§6 A Flow-Chart to Disprove the Keynesian Multiplier Effect Theory

Keynes' principle of effective demand based on the the Keynesian multiplier effect is mathematically wrong .

The Keynesian multiplier effect does not exist.

Yuichiro Hayashi, Y. Hayashi

July 1, 2004  Publication of the first draft 

  "The General Theory of Employment, Interest and Money" was published in 1935-36 by John Maynard Keynes. From that time, the Keynesian multiplier effect theory has constituted one of the bedrocks of current economics. For several decades, the theory has been one of the main themes not only in textbooks but also in the real economy. Current national economic policies are largely affected by it. Every book and web-site relating to economy is, except for a few objections, described based on the principle of effective demand which is connected to the investment multiplier effect( Fig.3-3 ) .

  The objections against Keynes's multiplier effect theory are either arguments without mathematical disproof* or mentions that the investment multiplier 1/ (1 - MPC) exists but  its effect is small. 

  * On 11 July, 2009, the author suddenly received an e-mail from Dr. Brian Chapman in Australia. According to the e-mail, he also has researched errors on Keynes' multiplier effect theory for long years, and he found my website on the evening before. Refer to the following websites. His conclusion that Keynes' multiplier effect formula is wrong is the same as mine. It was verified mathematically and independently of my research. 

http://www.qedinteractive.com.au/DissEcon.html 

http://www.qedinteractive.com.au/Chapman%20Keynesian%20Multiplier%20Paper%20for%20ATEC%202009%20Revised.pdf

  See the equations in [Question-18]. The basic equations of the present discussion are written only on 2 lines:Y₂ = C₂ + I₂ , ΔY₂ = ΔC₂ + ΔI₂, ΔC₂ = ΔC₁ = a _K ΔY₁, ΔY₂ =ΔY₁, ΔY₁ - a _K ΔY₁ = ΔI₂, ΔY₁ = ΔI₂  / (1 - a _K). If a _K =0.6, we have ΔY ₁= 2.5ΔI₂.

  Fig.3-3 looks beautiful. Both the above equation and the one in which ΔI₂  is replaced by ΔI₂+ ΔNX₂ + ΔG₂ in place of ΔI₂, are surely identities. Any error is nowhere to be found. Everybody has acknowledged scholastically (mathematically) the truth of the multiplier effect theory for decades. Even if anyone doubts the effect of the Keynesian multiplier in a depression, nobody can disprove it mathematically and presents a new theory displacing it.   

  Although this author recognize the existence of Keynes's investment multiplier as an equation, he insists that it dose not mean the the multiplier effect expected by Keynes, i.e. that the investment multiplier effect does not exist mathematically. Even I doubted the truth of my argument at first. I shall  present my questions here and answer them. 

   The Keynesian multiplier effect chart is shown in Fig.3-3.

Fig.3-3 Keynesian multiplier effect chart

(1)  On the one hand, the notation ΔY₂ is used for showing the sum of the incremental components in GDP in the reference year as shown in Eq.(3-37). On the other hand,  the notation ΔY₁ is also used in order to show the equivalent value to, on Y1 coordinate, for example, only ΔG₂. Unless otherwise stated, the former definition is adopted:

Δ Y2 = Δ C 2 + Δ I2 + Δ NX2 + Δ G 2

(3-37)

,where, for simplicity of notations, Δ government consumption = 0 is assumed.

(2)  Readers should know that the partial 'gross value added (GVA)' is a person's economic activity value which is added to valueless resources. Corporation is included in the word person.  All economic activities including producing, earning income and expending, are expressed by the economic those of partial GVA. As is well known, the GVA is measured from various economic aspects. For example, ΔC₂ (consumption goods) is a part of ΔY₂ which is measured in the economic aspect Y₂ (final products). It has the name consumption (goods ), in which its partial GVA is collected in the name of consumption. 

 (3)  Readers should recognize that each component in the right side of Eq.（3-37）is a realized final product within a certain time period ( or that which will be realized within a period between now and a future day ). It should not be allowed, in the equation, that one product is already realized and another is not yet produced and will only be realized on a certain future day. If the readers intend to extend the period from the end of the present period to a future day, they should adjust all component data in the equation to that future data.

(4)  Readers should recognize that the equations ΔY₁ (incomes) = ΔY₂  (sales of final products) = ΔY₃(all costs less purchase of intermediate products + profits) will hold in a yearly period in economic theories. Furthermore, when we have the equations, ΔY₁=∑components₁, ΔY₂=∑components₂ and ΔY₃=∑components₃,  the readers should recognize that, for example, the equation ∑components1 = ∑components₂ holds, and we must not subtract any components from only one side of this equation. 

(5)  Readers have recognized economic characteristics of ΔY₁, ΔY₂ and ΔY₃. Each indicator has its components( parts) respectively. For example, ΔY₃ has compensation of employees, operating surplus etc,  and ΔY₂ has final products of secter-1, secter-2 - - -  industries. The readers should recognize that each component( e.g. ΔC₂ ) in one aspect ( e.g. ΔY₂ ) can be conceptually  expressed with a composition of another aspect's components ( e.g. compensation of employees and operating surplus, - - -etc.), if a theoretical means is established. This is the concept adopted in the input-output table. In Chapter2, the new matrix equation, [vector of ΔY₂ (GDP) components] =[transformation matrix ] · [vector of ΔY₃ (GVA) components] is presented by the author. Consequently, the multiplier effect concept( the second impact effect by an incremental income) is not essentially included in the input-output table.

(6)  Final products, incomes and expenditures are simultaneously realized within one year. If a government's additional expenditure ΔG₁( = ΔG₂ ) is executed, the money for ΔG₁ has been obtained from the incomes within the one year, via transfer or borrowing. Note that recipients of ΔG₃ ( costs, such as employees compensation) do not necessarily correspond to tax payers or fund providers. 

(7)  As far as we allow to adopt the Assumption 5 (Δ I ₂+Δ NX₂=0 or Δ I ₂  = Δ NX₂ = 0 ) shown in §3, readers will surely recognize that the equation ΔY₁ = ΔG₂ +ΔC₂ holds at the Keynesian cross ( this is really at any time). The readers know the solution of the income ΔY₁ = ΔG₂  / (1 - a_K) known as the Keynesian multiplier effect. The equations ΔG2  / (1 - a_K) = ( ΔY₁ - ΔC₂)  / (1 - ΔC₂  / ΔY₂ ) = ΔY₂ = ΔG₂ +ΔC₂ hold. 

  The Keynesian multiplier effect equation is understood in such a manner that the multiplied income ΔY₁ = ΔG₂  / (1 - a_K) will be obtained  some day in the future naturally or in an induced manner, by the execution of ΔG₂. The above-mentioned equations show, however, that after all the government should  realize ΔY₂ = ΔG₂ + ΔC₂, which is a product or the breakdown of the product, in order to realize income ΔY₁ = ΔG₂  / (1 - a_K) within an accounting period in which ΔY₁ and ΔY₂ are measured.

  To tell the truth, we have overlooked the fact that ΔC₂ is already generated simultaneously with the execution of ΔG₂ in Fig.3-3  over a period of years (*).  Fig.3-3 must be exactly expressed as shown in Fig 4-6 by putting ΔC₂ and correcting ΔY₂ = ΔG₂ into ΔY₂ = ΔG₂ + ΔC₂ in Fig.3-3. 

[* It will be shown later that this results from the equilibrium theory method. ] 

Fig 4-6 Corrected Keynesian multiplier effect chart

  However, it is not the government's ( or Keynesians') purpose to create, naturally or in an induced manner, ΔY₁ (= ΔG₂ +ΔC₂) from only ΔG₂. Consequently, Fig.4-6 is mathematically correct, under Assumption 5, but it is also  incorrect/ flawed for only expenditure ΔG₂. In addition, even if we allow that Fig 4-6 shows the correct relation for both ΔG₂ and ΔC₂, we can't, by any means, include ΔC₂ = 0. Thus, Fig.3-3 is doubly incorrect/ flawed.

(8)  Someone may say, "We know that we can't create income ΔY₁ = ΔG₂ +ΔC₂ from only expenditure ΔG₂ in a short time. We expect that ΔC₂ will be obtained some day in the future according to the 'propensity to consume' or it will be induced by consumer confidence". This claim is wrong( incorrect/ flawed). The consumption ΔC₂ which is expected to be induced in the future is already finished as it is written in national economic accounts in the year when ΔG₂ is executed.  In one year, ΔG₂ had had an effect on the economy. Very much so! 

  The relationship between output (final products ) and the corresponding input( GVA) is shown in Fig.3-14. In a business depression, private investment will decrease.  ΔG₂ due to borrowing might have been a partial firms' profit or a partial employees compensation, both of which should have decreased, if ΔG₂ due to borrowing had not been executed.

Fig 3-14 Input-Output Table chart

(9)  Readers should know that the ΔG₂-Y₁ problem shown in Fig.3-3 is slightly different from the problem shown in [Question-18] in §4. I could not strictly discriminate between them at first. In the former case,  ΔI₂ = ΔNX₂ = 0 is assumed, but in the latter case,  ΔG₂ = ΔNX₂ = 0 or, ΔI₂ including ΔG₂ and ΔNX₂, is assumed.

(10)  Have readers understood the explanation mentioned above? In fact, this explanation also has a large error. In short, both  Fig.3-3 and Fig.4-6 are absolutely wrong (incorrect/ flawed ),  regardless of the reason that we can't include ΔC₂ =0 in these figures. These two Keynesian figures were obtained by adopting Assumption 5 ( Δ I₂ = Δ NX₂ = 0). Therefore, we obtained ΔY₁ = ΔG₂ / (1 - a_K). In fact, we can't adopt Assumption 5. The reason is that ΔY₁ = ΔG₂  / (1 - a_K - a_I - a_NX), where a_I = 'marginal propensity for private firms to invest', should hold. If we take a_I into consideration, the Keynesian multiplier will become about 10 times( see [Answer-24].).

  This will be explained in a little more detail. We will consider two cases: Case-1······ the analysis under the condition that Assumption 5 ( Δ I₂ = Δ NX₂ = 0) does not exist; Case-2······ the analysis under the condition that Assumption 5 ( Δ I₂ = Δ NX₂ = 0) is adopted. The marginal propensity to consume is defined as a_K = ΔC₂ / ΔY₂. If we suppose a_K = 0.6, the equation (Δ I₂ +Δ G₂+Δ NX₂ ) /ΔY₂ = 0.4 holds with Case-1.

  With Case-2, ΔG₂/ΔY₂ = 0.4 must hold when a_K = 0.6. This is impossible in the real economy. The real amount of ΔG₂ /ΔY₂ will be close to a statistical amount of G₂ /Y_2. If we suppose ΔG₂/ΔY₂ = 0.1, then, the chart, which includes both ΔG₂ and ΔY₂, will have to be drawn in such a manner that the amount of ΔG₂ is about 1/10 of ΔY₂, i.e. ΔY₂ =10ΔG₂.

   See Fig.4-6. The ratio of ΔY₂ to ΔG₂ is drawn as 1 to 0.4. This resulted from neglecting the existence of both Δ I₂ and Δ NX₂. Fig.4-6 is absolutely wrong( incorrect/ flawed). This has occurred when the amount of Δ I₂ +Δ NX₂ has been taken away from only the right side of Eq.(3-37) (the amount Δ C₂ + ΔI₂ is a part of the right-hand side of Eq. (3-37)), although the definition of ΔY₁ doesn't change ( ΔY₁(=ΔY₂) is the whole of the right-hand side of Eq. (3-37)). This has brought  ΔY₂ ≠ΔY₁ which doesn't satisfy the Keynesian cross condition. Therefore, an error in the Keynesian multiplier effect theory has originated in the adoption of the assumption, Δ I₂ = Δ NX₂ = 0. [ This sentence on 3 lines might be rewritten. ]

  We find that such a discrepancy between the right-hand side and the left hand side was not caused, if we had defined Y₁ as Y₁ =C₂ + G₂ from the beginning. However, we might make an unrealistic theory with the assumption that private investment doesn't exist. Consider a system which consists of components x_i ( i =1,2,·  · ). The whole is included as one of components. Suppose that the change of a certain component x_k is not effected by small changes of the other components. For example, this is the case such that only the investment is a function of an interest rate. When we make a differential analysis to the components not relating to the interest rate, we shall easily understand that both the investment itself and this amount in the whole must be taken away at the same time.

  Furthermore, it is known today, from the break-even analysis for an income statement, that we have the two types of wages: One is proportionally variable wages to the quantity of the production; The other is fixed type wages to it. Consider the case where one wishes to analyze a problem which consists of a production( = sales), costs with wages and a profit.  If we make a differential analysis to this problem without special attention, the fixed type wages might be taken away from the analysis. These are the dreadful aspects in differential analyses. We must review whether these treatments might have been done in other theoretical, economic analyses. [This paragraph would be reconsidered later.]

(11)  Let a( t) be a nonlinear function of time t. We define notations as: a_K( t)= ΔC₂( t)  / ΔY₁( t)  a_I( t) = Δ I ₂( t) / Δ Y₁( t),  a_NX( t) = ΔNX₂( t) / ΔY₁( t) and  a_G( t) = ΔG ₂( t) / ΔY₁( t), where a_K( t) or a_I( t) etc. is the marginal propensity of expenditure to each good. Since a_K( t)  =  ΔC₂ ( t) / Δt  / (ΔY₁( t)/ Δt) at any t, we have 

aK( t)  +  aI( t)   +  aNX( t)  +  aG( t)  =1

(3-R7)

  Multiplying both sides of Eq. (3-R7) by ΔY( t) ( = ΔY₁( t) = ΔY₂( t) ) again gives

ΔY( t) = aK( t)ΔY( t)  +  aI( t) ΔY( t)  +  aNX( t)ΔY( t)  +  aG( t)ΔY( t)

(3-R7-1)

  Thus we obtain, for example, from Eq. (3-R7-1), the following equations:

ΔY( t) = (Δ I ( t)  +ΔNX( t)  + ΔG( t) ) / (1 - aK( t) )

(3-R7-2)

ΔY( t) =  ( ΔC ( t) +Δ I ( t) ) / (1 - aNX( t) - aK( t) )

(3-R7-3)

Eq.  (3-R7-2) expresses the following two equations:

(Δ I ( t)  + Δ NX( t)  + ΔG( t) ) : ΔY( t) = (1 - aK( t) ) : 1

(3-R7-4)

ΔC ( t) : ΔY( t) = aK( t) : 1

(3-R7-5)

  That is to say, both Eq.(3-R7-4) and Eq.(3-R7-5) as well as Eq.(3-R7) express the proportions of the parts ΔC ( t), Δ I ( t), Δ NX( t) and ΔG( t) to the whole ΔY( t).  For that reason, Eq. (3-R7-2) is an identity.

  In addition, assume that indirect taxes = 0 to simplify notations. ΔGVA is shown as: ΔGVA = ΔD + Δπ + ΔW  + ΔT - ΔT = Δπ + ΔW where D is depreciation, π is operating surplus, W is employees compensation and T is transfers including direct taxes. From the relationship between GVA and GDP, we have, omitting the symbol ( t),:

ΔY =  (Δπ + ΔW ) + (ΔC + ΔI +  ΔNX + ΔG) - (Δπ + ΔW)+ (ΔT - ΔT)    =  (Δπ+ ΔW ) + {ΔI + (ΔG - ΔT) +ΔNX} - (Δπ +ΔW - ΔT - ΔC)

(3-R7-6)

  As mentioned in (10), why can't we adopt the assumption, Δ I₂ = ΔNX₂ = 0 ? The reason is that Eq.(3-37) should hold between the incremental five goods, ΔY₂, ΔC₂, Δ I ₂, ΔG₂ and ΔNX₂. This is the same as Eq.(3-R7) holds. This means that each good should be generated as follows:

(a)　It is generated in such a manner that Eq.(3-37) or Eq.(3-R7) is satisfied;

(b)  It is by nature independent of each other. However it is generated in a half mixed way of both dependence and independence, observing with each other and being affected by each other;

(c) In Eq.(3-R7-6), it is generated satisfying the following constraint condition:

ΔI + (ΔG - ΔT) +ΔNX = Δπ +ΔW - ΔT - ΔC

(3-R7-7);

(d)  It is not necessarily satisfied that each term in Eq. (3-R7) equals always 0;

(e)  The relationship between each economic term in the equations concerning the three economic aspects is realized in such a manner that it satisfies the break-even chart described in (19) below, reflecting people's will.

  Consequently, the following economic phenomena will often occur in theory: for example, no amount of an increase ΔG2 can give an increase ΔY2; a little amount of increase ΔG2 can generate the other goods largely.This is the basic principle in human activities.  

(12) If we adopt the next assumption:

Assumption7: ΔG₂ = ΔNX₂ = 0

, we obtain a figure in which ΔI₂ is replaced in place of ΔG₂ in Fig 4-6. In this case, the very problem of the additional expenditure by a government dose not hold because ΔG₂ = 0.

(13)  If we adopt the following assumption: 

Assumption8: ΔG₂ +ΔNX₂ is included in ΔI_T 

,we obtain ΔY₂ = ΔC₂ + ΔI_T. This case is like the one taken up in (12). The figure for this economic model is gotten by replacing ΔI_T in place of ΔG₂ in Fig 4-6, which is shown in Fig.5-5 below.

Fig 5-5 Keynes’ｓ original chart

  This economic model is used in J. M. Keynes' original work（, although he didn't show its model chart). In this regard, he thinks that ΔＣ₂ is naturally provided proportionately to ΔI_T someday without determining the time period.  We shall name Fig 5-5  'Keynes' original chart'.

  In Keynes' original chart, the Keynesian cross condition is satisfied and so the MPC exists. Does  any strange result occur when we use the chart to analyze the actual economic phenomena?

  To conclude ( see 5.3 and 5.4 in the next section), we can't use a chart like Keynes' original chart in order to analyze economies over time if time changes. Keynes' original chart model is a two-degrees of freedom system in which the variables are both C₂ and I₂, and Y₂ is a dependent variable. Just when Keynes assumed the existence of the MPC, one constraint condition is added to the system to create a one-degree of freedom system.

  J.M.Keynes claims the multiplier effect ΔY₁＝ΔI_T/ (1 - MPC). Then, ΔI_T must be the  only remaining variable in the one-degree of freedom system. ΔI_T should be given as a variable which doesn't relate to the passage of time at the beginning time point before the execution of ΔY₂. ΔC₂ depends on ΔI_T, and so both ΔC₂ and ΔI_T  don't relate to the passage of time. Thus, the economy shown by Keynes' original chart doesn't relate to the passage of time . This relationship is explained in detail in Fig.5-4 in the previous section. By the way, we can select ΔC₂ as a remaining variable, when we obtain ΔY₁=ΔC₂ / MPC. Analogously, ΔY₁ depends on ΔC₂, and both of them don't relate to the passage of time.

Fig 5-4 One-degree of freedom system

  By the way, we can select ΔC₂ as the remaining variable, when we obtain ΔY₁=ΔC₂/ MPC. Analogously, ΔY1 depends on ΔC₂, and both of them don't relate to the passage of time. We have a system which consists of ΔY₁(=ΔY₂), ΔC₂ and ΔI_T , That the system is a one-degree of freedom system is as follows: the actions of the system is determined uniquely only by the action of the whole ΔY₂, and ΔC₂ and ΔI_T are the constant proportions to the whole ΔY₂, even though each variable is a function of time.

(14)  In Keynes' original chart theory, the whole ΔY₂ , i.e. both ΔC₂ and ΔI_T doesn't change with time but rather by the uncontrollable fluctuations of people's decisions. Furthermore, ΔI_T must always be provided before the finished change ΔY₂. In short, the economy expressed by Keynes' original chart is one where both IT  and C₂ don't relate to the passage of time. This society is one where ergodecity is assumed. This might be similar to a preindustrial society. 

  This economic model might be appropriate in analyzing a distribution problem of economic goods or incomes, but it is not fitted  for analyzing a dynamic economic problem of an unsteady economic situation in a capitalist society. This viewpoint is important in particular when analyzing big problems of the capitalist country in an unsteady state of economy, most of all, an unemployment problem.

(15)  A large shortcoming in Keynes' original chart theory is that the MPC condition is broken just when we intend to supply ΔI_T . To resolve this problem, we must simultaneously and instantly provide ΔC₂ (=ΔI_T∙ MPC /(1-MPC) according to the supply ΔI_T which doesn't relate to the passage of time.

(16)  The next large shortcoming in Keynes' original chart theory occurs just when ΔG₂ is supplied in order to resolve an unemployment problem. This will be explained using  Fig.5-6. Let ΔI_T be ΔI_T = ΔI₂ + ΔNX₂ + ΔG₂. When we select ΔG₂ as a variable corresponding to the one-degree of freedom in this analysis, we have to consider the roles of remaining ΔI₂ and ΔNX₂ at the same time, and have to take them into consideration in the analysis.

Fig 5-6

  By virtue of Keynes' original chart theory, the equation  ΔY₁= (ΔI₂₊ ΔNX₂ + ΔG₂) / (1 - MPC) can be derived. The MPC is defined for ΔI_T. Consequently, both ΔI₂ and ΔNX₂ as well as ΔC₂ have to be simultaneously and instantly provided with the production ΔG₂. This brings up a puzzling situation. Now that we can't produce ΔI₂ and ΔNX₂ instantly (private investment demand decreases), according to the supply ΔG₂, the multiplier effect theory has been used for analyzing an unemployment problem.  

  In Keynes' original chart theory, simultaneous and instant supplying of ΔI₂, ΔN_X, and ΔG₂ is sought according to the supply ΔG₂.

  Can readers claim that Keynes’s original chart theory is an effective ways to analyze the problems of a dynamic economy though this theory has such characteristics? If we can't, Keynes’s original chart theory will lose its effectiveness at least in regard to a dynamic economic problem analysis.

(17)  As mentioned above, the real economy will have many-degrees of freedom with respect to time.  However, the MPC is factually found by statistical observations of the real economy as shown in Fig. 3-11, in which a_k = MPC . By the way, it is said that we have two types in consumption lines. One is the Keynes type in which the consumption line is a linear one with a vertical axis-intercept. The other is the Kuznets type in which the line is linear through the origin. The line  will really be a nonlinear line through the origin. How can the MPC be observed in the real economy?

Fig 3-11 Corrected Keynesian multiplier effect model

  The author thinks that the MPC can be obtained only when we look at  the resultant moving of consumption in a many-degrees of freedom economic system in a stable economy. Each firm always endeavors to stay profitable. The MPC, derived from the consumption psychology of a community exists. Each consumer observes his present income and envisages his future life. Each entrepreneur observes the consumption behavior of people and envisages his future state. Everyone observes present business conditions and decides his daily behavior. 

  The moves of a consumption product and an investment product are similar to those of a right leg and a left leg. If a road ahead is bright and seems to be safe, their moves will appear to be constant. The walking in this case will look like as if both legs are one pair and the move is based on a one-degree of freedom system. However, if one takes a right step on a road ahead being dark and not safe, the left step may stop or back off after that. If one is in the middle of boxing, each leg will move freely and independently. In the latter two examples, both legs move in a two-degrees of freedom system. 

In a similar way, a consumption product and an investment product will move in the stable economy in such a manner that the MPC is kept constant. However, in an unsteady economy with respect to time, each product will probably move independently by free-will, and the original nature of products, i.e. independence will appear. 

(18) Just as the Keynesian multiplier 1/(1- MPC) doesn't exist, the tax multiplier MPC/ (1 - MPC) doesn't exist also. This stands to reason because taxes are partial transfers between income group. See the explanation on the taxes problem in 5.1.

(19) In fact, there exists a more fundamental defect in the Keynesian multiplier effect theory, although this defect is not a mathematical error. As described in the previous section, there are two types of charts when we analyze economic problems. One is a time-series( or proportion) type input-output chart shown in Fig.5-1, and the other is a break-even type chart shown in Fig. 5-2.

Fig 5-1 Time-series type input-output chart

  In the time-series type chart shown in Fig.5-1, the horizontal axis shows realized, cumulative production output. The vertical axis shows components of realized, cumulative GVA( input). The time-series amount can be a monthly cumulative one, or a yearly final one to show it over a period of years.

  The characteristics of the time-series type chart are as follows:

(1) The band width of operating surplus( 'profit' in Fig.5-1) shows a part proportion of the whole, Y1. This holds for other components.

(2) As the operating surplus is a function of one variable, Y₂, Y₂  is uniquely determined for a given operating surplus, if the economy is in a stable condition.

(3)  Incremental GVA, ΔY is expressed as follows: ΔY = Δπ +ΔW +ΔD, where π = operating surplus, W = employees compensation and D = depreciation. As D is determined by the quantity of  depreciable assets produced in prior periods, we have ΔD＝0. Consequently, when we make an economic, analytic equation with the economic model  shown in Fig 5-1, only the directly proportional equation between ΔY and ΔW emerges because π is directly proportional to Y in the chart.  [ This paragraph might be rewritten later. ]

  However, W is in truth, expressed as follows: W = W_V + W_F, where W_V = the component, in W, which is proportional to Y, and W_F = the one, in W, which is not proportional to Y, and so W is a two variables function. In business fluctuations, W_F is not a constant but a variable which is determined by a management executive's will according to business conditions. He changes both W_V and W_F in such a manner that π/Y is held constant or π＞0 is kept even in bad business conditions. In analyzing a business fluctuation problem, one must not assume ΔW_F= 0. The analysis for the  unemployment problem is the very one for the ΔW_F. 

  In the break-even type chart shown in Fig. 5- 2, only the end point shows the realized GVA (input) and the realized output( products).  In the figure, GVA is expressed as income.

Fig 5-2 Break-even type chart

  The characteristics of this chart are as follows:

(1) Any point on the intermediate position of the 45 degrees line connecting the origin to the end point shows an imaginary coordinate point( imaginary output, imaginary input).

(2) The operating surplus( 'Profit' in Fig.5-2) is a function of 3 variables, the whole final products Y₂ , a variable cost ratio and fixed costs in GVA. If we take stocks into consideration in this type of chart, we can draw the 'break-even line' as described in §1 in 'Management Accounting'.

  This break-even type chart includes the break-even point which is a result of the relationship between input and output. In business management, business entrepreneurs always manage the relationship between input( costs and profit) and output( sales) .

  By the way, the shape of the graph in Fig 5- 2 always changes a little in an accounting period, because the expenses which are regarded as fixed costs at the beginning of the period, always fluctuates according to both business managers' and( for wages)  governmental( for depreciation) will.

  As Y₂ is placed at the same position in the both charts, the concepts of these two charts are prone to be misunderstood. However these two charts show different concepts. 

  The Keynesian's consumption function shown in Fig.3-3 is derived from the time-series type chart concept. What kind of information can we expect  for the additional government expenditure ΔG  from this chart? The answer will be that the yearly average proportion of ΔG( revised additional amount)  to G (amount on an initial budget) can be calculated. The income change corresponding to ΔG is, as a matter of course, the same amount as ΔG. However, the total income doesn't change mathematically, because ΔG is products which resulted from transfers between some people( or firms) and other people( or firms).

  As the national accounts results from all firms' income statements, the national economy shows integrated individual firm's movement. This movement of the economy is Fig.3-13. The point K is the only point where profit=0. Both F^V and F^F are independent variables. This is the real economy. In the real economy, both the Walrasian equilibrium and the Keynesian multiplier effect can never exist.   

Fig 3-13 Break-even type input-output chart

  The incremental part enclosed with the point K and K' is the same as Fig.3-14, although the both appearances are a little different. 

 

Fig. 3-14 Relationship between Components in Δ G

  The description in this part (19) might concern being right or wrong( incorrect/ flawed) for the profit maximization equation, Δπ = ΔY₂ - (ΔW( wages) + ΔD（ depreciation））=0, by the classical school( as used in Keynes’s theory ), which gives the first postulate ： wage equals marginal product of labor. The reason is as follows: (1) Costs including wages consist of both fixed and variable costs. The fixed costs are not constants but a kind of variable, which is determined by human will and changes without relating to final products quantity, whilst the variable costs change in proportion to the quantity; (2) The equation Δπ = 0 means π equals a constant, so that it may not be the profit maximization equation. 

  Within an accounting period when depreciation is almost regarded as constant, we have, from Eq.(3-40), Δπ = (1 - a _V) ΔY - ΔW_F, where a _V = W_V / Y ( variable wage cost ratio), ΔW_F = incremental, fixed wage cost and ΔY = ΔW_V + ΔW_F. The equation Δπ / ΔY＝0 means that π is constant, so that the wage line is parallel to the 45 degrees line.  Consequently, the equation Δπ =0 is not the profit maximization equation but that the equation Y = W + π (const.)+ D( const.)  i.e. ΔY = ΔW is assumed. This is impossible depending on the break-even chart which shows a real management. In any age entrepreneurs pursue the realization of the largest profit margins or the minimum losses through improvement of human abilities and increase or decrease of the ΔW_F.   It is of my current opinion that the profit maximization equation does not exist in production, even though ensuring of profit does. Therefore, the author is suspicious of the first postulate, so it is possible that my stance for this part may change after consideration. [The description of this paragraph is not final and will be corrected. Using the notation π(const.) might be incorrect. ]

  Although both employees and employers are content with a wage level, the author agrees with Keynes’ deduction that an unemployment situation exists regardless of the content. That is to say, the labor forces in enterprises are not determined by a supply-demand relationship between employees and employers with a parameter of wage, but are actually determined at the crossing point between final products supply and effective demand. 

  See Fig 3-13 which expresses the break-even type Δ G-Y chart. In this figure, the above paragraph states that the labor forces which are included in the costs on the vertical axis, are determined in such a way that all the costs plus a profit are satisfied with the final product demand on the horizontal axis. Despite the erroneous assumption: Y = W + π( const.)+ D( const.) adopted in the theory, the correct deduction was reached over the assumption. Nowadays this is only to be expected. However, this deduction is amazing when we consider that the break-even type profit chart had not yet spread( and perhaps still hasn't even today) over the economics field in the days of the work "The General Theory of Employment, Interest and Money". 

(20)   (1) When initial investment spending ΔI is implemented, investment demand rises by ΔI, which generates production by ΔI. The increase in production changes into income by ΔI according to GVA. (2) This income increase ΔI in turn raises consumption demand by MPC·ΔI,  which generates production by MPC·ΔI. This production changes into income by MPC·ΔI. (3) This income MPC·ΔI raises consumption, production and income by MPC²·ΔI as well. (4) This feedback continues indefinitely, which results in the multiplier effect : ΔY = ΔI + MPC·ΔI + MPC²·ΔI + · · · = ΔI / (1-MPC).  

  At each step, the following equations hold: increase in consumption demand = increase in consumption products  = increase in income from production. There appears to be no contradiction anywhere. This logic is that which has cemented the multiplier effect theory in modern economics. It is extremely difficult to simply explain the mathematical errors in this logic, so this has resulted in an extensive explanation. 

  Before J.M.Keynes, there had never been another person to consider this line of logic. Persons with no grounding in economics will not generally know of this logic. People in general tend to view economies as follows: products( as inventories) are manufactured depending on the quantity of labor, with labor being defined as employees, plants and equipment. They don't see a profit until a product is sold( really after a time period has passed).  In short, they think that no working or selling will result in no manufacture of products, hence no income. Many also believe that real economies are based on addition and subtraction. In fact, this belief is correct.  

  As is generally known, the logic of the multiplier effect uses the following Taylor series expansion: 

1/(1 - x) = 1 + x + x2 + · · ·             [ |x| < 1 ],    x = MPC

(6-1)

x/(1 - x) = x ( 1 + x + x2 + · · · )

(6-2)

  The logic by Taylor series is reduced to the following problem, as shown in Fig. 6-1. Can initial investment expenditure ΔI (fixed value) generate automatically consumption ΔC_CR  (variable value)? This question is by nature the same as,  can we  automatically get ΔY_CR = ΔC_CR/ MPC, if we give initial consumption ΔC_CR ?

Fig.  6-1

  First, we mention about whether the multiplier effect equation ΔY₁ = ΔG₂ / ( 1 - MPC)  by  government investment expenditure ΔG₂ holds. As is explained in (10), since the Keynesian cross condition doesn't hold in this equation, this is a mathematically wrong (incorrect/ flawed) equation. Consequently, a measure against unemployment by a government due to this equation is mathematically meaningless , so from now on, this equation will not be considered. 

    The problem is the equation ΔY₁ =  ΔI_T / (1 - MPC) which appears in Keynes' original chart shown in Fig 5-5. ΔI_T is highly correlated with private investment.

Fig 5-5 Keynes’ｓ original chart

  This equation claims ΔY₂ = ΔI_T / (1 - MPC) because ΔY₁ = ΔY₂. This equation is mathematically correct  as a relational expression between the symbols. However, this relationship only holds  when consumption products ΔC₂ are supplied and purchased, continuously and simultaneously, keeping the proportion MPC ( = ΔC₂/ ΔY₂). It is understood, from Fig 5-5, that we eventually acquire only income, ΔI_T,   which is equivalent to the initial investment expenditure ΔI_T.

  So this raises the question, how should the sentences in the first paragraph be rewritten? The first sentence "(1) When initial investment spending ΔI · · · · according to GVA." means that ΔI₂ (product by firms and its purchase by others) = ΔI₃ (cost and profit in firms) = ΔI₁ ( ΔGVA corresponding to ΔI₃ ). This is correct as it stands now.

  For the sake of decreasing symbols, we will not consider government finance here. This operation doesn't give logical inconsistency because the ΔG₂ problem is not considered here(ΔG₂ = 0 ), and the multiplier effect problem is the one between incremental components of GVA. What are the resources( inputs) of ΔI ? Those are capital depreciation ΔD (=0) and saving ΔS = Δπ_I + Δπ_C + ΔW_I + ΔW_C - ΔC, where π= profit, W = wages; subscripts I and C show that the quantity with each symbol belongs to the symbol products ( f.e. C). 

  In fact, when we say  "initial investment spending ΔI is implemented", the sentence "only initial investment product ΔI is produced, while consumption product ΔC is not produced" is requested as an implicit premise. Then,  we have ΔC = Δπ_C = ΔW_C = 0. This results in ΔS = Δπ_I + ΔW_I.  Under this premise, as a matter of course,  consumption doesn't increase because all the incremental incomes( = incremental wages + incremental profits) are expended in order to produce ΔI. Consequently, if we rewrite the first sentence as " only the initial investment products ΔI is produced and sold", the meaning of the latter sentence  becomes clear, and so the sentences (2), (3) and (4), which follow the first one, don't hold.  

  Although the author has stated, at the preceding description, that the sentence "only initial investment products ΔI is · · · as an implicit premise " is requested, Keynes' economic system, in the nature of the theory, can't allow the request. In Keynes' system, ΔC₂ with ΔI_T is one pair, and they can not be separated. 

  So in this case, the first paragraph should be rewritten as follows: if we ensure simultaneously the implementation of additional private investment( production and purchase) ΔI_T and additional private consumption ( production and sales) ΔC₂, at the proportion of MPC to 1 - MPC, in a certain time period, we can get additional inputs ΔC₁ = ΔC₂ and ΔI₁ = ΔI_T at the same proportion. The sentences regarding the multiplier effect (2), (3) and (4) don't follow the first sentence (1). If we could execute the measures of the first sentence , Keynes' economic system would be possible. However, even so, the multiplier effect doesn't exist. 

  That is to say, the above-mentioned clause 'When initial investment spending ΔI is implemented,' is not compatible with Keynes' multiplier effect theory. If an economic measure is like this, the MPC condition is not kept constant. If we change the clause into ' When additional investment and consumption ΔI + ΔC are  implemented,' the multiplier effect doesn't exist.

  Thus, we arrive at the conclusion that all the sentences in the first paragraph, '(1) When initial investment spending ΔI is implemented,  · · · (4)  · · ·  = ΔI / (1- MPC).' do not hold as far as the MPC condition is preserved.

  Let us return to the first. Is the following proposition correct?: if a government pay out cash benefits (fixed amount handout) ΔC_MONEY to people in order to stimulate consumption, investment fund ΔI_MONEY corresponding to a marginal propensity to save ( tax=0) will be induced. As a result, capital goods ΔI(=ΔI_MONEY) will be produced apart from the production of consumption goods ΔC（=ΔC_MONEY）. This does not hold in a depression. The reason is that the kinds of the consumption goods ΔC and the capital goods ΔI are different from each other. In a depression, plant and equipment including workers for both consumption goods and capital goods are in overcapacity conditions. Therefore, even if we assume that the cash benefits do not change into savings in the relevant year, although ΔC_MONEY can be transformed into the consumption goods ΔC, capital goods ΔI can not be produced. In addition, if consumption falls as much as ΔC_MONEY in the next year, the borrowed money ΔC_MONEY will remain. 

  Let us return to the first. Is the following proposition correct?: if a government pay out cash benefits (fixed amount handout) ΔC_MONEY to people in order to stimulate consumption, investment fund ΔI_MONEY corresponding to a marginal propensity to save ( tax=0) will be induced. As a result, capital goods ΔI(=ΔI_MONEY) will be produced apart from the production of consumption goods ΔC（=ΔC_MONEY）. This does not hold in a depression. The reason is that the kinds of the consumption goods ΔC and the capital goods ΔI are different from each other. In a depression, plant and equipment including workers for both consumption goods and capital goods are in overcapacity conditions. Therefore, even if we assume that the cash benefits do not change into savings in the relevant year, although ΔC_MONEY can be transformed into the consumption goods ΔC, capital goods ΔI can not be produced. In addition, if consumption falls as much as ΔC_MONEY in the next year, the borrowed money ΔC_MONEY will remain. 

  The logic of this part is very important. Consider an economic condition where: (a) it is in near full employment;  (b) plant and equipment are not in overcapacity; (c) business conditions are averagely on a profitable basis; (d) public capital goods are effectively useful. (e) assets held by financial institutions are sound. In the situation, an autonomous increase ΔC of private consumption goods can be inducement of a new investment fund ΔI_MONEY. If banks supply creative funds ΔI_MONEY to enterprises, ΔI_MONEY will be transformed into capital goods ΔI. Further, if wages are increased as much amount as maintaining the marginal propensity to consume being constant simultaneously around or immediately after the consumption, GDP will increase. If the wages are not increased, the facilities will be immediately in overcapacity. When many job seekers of low wages exist, another phenomenon will be produced.

  The economic processes of very good and very bad business conditions do not become a reversible process. The reason is that the preconditions described above are quite different from each other in both business conditions, so that  each different economic phenomenon occurs. That is to say, each economic phenomenon is irreversible one.  

  (21)  Commencing with Paul Samuelson’s multiplier–accelerator macro dynamic model*, several papers which developed the simple Keynes multiplier effect model, were presented. Through those papers, the above-mentioned treatment (20) holds. To simplify matters, the following equation is used.　

ｚ（ｔ）＝ｘ（ｔ）+ｙ（ｔ）

（6-3）

where time is denoted by t, C（ consumption） by ｘ, I（ investment） by y, and GDP by z.

  In this equation we have three variables ｘ, y, and z, where any of two can be independent variables and the remainder is a dependent variable.

  The following equation is added.

ｘ（ｔ）＝a · ｚ（ｔ - τ）

(6-4)

, where a is a constant and τ is a parameter which expresses a time lag. 

  If we add Eq.(6-4) to Eq.(6-3), the two independent variables automatically decrease to one. This results in one fixed or unique pattern of relationship between ｘ(t), y(t) and z(t), although the variables are indeterminate with respect to time and can move freely in relation to time.

  If we add the following equation:

ｙ（ｔ） = ｂ·（ｘ（ｔ） - ｘ（ｔ - τ））

(6-5)

, the movement of all variables becomes fixed and has a certain pattern with respect to time. Any form of equation such as a differential equation is allowable in place of Eq.(6-3) and Eq.(6-4), when the same treatment holds.

  The most important point is that Eq.(6-3) is always satisfied by the solutions of any two equations at any time by the definition of the economic terms. Eq.(6-3) shows thatｘ(t) and  y(t) are components of z(t), and the three volumes of them are already generated in a unique form at the same time. That is to say, if we don’t have an economic means that realizes ｘ(t), y(t) and z(t) surely and simultaneously, the economic model will not be able to be used as an economic policy. Solutions obtained by any three equations with initial conditions will be fixed to time, so economic movements after the initial conditions will have be determined. Given this solutions, any economic policy after the initial conditions will no longer be effective. In short, since the investment multiplier effect itself does not exist, these analytical methods are nonsignificant.

* Paul A. Samuelson, THE HISTORY OF ECONOMIC THOUGHT, WEBSITE, http://cepa.newschool.edu/het/home.htm. 

(22)   We shall consider the case where private-sector demand decreases. If we can increase government investment based on its economic policy for the decrease, the government investment will certainly be replaced with it even though the multiplier effect doesn’t exist. 

  We can try to imagine a nationalization process of private enterprises to state-owned ones. We assume, as a mater of course, that the concepts such as depreciation, sales and profit in the latter exist. The unemployment due to the decrease of the private-sector demand will not occur, if the following situations arise: Nonexistence of government borrowing increase; no competition against other countries and private-sector enterprises; ensuring sales by forcing; no business responsibility for losses.

  If this process is executed with an increase of government borrowing, the greatest obstacle to the unemployment relief is the absence of the business responsibility for losses. This is because we can’t clearly separate profit (loss) from depreciation recovered by sales. This can be understood from the following: On the one hand, in a piece of  equipment, a decrease of value due to physical deterioration as well as obsolescence i.e. a reduction in operating capacity is caused. On the other hand, a recovery of investment fund (cash purchase amount) is promoted. These two money values don’t necessarily coincide with each other in each accounting period. The loss corresponds to the impossibility of recovering the initial fund.

  If, through the government economic policy, the state-owned enterprises run losses for years, then the previous increase of government borrowing can be returned only by a new refinance. The point of this logic (the problem of the business responsibility for losses) is that the person, who can receive the reward from the capacity i.e. the recovery of the equipment expense by means of product sales, should be the owner of the equipment. In short, ｔhis results from the following: nongovernmental enterprises consider that the depreciation in business accounting through sales is a means of recovering of an investment cost, while government officials underneath think that the consumption of fixed capital( if the term depreciation is used) in social accounting is the cost for an imaginary government final consumption  (remaining value of capital goods with ages is pointless in accounting.).

(23)The economy in the one-degree of freedom system means that if people in an economic field moves toward a direction, other parties in the other field will automatically follows to the same direction. In Keynes’s multiplier effect theory, the generation of the final products（ = consumption goods + investment goods） should be in the one-degree of freedom system, because the MPC really exists statistically. Consequently, both consumption and investment should be generated simultaneously.

  However, such a movement between consumption and investment in the one- degree of freedom system naturally occurs in a stable economy with a few changes in economic growth, aside from the multiplier effect. The reason is that final products are equal to incomes. Relevant parties of consumption and investment (even in this field buyers and sellers exist) observe each other at any time. If one side convinces the cause of the other side’s activity, the former follows the latter.

  The following phenomenon possibly occurs: with a stable economic condition in a short run, an export ΔEX suddenly increases rapidly; a national income increases by just that much; a new domestic consumption is increasingly produced after the stimulation by the consumption; the domestic investment as （1-MPC）/MPC times of the consumption is executed; a consumption responds to the wages included in the investment again. In this manner, although the investment multiplier effect doesn't exist, we can say that a stimulation effect of investment to consumption, or the reverse might exist, but it doesn't always exist.

  Therefore, the author doesn’t deny the economic policy based on the principle of effective demand by Keynes because of my conclusion that the investment multiplier effect doesn’t exist. The government must actively address the national measures that cannot be achieved by individuals and the private sector in the system without an increase of public bonds. If the government doesn’t wish to undertake these economic measures, then that government has no place to exist.

(24)   Even if anyone looks at Keynse's investment multiplier effect formula or examines it, the formula appears to be surely an identical equation. Even the author sometimes lose the reason of the formula's error after a little time's passing. Important parts to know the error are: the item (10) in this section; in Fig. 9.3, §9, the correct expression, 5+ΔG= Y** = ( 2+ΔG)/（1-3/（5+ΔG））, giving ΔG=ΔG, presented by the author, on the other hand, the wrong expression, ΔG≠ ΔG/(1 - 3/5)=2.5ΔG, giving 1≠2.5, derived from the equilibrium theory. Additionally refer to the writing under Eq.(19) in Section 4, in the paper, Yuichiro, Hayashi, “Production Theory to Analyze Human Wisdom and Global Phenomena Operations”, on this website. Only a little mention concerning this problem is made.

Conclusion

  The error in the Keynesian multiplier effect theory results from the following 4 fundamental mistakes:

(a)   One is a misinterpretation of the flowing of partial GVA, including transfers( see 3. taxes problem in §5) in the three aspects, Y₁, Y₂,and Y₃, within a year. If we adopt the assumption, Δ I₂ = ΔNX₂ = 0, the partial output amount of Δ I₂ + ΔNX₂ is taken away from the total ΔY₂ , although ΔY₁ holds the total input. That is to say, the Keynesian cross condition ΔY₁= ΔY₂ is not satisfied.

(b)   The second is confused understanding of the two problems with the two different  assumptions; (i) ΔI₂ = ΔNX₂ = 0 and (ii) ΔG₂ = ΔNX₂ = 0.

(c)   The third is the fact that ΔC₂, which has been realized simultaneously with the execution of ΔG₂ ( or correctly ΔI₂ ), and is hidden from sight in  the Keynesian multiplier effect chart( Fig.3-3), has been overlooked. On this point, the author will consider the cause included in the equilibrium theory.

(d)   The fourth is as follows: Keynes' original chart model is a two-degrees of freedom system in which variables are both C₂ and I₂, and Y₂ is a dependent variable. Keynes didn't know that the system created a one-degree of freedom system just when he assumed the existence of the MPC. In this system, the amount ΔI_T ∙ MPC / (1 - MPC) = ΔC₂ must be provided simultaneously with the production ΔI_T in order to keep the MPC condition. When both ΔC₂ and ΔI₂ are fixedly connected to be an economic quantity with a one-degree of freedom, the series multiplier logic of production (= income) dose not hold. 

  Incidentally, Fig.4-7 is correct as a time series type input-output chart. However, Fig.4-7 has been made only for explanation, so such an economic phenomenon( independent increasing of only ΔG₂) will not occur in real-life. Fig.3-13 is correct as a break-even type input-output chart. 

 Fig.4-7 Correct time series type input-output chart

  When we look at final account data in  both a time-series type chart and a break-even type chart at the end point of one year, both data sets are the same. However, the former chart shows only data as a final result, but the latter chart shows the cause of the final result, as well as measures to manage the economy, in addition to the final data. When we want to analyze economic problems, including those of  unemployment, we must use the concept which is implied in the break-even type input-output chart.

  The errors in Keynes' multiplier effect theory resulted from several errors overlapping each other. Each error was not based on small mistakes but that relating to the fundamentals of conventional methods for analyzing economic problems. If readers don't advance step by step by checking the above-mentioned observations (1) - (21), understanding the error of the Keynesian theory as a whole may be difficult.

The end of the disproof

Postscript

  A part of descriptions has been deleted.

  In fact, there are two main causes for the result of the huge cumulative government  bonds in Japan. The one is as follows: Keynes's principle of effective demand theory, based upon the investment multiplier effect, is originally the theory which gives relief to the unemployment by a government; whereas, in Japan, this theory has been used as a basic theory grounding the means since a year for (1) economic  development, (2) overcoming a depression and (3) expanding domestic demand, going beyond its purpose; surprisingly, the basic formula  grounding this principle is itself wrong( incorrect/ flawed)! However, the error of the formula is the same all over the world. This abnormal state in Japan doesn't result from a mistaken economic policy such as applying a theory to a field beyond its application range, but comes from the Japanese national character of a single-mindedness that if people accept a thought, they never allow discussions about different views from various sectors nor thinking including disbelief.    

  On this subject, the followings are concerned:  Japanese remedies for the depressions resulted from the adoption of the floating exchange system by the U.S. in 1971 and the first oil shocks of 1973; the serious trade friction between Japan and the U.S., especially due to the rapid increase of automobile export from Japan to the .U.S. in 1980s; the announcement（1990） of public investments worth 430 billion yen over a ten-year period starting in 1991 for domestic demand expansion ; antidepression policy, with public investments, for handling the big depression occurred by the burst of the bubble economy at the start of 1991. 

  By the way, the government public investments expenditure of 2006 already went back to that of before the birth of the bubble economy. Consequently, the economic crisis problem due to the erroneous Keynes's investment multiplier effect formula is already over, apart from the repayment problem of the principal national funds.

  The other is as follows: the yearly social security costs in Japan, which are a typical consumption expenditure ( value decomposition of goods and services produced in a given year), has not been kept within the limit of the present tax revenue; then the deficit became consuming, of cumulative national savings, which can't be expected to recover (recovering by means of sales) in the future.

  This mistake doesn't originally relate to the advent of the aging society in Japan, but this results from a mistaken economics thought and political measures based on it, which have continued from the past. This is unrelated to Keynes’ multiplier effect theory but this results from a big mistake in economics.  Politicians mistakes are secondary, because politicians' economic understanding, including that of bureaucrats and media, doesn't surpass that of economist.

  Why has the mistaken thought, that people in the present and future generations should bear the senior welfare and pension costs of the past generation, endured and been allowed in economics for so many years? It is abdication of learning that if one might attribute its reason to the historical origin of the establishment of the pension system. Why haven’t economists advocated the logic that the value of human life must not be so high as to threaten the people’s livelihood ( i.e. life) both in the present and the future?

  According to the Keynesian’s systems of 3-accounts in the national economic accounting, the present yearly investment by people, including corporations, is only due to the present yearly saving (gross) by the people. Only the people that invested in the past receive the present yearly return of investments (repayment of debts). All the people, who didn’t invest in the past, accept no return in the present. Medical spending for human life has the same economic meaning as military spending for the fate of the nation. In addition, no creature has such a principle to ensure their survival into the future, and gives priority to their life rather than that of their offspring. Economists have responsibility for pursuit and argument of truth in economics. They need not accommodate themselves to the wish of the public.

  Concerning the foregoing descriptions, refer to the following website:　Current Japanese Fiscal Conditions and Issues to be Considered (2003)

  Let’s get back to the point. Anyone could give such easy explanations or questions from the beginning of Keynes’ work. Keynes himself understood the explanation more than anybody. At that time, all but he gave such explanations. In this economics situation, Keynes struggled to resolve the unemployment problem to find the revolutionary logic: a certain volume of investment generates the income of the investment multiplier times that of the investment; involuntary unemployment is generated through enterprises’ adjustment of the labor quantity due to inflexibility in labor prices; in order to rescue unemployment, public investment must be used to become the effective demand. Thus, all the economic minds around the world have admitted that the logic is ‘mathematically true’. From that point, nobody has been able to disprove the logic. 

  Joining the word ‘mathematical’ to the sentence ‘Anyone can reproduce a phenomenon based on a hypothesis ’ means ‘scientific’. Keynes’ multiplier effect theory has been ‘almost scientific’ for many years. Therefore, this theory is, even now, placed in almost every textbook of economics.  Thus, this theory is taught at a primary stage not because it doesn’t fit the economic reality or it is an old theory but because it is the most important foundation, supporting the modern economics, which has been conceded as mathematically true. The textbooks in which the multiplier effect theory is not placed, take precedence of the fact that the theory doesn’t fit the economic reality but they skirt the mathematical disproof of the theory. How do they explain the relationship between income and demand? 

  At this moment, any economist knows that the investment multiplier effect doesn’t fit the economic reality. However he does not dare to express a completely negative view against the multiplier effect theory based on a mathematical proof at the risk of his career as an economist. 

  In order to release the economy from the influence of the multiplier effect, it is imperative to disprove mathematically the multiplier effect theory, and to present a new theoretical method to analyze the unemployment problem in place of the multiplier effect theory. The reason is that firstly only the disproof of Keynes’ multiplier effect theory without presenting a new analytical method in place of it will return nowadays to the situation 70 years ago. 

  Secondly, such things as the economic fluctuations, instinctive economic remarks and the entire ordeal which followed the economic bubble burst in Japan, will have been forgotten a few generations later. Each economic measure executed in these years without the support of mathematical logic will be treated as one of many feasible measures in order to recover the result due to a failed economic policy. Still, after the years, remaining logic will be that that can still be admitted scientifically or mathematically true even in that year. The author can’t imagine how damages might be produced to the world economy, if the economic situation in the 1990s in Japan would reoccur in China or India.

  Keynes’ multiplier effect theory is not one that can be completely denied with simple explanations. No, the possible existence of the investment multiplier will not be able to be denied entirely in a certain economic situation. The reason is that the multiplier is like a fine thread (as the economics situation now stands), which connects production and demand. Thus, the MPC or the investment multiplier would remain in economic models into the future as a coefficient that is largely associated with economic group propensity or business executives’ will.

  A few courageous economists *（2） have denied for the multiplier effect theory at the risk of their careers as economists despite the cold reception and criticisms of the academic world, even though they couldn’t give mathematical disproof. The aforementioned behavior is undesirable with the view that it was not accompanied with a disproof on par with Keynes’ proof for the multiplier effect; alternatively, cited references of past nay-sayers was not given.

  Moreover, this behavior is discourteous to the following: Keynes himself; other famous economists; economists who have devoted their whole lives to the Keynesian theories; the preceding nay-sayers; university students over the world who are taught the theory even now; unemployed people, bankrupts, suicide victims and business managers in unfavorable business conditions. Incidentally, the recent sales of the authors’ company fell by a 1/3 in comparison to the previous average sales for several years.

  It has taken 11 years to reach here from the sudden idea of the managed gross profit chart theory. The author’s paper on that theory was presented to proceedings of a Japanese accounting academy several years ago. The original copy was ordered to be  largely rewritten more than once or twice. Its publish in the proceedings was finally rejected forever because its reading referees and the editorial board couldn’t ascertain whether the author’s solution, or Solomons’ solution appearing in accounting textbooks,   was true for the same accounting problem. Through constructing the disproof logic for Keynes’ multiplier effect theory, the author experienced, a number of times, feelings of hopelessness, because immediately after he thought his question had been solved, extremely difficult problems emerged one after another. The author himself has made all this effort. 

  Keynes is still the greatest economist in history through the creation of the principle of effective demand based on the investment multiplier effect theory and other outstanding achievements. Who but he could write the work ‘The General Theory of Employment, Interest and Money‘ and construct the fundamental framework of modern economics?

  Readers might think strange that the author has cited no earlier references for the multiplier effect theory throughout his formulation of the disproof. We have, in fact, both many research papers that intended to develop Keynes’ original theory, and a substantial number that intended to deny intuitively the multiplier effect. The latter references admit the existence of the investment multiplier, and state that the effect has declined for some reason or other. However, as far as the author knows, no study has existed, whose primary purpose is to mathematically disprove the logic. If such a study were to exist, and completely succeeded in its purpose, the multiplier effect theory would not appear in economics textbooks. Furthermore, such a large postwar economic growth in Japan and the huge quantities of national bonds (plus local bonds) might not have been realized.

  I went back and forth among many equations and charts correcting them. This web site shows my thought processes. I now find no small number of inconsistencies or duplicate descriptions within the sections. However, if I were to revise my descriptions consistently throughout, I would need much time. Such work will not change my conclusion that the Keynesian multiplier effect simply doesn't exist.

 In closing, I don't say that we don't need public investment but say that the Keynesian multiplier effect of public investment doesn't exist both mathematically and in accounting formulation. A public investment might trigger an upturn or recovery in the economy. However, the investment  might cause such an effect  but it will never produce the multiplier effect.

 The proper public investments are indispensable both for social rearrangement of incomes( or products) and economic growth. 

   Economically sound investments are the solid foundations for the people's future.

  Excessive consumption due to public funds is a malady that will eventually result in the economic demise of the nation.

Slight modification  2013, Aug. , 25

Research process

● The author finally reached the finding of Fig.4-6 on 13/Aug/2004. I needed 7 months to reach it from my first description of this section( after I had found the correct chart Fig.3-13). The reason that I so luckily reached there is that, firstly, I had the correct solution Fig.3-13, and that, secondly, I had had the knowledge of the managed gross profit chart theory.
● However, this finding of Fig.4-6 was useless(21/Dec/2004)! After all, the origin of the Keynesian's error resulted from the adoption of the assumption, Δ I2 = Δ NX2 = 0.
● Nevertheless, there existed a more fundamental defect in the Keynesian's multiplier effect theory, although this was not a mathematical error. The defect is that the Keynesians did not adopt the break-even type chart, but  that they adopted the time-series type chart (20/Jan/2005). Now I must add that luckily I  met the ΔG2-Y1 problem in another textbook before encountering the ΔI2-Y1problem in Keynes' original work. From Keynes' original work, we will find it almost impossible to detect errors for his logic for the multiplier. (10/Feb/2005)
● Besides, the author found that Keynes' original chart doesn't relate to the passage of time.   In addition, the MPC relationship in Keynes' original chart theory is broken just when we intend to add an incremental government expenditure to resolve an unemployment problem. (24/Feb/2005). In order to keep MPC constant, we have to provide both ΔI2 and ΔNX2 simultaneously with ΔG2 (4/Mar/2005)
● The author found an wrong( incorrect/ flawed) logic in the Taylor series expansion applied to the multiplier effect ( 1/Sep/2005).
● The author found that the erroneous derivation of Keynes's investment multiplier effect formula was made in the process of applying the equilibrium theory method. (Sept. 2006)
● In a good, both its planned demand function and actual supply function always equals in shape as far as a statistical, actual demand function is used. If this treatment is made for all goods, the equilibrium theory method that the cross points between aggregate demand and supply functions show the equilibrium points, doesn't hold. (20/1/2007)
● The author could finally express the error, included in Solomonds' theory, in a chart using monetary expression. It took 12 years from hitting on an idea of making a break-even chart for absorption costing. Anyway, the author never dreamed that the error in Solomonds' theory would relate to the error in Keynes's investment multiplier effect chart. The author knew nothing about economics while he was involved in Solomonds' theory.  We could not possibly achieve the complete disproof of Keynes's investment multiplier effect theory even in the future without the process of disproving Solomonds' theory. (31, Mar., 2007).
● The profit maximization does not hold in a time-series type production chart ( a proportion chart), because profit is almost proportional to Y. The profit maximization does not exist in a 45-degree break-even production chart. The reason is self-evident from the chart. (11, Apr., 2007)
● The equilibrium analysis method can not be applied to an analysis of national product accounts expressed with a 45-degree break-even chart type. （16, Apr. 2007)
● "Accounting System for Absorption Costing", a business method of accounting using absorption costing, including a method of drawing a break-even chart was admitted as a patent by the USPTO on 27, November, 2007.
● On September 7, 2008, the U.S. Treasury made capital infusion into Fannie Mae and Freddie Mac………Occurrence of the economic major incident……… Occurrence of unprecedented unemployment…………Fiscal stimuli. We face three problems:  (1) In national accounts, is it possible for financial debt( asset) to continuously increase without economic instability in theory? If such a situation does not theoretically hold or should not make hold in real economies in theory, this large drop of assets value will necessarily occur again sometime in the future in a different form from the direct cause of the latest financial crisis. (2)  For a country, the form of the account matrix concerning both GVA and GDP is already determined; a domestic demand type H matrix is determined for a domestic demand-led country; a foreign type H matrix is determined for a foreign demand-led country. Can, under the above condition, a new domestic (or foreign) demand stimulation policy assure the recovery of injected funds in a short period?  (3) We must build a new financial theory which bears practical execution: one which makes it possible to monitor the runaway of both the financial and insurance industries’products business, which have the analogous property of gambling in the world; markets must be able to shut off the effect of credit mutilation of financial products from a banking panic in production economies.
● Through the author’s research of the method of Leontief’s input-out table analysis written in economics textbooks, historical references and its explanatory documents by governmental institutions, he has noted the following: according to the input-out table analysis, when a demand, in an industrial sector, for incremental final products ΔY (monetary amount) is generated, the demand generates a multiplier effect to many other industrial sectors; the multiplier effect finally reaches the original sector itself; the effect continues on and on based upon the series expansion of Leontief’s inverse matrix; it is caused by the multiplier effect formula similar to Keynes’ investment multiplier effect in the principle of effective demand; these theories are internationally recognized.  Economic effects from the holding of Olympics and World expos have been analyzed with the above-mentioned method often being added e.g. Keynes’ investment multiplier effect; economic analytical methods, due to the multiplier effect concept, have become a universal standard basis in theoretical analyses for economic effect problems as a matter of course through mathematical formulations. For this reason, modern economic policy after Keynes and Leontief has been carried out with this concept.  For that reason, the author understands that the world would reject the denial of the multiplier effect concept, but as long as flawed theories remain orthodox, the public finances of all the countries in the world will rush into debt hell. To avoid such a future, denying the multiplier effect concept through only words will bring about no change. Only by mathematically disproving the multiplier effects included in Leontief’s input-out table analysis and Keynes’ investment multiplier effect formula will it be possible to effect change in economic policy. (1 Oct., 2009)
● There are controversies over the global economy. Apart from emotionally charged discussions, one must understand the economic meaning behind in the global economy. China (India as well from now) with their low wages and massive labor populations are now continuing to grow. Developed capitalist countries with high wages and labor populations are now in long recessions. The recessions are caused by a basic reason different from the causes of the global financial crisis. The advantage of foreign trade is that the two countries can grow both countries’ production competing against and complementing uncompetitive goods with each other. If the current economic situation continues thus, keeping each trade volume (monetary amount), the domestic industries in a high waged capitalist society will be crushed and unemployment will increase, as long as export industries cannot absorb employment population in the industries which would be excluded by the import goods; mass media will criticize the situation without understanding the cause; governmental spending will be increased indefinitely by politicians seeking election victory; as a result, governmental budget deficits will grow ad infinitum. However, if China or India implement capital advance to high waged countries directly and produce the same goods as their export goods, employing laborers in the other countries, this problem will soon be settled; mere lending of funds or purchasing of bonds will come to nothing, because production responsibility exists on the other side. Furthermore, if a low waged country raised wages by at least 3 or 4 times, the problem would, as a matter of course, similarly soon be settled. This problem is the same as one of currency exchange between countries.  In order to remedy the situation, it is necessary to reconstruct foreign trade theory targeting the maximization of GVA or labor population in the whole country; further, a worldwide strategy in cooperation with high waged countries all over the world is required. According to the author’s conjecture, its theoretical result will be the following: if a country with an infinite labor population of low wages and a country with a finite labor population of high wages competed with each other against the same kind of manufacturing and processing goods( less row materials) intending to increase both sides’export products keeping each monetary amount, the domestic industries in the high waged country would finally be crushed including all export industries. Without the need for difficult economic analyses, all producers who have competed with Chinese products and as a result have gone out of business and closed their domestic factories; and all laborers who have had wage cuts or have been made redundant in the world know the reason. In any country, a large proportion of the producing population (domestic laborers) is not associated with manufacturing export goods (their own country’s GVA) but is rather associated with purchasing import goods (other country’s GVA). After the US-Soviet Summit in Malta, the world economic situation which started with the Russian Revolution in 1917, changed; and a new war concerning economies and politics began in the world. This war would continue for several decades as long as the Chinese economy did not ruin itself. The world must surely recognize this situation.  China is such a country as the following: aside from its low wages; (1) its labor population is enormously large compared to its foreign trade partners from the view point of economic research subjects; (2) the relationship between the people and the nation is different from that in capitalist countries( the nation does not allow critical activities by its people against its wage systems); (3) it acts as an autocratic state in terms of economic interventions such as exchange operations etc.; (4) its land ownership system is different from capitalist countries in terms of land prices and a ban on domestic immigration; it can provide any goods desired by other countries( After the end of the cold war, other countries have, seeking profit, invested capital and skills in China, produced cheap goods and reimported the goods to their own countries. This is a typical example of the fallacy of composition.); (6) The Chinese technical level is excellent( they really launched a manned satellite). Once the same problem occurred between the USA and Japan. However, the Japanese labor population was small and Japan realized a high waged society. Japan advanced into the USA, setting up plants and hiring workers. There are high-tech industries which do not require much manpower and low-tech industries which require great manpower, in national industries. The low-tech industries will be crushed by imported goods produced in low waged countries. The high-tech industries will further promote laborsaving or will escape to lower waged countries. As a result the employment the total wages, will decrease. The production bases of the high-tech industries will be moved to the lower waged countries. This situation will continue over many decades all over the world.    Keep in mind that the author’s claim in this part of the paper has not been backed up with mathematical and accounting proof but is merely conjecture.(22 Jan., 2010)
● April, 2011.  Publiction of §2 Presentation of the Cash Flow Matrix (CF Matrix) Analysis for National Accounts and a Consideration of the Leontief Input Output Analysis Theory (PDF)
References
(1) John Maynard Keynes, "The General Theory of Employment, Interest and Money", 1997, GREAT MINDS SERIES, Prometheus Books.
(2) James C. W. Ahiakpor, "On the Mythology of the Keynesian Multiplier", http://www.geocities.com/ecocorner/intelarea/jmk7.html.
(3) Jochen Hartwig, "Trying to Make Sense of the Principle of Effective Demand", the seventh international post Keynesian workshop, 2002; URL,  Trying to Make Sense of the Principle of Effective Demand.
Last updated: 22 Jan., 2010