Past difficult problems related to Keynes's investment multiplier

§7 Past difficult problems relating to Keynes's investment multiplier

DCE Yuichiro Hayashi

October 16, 2006

The form of the Keynes's investment multiplier can be derived in economic terms other than investment products. Concerning this, a difficult question was presented by Oskar Ryszard Lange( ref. (1)) et al.

Let's follow Lange's contention, where Y= C+I, dC / ｄY= MPC, dI / ｄY= MPI, dS/dY = MPS（ S = Y - C: Saving ).

If investment I is a free( independent) variable, as the MPC( a constant) exists in the real economy, the following equation holds between incremental investment dI and incremental national income dY, from Keynes's multiplier logic:
dY = dI + MPC·dI + MPC²·dI + · · ·
= dI (1+ MPC+ MPC² + · · · )
= dI / (1 - MPC)	(7-1)
where
\| MPC \| < 1	(7-2)

Similarly, if consumption I is a free variable, the following equation holds:
dY = dC(1+ MPI+ MPI² + · · · )
= dC / (1 - MPI)	(7-3)

where
\| MPI \|<1	(7-4)

In Eq.(7-1), let dI₀denote an initial autonomous increment included in dI. The dI₀ changes into income dI₀.The income dI₀leads to new induced consumption (dC/dY)dI₀ and induced investment (dI/dY)dI₀, the sum of which is (dC/dY + dI/dY)dI₀. The sum changes into the next income, which induces products (dC/dY + dI/dY)²dI₀ · · ·.

As the sum of the induced incremental final products including dI₀ should be the sum of the induced income, the following holds:.
dY = dI₀ + (dC/dY + dI/dY)dI₀ + (dC/dY + dI/dY)²dI₀ + · · ·
= [1+ (dC/dY + dI/dY) + (dC/dY + dI/dY)²+ · · · ]dI₀
= dI₀/( 1 - (dC/dY + dI/dY) )	(7-5)
Similarly, this logic also holds in an initial autonomous, incremental consumption dC₀, thus we have:
dY = dC₀/( 1 - (dC/dY + dI/dY) )	(7-6)

Observing Eq.(7-5) and (7-6), we know that both the initial values dC₀and dI₀are increased by the same multiplier 1/( 1 - (dC/dY + dI/dY)) to become dY. That is to say, 'any given autonomous increment in expenditure has exactly the same effect upon national income, irrespective of whether the expenditure is for investment or for consumption ( Ref.(1), p.231)' . Comparing this result with Eq. (7-1) and (7-3), we are forced to conclude that the ' marginal compound propensity to spend' dE/dY=MPE defined by Eq.(7-8) exists, and Eq.(7-9) holds:
E = C + I	(7-7)
dE/dY = dC/dY + dI/dY
= MPE	(7-8)
dY/dE = 1/(1 - dE/dY)	(7-9)

The stability condition for Eq.(7-7) is as follows:
\| dE/dY \|<1	(7-10)

Eq.(7-10) contradicts the equation 1= dE/dY which is derived from Keynes's basic equation Y = C + I. To resolve the contradiction, the value [1 - dE/dY] is introduced and named the 'marginal reluctance to spend'. Then the stability condition Eq.(7-10) turns into the condition that the 'marginal reluctance to spend' [1 - dE/dY = (dY( income) - dE( spending))/dY] is positive, i.e., that the reluctance to spend is an increasing function of national income.

This condition is satisfied when dS/dY > dI/dY, that is, the marginal propensity to save dS/dY is greater than the marginal propensity to dI/dY. This relation indicates the familiar diagram derived from the equilibrium theory. However, this result contradicts Keynes's basic equation S = I.

Fig.7-1　Ref.(1）　ｐ.228

The contradiction of this formulation is due to the following: all people, including Lange, haven't acknowledged the fact that this economic system Y = C + I instantly changes into a one-degree of freedom system when the existence of the constant MPC is assumed, i.e. the separation of Y into C and I has become meaningless, and therefore they assume mistakenly that this system remains a two-degrees of freedom system composed of C and I variables. By the way, grounds for this argument, that this system is a one-degree of freedom system, are that C has a linear relationship with Y connected to MPC.　It is indifferent whether the type is Kuznets's one or Keynes's one.

It seems that the problems raised by Lange and by the others' pointing of its contradiction have remained unresolved. This will show that the difference between one-degree and two-degrees of freedom systems in the system Y= C + I has not been acknowledged until now. By the way, nobody, including entrepreneurs who most want to know, can find out the volume of the effective demand in advance in capitalistic societies based on free will. The reason is that it depends on the customer's will, and the circumstances of today's people always change and are different from those of tomorrow.

It is often described that Keynes's investment multiplier is applied to macro economic sectors, while the multisectoral multiplier, i.e., Leontief's inverse matrix, is applied to microscopic sectors, and so they are essentially the same as a multiplier. As the author of this website has explained in this document, Leontief's inverse matrix isn't equal to the multiplier effect but expresses the addition of both intermediate products and final products, or that of a number of intermediate business transactions reaching final products.

From the author's point of view, a closed economy is a kind of a one-degree of freedom system which is composed of both consumption and investment: they are connected with each other as if the connection is a one-degree of freedom; both Leontief's inverse matrix and author's values flowing matrix shown in the next section is, through supply and demand from the GVA to total products ( final products + intermediate consumption), expresses like an integrated constant which characterizes propensity or habit in a national style of living, culture and business practice. That is, these matrices are like a parameter which shows the differences of a culture or an economic structure between each country. In a firm, there exists a firm structure matrix between its own GVA and sales. To change the matrices largely is an economic structural reform or a firm one; to change them fundamentally is an economic revolution.

The additional intermediate consumption matrix（ the foregoing matrices) expresses efficiency of industry in the end, therefore it might relate to the quantity of final products( the volume of working places) although the author has not yet succeeded in clearly deriving the equations. This implies that various economic regulations and practices, produce their corresponding occupations (incomes) but they weaken their global competitiveness. The author feels that various secrets are hidden in this matrix including economic problems concerning economic growth and unemployment.

To avoid misunderstandings, this author would have to describe that he has analyzed the multiplier effect problems, in every section of this website, preserving the functional properties of C and I, in which C is an independent variable with a linear relationship with Y; I is also an independent variable which doesn't relate to a change of Y. These assumptions were not set by the author but by Keynes or Keynesians.

The author insists that C, I and Y are attributed to one independent variable to which they are connected with a determined proportion, because the MPC really exists in economy. That is to say, as far as Σ_iinput_i = Σ_ｊ output_ｊ holds in an economic system, Y=C + I, where MPC （a constant） = dC / dY or MPI （a constant）= dI / dY is statistically observed, the following equations hold: dY = dC / MPC = dI / (1 - MPC) = dI / MPI = dC / (1 - MPI). Additionally, the whole Y( both C and I), which is the single independent variable, consists of both a GVA part, which is generated in proportion to the volume of Y, and another GVA part, which is produced independently of it. By the way, it goes without saying that the MPC is a weakly-nonlinear function of Y, because the MPC is a statistic score which expresses an economic propensity of people.

(1) Oskar Ryszard Lange, The Theory of the Multiplier, 1943, Econometrica

16/Oct./2006　publication