Part 2 Economics
DCE Yuichiro Hayashi, Y. Hayashi
December 2003 
Chapter 1  Examination of Keynesian Multiplier Effect Theory
§1 Outline of the examination of the Keynesian multiplier effect theory 
Keynes' principle of effective demand based on the Keynesian multiplier effect is mathematically wrong.
The Keynesian multiplier effect doesn't exist.
In order to avoid complications on symbols, we shall consider a closed economy. Let Y  be the gross value added (GVA) in the national economic accounting. Furthermore,  the following notations are defined: Y3 = the gross value added; Y2 = demand( final products from the standpoint of customers) or supply( those from producers); Y1 = gross income( the value to which the GVA is distributed to firms, households and a government). 
The author adds the following definitions: Y3 = input in economic  activity ( costs and profit in final products）; Y2＝output in economic activity ( final products with stocks). Y1= both a broken-down value and the residual one, from the value Y2, which are carried into Y3 as its generation resources
By applying the managed gross profit theory to the input-output table in the national economic accounts, an input-output table chart shown in Fig. 2-7 has been derived. Fig. 2-7 shows the relationship between output( final product) and inputs( gross value added). In this chart, we can't find the 'marginal propensity to consume (MPC)',  

Fig. 2-7 Input-output table chart
where
Y= gross value added (=GVA),     (ε): it shows that data are those at the end of a yearly accounting period,    X final= final products,   Y (φ) = break-even products of Y,      P= operating surplus, 
FF= fixed costs in GVA,    FV= variable costs in GVA,   aV = variable cost ratio = FV / Y.
The coefficient a V corresponds to the variable cost ratio in a firm's break-even chart. A change in Y is the total change in both FV and [P+FF].
From the national economic accounts at the Keynesian cross, we have the following equations:

                    Y2 = C 2 + I 2 + NX 2 + G 2                                      (3-R1)

                    Y2 = Y1                                                                      (3-R2)
where Y1 = gross income, Y2 = final products, C2 = consumption, I 2 =  private investment, G2 = government expenditure, NX2 = exports - imports, 
At the new Keynesian cross after the remove from Y to Y + ΔY, we have:

                    ΔY2 =Δ C 2 + ΔI 2 + ΔNX 2 + ΔG 2                        (3-R3)

                    ΔY2 =Δ Y1                                                                 (3-R4)
From Eq. (3-R3), we have:

                   Δ C2 / ΔY2 + ΔI 2 / ΔY2+ ΔNX 2 / ΔY2+ ΔG 2 / ΔY2 = 1       (3-R5)

Substituting Eq. (3-R2) into Eq. (3-R5) gives:

                   Δ C2 / ΔY1 + ΔI 2 / ΔY1+ ΔNX 2 / ΔY1+ ΔG 2 / ΔY1 = 1       (3-R6)
We adopt the following assumption:

Assumption 1 ; The following statistical coefficients exist; aK =  MPC = Δ C 2 / ΔY1 ,  a I = Δ I 2 / Δ Y1,  aNX = ΔNX2 / ΔY1 and  aG = ΔG2 / ΔY1, where all the coefficients are constant. 
From Eq. (3-R6), between these coefficients, we have the following:

                   aK +  aI  +  aNX +  aG =1                                                (3-R7)
Eq. (3-R6) can be transformed to:
                  aKΔY1 + ΔG 2 + ΔI 2 + ΔNX2  = ΔY1                             (3-R8-1)
that is, 

                  ΔY1 = ΔG 2 / (1− aK) + (ΔI 2 + ΔNX 2 ) / (1- aK)           (3-R8-2)
Transforming Eq.(3-R7) gives:

                   aK +  aI + aNX +ΔG 2 / ΔY1 = 1                                      (3-R9-1)
that is, 

                  ΔY1 = ΔG 2 / (1 - aK -  aI - aNX )                                     (3-R9-2)
Eq. (3-R5),  Eq. (3-R6), Eq. (3-R8-2) and  Eq. (3-R9-2) are equivalent to each other under Eq. (3-R4)  and Assumption 1. 
Eq. (3-R7) can be explained in Fig. 4-3. 

Fig. 4-3 Meaning of aB or Relationships between coefficients
The multiplier effect by the Keynesian model is shown in Fig. 3-3.

Fig. 3-3 Keynesian multiplier effect chart
The multiplier effect equation by Keynesian model is as follows:
                              Δ Y1= Δ G2 / (1 - aK)                                       (3-13-1)
Eq.(3-13-1) is obtained by adding the following assumption, into  Eq.(3-R8-2),:
Assumption 2: ΔI 2  = ΔNX 2 = 0
Then, under Assumption1, Eq.(3-13-1) is equivalent to Eq.(3-11) which has been gotten by substituting Assumption 2 into Eq.(3-R1) .
                   ΔY2 = ΔC2 + ΔG2                                                       (3-11)
Eq.(3-11) expresses that ΔC2 and ΔG2 are already generated at the same time.
In fact, we have overlooked for a long time that ΔC2 is hidden from sight in Fig.3-3. Thus, Fig.3-3 is wrong in the expressing way of ΔY2 even if we allow the Assumption 2. Fig.3-3 must be drawn as shown in Fig. 4-6. We can't find the multiplier effect in Fig.4-6.

Fig. 4-6 Corrected Keynesian multiplier effect chart
As I2 is largely affected by a change of Y2 in the real economy,  (aI +aNX) is not zero in Eq.(3-R9-2). Hence, we can't adopt the Assumption 2, so Fig.4-6 doesn't hold. If we adopt the Assumption 2, aK will become a nonlinear function of Y1.  In addition, if we adopt the Assumption 2, the output amount corresponding to the partial outputs, Δ I2 and ΔNX2 is lost (the amount Δ C2 +ΔI2 is a part of the GVA)   from the total ΔY2 , although ΔY1 expresses the total input (ΔY1 is the whole of the GVA). That is to say, the Keynesian cross condition ΔY1= ΔY2 is not satisfied.  
However, this assumption is not the fundamental cause in errors in the Keynesian multiplier effect theory.　If an incremental final product is only ΔG2  in Eq.(3-R3) and Eq.(3-R4) , that is, ΔG2 = ΔI2 = ΔNX2 = 0, the following equation should hold:
                   ΔG2 = ΔY2 = ΔY1                                               (1-1)
Despite Eq.(1-1), ΔC2 is already generated with ΔG2 in Fig. 4-6. Consequently, Fig.4-6 is wrong for only ΔG2. For only ΔG2, Fig. 4-6 must be correctly replaced by Fig.4-7. 

Fig.4-7 Correct  ΔG-Y chart
When ΔG2 and ΔNX2 are included in ΔIT, this economic model is used in J. M. Keynes' original work, although he didn't show its model chart. When we express aB as IT and ΔG2 as ΔIT in Fig.4-6( above), we can name this chart 'Keynes' original chart' which is shown in Fig.5-5( below).

 Fig.5-5  Keynes' original chart
 
According to the author's consideration made in 5.3, §5, we can't use Keynes' original chart theory for analyzing an economy which changes with respect to time. 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. ΔC2 depends on ΔIT, and so both ΔC2 and ΔIT don't relate to the passage of time. This relationship is shown in Fig.5-4( below) where Δx = ΔC2, Δy = ΔIT, and Δz = ΔY2.

Fig.5-4 One-degree of freedom system
 
The most important shortcoming in Keynes' original chart theory is that the MPC relationship is broken off in capitalist free economy countries just when we intend to add an incremental government expenditure to resolve an unemployment problem. To resolve this problem, we must simultaneously and surely provide goods ΔC2, ΔI 2 and ΔNX2　according to the incremental demand ΔG2 which doesn't relate time passing .  
 
However, the MPC is factually found by statistical observations of the real economy as shown in Fig. 3-1. 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 data of consumption in a many degrees of freedom in a stationary economy. By virtue of both endeavoring to keep each firm profitable and the national propensity to consume, the MPC would have been kept constant in the stationary economy.
Fig. 2-7( above) is equivalent to Fig.3-13( below) which has been gotten in such a manner that ΔG2 increaseｓ from Fig.2-7.

 Fig.3-13 Correct  ΔG-Y chart
The author claims the following:
(1)  Fig. 3-3 which shows the Keynesian multiplier effect is wrong. Fig. 3-3 must be correctly drawn as shown in Fig. 4-6. In Fig. 4-6, we can't find the multiplier effect. However, in Fig. 4-6, ΔC2 is already generated with ΔG2, so Fig. 4-6 is also wrong (incorrect/ flawed) for only ΔG2. Nevertheless, Fig. 4-6 is absolutely wrong, because the Keynesian-cross condition ΔY2 = ΔY1 is not satisfied. Consequently, the principle of effective demand by J.M.keynes is clearly wrong as far as we use Eq.(3-13-1) as a means of solving an unemployment problem. 
(2) When ΔG2 and ΔNX2 are included in ΔIT, this economic model is used in J. M. Keynes' original work. As Keynes' original chart model is a one-degree of freedom system, both ΔC2 and ΔIT do not relate to the passage of time . Consequently, ΔC2 according to ΔIT must be simultaneously provided to keep the MPC condition when we  provide ΔIT in order to resolve an unemployment problem. That is to say, both ΔIT and ΔC2 are in one pair, and their activities can't be separated. Keynes' original chart theory doesn't help when analyzing an unemployment problem.　　　
(3)  For the Keynesian multiplier effect chart which belongs to the time series type chart, Fig.4-7 is correct. However, Fig.3-13 which belongs to the break-even type chart is correct for analyzing economic problems.
(4) A new matrix equation connecting each component of the gross value added to that of the final products has been presented in Chapter 2 in order to explain the validity of Fig.2-7 by Leontief's input-output table. 
In conclusion, the Keynesian multiplier effect theory Is mathematically wrong. 
Dec. 2003 Yuichiro Hayashi, http://www11.plala.or.jp/yuichiro-h/
Slight modification Aug. 2008