Keynes’s Fundamental Equations From The Treatise on Money

The symbols used in A Treatise on Money are somewhat different from the General Theory  and for those who are familiar with GT but not the Treatise are a little confusing at first.

See both Keynes’s A Treatise on Money vol.1 ch.10 and Robert Dimand’s work The Origin of the Keynesian Revolution and his chapter on the Treatise and the fundamental equations. See also Mario Seccareccia’s working paper,”Aspects of a new conceptual integration of Keynes’s Treatise on Money and the General Theory:Logical Time Units and Macroeconomic Price Formation, April 2002;  and Fausto Vicarelli, Keynes the Instability of Capitalism,Philadelphia: University of Pennsylvania press, 1984, pp. 65 ff; and A. Asimakopulos, Keynes’s General Theory and Accumulation, Cambridge :Cambridge University Press, 1991. pp. 13-17.

These symbols are as follows:

O total output of  goods in a physical sense per unit of time. So that O =R+C

R volume of liquid consumption goods and services flowing on to the market per unit of time.

C the net increase of Investment goods in physical units.

E  earnings of community or factors of production per unit of time or cost of production

S savings that is difference between money income and money expenditure on current consumption  E-R =S

I’ cost of production of new investment that is the current output of investment goods

I value in price terms of the current new output of investment goods

P price level of liquid consumption goods and services,  the reciprocal of the purchasing power of money

P’ the price level of investment goods

π price level of output as a whole

Q1 windfall profits  in consumption goods sector

Q2 windfall profits in investment goods sector

Q windfall profits for the economy as a whole ; not included in E or S so that Q=Q1+Q2

W rate of earnings per unit of human effort so that its inverse 1/W is the labour power of money.

W1 rate of earnings per unit of output E/O that is the rate of efficiency earnings and

e the co-efficient of efficiency such that W=eW1 so that e measures units of output per unit of human labour. So, for example, assume the wage is $10 per hour and five units of output are produced in an hour.Hence $10/5 =$ 2 the amount of labour cost in a unit of output. and e =5 so that eW1 =$10 the hourly wage.

M1 income deposits, cash held  by individuals

M2 business deposits cash deposits held by entrepreneurs

M3 savings deposits, held as investments , not to finance expenditures

Vi velocity of circulation of Mi  i =1,2, 3

L foreign lending private capital account

B foreign balance current account

G gold outflow or other balancing transactions G=L-B

Since P= is price level of liquid consumption goods, P.R= current expenditure on consumption goods and  E. C/O = I’ is cost of production of new investment.

Since expenditure on consumption goods equals difference between income and savings we have   P.R=E-S = E/O(R+C)-S = E/O.R + I’-S

(  Note that O=R+C so the expression E/O(R+C) reduces to E and since I’= E. C/O or E/O. C) we can substitute I’ for C and the expression becomes E/O .R +I’-S)

So P= E/O + I’-S/R  (1)

(i.e. I’-S all over R.  Keynes has divided each side by R to arrive at this. )              This is his first fundamental equation.

It states that the price level of liquid consumption goods is the outcome of  total earnings  per unit of output plus any differences between investment costs of production and savings in comparison to the flow of liquid consumption goods.

Dimand points out that this links windfall profits to entrepreneurs that go beyond their normal remuneration since I can exceed I’.  That is the cost of production of new investment is less than the value of it when sold on the market. It is this possibility for windfall profits that disappears in the General Theory(GT) when Keynes redefines income, savings and investment.Instead Keynes has the output level readjust to bring investment into equality with savings in the GT. See his discussion of windfalls and his redefinition of income,  savings and inverstment GT pp.52-61 and pp.60-61 in particular.

Friedrich A. Hayek was sharply critical of Keynes definition of terms  and his choice of units ( see his Reflections on the Pure Theory of Money  Hayek argues that Keynes operates with far too broad and ambivalent definitions of profits, for example which Hayek describes as ” a peculiarity which is likely to prove a stumbling block to most readers, the concept of entrepreneur’s profits. These are expressly excluded from the category of money income, and form a separate category of their own”    Furthermore,he objects to the way in which Keynes calculates the prices of finished products.

“he treats the process of the current output of consumption goods as an integral whole in which only the prices obtained at the end for the final products and the prices paid at the beginning for the factors of production have any bearing on its profitableness. He seems to think that sufficient account of any change in the relative supply (and therefore in the value) of intermediate products in the successive stages of that process is provided for by his concept of (positive or negative) investment, i.e., the net addition to (or diminution from) the capital of the community. But this is by no means sufficient if only the total or net increment (or decrement) of investment goods in all stages is considered and treated as a whole, and the possibility of fluctuations between these stages is neglected; yet this is just what Mr. Keynes does. ”   Keynes responded to Hayek’s critique in an article in Economica published in November , 1931. (See Vol 13, JMK: The General theory and After Part 1 Preparation pp 243-256. He rejected Hayek’s criticisms by reviewing his choice of terms and definitions, clarifying them as best he could and concluding that Hayek lacked good will toward the author and was in fact wedded to is own theory to such an extent that he could not appreciate Keynes’s work. Keynes went on to scathingly review Hayek’s own work Prices and Production  Keynes argued that the book seemed to him “as one of the most frightful muddles he had ever read”…It is an extraordinary example of how starting with a mistake a remorseless logician can end up in Bedlam. ” (p.252) see vol. 13, JMK: The General theory and After Part 1 Preparation, p.252. after exchanges like this it is not surprising that Keynes and Hayek became foes of one another , at least at the level of theory .

After Keynes derives his first fundamental equation he introduces the possibility of wage push inflation. Let W be the rate of earnings per unit of human effort, W1 the rate of earnings per unit of output and e the coefficient of efficiency so as above W=e.W1 then  rewriting eq.1

P=W1 + I’-S/R (all over R) (2)   =  1/e.W + I’-S/R  (3)  Keynes writes of this equation the price level of consumption goods -the inverse of the purchasing power of  money consists of these two terms. The first represents efficiency earnings or the cost of production ;the second is positive, negative or zero accordingly as the cost of new investment exceeds,is less than or equals the volume of current savings. So the stability of the purchasing power of money requires that efficiency earnings should be constant and that the cost of new investment should be equal to the volume of current savings.  Hence it is possible that even if efficiency earnings are stable investment savings disproportionalities can result in price pressure. As Keynes states:”the division of output between investment and goods for consumption is not necessarily the same as the division of income between savings and the expenditure on consumption. For workers are paid just as much when they are producing for investment as when they are producing for consumption; but having earned their wages, it is they who must please themselves whether they spend or refrain from spending them on consumption. Meanwhile, the entrepreneurs have been deciding quite independently in what proportions they shall produce the two categories of output.” Treatise vol. one ch.10 p.136   The price level of consumption  goods is determined independently of the price level of investment goods. Keynes then assumes that the price level of investment goods is given and he shows how the price level of output as a whole can be determined.

Let P’ be the price of new investment goods, π the price level of output as a whole, and I as distinct from I’ , the market value as distinct from I’ the cost of production of the increment of new capital goods.

Hence π =P.R+ P’.C/O (all over O)   = (E-S)+I/O (all over O) since I=P’C and P.R=E-S.

which is equal to E/O + I-S/O (4) which is the second of the fundamental equations.

Equation 4 can be rewritten as π = W1  + I-S/O  (5)     =1/e.W + I-S/O   (6)

Since W1 = E/O and 1/eW = W1.

Now Keynes turns to profits.

He lets Q1 be profit on the production and sale of consumption goods where profit is defined as a positive windfall that exceeds normal entrepreneurial remuneration. Recall that Keynes is using a Marshallian market model where perfect competition is the norm and windfalls are exceptional except where the business cycle creates them.  Q2 the corresponding profit on investment goods and Q the total profit.  then Q1 = P.R – E/O .R  =E-S – (E-I’)   = I’-S   (7)  So Q1 as Dimand explains is “the excess of the market value of the output of consumer goods over the earnings of the factors of production employed in that sector. ” (p.25)  P.R is the market value of consumer goods and E/O .R is the earnings of the factors of production in that sector.  Q2 = I-I’  and Q = Q1 + Q2   =I-S  (8) since Q1 + Q2 = I-I’ + I’ – S .

So profits on the production and sale of consumption goods are equal to the difference between the cost of new investment and savings and are negative when savings exceed the cost of new investment. Total profits on total output equal the difference between the value of new investment and savings, being negative when savings exceed the value of new investment.  Keynes then states equations 2 and 5 can be reworked as   P = W1 + Q1/R  (9)    and π = W1  + Q/O  (10)

Keynes concludes that equations 9 and 10 state “the price of consumption goods is equal to the rate of earnings of the factors of production plus the rate of profit per unit of output of consumption goods; and correspondingly with output as a whole.”

Keynes admits that these equations are identities “truisms which tell us nothing in themselves. ” But he believed that once fleshed out with data and analysis from the real economy they would help trace cause and effect in building a monetary theory of production. The circus of his younger followers in their critique of the treatise and in particular the notion of the widows’ cruse as somehow applying to entrepreneurial profits remaining undepleted by consumption but becoming a Danaid jar when entrepreneurs saved rather than consumed were convinced that Keynes needed to go further. In the critique offered by Austin Robinson according to Skidelsky,  Keynes is faulted for not having considered the problem of falling output and general depression which makes the widow’s cruse invalid.(See R.Skidelsky , JMK vol 2 The Economist as Saviour 1920-1937, p 447-448.) As Asimakopulos argues Keynes’s statement about the widows cruse and his fundamental equations “assume constant output, clearly not a basis for a general theory.” (p.17 , Keynes’s G.T.) However, Asimakopulos also admits that Keynes does not always assume constant output through out the Treatise. This was precisely the point that Keynes himself made in response to Robinson’s critique.

Asimakopulos also points out that because  Keynes defines profits as windfalls in the Treatise unlike in the GT savings will not equal invstment as it does in the GT where savings always adjust through the means of adjusting the total output to equal investment. “The actual sales proceeds of firms could differ from what the value of  the output produced would have been if equilibrium prices had prevailed. the differences between these sums represent windfall profit(if positive) or losses(if negative) and Keynes excludes these items from his definition of income. ” hence the difference in the treatise between savings and investment ”because saving is defined as the value of this notional income(income that includes normal profits whether they are obtained or not) minus actual consumption expenditures, while investment is represented by actual investment expenditures.” Because sales equals the sum of investment and consumption expenditures, investment equals these proceeds minus consumption. “A difference between savings and investment thus arises under these definitions whenever the value of Keynes’s notional income differs from actual sales proceeds. Investment exceeds (falls short of)saving by the amount of the windfall profits(losses) that occur when market forces result in actual prices for output that are higher(lower) than normal prices.” (Asimakopulos,  pp.13-14)  For Keynes, then in the Treatise, was trying to develop a method for analysing how fluctuations in price as well as in output could result from profit fluctuations that were traceable to differences in savings and investment.

In addition  Keynes was already trying in the treatise  to establish the virtue of consumption but hadn’t yet made the total transition to his theory of aggregate demand and the pivitol role of investment. This he soon turned to in the writing of his General theory. Nevertheless, despite its flaws the Treatise clearly establishes that Keynes had a sophisticated theory of the operations of the price system and the possible sources of inflation  and underemployment  which he incorporated in the GT in his chapter on prices showing that profit push, wage push and supply demand bottlenecks could establish inflationary pressure well before the point of full employment. The fact that the monetarist camp and the neo classical Keynesians had paid scant attention to the treatise and to the chapter on prices in the GT meant that when stagflation occurred they were unaware that Keynes had anticipated such a possibility  more than forty years before.

Keynes concludes his chapter on the fundamental equations by relating them to the quantity theory of money that he still was operating with although attempting to move beyond in the Treatise. He first explains that when savers decide the proportion of their money income that they wish to save as opposed to consume they must make a decision as to what proportion of their savings to hold in the form of bank deposits versus securities or cash or other forms of loan capital. He refers to this decision as a choice between” ‘ hoarding’ and’ investing.’ ” (p.141)

It is here that he begins to develop his theory of liquidity preference and his explanation  of the role of interest and the inverse price of securities that later plays an important role in his notion of the liquidity trap in the GT.  (p.142) He goes on to argue that price level of output as a whole and the amount of total profit depends upon 4 factors: the rate of saving, the cost of new investment, the bearishness of the public and the volume of savings deposits (p.143-144) He reduces these to ” the excess of savings over the cost of investment and the excess of bearishness on the part of the public as is unsatisfied by the creation of deposits by the banking system.”

So already it is clear that expectations play an important role in Keynes’s description of the workings of the price and output system. Keynes also suggests that the original quantity theory is in need of modification as a explicandum of price change. “If the volume of saving becomes unequal to the cost of new investment, or if the public disposition toward securities takes a turn …in the bullish or bearish direction, then the fundamental price levels can depart from their equilibrium values without any change having occurred in the quantity of money or in the velocities of circulation. …(this is true in the theoretical world) In the actual world a change in anything is likely to be accompanied by some change in everything else. But even so the degrees of change in the quantity of money, the velocities of circulation, and the volume of output will not be related in any definite or predictable ratio to the degrees of change in the fundamental price levels. Indeed this is notoriously the case at the acute phases of a credit cycle.” (p.147)

He concludes the chapter by reverting to a quantity theory style restatement of his conclusions.  In the case of equilibrium where I-I’ = S then if M1 is total income deposits +V1 their velocity then

π = P =M1V1/O    and if M1,M2, M3 are the totals of income deposits, business deposits and savings deposits and M total deposits then

P = V1(M-M2-M3)/O and then let w be the proportion of cash deposits to total deposits and V the average velocity of the cash deposits where V1 and V2 are the velocities of income deposits and business deposits  then   M1+M2= wM

and M1V1 + M2V2 =wM.V    then

M1=  M. (w(V2-V)/V2-V1    and  M1V1 = M(wV1(V2-V)/V2-V)  so  P=M/O . wV1(V2-V)/V2-V1

He notes  that P.O= M1V1   is similar to Irving Fisher’s P.T = M. V  except that O is output rather than transactions and M1V1 represent income deposits and their velocity  but M,V are cash deposits and their velocity.

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About haroldchorneyeconomist

I am Professor of political economy at Concordia university in Montréal, Québec, Canada. I received my B.A.Hons (econ.&poli sci) from the University of Manitoba. I also completed my M.A. degree in economics there. Went on to spend two years at the London School of Economics as a Ph.D. student in economics and then completed my Ph.D. in political economy at the University of Toronto. Was named a John W.Dafoe fellow, a CMHC fellow and a Canada Council fellow. I also was named a Woodrow Wilson fellow in 1968 after completing my first class honours undergraduate degree. Worked as an economist in the area of education, labour economics and as the senior economist with the Manitoba Housing and Renewal Corporation for the Government of Manitoba from 1972 to 1978. I also have worked as an economic consultant for MDT socio-economic consultants and have been consulted on urban planning, health policy, linguistic duality and public sector finance questions by the governments of Manitoba, Saskatchewan,the cities of Regina and Saskatoon, Ontario and the Federal government of Canada. I have also been consulted by senior leaders of the British Labour party, MPs from the Progressive Conservative party, the Liberal party and the New Democrats on economic policy questions. Members of the Government of France under the Presidency of Francois Mitterand discussed my work on public sector deficits. I have also run for elected office at the municipal level. I first began to write about quantitative easing as a useful policy option during the early 1980s.
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