A charting method for an economic fluctuation presented by the author

§8 A charting method for an economic fluctuation presented by the author
DCE Yuichiro Hayashi

October 16, 2006

Before examining the equilibrium theory method, the author will present his idea of the charting of economic activities including money and credit. We shall not take the transactions of both lands and stocks into consideration, because we now argue the changes of the GVA.

According to the cash flow 3-accounts matrix, for the national accounts, which will be explained in §1, chapter 3(Japanese), the cash flow [account items ±money and credit items, in the accrual basis] consists of a debit and credit pair, where the debit shows cash output, and the credit represents its input. The number of accounts is four: production, transfer-income, income-expenditure, business transaction. The sum of the debit equals that of credit in each of the 4 accounts. The reason is that the debit (cash outflow) = the credit (cash inflow) in the production account, and the three accounts other than the production one exist in the cash flow pass ways linked to each other. We regard one of the debit and credit pairs as a vector, and name it 'the cash flow vector' in the national accounts. We denote Y_Γ the representative vector of the 4 vectors. The vector Y_Γcan be any of the 4 vectors.

We shall set up a two-dimensional Cartesian coordinate system as shown in Fig.8-1. We put vector Y_Γon the 45-degrees line with the tail at the origin. We put each vector on each side of the square, which is denoted as Y_Γ1, Y_Γ2, Y_Γ3 or Y_Γ4respectively. The order of the vector number is useful if we assign 1 to production, and 3 to the income-expenditure. In Fig.8-1, we denote Y_Γ^the first equilibrium vector, ΔY_Γthe incremental vector, and Y_Γ^* the equilibrium vector after a change.


Fig.8-1

It should be noted that the Y_Γ vector on the 45-degrees line doesn't relate to the length of the diagonal line of the square. Moreover, it is not necessary that the Y_Γ vector should be put on the 45-degrees line. That is to say, the Y_Γ vector can be positioned at a slope line with any angle, if the tail of the vector is at the origin. Whether each vector Y_Γ1～Y_Γ4is parallel to each other or mutually-perpendicular is simply for the ease of graph plotting.

Fig.8-1 can be expressed as Fig.8-2. This chart shows more explicitly that a cash input into an economic activity field corresponds to a cash output from another field.

Fig.8-2

From the author's point of view, a macroeconomic change can be expressed by the expansion and contraction of vector Y_Γ. Then, the macroeconomic change is the same as those of the square in Fig.8-1 or of the circle in Fig.8-2. In addition, an actual economic vector is made of a saved asset vector plus Y_Γ.

A stable economic system composed of Y_Γi（i=1～4） vectors is a one-degrees of freedom system which means that the MPC doesn't change. If the economy is in a unstable condition where the MPC fluctuates, the economy system can obtain a certain number n - degrees of freedom more than one. In comparison, an economy is a one-degree of freedom system due to the following: when we investigate the relationship between ΔC and ΔY, we can observe a linear relationship between them, so this adds one constraint equation to the relationship of the variables ΔC, ΔY and [ΔY - ΔC] . The condition where the MPC exists means that both ΔC and [ΔY-ΔC] are always produced at the same time with a constant ratio.

We denote Y_Γ1[debit] the debit, and Y_Γ1[credit] in the production cash flow vector Y_Γ1. Since each firm has its break-even chart and associated cash flow vector, and with it, it is estimated that Y_Γ1 [Debit , Credit ], which is a set of each firm's cash flow vector, comprises of both a variable vector and a fixed vector. In this variable vector, the intermediate consumption P will play an important role. Then we denote Y_Γ^Vthe variable part, and Y_Γ^F the fixed one in Y_Γ. From the business input-output table matrix, we can estimate that both the variable and fixed parts exist in all the vectors Y_Γ1～Y_Γ4. Each sum of components of a variable or fixed part has the same economic value in each of the 4 economic activity fields.
ν=　tanθ
= l Y_Γ^V l / l Y_Γl	(8-1)

This relationship is shown in Fig.8-3. We name the ratio of the the variable part to that of the whole, in Y_Γ, 'the cash flow variable ratio' with the notation ν. The ratio ν is given in the following:


Fig.8-3　Y_Γ1vector

By the way, it should be noted that the subscripts i used in Y_iand Y_Γiare not compatible to each other, because Y_iis one vector of debit or credit, whereas Y_Γihas a vector pair which consists of debit and credit.

16/Oct./2006　Publication 18/Dec./2007 Modification