increases in income will tend to decrease the MPC, which in turn will decrease the multiplier. We also have to account for distributional forces: entrepreneurs might accrue a disproportional amount of new income, and their MPC may be lower than that of workers. Also, the unemployed may be living on “negative saving” (i.e. consuming their savings) and when they are re-employed their MPC might fall as they try to replenish their savings or repay loans.

Remember what Keynes believes the implications of the multiplier are: it is what explains the differences in magnitudes between the consequences on investment and the consequences on employment as a whole.

IV.

Now that we’ve dealt with possible alterations in net investment, another case we have to deal with is: what if changes in investment aren’t “foreseen sufficiently in advance” by consumer good industries? This introduces us to the concept of time lags, where an unforeseen increase in capital goods production will manifest the multiplication of aggregate demand over time. However, notes Keynes, an unforeseen increase in the production of capital goods will only gradually increase aggregate investment and it may cause the MPC to deviate and then finally return back to normal. But, the theory of the multiplier still applies — despite the fact that it’s an instantaneous relationship — in that the effect is equal to the increment in investment times its value; this relationship holds again when a new increment in aggregate investment occurs due to the time-lag.

To illustrate the point, let’s assume that an increase in capital goods production is entirely unforeseen, such that there is no increase in the production of consumer goods. Income earners in the capital goods industries will increase consumption, raising the prices of these goods and increasing the incomes of profit earners, who have a lower MPC, and depleting the existing stocks of consumer items. There is therefore a reduction in MPC and the multiplier, meaning that increases in aggregate investment is less than the total increase in investment in the capital goods industry. Everything balances out, though, when the consumer goods industries replenish their stocks to meet the increase in demand, the MPC rises, and there is an increase in aggregate investment bringing it to the level of former production of capital goods.

Keynes writes that this concept of the time-lag does play a role in his business cycle theory, but is inconsequential with regards to the validity of the multiplier theory. Also, unless the consumer goods industries are fulling employing their fixed capital, there’s no reason to assume that there will be a great time lag between capital good production and consumer good production.

V.

Let’s explore the relationship between the marginal propensity to consume and the average propensity to consume.

Assume that in a community where 5 million workers are employed 100% of income is consumed. The output of +100,000 workers would lead to a consumption of 99%; +100,000 workers 98%; +100,000 workers 97%’; etc. 10,000,000 represents full employment. The multiplier at the margin is 100/n when 5,000,000 + 100,000n workers are employed; further, n(n + 1)/2(50 + n) is invested. So, when 5,200,000 men are employed, the multiplier is 100/2 = 50; investment is 2(2 + 1)/2(50 + 2) = .06 (rounded up). Let’s say that investment falls by two-thirds; employment would only fall by around 2%.

Yet, when 9 million workers are employed the marginal multiplier is 100/([9m – 5m]/100k = 40) = 2½. Much lower MPC, much lesser fluctuations in employment, right? Wrong. Investment is now at 40(40 + 1)/2(50 + 40) = 1640/180 = ~9% (rounded down) of total income. So, if investment falls by two thirds then employment will fall by 44%!

The ratio of the proportional change in total demand to the proportional change in investment is: (∆Y/Y)/(∆I/I) = (∆Y/Y)([Y-C]/[∆Y – ∆C]) = (I – C/Y)/(1 – dC/dY).

All of this leads Keynes to some conclusions: (a) public works pay for themselves in countries with high unemployment and high MPC, but not where an economy is approaching full employment, and (b) as an economy approaches full employment, increases in investment will garner fewer and fewer increases in employment. Also, in an interesting application to the Great Depression in the United States, Keynes suggests that the low MPC he computes might be due to high corporate savings.

VI.

As established in chapter two, if there exists involuntary unemployment it means that the marginal disutility of labor is less than the utility of the marginal product. This suggests that even “wasteful” government spending may increase wealth (this is where the infamous “pyramids” and hole digging comments are made). So, the government could bury bank notes underground and employ people to dig them up and positively create wealth, even though alternative forms of investment might be preferred (e.g. building houses). The comment on digging holes is actually an analogy to gold mining, which Keynes notes is often considered a positive endeavor, but really not that different from digging holes for money. He notes that periods during which mining is high are periods of growth, yet where there is no gold available usually there is stagnant growth. What all of this is actually is is a somewhat sarcastic discussion of how people find such unproductive ventures to add to wealth, but yet oppose more sensible projects.

Two pyramids, two masses for the dead, are twice as good as one; but not so two railways from London to York.

— p. 131.