§4 Answers to My Questions for My Claim that the Keynesian Multiplier Effect Theory is Wrong

DCE Yuichiro Hayashi
Keynes' principle of effective demand based on the Keynesian multiplier effect is mathematically wrong.
The Keynesian multiplier effect doesn't exist.
Please forgive me that some beginner's questions in economics might be included in this section. However, all the questions relate to the Keynesian multiplier effect theory. I intend the general public and science-technology majored people besides economic specialists for readers. I also hope readers will forgive me that some inconsistencies between equations, figure numbers and assumption numbers are found in the previous, this and the following sections, because my theory is still now under building.
Note that the word 'good' always includes both a physical product and a service.
"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 described based on the principle of effective demand which is connected to the multiplier effect( Fig.3-3 ) excepting objections without mathematical disproof. 

Fig.3-3
All the discussions which doubt Keynes's multiplier effect,  recognize the existence of the investment multiplier 1/ (1 - MPC) and argue whether its effect is large or small. 
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. 
[Question-1] Consider each firm's income statement. In the statement, we have each break-even sales X i　(φ）. If we add the all  firms' income statements, we will have a break-even sales X(φ） for the final goods of  all the firms. Is the break-even sales X(φ） equal to the break -even sales constituted with FF and FV shown in Fig.11?
[Answer-1] Please see from Fig.2-1 to Fig. 2-8. Fig.2-1 holds in a single firm's accounting, therefore it will hold when all firms' figures are added together. Thus, the summation of X i(φ） will correspond with X final (φ） in Fig. 2-8. Fig. 3-12 also answers this question.
[Question-2]  Public investment must be a part of final product. Will it not then induce intermediate products by any multiplier effect?
[Answer-2] Public investment( demand) is surely not an intermediate product but a part of the final product. It is contracted with a first firm. The first firm will contract a part of the first contract with a second firm, and the following firms will follow suit. The volume of transactions may often increase by reselling. In this process, apparent monetary  volume of transactions induced by the first contract will increase. 
However,  the essential goods value, that is gross value added,  in the first contract is divided and distributed between various parties. The remaining value in each firm is merely a share of the essential value of the first contract. Thus, the first contract can never induce more value than the first, essential value.  
[Question-3]  We will define an intermediate product ratio a m, that is a m = Intermediate product / Total product. Does a m have another multiplier effect on economy in place of the Keynesian multiplier effect, even though the Keynesian multiplier effect theory is wrong?
[Answer-3] The problem of the  intermediate product is the same as that of the internal transaction problem found in firms' accounting.  External sales and  fixed assets finally produced by the firm itself responds to final product in national accounts. The amount of internal transactions do not affect the income statement by set-offing.  Hence, a m is unrelated to the relationship between final product and inputs. 
[Question-4]   May we be allowed to say that multiplier effect  Δ G / (1 - a V) exists in the future, even if the effect is small?
[Answer-4]  No, we are not allowed. See Δ G = Δ P +Δ FV + Δ FF. The right side is income. After the contract ΔG is signed, needed costs will be generated during the period of the contract. If expenditure ΔG is a consumption, ΔG will be soon finished. Sooner or later, expenditure ΔG is distributed and changes into income ΔG in the national accounts. Income ΔG is consumed, saved, or taxed within the present one year. In the next year, a new chart of Fig. 3-12 will be drawn independently of the previous year without the multiplier effect. ΔG has no effect over its value in the future. The economic growth problem is a different one from the multiplier effect. Refer to [Question-21] for this question. 
[Question-5]   Doesn't the infinite geometric series effect 1 / (1 - a V) = 1+ aV +a2V + · · · exist in the future, even if aV is a small value?
[Answer-5] This shape often appears in input-output problems in economics or accounting. Fig.4-1 shows the break-even chart where notations are as follows: XCR = critical output = critical input, V = V(X) = variable input, C = constant input, a = V(XCR) /XCR = tanα. 
  

 
                                          Fig.4-1 Break-even chart
We can easily obtain:
          1/(1 - a) = XCR/(XCR - V) 
                       = XCR/C
                       = ↑CD/↑AB
That is to say, 1/(1 - v) is a magnification from the constant C to the critical input XCR.
Incidentally, we have:
           ↑DE/↑AB = V / C
                            =  v ∙ XCR / C
                      = a / (1 - a) 
The expression, X=constant /  (1 - variable cost ratio), is merely another expression of output X = input X. The shape of 1 / (1 -  a V) is also found in Fig. 3-13 in my theory. There is no profound meaning in the series expression.  The shape of the infinite geometric series for 1 / (1  -  aV) is a result of sheer luck.
[Question-6]  Is there any value connecting to the future, included in the input-output table chart?
[Answer-6] There is no connection to the future in the input-output table chart. It only shows the result of past economic activity. Refer to the words 'domain' and 'range' in [Question-18] for this question. In addition, refer to [Question-23].
[Question-7]  Is there any possibility that variable cost ratio a V takes a negative sign? 
[Answer-7] Variable cost ratio a V itself takes a positive value, because a V = variable input energy / output. However, a V Δ Y will take a negative value, if Δ Y is negative.
[Question-8]  What is the relationship between the input-output table chart shown in Fig. 3-13 and the time series output chart shown in Fig.3- 8?
[Answer-8] These relationships can be expressed as shown in Fig.4-2.

Fig. 4-2 Relationships between input-output table chart and time series output chart 
[Question-9]  Doesn't the consumption function have any effect on the economy?
[Answer-9] Consumption will have a large effect on economic growth. This description claims that consumption is very important for the economy and the income multiplier effect by additional government expenditures with MPC does not exist. Refer to [Question-28].
[Question-10]  Will ΔY increase, if costs, ΔFF and  ΔFV increase by the relation: Δ Y = ΔP +ΔFV + ΔFF?
[Answer-10] Yes, given an economy where evaluation of goods is internally made, and exchanging of goods is ensured. Profit ( a part of injected energy) is a result of external evaluation in competition. If there is no guarantee of goods exchanging, there is a possibility that  Δ Y itself decreases.
[Question-11]  Does my theory deny the Principle of Effective Demand by J.M.Keynes?
[Answer-11] I don't deny his logic of involuntary unemployment generation. However, I deny the use of the equation ΔY1 = ΔG2  / (1 - MPC) as a means of solving the unemployment problem, because the equation is incorrect. Please refer to ' Chapter 3, The General Theory of Employment, Interest and Money'.  Read the descriptions of summing up in part II, comparing it with Fig. 1 as well as Fig. 13. Keynes says that the consumption function curve shows convex upward, i.e. the MPC gradually decreases as income increases. The under-mentioned description may be right, but we can't expect the multiplier effect of D2 to fill the gap between the labor supply and the goods demand. This will be explained later. I have not yet a solution to an unemployment problem.
His theory is explained using Fig. 13. Inputs mean community's aggregate income including internal labor costs, depreciation, profits and others( they come to the former three terms in the final stage). The aggregate income is a function of the volume of employment N. The amount of labor N is a function of effective demand D which is the sum of consumption D1 and investment D2. Consumption depends on psychological characteristics of the community ( propensity to consume). D is a function of labor N, that is D1+ D2 = D = Φ（N). D1 is, of course, a function of N, which is written as D1 = Χ（N) depending on the propensity to consume. Thus, we have Φ（N) - Χ（N) = D2. When employment N increases, D1 will increase, but not by so much as D, then a gap between labor supply and goods demand will occur. This is the core of Keynes's theory.  FF and  FV  in my theory correspond to the above-mentioned inputs. ( References  (1), (2) ). Profits are also included in FF and FV .  
[Question-12] Does this theory give any route to resolve  the unemployment problem intended by Keynes?
[Answer-12] National accounts are connected with past national production which excludes the unemployed. I have now no means of resolving the unemployment problem, however, Fig. 3-14 may partially relate to it.  In addition, a  balance sheet including unemployed and employed workers could possibly be related. 
[Question-13] Can we divide all costs F in the national production account into  FF and FV? 
[Answer-13]  I haven't  yet derived this method. The method of separating costs into variable costs and fixed costs in normal direct costing will be useful. A method of treating both inventories  and cost variance, formulated by me, will most likely be needed. 
[Question-14] Can the Keynesian multiplier explain the cause of economic growth?
[Answer-14] If we define I T2 = I 2 +  NX2 +  G2 , we obtain:

                     Δ Y 2 = Δ Y 1 = Δ Y = Δ I T2 / (1 - a K)                 (4-1)
It seems, from Eq. (4-1),  that Δ I T 2 causes economic growth Δ Y and the growth rate is 1/ (1 - a K). This is wrong. The principle of equivalent of three aspects holds at any period in a year or any  years. Eq. (4-1) expresses the following two equations:

                    Δ I T2 = (1 - a K) ΔY                                               (4-2)

                    Δ( Y - I T2) = a K ΔY                                               (4-3)
Adding Eq. (4-2) and Eq. (4-3) gives Δ Y = Δ Y. Two conclusions can be made, that this has no meaning or that another unknown cause of the economic growth would possibly exist.
[Question-15] What do you think autonomous consumption aB means ?
[Answer-15] *Consumption C is by nature a part of GDP Y and a nonlinear function ( apart from its cause) with respect to Y. The function C can be expressed with Taylor expansion at Yi with respect to Y. This gives C = Ci +C'i (Y - Yi )+ Ri, where Ci = C at Yi, C'i = the 1st derivative of C at Yi and Ri= remainder term. This is a linear equation. For each i, we have each equation. We can collect the equation data for many decades. If we handle the data statistically, we shall obtain the consumption function as shown in Fig. 4-3. Intercept a B appears when we approximate a nonlinear function, which go through the origin, by a linear function in the interval [0,Y1]　on the horizontal axis. 
.
                                        Fig. 4-3 Meaning of aB
* "In addition, it serves to warn us that the consumption function is probably curvilinear, originating at the origin and rising at a decreasing rate with respect to income.", William A. Schaffer, 2.4, Chapter 2, Regional Models of Income Determination: Simple-Economic-Base Theory, Regional Impact Models, July, 1999, http://www.rri.wvu.edu/WebBook/Schaffer/regionalGT.pdf
[Question-16] There is one case in which ΔY2 is affected by a change in Y2, that is ΔY2=ΔY2V,  and another in which it remains unaffected, that is ΔY2= ΔY2F. Is the chart shown in Fig.3-13 applicable to both cases? 
[Answer-16] Yes, it is. However, each part amount in ΔY3 will change. The reason is that profit is obtained from the result of the end of period. 
[Question-17] Is the  way of expressing the Δ G-Y chart limited between aspects Y2  and Y3? 
[Answer-17] No, it is not. We can select the other partner Y2 from  Y1, Y3 or even  Y2 itself. In the latter case,  ΔG is included in ΔFV, if this year's G is different from the last year's G.
[Question-18]  I have felt anxious again as to whether my theory is correct or not and I have reexamined textbooks of economics. They state as follows ( the suffixes have been added by me): 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. I knew later that this is an original economic model by J.M.Keynes.
This problem is different from the problem, shown in Fig.3-2, in which incremental expenditure ΔG2 is included with the assumption that I and NX are constant. I can't find a mathematical error in this process. Is the multiplier effect shown in the equations and in Fig. 3-3 correct after all? 
[Answer-18] I must organize again the economic and mathematical premises. 
(1) We must always discriminate an economic value as to whether it belongs to an input value (Y3 aspect) , to an output  value (Y2 aspect) or to an income including expenditure value ( Y1 aspect) when we consider an economic problem. 
When one good is exchanged between a seller and a buyer, the resulting monetary value is equal. The selling price of the good is a seller's expected value . A buyer's expected price is different from that. One transaction price is determined at the time of exchange. Why is the transaction price equal?  The reason is that a profit is included in the input value. The profit acts as a spacer or a filler which is used when erecting a building or setting up a machine in the right position. Inventory stocks play a similar role by the profit in Y2 aspect. 
I will mention in passing, that the equality of the prices between a seller and a buyer is possible, without profit, under the enforcement which denies free will. As a result, the explanation in which the equation Y2 = Y1 is given by the equilibrium between planned expenditures and actual ones is not so good. We can see a counter-example in regulated nations. The equality, Y2 = Y1 is realized by profit or enforcement. The  enforced prices include controlled and monopolistic ones.  
When we use words such as 'variable' or 'fixed' or 'increment', we must differentiate where the words are used in Y1 (income and expenditure), Y2 (product) or Y3 (cost and  profit) aspect. Increment ΔY2 in Y2 aspect is different from ΔY3 in Y3 aspect in meaning. When we have the equation ΔY2 = ΔY3 =ΔY1, it means that: (a) supplying (making) of consumption and investment goods; (b) obtaining and distributing of  incomes; (c) purchasing( consuming and investing) of goods and saving; (d) dying of both short life and long life goods, and both recovering ( refreshment or renewal) of people's working power and investment goods power.
In the (a) stage, old investment goods work cooperatively with people. In the (b) stage, firms receive costs of investment goods taking their places. In the (c) stage, credit is generated between sufficient and insufficient parties. These activities are simultaneously finished. 
(2) When we use increment sign Δ for  variables and make any incremental equation, we must always be aware of the difference between independent variables and dependent variables. We must further be aware of the domain of the independent variables, i.e., the set of all the independent variables (starting points in a function) , and the range of the dependent variables, i.e., the set of all the dependent variables (ending points in the function).  
(3) Each transaction among consumption goods and other goods occurs independently of each other. 
 In other words, when one consumption good is bought or sold, the transactions of investment goods and other goods do not relate to the consumption good transaction.
(4) The consumption function has resulted from statistical observations between averaged, past, yearly gross value added GVAi ( i=1,2, , ,) and consumptions Ci ( i=1,2, , ,). This means that each instant transaction does not necessarily follow the propensity to expend. 
(5) Each good sold in Y2 aspect is connected to each input in Y3 aspect in only one clearly defined path. The reason is that  the national accounts are made by a summation of each income statement  in firms. Each good can be connected to each cost and profit by cost accounting in one firm.  This fact in the national accounts will be explained in the next section. 
From the above-mentioned perspective, we shall reexamine the example in the textbooks. What is the domain in this problem? The domain is the set of all the time-series humans' inputs which are labor forces and intellectual powers injected into products. In other words, the domain is the set of each divided gross value added ΔY3.  Notice that the depreciation and the profit originally result from  humans' inputs (values recognized or accepted by the others). The range is the set of all the divided time-series products ΔY2 and incomes( or expenditures) ΔY1. 
The domain and the range closely relate to the time-interval. We shall assume as follows: (1) the consumption function is defined from 50 years ago to the end of the last year, (2) the shape of the consumption function is applicable to this one year. By the latter assumption, generality will not be lost in this problem. 
The condition where we have equality for Y1, Y2 and Y3 , is that we measure the value of Y(= GDP) in each aspect at the same time. If we regard the same time as the end of this year, generality will not be lost. Consequently, the domain and the range in this problem are defined within an accounting period( one year), i.e., this one year.  
When we consider ΔY2 in Y2 aspect, one problem occurs. In which time interval has ΔY2  been, or will be, generated?  As a matter of course, by the definition of the domain, ΔY2 is included in this one year time interval. Thus, Y2+ΔY2,  Y3+ΔY3 and Y1+ΔY1 are included in this one year.
The example description should be changed as follows. Y2 + ΔY2 = Y3+ ΔY3 = Y1+ ΔY1,  ΔY2 = ΔY3 = ΔY1 and Y2 = Y3 = Y1.  This results from the fact that the domain and the range are within this year and the transaction prices are equal. 
We make the following assumption: incremental consumption function ΔC1 = aKΔY1  is given in the year, where aK is the MPC and is a constant coefficient. This expresses that the resulting equations, which will be derived from now on, hold only when ΔC1 is really produced and expended as much as aK times of ΔY1 and aK is constant through the year. The reason is that the definition of consumption goods is that they are really produced and expended within one year.
As a matter of course, we have, at the same time,  ΔI1 = (1 - aK)ΔY1 = aIΔY1, ΔY1 = ΔY2,  ΔC1 = ΔC2 and ΔI1 = ΔI2. Hence, we have ΔY1 = ΔY2 = ΔC2 + ΔI2 = ΔC1 + ΔI2 = aKΔY1+ ΔI2. This shows that both ΔC2 and ΔI2 are already produced, ΔY2 is transformed into ΔY3,  ΔY3 into ΔY1,  and ΔY1 into ΔY2 . Accordingly, ΔY2 (both ΔC2 and  ΔI2) is already sold or ΔY1 is already expended. As a result, we obtain ΔY1 = ΔI 2 / (1 - a K). 
However, the equation ΔY1 = ΔI 2 / (1 - a K) does not express the multiplier effect, because we have, at the same time, ΔI 2 = ΔI1, ΔC 2 = ΔC1 and ΔC 2+ ΔI 2 = ΔC1 +ΔI1. These equations merely express that ΔC 2 and ΔC 1 can be replaced by ΔC 3 (the gross value added) as well as in  ΔI 2 , ΔI 1 and ΔI 3. 
The equation ΔY1 = ΔI 2 / (1 - aK) expresses that products ΔC 2 and ΔI 2 in one year are produced statistically with the proportion of aK to (1 - aK) and the total is ΔY1.  As a matter of course, each good in C 2 and I 2 is really produced independently of each other.
In general transactions in the real economy, in a daily or monthly period in one year,  transactions in C 2 do not necessarily relate to the consumption function. For example, we have the case where we have only ΔI 2 without ΔC 2, that is ΔC 2 = 0, when  ΔY2 = ΔY1 = ΔI 2.
In this case, we always have in the short period, ΔC2 = a'K ΔY1,  ΔI2 =  (1 - a'K) ΔY1 = a'I ΔY1,  a'K + a'I =1, where a'K is a nonlinear function of Y1. In this manner, we can't generally expect that aK (MPC) is constant in the real economy. Even in this case, the equations ΔC2 + ΔI2 = ΔC3 + ΔI3 = ΔC1 + ΔI1 always hold. We can't find the multiplier effect. It does not exist.
The equation ΔY2 = ΔC2 + ΔI2 ( [Question-18] ) means, that both ΔC 2 and ΔI2 have been produced. The equations ΔY2 = ΔY1 = ΔC1 + ΔI1 means that the  income, which is equivalent to ΔC1 and ΔI1,  have been gained by consumers and firms. The equation ΔY2 = ΔY1, means that a part of  ΔY1 has been  transferred as taxes to a government and expended in ΔY2. Therefore, if an incremental production or expenditure is only ΔI2, income is only ΔI1 (= ΔI2 )and consumption ΔC1 is not realized.  Thus, the present question and answer will be as follows. From the next questions, readers will know that the error in the Keynesian multiplier effect theory exists in the point that  it neglects or doesn't take into consideration simultaneity in economic actions such as producing, consuming and saving (see [Question-23]). 
[Question-19-1]  We denote symbols as follows: Y2 = production goods = C2(consumption goods) + I 2 (investment goods); Y3 = costs and profits included in the goods = C3 + I3; Y1= gross income =  C1(purchase of C2) + I1(purchase of I 2) = C1(consumed goods value) + GS( gross saving).  By the principle of equivalent of three aspects, it is assured at any time that Y1 = Y2 = Y3 for variables Y1, Y2 and Y3 ; C1 = C2 = C3; I1 = I2 = I3; I1=GS. We have the statistical result that C1 / Y1 = MPC( the intercept has been neglected) and I1 / Y1 = GS / Y1 = 1 - MPC.
What amount of income increase when additional investment ΔI2 is demanded from Y1 level or supplied from  Y2 level ? What effect does the demand have on the shape of the consumption function?
[Answer-19-1] The question gives Y1+ΔY1 = C2 + I2 + ΔI2. Then we have ΔY1 = ΔI2. If ΔI2 is small, the effect is small. If ΔI2 is large, MPC will become MPC = ( Y1+ΔI2 - (I2+ΔI2)) / (Y1+ ΔI2)=( Y1 - I1) / (Y1 + ΔI1) = C1 / (Y1 + ΔI1).
[Qestion-19-2] What amount of income increase when additional consumption and investment  (ΔC2+ΔI2) is demanded or supplied keeping MPC constant? What effect does the demand have on the shape of the consumption function?
[Answer-19-2] Y2 + ΔY2 = Y1+ ΔY1. Then we have ΔY2 = ΔC2 +Δ I2 = Δ C1 + Δ I1 = ΔY1. We already have ΔC2 = MPC∙ΔY1. Further we have ΔI2 = (1 - MPC) ΔY1.These underlined two equations only show that  ΔY1= ΔY2. Under this condition the shape of the consumption function remains unchanged.      
In addition, [Qestion-19-1] is the case where ΔC2 =Δ C1= 0 in [Qestion-19-2]. In a similar manner, ΔC2 = ΔC1=ΔY1= ΔY2  (aK=1) holds when Δ I1 = incremental gross saving = Δ I2 = 0. This latter case might occur in a mere instant in the national economy, or in a family or small local area, because ΔC or ΔI can express incremental consumption or investment for one good trade, a big transaction; in daily to yearly or local to national economic actions. However, MPC(<1) = ΔC / ΔY is obtained statistically in the national economic accounts for one year period. 
[Question-20]  Can we extend the fiscal period from 1 to 5 years which includes the future?
[Answer-20] Yes, we can, if we could make the national accounts in 5 years. However, nobody can predict that the value of aK will remain the same.  If incremental investment goods can give efficiency and confidence for the future, the marginal propensity to consume will be kept or rather increase, vice versa. 
[Question-21] Do you say that the main cause in the Keynesian model's errors has resulted from the difference between the figures (a) and (b) in  Fig.4-4?
[Answer-21] No, I don't. Both figures show the incremental, simultaneous production of ΔC2 and ΔI2.  In both cases, we have ΔC2 = aKΔY2 and ΔI2 = (1 - aK)ΔY2. Both cases will occur in the interval ΔY2. The difference between ΔC2 and ΔI2 is mainly the product live-spans.  In addition, if I2 is always a fixed value in Y2, we have ΔC2 = ΔY2 and ΔI2 =0, then we can't draw such a figure as (a). The figure will be corrected as ΔI2 =0 and ΔY2 = ΔC2 in the figure (a). The figure (a) shows that ΔI2 is the increment of Y2 which is not affected by a change of Y2. However, the relationships between ΔI2, ΔC2,  I2, C2  and ΔY2  in both figures are same, on the vertical line, at Y1 +ΔY1on the horizontal axis.
The difference between the figures (a) and (b)  in Fig. 4-4 is not an essential problem. The figure (b) in Fig.4-4 is correct for both productions ΔC2 and ΔI2, and wrong only for production ΔI2. I claim that if incremental supply (vertical axis) is solely ΔI2, incremental income( horizontal axis) is equal to ΔY2 (= ΔI2), as shown in  Fig. 4-5. 

                       (a)                                                                           (b)

Fig.  4-4

Fig. 4-5 

 In ΔG-Y chart problem, ΔG is originally an income for employees and firms to become a transfer to a government. ΔG does not belong to ΔC by definition. If ΔG is financed by a  gov't bond, this monetary value has been resulted from the income not consumed, i.e., saving.  ΔG changes into an expenditure and becomes costs and profits in firms' production. This flow occurs simultaneously.  In this flow, only the monetary value ΔG flows. Consequently, Fig. 18 is wrong  for only  ΔG. Refer to also [Question-19-1].
[Question-22] Is the Keynesian multiplier effect chart Fig.(3) right or wrong for an incremental expenditure ΔG2, if we allow the Assumption Δ I2 = Δ NX2 = 0?
[Answer-22]  We have the four basic equations, from the national economic accounts,:
                   Y2 (Demand) = Y1 (income)                                        (4-4)
                   Y2 = C 2 + I 2 + NX 2 + G2                                            (3-2)
                  ΔY2 = ΔC2 + ΔI2 + ΔNX2 + ΔG2                                 (3-37)
                  ΔY2 = Δ Y1                                                                      (4-5)
Keynesians assume that I2 + NX2 = a constant, in the time interval from K to K'. Then, we have:
Assumption-5: Δ I 2  = Δ NX 2 = 0                                                           
Substituting Assumption-5 into Eq.(37) gives:
                 Δ Y2 = Δ C2 + Δ G2                                                        (3-11)
Eq.(3-11) holds whether the consumption function exists or not under the assumption (I 2 + NX2)  is constant.
In Fig.3-3,  the Keynesian cross point K should basically express Eq.( 4-4) and Eq.(3-2). The point K has moved to the new point K'. Then, at K', both Eq. (3-37) and Eq. (4-5) should hold. Note that Δ C2, ΔI2, ΔNX2 and ΔG2 have been realized( produced) and incremental income ΔY1 has been received at K'.  
We have MPC( = aK ) in the interval from K to K', that is: 
                ΔC2 =  aK ∙ΔY1                                                                 (3-12)
When we regard ΔY1(=ΔY2) as a whole, Eq. (3-12) shows the part ratio of ΔC2.
When we see Eq.(3-37), the left side is a whole and the right side is a sum of parts Δ C2, Δ I2, Δ NX2 and Δ G2. From descriptions from Eq.(3-14) to Eq.(3-17), we have known how to express other parts when one part ratio to the whole is already known. That is:       
                  ΔI2 + ΔNX2 + Δ G 2 = (1 - aK )Δ Y1                              (3-R8)
This is the final equation. This is equivalent to Eq.(37).
Substituting Assumption-5 into (3-R8) gives: 
                 ΔG2 = (1 - aK )Δ Y1                                                         (3-13-1)
Thus, both Eq.(3-11)  and  Eq. (3-13-1) are equivalent to each other when we have Eq.(4-5) and Eq.(3-12) under Assumption-5. 
This process shows the following important facts: (1) Keynesians have substituted Assumption-5 into Eq.(3-37), so the Keynesian chart Fig.(3-3) is a special case of Fig. 4-3 under Assumption-5.  (2) both ΔC2 and ΔG2 are already produced and they are already received as incomes.  
Does ΔC2 remain in Fig.3-3?  The fact is that ΔC2 remains in Fig.3-3.  Fig.3-3 has been obtained by placing ΔC2 and correcting ΔY2 = ΔG2 into ΔY2 = ΔG2 + ΔC2 in Fig.3-3. The equation ΔY2 = ΔG2 + ΔC2 remains on the vertical axis after the remove K→K'. In any  Keynesian multiplier effect chart,  ΔC2 has never been drawn as shown in Fig.4-6. We could not see it as shown in Fig.3 for a long time. We have overlooked that ΔC2 is hidden from sight in Fig.3-3. Thus, it is concluded that Fig.3-3 is incorrect in the expressing way of ΔY2 even if we allow the Assumption-5. 

Fig. 4-6 Keynesian multiplier effect chart-2

Even if  the relation ΔY2 = ΔG2 + ΔC2 is placed as shown in Fig. 4-6, this chart is inadequate for only ΔG2. The operation placing ΔC2 = 0 in this chart is impossible. In conclusion, Fig.4-6 is wrong for only ΔG2. Fig.3-3  has double mistakes for only ΔG2!
As far as Keynesians' claim for the multiplier effect is based on or equivalent to Fig.3, their claim is mathematically wrong.
[Question-23]  I can't understand why the following economic actions occur at the same time: (1) producing consumption goods and  producing investment machines to make former goods; (2) making goods and obtaining incomes for their returns; (3) making goods and purchasing them.   
[Answer-23] The simultaneity of the economic actions are based upon human's cooperative working habit, mutual trust and society rules. These result from human instinct or the history of mankind. The simultaneity of economic actions in the division of a labor system society is assured under a trustworthy society or  financial institutional and law systems.   
Workers in a firm work to make goods. Workers' wages are paid whether the goods are sold or not.  Inventories (as final goods) are goods produced, not sold. The wages for the Inventories will have been paid before selling. The wages are deposited in banks. As the inventories are not sold, the wages are not yet spent to become temporary deposits. The deposits will already be borrowed by the firm, then the borrowing would have been used for the former workers' wages. This process shows that the depositing and the expending of inventories goes simultaneously with each other. This flow is assured by trust between the workers and the firm, and the firm's confidence in marketability for the goods, i.e. belief. Both the trust and the belief are included in believing.   
The short life-span goods work or for short time periods for humans as consumption goods, and most goods of them die within a year. This is as if foods are digested and change into nutrition for human activity. The long life-span goods work for humans for the depreciation periods repaying a common fund or recovering an initial equity capital. The all goods work or act simultaneously and cooperatively with human actions.　
These behaviors are assured by humans' mutual and self confidence, only when the simultaneity is realized. 
There is no simultaneity in conveyance of information, because objects of information are facts or doubts in a visible world, and it is located eccentrically. On the other hand, the simultaneity is in the mutual trust, because the objects of the trust exist in an invisible world (in people's  minds) like in faith, if the trust is taught among all the people and they have consciousness of impairing trust being shame or crime. 
Faith is conviction, so there is no doubt and a time-lag in it. The trust is an invisible rule to realize the simultaneity in a society.  If the mutual trust were ruined, and doubts governed people's minds, the simultaneity will not be assured, when a transaction will be separated into its cause and result, and the next step will not start until we check and know  the result. The trust system i.e. credit system is reinforced by banks' monitoring and examination functions.
Consequently, in order to maintain the trust (credit) system, a guarantee system or a redress system to assure the simultaneity in economic actions must be prepared in advance. 
The trust or the credit means 'believing words', i.e., believing a promise, a vow, or a contract etc.
[Question-24] We have the equation ΔY1 = ΔG2  / (1 - a K) or ΔY1 = ΔI2  / (1 - a K) . If a K =0.6, we have ΔY1= 2.5ΔI2 or ΔY 1= 2.5ΔG2. What do you interpret these equations? 
[Answer-24]  First, ΔY1 = ΔI2  / (1 - a K) is taken up. In this equation, we assume that I2 is the sum of all components in Y2 other than C2.  The equation ΔY1= 2.5ΔI2 is equal to the equation ΔY1 / ΔI2 =1 / 0.4. Furthermore, we have ΔY1 / ΔC2 =1 / 0.6. This is only to be expected. The latter is another expression of the MPC.
Second, ΔY1 = ΔG2  / (1 - a K) is taken up. It should be noted that this equation has been gotten by setting the assumption ΔI2 = ΔNX2 = 0. In a similar manner shown above, we have ΔY1 / ΔG2 =1 / 0.4; ΔY1 / ΔC2 =1 / 0.6. This is mathematically correct under the assumption ΔI2 = ΔNX2 = 0, but this assumption is wrong. See Fig.17. The equation ΔG2 / ΔY1 = ΔG1 / ΔY1  = ΔG1 / (ΔC1 + ΔI 1 + ΔNX1 + ΔG1) in one year period should be hold. Note that ΔG2 is executed through a year. This will be called the 'marginal propensity for government to expend'. The ratio will be about ΔG2 / ΔY1 = 1/10 e.g. ΔY1= 10ΔG2. ΔY1 in this paragraph means a total incremental gross income within one year comparing to the last year's Y1. In this meaning, we can't adopt the assumption ΔI2 = ΔNX2 = 0, because within the time period when ΔG2 is executed and ΔY2 is generated, as a matter of fact, ΔI2 and ΔNX2 are also generated. 
For each contract between the government and a firm, the equations ΔG2 = ΔG3 = ΔG1 always hold within the contract period. In this case, ΔY1= ΔG2  is used for only ΔG2 in order to insist that an equivalent income to additional expenditure ΔG2 is generated on Y1 coordinate with the execution ΔG2, and the income amount of ΔY1 = ΔG2. Fig.19 uses ΔY1 in this meaning.
References (1), (3)
[Question-25] What do you think the economic effect of increase or cut  in taxes is?
[Answer-25]  As an increase in government expenditure doesn't have a multiplied effect on income, an increase or cut in taxes doesn't have a multiplied effect on income or production.; that is to say, the tax multiplier - MPC / (1-MPC) doesn't exist. See the next section §5.
[Question-26]  What do you think about basic equations used in quantitative economics models?
[Answer-26] According to my theory, we have to distinguish the aspect of an economic value. If the incremental economic value is given as a business contract Δ G, that is in Y 2 aspect, we only have Δ Y 2 = Δ G. At the same time,  Δ G will be divided into inputs (Y 3 aspect) in firms and will be transformed into incomes ( Y 1 aspect) . We merely have Δ Y2 = Δ G 2 = Δ Y3 = Δ Y1 in accordance with the simultaneity of economic action. We can't find any other equation than these. The consumption function is obtained only by statistical observations. 
A human action is essentially one action which is expressed in several aspects. It depends upon both reason and feelings. We can't surely predict the next action from a human's present action. There is no assured or theoretical endogenous relationship between a human's present and the next action. If there is any relationship, it will be an empirical rule. However, once the empirical rule is disclosed, a new counteraction for the rule will occur.
As far as the basic equations used in the quantitative models is based on the multiplier effect theory, I doubt them. If coefficients in a quantitative basic equation are determined by statistical results, the resulting equation, as a matter of course, will fit to the statistical results.  
[Question-27] Have you found the true cause of economic growth?
[Answer-27] No, I haven't. The fundamental cause of economic growth is based on a will to make one small good and to trade in it. Notice that one commercial good (e.g. a theater or air ticket etc.) is comprised of investment goods ( machines, buildings or airplanes etc.) and other costs, the investment goods being allocated as applied costs. The good is suddenly generated as shown in Fig 3-13 (ΔG is regarded as one good or a contract), and then suddenly ceases. GDP is obtained by arithmetic addition( including subtraction) of each present product or contract, based on 'the principle of equivalent of three aspects'.   
The cause of production is, at the same time, the result of expenditure after income. A human activity is an organism's activity. On the one hand, an organism ( or a cell) is born, and on the other hand, another organism dies. A social good is like an organism. This is shown as ΔY3 ↔ ΔY2(Supply) ↔ ΔY2(Demand) ↔ Δ Y1 ↔ ΔY3 . This is one ring( cylinder, square pole etc.). The ring's height ( not diameter) is generated at the beginning of a year,  grows and degenerates everyday at an irregular speed on the 45 degree line in the Yi -Yj chart. People look at the ring from a front, back or side view. In the field of income disposal in this ring, people are not generally conscious of the existence of goods depreciation( or death).
Refer to [Answer-30]. No assured means to take a new step ΔY2( = ΔY3 =ΔY1) towards the future exists in a society of free will . The growth is realized by the every day human will to act.
If GVA increasingly grows every year, the economy will constantly grow. However, if any person could find an assured means of producing growth within the economy, depression would never occur in the future, and all countries, including developing ones, would develop equally. 
[Question-28]  What do you think about the economic effect of investment goods, whether it is private or government ones?  
[Answer-28] According to the resulting equations, there is no special difference between consumption goods and investment goods except those relating to foreign countries. The difference between them exists  mainly in their product life-spans, buyers( firms and consumers) and the necessity of debts( utilizing common funds). The boundary of consumption goods and investment goods is, by nature, not clear. Both goods are the same in that they are commercial goods for buyers, and they change into costs and profits for firms. For people, both goods are same in that they are the source of wages.
The working values within one year between a short life-span person and a long life-span person are equal. In a similar manner, the values of a consumption good and an investment good in one year are equal, if the two lost values in one year( although a durable consumption good lasts longer than one year ) are equal to each other.
Consequently, the economic effects by consumption and investment goods are equal. In the same way, the economic effects by both consumption and saving( gross) are same. The important thing is balance between consumption and investment according to conditions of a country.
[Question-29] We know that investment (net) is equal to saving (net) in a year. A good's value is equivalent to a partial GVA. Both investment and consumption are simultaneously realized. The marginal propensity to consume is almost unchangeable. Can't we therefore explain rapid GDP changes in business conditions?
[Answer-29] Manufacturing facilities live cooperatively with people. A machine has its capacity. For the average capacity, the machine is paid a standard cost( wage) per one day. In good business conditions, the machine works over its capacity. The compensation for the work quantity beyond the standard capacity is charge-free. This value of the over-work is distributed to a profit or to people. In bad business conditions, people must pay the standard cost to the machine despite its idleness( idle costs). This decreases profits or people's wages beyond necessity.     
This phenomenon is realized intuitively by entrepreneurs. See Fig.3 in "Outline of This Website". A profit is influenced by an incremental sale, allocation of fixed costs and a variable cost ratio, that is, a way of working . In fact, this phenomenon is observed in people. Coexistence between people and manufacturing facilities is one of the causes of the rapid GDP changes in the modern industrial society. If machines don't exist, economies will change more slowly than now.
[Question-30] What is the primary, theoretical difference between my claim and Keynesian's claim? 
[Answer-30] Firstly, I claim that Δ Y1 = Δ Y2 = Δ Y3 = Δ G in Fig. 1  by the principle of equivalent of three Aspects. Keynesian claims that if Δ Y3 = Δ Y2 in firms,   Δ Y2 = Δ G < Δ G / (1 - a K )= Δ Y1. 
Secondly, I claim that Δ G should be the total of Δ output, so ΔG should be placed in the left side of the equation, Δ output = Δ inputs. Keynesian claims that ΔG should be a part of the total output Δ Y2, so Δ G should be placed in the right side of the  equation, ΔY2 = ΔC 2 +( ΔI 2 + ΔNX 2) + ΔG 2 except for the problem of ( ΔI 2 + ΔNX 2) . This equation gives ΔG 2 < ΔY2. 
Thirdly, Keynesian claims that the multiplier 1 / (1 - a K)  for investment including NX2, is the cause of economic growth. I claim that the multiplier itself does not exist, and any assured means of producing  growth within the economy has not yet been found. If any assured means producing economic growth would be found, depression would not occur in the future, and all countries' economy including developing countries would grow equally.    
[Question-31] Where is the core for economics in my theory except for disproving  the Keynesian multiplier effect theory?

[Answer-31] The first core is in Fig.13/14. Fig. 13 has liberated the constraint that profit is constant in the Keynesian's ΔG-Y chart, so firms' profit Δ P and variable fixed cost Δ FF in the national accounts have become able to move freely on the 45° lines shown in Fig.13. By Fig.15, the competing relationship between Δ P and  Δ FF has been revealed. 

The second, is pointing out the importance of the role of Δ FF. My theory has denied the multiplier effect Δ G / (1 - a K). By definition, Δ FF is not proportional to the amount of Δ Y. As described in Subsection 2, Case-A, Y cannot move in organizations or societies where the concept of profit does not exist when  Δ FF = 0. Even if the concept of profit exists, the root of  dynamic change in economy exists firstly in the change Δ FF, whether it is positive or negative. Ratio a V expresses conventional techniques or procedures in social activities. In order to change a V, untiring social belief in structural reform is needed. The role which entrepreneurs and policy-makers should play in the economy has become increasingly important.
[Question-32]  What do you claim briefly for additional expenditure ΔG2 in conclusion?
[Answer-32] The Keynesian multiplier effect chart Fig. 3-3 is wrong in that ΔY2 = ΔG2 ,which should be corrected as ΔY2 = ΔG2+ΔC2 as shown in Fig.4-6. However, Fig. 4-6  is wrong for only ΔG2. Fig.3-13 as well as Fig.4-7 which has been changed from Fig.4-3 is also right (refer to Fig.4-5). In the latter figure, ΔC2 is not generated.

Fig.4-7 Another ΔG-Y chart

[Question-33] Don't economists including me confuse the following two income- production charts ?: 
 (1) the time series charts, for a yearly long time period, such as Fig.3-2( Keynesian's consumption function chart) and Fig.4-3
 (2) the input-output table chart, or the break-even chart such as Fig.13 in the previous section §2. 
[Answer-33] I don't confuse them. Keynesians probably confused them. Detailed explanation for this question will be given in the next section 3. §4.
[Question-34] Why did governments' active investment policies by borrowing supported by the principle of effective demand by the Keynesian multiplier effect historically and largely succeed in many countries? Why do we recently face collapses in these economic policies?    
[Answer-34] The answer for the former question is that any theory by nature belongs to a belief or a faith whether it is true or not. The faith is all the more strengthened if it is proved scientifically. Some investments brought a good result by chance. The success ignited people's minds. People found a small fire of hope for the future. The produced goods effectively and cooperatively worked for people beyond expectation . The fire grew to become a big fire. The big fire is economic confidence to produce goods and to consume or digest them.   
The answer for the latter question is that the Keynesian multiplier effect for the future doesn't exist. The fire went out. However, if the borrowing, people's common funds does not need to be repaid, no problems will occur. The reason is that the finished investments are real things, and the incomes provided by the investment are already received by people although it is not assured whether they effectively work for people in the future.  Only the debtor-creditor relationship remains.  By the way, the relationship belongs to the belief which is invisible in the physical world.     
[Question-35] What is the difference between a private investment and a public investment, which are executed by borrowing? Is it  economic efficiency?
[Answer-35] No, it is not aside from extreme cases. The credit resulted from the borrowing between a company and some citizens ( or a bank) is limited between the two. The credit between a nation and a citizen runs in the whole nation. (This answer might be rewritten in the future. 29/Oct/2005)
[Question-36]  What do you think about the principle of effective demand based on the Keynesian multiplier effect for an unemployment problem?
[Answer-36] If we intend to resolve an unemployment problem with the equation ΔG / (1 - aK )= Δ Y1, this attempt will not give a calculated volume of employment from the equation, because this equation is incorrect.  However, a volume of added employment will be expected  by ΔG ( more exactly, a volume of employees compensation included in ΔG ) with a government debt. By the way, I do not deny the importance of the effective demand( effective uses of saved money or purchasing power which can't find its uses as investment or consumption for some reason ).
The above-mentioned views excepting the parts of the multiplier effect theory will be examined and might be corrected from now poring over Keynes's original work.(9/Feb/2005)