Keynes, The General Theory: Chapter 10

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, Yw, 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 Yw outstrip increases in consumption (Cw), the following relationship holds true: Yw > Cw.  This allows us to define the marginal propensity to consume (MPC) as dCw/dYw.

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

∆Yw = ∆Cw + ∆Iw (i.e. the change in income is equal to the sum of the change in consumption and the change in investment).

∆Yw = k∆Iw, where 1 – 1/k is the MPCk 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 ∆Iw leads to change in employment ∆N2, the increase in total employment is ∆N = k’∆N2.

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 Yw).  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 ∆Yw/∆N and ∆Iw/∆N2.

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 ∆Cw is positive as Yw 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 ∆Sw is enough to cover the previous ↑I.  In other words, in a two time period model, It would be funded by St+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?

  1. 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;
  2. 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;
  3. 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.

16 thoughts on “Keynes, The General Theory: Chapter 10

  1. Blue Aurora

    Once again, I am pleased that you reference the work of Dr. Michael Emmett Brady in your quest to understand the General Theory. I’m also glad that you catch the sarcasm in Keynes’s voice regarding the pyramids and the railroads.

    How does this chapter of the General Theory rank in your opinion, though?

    Reply
    1. Jonathan Finegold Catalán Post author

      If it wasn’t so hard to get through (I’m going to definitely have to re-read the book), it would be my favorite so far even. I’ll hold my opinion, though, until I get through the other fourteen or so chapters.

      Reply
    2. Jonathan Finegold Catalán Post author

      What I’m interested in is how many of Keynes’ peers believed there is a direct relationship between consumption and investment opportunities.

      Reply
  2. Raoul

    It seems to me that the « multiplier » story results from a mere confusion between a descriptive relation and a causal one, and that it hinges on the fallacy according to which the relationship between the whole and the part doesn’t vary when you change the part, so that if you make the part bigger, not only the whole grows, but it grows more quickly than the part.

    For instance, if you have earned 100$, you may put 80 in your Right pocket, and 20 in your Left pocket. Thus, you have:

    Income = Right pocket + Left pocket
    Income = 0.8 x Income + Left pocket
    0.2 x Income = Left pocket
    Income = 5 x Left pocket

    Then, you say that “5” is your “Left pocket multiplier”. Next step, you decide to put in your Left pocket 40 instead of 20, in order that your income becomes 5 x 40 = 200.

    I think this scheme is correct on the basis of Keynes’ reasoning. Do you think I’m wrong?

    Reply
    1. Jonathan Finegold Catalán Post author

      Like I wrote in response to ‘Blue Aurora,’ I’m going to have to re-read the chapter/book; Keynes’ logic is difficult to follow, maybe because his causal view is different from mine. I have a few articles on the multiplier that I plan to read, including R.F. Kahn’s — maybe it’ll give me a better idea of the theory.

      The right/left pocket analogy I don’t think works, because Keynes is assuming a specific relationship between investment and consumption. The government employs 100 people building a pyramid; these 100 people are paid $1,000 each, of which they spend ½ on consumption (2x multiplier). There is an increase in consumption, swelling employment in these industries and in turn increasing demand for further investment.

      Reply
      1. Raoul

        Thanks for your answer, but I’m not sure to see of what “specific relationship between investment and consumption” you talk.

        Is it the MPC or the “definite ratio” resulting from the working of the “multiplier”?

        If the former, my point is precisely that Keynes unduly assumes that the MPC doesn’t change; if the latter, I find the idea that such a definite ratio exists is pretty fantastic.

        Reply
        1. Jonathan Finegold Catalán Post author

          The multiplier needs the MPC to be derived, so these are somehow causally related. In any case, I don’t think Keynes assumes the MPC will remain unchanged — he notes that there might be effects which may undermine the working of the multiplier; in fact, as society approaches full employment Keynes assumes the multiplier will fall, since so will consumption.

          What your pocket analogy doesn’t include is how each pocket affects employment. The fact is that consumption and investment isn’t the same as right and left pocket. Keynes sees the purpose of each category to be different. What drives investment is consumption, and in turn investment can increase consumption by increasing incomes.

          Reply
          1. Raoul

            The “multiplier” and the MPC are indeed causally related in the same sense as, for instance, if you want to calculate 3×14, you need perhaps to calculate first 3X10, then 3×4, and finally 3×10 + 3×4.

            In this sense, Mises writes “Logic and mathematics deal with an ideal system of thought. The relations and implications of their system are coexistent and interdependent. We may say as well that they are synchronous or that they are out of time. A perfect mind could grasp them all in one thought. Man’s inability to accomplish this makes thinking itself an action, proceeding step by step from the less satisfactory state of insufficient cognition to the more satisfactory state of better insight. But the temporal order in which knowledge is acquired must not be confused with the logical simultaneity of all parts of an aprioristic deductive system.” (HA, V, 1).

            Conversely, to take an easy example, the MPC itself is not causally related with the income or the investment. It only describes how the incomes is shared among consumption and investment.

            Well, in my opinion, it’s the same between the MPC and the “multiplier”. Keynes writes “so that we can write ΔYw = k ΔIw, where 1 – (1/k) is equal to the marginal propensity to consume.” In this mathematical relationship, there’s no room for any causality. The “multiplier” is directly defined as the reciprocal of the propensity to save. That’s probably why Keynes writes “…the logical theory of the multiplier, which holds good continuously, without time-lag, at all moments of time…”.

            Now, it’s sure Keynes says the MPC will change afterwards with the changes in employment, but that’s another story.

            I tried to develop this idea here: http://laissezlesfaits.wordpress.com/2012/04/20/the-multiplier-concept-must-be-entirely-discarded/

          2. Jonathan Finegold Catalán Post author

            I don’t understand your objection, nor do I understand your multiplication analogy. The multiplier is related to the MPC in the sense that Keynes believes that the reason to employ stems from the prospective yield of employment. This yield is, in part, decided by the propensity to consume. This isn’t just contained in the chapter of the multiplier, but makes up the essence of the entirety of The General Theory.

            In order to know 1 – (1/k), btw, you need to know the MPC. So, in order to know what k — or the multiplier is — you need to know the MPC; and the MPC is related to the MPS, in the sense that income is split between consumption and saving.

            You can’t just look at the mathematical equations and then assume there’s no prior causal logic to it. This is what The General Theory is. You can say the multiplier is wrong, but I’m having trouble seeing your critique as the right one.

  3. Raoul

    I think it’s an understatement to say that “The multiplier is related to the MPC in the sense that Keynes believes that the reason to employ stems from the prospective yield of employment”. The “multiplier” is uniquely determined by the MPC (or MPS), and that’s precisely the problem. Keynes is asserting that so strange thing that you can (i) by an apriori reasoning (yes, Keynes, here, is apriorist…) (ii) determine the quantitative effect of an economic event.

    The relation between the employment and the prospective yield of employment is of the essence of the entirety of the General Theory, but the multiplier formula—“so that we can write ΔYw = k ΔIw, where 1 – (1/k) is equal to the marginal propensity to consume”—is itself quite unsupported. Keynes discusses the reasons which are likely to vary at the margin the effects of the “multiplier” (namely, the leakages, the crowding-out…), but he doesn’t discuss the formula itself; he merely asserts it.

    Reply
    1. Jonathan Finegold Catalán Post author

      The ‘k’ in that formula refers to the “investment multiplier.” The “employment multiplier” is k’. Keynes write that k =/= k’, but for the sake of explaining his logic he’ll assume they are equal. But, that k =/= k’ is implicit in the fact that different firms face different supply functions; i.e. the amount of employment per firm can’t be quantified on the basis of just knowing the investment multiplier k. I do agree there’s a problem here, but it’s in the subtle rejection of the idea that we can know a quantitative relationship between the employment of ‘N’ men and the resulting output. In other words, maybe this is a subtle confession that measuring everything in wage-units and terms of employment can’t provide you with a quantitative relationship between investment, employment, and output.

      About the derivation of the equation, I think the first half of the sentence is just as important as the second half. An increase in income, or an increment in output, is distributed between consumption and investment. Therefore ΔY = ΔC + ΔI. It seems to me that the causality reads left to right. We know that the MPC is dC/dY, and from that we can deduce the investment multiplier. What is confusing is that in the equation ΔY = kΔI, the causality reads right to left.

      What explains the causality, I think, is Kahn’s “employment multiplier.” +ΔI leads to an increase of ΔY, by kΔI. This is distributed between consumption and investment. Consumption will rise, stimulating output “until the new level (and distribution) of incomes provides a margin of saving sufficient to correspond to the increased investment.” This takes us back to chapter 7, where he critiques theories of over-investment, since the increased output would lead to increased incomes, and therefore increased savings.

      Reply
  4. Blue Aurora

    Unfortunately, I haven’t read the economics literature of that decade very well. I don’t know what Marshall, Robertson, Edgeworth, Pigou, or Hawtrey would have thought.

    However, of the lot, I do know that Robertson did admit to Keynes that he was “ill trained in mathematics”. Dr. Michael Emmett Brady covers that fact in this following paper that you might want to read.

    http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1406842

    He also has a paper on Keynes’s use of marginal productivity theory in the General Theory.

    http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1591764

    Reply
  5. MH

    Jeffrey M. Herbener has a good discussion of the MPC here : “Dissent on Keynes: A Critical Appraisal of Keynesian Economics” (chapter 4, “The Myths of the Multiplier and the Accelerator” pp 76-77, section “MPC”). In the next section, “Multiplier”, he writes :

    Furthermore, he claimed that the MPC (Marginal Propensity to Consume) “is of considerable importance because it tells us how the next increment of output will have to be divided between consumption and investment” (ibid.: 115 [emphasis added]). The multiplier (k) equals 1/1-MPC, and thus, he concluded, “it tells us that, when there is an increment of aggregate investment, income will increase by an amount which is k times the increment of investment” (ibid, [emphasis added]).

    To use k in support of “public works,” the multiplier must have the mathematical precision Keynes gives it (ibid.: 116). Yet this precision leads to logical absurdities. Three absurd cases exist, corresponding to three violations of Keynes’s pronouncement that 0 < MPC < 1. As shown above, there is no accounting principle that the MPC be bound in this way, and there is ample evidence that the MPC is not so bound (see Table 4.1)."

    http://menghusblog.files.wordpress.com/2012/05/myths-of-the-multiplier-and-the-accelerator-table-4-1.png

    He then argues that if MPC = 1, the multiplier will be infinite. If MPC exceeds 1, the multiplier will be negative. If the MPC is negative, “k” will be a positive fraction. He then goes on :

    Furthermore, granting that a multiplier exists, why should it equal 1/1-MPC? Why does the MPC determine additional spending in response to changing income? Even Keynes assumed that, when income increases, some of the additional saved funds go into investment spending. Since this additional investment is reacting to income, it should be included with the MPC. Moreover, given Keynes’s accounting assumptions (Y = C + I), all spending comes out of income. Thus, no source of funds exists to initiate the multiplier process except hoards (or fiat money). This result applies with equal force to government expenditures. There is no reason to treat them as initiating the multiplier process since they must also come out of income. Finally, there is no direction of cause and effect in the equation. And his system of equations (Y = C + I; C = a + bY; I = d) is recursive, that is, it is impossible to tell, mathematically, which variable changes first and which ones follow.

    Reply
    1. Jonathan Finegold Catalán Post author

      I’ll have to read that chapter (and the entire book). In the first excerpt, I agree with Herberner: how can there be an a priori straight forward relationship between investment and income? But, I think the second excerpt misses a bit of Keynes’ ideas.

      I’ll break it down,

      Why does the MPC determine additional spending in response to changing income? Even Keynes assumed that, when income increases, some of the additional saved funds go into investment spending. Since this additional investment is reacting to income, it should be included with the MPC.

      But, in this chapter Keynes is explicit that the rise in savings is to make up for the previous time period’s increase in investment. There’s no new net investment. It follows that all new net spending is the change in consumption. No less, it’s this positive change in consumption that provides new opportunities to invest, since Keynes perceives there to be a direct relationship between consumption and investment (compare this to the indirect relationship of the Ricardo Effect). And the equation’s causality is the verbal reasoning that makes up most of The General Theory.

      Regarding the initiation of the multiplier process, I don’t know for sure. It might be that Keynes adopted an endogenous money position — like that of modern post Keynesians and Schumpeter. I’m not sure if he explains this later in the book, but it seems implicit that he assumes a rising nominal income (not just a rising real income).

      Reply
  6. Blue Aurora

    Regarding whether Keynes believed in endogenous money or exogenous money, I think one would have to read his A Treatise on Money. The General Theory is partly supposed to be an evolution of thought from the Treatise on Money. In practice, I think we all can agree that the money supply itself is partially endogenous and partially exogenous.

    Reply

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